uuid int64 541B 3,299B | dataset stringclasses 1
value | text stringlengths 1 4.29M |
|---|---|---|
2,877,628,089,454 | arxiv | \section{Introduction}
Erd\H{o}s first used the probabilistic method in 1947 to prove the existence of so-called \textit{Ramsey graphs}, i.e., $n$-vertex graphs with no clique or independent set of size $O(\log n)$~\cite{Erdos:47}.
Since then, the probabilistic method has been a popular technique for demonstrating the existence of various combinatorial objects, while at the same time leaving a desire for explicit constructions.
Even Ramsey graphs currently lack an explicit construction despite the fact that random graphs are (unverifiably) Ramsey with high probability.
This aggravatingly common state of affairs has been described by Avi Wigderson as the problem of \textit{finding hay in a haystack}.
Over the last decade, a new combinatorial object has been actively studied in applied mathematics:
An $M\times N$ matrix $\Phi$ is said to satisfy the \textit{$(K,\delta)$-restricted isometry property (RIP)} if
\begin{equation*}
(1-\delta)\|x\|_2^2
\leq\|\Phi x\|_2^2
\leq(1+\delta)\|x\|_2^2
\end{equation*}
for every vector $x$ with at most $K$ nonzero entries.
A matrix which satisfies RIP is well-suited as a \textit{compressive sensor}:
If $\Phi$ is $(2K,\delta)$-RIP with $\delta<\sqrt{2}-1$, then any nearly $K$-sparse signal $x$ can be stably reconstructed from noisy measurements $y=\Phi x+e$~\cite{Candes:08}.
Impressively, the number of measurements $M$ required for stable reconstruction is much smaller than the signal dimension~$N$.
As we will discuss in the next section, there are several known random constructions of RIP matrices with $M=O_\delta(K\operatorname{polylog}N)$, meaning the number of measurements can effectively scale with the complexity of the signal.
Unfortunately, RIP matrices provide yet another instance of finding hay in a haystack.
Much like Ramsey graphs, RIP matrices are ubiquitous, but verifying RIP is NP-hard in general~\cite{BandeiraDMS:13}.
To summarize the state of the art, almost every explicit construction of RIP matrices uses the following recipe (see~\cite{ApplebaumHSC:09,DeVore:07,FickusMT:12}, for example):
\begin{itemize}
\item[1.] Let the columns $\{\varphi_n\}_{n=1}^N$ of $\Phi$ have unit norm and satisfy $|\langle \varphi_n,\varphi_{n'}\rangle|\leq \mu$ for every $n\neq n'$.
\item[2.] Observe, either by the Riesz--Thorin theorem or the Gershgorin circle theorem, that every $K\times K$ principal submatrix $A$ of $\Phi^*\Phi$ satisfies $\|A-I\|_2\leq(K-1)\mu$.
\item[3.] Conclude that $\Phi$ is $(K,\delta)$-RIP for $\delta=(K-1)\mu$.
\end{itemize}
This coherence-based method of estimating $\delta$ requires $K\ll 1/\mu$.
However, Welch~\cite{Welch:74} gives the following universal bound
\begin{equation*}
\mu^2\geq\frac{N-M}{M(N-1)},
\end{equation*}
which is $\geq1/(2M)$ whenever $M\leq N/2$.
As such, in Vinogradov notation we have
\begin{equation*}
M\gg \min\{1/\mu^2,N\}\gg\min\{K^2,N\},
\end{equation*}
meaning a coherence-based guarantee requires $M$ to scale with the \textit{square} of the complexity of the signal.
This inferiority to existing guarantees for random constructions compelled Terence Tao to pose the problem of explicitly constructing better RIP matrices on his blog~\cite{Tao:07}.
Since then, one explicit construction has managed to chip away at this power of two---Bourgain, Dilworth, Ford, Konyagin and Kutzarova~\cite{BourgainDFKK:11a,BourgainDFKK:11b} provided a new recipe, which is largely based on the following higher-order version of coherence:
\begin{definition}
We say a matrix $\Phi=[\varphi_1\cdots\varphi_N]$ has \textit{$(K,\theta)$-flat restricted orthogonality} if
\begin{equation*}
\bigg|\bigg\langle\sum_{i\in I}\varphi_i,\sum_{j\in J}\varphi_j\bigg\rangle\bigg|
\leq\theta(|I||J|)^{1/2}
\end{equation*}
for every disjoint pair of subsets $I,J\subseteq\{1,\ldots,N\}$ with $|I|,|J|\leq K$.
\end{definition}
\begin{theorem}[essentially Theorem~13 in~\cite{{BandeiraFMW:13}}, cf.~Lemma~1 in~\cite{BourgainDFKK:11a}]
\label{thm.fro to rip}
If a matrix $\Phi$ with unit-norm columns has $(K,\theta)$-flat restricted orthogonality, then $\Phi$ satisfies the $(2K,\delta)$-restricted isometry property with $\delta=150\theta\log K$.
\end{theorem}
Using this, Bourgain et al.\ leveraged additive combinatorics to construct an RIP matrix with $M=O_\delta(K^{2-\varepsilon})$ (according to~\cite{BourgainDFKK:11b}, $\varepsilon$ is on the order of $10^{-15}$); the reader is encouraged to read~\cite{Mixon:14} for a gentle introduction to this construction.
Since explicit constructions are evidently hard to come by, it is prudent to consider a relaxation of the hay-in-a-haystack problem:
\begin{problem}[Derandomized hay in a haystack]
Given $N$, $K$, $\delta$, and a budget $H$, what is the smallest $M$ for which there exists a random $M\times N$ matrix satisfying the $(K,\delta)$-restricted isometry property with high probability, but whose distribution only uses at most $H$ random bits?
\end{problem}
This derandomization problem is the focus of this paper.
The next section features a survey of the current literature.
In Section~3, we review our approach, namely, populating the matrix with consecutive Legendre symbols.
Our main result (Theorem~\ref{theorem_legendre}) gives that a random seed of $H=O_\delta(K\log K\log N)$ bits suffices to ensure that such a matrix satisfies $(2K,\delta)$-RIP with high probability; we also conjecture that no random seed is necessary (Conjecture~\ref{conj.main conjecture}).
Section~4 states a more general version of our main result---that any matrix with small-bias Bernoulli entries satisfies RIP with high probability (Theorem~\ref{thm.unbiased rip}); this is a natural generalization of a standard result with iid Bernoulli entries, and our main result follows from this one after verifying that random Legendre symbols have small bias.
We conclude in Section~5 with the proof of Theorem~\ref{thm.unbiased rip}.
\section{Prior work on derandomizing restricted isometries}
When $H=0$, the explicit construction of Bourgain et al.~\cite{BourgainDFKK:11a,BourgainDFKK:11b} holds the current record of $M=O_\delta(K^{2-\varepsilon})$, though this construction also requires $N\ll M^{1+\varepsilon}$.
If one is willing to spend $H=O_\delta(KN\log(N/K))$ random bits, then simply toss a fair coin for each entry of $\Phi$ and populate accordingly with $\pm1/\sqrt{M}$'s; this matrix is known to be $(K,\delta)$-RIP with high probability provided $M=O_\delta(K\log(N/K))$~\cite{BaraniukDDW:08,MendelsonPT:09}, and this is the smallest known $M$ for which there exists an $M\times N$ matrix satisfying $(K,\delta)$-RIP.
Far less entropy is required if one is willing to accept more structure in $\Phi$.
For example, toss $N$ independent weighted coins, each with success probability $M_0/N$, and then collect rows of the $N\times N$ discrete Fourier transform matrix which correspond to successful tosses; after scaling the columns to have unit norm, this random partial Fourier matrix will be $(K,\delta)$-RIP with high probability provided $M_0=O_\delta(K\log^3K\log N)$; see~\cite{CandesT:06,CheraghchiGV:13,RudelsonV:08}.
The entropy of this construction is $H=O(M_0\log(N/M_0))=O_\delta(K\log^3K\log^2N)$.
For another construction, this one requiring $N\leq M^2$, toss $M$ fair coins to populate the first column with $\pm1/\sqrt{M}$'s, and then take the remaining columns to be Gabor translates and modulates of this random seed vector; this matrix is $(K,\delta)$-RIP with high probability provided $M=O_\delta(K\log^2K\log^2N)$~\cite{KrahmerMR:14}, and uses entropy $H=M$.
In comparison with the random partial Fourier matrix, this construction has slightly more rows in exchange for a log factor of less entropy.
Another exchange can be made by modifying the partial Fourier construction:
Take a random partial Fourier matrix with $BM$ rows, partition the rows into subcollections of size $B$, toss $B$ fair coins for each subcollection and negate half of the rows accordingly, and then add the rows in each subcollection to produce a total of $M$ rows; after normalizing the columns, the resulting $M\times N$ matrix will be RIP with high probability provided $B=O(\log^{6.5} N)$ and $M=O_\delta(K\log^2K\log N)$~\cite{NelsonPW:14}.
Thanks to $B$, the entropy in this construction is $H=O_\delta(K\log^2K\log^{8.5}N)$, and so we save a log factor in $M$ in exchange for several log factors in $H$.
It should be noted that the preceding random matrix constructions were not developed with the specific intent of derandomization, but rather, they were developed to satisfy certain application-driven structural requirements (or to allow for fast matrix--vector multiplication) and derandomization came as a convenient byproduct.
Other constructions have also been proposed with the particular goal of derandomization.
For one example, start with a matrix of Bernoulli $0$'s and $1$'s, and then for each column, apply an error-correcting binary code to convert the column into a codeword and then change the $0$'s and $1$'s to $\pm1/\sqrt{M}$'s.
If the codebook covers $\{0,1\}^M$ in Hamming distance, then this matrix differs from the corresponding Bernoulli matrix of $\pm1/\sqrt{M}$'s in only a few entries, resulting in little loss in the $\delta$ for RIP.
This error-correction trick was introduced in~\cite{CalderbankJN:11}, which also identified various codes to derandomize Bernoulli matrices with $M=O_\delta(K\log(N/K))$.
Unfortunately, the total entropy for each of these cases is invariably $H=(M-\operatorname{polylog}M)N=\Omega(MN)$; the dual of concatenated simplex codes offers slightly more derandomization $H=(M-M^\varepsilon\log M)N=\Omega(MN)$, but at the price of $M=\Omega(K^{1+\varepsilon})$.
A particularly general RIP construction uses a \textit{Johnson--Lindenstrauss (JL) projection}, namely a random $M\times N$ matrix $\Phi$ such that for every $x\in\mathbb{R}^N$ of unit norm,
\begin{equation*}
\operatorname{Pr}\Big(\big|\|\Phi x\|_2^2-1\big|\geq \varepsilon\Big)
\leq e^{-cM\varepsilon^2}
\qquad
\forall \varepsilon>0.
\end{equation*}
JL projections are useful for dimensionality reduction, as they allow one to embed $P$ members of $\mathbb{R}^N$ in $M=O((\log P)/\varepsilon^2)$ dimensions while preserving pairwise distances to within a distortion factor of $\varepsilon$.
It was established in~\cite{BaraniukDDW:08} that JL projections are $(K,\delta)$-RIP with high probability provided $M=O_\delta(K\log(N/K))$.
Moreover, there are several random matrices which are known to be JL, for example, matrices of Bernoulli $\pm1/\sqrt{M}$'s~\cite{Achlioptas:03}.
Not surprisingly, there has also been work on derandomizing JL projections, though such projections inherently require $H=\Omega_\varepsilon(M)$; since $M<N$, then for every instance $\Phi_i$ of the random matrix $\Phi$, there exists a unit-norm vector $x$ in the null space of $\Phi_i$, and so the probability $p_i$ that $\Phi=\Phi_i$ is $\leq e^{-cM\varepsilon^2}$, implying
\begin{equation*}
H
=\sum_{i}p_i\log(1/p_i)
\geq\min_i\log(1/p_i)
\gg_\varepsilon M.
\end{equation*}
This reveals an intrinsic bottleneck in derandomizing JL projections as a means to derandomize RIP matrices.
Nevertheless, the most derandomized RIP matrices to date are JL projections.
These particular JL projections exploit $r$-wise independent Bernoulli random variables, meaning every subcollection of size $r$ is mutually independent.
We note that the standard construction of $n$ Bernoulli random variables which are $r$-wise independent employs an $r$-wise independent hash family, requires only $O(r\log n)$ random bits, and is computationally efficient.
As a consequence of Theorem~2.2 in~\cite{ClarksonW:09}, a Bernoulli $\pm1/\sqrt{M}$ matrix with $\Omega_\varepsilon(M)$-wise independent entries is necessarily JL, thereby producing an RIP matrix with $M=O_\delta(K\log(N/K))$ and $H=O_\delta(K\log(N/K)\log N)$.
This has since been derandomized further by Kane and Nelson~\cite{KaneN:10}, who multiply a particular sequence of these JL projections to produce a JL family that leads to RIP matrices with $M=O_\delta(K\log(N/K))$ and $H=O_\delta(K\log(N/K)\log(K\log(N/K)))$; at the moment, this is the most derandomized construction with this minimal value of $M$.
In the spirit of derandomizing JL, note that Krahmer and Ward~\cite{KrahmerW:11} proved that randomly negating columns of an RIP matrix produces a JL projection, establishing that an RIP construction with entropy $H$ leads to a JL family with entropy $H+N$; of course, this is sub-optimal if $K$ is much smaller than $N$.
\section{Our approach}
In this paper, we propose a derandomized version of the matrix of Bernoulli $\pm1/\sqrt{M}$'s, and our construction makes use of the Legendre symbol.
For any integer $a$ and odd prime $p$, the \textit{Legendre symbol of $a$ and $p$} is defined to be
\begin{equation*}
\bigg(\frac{a}{p}\bigg)
:=
\left\{
\begin{array}{rl}
1&\mbox{if $\exists x\in\{1,\ldots,p-1\}$ such that $a\equiv x^2\bmod p$}\\
-1&\mbox{if $\nexists x\in\{0,\ldots,p-1\}$ such that $a\equiv x^2\bmod p$}\\
0&\mbox{if $a\equiv 0\bmod p$}.
\end{array}
\right.
\end{equation*}
The Legendre symbol is known for its pseudorandom behavior, and it is historically associated with derandomization.
For example, Erd\H{o}s used the probabilistic method in~\cite{Erdos:63} to show that almost every way of directing edges in a complete graph forms something called a \textit{paradoxical tournament}, and Graham and Spencer~\cite{GrahamS:71} later showed that directing edges between $p$ vertices according to the Legendre symbol (namely, $i\rightarrow j$ precisely when $((i-j)/p)=1)$ provides an explicit construction of such a tournament.
We take inspiration from this success in applying the Legendre symbol to find hay in a haystack.
Our particular application will make use of the following theorem (historically attributed to Harold Davenport), which establishes higher-order random-like cancellations:
\begin{theorem}[cf.\ Theorem~1 in~\cite{MauduitS:97}]
\label{thm.legendre_pseudorandom}
There exists $p_0$ such that for every $p\geq p_0$ and integers $0<d_1<\cdots<d_k<p$ and $1\leq t\leq p-d_k$, we have
\begin{equation*}
\bigg|
\sum_{n=0}^{t-1}\bigg(\frac{n+d_1}{p}\bigg)\cdots\bigg(\frac{n+d_k}{p}\bigg)
\bigg|
\leq9kp^{1/2}\log p.
\end{equation*}
\end{theorem}
If the sum above ranged over all $n\in\mathbb{Z}/p\mathbb{Z}$, then we could invoke Weil's character sum estimate (Theorem 2C in~\cite{Schmidt:76}) to get $\leq kp^{1/2}$.
Instead, the sum is truncated so as to avoid a zero Legendre symbol, and so we are penalized with a log factor---this estimate is made possible by a version of the Erd\H{o}s--Tur\'{a}n inequality.
For one interpretation of this result, draw $X$ uniformly from $\{0,\ldots,p-k-1\}$.
Then all moments and mixed moments of the random Legendre symbols $\{((X+i)/p)\}_{i=1}^k$ have size $\leq (9kp^{1/2}\log p)/(p-k-1)$, which vanishes if $k=o(p^{1/2}/\log p)$.
As such, these random Legendre symbols have low correlation, but notice how little entropy we used.
Indeed, at the price of only $H<\log_2p$ random bits to pick $X$, we can enjoy as many as, say, $k=p^{1/3}$ bits of low correlation.
This quasi-paradox reveals something subtle about Bernoulli random variables:
While zero correlation implies independence (meaning the total entropy is the number of random variables), it is possible to produce many Bernoulli random variables of low correlation with very little entropy; as we discuss in the next section, this is related to packing many nearly orthogonal vectors in low-dimensional Euclidean space.
In this paper, we leverage the pseudorandomness identified in Theorem~\ref{thm.legendre_pseudorandom} to derandomize the matrix of Bernoulli $\pm1/\sqrt{M}$'s while maintaining RIP.
Our particular matrix construction uses the following recipe:
\begin{itemize}
\item[1.]
Given $N$, $K$ and $\delta$, pick $M$ and $H$ sufficiently large, and take $p$ to be some prime $\geq2^H+MN$.
\item[2.]
Draw $X$ uniformly from $\{0,\ldots,2^H-1\}$, and populate the entries of an $M\times N$ matrix $\Phi$ one column at a time with consecutive Legendre symbols $\{((X+i)/p)\}_{i=1}^{MN}$.
\item[3.]
Use Theorem~\ref{thm.legendre_pseudorandom} to show that $\Phi$ has flat restricted orthogonality, and therefore satisfies $(K,\delta)$-RIP by Theorem~\ref{thm.fro to rip}.
\end{itemize}
The reader should rest assured that the third step above is not obvious, but it summarizes our proof technique for the main result of this paper:
\begin{theorem}[Main Result]
\label{theorem_legendre}
Given $N$, $K$ and $\delta$, take
\begin{equation*}
M=(C_1/\delta^2)K\log^2 K\log N,
\qquad
H=C_2K\log((K/\delta)\log K)\log N
\end{equation*}
with $C_1$ and $C_2$ sufficiently large, and let $p$ denote any prime $\geq2^H+MN$.
Draw $X$ uniformly from $\{0,\ldots,2^H-1\}$, and define the corresponding $M\times N$ matrix $\Phi$ entrywise by the Legendre symbol
\begin{equation*}
\Phi[m,n]
:=\frac{1}{\sqrt{M}}\bigg(\frac{X+M(n-1)+m}{p}\bigg).
\end{equation*}
Then $\Phi$ satisfies the $(2K,\delta)$-restricted isometry property with high probability.
\end{theorem}
At this point, we explain how to efficiently construct this matrix.
Note that the output (i.e., the $M\times N$ matrix $\Phi$) is of length $MN\leq N^2$, and so consider a construction algorithm to be computationally efficient if it takes time polynomial in $N$.
The first task is to find a prime $p$ which is $\geq2^H+MN$.
By Bertrand's postulate, it suffices to search the integers between $2^H+MN$ and $2(2^H+MN)$, of which a fraction of about $1/\log(2^H+MN)=\Omega(1/H)$ are primes by the prime number theorem.
As such, we may randomly draw an integer in this interval, run the AKS primality test~\cite{AgrawalKS:04}, and repeat until we find a prime; this randomized algorithm will succeed in $\operatorname{poly}(H)=\operatorname{poly}(N)$ time with high probability.
(We note that deterministic alternatives to this approach have been studied, for example, in a polymath project~\cite{TaoCH:12}, but the best known algorithms of this form use superpolynomial time.)
Once the prime $p$ has been selected, we draw $X$ uniformly from $\{0,\ldots,2^H-1\}$, which requires $H=O(N)$ queries of a Bernoulli random number generator.
Finally, the entries of $\Phi$ are populated with $MN\leq N^2$ different Legendre symbols, and each of these can be calculated in $\operatorname{polylog}(p)=\operatorname{poly}(N)$ time, either by appealing to Euler's criterion (using repeated squaring with intermediate reductions modulo $p$) or by exploiting various identities of the Jacobi symbol (which equals the Legendre symbol whenever the ``denominator'' is prime)~\cite{Cohen:93}.
We now identify the shortcomings of our main result.
First, the number of measurements $M$ that scales like $K\log^2K\log N$ instead of the minimal $K\log(N/K)$.
We credit this (admittedly small) difference to our use of flat restricted orthogonality (i.e., Theorem~\ref{thm.fro to rip}) to demonstrate RIP; indeed, Theorem~14 in~\cite{{BandeiraFMW:13}} gives that using flat restricted orthogonality to prove RIP of Gaussian matrices, which are known to be RIP with minimal $M=O_\delta(K\log(N/K))$~\cite{CandesT:06}, leads to a scaling in $M$ that is identical to Theorem~\ref{theorem_legendre}, and so our result can perhaps be strengthened by somehow proving RIP ``directly.''
Next, we note that our construction uses entropy $H$ that scales like $K\log K\log N$, which is slightly more than the JL construction of Kane and Nelson~\cite{KaneN:10} that uses $O_\delta(K\log(N/K)\log(K\log(N/K)))$ random bits.
As we noted earlier, JL constructions of RIP matrices necessarily use at least $\Omega_\delta(M)=\Omega_\delta(K\log(N/K))$ random bits, whereas we believe our Legendre symbol construction can be derandomized quite a bit more:
\begin{conjecture}
\label{conj.main conjecture}
There exists a universal constant $C$ such that for every $\delta>0$, there exists $N_0>0$ and $P_0(N)=O(2^{\operatorname{poly}(N)})$ such that for every $(K,M,N)$ satisfying
\begin{equation*}
M\geq(C/\delta^2)K\log(N/K),
\qquad
N\geq N_0,
\end{equation*}
and for every prime $p\geq P_0$, the $M\times N$ matrix $\Phi$ defined entrywise by the Legendre symbol
\begin{equation*}
\Phi[m,n]
:=\frac{1}{\sqrt{M}}\bigg(\frac{M(n-1)+m}{p}\bigg)
\end{equation*}
satisfies the $(2K,\delta)$-restricted isometry property.
\end{conjecture}
At its heart, Conjecture~\ref{conj.main conjecture} is a statement about how well the Legendre symbol exhibits additive cancellations.
In particular, if one is willing to use flat restricted orthogonality as a proof technique for RIP, the statement is essentially a bound on incomplete sums of Legendre symbols, much like those investigated in~\cite{Chung:94}.
We had difficulty proving these cancellations, and so we injected some randomness to enable the application of Theorem~\ref{thm.legendre_pseudorandom}.
Since Conjecture~\ref{conj.main conjecture} could very well be beyond the reach of modern techniques, we pose the following weaker and more general problem, whose solution would be a substantial advance in the derandomization of RIP (since this level of derandomization is not achievable with JL projections):
\begin{problem}[Breaking the Johnson--Lindenstrauss bottleneck]
\label{problem.JL bottleneck}
Find a construction of $M\times N$ matrices which satisfies the $(K,\delta)$-restricted isometry property with high probability whenever $M=O_{\delta}(K\operatorname{polylog}N)$ and uses only $H=o_{\delta;N\rightarrow\infty}(K\log(N/K))$ random bits.
\end{problem}
\section{The main result}
In this section, we reduce our main result to a more general (and independently interesting) result, which requires the following definition:
\begin{definition}
Let $\{X_i\}_{i=1}^n$ be a sequence of Bernoulli random variables taking values in $\{\pm1\}$.
We say $\{X_i\}_{i=1}^n$ is \textit{$\varepsilon$-biased} if
\begin{equation}
\label{eq.epsilon biased}
\bigg|\mathbb{E}\prod_{i\in I}X_i\bigg|\leq\varepsilon
\end{equation}
for every nonempty $I\subseteq\{1,\ldots,n\}$.
\end{definition}
This notion of bias was introduced by Vazirani in~\cite{Vazirani:86}.
We will use bias as a means for derandomizing RIP matrices.
In particular, we will show that instead of using iid Bernoulli entries, it suffices to have small-bias Bernoulli entries for RIP, and then we will apply Theorem~\ref{thm.legendre_pseudorandom} to verify that random Legendre symbols have sufficiently small bias (even though the required entropy is small).
The idea that small-bias random variables can be used for derandomization is not new; indeed, one of their primary applications is to reduce the number of random bits needed for randomized algorithms~\cite{NaorN:90}.
As promised, the following generalizes the RIP result for matrices with independent Bernoulli entries to matrices with small-bias Bernoulli entries:
\begin{theorem}
\label{thm.unbiased rip}
Fix $C=5760000$.
There exists $N_0$ for which the following holds:
Given $N\geq N_0$, $K\geq1$ and $\delta>0$, take $M=(C/\delta^2)K\log^2 K\log N$ and pick
\begin{equation*}
\varepsilon
\leq\exp\Big(-40K\log((150/\delta)K\log K)\log N\Big).
\end{equation*}
Then any $M\times N$ matrix populated with $\varepsilon$-biased Bernoulli random variables (scaled by $1/\sqrt{M}$) satisfies the $(2K,\delta)$-restricted isometry property with probability $\geq1-2N^{-2K}$.
\end{theorem}
We note that the constants in this theorem have not been optimized.
Also, taking $\varepsilon=0$ recovers the iid Bernoulli result, though with extra log factors appearing in $M$.
We will prove this result in the next section.
For now, we establish that Theorem~\ref{thm.unbiased rip} implies our main result (Theorem~\ref{theorem_legendre}).
Drawing $X$ uniformly from $\{0,\ldots,2^H-1\}$, we bound the bias of the Legendre symbols $\{((X+i)/p)\}_{i=1}^{MN}$ using Theorem~\ref{thm.legendre_pseudorandom}:
\begin{equation*}
\bigg|\mathbb{E}\prod_{i\in I}\bigg(\frac{X+i}{p}\bigg)\bigg|
=\bigg|\frac{1}{2^H}\sum_{x=0}^{2^H-1}\prod_{i\in I}\bigg(\frac{x+i}{p}\bigg)\bigg|
\leq\frac{1}{2^H}|I|p^{1/2}\log p.
\end{equation*}
Since $|I|\leq MN$ and further $p\leq2(2^H+MN)\leq4\cdot2^H$ by Bertrand's postulate, we have
\begin{equation*}
\frac{1}{2^H}|I|p^{1/2}\log p
\leq\frac{1}{2^H}MN(4\cdot2^H)^{1/2}(\log 4+H\log 2)
\leq 4MNH\cdot 2^{-H/2}
\leq 4N^2\cdot2^{-H/3}.
\end{equation*}
As such, the random Legendre symbols we use to populate $\Phi$ are $(4N^2\cdot2^{-H/3})$-biased.
To use Theorem~\ref{thm.unbiased rip}, it remains to verify that this bias is sufficiently small:
\begin{equation*}
\log 4+2\log N-\frac{H}{3}\log 2\leq-40K\log((150/\delta)K\log K)\log N,
\end{equation*}
which is indeed satisfied by our choice of $H$ (i.e., taking $C_2$ to be sufficiently large).
Of course, we are not the first to use the Legendre symbol to produce small-bias random variables (for example, see~\cite{AlonGHP:92,Peralta:90}).
Also, there are several other constructions of small-bias random variables (e.g.,~\cite{AlonGHP:92,Ben-AroyaT:09,NaorN:90}), but these do not naturally lead to a conjecture such as Conjecture~\ref{conj.main conjecture}.
Interestingly, a stronger version of the Chowla conjecture~\cite{Chowla:65} implies that a randomly seeded portion of the Liouville function also produces small-bias random variables, and so one might pose a corresponding version of Conjecture~\ref{conj.main conjecture}.
Unfortunately, a large class of small-bias random variables cannot be used in conjunction with Theorem~\ref{thm.unbiased rip} to break the Johnson--Lindenstrauss bottleneck (Problem~\ref{problem.JL bottleneck}).
To see this, we first make an identification between small-bias random variables and linear codes.
A \textit{linear code} $\mathcal{C}\subseteq\mathbb{F}_2^q$ is a subspace of \textit{codewords}.
The $n\times q$ \textit{generator matrix} $G$ of an $n$-dimensional linear code $\mathcal{C}$ has the property that $\mathcal{C}=\{xG:x\in\mathbb{F}_2^n\}$, i.e., the rows of $G$ form a basis for $\mathcal{C}$.
The \textit{weight} of a codeword is the number of entries with value $1$.
\begin{proposition}[cf.~\cite{AlonGHP:92,Ben-AroyaT:09,NaorN:90}]
\
\begin{itemize}
\item[(a)]
Let $G$ be the generator matrix of an $n$-dimensional linear code in $\mathbb{F}_2^q$ such that every nonzero codeword has weight between $(1-\varepsilon)q/2$ and $(1+\varepsilon)q/2$.
Randomly sample $j$ uniformly over $\{1,\ldots,q\}$ and take $X_i:=(-1)^{G_{ij}}$ for every $i=1,\ldots,n$.
Then $\{X_i\}_{i=1}^n$ is $\varepsilon$-biased.
\item[(b)]
Let $\{X_i\}_{i=1}^n$ be an $\varepsilon$-biased sequence drawn uniformly from some multiset $\mathcal{X}\subseteq\{\pm1\}^n$ of size~$q$ ($\mathcal{X}$ is called an \textit{$\varepsilon$-biased set}).
For each $x\in\mathcal{X}$, consider the corresponding $g\in\mathbb{F}_2^n$ such that $x_i=(-1)^{g_i}$ for every $i=1,\ldots,n$.
Then the matrix $G$ whose columns are the $g$'s corresponding to $x$'s in $\mathcal{X}$ is the generator matrix of an $n$-dimensional linear code in $\mathbb{F}_2^q$ such that every nonzero codeword has weight between $(1-\varepsilon)q/2$ and $(1+\varepsilon)q/2$.
\end{itemize}
\end{proposition}
To see the significance of this proposition, consider the $n$-dimensional linear code $\mathcal{C}\subseteq\mathbb{F}_2^q$ corresponding to an $\varepsilon$-biased set of size $q$.
For each codeword $c\in\mathcal{C}$, define a unit vector $v_c\in\mathbb{R}^q$ whose entries have the form $(-1)^{c_i}/\sqrt{q}$.
It is easy to verify that $|\langle v_c,v_{c'}\rangle|\leq\varepsilon$ whenever $c\neq c'$, and so the Welch bound~\cite{Welch:74} gives
\begin{equation*}
\varepsilon^2
\geq\frac{2^n-q}{q(2^n-1)}.
\end{equation*}
For our application, we have $n=MN$ and $q=2^H$.
If $H\leq MN-1$ (i.e., $q\leq2^{n-1}$), then the Welch bound implies $\varepsilon^2\geq1/(2q)=2^{-(H+1)}$.
This gives the following result:
\begin{proposition}
Every $\varepsilon$-biased set $\mathcal{X}\subseteq\{\pm1\}^n$ has entropy $H\geq\min\{\log(1/\varepsilon),n-1\}$.
\end{proposition}
As such, applying Theorem~\ref{thm.unbiased rip} with any $\varepsilon$-biased set requires $H \gg_\delta K\log K\log N$.
In this sense, the Legendre construction in Theorem~\ref{theorem_legendre} is optimal.
On the other hand, no $\varepsilon$-biased set can be used with Theorem~\ref{thm.unbiased rip} to break the Johnson--Lindenstrauss bottleneck.
\section{Proof of Theorem~\ref{thm.unbiased rip}}
We will first show that $\Phi$ has $(K,\theta)$-flat restricted orthogonality, and then appeal to Theorem~\ref{thm.fro to rip} to get the $(2K,\delta)$-restricted isometry property.
To this end, fix a disjoint pair of subsets $I,J\subseteq\{1,\ldots,N\}$ with $|I|,|J|\leq K$.
We seek to bound the following probability:
\begin{equation}
\label{eq.prob to bound}
\operatorname{Pr}\left[\bigg|\bigg\langle\sum_{i\in I}\varphi_i,\sum_{j\in J}\varphi_j\bigg\rangle\bigg|>\theta(|I||J|)^{1/2}\right]
=\operatorname{Pr}\left[\bigg|\sum_{i\in I}\sum_{j\in J}\sum_{m=1}^M\Phi[m,i]\Phi[m,j]\bigg|>\theta(|I||J|)^{1/2}\right]
\end{equation}
Applying a version of Markov's inequality then gives
\begin{equation}
\label{eq.prob to bound 2}
\eqref{eq.prob to bound}
\leq\frac{1}{\big(\theta(|I||J|)^{1/2}\big)^q}\mathbb{E}\left[\bigg(\sum_{i\in I}\sum_{j\in J}\sum_{m=1}^M\Phi[m,i]\Phi[m,j]\bigg)^q\right],
\end{equation}
for some even integer $q$ which we will optimize later.
Observe that, because $q$ is even, the absolute value inside the expectation was not needed.
For now, we expand the product of sums and use linearity of expectation to get
\begin{equation*}
\eqref{eq.prob to bound 2}
=\frac{1}{\big(\theta(|I||J|)^{1/2}\big)^q}
\sum_{(i_1,j_1,m_1)\in I\times J\times [M]}\cdots\sum_{(i_q,j_q,m_q)\in I\times J\times [M]}
\mathbb{E}\bigg[\prod_{v=1}^q\Phi[m_v,i_v]\Phi[m_v,j_v]\bigg].
\end{equation*}
Next, since \eqref{eq.prob to bound 2} is nonnegative, the triangle inequality gives
\begin{equation*}
\eqref{eq.prob to bound 2}
=|\eqref{eq.prob to bound 2}|
\leq\frac{1}{\big(\theta(|I||J|)^{1/2}\big)^q}
\sum_{\{(i_v,j_v,m_v)\}_{v=1}^q\in(I\times J\times [M])^q}
\left|\mathbb{E}\bigg[\prod_{v=1}^q\Phi[m_v,i_v]\Phi[m_v,j_v]\bigg]\right|.
\end{equation*}
Most of the terms in the sum take the form of \eqref{eq.epsilon biased}, and so we can bound their contributions accordingly.
Each of the remaining terms has the property that all of its factors appear an even number of times in the product, and we will control the contribution of these terms by establishing how few they are.
With this in mind, we now define an index subset $S\subseteq(I\times J\times [M])^q$ (think ``surviving'' indices).
Explicitly, $\{(i_v,j_v,m_v)\}_{v=1}^q\in S$ if for every $v$, each of the sets
\begin{equation*}
\{v'\in[q]:(i_{v'},m_{v'})=(i_{v},m_{v})\},
\qquad
\{v'\in[q]:(j_{v'},m_{v'})=(j_{v},m_{v})\}
\end{equation*}
has an even number of elements.
We are able to use $\varepsilon$-bias to control all of the terms not in $S$.
Indeed, let $\{(i_v,j_v,m_v)\}_{v=1}^q\in S^\mathrm{c}$. Then there are $t\geq1$ distinct entries of $\Phi$ that appear in the product in (\ref{eq.prob to bound 2}) an odd number of times (observe that $I$ and $J$ are disjoint and hence an entry can not be simultaneously of the form $\Phi[m_v,i_v]$ and $\Phi[m_v,j_v]$).
Since there are a total of $2q$ entries of $\Phi$ in the product and the square of any entry is $1/M$, we can apply $\varepsilon$-bias to deduce that
\begin{equation}\label{eq_chowlatermsbound}
\left|\mathbb{E}\bigg[\prod_{v=1}^q\Phi[m_v,i_v]\Phi[m_v,j_v]\bigg]\right|
\leq M^{t/2-q}\frac{\varepsilon}{M^{t/2}}
=M^{-q}\varepsilon.
\end{equation}
Overall, the sum in our bound becomes
\begin{align*}
&\sum_{\{(i_v,j_v,m_v)\}_{v=1}^q\in S}\left|\mathbb{E}\bigg[\prod_{v=1}^q\Phi[m_v,i_v]\Phi[m_v,j_v]\bigg]\right|
+\sum_{\{(i_v,j_v,m_v)\}_{v=1}^q\in S^\mathrm{c}}\left|\mathbb{E}\bigg[\prod_{v=1}^q\Phi[m_v,i_v]\Phi[m_v,j_v]\bigg]\right|\\
&\qquad\qquad \leq M^{-q}\left(|S|+\varepsilon|S^\mathrm{c}|\right),
\end{align*}
where the inequality applies the fact that $|\Phi[m,n]|=1/\sqrt{M}$ to each term from $S$ and (\ref{eq_chowlatermsbound}) to each term from $S^\mathrm{c}$.
At this point, we wish to bound $|S|$.
For each $s=\{(i_v,j_v,m_v)\}_{v=1}^q\in S$, there exist two perfect matchings of $\{1,\ldots,q\}$, say $\mathcal{M}_1$ and $\mathcal{M}_2$, such that
\begin{equation*}
(i_{v},m_{v})=(i_{v'},m_{v'})~\forall\{v,v'\}\in \mathcal{M}_1,
\quad
(j_{v},m_{v})=(j_{v'},m_{v'})~\forall\{v,v'\}\in \mathcal{M}_2.
\tag{$P[s,\mathcal{M}_1,\mathcal{M}_2]$}
\end{equation*}
(Here, we use the phrase ``perfect matching'' so as to convey that $\{1,\ldots,q\}$ is partitioned into sets of size $2$.
Let $\mathcal{M}(q)$ denote all such perfect matchings.)
We now start our bound:
\begin{align*}
|S|
&=\#\Big\{s\in(I\times J\times [M])^q:\exists\mathcal{M}_1,\mathcal{M}_2\in\mathcal{M}(q)\mbox{ such that }P[s,\mathcal{M}_1,\mathcal{M}_2]\Big\}\\
&\leq\#\Big\{(s,\mathcal{M}_1,\mathcal{M}_2):P[s,\mathcal{M}_1,\mathcal{M}_2]\Big\}\\
&=\sum_{\mathcal{M}_1\in\mathcal{M}(q)}\sum_{\mathcal{M}_2\in\mathcal{M}(q)}\#\Big\{s:P[s,\mathcal{M}_1,\mathcal{M}_2]\Big\}.
\end{align*}
Next, given a perfect matching $\mathcal{M}\in\mathcal{M}(q)$, let $C(\mathcal{M})$ denote the set of $q$-tuples $\{m_v\}_{v=1}^q$ satisfying $m_{v}=m_{v'}$ whenever $\{v,v'\}\in \mathcal{M}$ (these are the $q$-tuples which consistently ``color'' the matching).
Note that the $q$-tuple of $m_v$'s from $s=\{(i_v,j_v,m_v)\}_{v=1}^q$ satisfying $P[s,\mathcal{M}_1,\mathcal{M}_2]$ lies in both $C(\mathcal{M}_1)$ and $C(\mathcal{M}_2)$.
Moreover, the $q$-tuple of $i_v$'s from $s=\{(i_v,j_v,m_v)\}_{v=1}^q$ satisfying $P[s,\mathcal{M}_1,\mathcal{M}_2]$ assigns members of $I$ to the $q/2$ pairs $\{v,v'\}\in\mathcal{M}_1$, meaning the $q$-tuple is one of $|I|^{q/2}$ such possibilities; similarly, there are $|J|^{q/2}$ possibilities for the $q$-tuple of $j_v$'s.
As such, we can continue our bound:
\begin{equation*}
|S|
\leq|I|^{q/2}|J|^{q/2}\sum_{\mathcal{M}_1\in\mathcal{M}(q)}\sum_{\mathcal{M}_2\in\mathcal{M}(q)}|C(\mathcal{M}_1)\cap C(\mathcal{M}_2)|.
\end{equation*}
At this point, we note that $\sum_{\mathcal{M}_2\in\mathcal{M}(q)}|C(\mathcal{M}_1)\cap C(\mathcal{M}_2)|$ does not depend on $\mathcal{M}_1$, since any permutation of $\{1,\ldots,q\}$ that sends one version of $\mathcal{M}_1$ to another can be viewed as a permutation acting on $\mathcal{M}_2$ (thereby merely permuting the summands).
As such, we can express our bound in terms of a ``canonical matching'' $\mathcal{M}_0:=\{\{1,2\},\{3,4\},\ldots,\{q-1,q\}\}$:
\begin{equation*}
|S|
\leq|I|^{q/2}|J|^{q/2}|\mathcal{M}(q)|\sum_{\mathcal{M}_2\in\mathcal{M}(q)}|C(\mathcal{M}_0)\cap C(\mathcal{M}_2)|.
\end{equation*}
Note that $|\mathcal{M}(q)|=(q-1)!!$ since there are $q-1$ possible members to match with $1$, and then $q-3$ members remaining to match with the smallest free element, etc.
It remains to determine the sum over $\mathcal{M}(q)$.
To parse this sum, consider a graph of $q$ vertices with edge set $\mathcal{M}_0\cup\mathcal{M}_2$.
Then $|C(\mathcal{M}_0)\cap C(\mathcal{M}_2)|$ counts the number of ways of coloring the vertices using $M$ colors in such a way that each component (which is necessarily a cycle) has vertices of a common color.
We will use this interpretation to prove the following claim:
\begin{equation*}
\sum_{\mathcal{M}_2\in\mathcal{M}(q)}|C(\mathcal{M}_0)\cap C(\mathcal{M}_2)|
=\frac{(M+q-2)!!}{(M-2)!!}.
\end{equation*}
To proceed, start with a graph $G$ of $q$ vertices, and let the edge set be $\mathcal{M}_0$.
Let $A$ denote the ``available'' set of vertices, and initialize $A$ as $\{1,\ldots,q\}$.
We will iteratively update the pair $(G,A)$ as we build all possible matchings $\mathcal{M}_2$ and colorings.
At each step, denote $k:=|A|$.
Take the least-numbered available vertex and draw an edge between it and any of the other $k-1$ available vertices.
Add this edge to $\mathcal{M}_2$ and remove both vertices from $A$.
If this edge completes a cycle in $G$, then color that cycle with any of the $M$ colors.
Overall, in this step you either pick one of $k-2$ vertices which do not complete the cycle, or you complete the cycle and pick one of $M$ colors; this totals to $M+k-2$ choices at each step.
Since the number $k$ of available vertices decreases by $2$ at each step, the product of these totals corresponds to the claim.
At this point, we summarize the status of our bound:
\begin{align}
\nonumber
\operatorname{Pr}\left[\bigg|\bigg\langle\sum_{i\in I}\varphi_i,\sum_{j\in J}\varphi_j\bigg\rangle\bigg|>\theta(|I||J|)^{1/2}\right]
&\leq\frac{1}{\big(M\theta(|I||J|)^{1/2}\big)^q}\bigg(|S|+\varepsilon|S^\mathrm{c}|\bigg)\\
\label{eq.two terms to bound}
&\leq\frac{(q-1)!!(M+q-2)!!}{(M\theta)^q(M-2)!!}+\frac{\varepsilon|I|^{q/2}|J|^{q/2}}{\theta^q},
\end{align}
where the last inequality applies our bound on $|S|$ along with $|S^\mathrm{c}|\leq|(I\times J\times [M])^q|=(|I||J|M)^q$.
To continue, we will bound the terms in \eqref{eq.two terms to bound} separately.
When $M$ is even (otherwise, $M-1$ is even and similar bounds will apply), the first term has the following bound:
\begin{align*}
\frac{(q-1)!!(M+q-2)!!}{(M\theta)^q(M-2)!!}
&\leq\frac{q!!(M+q-2)!!}{(M\theta)^q(M-2)!!}\\
&\leq\frac{2^q((\frac{q}{2})!)^2}{(M\theta)^q}\binom{\frac{M}{2}+\frac{q}{2}-1}{\frac{q}{2}}
\leq\frac{2^qe^2(\frac{q/2+1}{e})^{q+2}}{(M\theta)^q}\bigg(e\cdot\frac{\frac{M}{2}+\frac{q}{2}-1}{\frac{q}{2}}\bigg)^{q/2},
\end{align*}
where the last step uses Stirling-type bounds on both the factorial and the binomial coefficient.
Next, we combine like exponents to get
\begin{align*}
\frac{(q-1)!!(M+q-2)!!}{(M\theta)^q(M-2)!!}
&\leq e^2\Big(\frac{q/2+1}{e}\Big)^2\bigg(\frac{4}{(M\theta)^2}\Big(\frac{q/2+1}{e}\Big)^2\cdot e\cdot\Big(\frac{M}{q}+1-\frac{2}{q}\Big)\bigg)^{q/2}\\
&\leq q^2\bigg(\frac{4}{(M\theta)^2}\Big(\frac{q}{e}\Big)^2\cdot e\cdot\frac{2M}{q}\bigg)^{q/2},
\end{align*}
where the last step assumes $q\geq 2$ (given our choice of $q$ later, this will occur for sufficiently large $N$) so that $q/2+1\leq q$ and also assumes $q\leq M$ (which will follow from the fact that $\delta\leq 1$) so that $M/q+1\leq 2M/q$.
At this point, we take $q=M\theta^2/8$ (or more precisely, take $q$ to be the smallest even integer larger than $M\theta^2/8$) to get
\begin{equation*}
\frac{(q-1)!!(M+q-2)!!}{(M\theta)^q(M-2)!!}
\leq \Big(\frac{M\theta^2}{8}\Big)^2\cdot e^{-M\theta^2/16}.
\end{equation*}
This can be simplified by noting that $M=(C/\delta^2)K\log^2K\log N$ and $\theta=\delta/(150\log K)$ together give $M\theta^2=256K\log N$:
\begin{equation}
\label{eq.first term bound}
\frac{(q-1)!!(M+q-2)!!}{(M\theta)^q(M-2)!!}
\leq \operatorname{exp}\bigg(2\log\Big(\frac{M\theta^2}{8}\Big)-\frac{M\theta^2}{16}\bigg)
\leq\operatorname{exp}\Big(-8K\log N\Big),
\end{equation}
for sufficiently large $N$.
Next, we bound the second term in \eqref{eq.two terms to bound}:
\begin{equation}
\label{eq.second term bound}
\frac{\varepsilon|I|^{q/2}|J|^{q/2}}{\theta^q}
\leq\varepsilon(K/\theta)^q
=\varepsilon((150/\delta)K\log K)^q
\leq\exp\Big(-8K\log((150/\delta)K\log K)\log N\Big)
\end{equation}
Now we can combine the bounds \eqref{eq.first term bound} and \eqref{eq.second term bound} to get
\begin{align*}
&\operatorname{Pr}\left[\bigg|\bigg\langle\sum_{i\in I}\varphi_i,\sum_{j\in J}\varphi_j\bigg\rangle\bigg|>\theta(|I||J|)^{1/2}\right]\\
&\qquad\leq \operatorname{exp}\Big(-8K\log N\Big)+\exp\Big(-8K\log((150/\delta)K\log K)\log N\Big)\\
&\qquad\leq 2\operatorname{exp}\Big(-8K\log N\Big).
\end{align*}
At this point, we are ready to show that $\Phi$ has the $(2K,\delta)$-restricted isometry property with high probability.
This calculation will use a union bound over all possible choices of disjoint $I,J\subseteq\{1,\ldots,N\}$ with $|I|,|J|\leq K$, of which there are
\begin{equation*}
\sum_{|I|=1}^K\sum_{|J|=1}^K\binom{N}{|I|}\binom{N-|I|}{|J|}
\leq K^2\binom{N}{K}^2
\leq K^2\Big(\frac{eN}{K}\Big)^{2K}
\leq\operatorname{exp}\Big(4K+2K\log(N/K)\Big).
\end{equation*}
With this, we apply a union bound to get
\begin{align*}
\operatorname{Pr}\Big[\mbox{$\Phi$ is not $(2K,\delta)$-RIP}\Big]
&\leq\operatorname{Pr}\Big[\mbox{$\Phi$ does not have $(K,\theta)$-FRO}\Big]\\
&\leq\operatorname{exp}\Big(4K+2K\log(N/K)\Big)\cdot\operatorname{Pr}\left[\bigg|\bigg\langle\sum_{i\in I}\varphi_i,\sum_{j\in J}\varphi_j\bigg\rangle\bigg|>\theta(|I||J|)^{1/2}\right]\\
&\leq2\operatorname{exp}\Big(4K+2K\log(N/K)-8K\log N\Big)\\
&\leq2\operatorname{exp}\Big(-2K\log N\Big),
\end{align*}
which completes the result.
\section*{Acknowledgments}
The authors thank Prof.\ Peter Sarnak for insightful discussions.
A.\ S.\ Bandeira was supported by AFOSR Grant No.\ FA9550-12-1-0317, and M.\ Fickus and D.\ G.\ Mixon were supported by NSF Grant No.\ DMS-1321779.
The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
|
2,877,628,089,455 | arxiv | \section{Introduction}
Knowledge graphs (KGs) store structured information of real-world entities and facts.
A KG usually consists of a collection of triplets. Each triplet $(h, r, t)$ indicates that head entity $h$ is related to tail entity $t$ through relationship type $r$.
Nonetheless, KGs are often incomplete and noisy.
To address this issue, researchers have proposed a number of KG completion methods to predict missing links/relations in KGs \cite{bordes2013translating, trouillon2016complex, yang2015embedding, sun2019rotate, kazemi2018simple, zhang2019quaternion, galarraga2015fast, yang2017differentiable, ho2018rule, zhang2019iteratively, sadeghian2019drum}.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\includegraphics[width=\textwidth]{figures/kg_1.pdf}
\caption{Consider we aim to predict whether \rl{Ron Weasley} or \rl{Hedwig} is a \rl{Pet of} \rl{Harry Potter}. Both entities have the same relational path (\rl{Lives with}) to \rl{Harry Potter} but they have distinct relational context: \rl{Ron Weasley} has $\{\rl{Brother of}, \rl{Lives with}\}$, while \rl{Hedwig} has $\{\rl{Bought}, \rl{Lives with}\}$. Capturing the relational context of entities allows our model to make a distinction between \rl{Ron Weasley}, who is a person, and \rl{Hedwig}, which is an owl.}
\label{fig:kg_1}
\end{subfigure}
\begin{subfigure}[b]{0.47\textwidth}
\includegraphics[width=\textwidth]{figures/kg_2.pdf}
\caption{Two head entities \rl{Hermione Granger} and \rl{Draco Malfoy} have the same relational context $\{\rl{Occupation}, \rl{House}\}$, but different relational paths to the tail entity \rl{Harry Potter} \{(\rl{House}, \rl{House}), (\rl{Occupation}, \rl{Occupation})\} vs. \{(\rl{Occupation}, \rl{Occupation})\}, which allows our model to predict friendship between \rl{Harry Potter} and \rl{Hermione Granger} vs. \rl{Draco Malfoy}.}
\label{fig:kg_2}
\end{subfigure}
\caption{(a) Relational context of an entity and (b) relational paths between entities. Our model is able to capture both.}
\label{fig:intro}
\end{figure}
In general, relation types are not uniformly distributed over a KG but spatially correlated with each other.
For example, the neighboring relations of ``\rl{graduated from}'' in the KG are more likely to be ``\rl{person.birthplace}'' and ``\rl{university.location}'' rather than ``\rl{movie.language}''.
Therefore, for a given entity pair $(h, t)$, characterizing the relation types of neighboring links of $h$ and $t$ will provide valuable information when inferring the relation type between $h$ and $t$.
Inspired by recent success of graph neural networks \cite{kipf2017semi, hamilton2017inductive, xu2019powerful}, we propose using \textit{message passing} to capture the neighborhood structure for a given entity pair.
However, traditional message passing methods usually assume that messages are associated with nodes and messages are passed from nodes to nodes iteratively, which are not suitable for KGs where edge features (relation types) are more important.
\xhdr{Relational message passing}
To address the above limitation, we propose \textit{relational message passing} for KG completion.
Unlike traditional node-based message passing, relational message passing \textit{only} considers edge features (relation types), and passes messages of an edge directly to its neighboring edges.
Note that since relational message passing only models relations rather than entities, it brings three additional benefits compared with existing knowledge graph embedding methods \cite{bordes2013translating, trouillon2016complex, yang2015embedding, sun2019rotate, kazemi2018simple, zhang2019quaternion}:
(1) it is \textit{inductive}, since it can handle entities that do not appear in the training data during inference stage;
(2) it is \textit{storage-efficient}, since it does not calculate embeddings of entities; and
(3) it is \textit{explainable}, since it is able to provide explainability for predicted results by modeling the correlation strength among relation types.
However, a potential issue of relational message passing is that its computational complexity is significantly higher than node-based message passing (Theorem \ref{thm:2}).
To solve this issue, we propose \textit{alternate relational message passing} that passes relational messages between nodes and edges \textit{alternately} over the KG.
We prove that alternate message passing scheme greatly improves time efficiency and achieves \textit{the same order of computational complexity} as traditional node-based message passing (Theorem \ref{thm:1} and \ref{thm:3}).
\xhdr{Relational context and relational paths}
Under the alternate relational message passing framework, we explore two kinds of local subgraph topology for a given entity pair $(h, t)$ (see Figure \ref{fig:intro} for an illustrating example):
(1) \textit{Relational context}.
It is important to capture the neighboring relations of a given entity in the KG, because neighboring relations provide us with valuable information about what is the nature or the ``type'' of the given entity (Figure \ref{fig:kg_1}).
Many entities in KGs are not typed or are very loosely typed, so being able to learn about the entity and its context in the KG is valuable.
We design a multi-layer relational message passing scheme to aggregate information from multi-hop neighboring edges of $(h, t)$.
(2) \textit{Relational paths}.
Note that modeling only relational context is not able to identify the relative position of $(h, t)$.
It is also important to capture the set of relational paths between $(h, t)$ (Figure \ref{fig:kg_2}).
Here different paths of connections between the entities reveal the nature of their relationship and help with the prediction.
Therefore, we calculate all relational paths connecting $h$ and $t$ in the KG and pass relational messages along these paths.
Finally, we use an attention mechanism to selectively aggregate representations of different relational paths, then combine the above two modules together for relation prediction.
\xhdr{Experiments}
We conduct extensive experiments on five well-known KGs as well as a new KG proposed by us, \textit{DDB14 dataset}.
Experimental results demonstrate that our proposed model \textsc{PathCon}\xspace (short for relational PATHs and CONtext) significantly outperforms state-of-the-art KG completion methods, for example, the absolute Hit@1 gain over the best baseline is $16.7\%$ and $6.3\%$ on WN18RR and NELL995, respectively.
Our ablation studies show the effectiveness of our approach and demonstrate the importance of relational context as well as relational paths.
Our method is also shown to maintain strong performance in inductive KG completion, and it provides high explainability by identifying important relational context and relation paths for a given predicted relation.
\xhdr{Contributions}
Our key contributions are listed as follows:
\begin{itemize}
\item We propose \textit{alternate relational message passing} framework for KG completion, which is \textit{inductive}, \textit{storage-efficient}, \textit{explainable}, and \textit{computationally efficient} compared with existing embedding-based methods;
\item Under the proposed framework, we explore two kinds of subgraph topology: \textit{relational context} and \textit{relational paths}, and show that they are critical to relation prediction;
\item We propose a new KG dataset DDB14 (Disease Database with 14 relation types) that is suitable for KG-related research.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.42\textwidth]{figures/model}
\caption{An example of \textsc{PathCon}\xspace considering both the relational context within 2 hops of the head and the tail entities (denoted by red edges) and relational paths of length up to 3 relations that connect head to tail (denoted by green arrows). Context and paths are captured based on relation types (not entities) they contain. By combining the context and paths \textsc{PathCon}\xspace predicts the probability of relation $r$.}
\label{fig:model}
\end{figure}
\section{Problem Formulation}
\label{sec:problem_formulation}
Let $\mathcal G = (\mathcal V, \mathcal E)$ be an instance of a knowledge graph, where $\mathcal V$ is the set of nodes and $\mathcal E$ is the set of edges.
Each edge $e$ has a relation type $r \in \mathcal R$.
Our goal is to predict missing relations in $\mathcal G$, i.e., given an entity pair $(h, t)$, we aim to predict the relation of the edge between them.\footnote{Some of the related work formulates this problem as predicting the missing tail (head) entity given a head (tail) entity and a relation. The two problems are actually reducible to each other: Given a model $\Phi (\cdot | h, t)$ that outputs the distribution over relation types for an entity pair $(h, t)$, we can then build a model $\Gamma (\cdot | h, r) = \textsc{SoftMax}_t \left( \Phi (r | h, t) \right)$ that outputs the distribution over tail entities given $h$ and $r$, and vice versa. Since the two problems are equivalent, we only focus on relation prediction in this work.}
Specifically, we aim to model the distribution over relation types given a pair of entities $(h, t)$: $p (r | h, t)$.
This is equivalent to modeling the following term
\begin{equation}
\label{eq:bayes}
p (r | h, t) \propto p (h, t | r) \cdot p (r)
\end{equation}
according to Bayes' theorem.
In Eq. (\ref{eq:bayes}), $p (r)$ is the prior distribution over relation types and serves as the regularization of the model.
Then the first term can be further decomposed to
\begin{equation}
\label{eq:bayes2}
p (h, t | r) = \frac{1}{2} \Big( p (h | r) \cdot p (t | h, r) + p (t | r) \cdot p (h | t, r) \Big).
\end{equation}
Eq. (\ref{eq:bayes2}) sets up the guideline for designing our model.
The term $p (h | r)$ or $p (t | r)$ measures the likelihood of an entity given a particular relation.
Since our model does not consider the identity of entities, we use an entity's \textit{local relational subgraph} instead to represent the entity itself, i.e., $p \big( C(h) | r \big)$ and $p \big( C(t) | r \big)$ where $C(\cdot)$ denotes the local relational subgraph of an entity.
This is also known as \textit{relational context} for $h$ and $t$.
The term $p (t | h, r)$ or $p (h | t, r)$ in Eq. (\ref{eq:bayes2}) measures the likelihood of how $t$ can be reached from $h$ or the other way around given that there is a relation $r$ between them.
This inspires us to model the \textit{relational paths} between $h$ and $t$ in the KG.
In the following we show how to model the two factors in our method and how they contribute to relation prediction.
\begin{table}[t]
\centering
\small
\setlength{\tabcolsep}{10pt}
\begin{tabular}{c|c}
\hline
Symbol & Description \\
\hline
$h, t$ & Head entity and tail entity \\
$r$ & Relation type \\
$s_e^i$ & Hidden state of edge $e$ at iteration $i$ \\
$m_v^i$ & Message of node $v$ at iteration $i$ \\
$\mathcal N(e)$ & Endpoint nodes of edge $e$ \\
$\mathcal N(v)$ & Neighbor edges of node $v$ \\
$s_{(h,t)}$ & Context representation of the entity pair ($h$,$t$) \\
$s_{h\to t}$ & Path representation of all paths from $h$ to $t$ \\
$\alpha_P$ & Attention weight of path $P$ \\
$\mathcal P_{h\rightarrow t}$ & Set of paths from $h$ to $t$ \\
\hline
\end{tabular}
\vspace{0.05in}
\caption{Notation used in this paper.}
\vspace{-0.2in}
\label{table:symbols}
\end{table}
\section{Our Approach}
In this section, we first introduce the relational message passing framework, then present two modules of the proposed \textsc{PathCon}\xspace: relational context message passing and relational path message passing.
Notations used in this paper are listed in Table \ref{table:symbols}.
\subsection{Relational Message Passing Framework}
\label{sec:framework}
\textbf{Traditional node-based message passing}.
We first briefly review traditional node-based message passing method for general graphs.
Assume that each node $v$ is with feature $x_v$.
Then the message passing runs for multiple timesteps over the graph, during which the hidden state $s^i_v$ of each node $v$ in iteration $i$ is updated by
\begin{align}
m^i_v =& A \left( \big \{ s^i_u \big \}_{u \in \mathcal N (v)} \right), \label{eq:node_based_mp_1} \\
s^{i+1}_v =& U \left( s^i_v, m^i_v \right), \label{eq:node_based_mp_2}
\end{align}
where $m^i_v$ is the message received by node $v$ in iteration $i$, $\mathcal N(v)$ denotes the set of neighbor nodes of $v$ in the graph, $A(\cdot)$ is message aggregation function, and $U(\cdot)$ is node update function.
The initial hidden state $s^0_v = x_v$.
The above framework, though popular for general graphs and has derived many variants such as GCN \cite{kipf2017semi}, GraphSAGE \cite{hamilton2017inductive}, and GIN \cite{xu2019powerful}, faces the following challenges when applied to knowledge graphs:
(1) Unlike general graphs, in most KGs, edges have features (relation types) but nodes don't, which makes node-based message passing less natural for KGs.
Though node features can be set as their identities (i.e., one-hot vectors), this will lead to another two issues:
(2) Modeling identity of nodes cannot manage previously unseen nodes during inference and fails in inductive settings.
(3) In real-world KGs, the number of entities are typically much larger than the number of relation types, which requires large memory for storing entity embeddings.
\xhdr{Relational message passing}
To address the above problems, a natural thought is to perform message passing over edges instead of nodes:
\begin{align}
m^i_e =& A \left( \big \{ s^i_{e'} \big \}_{e' \in \mathcal N (e)} \right), \label{eq:relational_mp_1} \\
s^{i+1}_e =& U \left( s^i_e, m^i_e \right), \label{eq:relational_mp_2}
\end{align}
where $\mathcal N(e)$ denotes the set of neighbor edges of $e$ (i.e., edges that share at lease one common end-point with $e$) in the graph, and $s^0_e = x_e$ is the initial edge feature of $e$, i.e., the relation type.
Therefore, Eqs. (\ref{eq:relational_mp_1}) and (\ref{eq:relational_mp_2}) are called \textit{relational message passing}.
Relational message passing avoids the drawbacks of node-based message passing, however, it brings a new issue of computational efficiency when passing messages.
To see this, we analyze the computational complexity of the two message passing schemes (proofs are given in Appendix \ref{sec:proof_1} and \ref{sec:proof_2}):
\begin{theorem}[Complexity of node-based message passing]
\label{thm:1}
Consider a graph with $N$ nodes and $M$ edges.
The expected cost of node-based message passing (Eqs. (\ref{eq:node_based_mp_1}) and (\ref{eq:node_based_mp_2})) in each iteration is $2M + 2N$.
\end{theorem}
\begin{theorem}[Complexity of relational message passing]
\label{thm:2}
Consider a graph with $N$ nodes and $M$ edges.
The expected cost of relational message passing (Eqs. (\ref{eq:relational_mp_1}) and (\ref{eq:relational_mp_2})) in each iteration is $N \cdot {\rm Var}[d] + \frac{4M^2}{N}$, where ${\rm Var}[d]$ is the variance of node degrees in the graph.
\end{theorem}
\xhdr{Alternate relational message passing}
According to the above theorems, the complexity of relational message passing is much higher than node-based message passing, especially in real-world graphs where node distribution follows the power law distribution whose variance (${\rm Var}[d]$) is extremely large due to the long tail.
To reduce the redundant computation in relational message passing and improve its computational efficiency, we propose the following message passing scheme for KGs:
\begin{align}
m^i_v =& A_1 \left( \big \{ s^i_{e} \big \}_{e \in \mathcal N (v)} \right), \label{eq:alternate_mp_1} \\
m^i_e =& A_2 \left( m^i_v, m^i_u \right), \ v, u \in \mathcal N(e), \label{eq:alternate_mp_2} \\
s^{i+1}_e =& U \left( s^i_e, m^i_e \right). \label{eq:alternate_mp_3}
\end{align}
We decompose edge aggregation in Eq. (\ref{eq:relational_mp_1}) into two steps as Eqs. (\ref{eq:alternate_mp_1}) and (\ref{eq:alternate_mp_2}).
In Eq. (\ref{eq:alternate_mp_1}), for each node $v$, we aggregate all the edges that $v$ connects to by an aggregation function $A_1(\cdot)$ and get message $m^i_v$, where $\mathcal N(v)$ denotes the set of neighbor edges for node $v$.
Then in Eq. (\ref{eq:alternate_mp_2}), we obtain message $m^i_e$ of edge $e$ by aggregating messages from its two end-points $v$ and $u$ using function $A_2(\cdot)$, where $\mathcal N(e)$ denotes the set of neighbor nodes for edge $e$.
The hidden state of edge $e$ is finally updated using the message $m^i_e$ as in Eq. (\ref{eq:alternate_mp_3}).
An intuitive understanding of alternate relational message passing is that nodes here serve as ``distribution centers'' that collect and temporarily store the messages from their neighbor edges, then propagate the aggregated messages back to each of their neighbor edges.
Therefore, we call Eqs. (\ref{eq:alternate_mp_1})-(\ref{eq:alternate_mp_3}) \textit{alternate relational message passing}, as messages are passed alternately between nodes and edges.
The complexity of alternate relational message passing is given as follows (proof is given in Appendix \ref{sec:proof_3}):
\begin{theorem}[Complexity of alternate relational message passing]
\label{thm:3}
Consider a graph with $N$ nodes and $M$ edges.
The expected cost of alternate relational message passing (Eqs. (\ref{eq:alternate_mp_1})-(\ref{eq:alternate_mp_3})) in each iteration is $6M$.
\end{theorem}
From Theorem \ref{thm:3} it is clear to see that alternate relational message passing greatly reduces the time overhead and achieves the same order of complexity as node-based message passing.
\xhdr{Remarks}
We present the following two remarks to provide more insight on the proposed framework:
\begin{remark}[Relationship with belief propagation]
\label{remark:1}
Alternate relational message passing is conceptually related to belief propagation (BP) \cite{yedidia2003understanding}, which is also a message-passing algorithm that passes messages between nodes and edges.
But note that they are significantly different in:
(1) application fields.
BP is used to calculate the marginal distribution of unobserved variables in a graphical model, while our method aims to predict the edge type in KGs;
(2) the purpose of using edge-node alternate message passing.
BP uses this because of the special structure of factor graphs, while we use this to reduce the computational overhead.
\end{remark}
\begin{remark}[Utilizing node features]
\label{remark:2}
Though our proposed framework is claimed to only use edge features, it can be easily extended to the case where node features are present and assumed to be important, by additionally including the feature vector of node $v$ in Eq. (\ref{eq:alternate_mp_1}), i.e., $m^i_v = A_1 \left( \big \{ s^i_{e} \big \}_{e \in \mathcal N (v)}, x_v \right)$, where $x_v$ is the feature of node $v$.
As long as node features do not contain node identities, our proposed framework is still inductive.
We do not empirically study the performance of our method on node-feature-aware cases, because node features are unavailable for all datasets used in this paper.
We leave the exploration of this extension to future work.
\end{remark}
\subsection{Relational Context}
\label{sec:context}
For a KG triplet $(h, r, t)$, relational context of $h$ and $t$ is usually highly correlated with $r$.
For example, if $r$ is ``graduated from'', it's reasonable to guess that the surrounding relations of $h$ are ``person.birthplace'', ``person.gender'', etc., and the surrounding relations of $t$ are ``institution.location'', ``university.founder'', ``university.president'', etc.
Therefore, the context of $h$ and $t$ will provide valuable clues when identifying the relation type of the edge between them, and here we use the proposed message passing method to learn from relational context.
Denote $s_e^i$ as the hidden state of edge $e$ in iteration $i$, and $m_v^i$ as the message stored at node $v$ in iteration $i$.
We instantiate the alternate relational message passing in Eqs. (\ref{eq:alternate_mp_1})-(\ref{eq:alternate_mp_3}) to learn the representation of each edge:
\begin{align}
m^i_v =& \sum\nolimits_{e \in \mathcal N (v)} s^i_{e}, \label{eq:context_mp_1} \\
s^{i+1}_e =& \sigma \left( \left[ m^i_v, m^i_u, s^i_e \right] \cdot W^i + b^i \right), \ v, u \in \mathcal N(e),\label{eq:context_mp_2}
\end{align}
where $[\cdot]$ is the concatenation function, $W^i$, $b^i$, and $\sigma(\cdot)$ are the learnable transformation matrix, bias, and nonlinear activation function, respectively.\footnote{We shall discuss other implementations of Eqs. (\ref{eq:alternate_mp_1})-(\ref{eq:alternate_mp_3}) in Section \ref{sec:design_alternatives} and examine their performance in experiments.}
$s_e^0 = x_e$ is initial feature of edge $e$, which can be taken as the one-hot identity vector of the relation type that $e$ belongs to.\footnote{In cases where relation types have names, initial features can also be bag-of-words (BOW) or sentence embeddings learned by language models like BERT \cite{devlin2018bert}.
We shall investigate the performance of different initial feature types in experiments.}
Relational context message passing in Eqs. (\ref{eq:context_mp_1}) and (\ref{eq:context_mp_2}) are repeated for $K$ times.
The final message $m_h^{K-1}$ and $m_t^{K-1}$ are taken as the representation for head $h$ and tail $t$, respectively.
We also give an illustrative example of relational context message passing in Figure \ref{fig:model}, where the red/pink edges denote the first-order/second-order contextual relations.
\subsection{Relational Paths}
\label{sec:path}
We follow the discussion in Section \ref{sec:problem_formulation} and discuss how to model the term $p(t | h, r)$ or $p(h | t, r)$.
Note that we do not consider node/edge identity in relational context message passing, which leads to a potential issue that our model is not able to identify the relative position between $h$ and $t$ in the KG.
For example, suppose for a given entity pair $(h, t)$, $h$ is surrounded by ``person.birthplace'', ``person.gender'', etc., and $t$ is surrounded by ``institution.location'', ``university.founder'', ``university.president'', etc.
Then it can be inferred that $h$ is probably a person and $t$ is probably a university, and there should be a relation ``graduated\_from'' between them because such a pattern appears frequently in the training data.
However, the person may have no relationship with the university and they are far from each other in the KG.
The reason why such false positive case happens is that relational context message passing can only detect the ``type'' of $h$ and $t$, but is not aware of their relative position in the KG.
To solve this problem, we propose to explore the connectivity pattern between $h$ and $t$, which are represented by the paths connecting them in the KG.
Specifically, a raw path from $h$ to $t$ in a KG is a sequence of entities and edges: $h(v_0) \xrightarrow{e_0} v_1 \xrightarrow{e_1} v_2 \cdots v_{L-1} \xrightarrow{e_{L-1}} t(v_L)$, in which two entities $v_i$ and $v_{i+1}$ are connected by edge $e_i$, and each entity in the path is unique.\footnote{Entities in a path are required to be unique because a loop within a path does not provide additional semantics thus should be cut off from the path.}
The corresponding relational path $P$ is the sequence of relation types of all edges in the given raw path, i.e., $P = \left( r_{e_0}, r_{e_1}, ..., r_{e_{L-1}} \right)$, where $r_{e_i}$ is the relation type of edge $e_i$.
Note that we do not use the identity of nodes when modeling relational paths, which is the same as for relational context.
Denote $\mathcal P_{h \rightarrow t}$ as the set of all relational paths from $h$ to $t$ in the KG.
Our next step is to define and calculate the representation of relational paths.
In \textsc{PathCon}\xspace, we assign an independent embedding vector $s_P$ for each relational path $P \in \mathcal P_{h \rightarrow t}$.\footnote{Other methods for calculating path representations are also possible. We shall discuss them in Section \ref{sec:design_alternatives}.}
A potential concern here is that the number of different paths increases exponentially with the path length (there are $|r|^k$ $k$-hop paths), however, in practice we observe that in real-world KGs most paths actually do not occur (e.g., only 3.2\% of all possible paths of length 2 occur in FB15K dataset), and the number of different paths is actually quite manageable for relatively small values of $k$ ($k\le4$).
An illustrative example of relational paths is shown in Figure \ref{fig:model}, where the two green arrows denote the relational paths from head entity $h$ to tail entity $t$.
\subsection{Combining Relational Context and Paths}
\label{sec:model}
For relational context, we use massage passing scheme to calculate the final message $m_h^{K-1}$ and $m_t^{K-1}$ for $h$ and $t$, which summarizes their context information, respectively.
$m_h^{K-1}$ and $m_t^{K-1}$ are further combined together for calculating the context of $(h, t)$ pair:
\begin{equation}
\label{eq:context}
s_{(h, t)} = \sigma \left( \left[ m^{K-1}_h, m^{K-1}_t \right] \cdot W^{K-1} + b^{K-1} \right),
\end{equation}
where $s_{(h, t)}$ denotes the context representation of the entity pair $(h, t)$.
Note here that Eq. (\ref{eq:context}) should only take messages of $h$ and $t$ as input without their connecting edge $r$, since the ground truth relation $r$ should be treated unobserved in the training stage.
For relational paths, note that there may be a number of relational paths for a given $(h, t)$ pair, but not all paths are logically related to the predicted relation $r$, and the importance of each path also varies.
In \textsc{PathCon}\xspace, since we have already known the context $s_{(h, t)}$ for $(h, t)$ pair and it can be seen as prior information for paths between $h$ and $t$, we can calculate the importance scores of paths based on $s_{(h, t)}$.
Therefore, we first calculate the attention weight of each path $P$ with respect to the context $s_{(h, t)}$:
\begin{equation}
\label{eq:weights}
\alpha_P = \frac{\exp \left( {s_P}^\top s_{(h, t)} \right)}{\sum_{P \in \mathcal P_{h \rightarrow t}} \exp \left( {s_P}^\top s_{(h, t)} \right)},
\end{equation}
where $\mathcal P_{h \rightarrow t}$ is the set of all paths from $t$ to $t$.
Then the attention weights are used to average representations of all paths:
\begin{equation}
\label{eq:attention}
s_{h \rightarrow t} = \sum\nolimits_{P \in \mathcal P_{h \rightarrow t}} \alpha_P s_P,
\end{equation}
where $s_{h \rightarrow t}$ is the aggregated representation of relational paths for $(h, t)$.
In this way, the context information $s_{(h, t)}$ is used to assist in identifying the most important relational paths.
Given the relational context representation $s_{(h,t)}$ and the relational path representation $s_{h \rightarrow t}$, we can predict relations by first adding the two representation together and then taking softmax as follows:
\begin{equation}
p(r|h,t)=\textsc{SoftMax} \left( s_{(h,t)} + s_{h\to t} \right).
\end{equation}
Our model can be trained by minimizing the loss between predictions and ground truths over the training triplets:
\begin{equation}
\label{eq:loss}
\min \mathcal L = \sum_{(h, r, t) \in \mathcal D} J \big( p(r|h, t), \ r \big),
\end{equation}
where $\mathcal D$ is the training set and $J(\cdot)$ is the cross-entropy loss.
It is worth noticing that the context representation $s_{(h, t)}$ plays two roles in the model:
It directly contributes to the predicted relation distribution, and it also helps determine the importance of relational paths with respect to the predicted relation.
\subsection{Discussion on Model Explainability}
\label{sec:explain}
Since \textsc{PathCon}\xspace only models relations without entities, it is able to capture pure relationship among different relation types thus can naturally be used to explain for predictions.
The explainability of \textsc{PathCon}\xspace is two-fold:
On the one hand, modeling relational context captures the correlation between contextual relations and the predicted relation, which can be used to indicate important neighbor edges for the given relation.
For example, ``institution.location'', ``university.founder'', and ``university.president'' can be identified as important contextual relations for ``graduated from''.
On the other hand, modeling relational paths captures the correlation between paths and the predicted relation, which can indicate important relational paths for the given relation.
For example, (``schoolmate of'', ``graduated from'') can be identified as an important relational path for ``graduated from''.
It is interesting to see that the explainability provided by relational paths is also connected to first-logic logical rules with the following form:
\begin{equation*}
B_1(h, x_1) \wedge B_2 (x_1, x_2) \wedge \cdots \wedge B_L(x_{L-1}, t) \Rightarrow r(h, t),
\end{equation*}
where $\bigwedge B_i$ is the conjunction of relations in a path and $r(h, t)$ is the predicted relation.
The above example of relational path can therefore be written as the following rule:
\begin{equation*}
\begin{split}
&(h, \ \textsf{schoolmate of}, \ x) \wedge (x, \ \textsf{graduated from}, \ t) \\
\Rightarrow & (h, \ \textsf{graduated from}, \ t).
\end{split}
\end{equation*}
Therefore, \textsc{PathCon}\xspace can also be used to learn logical rules from KGs just as prior work \cite{galarraga2015fast, yang2017differentiable, ho2018rule, zhang2019iteratively, sadeghian2019drum}.
\subsection{Design Alternatives}
\label{sec:design_alternatives}
Next we discuss several design alternatives for \textsc{PathCon}\xspace.
In our ablation experiments we will compare \textsc{PathCon}\xspace with the following alternative implementations.
When modeling relational context, we propose two alternatives for context aggregator, instead of the Concatenation context aggregator in Eqs. (\ref{eq:context_mp_2}) and (\ref{eq:context}):
\xhdr{Mean context aggregator}
It takes the element-wise mean of the input vectors, followed by a nonlinear transformation function:
\begin{equation}
s^{i+1}_e = \sigma \left( \frac{1}{3} \big( m^i_v + m^i_u + s^i_e \big) W + b \right), \ v, u \in \mathcal N(e),
\end{equation}
The output of Mean context aggregator is invariant to the permutation of its two input nodes, indicating that it treats the head and the tail equally in a triplet.
\xhdr{Cross context aggregator}
It is inspired by combinatorial features in recommender systems \cite{wang2019multi}, which measure the interaction of unit features (e.g., AND(gender=\textsf{female}, language=\textsf{English})).
Note that Mean and Concatenation context aggregator simply transform messages from two input nodes separately and add them up together, without modeling the interaction between them that might be useful for link prediction.
In Cross context aggregator, we first calculate all element-level pairwise interactions between messages from the head and the tail:
\begin{equation}
m^i_v {m^i_u}^\top =
\begin{bmatrix}
{m^i_v}^{(1)} {m^i_u}^{(1)} & \cdots & {m^i_v}^{(1)} {m^i_u}^{(d)}\\
\cdots & & \cdots\\
{m^i_v}^{(d)} {m^i_u}^{(1)} & \cdots & {m^i_v}^{(d)} {m^i_u}^{(d)}
\end{bmatrix},
\end{equation}
where we use superscript with parentheses to indicate the element index and $d$ is the dimension of $m^i_v$ and $m^i_u$.
Then we summarize all interactions together via flattening the interaction matrix to a vector then multiplied by a transformation matrix:
\begin{equation}
s^{i+1}_e = \sigma \left( {\rm flatten} \big( m^i_v {m^i_u}^\top \big) W_1^i + s^i_e W_2^i + b^i \right), \ v, u \in \mathcal N(e).
\end{equation}
It is worth noting that Cross context aggregator preserves the order of input nodes.
\xhdr{Learning path representation with RNN}
When modeling relational paths, recurrent neural network (RNN) can be used to learn the representation of relational path $P = (r_1, r_2, ...)$:
\begin{equation}
s_P = {\rm RNN} \left( r_1, r_2, ... \right),
\end{equation}
instead of directly assigning an embedding vector to $P$.
The advantage of RNN against path embedding is that its number of parameters is fixed and does not depend on the number of relational paths.
Another potential benefit is that RNN can hopefully capture the similarity among different relational paths.
\xhdr{Mean path aggregator}
When calculating the final representation of relational paths for $(h, t)$ pair, we can also simply average all the representations of paths from $h$ to $t$ instead of the Attention path aggregator in Eqs. (\ref{eq:weights}) and (\ref{eq:attention}):
\begin{equation}
s_{h \rightarrow t} = \sum\nolimits_{P \in \mathcal P_{h \rightarrow t}} s_P.
\end{equation}
Mean path aggregator can be used in the case where representation of relational context is unavailable, since it does not require attention weights as input.
\section{Experiments}
In this section, we evaluate the proposed \textsc{PathCon}\xspace model, and present its performance on six KG datasets.
\subsection{Experimental Setup}
\textbf{Datasets}.
We conduct experiments on five standard KG benchmarks: \textbf{FB15K}, \textbf{FB15K-237}, \textbf{WN18}, \textbf{WN18RR}, \textbf{NELL995}, and one KG dataset proposed by us: \textbf{DDB14}.
\textbf{FB15K} \cite{bordes2011learning} is from Freebase \cite{bollacker2008freebase}, a large-scale KG of general human knowledge.
\textbf{FB15k-237} \cite{toutanova2015observed} is a subset of FB15K where inverse relations are removed.
\textbf{WN18} \cite{bordes2011learning} contains conceptual-semantic and lexical relations among English words from WordNet \cite{miller1995wordnet}.
\textbf{WN18RR} \cite{dettmers2018convolutional} is a subset of WN18 where inverse relations are removed.
\textbf{NELL995} \cite{xiong2017deeppath} is extracted from the 995th iteration of the NELL system \cite{carlson2010toward} containing general knowledge.
In addition, we present a new dataset \textbf{DDB14} that is suitable for KG-related tasks.
DDB14 is collected from Disease Database\footnote{\url{http://www.diseasedatabase.com}}, which is a medical database containing terminologies and concepts such as diseases, symptoms, drugs, as well as their relationships.
We randomly sample two subsets of 4,000 triplets from the original one as validation set and test set, respectively.
The statistics of the six datasets are summarized in Table \ref{table:statistics}.
We also calculate and present the mean and variance of node degree distribution (i.e., $\mathbb E[d]$ and ${\rm Var}[d]$) for each KG.
It is clear that ${\rm Var}[d]$ is large for all KGs, which empirically demonstrates that the complexity of relational message passing is fairly high, thus alternate relational message passing is necessary for real graphs.
\begin{table}[t]
\centering
\small
\setlength{\tabcolsep}{2pt}
\begin{tabular}{c|cccccc}
\hline
& FB15K & FB15K-237 & WN18 & WN18RR & NELL995 & DDB14 \\
\hline
\#nodes & 14,951 & 14,541 & 40,943 & 40,943 & 63,917 & 9,203 \\
\#relations & 1,345 & 237 & 18 & 11 & 198 & 14 \\
\#training & 483,142 & 272,115 & 141,442 & 86,835 & 137,465 & 36,561 \\
\#validation & 50,000 & 17,535 & 5,000 & 3,034 & 5,000 & 4,000 \\
\#test & 59,071 & 20,466 & 5,000 & 3,134 & 5,000 & 4,000 \\
$\mathbb E[d]$ & 64.6 & 37.4 & 6.9 & 4.2 & 4.3 & 7.9 \\
${\rm Var}[d]$ & 32,441.8 & 12,336.0 & 236.4 & 64.3 & 750.6 & 978.8 \\
\hline
\end{tabular}
\vspace{0.05in}
\caption{Statistics of all datasets. $\mathbb E[d]$ and ${\rm Var}[d]$ are mean and variance of the node degree distribution, respectively.}
\label{table:statistics}
\vspace{-0.2in}
\end{table}
\xhdr{Baselines}
We compare \textsc{PathCon}\xspace with several state-of-the-art models, including \textbf{TransE} \cite{bordes2013translating}, \textbf{ComplEx} \cite{trouillon2016complex}, \textbf{DistMult} \cite{yang2015embedding}, \textbf{RotatE} \cite{sun2019rotate}, \textbf{SimplE} \cite{kazemi2018simple}, \textbf{QuatE} \cite{zhang2019quaternion}, and \textbf{DRUM} \cite{sadeghian2019drum}. The first six models are embedding-based methods, while DRUM only uses relational paths to make prediction.
The implementation details of baselines (as well as our method) is provided in Appendix \ref{sec:implementation}.
We also conduct extensive ablation study and propose two reduced versions of our model, \textbf{\textsc{Con}} and \textbf{\textsc{Path}}, which only use relational context and relational paths, respectively, to test the performance of the two components separately.
The number of parameters of each model on DDB14 are shown in Table \ref{table:params}.
The result demonstrates that \textsc{PathCon}\xspace is much more storage-efficient than embedding-based methods, since it does not need to calculate and store entity embeddings.
\begin{table}[h]
\centering
\small
\setlength{\tabcolsep}{2pt}
\begin{tabular}{c|ccccccc}
\hline
Method & TransE & ComplEx & DisMult & RotatE & SimplE & QuatE & \textsc{PathCon}\xspace \\
\hline
\#param. & 3.7M & 7.4M & 3.7M & 7.4M & 7.4M & 14.7M & 0.06M \\
\hline
\end{tabular}
\vspace{0.05in}
\caption{Number of parameters of all models on DDB14.}
\vspace{-0.2in}
\label{table:params}
\end{table}
\begin{table*}[t]
\centering
\small
\setlength{\tabcolsep}{2pt}
\begin{tabular}{c|ccc|ccc|ccc|ccc|ccc|ccc}
\hline
& \multicolumn{3}{c|}{FB15K} & \multicolumn{3}{c|}{FB15K-237} & \multicolumn{3}{c|}{WN18} & \multicolumn{3}{c|}{WN18RR} & \multicolumn{3}{c|}{NELL995} & \multicolumn{3}{c}{DDB14} \\
& \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c|}{Hit@3} & \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c|}{Hit@3} & \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c|}{Hit@3} & \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c|}{Hit@3} & \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c|}{Hit@3} & \multicolumn{1}{c}{MRR} & \multicolumn{1}{c}{Hit@1} & \multicolumn{1}{c}{Hit@3} \\
\hline
TransE & 0.962 & 0.940 & 0.982 & 0.966 & 0.946 & 0.984 & 0.971 & 0.955 & 0.984 & 0.784 & 0.669 & 0.870 & \underline{0.841} & \underline{0.781} & \underline{0.889} & \underline{0.966} & \underline{0.948} & 0.980 \\
ComplEx & 0.901 & 0.844 & 0.952 & 0.924 & 0.879 & 0.970 & \underline{0.985} & \underline{0.979} & \underline{0.991} & 0.840 & 0.777 & 0.880 & 0.703 & 0.625 & 0.765 & 0.953 & 0.931 & 0.968 \\
DistMult & 0.661 & 0.439 & 0.868 & 0.875 & 0.806 & 0.936 & 0.786 & 0.584 & 0.987 & 0.847 & \underline{0.787} & 0.891 & 0.634 & 0.524 & 0.720 & 0.927 & 0.886 & 0.961 \\
RotatE & 0.979 & 0.967 & 0.986 & 0.970 & 0.951 & 0.980 & 0.984 & \underline{0.979} & 0.986 & 0.799 & 0.735 & 0.823 & 0.729 & 0.691 & 0.756 & 0.953 & 0.934 & 0.964 \\
SimplE & \underline{0.983} & \underline{0.972} & \underline{0.991} & 0.971 & 0.955 & 0.987 & 0.972 & 0.964 & 0.976 & 0.730 & 0.659 & 0.755 & 0.716 & 0.671 & 0.748 & 0.924 & 0.892 & 0.948 \\
QuatE & \underline{0.983} & \underline{0.972} & \underline{0.991} & \underline{0.974} & \underline{0.958} & \underline{0.988} & 0.981 & 0.975 & 0.983 & 0.823 & 0.767 & 0.852 & 0.752 & 0.706 & 0.783 & 0.946 & 0.922 & 0.962 \\
DRUM & 0.945 & 0.945 & 0.978 & 0.959 & 0.905 & 0.958 & 0.969 & 0.956 & 0.980 & \underline{0.854} & 0.778 & \underline{0.912} & 0.715 & 0.640 & 0.740 & 0.958 & 0.930 & \underline{0.987} \\
\hline
\tabincell{c}{\textsc{Con}} & \tabincell{c}{0.962\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.934\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.988\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.978\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.961\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.995}\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.960\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{0.927\\[-1.2ex] \footnotesize $\pm$ 0.005} & \tabincell{c}{0.992\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.943\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{0.894\\[-1.2ex] \footnotesize $\pm$ 0.004} & \tabincell{c}{0.993\\[-1.2ex] \footnotesize $\pm$ 0.003} & \tabincell{c}{0.875\\[-1.2ex] \footnotesize $\pm$ 0.003} & \tabincell{c}{0.815\\[-1.2ex] \footnotesize $\pm$ 0.004} & \tabincell{c}{0.928\\[-1.2ex] \footnotesize $\pm$ 0.003} & \tabincell{c}{0.977\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.961\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.994\\[-1.2ex] \footnotesize $\pm$ 0.001} \\
\tabincell{c}{\textsc{Path}} & \tabincell{c}{0.937\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.918\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.951\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.972\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.957\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.986\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.981\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.971\\[-1.2ex] \footnotesize $\pm$ 0.005} & \tabincell{c}{0.989\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.933\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.897\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.961\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.737\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.685\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{0.764\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{0.969\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{0.948\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.991\\[-1.2ex] \footnotesize $\pm$ 0.000} \\
\textsc{PathCon}\xspace & \tabincell{c}{\textbf{0.984}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.974}\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{\textbf{0.995}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.979}\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{\textbf{0.964}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{0.994\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.993}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.988}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.998}\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{\textbf{0.974}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.954}\\[-1.2ex] \footnotesize $\pm$ 0.002} & \tabincell{c}{\textbf{0.994}\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{\textbf{0.896}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.844}\\[-1.2ex] \footnotesize $\pm$ 0.004} & \tabincell{c}{\textbf{0.941}\\[-1.2ex] \footnotesize $\pm$ 0.004} & \tabincell{c}{\textbf{0.980}\\[-1.2ex] \footnotesize $\pm$ 0.000} & \tabincell{c}{\textbf{0.966}\\[-1.2ex] \footnotesize $\pm$ 0.001} & \tabincell{c}{\textbf{0.995}\\[-1.2ex] \footnotesize $\pm$ 0.000} \\
\hline
\end{tabular}
\vspace{0.05in}
\caption{Results of relation prediction on all datasets. Best results are highlighted in bold, and best results of baselines are highlighted with underlines.}
\label{table:result_1}
\vspace{-0.1in}
\end{table*}
\xhdr{Evaluation Protocol}
We evaluate all methods on relation prediction, i.e., for a given entity pair $(h, t)$ in the test set, we rank the ground-truth relation type $r$ against all other candidate relation types.
It is worth noticing that most baselines are originally designed for head/tail prediction, therefore, their negative sampling strategy is to corrupt the head or the tail for a true triple $(h, r, t)$, i.e., replacing $h$ or $t$ with a randomly sampled entity $h'$ or $t'$ from KGs, and using $(h', r, t)$ or $(h, r, t')$ as the negative sample.
In relation prediction, since the task is to predict the missing relation for a given pair $(h, t)$, we modify the negative sampling strategy accordingly by corrupting the relation $r$ of each true triplet $(h, r, t)$, and use $(h, r', t)$ as the negative sample where $r'$ is randomly sampled from the set of relation types.
This new negative sampling strategy can indeed improve the performance of baselines in relation prediction.
We use \textbf{MRR} (mean reciprocal rank) and \textbf{Hit@1, 3} (hit ratio with cut-off values of 1 and 3) as evaluation metrics.
\subsection{Main Results}
\textbf{Comparison with baselines}.
The results of relation prediction on all datasets are reported in Table \ref{table:result_1}.
In general, our method outperforms all baselines on all datasets.
Specifically, the absolute Hit@1 gain of \textsc{PathCon}\xspace against the best baseline in relation prediction task are $0.2\%$, $0.6\%$, $0.9\%$, $16.7\%$, $6.3\%$, and $1.8\%$ in the six datasets, respectively.
The improvement is rather significant for WN18RR and NELL995, which are exactly the two most sparse KGs according to the average node degree shown in Table \ref{table:statistics}.
This empirically demonstrates that \textsc{PathCon}\xspace maintains great performance for sparse KGs, and this is probably because \textsc{PathCon}\xspace has much fewer parameters than baselines and is less prone to overfitting.
In contrast, performance gain of \textsc{PathCon}\xspace on FB15K is less significant, which may be because the density of FB15K is very high so that it is much easier for baselines to handle.
In addition, the results also demonstrate the stability of \textsc{PathCon}\xspace as we observe that most of the standard deviations are quite small.
Results in Tables \ref{table:result_1} also show that, in many cases \textsc{Con} or \textsc{Path} can already beat most baselines.
Combining relational context and relational paths together usually leads to even better performance.
\xhdr{Inductive KG completion}
We also examine the performance of our method in inductive KG completion.
We randomly sample a subset of nodes that appears in the test set, then remove these nodes along with their associated edges from the training set.
The remaining training set is used to train the models, and we add back the removed edges during evaluation.
The evaluation transforms from fully conductive to fully inductive when the ratio of removed nodes increases from 0 to 1.
The results of \textsc{PathCon}\xspace, DistMult, and RotatE on relation prediction task are plotted in Figure \ref{fig:inductive}.
We observe that the performance of our method decreases slightly in fully inductive setting (from 0.954 to 0.922), while DistMult and RotatE fall to a ``randomly guessing'' level.
This is because the two baselines are embedding-based models that rely on modeling node identity, while our method does not consider node identity thus being naturally generalizable to inductive KG completion.
\begin{figure*}
\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/inductive.pdf}
\vspace{-0.2in}
\caption{Results of inductive KG completion on WN18RR.}
\label{fig:inductive}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/layers.pdf}
\vspace{-0.2in}
\caption{Results of \textsc{PathCon}\xspace with different hops/length on WN18RR.}
\label{fig:layers}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/neighbor_agg.pdf}
\vspace{-0.2in}
\caption{Results of \textsc{Con} with different context aggregators.}
\label{fig:neighbor_agg}
\end{minipage}
\end{figure*}
\begin{figure}
\begin{minipage}[t]{0.47\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/path_agg_repr.pdf}
\vspace{-0.2in}
\caption{Results of \textsc{PathCon}\xspace with different path representation types and path aggregators on WN18RR.}
\label{fig:path_agg_updater}
\end{minipage}
\hfill
\begin{minipage}[t]{0.46\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/edge_feature.pdf}
\vspace{-0.2in}
\caption{Results of \textsc{Con}, \textsc{Path}, and \textsc{PathCon}\xspace with different initial features of relations on NELL995.}
\label{fig:edge_feature}
\end{minipage}
\end{figure}
\subsection{Model Variants}
\label{sec:model_variants}
\textbf{The number of context hops and maximum path length}.
We investigate the sensitivity of our model to the number of context hops and maximum path length.
We vary the two numbers from 0 to 4 (0 means the corresponding module is not used), and report the results of all combinations (without (0, 0)) on WN18RR in Figure \ref{fig:layers}.
It is clear to see that increasing the number of context hops and maximum path length can significantly improve the result when they are small, which demonstrates that including more neighbor edges or counting longer paths does benefit the performance.
But the marginal benefit is diminishing with the increase of layer numbers.
Similar trend is observed on other datasets too.
\xhdr{Context aggregators}
We study how different implementations of context aggregator affect the model performance.
The results of Mean, Concat, and Cross context aggregator on four datasets are shown in Figure \ref{fig:neighbor_agg} (results on FB15K and WN18 are omitted as they are similar to FB15K-237 and WN18RR, respectively).
The results show that Mean performs worst on all datasets, which indicates the importance of node orders when aggregating features from nodes to edges.
It is also interesting to notice that the performance comparison between Concat and Cross varies on different datasets:
Concat is better than Cross on NELL995 and is worse than Cross on WN18RR, while their performance is on par on FB15K-237 and DDB14.
However, note that a significant defect of Cross is that it has much more parameters than Concat, which requires more running time and memory resource.
\xhdr{Path representation types and path aggregators}
We implement four combinations of path representation types and path aggregators: Embedding+Mean, Embedding+Attention, RNN+Mean, and RNN+Attention, of which the results are presented in Figure \ref{fig:path_agg_updater}.
Different from context aggregators, results on the six datasets are similar for path representation types and path aggregators, so we only report the results on WN18RR.
We find that Embedding is consistently better than RNN, which is probably because the length of relational paths are generally short (no more than 4 in our experiments), so RNN can hardly demonstrate its strength in modeling sequences.
The results also show that Attention aggregator performs slightly better than Mean aggregator.
This demonstrates that the contextual information of head and tail entities indeed helps identify the importance of relational paths.
\xhdr{Initial edge features}
Here we examine three types of initial edge features: identity, BOW, and BERT embedding of relation types.
We choose to test on NELL995 because its relation names consist of relatively more English words thus are semantically meaningful (e.g., ``organization.headquartered.in.state.or.province'').
The results are reported in Figure \ref{fig:edge_feature}, which shows that BOW features are slightly better than identity, but BERT embeddings perform significantly worse than the other two.
We attribute this finding to that
BERT embeddings are better at identifying semantic relationship among relation types, but our model aims to learn the mapping from BERT embeddings of context/paths to the identity of predicted relation types.
In other words, BERT may perform better if the predicted relation types are also represented by BERT embeddings, so that this mapping is learned within the embedding space.
We leave the exploration as future work.
\subsection{Case Study on Model Explainabilty}
We choose FB15K-237 and DDB14 as the datasets to show the explainability of \textsc{PathCon}\xspace.
The number of context hops is set to 1 and the maximum path length is set to 2.
When training is completed, we choose three relations from each dataset and list the most important relational context/paths to them based on the transformation matrix of the context/path aggregator.
The results are presented in Table \ref{table:rule}, from which we find that most of the identified context/paths are logically meaningful.
For example, ``education campus of'' can be inferred by ``education institution in'', and ``is associated with'' is found to be a transitive relation.
In addition, more visualized results and discussion on DDB14 dataset are included in Appendix \ref{sec:ddb14_explain}.
\begin{table*}[t]
\centering
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{c|c|c|c}
\hline
& Predicted relation & Important relational context & Important relational paths \\
\hline
\multirow{3}{*}{FB15K-237} & award winner & award honored for, award nominee & (award nominated for), (award winner, award category) \\
& film written by & film release region & (film edited by), (film crewmember) \\
& education campus of & education major field of study & (education institution in) \\
\hline
\multirow{3}{*}{DDB14} & may cause & may cause, belongs to the drug family of & (is a risk factor for), (see also, may cause) \\
& is associated with & is associated with, is a risk factor for & (is associated with, is associated with) \\
& may be allelic with & may be allelic with, belong(s) to the category of & (may cause, may cause), (may be allelic with, may be allelic with) \\
\hline
\end{tabular}
\vspace{0.05in}
\caption{Examples of important context/paths identified by \textsc{PathCon}\xspace on FB15K-237 and DDB14.}
\label{table:rule}
\vspace{-0.2in}
\end{table*}
\section{Related Work}
\subsection{Knowledge Graph Completion}
KGs provide external information for a variety of downstream tasks such as recommender systems \cite{wang2018dkn, wang2018ripplenet, wang2019exploring} and semantic analysis \cite{wang2018shine}.
Most existing methods of KG completion are based on embeddings, which normally assign an embedding vector to each entity and relation in the continuous embedding space and train the embeddings based on the observed facts.
One line of KG embedding methods is \textit{translation-based}, which treat entities as points in a continuous space and each relation translates the entity point.
The objective is that the translated head entity should be close to the tail entity in real space \cite{bordes2013translating}, complex space \cite{sun2019rotate}, or quaternion space \cite{zhang2019quaternion}, which have shown capability to handle multiple relation patterns and achieve state-of-the-art result.
Another line of work is \textit{multi-linear} or \textit{bilinear models}, where they calculate the semantic similarity by matrix or vector dot product in real \cite{yang2015embedding} or complex space \cite{trouillon2016complex}.
Besides, several embedding-based methods explore the architecture design that goes beyond point vectors \cite{socher2013reasoning, dettmers2018convolutional}.
However, these embedding-based models fail to predict links in inductive setting, neither can they discover any rules that explain the prediction.
\subsection{Graph Neural Networks}
Existing GNNs generally follow the idea of neural message passing \cite{gilmer2017neural} that consists of two procedures: propagation and aggregation.
Under this framework, several GNNs are proposed that take inspiration from convolutional neural networks \cite{duvenaud2015convolutional, hamilton2017inductive, kipf2017semi, wang2020unifying}, recurrent neural networks \cite{li2015gated}, and recursive neural networks \cite{bianchini2001processing}.
However, these methods use node-based message passing, while we propose passing messages based on edges in this work.
There are two GNN models conceptually connected to our idea of identifying relative position of nodes in a graph.
DEGNN \cite{li2020distance} captures the distance between the node set whose representation is to be learned and each node in the graph, which is used as extra node attributes or as controllers of message aggregation in GNNs.
SEAL \cite{zhang2018link} labels nodes with their distance to two nodes $a$ and $b$ when predicting link existence between $(a, b)$.
In contrast, we use relational paths to indicate the relative position of two nodes.
Researchers also tried to apply GNNs to knowledge graphs.
For example, Schlichtkrull \textit{et al.} \cite{schlichtkrull2018modeling} use GNNs to model the entities and relations in KGs, however, they are limited in that they did not consider the relational paths and cannot predict in inductive settings.
Wang \textit{et al.} \cite{wang2019knowledge_a, wang2019knowledge_b} use GNNs to learn entity embeddings in KGs, but their purpose is to use the learned embeddings to enhance the performance of recommender systems rather than KG completion.
\section{Conclusion and Future Work}
We propose \textsc{PathCon}\xspace for KG completion.
\textsc{PathCon}\xspace considers two types of subgraph structure in KGs, i.e., contextual relations of the head/tail entity and relational paths between head and tail entity.
We show that both relational context and relational paths are critical to relation prediction, and they can be combined further to achieve state-of-the-art performance.
Moreover, \textsc{PathCon}\xspace is also shown to be inductive, storage-efficient, and explainable.
We point out four directions for future work.
First, as we discussed in Remark \ref{remark:2}, it is worth studying the empirical performance of \textsc{PathCon}\xspace on node-feature-aware KGs.
Second, as we discussed in Section \ref{sec:model_variants}, designing a model that can better take advantage of pre-trained word embeddings is a promising direction;
Third, it is worth investigating why RNN does not perform well, and whether we can model relational paths better;
Last, it is interesting to study if the context representation and path representation can be assembled in a more principled way.
\xhdr{Acknowledgements}
This research has been supported in part by DARPA, ARO, NSF, NIH, Stanford Data Science Initiative, Wu Tsai Neurosciences Institute, Chan Zuckerberg Biohub, Amazon, JPMorgan Chase, Docomo, Hitachi, Intel, JD.com, KDDI, NVIDIA, Dell, Toshiba, Visa, and UnitedHealth Group.
\bibliographystyle{ACM-Reference-Format}
|
2,877,628,089,456 | arxiv | \section{Introduction}
The Green Bank North Celestial Cap (GBNCC; \citealt{slr+14}) pulsar survey began in 2009 and, when complete, will cover the entire sky accessible to the 100\,m Robert C. Byrd Green Bank Telescope (GBT; $\delta\geq-40\degr$, or 85\% of the celestial sphere) at 350\,MHz. As of mid-2019, the survey is 85\% complete and 161 pulsars have been discovered, including 25 millisecond pulsars (MSPs) and 16 rotating radio transients \citep[RRATs;][]{mll+06}. Timing solutions for these discoveries have been published in \cite{slr+14}, \cite{kkl+15}, \cite{kmlk+18}, \cite{lsk+18}, and \citet{acd+19}, and more are forthcoming. As such, this constitutes one of the largest and most uniform pulsar surveys to date.
In addition to the newly discovered pulsars, the uniform coverage of GBNCC allows a robust re-assessment of the known pulsar population with reliable flux density measurements. Here we present a detailed search for all known pulsars in the GBNCC footprint.
We find that 572 previously published pulsars and 98 unpublished pulsars have been re-detected by the survey pipeline and visually confirmed, comprising 670 detections in total, the largest low-frequency, single-survey sample. Similar to previous efforts based on results from the Parkes Multibeam Pulsar Survey (PMPS) and the Pulsar Arecibo L-band Feed Array (PALFA) survey \citep[e.g., see][]{lfl+06,slm+14,lbh+15}, we conduct a detailed analysis of the GBNCC pulsar survey and compare its sensitivity with that of other surveys in overlapping regions of sky. Flux densities at 350\,MHz ($S_{350}$) are presented for all detections, as well as
pulse widths and profiles.
In \S\ref{sec:methods}, we outline the process used to generate a comprehensive list of pulsars as well as predicting and measuring signal-to-noise ratios (S/N) of detections in the survey. In \S\ref{sec:results}, we present the recovered S/N and flux density measurements for all detected pulsars as well as measurements of pulse width, dispersion measure, and spectral index. We also present the profiles for all of these pulsars. In \S\ref{sec:sensitivity}, we discuss how the GBNCC survey is performing compared to expectations and RFI characteristics of the survey, and remark on interesting detections and notable non-detections. We also discuss the implications of our results for the Galactic pulsar population. Finally, in \S\ref{sec:conc}, we summarize the main conclusions of this analysis.
\section{Sample Assembly and Data Reduction}
\label{sec:methods}
The GBNCC data set as of late fall 2018 included $\sim$108,000 120\,s pointings, each tagged with a unique beam number. Each dual-polarization observation was taken with the GBT over the past $\simeq$10 years. The survey utilizes the GUPPI backend, with a sampling time of 82\,$\mu$s and 100\,MHz of bandwidth centered at 350\,MHz \citep[for more information on the observing setup for the GBNCC survey, see][]{slr+14}. We began by organizing a comprehensive list of all known pulsars with parameters that were available for use, whether they were published or not. By utilizing the Australia Telescope National Facility (ATNF) pulsar catalog\footnote{\url{http://www.atnf.csiro.au/research/pulsar/psrcat}} \citep[v1.59,][]{mhth05}, we amassed the bulk of the sources from the list of all published pulsars and their positions on the sky as well as their spin parameters and other relevant quantities (dispersion measure, etc.). Discovery parameters are also available for additional pulsars that have not been published but were detected in a number of other recent or ongoing surveys. Many of these surveys, including AODrift \citep{dsm+13}, SUPERB \citep[][Spiewak et al., 2019, in prep.]{kbj+18}, GBT 350\,MHz Drift \citep{blr+13}, PALFA \citep{cfl+06,lbh+15}, LOTAAS \citep{scb+19}, and HTRU-South \citep{kjs+10} include pulsars that are in GBNCC survey area, and so were included in the list. More information on these surveys is included in Table \ref{tbl:surv}. Furthermore, we included the list of pulsars that had been discovered in the search pipeline for the GBNCC survey. We then limited this list to pulsars within the range of the survey, i.e., pulsars with $\delta>-40\degr$. In total, this list contained 2299 pulsars. We determined which pulsars were within 30\arcmin\ (FWHM of GBT at 350\,MHZ) of completed GBNCC pointings, adjusting when necessary to compensate for large ($>$30\arcmin) uncertainties in pulsar position. This reduced the total number of pulsar candidates to 1413. We could then match each pulsar with the GBNCC beams closest to its position before beginning to process the data.
Radio frequency interference (RFI) excision is the first step of GBNCC data analysis, and is done primarily with the \texttt{rfifind} tool from the \texttt{PRESTO}\footnote{\url{http://www.cv.nrao.edu/~sransom/presto/}} pulsar data analysis software package \citep{smr+01} as described in \S3.1 of \citet{slr+14}. We also performed an analysis of the \texttt{rfifind} output files spanning the lifetime of the GBNCC survey up to late 2018 (roughly 83\% of the total survey) to characterize the effects of RFI over the course of the survey. These files contain information about which frequency channels were masked due to RFI for every 120 second scan in the survey. For a particular scan, the effective bandwidth $\Delta\nu$ is the total 100\,MHz bandwidth of the GBT 350\,MHz receiver multiplied by the ratio of unmasked to total channels for that scan, minus an additional 20\,MHz for rolloff.
In some cases, the \texttt{rfifind} masks were insufficient to remove additional RFI that was either narrow in frequency space or brief in time. The latter often appears as a very bright burst at $\sim$0 DM for portions of the observation. To mitigate this, we employed some additional narrowband flagging in the \texttt{PRESTO} \texttt{prepfold} command as well as removing corrupted portions of the scan in the time-domain. Note that these changes also alter the values for $\tau_{\rm obs}$ and $\Delta\nu$ which consequently change the measured S/N for a given observation. For this reason, we calculate the fraction of data points from the observations that were not omitted in processing and multiply the total bandwidth by this fraction.
After removal of RFI, we dedispersed and folded the observations at each pulsar's rotational period and integrated the profiles to obtain a single average profile for each observation. For the vast majority of sources included in this analysis, a precise ephemeris from the ATNF catalog was used to perform the folding. In all other cases, only the discovery parameters (period, DM and, if known, period derivative) were used. We also repeated this process while allowing dispersion measure to vary and, in some cases, also allowing variations in period and period derivative. This second iteration allows for fine-tuning previously published parameters at the cost of potentially finding bright RFI, which will often occur when attempting to detect low-DM pulsars as sources of RFI have DM = 0\,pc\,cm$^{-3}$. The 120 second observation times utilized in the GBNCC survey limit sensitivity to period refinement, so fitting for period was only used to increase the S/N of detections of pulsars for which only discovery parameters were used, and no further timing analysis was done as a part of this study. All folded data were visually inspected to determine likelihood of an actual detection. In cases where RFI still existed in the data, we removed high order ($>$5) polynomials from the off-pulse regions of the profile. With folded profiles, we calculated a measured signal-to-noise ratio (S/N) \citep{lk04},
\begin{equation}
{\rm S/N}_{\rm meas} = \sum_{i=0}^{N_{\rm bin}} \frac{p_i-\bar{p}_{\rm off}}{\sigma_{\rm off} \sqrt{W N_{\rm bin}/P}}~\gamma,
\label{eq:snm}
\end{equation}
where $N_{\rm bin}$ is the number of bins across the pulse profile,
$p_i$ is the value of bin $i$,
$\bar{p}_{\rm off}$ is the mean of the off-pulse bins,
$\sigma_{\rm off}$ is the standard deviation of the off-pulse bins, $W$ is the on-pulse width in seconds, $P$ is the pulsar spin period in seconds, and $\gamma$ is a correction factor. When continuous signals are assigned to a finite number of bins in the profile during the folding process in \texttt{PRESTO}, their intensity is ``smeared" over the neighboring bins, resulting in correlations in the bins' intensities. This correction, dubbed $\gamma$, depends on the sampling time and the number of bins in the profile, which (for this study) is dependent on the pulsar spin period. Typical values are close to 0.95. The number of bins $N_{\rm bin}$ was determined by the pulsar period as follows: profiles for pulsars with periods shorter than 1.7\,ms had 28 bins, periods shorter than 10\,ms had 50 bins, periods shorter than 50\,ms had 128 bins, and all others had 200 bins. This prescription retains sensitivity to long-period pulsars but avoids bin widths corresponding to time intervals smaller than the sampling time of 82\,$\mu$s. Pulse widths were determined with a standard process. First, sigma-clipping was used to find the off-pulse region. Then, the peak value above the noise floor was identified, and bins on either side of the peak were added to the on-pulse width. This process was repeated, adding bins on the sides of the peak until we reached bins within 2$\sigma$ of the mean of the noise. The edges of the pulse were found by fitting lines to the two bins on either side of the pulse and finding the fraction of the outermost bins that were above the noise floor. At this point, we consider the full on-pulse width to be determined. Each profile was then checked by eye, and corrections to the on-pulse region were made. Any components of the pulse width that were distinct from the main pulse were determined using the same algorithm. To determine the sensitivity of uncertainties in S/N from the choice of the number of on-pulse bins, noisy Gaussian pulses were simulated and various width choices were used to measure the fractional error on S/N. From this test, it was found that on-pulse widths that exceed at least one $\sigma$ beyond the Gaussian mean were sufficient to greatly reduce the fractional uncertainty on S/N. Beyond this, adding bins had little effect on this fractional uncertainty - so, pulse widths were chosen to encompass all of the pulse visible above the noise. In some cases, additional RFI features were removed prior to the determination of $W$ to minimize errors in $W$ and S/N (see \S\ref{sec:rfi}).
Characteristic measurements of pulse width include measurements at both 50\% and 10\% of the pulse profile's maximum amplitude (hereafter $W_{50}$ and $W_{10}$, respectively). These widths are dependent on both pulse period and observing frequency, so measurements at 350\,MHz help to fill out the low-frequency regime for a wide range of pulse periods. However, the noise floor in some pulsars limits the ability to determine $W_{10}$ robustly. Note also that $W_{50}$ and $W_{10}$ are distinct from $W$, which includes all bins that contain the pulse signal, and so $W$ is generally slightly larger than $W_{10}$.
The expected S/N of a pulsar can be estimated as \citep{dtw+85,lk04}
\begin{equation}
{\rm S/N}_{\rm exp} = \frac{ S_{350}G\sqrt{N_{\rm pol}\tau_{\rm obs}\Delta\nu}} {T_{\rm sys}\beta} \sqrt{\frac{P-W}{W}} f(\theta),\label{eq:sne}
\end{equation}
where $S_{350}$ is the flux density at 350\,MHz,
$G=2$\,K/Jy is the gain of the Green Bank Telescope \citep{slr+14},
$N_{\rm pol}=2$ is the number of polarizations recorded,
$\tau_{\rm obs}=120$\,s is the length of the observation,
$\Delta\nu$ is the bandwidth in MHz after removing RFI (see \S\ref{sec:rfi}),
$T_{\rm sys}$ is the system temperature (including the sky temperature at the source position, receiver temperature $\simeq$ 20\,K, and CMB temperature $\simeq$ 3\,K),
$\beta\simeq1.1$ is an instrument-dependent correction factor due to downsampling the data to 2 bits \citep{lk04},
and $f(\theta)$ is a radial Gaussian factor accounting for sensitivity degradation as a function of angular offset from the center of the circular beam $\theta$. The sky temperature in the direction of each pulsar was determined by using the measurements made by \cite{hks+81} for the beam positions, scaled to 350\,MHz using with the spectral index therein, $-$2.6.
Where possible we use flux densities at other frequencies and previous measurements of spectral index ($\alpha$, with $S_\nu \propto \nu^{\alpha}$) from the ATNF catalog to determine an expected flux density at 350\,MHz and the expected S/N \citep{mhth05}. In cases where there was no published value for $\alpha$ but flux densities at both 400\,MHz and 1400\,MHz were published, we determine a spectral index using a simple power law. In all other cases, we assume a spectral index of $-1.4$ \citep{blr+14} to estimate the flux density at 350\,MHz. We also calculate the measured flux density of each pulsar by inverting Equation \ref{eq:sne} and using measured values for S/N (determined from Equation \ref{eq:snm}) and pulse width. Comparing the expected flux density to our measurements can both roughly confirm our current models for pulsar emission as well as aid in explaining non-detections.
\section{Pulsar Flux Density Census at 350\,MHz}
\label{sec:results}
\begin{figure*}[ht]
\centering
\plotone{psr_map_Gal_012820.pdf}
\caption{Sky map with pulsars from overlapping surveys, plotted in Galactic coordinates as a Mollweide projection. The shaded regions indicate completed GBNCC observations. Detected pulsars from the ATNF catalog and pulsars that were detected using discovery parameters from overlapping surveys are differentiated by marker type, with green plus symbols indicating pulsars from the catalog and red triangles indicating pulsars from the surveys listed in Table \ref{tbl:surv}. Pulsars that were not detected are plotted as blue ``x" symbols.}\label{fig:skymap}
\end{figure*}
We detected 670 pulsars out of a total of 1413 in the survey area, and these detections are listed in Table \ref{tbl:det} in Appendix 1. For all following analysis, the beams corresponding to the brightest detections (highest S/N) were used, as these are most likely to represent the pulsars' flux density. Along with pulsar names, we provide several relevant quantities: dispersion measure from searching with \texttt{PRESTO} \citep{smr+01}, MJD of the brightest detection, angular offset from the center of the beam, $W_{50}$, $W_{10}$ (when S/N was large enough), detection S/N, 350\,MHz flux density measured from the GBNCC data, and measured spectral index $\alpha$ (see \S\ref{sec:spindx}). Uncertainties on the S/N and flux densities were calculated using standard error propagation from equations \ref{eq:snm} and \ref{eq:sne} and uncertainties on bandwidth, temperature, and $\theta$ of 5\,MHz, 10\,K, and 0.5 degrees, respectively. Among these are 66 millisecond pulsars (MSP), defined here as pulsars with spin periods shorter than 30\,ms. The integrated pulse profiles for all of the brightest detections are shown in Figures \ref{fig:prof0} $-$ \ref{fig:prof11} along with pulsar names, dispersion measure, and flux density. Figure \ref{fig:skymap} shows all detected pulsars plotted by their Galactic positions, and different markers indicate whether or not the pulsars were from the ATNF catalog or were a part of one of the other survey lists mentioned above.
\subsection{Comparison Between the GBNCC and Overlapping Pulsar Surveys}
\begin{deluxetable*}{lcccc}
\tablecaption{Pulsar Survey Comparison \label{tbl:surv}}
\tablehead{
\colhead{Survey} & \colhead{Central Frequency} & \colhead{Limiting Flux Density\tablenotemark{a}} & \colhead{Detections\tablenotemark{b}} & \colhead{Reference} \\
& \colhead{(MHz)} & \colhead{(mJy)} & & }
\startdata
AODrift \dotfill & \phn327 & 0.59 & 7/13 & \cite{dsm+13}\\
HTRU$-$S (low-lat) \dotfill & 1352 & 0.40 & 0/9\phn & \cite{kjs+10}\\
HTRU$-$S (med-lat) \dotfill & 1352 & 0.95 & 3/27 & \cite{kjs+10}\\
HTRU$-$S (high-lat) \dotfill & 1352 & 1.2\phn & 1/8\phn & \cite{kjs+10}\\
SUPERB \dotfill & 1352 & 0.4\phn & 2/15 & \cite{kbj+18}, Spiewak et al. (in prep)\\
LOTAAS \dotfill& \phn134 & 0.63 & 10/39\phn & \cite{scb+19} \\
PALFA \dotfill& 1400 & 0.23 & 0/29 & \cite{lbh+15} \\
GBT350 \dotfill & \phn350 & 0.59 & 3/6\phn & \cite{blr+13} \\
GBNCC \dotfill & \phn350 & 0.70 & 72/72\phn & \cite{slr+14} \\
\enddata
\tablenotetext{a}{Averaged over the survey area and scaled to 350\,MHz.}
\tablenotetext{b}{Number of detections of pulsars from this survey by GBNCC/number of pulsars from this survey within the GBNCC survey area.}
\tablecomments{Information about individual detections is reported in Table \ref{tbl:det}.}
\end{deluxetable*}
Out of the 210 pulsars with discovery parameters that are not currently listed in the ATNF catalog, 98 were detected. Names, central frequencies, scaled limiting flux densities, and the ratio of detected to processed pulsars are given for each survey in Table \ref{tbl:surv}. It should be noted that there are many pulsars from these surveys (excluding GBNCC) in regions of the sky where the GBNCC survey has yet to observe, and so they may be detected in the future; these pulsars are not included in the counts listed in Table \ref{tbl:surv}. Three of these surveys (SUPERB, HTRU-S, PALFA) were conducted at higher frequencies,, where average sky temperature (especially near the Galactic plane) is much lower. This reason and the increased sensitivity to high DM pulsars at high frequency is useful for diagnosing missed detections. Because these pulsars have neither published flux densities nor spectral indices, reasons for missed detections cannot be determined more robustly than those due to sky temperature, position relative to the survey, extreme nulling/intensity variation, and high DM/short periods. It is also possible that for some of these pulsars, the discovery parameters may not be precise enough to be found in this analysis.
The most surprising missed detections come from the GBT350, AODrift, and LOTAAS surveys, which all have comparable sensitivities and frequencies. In an effort to explain why these pulsars were missed, all of the discovery plots were checked against our results, and acceleration searches were run. Three pulsars (J0100+69 and J0121+14 from LOTAAS, and J1854+36 from AODrift) that were originally missed were found on the second trial, as the DM used in the first run was not close enough to the DM at which the pulsar was discovered. For the majority of pulsars that were not detected after re-running the pipeline, the discoveries were quite dim. The LOTAAS survey also has much longer integration times (60 minutes), which significantly improves the chances of the survey detecting pulsars which may be faint and/or nulling. When checking the discovery plots, it became clear that both of these effects were common to many of the missed pulsars. Some pulsars even appeared to exhibit nulling with `off' times as large as 100 seconds. Nulling behavior was also seen in many cases for the AODrift survey. For the GBT350 missed pulsars, all three of those that were missed were faint, and several GBNCC beams in which the pulsars were most likely to be found had RFI that spanned the entire 100\,MHz band.
Eight binary pulsars that were originally discovered in the GBNCC survey were not detected in the first pass of this pipeline. These pulsars required acceleration searches, which are automatically performed as a part of the search pipeline, but not here. As a part of the missed pulsar analysis, we ran an additional acceleration search using \texttt{ACCELSEARCH} from within the \texttt{PRESTO} package, and they were all detected. We also reprocessed data for 15 binary pulsars from the ATNF catalog with short ($\leq$ 0.5 day) orbital periods that were not detected in the first pass using acceleration searches; none of these were detected.
Pulsars with long periods (greater than 2.5\,s) were also followed up with a search for single pulses. Because these pulsars would only be observed for at most 48 pulses, non-detections are more common. To address this, we implemented \texttt{single\_pulse\_search.py} from the \texttt{PRESTO} package, which searches a range of dispersion measures to find bright single pulses in the data and characterize them by their S/N. In this way, a pulsar that is not detected via a periodicity search may be found by individual pulses. However, we were still unable to find these pulsars using this method.
\subsection{Spectral Indices}
\label{sec:spindx}
\begin{deluxetable}{cccc}
\tablecaption{Broken Powerlaw Spectral Indices \label{tbl:spindx}}
\tablehead{\colhead{PSR} & \colhead{$\alpha_{l}$\tablenotemark{a}} & \colhead{$\alpha_{h}$\tablenotemark{a}} & \colhead{Break Frequency} \\
& & & (MHz)}
\startdata
J0034-0534 & 0.6(3) & $-$3.1(2) & 181 \\
J0218+4232 & 1.15(7) & $-$2.7(4) & 149 \\
J1900-2600 & 0.2(4) & $-$1.89(15) & 204 \\
J2002+4050 & 0.2(16) & $-$1.51(18) & 378 \\
\enddata
\tablenotetext{a}{Spectral indices below ($\alpha_l$) and above ($\alpha_h$) the break.}
\tablecomments{Quantities in parentheses are uncertainties in the last digit. See Figure~\ref{fig:spindx} for the corresponding plots.}
\end{deluxetable}
Many previously published spectral indices were determined from flux measurements from high-frequency surveys \citep[e.g., see][]{jsk+18}. Therefore, low frequency surveys like the GBNCC survey provide more stringent constraints on these calculations. Results from this analysis are listed in Table \ref{tbl:det}.
The majority of the pulsars in this data set follow a single power law, or do not have enough ($>$2) flux density measurements to fit multiple power law functions. However, there are a small number of cases where the emission is better fit by a broken power law, defined instead as a piecewise function composed of two power laws.
All 339 pulsars for which we measured spectral index had three or more flux measurements (including our 350\,MHz measurements) and were checked by eye to determine whether or not a broken line fit was appropriate. Four pulsars fit these criteria. For these pulsars, we fit two lines, one for high frequency flux density measurements and one for low frequency. The breaking point for the power law was determined by finding the maximum change in the derivative of flux density with respect to frequency. A similar analysis was done in \cite{mkb+17}. Plots of these cases are provided in Figure \ref{fig:spindx} with both indices included. These plots also display the best-fit line to all measured flux densities. The measured values of $\alpha_{l}$ and $\alpha_{h}$ are reported in Table \ref{tbl:spindx}.
\begin{figure*}[ht]
\epsscale{1.}
\plotone{broken_spindx.pdf}
\caption{Pulsars with broken power-law spectral indices. We plot all available measurements of flux density in the ATNF catalog as well as the 350\,MHz measurements made in this study against observing frequency. We fit two disjoint lines to the low- and high-frequency measurements (orange solid lines). The red dashed line indicates the frequency of the turnover in the spectrum, determined by finding the point at which the two lines match up. Information for these measurements is presented in Table \ref{tbl:spindx}.}
\label{fig:spindx}
\end{figure*}
\subsection{Comparison of Dispersion Measure with Catalog Values}
The relatively low frequency of the GBNCC survey allows much higher precision DM measurements than typical 1400-MHz surveys, as dispersion across the band scales as $\nu^{-2}$. As pulses propagate through the interstellar medium, this dispersion results in a frequency dependent delay that smears out the arrival time of the pulse. Tools within the \texttt{PRESTO} package adjust for this, shifting the low frequency portion of the signal back in time to line up the pulse across the band. Using the \texttt{dmsearch} flag contained within the \texttt{PRESTO} command \texttt{prepfold}, we processed each of the pulsars and recovered more accurate values of DM. The program adjusts for dispersion and then folds the data at the pulsar's period to line up the pulses in both time and frequency. When \texttt{dmsearch} is off, the program does not tune the DM to maximize S/N; otherwise, the DM which aligns the pulses in frequency is returned as a new DM. In some cases, RFI caused the DM searching algorithm to return erroneous values for DM, and so we were unable to refine the dispersion measure. For these pulsars, we include the previously published DM in Table \ref{tbl:det} and mark them with a double dagger. More often, we were able to improve upon the previously published values of DM. Most of the discrepancies were small, but in some cases, our more precise DM measurement differed from the previous value significantly. For the pulsars with significant changes to their previously catalogued DM, we followed up with \texttt{TEMPO}\footnote{\url{http://tempo.sourceforge.net}} (maintained and distributed by Princeton University and the Australia Telescope National Facility). We split each detection into four subbands and created precise pulse times-of-arrival (TOAs) which can then be utilized to fit for DM. This method provides marginally more precise measurements, and so was only performed on pulsars with significant changes to previous DM measurements ($\geq$3$\sigma$). All newly measured DMs are presented in Table \ref{tbl:det}, and Table \ref{tbl:dm} highlights the pulsars which were followed up with \texttt{TEMPO} timing.
\section{Survey Sensitivity}
\label{sec:sensitivity}
\begin{figure*}
\plotone{GBNCC_senscurve_012820.pdf}
\caption{Flux density sensitivity in the GBNCC as a function of pulse period. Assuming a duty cycle of 6\% and an average unmasked bandwidth of 67\,MHz (which incorporates a 20\,MHz rolloff in the bandpass), we plot the predicted lower limit on the flux density of detectable pulsars for dispersion measures of 20, 50, 100, 150, 200, and 300\,pc\,cm$^{-3}$. To determine the sky temperature for the curves, we found the average sky temperature as a function of DM using the sky temperatures at the positions of all detected pulsars. We then drew from this function the temperatures at each DM for which a curve is plotted. For the above DMs, the function returns 95, 126, 171, 208, 237, and 273\,K. We glean the minimum detectable S/N for the survey by matching the curves to the faintest detection. This was found to be $\sim$3.8. Higher DM pulsars are more susceptible to smearing, and so the likelihood of detection is decreased for high DM, short period pulsars. We also plot both the detections (plus symbols) and non-detections (triangles), which are colored by their DM.}\label{fig:senscurve}
\end{figure*}
\begin{figure}
\epsscale{1.3}
\plotone{snm_vs_sne_120919.pdf}
\caption{Measured S/N vs.\ expected S/N for detections in the GBNCC survey. Extrinsic contributions to expected S/N include system temperature, telescope gain, scintillation, and offset from the beam center (newer pulsars without full timing solutions may have significant uncertainties in position). Errors in these quantities, previous flux measurements, and spectral indices increase the spread about unity, as does variable pulsar emission, i.e. nulling. }\label{fig:sn_mve}
\end{figure}
\subsection{Efficiency of GBNCC Survey}\label{subsec:efficiency}
In total, there were 5633 unique beams analyzed, yielding 1328 unique detections of the 670 pulsars. Given that there were 102948 beams that had been observed at the beginning of this project, this corresponds to an average number of detections per beam of $\sim$\,0.013 (0.063 detections per square degree), and $\sim$\,0.38 detections per hour of observing.
The ability to detect pulsars at 350\,MHz is limited most stringently by sky temperature and scattering in the interstellar medium (which correlates with dispersion). The expected S/N for detections is inversely proportional to system temperature, which is dominated by sky temperature near the Galactic plane. At 350\,MHz, this effect is quite significant, with temperatures approaching 1000\,K in this region. Scattering is especially detrimental in the detection of pulsars with short periods, as even a few milliseconds of smearing can eliminate the pulse entirely. Given a particular spin period and the estimated DM smearing, we can estimate the minimum flux density that will be detected by the survey. This relationship comes from solving Equation \ref{eq:sne} for flux density and assuming both an average sky temperature and duty cycle for the pulsars in the survey. Plotted in Figure \ref{fig:senscurve} are curves corresponding to a number of trial values of DM, showing the sensitivity floor at those values. Because DM and sky temperature are correlated, we determined the average sky temperature for each curve that is plotted, resulting in an increase in minimum detectable signals for higher DM pulsars. Also plotted are flux density measurements for detections made by this survey and expected flux density measurements for the pulsars which were not successfully detected. The colors in the plot correspond to the dispersion measure of each pulsar, showing how pulsars that may be intrinsically bright enough to be detected can still be missed because of dispersive smearing and/or scattering. The minimum flux density expected to be measured in the survey (regardless of spin period) can be determined to be the asymptotic value of the DM curve corresponding to the faintest detection. This value is directly proportional to the minimum S/N which results in a detection, hereafter ${\rm S/N}_{\rm cut}$, which was found to be $\sim$\,3.8. For all detections, we plot both the expected S/N at 350\, MHz as well as the measured S/N of the detection. These are plotted in Figure \ref{fig:sn_mve} along with a line marking unity. There is a large spread about this line, due mostly to stochastic noise sources in the data (telescope noise, temperature fluctuations, scintillation, and variable pulsar emission). When examining these results, several of the more significant outliers were analyzed in closer detail. One of the three significant outliers in the lower right portion of the plot was found to be a new nulling candidate, and the other two were initially labeled as possible nullers that could not be verified without higher resolution observations.
Low frequency observations can result in significant deterioration of the pulse due to scattering and scintillation effects, as residual dispersive time delay within a frequency channel with finite width increases as $\nu^{-3}$ and scattering roughly as $\nu^{-4}$ \citep{lk04}. Both of these phenomena result in a broadened pulse and subsequently a reduction in S/N.
To shed light on the causes for some of the missed pulsars, we calculate the expected S/N using information from both the catalog and information about the beams in which we expect to detect them. We predict flux density at 350\,MHz calculated as described in \S 2, determine the masked fraction of the closest beam to the pulsar's position (when measured), and determine $T_{sys}$ for the corresponding sky position. To determine $W$, we fit a line to our measurements of $W_{10}$ as a function of spin period and draw from this function. This allows for a measurement of the spin period-pulse width relation at 350\,MHz, supplementing previous measurements at other frequencies. This best fit line was measured to be $W_{10}$ = $18.5\degr(4)P^{0.270(10)}$, which is consistent with the relation determined in \cite{jk19} modulo a frequency-dependent scaling factor \citep[for a more in-depth analysis, see][]{cw14}. This fit is shown in Figure \ref{fig:pvw}.
\begin{figure}[ht]
\centering
\epsscale{1.2}
\plotone{pvw.pdf}
\caption{Pulse width at 10\% of the pulse maximum as a function of spin period. The solid line shows the line of best fit through the data, described by $W_{10}$ = $18.5\degr(4)P^{0.270(10)}$. The dashed line shows the minimum bin width as a function of period, as described in \S \ref{sec:methods}. }
\label{fig:pvw}
\end{figure}
After drawing widths from either the catalog or the above function (based on availability of previous measurements of $W_{10}$ near 350\,MHz), we determined the expected S/N for all non-detections. These are plotted along with the measured S/N for all of the detections in Figure \ref{fig:sn_hist}. The detections have been divided between those found from the catalog and those discovered by the GBNCC survey, and non-detections are divided based on Galactic latitude. These divisions allow for direct comparison between the survey's ability to detect pulsars blindly as well as the limits placed on the survey by high temperatures and scattering near the Galactic plane. Included in the plot are three different S/N cutoffs placed during different stages of the survey. The least stringent cutoff of S/N = 12 comes from \cite{slr+14}, where it was used as an estimated cutoff for detection to predict the survey's sensitivity. At this S/N, $\simeq$75\% of non-detections are not expected to be detected. Pulsars close to the plane generally have lower S/N as the temperature is so high, while pulsars outside of the plane generally have smaller DM and temperature but more scintillation. The two detection curves show that the GBNCC is sensitive to intrinsically fainter pulsars, as the histogram is skewed toward lower measured S/N than those from the catalog. Note that there was one pulsar discovered by the GBNCC search pipeline with S/N = 5.98, which is the bin to the left of the search S/N cutoff.
\begin{figure*}[ht]
\epsscale{1.2}
\centering
\plotone{sn_hist.pdf}
\caption{Histograms of measured S/N for detections and expected S/N for non-detections. Detections are differentiated by GBNCC discovery/catalog pulsars (green/cyan lines), and non-detections by distance from the Galactic plane (the red line indicates pulsars that are within 10$\degr$ from the plane, and the orange line indicates pulsars outside of this region). The dashed lines indicate three different S/N cutoffs: the first line, in black, show the minimum S/N detected in the survey; the second, in grey, indicates the significance down to which candidates are folded in the GBNCC search pipeline; and the third, in blue, shows the predicted S/N limit used in \cite{slr+14} to predict sensitivity of the survey.}
\label{fig:sn_hist}
\end{figure*}
In Figure \ref{fig:p0vsdm}, we plot all pulsars' periods against their dispersion measure. Each point's color and shape describe whether or not the pulsar was detected, and if not, whether we expect to have detected it. Missed detections that were unexpected are plotted with point sizes reflecting the expected flux density (calculated as described in \S\ref{sec:methods}) normalized by the value of the effective sensitivity curve for that pulsar, so larger points indicate pulsars with expected flux density much higher than the minimum detectable flux density at the pulsar's position.
\begin{figure*}[ht]
\epsscale{1}
\plotone{p0vsDM_120919.pdf}
\caption{Period vs.\ DM for all included pulsars. Blue symbols indicate detections made by the survey, and red symbols indicate non-detections. Red triangles indicate missed pulsars that were not expected to be detected, in that they lie below the expected sensitivity of the survey. Red circles indicate missed pulsars that lie above their expected sensitivity, and so were unexpected non-detections (see \S\ref{subsec:efficiency} for details). Blue circles indicate detections that were expected, and blue x symbols indicate detection of pulsars with expected flux densities that were below our sensitivity limit. The area of these points is given by the ratio of expected flux density to the limiting flux density at the pulsar's position.}
\label{fig:p0vsdm}
\end{figure*}
In total, there are 116 undetected pulsars plotted in Figure \ref{fig:p0vsdm} that have been classified as ``unexpected" by the logic above. Many of these pulsars are quite close to the sensitivity line, and so small errors in other flux density measurements and spectral indices may change them to ``expected." Because the effective sensitivity curve includes temperature and bandwidth (RFI, by proxy) information, reasons for missed detections are limited to effects that are harder to characterize. The most likely contributors include scintillation, abnormal pulsar behavior (i.e. nulling), and imprecise previous measurements of pulsar parameters resulting in inflated expected flux densities. Scintillation depends on DM \citep{cl91}, with increased timescales for smaller DM. Many of the non-detected pulsars that are outside of the Galactic plane are in this low-DM high-scintillation regime, and are likely to have been obscured (the expected number of scintles in the observation are on the order of $\sim$10). Many of the other missed detections, especially those from surveys with comparable limiting fluxes, were inspected individually. Many of these were obscured by significant RFI across the band. For example, PSR J0108$-$1431 (spin period of $\simeq0.81$\,s and DM of 2.38\,pc\, cm$^{-3}$, to the right of the bottom center of Figure \ref{fig:p0vsdm}) should be easily detected but was obscured by RFI. When examining a number of the other sources, it was found that many of the published spectral indices came from a 1400\,MHz study conducted by \cite{hwx+17}, and were unusually steep. This steepness results in high expected values of flux at 350\,MHz, which are not reflected in our results.
\subsection{RFI Analysis}\label{sec:rfi}
To visualize how RFI affects the efficiency of the survey, we determined the limiting flux density for each beam based on a S/N cutoff of 3.8, the temperature at the sky position of the beam, and the bandwidth available after RFI excision. Figure \ref{fig:flux_lim_hist} displays a histogram of the beams by their limiting flux, and Figure \ref{fig:flux_lim_map} shows these same data projected onto their sky positions. The sky map depicts a few important characteristics of the survey: the most obvious is the decreased sensitivity near the Galactic plane, but also visible are many individual pointings within the completed regions where significant RFI masking has reduced sensitivity. To mitigate this, these beams will be scheduled for reobserving. There is a small discrepancy between the number of observed beams displayed in Figures \ref{fig:skymap} and \ref{fig:flux_lim_map} due to a backlog of data which has yet to processed, and so mask fractions have not been determined for these beams.
\begin{figure}
\epsscale{1.2}
\plotone{GBNCC_fluxdensity_limit_hist.pdf}
\caption{Cumulative histogram of limiting flux density for GBNCC. The mean and median limiting flux densities in the histogram 0.74\,mJy and 0.62\,mJy, and the values range from 0.42\,mJy to 47.\,mJy. All flux density values are given in mJy.}
\label{fig:flux_lim_hist}
\end{figure}
\begin{figure*}
\plotone{limiting_flux_skymap.pdf}
\caption{Sky map of GBNCC beams, colored by limiting flux density. The map is plotted in Galactic coordinates on a Mollweide projection, and the flux density is given in mJy.}
\label{fig:flux_lim_map}
\end{figure*}
\subsection{Nulling/Mode-Changing Candidates}
The large set of data analyzed in this study as well as the ``by-eye" verification of all detections allowed for easy identification of potential nulling/mode-changing candidates in the results. This way, we are sensitive to nulling timescales between that of the pulsar spin period and the observation time (120 seconds). These cases were first identified by the appearance of missing pulses in the time-phase plots from processing using the \texttt{PRESTO} package. When a pulsar was noted as a candidate, we followed up using the \texttt{dspsr}\footnote{\url{http://dspsr.sourceforge.net/index.shtml}} package. We folded the time series data in 10 second integrations, zapped remaining RFI by hand, and integrated across frequency using the \texttt{pav} and \texttt{pam} commands within \texttt{PSRCHIVE}\footnote{\url{http://psrchive.sourceforge.net/index.shtml}}. When it was possible to discern on- and off-pulse regions by eye (i.e. significant changes in intensity for some rotations), the candidates were considered likely to be nulling. Some pulsars exhibited behavior similar to mode-changing, where multiple components of the averaged profile were found to be on during different portions of the observation. These pulsars were not treated differently than other nulling candidates $-$ we folded for single pulses to determine the likelihood that different components were visible. All of these sources will be followed up in later works regarding these data. In total, 223 pulsars were found to exhibit intensity variations similar to nulling or mode-changing during their observations, 62 of which have not previously been found to to do so. These candidates' names are marked in Table \ref{tbl:det} with an asterisk.
\subsection{The Galactic Pulsar Population}
Given its overall sky coverage and the large number of pulsar detections reported here (670), the GBNCC survey will play an important role in future understanding of the Galactic pulsar population. To date, the GBNCC survey has detected 571 non-recycled (long-period) pulsars in the Galactic field and 70 Galactic MSPs, which have undergone recycling and have spin periods, $P<30$\,ms. Remaining detections are either associated with globular clusters (3) or are recycled pulsars with spin periods, $P>30$\,ms (26), and have been intentionally ignored for the following analysis, since our current models do not adequately describe the features of this sub-population.
To estimate expected numbers of non-recycled/millisecond pulsar detections in the GBNCC survey, Galactic populations were simulated using {\sc PsrPopPy2}\footnote{https://github.com/devanshkv/PsrPopPy2}, a more recent and currently-maintained version of {\sc PsrPopPy} \citep[][and citations within]{blr+14}. Pulsar populations were generated using {\sc PsrPopPy2}'s {\tt populate} function, which simulates pulsars by drawing parameters from predefined distributions until some condition is met. Due to its large sample size, population estimates from the Parkes Multibeam Pulsar Survey (PMPS) provide the best-known sample parameters. For this reason, these results were used to set a limit on the number of pulsars simulated by {\tt populate}. For the non-recycled pulsar population, pulsars were generated until a synthetic PMPS ``detected" 1038 sources; for MSPs, the desired population size was set to 30,000 sources. Specific parameters defining pulsars' Galactic radial distribution, as well as scale height, spin period, luminosity, and duty cycle can be found in \citet{slm+14}. However, an updated model for the MSP $P$-distribution \citep{lem+15} was implemented in simulations here.
Synthetic surveys were conducted with 100 realizations each of the Galactic non-recycled/millisecond pulsar populations using {\tt survey} and a GBNCC model file, including survey parameters identical to those presented in \S\ref{sec:methods} and lists of completed/remaining GBNCC pointing positions. In the first round of simulations, we fixed the S/N limit for detections to ${\rm S/N}_{\rm cut}=3.8$ (as determined in \S\ref{subsec:efficiency}). This simulation predicted 1442/126 simulated detections for non-recycled/millisecond pulsar populations, respectively (on average; compared to 571/70 actual detections). We then fixed the number of simulated non-recycled/millisecond pulsar detections to their actual values (571/70) and found nominal S/N thresholds for each sub-population, ${\rm S/N}_{\rm cut}=15.3/9.1$. The discrepancies between simulated and actual yields suggest uncertainties in population parameters informed primarily by the PMPS survey, which targeted the Galactic plane and was conducted at 1.4\,GHz. Population parameters determined by these previous surveys produce over-estimates for GBNCC pulsar yields. As an all-sky, low-frequency search, the GBNCC survey (when complete) will be a valuable counterpoint to further refine non-recycled/millisecond pulsar population parameters. As we will show below, positional and rotational parameters of the simulated populations do not match the detected population when these thresholds are set.
\begin{deluxetable}{lcccc}
\tablecaption{K-S test statistics and $p$-values resulting from comparisons between actual/simulated parameter distributions for non-recycled/millisecond pulsars. In cases where the $p$-value is $<1\%$, the null hypothesis (that the two distributions are the same) is rejected.\label{tbl:ks}}
\tablehead{\colhead{Parameter} & \multicolumn{2}{c}{Normal\tablenotemark{a}}& \multicolumn{2}{c}{MSP\tablenotemark{b}}\\
\cline{2-3} \cline{4-5}
& \colhead{K-S} & \colhead{$p(\%)$} & \colhead{K-S} & \colhead{$p(\%)$}
}
\startdata
Spin Period ($P$) & 0.20 & $\ll1$ & 0.14 & \phn$10$ \\
Dispersion Measure (DM) & 0.21 & $\ll1$ & 0.26 & $<1$ \\
Flux Density ($S_{350}$) & 0.13 & $\ll1$ & 0.21 & $<1$ \\
Galactic Latitude ($b$) & 0.04 & \phn\phn$41$ & 0.17 & \phn\phn$3$ \\
\enddata
\tablenotetext{a}{For simulated non-recycled pulsars, ${\rm S/N}_{\rm cut}=15.3$.}
\tablenotetext{b}{For simulated MSP population, ${\rm S/N}_{\rm cut}=9.1$.}
\end{deluxetable}
To test the validity of underlying non-recycled/millisecond pulsar populations, we compared cumulative distribution functions (CDFs) of simulated ({\tt sim}) pulsar parameters ($P$, DM, $S_{350}$, and $b$) with those of the actual ({\tt act}) detections using a {\tt scipy} implementation of the 2-sample Kolmogorov–Smirnov (K-S) test. For each parameter, the K-S test statistic and $p$-value were computed over a range of ${\rm S/N}_{\rm cut}$. When $p<1\%$, the null hypothesis (that {\tt act}/{\tt sim} parameters are drawn from the same underlying distribution) is rejected. Figures \ref{fig:norm_cdfs} and \ref{fig:msp_cdfs} illustrate these comparisons for non-recycled and millisecond pulsar population parameter distributions, and Table \ref{tbl:ks} summarizes K-S test results when the nominal ${\rm S/N}_{\rm cut}$ values for non-recycled/millisecond pulsar sub-populations (15.3/9.1) are implemented, though we measured these p-values for a range of imposed ${\rm S/N}_{\rm cut}$ values.
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{NORM-Hists+CDFs.pdf}
\caption{Normalized histograms showing comparisons between (a) spin period, $P$, (b) Galactic latitude, $b$, (c) flux density, $S_{350}$, and (d) dispersion measure, DM, distributions for simulated non-recycled pulsars (blue) and actual detections (orange). The rightmost panel in each row compares actual/simulated CDFs for each parameter. K-S tests comparing these CDFs (see Table \ref{tbl:ks} for details) show disagreement between {\tt act}/{\tt sim} $P$, $S_{350}$, and DM distributions, but $p=41\%$ for $b$ distributions.}
\label{fig:norm_cdfs}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{MSP-Hists+CDFs.pdf}
\caption{Normalized histograms showing comparisons between (a) spin period, $P$, (b) Galactic latitude, $b$, (c) flux density, $S_{350}$, and (d) dispersion measure, DM, distributions for simulated millisecond pulsars (blue) and actual detections (orange). The rightmost panel in each row compares actual/simulated CDFs for each parameter. K-S tests comparing these CDFs (see Table \ref{tbl:ks} for details) show disagreement between {\tt act}/{\tt sim} $S_{350}$ and DM distributions, but distributions for $b$ and $P$ have $p=3\%$ and $10\%$, respectively.}
\label{fig:msp_cdfs}
\end{figure*}
Comparing {\tt act}/{\tt sim} parameters for the non-recycled pulsar population, we find broad agreement between $b$ distributions, regardless of ${\rm S/N}_{\rm cut}$. Results for other non-recycled pulsar parameters in Table \ref{tbl:ks} show significant inconsistencies between {\tt act}/{\tt sim} samples. DM distributions are clearly different for ${\rm S/N}_{\rm cut}>4$, likely due to an over-abundance of low-DM simulated detections. For ${\rm S/N}_{\rm cut}=15.3$, we find twice as many {\tt sim} detections with ${\rm DM}<35$\,pc\,cm$^{-3}$. Presumably due to the prevalence of nearby {\tt sim} sources, this sample also has a larger fraction of high-flux density sources, so $S_{350}$ distributions are statistically different for ${\rm S/N}_{\rm cut}=15.3$. However, there is a small window ($10.25<{\rm S/N}_{\rm cut}<12.25$) where {\tt act}/{\tt sim} $S_{350}$ distributions become statistically similar, with $p>1\%$. The null hypothesis is rejected for $P$ due to {\tt act}/{\tt sim} log-normal distributions having different mean values: $\langle\log P_{\rm act}\rangle = 2.88$ versus $\langle\log P_{\rm sim}\rangle = 2.72$ (see Figure \ref{fig:norm_cdfs}). This discrepancy persists, regardless of chosen ${\rm S/N}_{\rm cut}$.
Because the simulated versions of the non-recycled pulsar population were primarily informed by PMPS \citep[e.g.][]{lfl+06}, which was conducted at 1.4\,GHz and exclusively covered regions of sky near the Galactic plane ($|b| < 5\,\degr$), we expect there to be bias toward highly dispersed pulsars near the plane. Due to more uniform sky coverage and \--- near the Galactic plane \--- higher sky temperatures and more significant scattering at 350\,MHz, the majority of GBNCC detections ($67\%$) are away from the plane ($|b| > 5\,\degr$). Young pulsars are typically born in the plane and tend to be found nearby, therefore GBNCC's reduced sensitivity to low-latitude sources means that relatively few detections are young pulsars. The $P$-$\dot{P}$ diagram in Figure \ref{fig:ppdot} nicely illustrates this shortage of pulsars detected with characteristic ages, $\tau \leq 1$\,Myr. By imposing an age cutoff on non-recycled pulsars in the ATNF catalog, $\tau > 1$\,Myr, the resulting simulated spin period distribution is statistically similar to that of GBNCC detections (K-S $p>1\%$). This selection effect accounts for the apparent differences between {\tt act}/{\tt sim} $P$-distributions, but can not explain discrepancies in $S_{350}$ and DM distributions for non-recycled pulsars.
K-S tests comparing {\tt act}/{\tt sim} parameter distributions for the MSP population show better agreement (see Table \ref{tbl:ks} and Figure \ref{fig:msp_cdfs}). For MSPs, selection effects based on Galactic latitude and spin period do not come into play since MSPs are more isotropically distributed and model parameters for this sub-population are based on results from multiple Parkes Telescope surveys (see \cite{lem+15}, and references therein). For these reasons, the simulated population's spin periods are statistically similar to the sample detected by GBNCC. This conclusion does not change, regardless of the chosen ${\rm S/N}_{\rm cut}$ value. For $b$, the null hypothesis is still not rejected by our criteria ($p<1\%$). Based on the $b$ histograms themselves, there appears to be an absence of detections in the {\tt act} sample in/near the Galactic plane, which is not the case for {\tt sim} sources. The null hypothesis is rejected for $S_{350}$ due to the over-abundance of high-flux-density sources in the {\tt sim} sample compared to those present in the {\tt act} sample. Median flux densities for {\tt act}/{\tt sim} detections are 4.9/6.8\,mJy respectively. Comparing {\tt act}/{\tt sim} DM distributions, the {\tt sim} sample consists of a higher fraction of high-DM MSPs and 12\% of simulated detections have DMs in excess of the {\tt act} maximum value, $104.5$\,pc\,cm$^{-3}$. This is likely related to the bias toward high-flux-density detections noted in $S_{350}$ for {\tt sim} MSPs mentioned above.
Based on discrepancies between predicted yields from simulations and actual numbers of detections by the GBNCC survey, it appears that model parameters need to be further refined in order to generate more realistic Galactic pulsar populations in the future. For now, we proceed with nominal ${\rm S/N}_{\rm cut}$ values in order to estimate the GBNCC survey's future yields. In the remaining $\approx$21,000 pointings, we expect an additional 160/16 non-recycled/millisecond pulsar detections, or $\approx60/5$ discoveries, accounting for detectable known pulsars in regions of sky remaining \citep{mhth05}.
\begin{figure}
\epsscale{1.2}
\plotone{ppdot.pdf}
\caption{Period vs.\ period derivative for pulsars in GBNCC survey area. Shown in grey are pulsars that were not detected, and blue plus symbols show detections.}
\label{fig:ppdot}
\end{figure}
\section{Conclusions}
\label{sec:conc}
We have provided all detections of currently known pulsars that exist within the area of the 350\,MHz GBNCC pulsar survey and performed some preliminary analysis of the resulting data set. Specifically, we have provided new flux density and pulse width measurements as well as pulse profiles for the 670 detections. When possible, we used our flux density measurements with previous measurements at different frequencies to refine spectral index. We also made a measurement of the spin period-pulse width relation, observing a powerlaw correlation of the form $W_{10} \propto P^{-0.27}$. The low frequency of the survey provides increased sensitivity to dispersion, allowing for more precise measurements of DM for many pulsars that have only been measured in high frequency surveys. Using all of this information, we have made quantitative measurements of the survey's efficacy and the RFI environment at 350\,MHz, with a minimum detectable S/N of $\sim$3.8 and a mean limiting flux density of 0.74\,mJy. These measurements have allowed us to make realistic predictions about the survey's yield when complete based on the detectability of known pulsars in the dataset, and we expect to detect on the order of 160 non-recycled pulsars and 15 MSPs. The simulations from which these expectations come uncovered discrepancies in DM, spin period, and spatial distribution in the Galaxy for the simulated populations which will be addressed in a future study. Combing through the data following processing has brought many interesting characteristics of pulsars in the survey to light, including 223 pulsars exhibiting evidence of variable intensities suggestive of nulling/mode-changing and 4 showing evidence for broken power-law spectral energy distributions. These kinds of qualitative observations pave the way for follow-up quantitative analyses of these data and the remaining beams that will be observed in the next few years.
\section*{Acknowledgements}
We thank our anonymous referee for their suggestions and guidance. This work was supported by the NANOGrav Physics Frontiers Center, which is supported by the National Science Foundation award 1430284. The Green Bank Observatory is a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc.
R.S.\ acknowledges support through the Australian Research Council grant FL150100148.
WF acknowledges the WVU STEM Mountains of Excellence Graduate Fellowship. MM and MS acknowledge the National Science Foundation OIA award number 1458952. JvL acknowledges funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 617199 (`ALERT'), and from Vici research programme `ARGO' with project number 639.043.815, financed by the Netherlands Organisation for Scientific Research (NWO). VMK acknowledges the NSERC Discovery Grant, the Herzberg Award, FRQNT and CRAQ, Canada Research Chairs, CIFAR and the Webster Foundation Fellowship, and the Trottier Chair in Astrophysics and Cosmology. Computations were made on the supercomputer Guillimin at McGill University\footnote{\url{www.hpc.mcgill.ca}}, managed by Calcul Quebec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQuebec, RMGA and the Fonds de recherche du Quebec - Nature et technologies (FRQ-NT). The CyberSKA project was funded by a CANARIE NEP-2 grant. PC acknowledges the FRQNT Doctoral Research Award. SMR is a CIFAR Senior Fellow. Pulsar research at UBC is supported by an NSERC Discovery Grant and by the Canada Foundation for Innovation.
\software{\texttt{Astropy} \citep{astropy:2018}, \texttt{PRESTO} \citep{smr+01},
\texttt{PsrPopPy2} \citep{blr+14},
\texttt{SciPy} \citep{scipy},
\texttt{NumPy} \citep{numpy},
\texttt{dspsr} \citep{sb11},
\texttt{PSRCHIVE} \citep{hsm04},
\texttt{TEMPO} (\url{http://tempo.sourceforge.net/})}
\facilities{Robert C. Byrd Green Bank Telescope (GBT)}
\bibliographystyle{apj}
|
2,877,628,089,457 | arxiv |
\section{Introduction}
\subsection{Hypergraph Turán Problems}
Given a $k$-graph $H$, write $\pi\left(n, H\right)$ for the maximum density of $k$-uniform edges among $H$-free hypergraphs with size $n$, and let $\pi \left(H \right) = \lim_{n \rightarrow \infty} \pi\left(n, H\right)$. It is known that the limit always exists. Let $K^{(k)}_n$ be the complete $k$-graph with $n$ vertices. A landmark result by Turán determined the values $\pi\left(n, K^{(2)}_r\right)$ exactly, with the unique graphs attaining the maximum.
\begin{theorem}[\cite{turan_original}]
$\pi\left(n, K^{(2)}_r\right)$ is uniquely attained at the balanced complete $(r-1)$-partite graph on $n$ vertices.
\end{theorem}
Following this, Erdős and Stone found more generally the value $\pi(H)$ for all graph $H$.
\begin{theorem}[\cite{erdos_stone}] Suppose $H$ is a graph with chromatic number $\chi(H)$, then
\begin{equation*}
\pi(H) = 1 - \frac{1}{\chi(H)-1}.
\end{equation*}
\end{theorem}
The corresponding question, when $k>2$, is still open and seems to be much more difficult. There are sporadic results for various $k$-graphs, but no $\pi\left(K^{(k)}_r\right)$ value is known. The best general upper bound comes from de Caen.
\begin{theorem}[\cite{de_caen_bound}]
\begin{equation*}
\pi\left(n, K^{(k)}_r\right) \leq 1 - \left( 1 + \frac{r-k}{n-r+1} \right) \frac{1}{\binom{r-1}{k-1}}.
\end{equation*}
\end{theorem}
For an extensive survey, focusing on the $\pi\left(n, K^{(k)}_r\right)$ problem, with various lower and upper bounds, see \cite{sido_survey}. More recent coverage of the question with different $k$-graphs can be found in \cite{keevash_survey}.
\subsection{Generalized Turán Problems}
As a possible generalization of the Turán question, one can ask the maximum density of a given $k$-graph $F$, instead of the $k$-edges. For $F, H$ given $k$-graphs, write $\pi(n, F, H)$ for the maximum density of $F$ among $n$ sized $H$-free $k$-graphs and use $\pi (F, H) = \lim_{n \rightarrow \infty} \pi(n, F, H)$. For complete graphs, this was initially investigated by Erdős \cite{erdos_generalized_1, erdos_generalized_2}.
\begin{theorem}[\cite{erdos_generalized_1}]
For $2 \leq g < r$ integers $\pi\left(n, K^{(2)}_g, K^{(2)}_r\right)$ is uniquely attained at the balanced complete $(r-1)$-partite graph on $n$ vertices.
\end{theorem}
Note that this gives asymptotically that $\pi\left(K^{(2)}_g, K^{(2)}_r\right) = \prod_{m=2}^{g} \left(1 - \frac{m-1}{r-1} \right)$. The generalized Turán problem for graphs was systematically investigated by Alon and Shikhelman, obtaining a result similar to Erdős-Stone.
\begin{theorem}[\cite{alon_generalized_erdos_stone}]
For any graph $H$, with chromatic number $\chi(H)$, the following holds
\begin{equation*}
\pi\left(K^{(2)}_g, H\right) = \prod_{m=2}^{g} \left(1 - \frac{m-1}{\chi(H)-1} \right).
\end{equation*}
\end{theorem}
In addition, \cite{alon_generalized_erdos_stone} investigates degenerate generalized Turán questions -- the rate of convergence of $\pi(n, F, H)$ when $\pi(F, H) = 0$. \cite{deg_gen_hyp_turan} finds various bounds for several degenerate generalized hypergraph Turán problems.
The generalized Turán problem for complete $k$-graphs corresponds with the separation of different layers of the boolean hypercube using a $k$-CNF. This idea appears for example in \cite{application_sidorenko} and will be further explored in a different paper. \cite{application_vcdim} gives new insights into the set of satisfying assignments of CNFs using a variant of the VC dimension. Bounds on this variant of the VC dimension turn out to be equivalent to a generalized Turán-type conjecture.
\subsection{Flag Algebras}
The theory of flag algebras \cite{razb_flag_algebras} provides a systematic approach to studying extremal combinatorial problems and the tools available for solving them. It gives a common ground for combinatorial ideas, by expressing them as linear operators, acting between flag algebras. Linearity means the different techniques can be easily combined with linear programming/linear algebra.
A large part of the theory can be automated with state-of-the-art optimization algorithms, providing spectacular improvements in density bounds. There has been significant progress in the famous tetrahedron problem \cite{de_caen_bound, tetra_previous} with the previous best bound being $\pi\left( K^{(3)}_4 \right) \leq \frac{3+\sqrt{7}}{12} < 0.59360$ while flag algebraic calculations improved it to the following bound: \begin{theorem}[\cite{razb_4_vertex} verified in \cite{flagmatic, baber_thesis}]
$\pi\left( K^{(3)}_4 \right) \leq 0.56167.$
\end{theorem} Note that the best known lower bound is $5/9 \leq \pi\left( K^{(3)}_4 \right)$. For excluded $K^{(3)}_5$ the calculations give the following: \begin{theorem}[\cite{baber_thesis}]
$\pi\left( K^{(3)}_5 \right) \leq 0.76954$
\end{theorem} with best known lower bound $3/4 \leq \pi\left( K^{(3)}_5 \right)$.
For a list of results provided by flag algebraic calculations, see \cite{razb_4_vertex, flagmatic}. The power of flag algebra has been illustrated in a wide range of other combinatorial questions \cite{ramsey_flag, permutation_flag}. Unfortunately, the computer-generated proofs lack insight and scale-ability compared to classical, hand-crafted arguments. They often only work in a small enough parameter range (for example, bounding $\pi\left(H\right)$ for $H$ with at most $7$ vertices). A survey by Razborov \cite{flag_interim_report} calls such applications plain.
One of the goals of this paper is to show that the powerful plain flag algebra method, in this case, can be performed by hand, resulting in a general and scale-able theorem. The main ideas and proof steps, therefore, correspond with a plain application of flag algebra and were heavily inspired by it. During the proofs, relevant parts of the flag algebra theory will be highlighted. While the asymptotic result can be fully proved with flag algebraic manipulations, the bound with finite $n$ is only attainable with a more precise bounding of the errors. The theory is not explained here, for a quick introduction see \cite{flag_first_glance} or the original text \cite{razb_flag_algebras}.
\subsection{Overview of the Result}
In this paper, the generalized Turán problem for complete hypergraphs will be investigated, with the following contribution:
\begin{theorem}\label{main_theorem}
For integers $1 < k \leq g < r$ and any $n>(r-1) \left(1 + \left(\frac{(r-k)}{k-1}\right)^2 \right),$
\begin{equation*}
\begin{gathered}
\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right) \leq \\
\leq \left(1 + \frac{(r-1) (r-k)^2}{(k-1)^2 n-(r-1) \left(2 k^2-2 k (r+1)+r^2+1\right)} \right)\prod_{m=k}^{g} \left(1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right).
\end{gathered}
\end{equation*}
\end{theorem}
Note this means asymptotically that
\begin{corollary}
For integers $1 < k \leq g < r$
\begin{equation*}
\pi\left(K^{(k)}_{g}, K^{(k)}_r\right) \leq \prod_{m=k}^{g} \left(1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right).
\end{equation*}
\end{corollary}
The asymptotic bound is known to be tight when $k=2$, with the matching, balanced $(r-1)$-partite construction. Additionally, it agrees with the best-known general hypergraph Turán bound by de Caen \cite{de_caen_bound} which is conjectured to not be tight.
\cite{sido_survey} describes various lower bound constructions for the $2 < k = g$ case. A simple construction (when $k-1$ divides $r-1$) splits the vertex set into $\frac{r-1}{k-1}$ equal groups and includes each $k$ set that is not fully contained in a group. Using $l = \frac{r-1}{k-1}$ and an inclusion-exclusion calculation, this gives the asymptotic bound
\begin{equation*}
\sum_{s=0}^{\lfloor g/k \rfloor} (-1)^{s} \binom{l}{s} \sum_{\substack{k \leq i_1, \dots, i_s \\ i_1+ \dots + i_s \leq g}} \binom{g}{i_1, \dots, i_s, g-i_1- \dots - i_s}l^{-i_1- \dots - i_s} \leq \pi\left( K^{(k)}_g, K^{(k)}_r\right).
\end{equation*}
While it is known that this construction is not optimal when $k=g$, in the $k \ll g$ regime, where $K^{(k)}_g$ appears more if the edges are "grouped", it provides a stronger bound. \Cref{pi345_section} shows that this is asymptotically the best construction for $\pi \left( K^{(3)}_4, K^{(3)}_5\right)$. In general $g=r-1$ gives
\begin{equation*}
\begin{split}
& \ \sqrt{ 2 \pi r} \ \ e^{\frac{\log(2 \pi k)r}{2k}} \approx \\[2ex]
\approx & \ \binom{r-1}{k-1, k-1, \dots , k-1} l^{-(r-1)} \leq \\[2ex]
\leq & \ \pi\left(K^{(k)}_{r-1}, K^{(k)}_{r}\right) \leq \\[2ex]
\leq & \ \prod_{m=k}^{r-1} \left( 1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right) \leq \\[2ex]
\leq & \ e^{(k-r)/k}.
\end{split}
\end{equation*}
Given a hypergraph $G$, write $d(H, G)$ for the induced density of $H$ in $G$. The main tool used in the proof of \cref{main_theorem} is:
\begin{lemma}\label{main_lemma}
For all $0 \leq x$ and integers $k \leq m < n$, if $G$ is an $n$ vertex $k$-graph then
\begin{equation*}
\begin{matrix*}[l]
0 \geq & \left(- \frac{1 - \frac{k-1}{m}}{x}\right) & d\left(K^{(k)}_{m+1}, G\right) & + \\ & \left(2 - \frac{k-1}{mx} - \frac{1}{(n-m)x}\right) & d\left(K^{(k)}_m, G\right) & + \\ & (-x) & d\left(K^{(k)}_{m-1}, G\right). &
\end{matrix*}
\end{equation*}
\end{lemma}
Note that the densities $d\left(K^{(k)}_m, G\right)$ only appear linearly in the expression. For different $k, g, r$ parameters, a convex combination of the expressions appearing in \cref{main_lemma}, with suitable $x, m$ values substituted in yields \cref{main_theorem}. As a comparison, \cite{de_caen_bound} utilizes similar ideas, but with a more complicated (non-linear) expression,
\begin{equation*}
f_{m+1} \geq \frac{m^2 f_m}{(m-k+1)(n-m)} \left(\frac{f_m (n-m+1)}{f_{m-1} m} - \frac{(k-1)(n-m) + m}{m^2} \right)
\end{equation*} where $f_m = d\left(K^{(k)}_m, G\right)$ for short.
\subsection{Outline of the Paper}
\Cref{notation_section} summarizes the important notations and conventions throughout the paper. The proof of \cref{main_theorem} is included in \cref{main_theorem_section} using two important components: \cref{main_lemma}, which is proved in \cref{lindens_section}; and a technical calculation (\cref{tridiag_lemma}), that is included in \cref{tridiag_section}. The short \cref{pi345_section} includes a certificate for $\pi\left(K^{(3)}_4, K^{(3)}_5\right) = 3/8$. The paper finishes with a few concluding remarks in \cref{outro_section}, the limitations of this approach and possible directions.
\section{Notation and Conventions}\label{notation_section}
\subsection{Basic Notation}
For a set $V$, the collection of subsets with size $k$ is denoted by $\binom{V}{k}$. The hypergraphs are identified with their edge sets; $G \subseteq \binom{V(G)}{k}$ is a $k$-graph with $V(G)$ vertex set. $K_n^{(k)}$ is the complete $k$-graph with $n$ vertices. For $S \subseteq V(G)$ the induced sub-hypergraph is $G \! \upharpoonright_S = G \cap \binom{S}{k}$. Hypergraph isomorphism is represented by $G \simeq H$. $\mathcal{H}^{(k)}_n$ is the collection of non-isomorphic $k$-graphs having $n$ vertices.
Bold symbols indicate random variables. The uniform distribution from a set $V$ is represented by $\operatorname{Unif}(V)$. The density of $H$ in $G$ is defined to be $d(H, G) = \mathbb{P} \left[ G \! \upharpoonright_{\mathbf{S}} \simeq H \right]$ where $\mathbf{S} \sim \operatorname{Unif}\binom{V(G)}{|H|}$. Notice that this is the induced density, corresponding more with the flag algebraic approach, rather than the classical sub-hypergraph inclusion (referenced in the introduction). Write $d_s(H, G) = \mathbb{P} \left[ G \! \upharpoonright_{\mathbf{S}} \simeq F, H \subseteq F \right]$ for the classical inclusion with the same $\mathbf{S} \sim \operatorname{Unif}\binom{V(G)}{|H|}$. When $H$ is a complete hypergraph, the two notions are equivalent. The generalized Turán problem is to determine the value $$\pi\left(n, F, H\right) = \max\left\{d_s(F, G) \ : \ G \in \mathcal{H}_n, \quad d_s(H, G) = 0 \right\}.$$ The asymptotic problem asks $\pi(F, G) = \lim_{n \rightarrow \infty } \pi(n, F, G)$ (it is known that the limit always exists).
The quantity $$x_{m, r}^{(k)} = 1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}}$$ will be important, these are the terms appearing in the product.
\subsection{Flag Notation}
The hypergraphs represent the corresponding flags with empty type. $T^{(k)}_n$ is the complete type with $n$ vertices and all $k$-uniform edges. Type is indicated as a superscript. In particular $K^{(k), T^{(k)}_m}_{n}$ is the unique complete flag on $n$ vertices with a type having $m$ vertices. For a type $T$, $T$ also represents the flag with type $T$ and no extra vertices/edges. The averaging operator, transforming a $T$-typed flag $F^T$ into a flag with empty type is $\left\llbracket F^T \right\rrbracket_T$.
\subsection{Conventions}
In most of the proofs, the symbol $k$ is fixed, and the appearing statements concern $k$-graphs. For this reason, $k$ superscripts from the notations are often dropped. Additionally, $g, \ r, \ n$ symbols are reserved. They are integer parameters of the main question; determining the value of $\pi(n, K_g, K_r)$.
\section{Proof of Main Theorem}\label{main_theorem_section}
In this short section \cref{main_theorem} will be proved with the use of \cref{main_lemma} and \cref{tridiag_lemma}, a technical result included in \cref{tridiag_section}. Let's recall the main theorem, with the introduced $x^{(k)}_{m, r}$ notation.
\begin{reptheorem}{main_theorem}
Given integers $1 < k \leq g < r$ and $n>(r-1) \left(1 + \left(\frac{(r-k)}{k-1}\right)^2 \right)$, then
\begin{equation*}
\begin{gathered}
\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right) \\ \leq \left(1 + \frac{(r-1) (r-k)^2}{(k-1)^2 n-(r-1) \left(2 k^2-2 k (r+1)+r^2+1\right)} \right) \prod_{m=k}^{g} x^{(k)}_{m, r}.
\end{gathered}
\end{equation*}
\end{reptheorem}
In the following, the superscript $k$ is dropped from the notations for easier readability. The following claim gives bounds on the $x_{m, r}$ values. It follows easily after expanding the definition of $x_{m, r}$, therefore the proof is not included.
\begin{claim}\label{x_positive_claim}
When $k-1 \leq m \leq r$, $$0 \leq x^{(k)}_{m, r} \leq 1,$$ with equality at $m=r$ and $m = k-1$ respectively. The smallest nonzero value is $x_{r-1, r} = \frac{k-1}{r-k}$
\end{claim}
\begin{proof}[Proof of \cref{main_theorem}]
Choose any $G \in \mathcal{H}_{n}$ (irrespective of the value of $d(K_{r}, G)$), with $$n>(r-1) \left(1 + \left(\frac{(r-k)}{k-1}\right)^2 \right),$$ and for short write $f_m = d\left(K_{m}, G\right)$. In the range $m \in \{k, k+1, ..., r-1\}$, $x_{m, r}$ is positive, therefore \cref{main_lemma}, with $x=x_{m, r}$ holds.
\begin{equation*}
0 \geq - \frac{1 - \frac{k-1}{m}}{x_{m, r}}f_{m+1} + \left(2 - \frac{k-1}{m x_{m, r}} - \frac{1}{(n-m)x_{m, r}} \right) f_{m} - x_{m, r} f_{m-1}
\end{equation*}
The value $(n-m)x_{m, r}$ is minimal at $m=r-1$. Use $E_m$ for the above expression but $\frac{1}{(n-m)x_{m, r}}$ replaced with $\frac{r-k}{(n-r+1)(k-1)}$.
\begin{equation*}
E_{m} = - \frac{1 - \frac{k-1}{m}}{x_{m, r}}f_{m+1} + \left(2 - \frac{k-1}{m x_{m, r}} - \frac{r-k}{(n-r+1)(k-1)} \right) f_{m} - x_{m, r} f_{m-1}
\end{equation*}
The replacement decreases the value, giving that each $E_{m}$ is still non-positive. This gives that any $\delta_{k}, \delta_{k+1}, ..., \delta_{r-1}$ sequence with all $0 \leq \delta_m$ results in $0 \geq \sum_{m=k}^{r-1} \delta_{m} E_{m}$. For a lower bound it is enough to find coefficients $0 \leq \delta_m$ satisfying \begin{equation} \label{main_theorem_equation}
0 \geq \sum_{m=k}^{r-1} \delta_m E_{m} = - \delta_k x_{k, r} f_{k-1} + f_g - \delta_{r-1} \frac{1-\frac{k-1}{r-1}}{x_{r-1, r}} f_r.
\end{equation}
With the assumption that $f_r = d(K_r, G) = 0$ and the simple observation that $f_{k-1} = d(K_{k-1}, G) = 1$, one can deduce from \cref{main_theorem_equation} that $\pi\left(n, K_g, K_r \right) \leq \delta_k x_{k, r}$. Notice that finding $\delta_m$ corresponds with solving \cref{main_theorem_equation}, a system of linear equations. The technical \cref{tridiag_section} includes a way to approximate this system of linear equations. The following lemma summarizes the result, concluding the proof of \cref{main_theorem}.
\begin{lemma}\label{tridiag_lemma}
If $n>(r-1) \left(1 + \left(\frac{(r-k)}{k-1}\right)^2 \right)$ then the solution to the linear equations \cref{main_theorem_equation} satisfies that $\delta_m \geq 0$ and that $$\delta_k = \left(1 + \frac{(r-1) (r-k)^2}{(k-1)^2 n-(r-1) \left(2 k^2-2 k (r+1)+r^2+1\right)} \right) \prod_{m=k+1}^g x_{m, r}.$$
\end{lemma}
\end{proof}
\section{Linear Density Relations}\label{lindens_section}
This section covers the main combinatorial calculations involved in the proof of \cref{main_lemma}. The main purpose of \cref{main_lemma} is to act as a building block. Not only the validity is easier to verify, but it also involves the densities $d(K_m, G)$ linearly, therefore it can be combined easily; as illustrated in \cref{main_theorem}. The small claims in this section follow closely core results from the flag algebra theory. The connection will be highlighted in \cref{flag_claims_remark}. The connection between the plain flag algebra application and this proof is discussed in \cref{flag_sdp_remark}.
In the upcoming proofs, the $k$ superscript will not be included. $S$ is any subset of $V(G)$, while $S_m$ is an $m$ element subset of $V(G)$. Write $q(S)$ for the indicator function that is $1$ when $S$ induces a complete hypergraph in $G$ and $0$ otherwise. $l(S)$ is the number of $v \in V(G) \setminus S$ where $S+v$ is complete in $G$. The corresponding probability is $r(S_m) = \frac{l(S_m)}{n-m}$. Similarly $rr(S)$ is the probability that two different vertex extensions are both complete. $$rr(S_m) = \frac{\binom{l(S_m)}{2}}{\binom{n-m}{2}}$$
When $H \in \mathcal{H}_n$, then $s(H)$ is used for the size of the intersection of non-edges in $H$. In particular $s(K_n) = n$ and $s(K_n^-) = k$ where $K_n^-$ represents the hypergraph on $n$ vertices that has exactly one edge missing.
In the proof of \cref{main_lemma}, the fact, that $q(S)(r(S)-x)^2$ is always positive, will be exploited. Understanding the terms in the square is done by the following short claims. First, $r(S)^2$ and $rr(S)$ are related.
\begin{claim}\label{ppp_claim}
$$r(S_{m-1})^2 \leq rr(S_{m-1}) + r(S_{m-1}) \frac{1}{n-m}$$
\end{claim}
This simply follows from \begin{equation*}
r(S_{m-1})^2 - rr(S_{m-1}) = r(S_{m-1}) \left(\frac{l(S_{m-1})}{n-m+1} - \frac{l(S_{m-1})-1}{n-m}\right) \leq r(S_{m-1}) \frac{1}{n-m}.
\end{equation*}
Second, linear equality between densities is shown.
\begin{claim}\label{chain_rule_claim}
Suppose $m \leq l \leq n$ with $F \in \mathcal{H}_{m}$ and $G \in \mathcal{H}_n$ then
\begin{equation*}
d(F, G) = \sum_{H \in \mathcal{H}_{l}} d(F, H) d(H, G),
\end{equation*}
in particular
\begin{equation*}
d(K_m, G) = \sum_{H \in \mathcal{H}_{m+1}} \frac{s(H)}{m+1} d(H, G).
\end{equation*}
\end{claim}
\begin{proof}
Note that a uniform $m$ sized subset of $V(G)$ can be sampled by first choosing $\mathbf{S}_l \sim \operatorname{Uniform}\binom{V(G)}{l}$ and then $\mathbf{S}_m \sim \operatorname{Uniform}\binom{\mathbf{S}_l}{m}$. The claim follows from the law of total probability. The events $\{G \! \upharpoonright_{\mathbb{S}_l} \simeq H \ : \ H \in \mathcal{H}_l\}$ partition the probability space, therefore \begin{equation*}
\begin{split}
d(F, G) = & \mathbb{P}\Big[ G \! \upharpoonright_{\mathbf{S}_m} \simeq F \Big] \\
= & \sum_{H \in \mathcal{H}_l} \mathbb{P}\Big[ G \! \upharpoonright_{\mathbf{S}_m} \simeq F \ \Big| \ G \! \upharpoonright_{\mathbf{S}_l} \simeq H \Big] \mathbb{P}\Big[ G \! \upharpoonright_{\mathbf{S}_l} \simeq H \Big] \\
= & \sum_{H \in \mathcal{H}_l} d(F, H) d(H, G).
\end{split}
\end{equation*}
The special case follows from $d(K_m, H) = \frac{s(H)}{m+1}$ when $H \in \mathcal{H}_{m+1}$.
\end{proof}
In the proof of \cref{main_lemma}, $S_{m-1}$ is chosen uniformly from the possible $m-1$ sized sets. The final claim connects the expected values arising in the terms of $q(S)(p(S_{m-1})-x)^2$ with densities of various $k$-graphs in $G$.
\begin{claim}\label{exp_claim} Suppose $\mathbf{S}_{m-1}$ is chosen uniformly randomly from the set $\binom{V(G)}{m-1}$ then
\leavevmode
\begin{enumerate}
\item $$\mathbb{E} \left[ q(\mathbf{S}_{m-1})r(\mathbf{S}_{m-1}) \right] = d\left(K_{m}, G\right), $$
\item $$\mathbb{E} \left[ q(\mathbf{S}_{m-1}) rr(\mathbf{S}_{m-1}) \right] = \sum_{H \in \mathcal{H}_{m+1}} \frac{\binom{s(H)}{2}}{\binom{m+1}{2}} d\left(H, G\right).$$
\end{enumerate}
\end{claim}
\begin{proof}
Similar to the proof of \cref{chain_rule_claim}, first choosing $\mathbf{S}_l \sim \operatorname{Uniform}\binom{V(G)}{l}$ and then $\mathbf{S}_{m-1} \sim \operatorname{Uniform}\binom{\mathbf{S}_l}{m-1}$ results in uniformly distributed $\mathbf{S}_{m-1}$. Note that $r(\mathbf{S}_{m-1})$ and $rr(\mathbf{S}_{m-1})$ corresponds with choosing $1$ and $2$ additional vertices accordingly, and then checking a condition on the extended set.
In particular, for $\mathbf{S}_m, \mathbf{S}_{m-1}$ pair, let $R\left(\mathbf{S}_m, \mathbf{S}_{m-1}\right)$ be the event that $G \! \upharpoonright_{\mathbf{S}_m} \simeq K_m$ (and therefore $G \! \upharpoonright_{\mathbf{S}} \simeq K_{m-1}$). The claim follows from the law of total expectation, by conditioning on the shape of $G \! \upharpoonright_{\mathbf{S}_m}$
\begin{equation*}
\begin{split}
& \mathbb{E} \left[ q(\mathbf{S}_{m-1})r(\mathbf{S}_{m-1}) \right] \\
& \qquad = \sum_{H \in \mathcal{H}_{m}} \mathbb{P}\left[ R(\mathbf{S}_m, \mathbf{S}_{m-1}) \ \large| \ G \! \upharpoonright_{\mathbf{S}_m} = H \right] \mathbb{P}\left[ G \! \upharpoonright_{\mathbf{S}_m} = H \right] \\
& \qquad = d(K_m, G).
\end{split}
\end{equation*}
Since only $H = K_{m}$ contains a suitable $m-1$ sized subset that satisfies $R_{\mathbf{S}_m, \mathbf{S}_{m-1}}$.
For the second part, write $RR(\mathbf{S}_{m+1}, \mathbf{S}_{m-1})$ for the event that there are two copies of $K_{m}$ inside $G \! \upharpoonright_{\mathbf{S}_{m+1}}$ intersecting exactly at $\mathbf{S}_{m-1}$ (again this implies $G \! \upharpoonright_{\mathbf{S}_{m-1}} = K_{m-1}$). The calculation in this case gives
\begin{equation*}
\begin{split}
& \mathbb{E} \left[ q(\mathbf{S}_{m-1})rr(\mathbf{S}_{m-1}) \right] \\
& \qquad = \sum_{H \in \mathcal{H}_{m+1}} \mathbb{P}\left[ RR(\mathbf{S}_{m+1}, \mathbf{S}_{m-1}) \ \large| \ G \! \upharpoonright_{\mathbf{S}_{m+1}} = H \right] \mathbb{P}\left[ G \! \upharpoonright_{\mathbf{S}_{m+1}} = H \right] \\
& \qquad = \sum_{H \in \mathcal{H}_{m+1}} \frac{\binom{s(H)}{2}}{\binom{m+1}{2}} d\left(H, G\right)
\end{split}
\end{equation*}
since in a given $G \! \upharpoonright_{\mathbf{S}_{m+1}} \simeq H$, a randomly chosen $\mathbf{S}_{m-1}$ satisfies $RR(\mathbf{S}_{m+1}, \mathbf{S}_{m-1})$ with probability $\frac{\binom{s(H)}{2}}{\binom{m+1}{2}}$.
\end{proof}
\begin{remark}\label{flag_claims_remark}
The above claims all correspond to parts of the general flag algebra theory \cite{razb_flag_algebras}.
\begin{enumerate}
\item \cref{chain_rule_claim} corresponds to the chain rule (Lemma 2.2). In the language of flags it gives $$K_m = \sum_{H \in \mathcal{H}_{m+1}} \frac{s(H)}{m+1} H.$$
\item \cref{ppp_claim} corresponds to products (Lemma 2.3). It is more or less equivalent with $$p\left(K_m^{T_{m-1}}; G^{T_{m-1}}\right)^2 - p\left(K_m^{T_{m-1}}, K_m^{T_{m-1}}; G^{T_{m-1}}\right) = O(|G|^{-1}).$$
\item \cref{exp_claim} corresponds to averaging (Theorem 2.5). It is a restatement of \begin{equation*}
\left\llbracket K_m^{T_{m-1}} \right\rrbracket_{T_{m-1}} = K_m
\end{equation*}
and \begin{equation*}
\left\llbracket \left( K_m^{T_{m-1}} \right)^2 \right\rrbracket_{T_{m-1}} = \sum_{H \in \mathcal{H}_{m+1}} \frac{\binom{s(H)}{2}}{\binom{m+1}{2}} H.
\end{equation*}
\end{enumerate}
\end{remark}
With these claims, the main lemma follows easily.
\begin{replemma}{main_lemma}
For all $x>0$ and integers $k \leq m < n$, if $G \in \mathcal{H}_n$ then the following holds
\begin{equation*}
\begin{matrix*}[l]
0 \geq & \left(- \frac{1 - \frac{k-1}{m}}{x}\right) & d\left(K_{m+1}, G\right) & + \\ & \left(2 - \frac{k-1}{mx} - \frac{1}{(n-m)x}\right) & d\left(K_m, G\right) & + \\ & (-x) & d\left(K_{m-1}, G\right). &
\end{matrix*}
\end{equation*}
\end{replemma}
\begin{proof}
When $x \in \mathbb{R}$ the quantity $q(S)(r(S) - x)^2$ is always non-negative. Therefore choosing $\mathbf{S}_{m-1} \sim \operatorname{Uniform}\binom{V(G)}{m-1}$ the following is true
$$0 \leq \mathbb{E} \left[ q(\mathbf{S}_{m-1}) (r(\mathbf{S}_{m-1}) - x)^2 \right].$$ Expanding the terms and applying \cref{ppp_claim} gives
\begin{equation*}
0 \leq \mathbb{E} \left[ q(\mathbf{S}_{m-1})rr(\mathbf{S}_{m-1}) +\left(\frac{1}{n-m} - 2x\right)q(\mathbf{S}_{m-1})r(\mathbf{S}_{m-1}) + x^2 q(\mathbf{S}_{m-1}) \right].
\end{equation*}
The substitution from \cref{exp_claim} yields the following expression, without expected values:
\begin{equation*}
0 \leq \sum_{H \in \mathcal{H}_{m+1}} \frac{\binom{s(H)}{2}}{\binom{m+1}{2}} d\left(H, G\right) + \left(\frac{1}{n-m} - 2x\right) d\left(K_{m}, G\right) + x^2 d\left(K_{m}, G\right).
\end{equation*}
Notice that $s(H)$ is maximal on $K_{m+1}$, otherwise it is at most $k$. This observation gives
\begin{equation*}
\begin{split}
0 & \leq \frac{k-1}{m} \sum_{^{H \in \mathcal{H}_{m+1}}_{H \neq K_{m+1}}} \frac{s(H)}{(m+1)} d\left(H, G\right) \\ & \qquad + d\left(K_{m+1}, G\right) \\ & \qquad + \left(\frac{1}{n-m} - 2x\right) d\left(K_{m}, G\right) \\ & \qquad + x^2 d\left(K_{m-1}, G\right).
\end{split}
\end{equation*}
Finally expanding $\frac{k-1}{m} d(K_{m}, G)$ using \cref{chain_rule_claim} results in
\begin{equation*}
\begin{matrix*}[l]
0 \leq & \left(1 - \frac{k-1}{m}\right) & d\left(K_{m+1}, G\right) & + \\ & \left(\frac{k-1}{m} + \frac{1}{(n-m)} - 2x\right) & d\left(K_m, G\right) & + \\ & x^2 & d\left(K_{m-1}, G\right). &
\end{matrix*}
\end{equation*}
Since $x \geq 0$, note that \cref{main_lemma} is a $-1/x$ multiple of the above, and the proof is complete.
\end{proof}
\begin{remark}\label{flag_sdp_remark}
This lemma can be easily stated as \begin{equation*}
0 \leq \left(1 - \frac{k-1}{m}\right) K_{m+1} + \left(\frac{k-1}{m} - 2x\right) K_m + x^2 K_{m-1}
\end{equation*} in the language of flags. The proof uses the expansion of the simple square
\begin{equation*}
0 \leq \left\llbracket \left( K_m^{T_{m-1}} - xT_{m-1} \right)^2 \right\rrbracket_{T_{m-1}}
\end{equation*} In a plain application of flag algebra, the computer finds a conic combination of squares, similar to the above expression. \Cref{main_lemma} provides squares in a form that is easy to handle later (they only involve a small number of $d(K_m, G)$ values). The following section shows that the target expression \begin{equation*}
K_g \leq \prod_{m=k}^g x^{(k)}_{m, r} + c K_r
\end{equation*} for some $c$ constant, lies in the conic combination of the squares.
\end{remark}
\section{\texorpdfstring{$\pi\left(K^{(3)}_4, K^{(3)}_5\right) = 3/8$}{pi K3,4, K3,5 = 3/8}}\label{pi345_section}
The combination of more sophisticated, but still simple squares can provide tight bounds for $\pi\left(K^{(3)}_4, K^{(3)}_5\right) = 3/8$. For an easier description of the flags, consider the complement question. If $E_n$ is the $3$-graph with $n$ vertices and no edges then $$\pi\left(K^{(3)}_4, K^{(3)}_5\right) = \lim_{n \rightarrow \infty} \max \left\{ d(E_4, G) \ : \ G \in \mathcal{H}^{(3)}_n, \ \ d(E_5)=0 \right\}.$$
Use $P_n$ for the corresponding type with $n$ vertices and no edges. Note $E_n^{P_m}$ is unique for any $m<n$ pair. Define the further flags:
\begin{enumerate}
\item $L^{P_2}_{a}$ is the flag with vertex set $\{0, 1, 2, 3\}$, edge set $\{(0, 2, 3)\}$ and type formed from the vertices $0, 1$. Similarly, write $L^{P_2}_{b}$ for the flag with the same vertex set and type but $\{(1, 2, 3)\}$ edge set.
\item Use $M^{P_3}_a$ for the flag with vertices $\{0, 1, 2, 3\}$, edges $\{(1, 2, 3)\}$ and type from $0, 1, 2$. Symmetrically, with the same vertex and type set use $M^{P_3}_b$ for the edge set $\{(0, 2, 3)\}$. And $M^{P_3}_c$ for the edge set $\{(0, 1, 3)\}$.
\item $N^{Q_4}$ is the flag with vertices $\{0, 1, 2, 3, 4\}$, edges $\{(0, 1, 2)\}$ and type formed by $0, 1, 2, 3$. Note that $Q_4$ does not agree with any of the $T_n$ or $P_n$ types.
\item $O^{T_4}_{a}$ has vertex set $\{0, 1, 2, 3, 4\}$, edge set $\{(0, 1, 4)\}$ and type formed from $0, 1, 2, 3$. Additionally write $O^{T_4}_{b}$ for the flag with the same vertex and type set but $\{(2, 3, 4)\}$ edges.
\end{enumerate}
\input{illustrations/pi345}
Then the following inequality holds on $E_5$-free hypergraphs:
\begin{equation}\label{pi345_equation}
\begin{split}
0 \leq \ \ \frac{2}{3} & \left\llbracket \left( E_3^{P_1} - \frac{3}{4} P_1 \right)^2 \right\rrbracket_{P_1} +
\frac{1}{6} \left\llbracket \left( L^{P_2}_a - L^{P_2}_b \right)^2 \right\rrbracket_{P_2} + \\
\frac{13}{12} & \left\llbracket \left( M^{P_3}_a + M^{P_3}_b + M^{P_3}_c - \frac{1}{2} P_3 \right)^2 \right\rrbracket_{P_3} +
\frac{11}{12} \left\llbracket \left( E_4^{P_3} - \frac{1}{2} P_3 \right)^2 \right\rrbracket_{P_3} + \\
2 & \left\llbracket \left( N^{Q_4} - \frac{1}{2} Q_4 \right)^2 \right\rrbracket_{Q_4} +
\frac{1}{2} \left\llbracket \left( O^{T_4}_a - O^{T_4}_b \right)^2 \right\rrbracket_{P_4} \leq \quad \frac{3}{8} - E_4.
\end{split}
\end{equation}
So far the only verification of \cref{pi345_equation} requires a tedious (computer assisted) checking of all the $2102$ hypergraphs in $\mathcal{H}^{(3)}_6$ without $E_5$. This can be found in the supplement. The corresponding lower bound is attained at $G_n = K^{(3)}_{\lfloor n/2 \rfloor} \bigsqcup K^{(3)}_{\lceil n/2 \rceil}$. Note that $d(E_5, G_n)=0$ while $\lim_{n \rightarrow \infty} d(E_4, G_n) = \frac{3}{8}$.
\section{Concluding Remarks}\label{outro_section}
This paper investigated a natural extension of the generalized Turán problem to hypergraphs. The result matches the best-known general bounds for $k$-graphs but fails to provide tight bounds when $k>3$.
The main combinatorial insight comes from the simple inequality \begin{equation}\label{flag_square_equation}
0 \leq \left\llbracket \left( K_m^{T_{m-1}} - xT_{m-1} \right)^2 \right\rrbracket_{T_{m-1}},
\end{equation} combined with a close approximation of $\left\llbracket \left( K_m^{T_{m-1}} \right)^2 \right\rrbracket_{T_{m-1}}$. As shown in the paper, the convex combination of these squares includes difficult results for the generalized hypergraph Turán problem.
The long list of questions improved by the plain flag algebraic method indicates that finding more sophisticated squares can greatly improve the available density bounds. It would be interesting to identify other families of simple linear density relations (like the one described in \cref{main_lemma}) whose conic combination includes new bounds for extremal hypergraph problems, even better if the bounds are tight. The provided certificate for $\pi\left(K^{(3)}_4, K^{(3)}_5\right) = 3/8$ can perhaps be generalized to larger cases. It is interesting that for the smallest $k, g, r$ tuple, which is not already known ($k=2$) and is not a classical hypergraph Turán problem ($k=g$), the exact solution follows from flag algebraic calculations. It also highlights the limitations of computer assisted searches: calculations for problems with higher parameters are infeasible.
\subsection{Finding Squares}
There is an easy to describe reason why \cref{flag_square_equation} fails to provide tight bounds for $k$-graphs where $k>2$ but is asymptotically exact when $k=2$. The extremal configuration for $\pi\left(n, K^{(2)}_g, K^{(2)}_r\right)$ is a unique balanced $(r-1)$-partite graph, call it $G_r(n)$, and say $G_r^{T_{m-1}}(n)$ is the same structure with a complete $(m-1)$-tuple marked as a type. Any choice of $T_{m-1}$ results in the same $\lim_{n \rightarrow \infty} d\left(K^{T_{m-1}}_m, G_r^{T_{m-1}}(n)\right)$ value, which is $x^{(2)}_{m, r} = 1 - \frac{m-1}{r-1}$. In contrast, the conjectured optimal constructions when $k>2$ give different values for different $T_{m-1}$ choices, therefore no $x \in \mathbb{R}$ exists with $$\left\llbracket \left( K_m^{T_{m-1}} - xT_{m-1} \right)^2 \right\rrbracket_{T_{m-1}} = 0$$ on the conjectured optimal constructions. This slackness gives the difference between the conjectured optimal constructions and the proved bounds here. The values $x^{(k)}_{m, r}$ are chosen optimally, any asymptotically significant improvement must utilize a different combinatorial insight.
\section{The Associated Tridiagonal Matrix}\label{tridiag_section}
\begin{definition}
Given $k < r$ integers, the problem has an associated tridiagonal matrix $D^{(k)}_r$ with entries $d_{l, m}$ indexed by the range $k \leq l, m < r$ \begin{equation*}
d_{l, m} = \begin{cases}
-x^{(k)}_{m, r} & \text{ if } \ l=m-1\\
\\
2 - \frac{k-1}{mx^{(k)}_{m, r}} & \text{ if } \ l=m \\
\\
-\frac{1 - \frac{k-1}{m}}{x^{(k)}_{m, r}} & \text{ if } \ l=m+1\\
\\
0 & \text{ otherwise}
\end{cases}
\end{equation*}
\end{definition}
Write $\epsilon = \frac{r-k}{(n-r+1)(k-1)}$ and notice that $D^{(k)}_{r} - \epsilon I$ has column values equal to the coefficients in $E_{m}$. With the new notation, recall \cref{tridiag_lemma}, which was used to finish the proof of \cref{main_theorem}.
\begin{replemma}{tridiag_lemma}
Let $\Delta^{(k)}_{r}(\epsilon) = \left(D^{(k)}_{r} - \epsilon I \right)^{-1}$ be the inverse of the associated tridiagonal matrix, with entries $\delta_{m, g}(\epsilon)$. If $0 \leq \epsilon < \frac{k-1}{(r-1)(r-k)}$, then the values $\delta_{m, g}(\epsilon)$ are all positive, and $$\delta_{k, g}(\epsilon) \leq \frac{1}{1-\epsilon \frac{(r-1)(r-k)}{k-1}} \prod_{m=k}^g x_{m+1, r}$$
\end{replemma}
Notice that $\epsilon=0$ corresponds with the inverse of $D_r^{(k)}$, and the asymptotic question when $n \rightarrow \infty$. This section is devoted to the proof of \cref{tridiag_lemma}, but it is illuminating and helpful for the proof to first calculate the inverse of $D_r^{(k)}$.
\subsection{The inverse of \texorpdfstring{$D^{(k)}_r$}{Dk,r}}
\begin{lemma}\label{simple_tridiag_lemma}
Let $\Delta^{(k)}_{r} = \left(D^{(k)}_{r}\right)^{-1}$ be the inverse of the associated tridiagonal matrix with entries $\delta_{m, g}$. Then $\delta_{m, g}$ are all positive and $$\delta_{k, g} = \prod_{m=k+1}^{g} x^{(k)}_{m, r}.$$
\end{lemma}
In the following proofs, the $k$ superscripts are omitted to increase readability. The complete inverse can be calculated following the method described in \cite{tridiagonal_inverse}. The value $\theta_m$ represents the determinant of the rows and columns indexed by the set $\{k, k+1, ..., m\}$, while $\phi_m$ is the determinant for the rows and columns indexed by $\{m, m+1, ..., r-1\}$.
From the cofactor calculation of determinants and inverses, the following claim is easy to verify.
\begin{claim}\label{determinant_long_claim}\leavevmode
\begin{enumerate}
\item For all $k \leq m < r$ the induction $$\theta_m = d_{m, m} \theta_{m-1} - d_{m-1, m} d_{m, m-1} \theta_{m-2}$$ holds with initial values $\theta_{k-1} = 1$ and $\theta_{k-2} = 0$
\item For all $k \leq m < r$ the reverse induction $$\phi_{m} = d_{m, m} \phi_{m+1} - d_{m+1, m} d_{m, m+1} \phi_{m+2}$$ holds with initial values $\phi_{r} = 1$ and $\phi_{r+1} = 0$
\item For all $k \leq m < r$ the determinant can be calculated $$\operatorname{Det}\left(D_r\right) = \theta_{r-1} = \phi_{k} = \theta_m \phi_{m+1} - d_{m, m+1} d_{m+1, m} \theta_{m-1} \phi_{m+2}$$
\item The entries of the inverse matrix are \begin{equation*}
\delta_{m, g} = \frac{(-1)^{m+g+r-k}}{\operatorname{Det}(D_r)}\begin{cases}
\theta_{m-1} \phi_{g+1} \prod_{i=m}^{g-1} d_{i, i+1} & \ \text{if } m \leq g \\
\ \\
\theta_{g-1} \phi_{m+1} \prod_{i=g}^{m-1} d_{i+1, i} & \ \text{otherwise}
\end{cases}
\end{equation*} In the corresponding $k \leq m, g <r$ range.
\end{enumerate}
\end{claim}
First the $\phi_{m}$ values will be calculated using the recursive expression above.
\begin{claim}\label{phi_claim}
$\phi_{m} = 1$ in the range $k \leq m \leq r$
\end{claim}
\begin{proof}
By reverse induction. The claim holds for $m=r$ and notice $$\phi_{r-1} = d_{r-1, r-1} = 2 - \frac{\frac{k-1}{r-1}}{1- \frac{\binom{r-2}{k-1}}{\binom{r-1}{k-1}}} = 1$$
Using point 2 from \cref{determinant_long_claim} \begin{equation*}
\begin{split}
\phi_{m} = & d_{m, m} - d_{m+1, m} d_{m, m+1} \\ =& 2 - \frac{k-1}{mx_{m, r}} - \left( 1 - \frac{k-1}{m}\right) \frac{x_{m+1, r}}{x_{m, r}} \\
=& 2 - \frac{k-1}{m(1-u)} - \left( 1 - \frac{k-1}{m}\right) \frac{1-\frac{m
u}{m-k+1}}{1-u} \\ =& 1
\end{split}
\end{equation*}
Where $u = \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} = 1-x_{m, r}$ and $u\frac{m}{m-k+1} = \frac{\binom{m}{k-1}}{\binom{r-1}{k-1}}= 1-x_{m+1, r}$ substitution was used to simplify the calculation.
\end{proof}
This gives that $\operatorname{Det}\left(D_r\right) = \theta_{r-1} = \phi_{k} = 1$.
\begin{claim}\label{theta_claim}
In the $k-1 \leq m < r$ range, $\operatorname{Sign}(\theta_m) = 1$
\end{claim}
\begin{proof}
By induction, note that the claim holds for $\theta_{k-1} = 1$. Then using point 3 from \cref{determinant_long_claim} and the value for the determinant,
\begin{equation*}
\theta_m = 1 + d_{m, m+1}d_{m+1, m}\theta_{m-1}
\end{equation*}
Using \cref{x_positive_claim}, note that the values $d_{m, m+1} = -x_{m+1, r}$ and $d_{m+1, m} = -\frac{1 - \frac{k-1}{m}}{x_{m, r}}$ are both negative. Therefore their product; and by induction, $\theta_{m-1}$, are positive.
\end{proof}
\begin{proof}[Proof of \cref{simple_tridiag_lemma}]
It claims two things,
\begin{enumerate}
\item The entries $\delta_{m, g}$ are all positive: \cref{determinant_long_claim} point 4 gives that \begin{equation*}
\operatorname{Sign}(\delta_{m, g}) = (-1)^{m+g+r-k} \begin{cases} \prod_{i=m}^{g-1} \operatorname{Sign}(d_{i, i+1}) & \text{if } m \leq g \\
\prod_{i=g}^{m-1} \operatorname{Sign}(d_{i+1, i}) & \text{otherwise.}\end{cases}
\end{equation*} using $\operatorname{Sign}(\theta_m)=1$ from \cref{theta_claim} and that $\phi_m = 1$ from \cref{phi_claim}. Since all $d_{i, i+1}, d_{i+1, i}$ are negative, the inverse of $D_r$ has only positive entries.
\item $\delta_{k, g} = \prod_{m=k}^{g} x_{m+1, r}$: Again substituting the values $d_{i, i+1} = -x_{i+1, r}$ and $\operatorname{Det}(D_r) = \phi_{g+1} = \theta_{k-1} = 1$ into \cref{determinant_long_claim} point 4 gives the stated value for $\delta_{k, g}$.
\end{enumerate}
\end{proof}
Interestingly, the values $\delta_{k, g}$ are easy enough to calculate exactly. In contrast, $\delta_{m, g}$ requires the value of some $\theta_m$ which is difficult to find in general with the recursive expression. The sign of $\theta_m$ is easy to find, exactly what is needed for the proof.
\subsection{The inverse of \texorpdfstring{$D^{(k)}_r - \epsilon I$}{Dk,r - eps I}}
Consider the same calculation but with $D_r - \epsilon I$. The value $\phi_m(\epsilon)$ is the determinant for the rows and columns indexed by $\{m, m+1, ..., r-1\}$ of $D_r - \epsilon I$. The next claim shows that $\phi_m(\epsilon)$ is increasing in $m$ when $\epsilon>0$. A notation for the increments will be useful, write $\zeta_m(\epsilon) = \phi_{m+1}(\epsilon) - \phi_m(\epsilon)$.
\begin{claim}
When $k \leq m < r$ and $0 \leq \epsilon \leq \frac{k-1}{(r-1)(r-m)}$, $$0 \leq \zeta_m(\epsilon) \leq \epsilon \frac{r-1}{k-1} \left(1 - \left( 1 - \frac{k-1}{r-1} \right)^{r-m}\right)$$ and correspondingly $$1 \geq \phi_m(\epsilon) \geq 1 - \epsilon \frac{(r-1)(r-m)}{k-1}$$
\end{claim}
\begin{proof}
Use point 2 from \cref{determinant_long_claim}. The initial value is $\zeta_{r}(\epsilon) = 0$ and by reverse induction take
\begin{equation*}
\begin{split}
\phi_m(\epsilon) = & \left( d_{m, m} - \epsilon \right) \phi_{m+1}(\epsilon) - d_{m+1, m} d_{m+1, m} \phi_{m+2}(\epsilon) \\
= & \left( 2 - \frac{k-1}{mx_{m, r}} - \epsilon \right) \phi_{m+1}(\epsilon) - \left( 1 - \frac{k-1}{mx_{m, r}} \right) \left( \phi_{m+1}(\epsilon) + \zeta_{m+1}(\epsilon) \right) \\
= & \phi_{m+1}(\epsilon) - \zeta_{m+1}(\epsilon) \left(1 - \frac{k-1}{mx_{m, r}} \right) - y\phi_{m+1}(\epsilon).
\end{split}
\end{equation*}
Therefore \begin{equation}\label{zeta_equation}
\zeta_m(\epsilon) = \zeta_{m+1}(\epsilon) \left(1 - \frac{k-1}{mx_{m, r}} \right) + \epsilon \phi_{m+1}(\epsilon).
\end{equation}
Note that $$0 \leq d_{m, m+1} d_{m+1, m} = \left(1 - \frac{k-1}{mx_{m, r}} \right) \leq \left(1 - \frac{k-1}{r-1} \right)$$ in the $k \leq m < r$ range. As $\epsilon \leq \frac{k-1}{(r-1)(r-m)} < \frac{k-1}{(r-1)(r-m-1)}$ by reverse induction it holds that $0 \leq \phi_{m+1}(\epsilon)$ and $0 \leq \zeta_{m+1}(\epsilon)$ giving the required lower bound $0 \leq \zeta_{m}(\epsilon)$. This implies the upper bound $\phi_{m}(\epsilon) \leq 1$.
For the $\zeta_m(\epsilon)$ upper bound, in \cref{zeta_equation} bound each term: $m x_{m, r} \leq (r-1)$ and $\phi_{m+1}(\epsilon) \leq 1$. This gives the intermediate result $$\zeta_m(\epsilon) \leq \zeta_{m+1}(\epsilon) \left(1 - \frac{k-1}{r-1} \right) + \epsilon.$$ which, by iterated application and $\zeta_{r}(\epsilon) = 0$ initial value, implies $$\zeta_m(\epsilon) \leq \epsilon \frac{r-1}{k-1} \left(1 - \left( 1 - \frac{k-1}{r-1} \right)^{r-m}\right).$$ A summation formula for the upper and lower $\zeta_m(\epsilon)$ bounds combined with the initial $\phi_{r}(\epsilon) = 1$ value gives $$1 \geq \phi_m(\epsilon)\geq 1 - \epsilon \frac{(r-1)^2}{(k-1)^2} \left( \frac{(k-1)(r-m)}{r-1} + \left(1 - \frac{k-1}{r-1} \right)^{r-m} - 1\right).$$ This provides a tighter bound but for simplicity use $$1 - \epsilon \frac{(r-1)^2}{(k-1)^2} \left( \frac{(k-1)(r-m)}{r-1} + \left(1 - \frac{k-1}{r-1} \right)^{r-m} - 1\right) \geq 1 - \epsilon \frac{(r-1)(r-m)}{k-1}.$$
\end{proof}
This gives a simple linear bound for the determinant. When $0 < \epsilon < \frac{k-1}{(r-1)(r-k)}$, $$1 - \epsilon \frac{(r-1)(r-k)}{k-1} \leq \phi_k(\epsilon) = \operatorname{Det}(D_r - \epsilon I) \leq 1.$$ If $0 \leq \epsilon < \frac{k-1}{(r-1)(r-k)}$ then the determinant is strictly positive, bounding the smallest eigenvalue of $D_r$.
\begin{proof}[Proof of \cref{tridiag_lemma}]
Again it claims two things.
\begin{enumerate}
\item The entries $\delta_{m, g}(\epsilon)$ are all positive: By assumption, $\epsilon$ is smaller than the smallest eigenvalue of $D_r$. Therefore the expansion $$\left(D_r - \epsilon I \right)^{-1} = D_r^{-1} + \epsilon D_r^{-2} + \epsilon^2 D_r^{-3}... $$ holds. \Cref{simple_tridiag_lemma} shows that the entries in $D_r^{-1}$ (and in $D_r^{-i}$ for $1 > i$ correspondingly) are all positive, giving the required positivity of $\delta_{m, g}(\epsilon)$.
\item $$\delta_{k, g}(\epsilon) \leq \frac{1}{1-\epsilon \frac{(r-1)(r-k)}{k-1}} \prod_{m=k}^g x_{m+1, r}:$$ This follows from substituting the bounds $\phi_{m}(\epsilon) \leq 1$ and $1 - \epsilon \frac{(r-1)(r-k)}{k-1} \leq \operatorname{Det}(D_r - \epsilon I)$ into \cref{determinant_long_claim} point 4.
\end{enumerate}
\end{proof}
|
2,877,628,089,458 | arxiv | \section*{{\normalsize \bf #2}}\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\newblock{\hskip .11em plus .33em minus -.07em}
\sloppy
\sfcode`\.=1000\relax}
\let\endthebibliograph=\endlist
\newcommand{\lcorr}{{\Large \bf ![}\,}
\newcommand{\rcorr}{{\Large \bf ]!}\,}
\newcommand{\dist}{{\mathcal D}'({\mathbb{R}}^n)}
\newcommand{\sign}{{\rm sign}\, }
\newcommand{\test}{{\mathcal D}({\mathbb{R}}^n)}
\newcommand{\tombstone}{\hspace{0cm}\hspace*{\fill} \rule{3mm}{3mm}
\\[2mm]}
\newcommand{\euclid}{{\mathbb{R}}^n}
\newcommand{\re}{{\mathbb{R}}}
\newcommand{\R}{{\mathbb{R}}^n}
\newcommand{\cj}{{\mathcal{J}}}
\newcommand{\cs}{{\mathcal{S}}}
\newcommand{\cl}{{\mathcal{L}}}
\newcommand{\cf}{{\mathcal{F}}}
\newcommand{\cfi}{{\mathcal{F}}^{-1}}
\newcommand{\besovn}[3]{B_{{#2},{#3}}^{#1}({\mathbb R}^n)}
\newcommand{\besov}[3]{B_{{#2},{#3}}^{#1}({\mathbb R})}
\newcommand{\besovhom}[3]{\dot{B}_{{#2},{#3}}^{#1}({\mathbb R})}
\newcommand{\besovhomn}[3]{\dot{B}_{{#2},{#3}}^{#1}({\mathbb R}^n)}
\newcommand{\vp}{{\mathcal V}_p}
\newtheorem{theorem}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\input{tcilatex}
\begin{document}
\title{Complex interpolation of variable Triebel-Lizorkin spaces}
\author{Douadi Drihem \\
Department of Mathematics, \\
Laboratory of Functional Analysis and Geometry of spaces, \\
M'sila University, M'sila, Algeria\\
\texttt{douadidr@yahoo.fr}}
\date{\today }
\maketitle
\begin{abstract}
We study complex interpolation of variable Triebel-Lizorkin spaces,
especially we\ present the\ complex interpolation of $F_{p(\cdot
),q}^{\alpha }$ and $F_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}$ spaces.
Also, some limiting cases are given.
\noindent MSC classification: 46E35, 26B35.\newline
Key words and phrases: Triebel-Lizorkin spaces, Complex interpolation, Calde
\'{o}n products, Variable exponents.
\end{abstract}
\section{Introduction}
Interpolation of spaces have been a central topic in analysis, and are now
of increasing applications in many fields of mathematics especially harmonic
analysis and partial differential equations. For more details on this topic
we refer the reader to Bergh and L\"{o}fstr\"{o}m \cite{BL76}, and Triebel
\cite{T3}, where the complex interpolation for Besov and/or Triebel-Lizorkin
spaces are given. The main purpose of this paper is to establish the complex
interpolation for variable Triebel-Lizorkin spaces. Firstly we use the
so-called retraction method to present interpolation results in variable
Triebel-Lizorkin spaces $F_{p(\cdot ),q}^{\alpha }$. Secondly\ we shall
apply a method which has been used by \cite{FJ90}\ and\ \cite{SSV13}, where
we shall calculate the Calder\'{o}n products of associated sequence spaces.
Then, from an abstract theory on the relation between the complex
interpolation and the Calder\'{o}n product of Banach lattices obtained by
Calder\'{o}n \cite{Ca64}, Frazier, Jawerth \cite{FJ90}, Mendez, Mitrea \cit
{MM00} and Kalton, Maybororda, Mitrea \cite{KMM07}, we deduce the complex
interpolation theorems of these sequence spaces. Under some assumptions the
complex interpolation theorems for $F_{p(\cdot ),p(\cdot )}^{\alpha (\cdot
)} $\ are lifted by the $\varphi $-transforms characterization of variable
Triebel-Lizorkin spaces. Finally we will present and briefly discuss some
results concerning the complex interpolation for the spaces $F_{p(\cdot
),q(\cdot )}^{\alpha (\cdot )}$.
As usual, we denote by $\mathbb{R}^{n}$ the $n$-dimensional real Euclidean
space, $\mathbb{N}$ the collection of all natural numbers and $\mathbb{N
_{0}:=\mathbb{N}\cup \{0\}$. The letter $\mathbb{Z}$ stands for the set of
all integer numbers.\ The expression $f\lesssim g$ means that $f\leq c\,g$
for some independent constant $c$ (and non-negative functions $f$ and $g$),
and $f\approx g$ means $f\lesssim g\lesssim f$.\vskip5pt
By supp $f$ we denote the support of the function $f$ , i.e., the closure of
its non-zero set. If $E\subset {\mathbb{R}^{n}}$ is a measurable set, then
|E|$ stands for the (Lebesgue) measure of $E$ and $\chi _{E}$ denotes its
characteristic function.\vskip5pt
For $v\in \mathbb{Z}$ and $m=(m_{1},...,m_{n})\in \mathbb{Z}^{n}$, let
Q_{v,m}$ be the dyadic cube in $\mathbb{R}^{n}$, $Q_{v,m}:
\{(x_{1},...,x_{n}):m_{i}\leq 2^{v}x_{i}<m_{i}+1,i=1,2,...,n\}$. For the
collection of all such cubes we use $\mathcal{Q}:=\{Q_{v,m}:v\in \mathbb{Z
,m\in \mathbb{Z}^{n}\}$. For each cube $Q$, we denote by $x_{v,m}$ the lower
left-corner $2^{-v}m$ of $Q=Q_{v,m}$ and its side length by $l(Q)$.
Furthermore, we put $v_{Q}:=-\log _{2}l(Q)$, $v_{Q}^{+}:=\max (v_{Q},0)$ and
$\chi _{Q_{v,m}}=\chi _{v,m},v\in \mathbb{Z},m\in \mathbb{Z}^{n}$.\vskip5pt
The symbol $\mathcal{S}(\mathbb{R}^{n})$ is used in place of the set of all
Schwartz functions on $\mathbb{R}^{n}$.\ We define the Fourier transform of
a function $f\in \mathcal{S}(\mathbb{R}^{n})$ by
\begin{equation*}
\mathcal{F}(f)(\xi ):=(2\pi )^{-n/2}\int_{\mathbb{R}^{n}}e^{-ix\cdot \xi
}f(x)dx,\quad \xi \in \mathbb{R}^{n}.
\end{equation*
We denote by $\mathcal{S}^{\prime }(\mathbb{R}^{n})$ the dual space of all
tempered distributions on $\mathbb{R}^{n}$. For $v\in \mathbb{Z}$, $\varphi
\in \mathcal{S}(\mathbb{R}^{n})$ and $x\in \mathbb{R}^{n}$, we set
\widetilde{\varphi }(x):=\overline{\varphi (-x)}$, $\varphi
_{v}(x):=2^{vn}\varphi (2^{v}x)$, an
\begin{equation*}
\varphi _{v,m}(x):=2^{vn/2}\varphi (2^{v}x-m)=|Q_{v,m}|^{1/2}\varphi
_{v}(x-x_{v,m})\quad \text{if\quad }Q=Q_{v,m}.
\end{equation*}
The variable exponents that we consider are always measurable functions $p$
on $\mathbb{R}^{n}$ with range in $[c,\infty \lbrack $ for some $c>0$. We
denote the set of such functions by $\mathcal{P}_{0}$. The subset of
variable exponents with range $[1,\infty \lbrack $ is denoted by $\mathcal{P}
$. We use the standard notation $p^{-}:$=$\underset{x\in \mathbb{R}^{n}}
\text{ess-inf}}$ $p(x)$ and $p^{+}:$=$\underset{x\in \mathbb{R}^{n}}{\text
ess-sup }}p(x)$.
The variable exponent modular is defined by $\varrho _{p(\cdot )}(f):=\int_
\mathbb{R}^{n}}\rho _{p(x)}(\left\vert f(x)\right\vert )dx$, where $\rho
_{p}(t)=t^{p}$. The variable exponent Lebesgue space $L^{p(\cdot )}$\
consists of measurable functions $f$ on $\mathbb{R}^{n}$ such that $\varrho
_{p(\cdot )}(\lambda f)<\infty $ for some $\lambda >0$. We define the
Luxemburg (quasi)-norm on this space by the formula $\left\Vert f\right\Vert
_{p(\cdot )}:=\inf \big\{\lambda >0:\varrho _{p(\cdot )}\big(\frac{f}
\lambda }\big)\leq 1\big\}$. A useful property is that $\left\Vert
f\right\Vert _{p(\cdot )}\leq 1$ if and only if $\varrho _{p(\cdot )}(f)\leq
1$, see \cite{DHHR}, Lemma 3.2.4.
Let $p,q\in \mathcal{P}_{0}$. The Lebesgue-sequence space $L^{p(\cdot
)}(\ell _{q(\cdot )})$ is defined to be the space of all family of functions
$f_{v},v\geq 0$\ such tha
\begin{equation*}
\big\|\left( f_{v}\right) _{v\geq 0}\big\|_{L^{p(\cdot )}(\ell _{q(\cdot
)})}:=\big\|\big\|\left( f_{v}(x)\right) _{v\geq 0}\big\|_{\ell _{q(x)}
\big\|_{p(\cdot )}.
\end{equation*
It is easy to show that $L^{p(\cdot )}(\ell _{q(\cdot )})$\ is always a
quasi-normed space\ and it is a normed space, if $\min (p(x),q(x))\geq 1$\
holds point-wise.
We say that $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is \textit{locally }lo
\textit{-H\"{o}lder continuous}, abbreviated $g\in C_{\text{loc}}^{\log }$,
if there exists $c_{\log }(g)>0$ such tha
\begin{equation}
\left\vert g(x)-g(y)\right\vert \leq \frac{c_{\log }(g)}{\log
(e+1/\left\vert x-y\right\vert )} \label{lo-log-Holder}
\end{equation
for all $x,y\in \mathbb{R}^{n}$. We say that $g$ satisfies the log\textit{-
\"{o}lder decay condition}, if there exists $g_{\infty }\in \mathbb{R}$ and
a constant $c_{\log }>0$ such tha
\begin{equation*}
\left\vert g(x)-g_{\infty }\right\vert \leq \frac{c_{\log }}{\log
(e+\left\vert x\right\vert )}
\end{equation*
for all $x\in \mathbb{R}^{n}$. We say that $g$ is \textit{globally}-lo
\textit{-H\"{o}lder continuous}, abbreviated $g\in C^{\log }$, if it i
\textit{\ }locally log-H\"{o}lder continuous and satisfies the log-H\"{o
lder decay\textit{\ }condition.\textit{\ }The constants $c_{\log }(g)$ and
c_{\log }$ are called the \textit{locally }log\textit{-H\"{o}lder constant
and the log\textit{-H\"{o}lder decay constant}, respectively\textit{.} We
note that all functions $g\in C_{\text{loc}}^{\log }$ always belong to
L^{\infty }$. It is known that for $p\in C^{\log }$ we hav
\begin{equation}
\Vert \chi _{B}\Vert _{{p(\cdot )}}\Vert \chi _{B}\Vert _{{p}^{\prime }
(\cdot )}}\approx |B|. \label{DHHR}
\end{equation
Also
\begin{equation}
\Vert \chi _{B}\Vert _{{p(\cdot )}}\approx |B|^{\frac{1}{p(x)}},\quad x\in B
\label{DHHR1}
\end{equation
for small balls $B\subset {\mathbb{R}^{n}}$ ($|B|\leq 2^{n}$), with
constants only depending on the $\log $-H\"{o}lder constant of $p$ (see, for
example, \cite[Section 4.5]{DHHR}). Here ${p}^{\prime }$ denotes the
conjugate exponent of $p$ given by $1/{p(\cdot )}+1/{p}^{\prime }{(\cdot )
=1 $. \vskip5pt
Recall that $\eta _{v,m}(x):=2^{nv}(1+2^{v}\left\vert x\right\vert )^{-m}$,
for any $x\in \mathbb{R}^{n}$, $v\in \mathbb{N}_{0}$ and $m>0$. Note that
\eta _{v,m}\in L^{1}$ when $m>n$ and that $\big\|\eta _{v,m}\big\|_{1}=c_{m}$
is independent of $v$.
\subsection{Basic tools}
In this subsection we present some results which are useful for us. The
following lemma is from \cite[Lemma 6.1]{DHR}, see also \cite[Lemma 19
{KV122}.
\begin{lemma}
\label{DHR-lemma}Let $\alpha \in C_{\mathrm{loc}}^{\log }$ and let $R\geq
c_{\log }(\alpha )$, where $c_{\log }(\alpha )$ is the constant from
\mathrm{\eqref{lo-log-Holder}}$ for $\alpha $. The
\begin{equation}
2^{v\alpha (x)}\eta _{v,h+R}(x-y)\leq c\text{ }2^{v\alpha (y)}\eta
_{v,h}(x-y) \label{alpha-est}
\end{equation
with $c>0$ independent of $x,y\in \mathbb{R}^{n}$ and $v,h\in \mathbb{N
_{0}. $
\end{lemma}
The previous lemma allows us to treat the variable smoothness in many cases
as if it were not variable at all, namely we can move the term inside the
convolution as follows
\begin{equation*}
2^{v\alpha (x)}\eta _{v,h+R}\ast \left\vert f\right\vert (x)\leq c\text{
\eta _{v,h}\ast (2^{v\alpha (\cdot )}\left\vert f\right\vert )(x).
\end{equation*}
\begin{lemma}
\label{DHRlemma}Let $p,q\in C^{\log }$ with $1<p^{-}\leq p^{+}<\infty $ and
1<q^{-}\leq q^{+}<\infty $. For $m>n$, there exists $c>0$ such tha
\begin{equation*}
\left\Vert (\eta _{v,m}\ast f_{v})_{v}\right\Vert _{L^{p(\cdot )}(\ell
_{q(\cdot )})}\leq c\left\Vert (f_{v})_{v}\right\Vert _{L^{p(\cdot )}(\ell
_{q(\cdot )})}.
\end{equation*}
\end{lemma}
The proof is given in \cite[Theorem 3.2]{DHR}.
Now we introduce the following sequence space.
\begin{definition}
Let $p,q\in \mathcal{P}_{0}$ where $0<p^{+},q^{+}<\infty $ and let $\alpha
\mathbb{R}^{n}\rightarrow \mathbb{R}$. Then for all complex valued sequences
$\lambda :=\{\lambda _{v,m}\}_{v\in \mathbb{N}_{0},m\in \mathbb{Z
^{n}}\subset \mathbb{C}$ we defin
\begin{equation*}
f_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot
\right) }:=\big\{\lambda :\left\Vert \lambda \right\Vert _{f_{p\left( \cdot
\right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right) }}<\infty
\big\},
\end{equation*
wher
\begin{equation*}
\left\Vert \lambda \right\Vert _{f_{p\left( \cdot \right) ,q\left( \cdot
\right) }^{\alpha \left( \cdot \right) }}:=\Big\|\Big(\sum\limits_{m\in
\mathbb{Z}^{n}}2^{v(\alpha \left( \cdot \right) +\frac{n}{2})}\lambda
_{v,m}\chi _{v,m}\Big)_{v\geq 0}\Big\|_{L^{p(\cdot )}(\ell _{q(\cdot )})}
\end{equation*
and
\begin{equation*}
f_{\infty ,q}^{\alpha \left( \cdot \right) }:=\big\{\lambda :\left\Vert
\lambda \right\Vert _{f_{\infty ,q}^{\alpha \left( \cdot \right) }}<\infty
\big\},
\end{equation*
wher
\begin{equation*}
\left\Vert \lambda \right\Vert _{f_{\infty ,q}^{\alpha \left( \cdot \right)
}}:=\sup_{Q\in \mathcal{Q}}\frac{1}{|Q|^{1/q}}\Big(\su
\limits_{v=v_{Q}^{+}}^{\infty }\int\limits_{Q}\sum\limits_{m\in \mathbb{Z
^{n}}2^{v(\alpha \left( x\right) +\frac{n}{2})q}\left\vert \lambda
_{v,m}\right\vert ^{q}\chi _{v,m}(x)dx\Big)^{1/q}.
\end{equation*}
\end{definition}
Notice that the supremum can be taken respect to dyadic cubes with side
length $\leq 1$.
\begin{lemma}
\label{lamda-est}Let $\alpha \in C_{\mathrm{loc}}^{\log }$, $p,q\in C^{\log }
$, $0<p^{+},q^{+}<\infty $, $j\in \mathbb{N}_{0},m\in \mathbb{Z}^{n}$ and
x\in Q_{j,m}$. Let $\lambda \in f_{p\left( \cdot \right) ,q\left( \cdot
\right) }^{\alpha \left( \cdot \right) }$. Then there exists $c>0$
independent of $j$ and $m$ such tha
\begin{equation*}
|\lambda _{j,m}|\leq c\text{ }2^{-j(\alpha (x)-\frac{n}{p(x)}+\frac{n}{2
)}\left\Vert \lambda \right\Vert _{f_{p\left( \cdot \right) ,q\left( \cdot
\right) }^{\alpha \left( \cdot \right) }}.
\end{equation*}
\end{lemma}
\textbf{Proof.} Let $\lambda \in f_{p\left( \cdot \right) ,q\left( \cdot
\right) }^{\alpha \left( \cdot \right) },j\in \mathbb{N}_{0},m\in \mathbb{Z
^{n}$ and $x\in Q_{j,m}$. Using the fact that $2^{j(\alpha \left( x\right)
-\alpha \left( y\right) )}\leq c$ for any $x,y\in Q_{j,m}$, we obtai
\begin{eqnarray*}
2^{j\alpha \left( x\right) p^{-}}|\lambda _{j,m}|^{p^{-}}
&=&|Q_{j,m}|^{-1}\int_{Q_{j,m}}2^{j\alpha \left( x\right) p^{-}}|\lambda
_{j,m}|^{p^{-}}\chi _{j,m}(y)dy \\
&\leq &c|Q_{j,m}|^{-1}\int_{Q_{j,m}}2^{j\alpha \left( y\right)
p^{-}}|\lambda _{j,m}|^{p^{-}}\chi _{j,m}(y)dy.
\end{eqnarray*
Applying H\"{o}lder's inequality to estimate this expression by
\begin{equation*}
c|Q_{j,m}|^{-1}\big\|2^{j\alpha \left( \cdot \right) p^{-}}|\lambda
_{j,m}|^{p^{-}}\chi _{j,m}\big\|_{p/p^{-}}\left\Vert \chi _{j,m}\right\Vert
_{(p/p^{-})^{\prime }}\leq c\left\Vert \lambda \right\Vert _{f_{p\left(
\cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right)
}}^{p^{-}}\left\Vert \chi _{j,m}\right\Vert _{p/p^{-}}^{-1}2^{-j\frac{np^{-
}{2}},
\end{equation*
where we have used $\mathrm{\eqref{DHHR}}$. Therefore for any $x\in Q_{j,m}
\begin{equation*}
|\lambda _{j,m}|\leq c\text{ }2^{-j(\alpha (x)-\frac{n}{p(x)}+\frac{n}{2
)}\left\Vert \lambda \right\Vert _{f_{p\left( \cdot \right) ,q\left( \cdot
\right) }^{\alpha \left( \cdot \right) }},
\end{equation*
by $\mathrm{\eqref{DHHR1}}$, which completes the proof. \ $\square $
\begin{prop}
\label{prop1}\textit{Let }$\alpha \in C_{\mathrm{loc}}^{\log }$\textit{.
Then }$\lambda =\{\lambda _{v,m}\in \mathbb{C}\}_{v\in \mathbb{N}_{0},m\in
\mathbb{Z}^{n}}\in f_{\infty ,q}^{\alpha \left( \cdot \right) }$\textit{\ if
and only if for each \ dyadic cube }$Q_{v,m}$ there is a subset
E_{Q_{v,m}}\subset Q_{v,m}$ with $|E_{Q_{v,m}}|>|Q_{v,m}|/2$ (or any other,
fixed, number $0<\varepsilon <1$) such tha
\begin{equation*}
\Big\|\Big(\sum_{v=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}2^{v(\alpha
\left( \cdot \right) +\frac{n}{2})q}|\lambda _{v,m}|^{q}\chi _{E_{v,m}}\Big
^{1/q}\Big\|_{\infty }<\infty .
\end{equation*
Moreover, the infimum of this expression over all such collections
\{E_{Q_{v,m}}\}_{v,m}$ is equivalent to $\left\Vert \lambda \right\Vert
_{f_{\infty ,q}^{\alpha \left( \cdot \right) }}$.
\end{prop}
The proof is given in \cite{D7}. We will make use of the following
statement, see \cite{DHHMS}, Lemma 3.3.
\begin{lemma}
\label{DHHR-estimate}Let $p\in C^{\log }$ with $1\leq p^{-}\leq p^{+}<\infty
$. Then for every $m>0$ there exists $\gamma =e^{-2m/c_{\log }(1/p)}\in
\left( 0,1\right) $ such tha
\begin{equation*}
\Big(\frac{\gamma }{\left\vert Q\right\vert }\int_{Q}\left\vert
f(y)\right\vert dy\Big)^{p\left( x\right) }\leq \frac{1}{\left\vert
Q\right\vert }\int_{Q}\left\vert f(y)\right\vert ^{p\left( y\right) }dy+\min
\left( \left\vert Q\right\vert ^{m},1\right) g(x)
\end{equation*
for every cube $\mathrm{(}$or ball$\mathrm{)}$ $Q\subset \mathbb{R}^{n}$,
all $x\in Q\subset \mathbb{R}^{n}$ and all $f\in L^{p\left( \cdot \right)
}+L^{\infty }$\ with $\left\Vert f\right\Vert _{p\left( \cdot \right)
}+\left\Vert f\right\Vert _{\infty }\leq 1$, wher
\begin{equation*}
g(x):=\left( e+\left\vert x\right\vert \right) ^{-m}+\frac{1}{\left\vert
Q\right\vert }\int_{Q}\left( e+\left\vert y\right\vert \right) ^{-m}dy.
\end{equation*}
\end{lemma}
Notice that in the proof of this theorem we need only that
\begin{equation*}
\int_{Q}\left\vert f(y)\right\vert ^{p\left( y\right) }dy\leq 1
\end{equation*
and/or $\left\Vert f\right\Vert _{\infty }\leq 1$. Moreover if $\left\vert
Q\right\vert \leq 1$, we have $g(x)\leq c$ $\eta _{0,m}(x)$ for any $x\in Q
, where $c>0$ is independent of $x$ and $m$.
\section{Variable Triebel-Lizorkin spaces}
The definition of Triebel-Lizorkin spaces of variable smoothness and
integrability is based on the technique of decomposition of unity exactly in
the same manner as in the case of constant exponents. Select a pair of
Schwartz functions $\Phi $ and $\varphi $ satisf
\begin{equation}
\text{supp}\mathcal{F}\Phi \subset \overline{B(0,2)}\text{ and }|\mathcal{F
\Phi (\xi )|\geq c\text{ if }|\xi |\leq \frac{5}{3} \label{Ass1}
\end{equation
and
\begin{equation}
\text{supp}\mathcal{F}\varphi \subset \overline{B(0,2)}\backslash B(0,1/2
\text{ and }|\mathcal{F}\varphi (\xi )|\geq c\text{ if }\frac{3}{5}\leq |\xi
|\leq \frac{5}{3} \label{Ass2}
\end{equation
where $c>0$. It easy to see that $\int_{\mathbb{R}^{n}}x^{\gamma }\varphi
(x)dx=0$ for all multi-indices $\gamma \in \mathbb{N}_{0}^{n}$.
Now, we define the spaces under consideration.
\begin{definition}
\label{B-F-def}Let $\alpha :\mathbb{R}^{n}\rightarrow \mathbb{R}$ and
p,q\in \mathcal{P}_{0}$ with $0<p^{+},q^{+}<\infty $ . Let $\Phi $ and
\varphi $ satisfy $\mathrm{\eqref{Ass1}}$ and $\mathrm{\eqref{Ass2}}$,
respectively and we put $\varphi _{v}=2^{vn}\varphi (2^{v}\cdot )$. The
Triebel-Lizorkin space $F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}$\ is the
collection of all $f\in \mathcal{S}^{\prime }(\mathbb{R}^{n})$\ such that
\begin{equation}
\left\Vert f\right\Vert _{F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}}:=\big\
\left( 2^{v\alpha \left( \cdot \right) }\varphi _{v}\ast f\right) _{v\geq 0
\big\|_{L^{p(\cdot )}(\ell _{q(\cdot )})}<\infty , \label{B-def}
\end{equation
where $\varphi _{0}$ is replaced by $\Phi $.
\end{definition}
The Triebel-Lizorkin spaces with variable smoothness have first been
introduced in \cite{DHR} under much more restrictive conditions on $\alpha
(\cdot )$. If $p(\cdot )$, $q(\cdot )$ and $\alpha (\cdot )$ are constants,
then we derive the well known Triebel-Lizorkin spaces. Taking $\alpha \in
\mathbb{R}$ and $q\in (0,\infty ]$ as constants we derive the spaces
F_{p(\cdot ),q}^{\alpha }$ studied by Xu in \cite{Xu08} and \cite{Xu09}. The
spaces $F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}$ are independent of the
particular choice of the system $\left\{ \varphi _{v}\right\} _{v}$\
appearing in their definitions. Moreover, if $\alpha \in C_{\mathrm{loc
}^{\log }$ and\ $p,q\in \mathcal{P}_{0}^{\log }$ with $0<p^{-}\leq
p^{+}<\infty $ and $0<q^{-}\leq q^{+}<\infty $, then
\begin{equation}
\mathcal{S}(\mathbb{R}^{n})\hookrightarrow F_{p(\cdot ),q(\cdot )}^{\alpha
(\cdot )}\hookrightarrow \mathcal{S}^{\prime }(\mathbb{R}^{n}). \label{emb}
\end{equation}
\begin{definition}
\label{B-F-def2}Let $\alpha :\mathbb{R}^{n}\rightarrow \mathbb{R}$ and
0<q<\infty $. Let $\Phi $ and $\varphi $ satisfy $\mathrm{\eqref{Ass1}}$ and
$\mathrm{\eqref{Ass2}}$, respectively and we put $\varphi _{v}=2^{vn}\varphi
(2^{v}\cdot )$. The Triebel-Lizorkin space $F_{\infty ,q}^{\alpha (\cdot )}
\ is the collection of all $f\in \mathcal{S}^{\prime }(\mathbb{R}^{n})$\
such that
\begin{equation*}
\left\Vert f\right\Vert _{F_{\infty ,q}^{\alpha (\cdot )}}:=\sup_{Q\in
\mathcal{Q}}\frac{1}{|Q|^{1/q}}\Big(\sum\limits_{v=v_{Q}^{+}}^{\infty
}\int\limits_{Q}2^{v\alpha \left( x\right) q}\left\vert \varphi _{v}\ast
f(x)\right\vert ^{q}dx\Big)^{1/q}<\infty ,
\end{equation*
where $\varphi _{0}$ is replaced by $\Phi $.
\end{definition}
For more information about these function spaces, consult \cite{D6} and \cit
{D7}, with different notation. Notice that the supremum can be taken respect
to dyadic cubes with side length $\leq 1$. One of the key tools to prove the
interpolation property of the spaces is their $\varphi $-transforms
characterization, which transfers the problem from function spaces to their
corresponding sequence spaces. Let $\Phi $ and $\varphi $ satisfy,
respectively $\mathrm{\eqref{Ass1}}$ and $\mathrm{\eqref{Ass2}}$. By \cite
pp. 130--131]{FJ90}, there exist \ functions $\Psi \in \mathcal{S}(\mathbb{R
^{n})$ satisfying $\mathrm{\eqref{Ass1}}$ and $\psi \in \mathcal{S}(\mathbb{
}^{n})$ satisfying $\mathrm{\eqref{Ass2}}$ such that for all $\xi \in
\mathbb{R}^{n}
\begin{equation}
\mathcal{F}\widetilde{\Phi }(\xi )\mathcal{F}\Psi (\xi )+\sum_{j=1}^{\infty
\mathcal{F}\widetilde{\varphi }(2^{-j}\xi )\mathcal{F}\psi (2^{-j}\xi
)=1,\quad \xi \in \mathbb{R}^{n}. \label{Ass4}
\end{equation}
Furthermore, we have the following identity for all $f\in \mathcal{S
^{\prime }(\mathbb{R}^{n})$; see \cite[(12.4)]{FJ90
\begin{eqnarray*}
f &=&\Psi \ast \widetilde{\Phi }\ast f+\sum_{v=1}^{\infty }\psi _{v}\ast
\widetilde{\varphi }_{v}\ast f \\
&=&\sum_{m\in \mathbb{Z}^{n}}\widetilde{\Phi }\ast f(m)\Psi (\cdot
-m)+\sum_{v=1}^{\infty }2^{-vn}\sum_{m\in \mathbb{Z}^{n}}\widetilde{\varphi
_{v}\ast f(2^{-v}m)\psi _{v}(\cdot -2^{-v}m).
\end{eqnarray*
Recall that the $\varphi $-transform $S_{\varphi }$ is defined by setting
(S_{\varphi })_{0,m}=\langle f,\Phi _{m}\rangle $ where $\Phi _{m}(x)=\Phi
(x-m)$ and $(S_{\varphi })_{v,m}=\langle f,\varphi _{v,m}\rangle $ where
\varphi _{v,m}(x)=2^{vn/2}\varphi (2^{v}x-m)$ and $v\in \mathbb{N}$. The
inverse $\varphi $-transform $T_{\psi }$ is defined by
\begin{equation*}
T_{\psi }\lambda :=\sum_{m\in \mathbb{Z}^{n}}\lambda _{0,m}\Psi
_{m}+\sum_{v=1}^{\infty }\sum_{m\in \mathbb{Z}^{n}}\lambda _{v,m}\psi _{v,m},
\end{equation*
where $\lambda :=\{\lambda _{v,m}\}_{v\in \mathbb{N}_{0},m\in \mathbb{Z
^{n}}\subset \mathbb{C}$, see \cite{FJ90}.
Now we present the $\varphi $-transform characterization of these function
spaces, see \cite{DHR} and \cite{YZW151}.
\begin{theorem}
\label{phi-tran}Let $\alpha \in C_{\mathrm{loc}}^{\log }$ and $p,q\in
C^{\log }$ with $0<p^{+},q^{+}<\infty $. \textit{Suppose that }$\Phi $,
\Psi \in \mathcal{S}(\mathbb{R}^{n})$ satisfy\ $\mathrm{\eqref{Ass1}}$ and
\varphi ,\psi \in \mathcal{S}(\mathbb{R}^{n})$ satisfy\ $\mathrm{\eqref{Ass2
}$ such that $\mathrm{\eqref{Ass4}}$ holds. The operators $S_{\varphi
}:F_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot
\right) }\rightarrow f_{p\left( \cdot \right) ,q\left( \cdot \right)
}^{\alpha \left( \cdot \right) }$ and $T_{\psi }:f_{p\left( \cdot \right)
,q\left( \cdot \right) }^{\alpha \left( \cdot \right) }\rightarrow
F_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot
\right) }$ are bounded. Furthermore, $T_{\psi }\circ S_{\varphi }$ is the
identity on $F_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha
\left( \cdot \right) }$.
\end{theorem}
Notice that this theorem is true for $F_{\infty ,q}^{\alpha (\cdot )}$
spaces, with $\alpha \in C_{\mathrm{loc}}^{\log }$ and $0<q<\infty $, see
\cite{D6}.
\section{Complex interpolation}
In this section we study complex interpolation of variable Triebel-Lizorkin
spaces.
\subsection{Complex interpolation for the spaces $F_{p(\cdot ),q}^{\protec
\alpha }$}
In this subsection we study the complex interpolation of variable
Triebel-Lizorkin spaces $F_{p(\cdot ),q}^{\alpha }$. We use the so-called
retraction method which allows us to reduce the problem to the interpolation
of appropriate sequence spaces, see for instance the monographs [2, 17-18].
More information about complex interpolation of Besov and Triebel-Lizorkin
spaces of fixed exponents can be found in \cite{T3}, \cite{T95}, \cite{YYZ13}
and \cite{WSD15}. See \cite{NS12}\ for the complex interpolation (introduced
by Triebel \cite{T3}) of Besov spaces and Triebel-Lizorkin spaces with
variable exponents. See also \cite{AH} for the complex interpolation of
variable Besov spaces. Complex interpolation between variable Lebesgue
spaces and $BMO$\ (or Hardy spaces) is given in \cite{Ko09}.
Let $A_{0}:=\{z\in \mathbb{C}:0<\func{Re}z<1\}$ and $A:=\{z\in \mathbb{C
:0\leq \func{Re}z\leq 1\}$.
\begin{definition}
Let $(X_{0},X_{1})$ be an interpolation couple of Banach lattices. Define
\mathcal{F(}X_{0},X_{1})$ as the space of bounded analytic functions
g:A_{0}\rightarrow X_{0}+X_{1}$, which extend continuously to the\ closure $A
$, such that the functions $t\rightarrow g(j+it)$ are bounded continuous
functions into $X_{j},j=0,1$, which tend to zero as $\left\vert t\right\vert
\rightarrow \infty $. We endow $\mathcal{F(}X_{0},X_{1})$ with the norm
\begin{equation*}
\left\Vert g\right\Vert _{\mathcal{F(}X_{0},X_{1})}:=\max \big
\sup_{t}\left\Vert g(it)\right\Vert _{X_{0}},\sup_{t}\left\Vert
g(1+it)\right\Vert _{X_{1}}\big).
\end{equation*
Further, we define the complex interpolation space
\begin{equation*}
\lbrack X_{0},X_{1}]_{\theta }:=\{f\in X_{0}+X_{1}:f=g(\theta )\text{ for
some }g\in \mathcal{F(}X_{0},X_{1})\},\quad 0<\theta <1
\end{equation*
and
\begin{equation*}
\left\Vert f\right\Vert _{[X_{0},X_{1}]_{\theta }}:=\inf \{\left\Vert
g\right\Vert _{\mathcal{F(}X_{0},X_{1})}:g\in \mathcal{F(}X_{0},X_{1}),\quad
g(\theta )=f\}.
\end{equation*}
\end{definition}
Let $X$ be a complex Banach space. A function $f:\mathbb{R}^{n}\rightarrow X$
is said to be a simple function if it can be written a
\begin{equation*}
f=\sum_{j=1}^{N}a_{j}\chi _{A_{j}}
\end{equation*
with $a_{j}\in X$ and pairwise disjoint $A_{j}\subset \mathbb{R}^{n}$,
\left\vert A_{j}\right\vert <\infty $ $(j=1,...,N)$. Let $p\in \mathcal{P}$.
The Bochner-Lebesgue spaces with variable exponent $L^{p(\cdot )}(\mathbb{R
^{n},X)$ is the collection of all measurable functions $f:\mathbb{R
^{n}\rightarrow X$ endowed with the norm
\begin{equation*}
\left\Vert f\right\Vert _{L^{p(\cdot )}(\mathbb{R}^{n},X)}:=\inf \big\
\lambda >0:\varrho _{L^{p(\cdot )}(\mathbb{R}^{n},X)}\big(\frac{f}{\lambda
\big)\leq 1\big\}.
\end{equation*
The spaces $L^{p(\cdot )}(\mathbb{R}^{n},X)$ have been introduced by C.
Cheng and J. Xu \cite{CX13}.
Let $X_{0}$ and $X_{1}$, be two complex Banach spaces, both linearly and
continuously embedded in a linear complex Hausdorff space $\mathcal{A}$. Two
such Banach spaces are said to be an interpolation couple $(X_{0},X_{1})$.
\begin{theorem}
\label{Lebsgue-int}Let $0<\theta <1$. Let $p_{0},p_{1}\in \mathcal{P}$ with
1\leq p_{0}^{+},p_{1}^{+}<\infty $ . We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )}+\frac{\theta }
p_{1}(\cdot )}.
\end{equation*
Further let $(X_{0},X_{1})$ be an interpolation couple. The
\begin{equation*}
\lbrack L^{p_{0}(\cdot )}(\mathbb{R}^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R
^{n},X_{1})]_{\theta }=L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta }).
\end{equation*}
\end{theorem}
\textbf{Proof.} Our approach follows essentially \cite{BL76} and \cite{T95}.
It is not hard to see, that the space of simple functions is dense in
L^{p_{0}(\cdot )}(\mathbb{R}^{n},X_{0})\cap L^{p_{1}(\cdot )}(\mathbb{R
^{n},X_{1})$, and thus also in $L^{p_{0}(\cdot )}(\mathbb{R
^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1})]_{\theta }$ by \cite
Theorem 4.2.2]{BL76}. From now we consider only simple functions. Let
f(x)\neq 0$ be a simple function
\begin{equation}
f=\sum_{j=1}^{N}a_{j}\chi _{A_{j}} \label{simple-fun}
\end{equation
with $a_{j}\in X_{0}\cap X_{1}$ and pairwise disjoint $A_{j}\subset \mathbb{
}^{n}$, $\left\vert A_{j}\right\vert <\infty $ $(j=1,...,N)$. Let us prove
tha
\begin{equation*}
\left\Vert f\right\Vert _{[L^{p_{0}(\cdot )}(\mathbb{R}^{n},X_{0}),L^{p_{1}
\cdot )}(\mathbb{R}^{n},X_{1})]_{\theta }}\leq \left\Vert f\right\Vert
_{L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta })}.
\end{equation*
By the\ scaling argument, we see that it suffices to consider the case
\left\Vert f\right\Vert _{L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta
})}=1$. We put for $0\leq \func{Re}z\leq 1$
\begin{equation*}
g(x,z)=h(x,z)\left\Vert f(x)\right\Vert _{[X_{0},X_{1}]_{\theta }}^{\big
\frac{p(x)}{p_{1}(x)}-\frac{p(x)}{p_{0}(x)}\big)(z-\theta )},
\end{equation*
where $h\in \mathcal{F(}X_{0},X_{1})$ for fixed $x\in \mathbb{R}^{n}$,
h(x,\theta )=f(x)$ and $h(x,z)=h(y,z)$ if $x,y\in A_{j},j=1,...,N$ and $z$
is defined on the strip $0\leq \func{Re}z\leq 1$, $h(x,z)=0$ for any $x\in
\mathbb{R}^{n}-\cup _{j=1}^{N}A_{j}$ and
\begin{equation*}
\left\Vert h(x,it)\right\Vert _{X_{0}},\left\Vert h(x,1+it)\right\Vert
_{X_{1}}\leq \left\Vert f(x)\right\Vert _{[X_{0},X_{1}]_{\theta
}}+\varepsilon ,
\end{equation*
$\varepsilon $ is a given positive number. Using this we deriv
\begin{eqnarray*}
\varrho _{L^{p_{0}(\cdot )}(\mathbb{R}^{n},X_{0})}(g(it)) &=&\int_{\mathbb{R
^{n}}\left\vert h(x,it)\right\vert ^{p_{0}(x)}\left\Vert f(x)\right\Vert
_{[X_{0},X_{1}]_{\theta }}^{-\big(\frac{p(x)}{p_{1}(x)}-\frac{p(x)}{p_{0}(x)
\big)\theta p_{0}(x)}dx \\
&\leq &\int_{\mathbb{R}^{n}}\left\Vert f(x)\right\Vert
_{[X_{0},X_{1}]_{\theta }}^{p(x)}dx+\varepsilon ^{\prime }=1+\varepsilon
^{\prime }.
\end{eqnarray*
Similarly
\begin{equation*}
\varrho _{L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1})}(g(1+it))\leq
1+\varepsilon ^{\prime }.
\end{equation*
Thu
\begin{equation*}
\left\Vert g\right\Vert _{\mathcal{F(}L^{p_{0}(\cdot )}(\mathbb{R
^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1}))}\leq 1.
\end{equation*
This and $g(\cdot ,\theta )=f(\cdot )$\ we obtain\
\begin{equation*}
\left\Vert f\right\Vert _{[L^{p_{0}(\cdot )}(\mathbb{R}^{n},X_{0}),L^{p_{1}
\cdot )}(\mathbb{R}^{n},X_{1})]_{\theta }}\leq 1.
\end{equation*
Now let us prove tha
\begin{equation*}
\left\Vert f\right\Vert _{L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta
})}\leq \left\Vert f\right\Vert _{[L^{p_{0}(\cdot )}(\mathbb{R
^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1})]_{\theta }}.
\end{equation*
Let $f$ be a simple function of type $\mathrm{\eqref{simple-fun}}$. Further,
let $g(x,z)\in \mathcal{F(}X_{0},X_{1})$ where $g(x,\theta )=f(x)$. We will
prove that
\begin{equation*}
\left\Vert f\right\Vert _{L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta
})}\leq c\left\Vert g\right\Vert _{\mathcal{F(}L^{p_{0}(\cdot )}(\mathbb{R
^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1}))}.
\end{equation*
From Lemma 4.3.2 in \cite{BL76} and H\"{o}lder's inequality, we obtai
\begin{eqnarray*}
\left\Vert f(x)\right\Vert _{[X_{0},X_{1}]_{\theta }} &\leq &\Big(\frac{1}
1-\theta }\int_{R}\left\Vert g(x,it)\right\Vert _{X_{0}}\mu _{0}(\theta ,t)d
\Big)^{1-\theta } \\
&&\times \Big(\frac{1}{\theta }\int_{R}\left\Vert g(x,1+it)\right\Vert
_{X_{1}}\mu _{1}(\theta ,t)dt\Big)^{\theta },
\end{eqnarray*
where $\mu _{0}(\theta ,t)$ and $\mu _{1}(\theta ,t)$ are the Poisson
kernels for the strip $A_{0}$,
\begin{equation*}
\frac{1}{1-\theta }\int_{R}\mu _{0}(\theta ,t)dt=\frac{1}{\theta
\int_{R}\mu _{1}(\theta ,t)dt=1.
\end{equation*
Using H\"{o}lder's inequality with respect to the variable $t$\ we get
\begin{eqnarray*}
\left\Vert f(x)\right\Vert _{[X_{0},X_{1}]_{\theta }} &\leq &\Big(\frac{1}
1-\theta }\int_{R}\left\Vert g(x,it)\right\Vert _{X_{0}}^{p_{0}(x)}\mu
_{0}(\theta ,t)dt\Big)^{(1-\theta )/p_{0}(x)} \\
&&\times \Big(\frac{1}{\theta }\int_{R}\left\Vert g(x,1+it)\right\Vert
_{X_{1}}^{p_{1}(x)}\mu _{1}(\theta ,t)dt\Big)^{\theta /p_{1}(x)}.
\end{eqnarray*
Hence, again by H\"{o}lder's inequality with respect to the variable $x$\ we
ge
\begin{eqnarray*}
\left\Vert f\right\Vert _{L^{p(\cdot )}(\mathbb{R}^{n},[X_{0},X_{1}]_{\theta
})} &\lesssim &\Big\|\Big(\frac{1}{1-\theta }\int_{R}\left\Vert
g(x,it)\right\Vert _{X_{0}}^{p_{0}(\cdot )}\mu _{0}(\theta ,t)dt\Big
^{1/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot )}^{1-\theta } \\
&&\times \Big\|\Big(\frac{1}{\theta }\int_{R}\left\Vert g(x,1+it)\right\Vert
_{X_{0}}^{p_{1}(\cdot )}\mu _{0}(\theta ,t)dt\Big)^{1/p_{1}(\cdot )}\Big\
_{p_{1}(\cdot )}^{\theta }.
\end{eqnarray*
The first norm is bounded b
\begin{equation*}
\left\Vert g\right\Vert _{\mathcal{F(}L^{p_{0}(\cdot )}(\mathbb{R
^{n},X_{0}),L^{p_{1}(\cdot )}(\mathbb{R}^{n},X_{1}))}.
\end{equation*
Indeed
\begin{eqnarray*}
\int_{\mathbb{R}^{n}}\frac{1}{1-\theta }\int_{R}\Big\|\frac{g(x,it)}
\left\Vert g\right\Vert _{\mathcal{F}}}\Big\|_{X_{0}}^{p_{0}(x)}\mu
_{0}(\theta ,t)dtdx &=&\frac{1}{1-\theta }\int_{R}\int_{\mathbb{R}^{n}}\Big\
\frac{g(x,it)}{\left\Vert g\right\Vert _{\mathcal{F}}}\Big\
_{X_{0}}^{p_{0}(x)}dx\mu _{0}(\theta ,t)dt \\
&\leq &\frac{1}{1-\theta }\int_{R}\mu _{0}(\theta ,t)dt=1.
\end{eqnarray*
Similarly for the rest term. The proof of theorem is complete. \ $\square $
Now we present complex interpolation of variable\ Triebel-Lizorkin space
F_{p(\cdot ),q}^{\alpha }$. We use the so-called retraction method which
allows us to reduce the problem to the interpolation of appropriate sequence
spaces. We recall that a Banach space $X$ is called a retract of a Banach
space $Y$ if there are linear continuous operators $R:Y\rightarrow X$
(retraction ) and $S:X\rightarrow Y$ (co-retraction ) such that the
composition $RS$ is the identity operator in $X$.
First let us recall the Littlewood-Paley decomposition. Let $\Psi $\ be a
function\ in $\mathcal{S}(\mathbb{R}^{n})$\ satisfying $0\leq \Psi (x)\leq 1$
for all $x\in \mathbb{R}^{n}$, $\Psi (x)=1$\ for\ $\left\vert x\right\vert
\leq 1$\ and\ $\Psi (x)=0$\ for\ $\left\vert x\right\vert \geq 2$.\ We put
\mathcal{F}\varphi _{0}(x)=\Psi (x)$, $\mathcal{F}\varphi (x)=\Psi (\frac{x}
2})-\Psi (x)$\ and $\mathcal{F}\varphi _{v}(x)=\mathcal{F}\varphi (2^{1-v}x)$
for $v=1,2,3,....$ Then $\{\mathcal{F}\varphi _{v}\}_{v\in \mathbb{N}_{0}}$\
is a resolution of unity, $\sum_{v=0}^{\infty }\mathcal{F}\varphi _{v}(x)=1$
for all $x\in \mathbb{R}^{n}$.\ Thus we obtain the Littlewood-Paley
decomposition
\begin{equation*}
f=\sum_{v=0}^{\infty }\varphi _{v}\ast f
\end{equation*
of all $f\in \mathcal{S}^{\prime }(\mathbb{R}^{n})$ $($convergence in
\mathcal{S}^{\prime }(\mathbb{R}^{n}))$.\ We defin
\begin{equation*}
S(f)=(\varphi _{v}\ast f)_{v}
\end{equation*
and
\begin{equation}
R((f_{v})_{v})=\sum_{v=0}^{\infty }\omega _{v}\ast f_{v}, \label{serie}
\end{equation
where $\omega _{v}=\varphi _{v-1}+\varphi _{v}+\varphi _{v+1}.$
\begin{theorem}
\label{rect}Let $p\in C^{\log }$ with $1\leq p^{-}<\infty $, $1\leq q\leq
\infty $ and $\alpha \in \mathbb{R}$. Let $\{\mathcal{F}\varphi _{v}\}_{v\in
\mathbb{N}_{0}}$\ be a resolution of unity. Then $S$ is a co-retraction from
$F_{p(\cdot ),q}^{\alpha }$into $L^{p(\cdot )}(\ell _{q}^{\alpha })$ and $R$
is a corresponding retraction from $L^{p(\cdot )}(\ell _{q}^{\alpha })$ onto
$F_{p(\cdot ),q}^{\alpha }$.
\end{theorem}
\textbf{Proof.} The convergence of the series $\mathrm{\eqref{serie}}$ in
\mathcal{S}^{\prime }(\mathbb{R}^{n})$ can be obtained by Lemma 4.5 of \cit
{AH}, where one has to take into consideration $L^{p(\cdot )}(\ell
_{q}^{\alpha })\hookrightarrow \ell _{\infty }^{\alpha }(L^{p(\cdot )})$.
Clearly $R$ is a bounded linear operator from $F_{p(\cdot ),q}^{\alpha }$
into $L^{p(\cdot )}(\ell _{q}^{\alpha })$. Moreover, $RS$ is the identity
operator in $F_{p(\cdot ),q}^{\alpha }$. Using the support properties of
\mathcal{F}\varphi _{v}$ and $\mathcal{F}\omega _{v}$,
\begin{eqnarray*}
\varphi _{v}\ast R((f_{i})_{i}) &=&\sum_{i=v-2}^{v+2}\varphi _{v}\ast \omega
_{i}\ast f_{i} \\
&=&\sum_{k=-2}^{2}\varphi _{v}\ast \omega _{k+v}\ast f_{k+v},\text{ \ \ \
v=0,1,....
\end{eqnarray*}
We can estimate $\left\vert \varphi _{v}\ast \omega _{k+v}\ast
f_{k+v}\right\vert $ by
\begin{equation*}
(\mathcal{\eta }_{v}\mathcal{\ast }\left\vert f_{k+v}\right\vert ^{t})^{1/t}
\text{ \ }0<t<\min (p^{-},q),
\end{equation*
see \cite{D6}. Applying Lemma \ref{DHRlemma}
\begin{eqnarray*}
\left\Vert R((f_{i})_{i})\right\Vert _{F_{p(\cdot ),q}^{\alpha }} &\lesssim
&\sum_{k=-2}^{2}\Big\|\left( 2^{v\alpha }f_{k+v}\right) _{v\geq 0}\Big\
_{L^{p(\cdot )}(\ell _{q})} \\
&\lesssim &\big\|\left( 2^{v\alpha }f_{v}\right) _{v\geq 0}\big\
_{L^{p(\cdot )}(\ell _{q})}.
\end{eqnarray*
The proof of theorem is complete. \ $\square $
Now we are ready to formulate the main statement of this subsection.
\begin{theorem}
\label{Inter-rest}Let $0<\theta <1$. Let $p_{0},p_{1}\in C^{\log }$ with
1\leq p_{0}^{+},p_{1}^{+}<\infty $, $1\leq q_{0},q_{1}<\infty $ and $\alpha
_{0},\alpha _{1}\in \mathbb{R}$. We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )}+\frac{\theta }
p_{1}(\cdot )},\text{ }\frac{1}{q}:=\frac{1-\theta }{q_{0}}+\frac{\theta }
q_{1}}\text{ \ and \ }\alpha :=(1-\theta )\alpha _{0}+\theta \alpha _{1}.
\end{equation*
The
\begin{equation*}
\lbrack F_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}},F_{p_{1}(\cdot
),q_{1}}^{\alpha _{1}}]_{\theta }=F_{p(\cdot ),q}^{\alpha }
\end{equation*
holds in the sense of equivalent norms.
\end{theorem}
\textbf{Proof.} The embedding $\mathrm{\eqref{emb}}$ shows that
(F_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}},F_{p_{1}(\cdot ),q_{1}}^{\alpha
_{1}}) $ is an interpolation couple. By Theorem \ref{rect},
\begin{equation*}
\left\Vert f\right\Vert _{[F_{p_{0}(\cdot ),q_{0}}^{\alpha
_{0}},F_{p_{1}(\cdot ),q_{1}}^{\alpha _{1}}]_{\theta }}\approx \left\Vert
(\varphi _{v}\ast f)_{v}\right\Vert _{[L^{p_{0}(\cdot )}(\ell
_{q_{0}}^{\alpha _{0}}),L^{p_{1}(\cdot )}(\ell _{q_{1}}^{\alpha
_{1}})]_{\theta }}.
\end{equation*
Now by Theorem \ref{Lebsgue-int} the left-hand side is equivalent to
\left\Vert (\varphi _{v}\ast f)_{v}\right\Vert _{L^{p(\cdot )}(\ell
_{q}^{\alpha })}$. \ $\square $
\subsection{Complex interpolation for the spaces $F_{p(\cdot ),p(\cdot )}^
\protect\alpha (\cdot )}$ and some limiting cases}
In this subsection we present interpolation results in $F_{p(\cdot ),p(\cdot
)}^{\alpha (\cdot )}$\ spaces and some consequences,. We follow the approach
of Frazier and Jawerth \cite{FJ90}, see also \cite{SSV13} and \cite{YYZ13}.
We start by defining the Calder\'{o}n product of two Banach lattices. Let $
\mathcal{A},S
\mu
)$ be a $\sigma $-finite measure space and let $\mathfrak{M}$ be the class
of all complex-valued, $\mu $-measurable functions on $\mathcal{A}$. Then a
Banach space $X\subset \mathfrak{M}$ is called a Banach lattice of functions
if for every $f\in X$ and $g\in \mathfrak{M}$ with $|g(x)|\leq |f(x)|$ for
\mu $-a.e. $x\in X$ one has $g\in X$ and $\left\Vert g\right\Vert _{X}\leq
\left\Vert f\right\Vert _{X}$.
\begin{definition}
Let $(\mathcal{A},S
\mu
)$ be a $\sigma $-finite measure space and let $\mathfrak{M}$ be the class
of all complex-valued, $\mu $-measurable functions on $\mathcal{A}$. Suppose
that $X_{0}$ and $X_{1}$ are Banach lattices on $\mathfrak{M}$. Given
0<\theta <1$, define the Calder\'{o}n product $X_{0}^{1-\theta }\cdot
X_{1}^{\theta }$ as the collection of all functions $f\in \mathfrak{M}$
satisfyin
\begin{equation*}
\left\Vert f\right\Vert _{X_{0}^{1-\theta }\cdot X_{1}^{\theta }}:=\inf
\big\{\left\Vert g\right\Vert _{X_{0}}^{1-\theta }\left\Vert h\right\Vert
_{X_{1}}^{\theta }:|f|\leq |g|^{1-\theta }|h|^{\theta },\mu \text{-}a.e.,\
\left\Vert g\right\Vert _{X_{0}}\leq 1,\ \left\Vert h\right\Vert
_{X_{1}}\leq 1\big\}.
\end{equation*}
\end{definition}
\begin{remark}
Calder\'{o}n products have been introduced by Calder\'{o}n \cite{Ca64} (in a
little bit different form which coincides with the above one). Further
properties we refer to, Frazier and Jawerth \cite{FJ90}, Mendez and Mitrea
\cite{MM00}, Kalton, Mayboroda and Mitrea \cite{KMM07} and Yang, Yuan and
Zhuo \cite{YYZ13}.
\end{remark}
We need a few useful properties, see \cite{YYZ13}.
\begin{lemma}
\label{YYZ-Lemma}Let $(\mathcal{A},S
\mu
)$ be a $\sigma $-finite measure space and let $\mathfrak{M}$ be the class
of all complex-valued, $\mu $-measurable functions on $\mathcal{A}$. Suppose
that $X_{0}$ and $X_{1}$ are Banach lattices on $\mathfrak{M}$. Let
0<\theta <1$.
$\mathrm{(i)}$ Then the Calder\'{o}n product $X_{0}^{1-\theta }\cdot
X_{1}^{\theta }$ is a Banach space.
$\mathrm{(ii)}$ Define the Calder\'{o}n product $\widetilde{X_{0}^{1-\theta
}\cdot X_{1}^{\theta }}$ as the collection of all functions $f\in \mathfrak{
}$ satisfyin
\begin{equation*}
\left\Vert f\right\Vert _{\widetilde{X_{0}^{1-\theta }\cdot X_{1}^{\theta }
}:=\inf \big\{M>0:|f|\leq M|g|^{1-\theta }|h|^{\theta },\ \left\Vert
g\right\Vert _{X_{0}}\leq 1,\ \left\Vert h\right\Vert _{X_{1}}\leq 1\big\}.
\end{equation*
Then $\widetilde{X_{0}^{1-\theta }\cdot X_{1}^{\theta }}=X_{0}^{1-\theta
}\cdot X_{1}^{\theta }$ follows with equality of norms.
\end{lemma}
Now we turn to the investigation of the Calder\'{o}n products of the
sequence spaces $f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}$.
\begin{theorem}
\label{calderon-prod1}Let $0<\theta <1$. Let $p_{0},p_{1}\in C^{\log }$ with
$1\leq p_{0}^{+},p_{1}^{+}<\infty $ and $\alpha _{0},\alpha _{1}\in C_
\mathrm{loc}}^{\log }$. We pu
\begin{equation}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )}+\frac{\theta }
p_{1}(\cdot )}\text{ and }\alpha (\cdot ):=(1-\theta )\alpha _{0}(\cdot
)+\theta \alpha _{1}(\cdot ). \label{Th-Cond}
\end{equation
The
\begin{equation*}
\big(f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot )}\big)^{1-\theta
\big(f_{p_{1}(\cdot ),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}\big)^{\theta
}=f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}
\end{equation*
holds in the sense of equivalent norms.
\end{theorem}
\textbf{Proof}.
\textit{Step 1.} We shall prov
\begin{equation*}
\big(f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot )}\big)^{1-\theta
\big(f_{p_{1}(\cdot ),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}\big)^{\theta
}\hookrightarrow f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}.
\end{equation*
We suppose, that sequences $\lambda :=(\lambda _{j,m})_{j,m}$, $\lambda
^{i}:=(\lambda _{j,m}^{i})_{j,m},i=0,1$, are given and tha
\begin{equation*}
|\lambda _{j,m}|\leq |\lambda _{j,m}^{0}|^{1-\theta }|\lambda
_{j,m}^{1}|^{\theta }
\end{equation*
holds for all $j\in \mathbb{N}_{0}$ and $m\in \mathbb{Z}^{n}$. Let
\begin{equation}
g(x):=\Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}2^{j(\alpha
(x)+n/2)p(x)}|\lambda _{j,m}|^{p(x)}\chi _{j,m}(x)\Big)^{1/p(x)}.
\label{notation1}
\end{equation
Since,
\begin{eqnarray*}
&&2^{j(\alpha (x)+n/2)p(x)}|\lambda _{j,m}|^{p(x)}\chi _{j,m}(x) \\
&=&\left( 2^{j(\alpha _{0}(x)+n/2)}|\lambda _{j,m}|\chi _{j,m}(x)\right)
^{p(x)(1-\theta )}\left( 2^{j(\alpha _{1}(x)+n/2)}|\lambda _{j,m}|\chi
_{j,m}(x)\right) ^{p(x)\theta },
\end{eqnarray*
then, H\"{o}lder's inequality implies that $g(x)$ can be estimated b
\begin{eqnarray}
&&\Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\left(
2^{j(\alpha _{0}(x)+n/2)}|\lambda _{j,m}|\chi _{j,m}(x)\right) ^{p_{0}(x)
\Big)^{(1-\theta )/p_{0}(x)} \label{Second-est} \\
&&\times \Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\left(
2^{j(\alpha _{1}(x)+n/2)}|\lambda _{j,m}|\chi _{j,m}(x)\right) ^{p_{1}(x)
\Big)^{\theta /p_{1}(x)}. \notag
\end{eqnarray
Apply H\"{o}lder's inequality again but with conjugate indices $\frac
p_{0}(\cdot )}{(1-\theta )}$ and $\frac{p_{1}(\cdot )}{\theta }$, we obtai
\begin{equation*}
\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}\leq \big\
\lambda ^{0}\big\|_{f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot
)}}^{1-\theta }\big\|\lambda ^{1}\big\|_{f_{p_{1}(\cdot ),p_{1}(\cdot
)}^{\alpha _{1}(\cdot )}}^{\theta }.
\end{equation*
\textit{Step 2.} Now we turn to the proof o
\begin{equation*}
f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}\hookrightarrow \big(f_{p_{0}(\cdot
),p_{0}(\cdot )}^{\alpha _{0}(\cdot )}\big)^{1-\theta }\big(f_{p_{1}(\cdot
),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}\big)^{\theta }.
\end{equation*
We will use Lemma \ref{YYZ-Lemma}. Let the sequence $\lambda \in f_{p(\cdot
),p(\cdot )}^{\alpha (\cdot )}$ be given wit
\begin{equation*}
\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}\neq 0.
\end{equation*
We have to find sequences $\lambda ^{0}$ and $\lambda ^{1}$ such that
|\lambda _{j,m}|\leq M|\lambda _{j,m}^{0}|^{1-\theta }|\lambda
_{j,m}^{1}|^{\theta }$ for every $j\in \mathbb{N}_{0}$, $m\in \mathbb{Z}^{n}$
and
\begin{equation}
\big\|\lambda ^{0}\big\|_{f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot
)}}\lesssim 1\text{ \ and \ }\big\|\lambda ^{1}\big\|_{f_{p_{1}(\cdot
),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}}\lesssim 1 \label{Est-Int}
\end{equation
with some constant $c$ independent of $\lambda $. We follow ideas of the
proof of Theorem 8.2 in Frazier and Jawerth \cite{FJ90}, see also Sickel,
Skrzypczak and Vyb\'{\i}ral \cite{SSV13}. Se
\begin{equation*}
u(\cdot ):=p(\cdot )\theta \big(\tfrac{\alpha _{1}(\cdot )}{p_{0}(\cdot )}
\tfrac{\alpha _{0}(\cdot )}{p_{1}(\cdot )}\big)+\tfrac{n}{2}\big(\tfrac
p(\cdot )}{p_{0}(\cdot )}-1\big)
\end{equation*
and
\begin{equation*}
v(\cdot ):=p(\cdot )(1-\theta )\big(\tfrac{\alpha _{0}(\cdot )}{p_{1}(\cdot
}-\tfrac{\alpha _{1}(\cdot )}{p_{0}(\cdot )}\big)+\tfrac{n}{2}\big(\tfrac
p(\cdot )}{p_{1}(\cdot )}-1\big).
\end{equation*
We pu
\begin{equation*}
\lambda _{j,m}^{0}:=2^{ju(x_{j,m})}\Big(\tfrac{|\lambda _{j,m}|}{\big\
\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\Big
^{p(x_{j,m})/p_{0}(x_{j,m})},\quad x_{j,m}=2^{-j}m.
\end{equation*
Also, se
\begin{equation*}
\lambda _{j,m}^{1}:=2^{jv(x_{j,m})}\Big(\tfrac{|\lambda _{j,m}|}{\big\
\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\Big
^{p(x_{j,m})/p_{1}(x_{j,m})}.
\end{equation*
Observe tha
\begin{equation*}
|\lambda _{j,m}|\leq \big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha
(\cdot )}}\big(\lambda _{j,m}^{0}\big)^{1-\theta }\big(\lambda _{j,m}^{1
\big)^{\theta },
\end{equation*
which holds now for all pairs $(j,m)$.
\textbf{Estimation of }$\big\|\lambda ^{0}\big\|_{f_{p_{0}(\cdot
),p_{0}(\cdot )}^{\alpha _{0}(\cdot )}}$. Se
\begin{equation*}
I(\cdot ):=\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\Big
2^{j(\alpha _{0}(\cdot )+\frac{n}{2})}|\lambda _{j,m}^{0}|\chi _{j,m}\Big
^{p_{0}(\cdot )}.
\end{equation*
We use the local log-H\"{o}lder continuity of $\alpha _{0},\alpha _{1},p_{0}$
and $p_{1}$ to show tha
\begin{equation}
2^{ju(x)}\leq c\text{ }2^{ju(y)}\quad \text{and}\quad 2^{j(\alpha (x)-\frac{
}{p(x)})}\leq c\text{ }2^{j(\alpha (y)-\frac{n}{p(y)})}\text{, \ \ }x,y\in
Q_{j,m}, \label{est-lamda}
\end{equation
where $c>0$ is independent of $m\in \mathbb{Z}^{n}$ and $j\in \mathbb{N}_{0}
. Now $\Big(\tfrac{|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot
),p(\cdot )}^{\alpha (\cdot )}}}\Big)^{\frac{p(x_{j,m})}{p_{0}(x_{j,m})}}$\
can be rewritten u
\begin{equation*}
\Big(2^{j(\alpha (x_{j,m})+\frac{n}{2}-\frac{n}{p(x_{j,m})})}\tfrac{|\lambda
_{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}
\Big)^{\frac{p(x_{j,m})}{p_{0}(x_{j,m})}}2^{-j(\alpha (x_{j,m})+\frac{n}{2}
\frac{n}{p(x_{j,m})})\frac{p(x_{j,m})}{p_{0}(x_{j,m})}}.
\end{equation*
After applying Lemma \ref{lamda-est}, we get
\begin{equation}
2^{j(\alpha (x_{j,m})+\frac{n}{2}-\frac{n}{p(x_{j,m})})}\tfrac{|\lambda
_{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}
\lesssim 1. \label{lemma-cond}
\end{equation
Observe tha
\begin{eqnarray*}
2^{j(\alpha (x_{j,m})+\frac{n}{2}-\frac{n}{p(x_{j,m})})}\tfrac{|\lambda
_{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}
&=&\frac{1}{\left\vert Q_{j,m}\right\vert }\int_{Q_{j,m}}2^{j(\alpha
(x_{j,m})+\frac{n}{2}-\frac{n}{p(x_{j,m})})}\tfrac{|\lambda _{j,m}|}{\big\
\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}dy \\
&\lesssim &\frac{1}{\left\vert Q_{j,m}\right\vert }\int_{Q_{j,m}}2^{j(\alpha
(y)+\frac{n}{2}-\frac{n}{p(y)})}\tfrac{|\lambda _{j,m}|}{\big\|\lambda \big\
_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}dy
\end{eqnarray*
for any $m\in \mathbb{Z}^{n}$ and any $j\in \mathbb{N}_{0}$. Taking the
\frac{\sigma (x_{j,m})p(x_{j,m})}{p_{0}(x_{j,m})}$-power, with $\frac
p_{0}(\cdot )}{p(\cdot )}<\sigma (\cdot )<1$ and $\sigma \in C^{\log }$, by
\mathrm{\eqref{lemma-cond}}$ we can apply Lemma \ref{DHHR-estimate} and
obtain tha
\begin{eqnarray*}
&&\Big(2^{j(\alpha (x_{j,m})+\frac{n}{2}-\frac{n}{p(x_{j,m})})}\tfrac
|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha
(\cdot )}}}\Big)^{\frac{\sigma (x_{j,m})p(x_{j,m})}{p_{0}(x_{j,m})}} \\
&\lesssim &\frac{1}{\left\vert Q_{j,m}\right\vert }\int_{Q_{j,m}}\Big
2^{j(\alpha (y)+\frac{n}{2}-\frac{n}{p(y)})}\tfrac{|\lambda _{j,m}|}{\big\
\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\Big)^{\frac
\sigma (y)p(y)}{p_{0}(y)}}dy+2^{-jnh}\eta _{0,h}(x_{j,m}).
\end{eqnarray*
Applying again Lemma \ref{lamda-est}, the last term with $\frac{1}{\sigma
(x_{j,m})}$-power is bounded b
\begin{eqnarray*}
&&\frac{2^{\frac{1}{\sigma ^{-}}-1}}{\left\vert Q_{j,m}\right\vert
\int_{Q_{j,m}}\Big(2^{j(\alpha (y)+\frac{n}{2}-\frac{n}{p(y)})}\tfrac
|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha
(\cdot )}}}\Big)^{\frac{p(y)}{p_{0}(y)}}dy+2^{\frac{1}{\sigma ^{-}
-jnh+1}\eta _{0,h}(x_{j,m}) \\
&\lesssim &\eta _{j,\rho }\ast \Big(2^{j(\alpha (\cdot )+\frac{n}{2}-\frac{
}{p(\cdot )})}\tfrac{|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot
),p(\cdot )}^{\alpha (\cdot )}}}\chi _{Q_{j,m}}\Big)^{\frac{p(\cdot )}
p_{0}(\cdot )}}(x)+2^{-jnh}\eta _{0,h}(x)
\end{eqnarray*
for any $h,\rho >0$ and any $x\in Q_{j,m}$. Applying the second estimate of
\mathrm{\eqref{est-lamda}}$, the term $\Big(\tfrac{|\lambda _{j,m}|}{\big\
\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\Big)^{\frac
p(x_{j,m})}{p_{0}(x_{j,m})}}$ can be estimated b
\begin{equation*}
c\text{ }\eta _{j,\rho _{1}}\ast \Big(\tfrac{|\lambda _{j,m}|}{\big\|\lambda
\big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\chi _{Q_{j,m}}\Big)^
\frac{p(\cdot )}{p_{0}(\cdot )}}(x)+c\text{ }2^{-j(nh+d)}\eta _{0,h}(x),
\end{equation*
wher
\begin{equation*}
d=\big((\alpha +\frac{n}{2}-\frac{n}{p})\frac{p}{p_{0}}\big)^{-}\text{ and
\rho _{1}=\rho -c_{\log }((\alpha +\frac{n}{2}-\frac{n}{p})\frac{p}{p_{0}}).
\end{equation*
From the estimations above, for any $x\in Q_{j,m}
\begin{equation*}
2^{j(\alpha _{0}(x)+\frac{n}{2})}|\lambda _{j,m}^{0}|\lesssim \eta _{j,\rho
_{2}}\ast \digamma _{j,m}(x)+2^{-j(nh+d-(u^{+}+\alpha _{0}^{+}+\frac{n}{2
))}\eta _{0,h}(x),
\end{equation*
where the implicit positive constant not depending on $x,m$ and $j$, with
\begin{equation*}
\digamma _{j,m}=2^{j(\alpha _{0}(\cdot )+\frac{n}{2})+ju(\cdot )}\Big(\tfrac
|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha
(\cdot )}}}\Big)^{\frac{p(\cdot )}{p_{0}(\cdot )}}\chi _{Q_{j,m}},
\end{equation*
where
\begin{equation*}
\rho _{2}=\rho _{1}-c_{\log }(\alpha _{0})-c_{\log }(u).
\end{equation*
Hence $\big\|\left( I^{p_{0}(\cdot )}(\cdot )\right) ^{1/p_{0}(\cdot )}\big\
_{p_{0}(\cdot )}$ is bounded b
\begin{equation*}
c\Big\|\Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\left( \eta
_{j,\rho _{2}}\ast \digamma _{j,m}\right) ^{p_{0}(\cdot )}\Big
^{1/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot )}+c\big\|\eta _{0,h}\big\
_{p_{0}(\cdot )}\Big(\sum_{j=0}^{\infty }2^{-j(nh+d-(u^{+}+\alpha _{0}^{+}
\frac{n}{2}))p_{0}^{-}}\Big)^{\frac{1}{p_{0}^{-}}},
\end{equation*
where in the second estimate we used the embedding $L^{p(\cdot )}(\ell
^{p^{-}})\hookrightarrow L^{p(\cdot )}(\ell ^{p(\cdot )})$. By taking $h$
large enough such that $h>(\alpha _{0}^{+}+u^{+}+\frac{n}{2})/n-\frac{d}{n}$
the second term is bounded. Taking $\rho _{2}$ large enough and applying
Lemma\ \ref{DHRlemma} to estimate the first expression b
\begin{eqnarray*}
&&c\Big\|\Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\digamma
_{j,m}^{p_{0}(\cdot )}\Big)^{1/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot )} \\
&=&c\Big\|\Big(\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z
^{n}}2^{j(\alpha (\cdot )+\frac{n}{2})p(\cdot )}\Big(\tfrac{|\lambda _{j,m}
}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}}}\Big
^{q(\cdot )}\chi _{Q_{j,m}}\Big)^{1/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot )},
\end{eqnarray*
since $u(\cdot )+\alpha _{0}(\cdot )=\alpha (\cdot )\frac{p(\cdot )}
p_{0}(\cdot )}+\frac{n}{2}\big(\frac{p(\cdot )}{p_{0}(\cdot )}-1\big)$.
Therefore, $\big\|\left( I^{p_{0}(\cdot )}(\cdot )\right) ^{1/p_{0}(\cdot )
\big\|_{p_{0}(\cdot )}$ is bounded b
\begin{equation*}
c\Big\|\Big(\frac{g(\cdot )}{\big\|\lambda \big\|_{f_{p(\cdot ),p(\cdot
)}^{\alpha (\cdot )}}}\Big)^{p(\cdot )/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot
)}+c.
\end{equation*
This term is bounded, since
\begin{equation*}
\int_{\mathbb{R}^{n}}\Big(\frac{g(x)}{\big\|\lambda \big\|_{f_{p(\cdot
),p(\cdot )}^{\alpha (\cdot )}}}\Big)^{p(x)}dx\leq 1.
\end{equation*
\textbf{Estimation of }$\big\|\lambda ^{1}\big\|_{f_{p_{1}(\cdot
),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}}$. Replacing $\alpha _{0}$, $p_{0}$
and $u$ by $\alpha _{1}$, $p_{1}$ and $v$, respectively and this leads to
the desired inequality. \ $\square $
Notice that this theorem can be generalized to the case
0<p_{0}^{+},p_{1}^{+}<\infty $.
\begin{theorem}
\label{calderon-prod2}Let $0<\theta <1$ and $1\leq q_{0},q_{1}<\infty $. Let
$p_{0}\in C^{\log }$ with $1\leq p_{0}^{+}<\infty $ and $\alpha _{0},\alpha
_{1}\in C_{\mathrm{loc}}^{\log }$. We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )},\text{ }\frac{1}{q}:
\frac{1-\theta }{q_{0}}+\frac{\theta }{q_{1}}\text{\ \ and\ \ }\alpha (\cdot
):=(1-\theta )\alpha _{0}(\cdot )+\theta \alpha _{1}(\cdot ).
\end{equation*
The
\begin{equation*}
\big(f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )}\big)^{1-\theta }\big
f_{\infty ,q_{1}}^{\alpha _{1}(\cdot )}\big)^{\theta }=f_{p(\cdot
),q}^{\alpha (\cdot )}
\end{equation*
holds in the sense of equivalent norms.
\end{theorem}
\textbf{Proof.}
\textit{Step 1.} We shall prov
\begin{equation*}
\big(f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )}\big)^{1-\theta }\big
f_{\infty ,q_{1}}^{\alpha _{1}(\cdot )}\big)^{\theta }\hookrightarrow
f_{p(\cdot ),q}^{\alpha (\cdot )}.
\end{equation*
We suppose, that sequences $\lambda :=(\lambda _{j,m})_{j,m}$, $\lambda
^{i}:=(\lambda _{j,m}^{i})_{j,m},i=0,1$, are given and tha
\begin{equation*}
|\lambda _{j,m}|\leq |\lambda _{j,m}^{0}|^{1-\theta }|\lambda
_{j,m}^{1}|^{\theta }
\end{equation*
holds for all $j\in \mathbb{N}_{0}$ and $m\in \mathbb{Z}^{n}$. In $\mathrm
\eqref{notation1}}$ we replace $p(x)$ by $q$ and $\chi _{j,m}$ by $\chi
_{E_{Q_{j,m}}}$, with $E_{Q_{j,m}}\subset Q_{j,m}$ and
|E_{Q_{j,m}}|>|Q_{j,m}|/2$, we obtain $\mathrm{\eqref{Second-est}}$ with
q_{0}$ and $q_{1}$ in place of $p_{0}(x)$ and $p_{0}(x)$, respectively, and
\chi _{E_{Q_{j,m}}}$ in place of $\chi _{j,m}$. Estimate the second factor
by its $L^{\infty }$-norm and using Propositions \ref{prop1} we get \ $\big\
\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}\leq \big\|\lambda ^{0
\big\|_{f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )}}^{1-\theta }\big\
\lambda ^{1}\big\|_{f_{\infty ,q_{1}}^{\alpha _{1}(\cdot )}}^{\theta }$.
\textit{Step 2.} We prov
\begin{equation*}
f_{p(\cdot ),q}^{\alpha (\cdot )}\hookrightarrow \big(f_{p_{0}(\cdot
),q_{0}}^{\alpha _{0}(\cdot )}\big)^{1-\theta }\big(f_{\infty
,q_{1}}^{\alpha _{1}(\cdot )}\big)^{\theta }.
\end{equation*
Let the sequence $\lambda \in f_{p(\cdot ),q}^{\alpha (\cdot )}$ be given
wit
\begin{equation*}
\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}\neq 0.
\end{equation*
Let $\delta =-\frac{q}{q_{1}}$ and $\gamma (\cdot ):=\frac{p(\cdot )}
p_{0}(\cdot )}-\frac{q}{q_{0}}$. Observe that $\gamma $ is a constant
function. We follow ideas of the proof of Theorem 8.2 in Frazier and Jawerth
\cite{FJ90}. Set
\begin{equation*}
A_{\ell ,\gamma }:=\Big\{x\in \mathbb{R}^{n}:\Big(\frac{g(x)}{\big\|\lambda
\big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}}\Big)^{\gamma }>2^{\ell }\Big\},
\end{equation*
with $\ell \in \mathbb{Z}$. Obviously $A_{\ell +1,\gamma }\subset A_{\ell
,\gamma }$ for any $\ell \in \mathbb{Z}$. Now we introduce a (partial)
decomposition of $\mathbb{N}_{0}\times \mathbb{Z}^{n}$ by takin
\begin{equation*}
C_{\ell }^{\gamma }:=\{(j,m):|Q_{j,m}\cap A_{\ell ,\gamma }|>\frac{|Q_{j,m}
}{2}\text{ \ and \ }|Q_{j,m}\cap A_{\ell +1,\gamma }|\leq \frac{|Q_{j,m}|}{2
\},\quad \ell \in \mathbb{Z}.
\end{equation*
The sets $C_{\ell }^{\gamma }$ are pairwise disjoint, i.e., $C_{\ell
}^{\gamma }\cap C_{v}^{\gamma }=\emptyset $ if $\ell \neq v$. Let us prove
that $\lambda _{j,m}=0$ holds for all tuples $(j,m)\notin \cup _{\ell
}C_{\ell }^{\gamma }$. Let us consider one such tuple $(j_{0},m_{0})$ and
let us choose $\ell _{0}\in \mathbb{Z}$ arbitrary. First suppose that
(j_{0},m_{0})\notin C_{\ell _{0}}^{\gamma }$, then eithe
\begin{equation}
|Q_{j_{0},m_{0}}\cap A_{\ell _{0},\gamma }|\leq \frac{|Q_{j_{0},m_{0}}|}{2
\text{\quad or\quad }|Q_{j_{0},m_{0}}\cap A_{\ell _{0}+1,\gamma }|>\frac
|Q_{j_{0},m_{0}}|}{2}. \label{Cond11}
\end{equation
Let us assume for the moment that the second condition is satisfied. By
induction on $\ell $ it follow
\begin{equation}
|Q_{j_{0},m_{0}}\cap A_{\ell +1,\gamma }|>\frac{|Q_{j_{0},m_{0}}|}{2}\quad
\text{for all\quad }\ell \geq \ell _{0}. \label{Cond21}
\end{equation
Let $D:=\cap _{\ell \geq \ell _{0}}Q_{j_{0},m_{0}}\cap A_{\ell +1,\gamma }$.
The family $\{Q_{j_{0},m_{0}}\cap A_{\ell ,\gamma }\}_{\ell }$ is a
decreasing family of sets, i.e., $Q_{j_{0},m_{0}}\cap A_{\ell +1,\gamma
}\subset Q_{j_{0},m_{0}}\cap A_{\ell ,\gamma }$. Therefore, in view of
\mathrm{\eqref{Cond21}}$, the measure of the set $D$ is larger than or equal
to $\frac{|Q_{j_{0},m_{0}}|}{2}$. Hence $\varrho _{p(\cdot )}\big(\frac{g}
\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}}\big)$ is greater
than or equal t
\begin{equation*}
\int_{\mathbb{R}^{n}}\Big(\frac{g(x)}{\big\|\lambda \big\|_{f_{p(\cdot
),q}^{\alpha (\cdot )}}}\Big)^{p(x)}\chi _{Q_{j_{0},m_{0}}\cap A_{\ell
+1,\gamma }}(x)dx\geq 2^{\ell \frac{p^{-}}{\gamma }}|D|,\quad \ell \geq \max
(\ell _{0},0).
\end{equation*
Now $\varrho _{p(\cdot )}\big(\frac{g}{\big\|\lambda \big\|_{f_{p(\cdot
),q}^{\alpha (\cdot )}}}\big)$ is finite, since $\lambda \in f_{p(\cdot
),q}^{\alpha (\cdot )}$, letting $\ell $ tend to infinity and using that
|D|\geq \frac{|Q_{j_{0},m_{0}}|}{2}$, we get a contradiction. Hence, we have
to turn in $\mathrm{\eqref{Cond11}}$ to the situation where the first
condition is satisfied. We claim
\begin{equation*}
|Q_{j_{0},m_{0}}\cap A_{\ell ,\gamma }|\leq \frac{|Q_{j_{0},m_{0}}|}{2}\quad
\text{for all\ }\ell \in \mathbb{Z}.
\end{equation*
Obviously this yield
\begin{equation}
|Q_{j_{0},m_{0}}\cap A_{\ell ,\gamma }^{c}|>\frac{|Q_{j_{0},m_{0}}|}{2}\quad
\text{for all\ }\ell \in \mathbb{Z}, \label{Cond2.11}
\end{equation
again\ this follows by induction on $\ell $ using $(j_{0},m_{0})\notin \cup
_{\ell }C_{\ell }^{\gamma }$. Set $E=\cap _{\ell \geq \max (0,-\ell
_{0})}Q_{j_{0},m_{0}}\cap A_{-\ell ,\gamma }^{c}=\cap _{\ell \geq \max
(0,-\ell _{0})}h_{\ell }$. The family $\{h_{\ell }\}_{\ell }$ is a
decreasing family of sets, i.e., $h_{\ell +1}\subset h_{\ell }$. Therefore,
in view of $\mathrm{\eqref{Cond2.11}}$, the measure of the set $E$ is larger
than or equal to $\frac{|Q_{j_{0},m_{0}}|}{2}$. By selecting a point $x\in E$
we obtai
\begin{equation*}
2^{j_{0}\alpha (x)}|\lambda _{j_{0},m_{0}}|\leq g(x)\leq \big\|\lambda \big\
_{f_{p(\cdot ),q}^{\alpha (\cdot )}}2^{-\ell /\gamma }.
\end{equation*
Now, for $\ell $ tending to $+\infty $ the claim, namely $\lambda
_{j_{0},m_{0}}=0$, follows. We put, as in the proof of Theorem \re
{calderon-prod1}
\begin{equation*}
u(\cdot ):=q\theta \big(\tfrac{\alpha _{1}(\cdot )}{q_{0}}-\tfrac{\alpha
_{0}(\cdot )}{q_{1}}\big)+\tfrac{n}{2}\big(\tfrac{q}{q_{0}}-1\big)
\end{equation*
and
\begin{equation*}
v(\cdot ):=q(1-\theta )\big(\tfrac{\alpha _{0}(\cdot )}{q_{1}}-\tfrac{\alpha
_{1}(\cdot )}{q_{0}}\big)+\tfrac{n}{2}\big(\tfrac{q}{q_{1}}-1\big).
\end{equation*
If $(j,m)\notin \cup _{\ell \in \mathbb{Z}}C_{\ell }^{\gamma }$, then we
define $\lambda _{j,m}^{0}=\lambda _{j,m}^{1}=0$. Let $(j,m)\in C_{\ell
}^{\gamma }$. We set
\begin{equation*}
\lambda _{j,m}^{0}:=\lambda _{j,m,u,q_{0}}^{0,\frac{\delta }{\gamma
}:=2^{\ell +ju(x_{j,m})}\Big(\tfrac{|\lambda _{j,m}|}{\big\|\lambda \big\
_{f_{p(\cdot ),q}^{\alpha (\cdot )}}}\Big)^{q/q_{0}}.
\end{equation*
Also, se
\begin{equation*}
\lambda _{j,m}^{1}:=\lambda _{j,m,v,q_{1}}^{1,\frac{\delta }{\gamma
}:=2^{\ell \frac{\delta }{\gamma }+jv(x_{j,m})}\Big(\tfrac{|\lambda _{j,m}|}
\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}}\Big)^{q/q_{1}}.
\end{equation*
Observe tha
\begin{eqnarray*}
|\lambda _{j,m}| &\leq &\big\|\lambda \big\|_{f_{p(\cdot ),q(\cdot
)}^{\alpha (\cdot )}}2^{-\ell (1-\theta +\frac{\delta }{\gamma }\theta
)}2^{-j(u(x_{j,m})(1-\theta )+v(x_{j,m})\theta )}\big(\lambda _{j,m}^{0}\big
^{1-\theta }\big(\lambda _{j,m}^{1}\big)^{\theta } \\
&=&\big\|\lambda \big\|_{f_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}}\big
\lambda _{j,m}^{0}\big)^{1-\theta }\big(\lambda _{j,m}^{1}\big)^{\theta },
\end{eqnarray*
which holds now for all pairs $(j,m)$. As in the proof of Theorem \re
{calderon-prod1}, We will prove the following two inequalitie
\begin{equation*}
\big\|\lambda ^{0}\big\|_{f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot
)}}\lesssim 1\text{\quad and\quad }\big\|\lambda ^{1}\big\|_{f_{\infty
,q_{1}}^{\alpha _{1}(\cdot )}}\lesssim 1.
\end{equation*
\textbf{Estimation of }$\big\|\lambda ^{0}\big\|_{f_{p_{0}(\cdot
),q_{0}}^{\alpha _{0}(\cdot )}}$. We writ
\begin{equation*}
\sum_{j=0}^{\infty }\sum\limits_{m\in \mathbb{Z}^{n}}\Big(2^{j(\alpha
_{0}(\cdot )+\frac{n}{2})}|\lambda _{j,m}^{0}|\chi _{j,m}\Big
^{q_{0}}=\sum_{\ell =-\infty }^{\infty }\sum\limits_{(j,m)\in C_{\ell
}^{\gamma }}\cdot \cdot \cdot =:I.
\end{equation*
We use the local log-H\"{o}lder continuity of $\alpha _{0}$ and $\alpha _{1}$
to show tha
\begin{equation*}
2^{ju(x_{j,m})}\leq c\text{ }2^{ju(t)}\quad \text{and}\quad 2^{j\alpha
_{0}(x)}\leq c\text{ }2^{j\alpha _{0}(t)}\text{, \ \ }x,t\in Q_{j,m},
\end{equation*
where $c>0$ is independent of $\ell $ and $j$. Therefore,
\begin{eqnarray*}
2^{j(\alpha _{0}(x)+\frac{n}{2})+ju(x_{j,m})} &\lesssim &\frac{1}
|Q_{j,m}\cap A_{\ell ,\gamma }|}\int_{Q_{j,m}\cap A_{\ell ,\gamma
}}2^{j(\alpha _{0}(t)+\frac{n}{2})+ju(t)}dt \\
&\lesssim &\frac{1}{|Q_{j,m}|}\int_{Q_{j,m}\cap A_{\ell ,\gamma
}}2^{j(\alpha _{0}(t)+\frac{n}{2})+ju(t)}dt \\
&\lesssim &\eta _{j,h}\ast 2^{j(\alpha _{0}(\cdot )+\frac{n}{2})+ju(\cdot
)}\chi _{Q_{j,m}\cap A_{\ell ,\gamma }}(x),\text{ \ \ }j,m\in C_{\ell
}^{\gamma },
\end{eqnarray*
where $h>n$ and the implicit positive constant not depending on $x$, $\ell ,m
$ and $j$. Hence $\big\|\left( I^{q_{0}}(\cdot )\right) ^{1/q_{0}}\big\
_{p_{0}(\cdot )}$ is bounded b
\begin{equation*}
c\Big\|\Big(\sum_{\ell =-\infty }^{\infty }\sum\limits_{(j,m)\in C_{\ell
}^{\gamma }}\left( \eta _{j,h}\ast \digamma _{j,\ell ,m}\right) ^{q_{0}}\Big
^{1/q_{0}}\Big\|_{p_{0}(\cdot )},
\end{equation*
where
\begin{equation*}
\digamma _{j,\ell ,m}=2^{j(\alpha _{0}(\cdot )+\frac{n}{2})+ju(\cdot )+\ell
\Big(\tfrac{|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),q(\cdot
)}^{\alpha (\cdot )}}}\Big)^{\frac{q}{q_{0}}}\chi _{Q_{j,m}\cap A_{\ell
,\gamma }}.
\end{equation*
Applying Lemma\ \ref{DHRlemma} to estimate the last norm b
\begin{eqnarray*}
&&c\Big\|\Big(\sum_{\ell =-\infty }^{\infty }\sum\limits_{(j,m)\in C_{\ell
}^{\gamma }}\digamma _{j,\ell ,m}^{q_{0}}\Big)^{1/q_{0}}\Big\|_{p_{0}(\cdot
)} \\
&=&c\Big\|\Big(\sum_{\ell =-\infty }^{\infty }\sum\limits_{(j,m)\in C_{\ell
}^{\gamma }}2^{j(\alpha (\cdot )+\frac{n}{2})q+\ell q_{0}}\Big(\tfrac
|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}
\Big)^{q}\chi _{Q_{j,m}\cap A_{\ell ,\gamma }}\Big)^{1/q_{0}}\Big\
_{p_{0}(\cdot )},
\end{eqnarray*
since $u(\cdot )+\alpha _{0}(\cdot )=\alpha (\cdot )\frac{q}{q_{0}}+\frac{n}
2}\Big(\frac{q}{q_{0}}-1\Big)$. Observe tha
\begin{equation*}
2^{\ell }\leq \Big(\frac{g(x)}{\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha
(\cdot )}}}\Big)^{\gamma }
\end{equation*
for any $x\in Q_{j,m}\cap A_{\ell ,\gamma }$ and since $\gamma +\frac{q}
q_{0}}=\frac{p(\cdot )}{p_{0}(\cdot )}$, then $\big\|\left( I^{q_{0}}(\cdot
)\right) ^{1/q_{0}}\big\|_{p_{0}(\cdot )}$ is bounded b
\begin{eqnarray*}
&&c\Big\|\Big(\frac{g}{\big\|\lambda \big\|_{f_{p(\cdot ),q(\cdot )}^{\alpha
(\cdot )}}}\Big)^{\gamma }\Big(\frac{g}{\big\|\lambda \big\|_{f_{p(\cdot
),q(\cdot )}^{\alpha (\cdot )}}}\Big)^{q/q_{0}}\Big\|_{p_{0}(\cdot )} \\
&=&c\Big\|\Big(\frac{g}{\big\|\lambda \big\|_{f_{p(\cdot ),q(\cdot
)}^{\alpha (\cdot )}}}\Big)^{p(\cdot )/p_{0}(\cdot )}\Big\|_{p_{0}(\cdot )}.
\end{eqnarray*
Obviously, the last norm is less than or equal to one.
\textbf{Estimation of }$\big\|\lambda ^{1}\big\|_{f_{\infty ,q_{1}}^{\alpha
_{1}(\cdot )}}$. By Proposition \ref{prop1} with $E_{Q_{j,m}}^{\ell
}=Q_{j,m}\cap A_{\ell +1,\gamma }^{c}$,
\begin{equation*}
\big\|\lambda ^{1}\big\|_{f_{\infty ,q_{1}}^{\alpha _{1}(\cdot )}}\lesssim
\Big\|\Big(\sum_{\ell =-\infty }^{\infty }\sum\limits_{(j,m)\in C_{\ell
}^{\gamma }}2^{j(\alpha _{1}\left( \cdot \right) +\frac{n}{2})q_{1}}(\lambda
_{j,m}^{1})^{q_{1}}\chi _{E_{Q_{j,m}}^{\ell }}\Big)^{1/q_{1}}\Big\|_{\infty
}.
\end{equation*
Observe tha
\begin{equation*}
\lambda _{j,m}^{1}\lesssim 2^{\ell \frac{\delta }{\gamma }+jv(x)}\Big(\frac
|\lambda _{j,m}|}{\big\|\lambda \big\|f_{p(\cdot ),q}^{\alpha (\cdot )}}\Big
^{\frac{q}{q_{1}}}\leq 2^{jv(x)}\Big(\frac{g(x)}{\big\|\lambda \big\
_{f_{p(\cdot ),q}^{\alpha (\cdot )}}}\Big)^{-\frac{q}{q_{1}}}\Big(\frac
|\lambda _{j,m}|}{\big\|\lambda \big\|_{f_{p(\cdot ),q}^{\alpha (\cdot )}}
\Big)^{\frac{q}{q_{1}}}
\end{equation*
for any $x\in E_{Q_{j,m}}^{\ell }$ and any $(j,m)\in C_{\ell }^{\gamma }$.
Therefore, $\big\|\lambda ^{1}\big\|_{f_{\infty ,q_{1}}^{\alpha _{1}(\cdot
)}}\lesssim 1$.\ Hence, we complete the proof. $\ \ \square $
Notice that this theorem for $\alpha _{0}=\alpha _{1}=0$ and $q_{0}=q_{1}=2$
was proved by T. Kopaliani, \cite[Theorem 3.1]{Ko09}. This theorem can be
generalized to the case $0<p_{0}^{+},p_{1}^{+},q_{0},q_{1}<\infty $.
Suppose \ that $X_{0}$ and $X_{1}$ are Banach lattices on measure space
\left( \mathcal{M},\mu \right) $, and let $X=X_{0}^{1-\theta }\times
X_{1}^{\theta }$ for some $0<\theta <1$. Suppose that $X$ hus the propert
\begin{equation}
f\in X,\text{ \ }\left\vert f_{n}\left( x\right) \right\vert \leq \left\vert
f\left( x\right) \right\vert \text{, }\mu \text{-a.e., \ and \
\lim_{n\rightarrow \infty }f_{n}=f,\mu \text{-}a.e.\Longrightarrow
\lim_{n\rightarrow \infty }\left\Vert f_{n}\right\Vert _{X}=\left\Vert
f\right\Vert _{X}. \label{Ca-property}
\end{equation
Calder\'{o}n \cite[p. 125]{Ca64} then shows that $X_{0}^{1-\theta }\times
X_{1}^{\theta }=[X_{0},X_{1}]_{\theta }$.
Now we turn to the complex interpolation of the distribution spaces\
F_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}$.
\begin{theorem}
\label{Inter1}Let $0<\theta <1$. Let $p_{0},p_{1}\in C^{\log }$ with $1\leq
p_{0}^{+},p_{1}^{+}<\infty $ and $\alpha _{0},\alpha _{1}\in C_{\mathrm{loc
}^{\log }$. We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )}+\frac{\theta }
p_{1}(\cdot )}\text{ \ and \ }\alpha (\cdot ):=(1-\theta )\alpha _{0}(\cdot
)+\theta \alpha _{1}(\cdot ).
\end{equation*
The
\begin{equation}
\lbrack f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot )},f_{p_{1}(\cdot
),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}]_{\theta }=f_{p(\cdot ),p(\cdot
)}^{\alpha (\cdot )} \label{Int1}
\end{equation
an
\begin{equation}
\lbrack F_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot )},F_{p_{1}(\cdot
),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}]_{\theta }=F_{p(\cdot ),p(\cdot
)}^{\alpha (\cdot )} \label{Int2}
\end{equation
holds in the sense of equivalent norms.
\end{theorem}
\textbf{Proof. }Since $f_{p_{0}(\cdot ),p_{0}(\cdot )}^{\alpha _{0}(\cdot )}$
and $f_{p_{1}(\cdot ),p_{1}(\cdot )}^{\alpha _{1}(\cdot )}$ are Banach
lattices. Then, it suffices to use Calder\'{o}n's result to $X=f_{p(\cdot
),p(\cdot )}^{\alpha (\cdot )}$\ where the property $\mathrm
\eqref{Ca-property}}$ follows easily from the dominated convergence theorem.
Hence Theorem \ref{calderon-prod1} yields $\mathrm{\eqref{Int1}}$. Now
\mathrm{\eqref{Int2}}$ follows from $\mathrm{\eqref{Int1}}$ and Theorem \re
{phi-tran}. $\ \ \square $
Similarly we formulate the main statement on complex interpolation of
variable Triebel-Lizorkin spaces\ $F_{p(\cdot ),q}^{\alpha (\cdot )}$.
\begin{theorem}
\label{Inter2}Let $0<\theta <1$ and $1\leq q_{0},q_{1}<\infty $. Let
p_{0}\in C^{\log }$ with $1\leq p_{0}^{+}<\infty $ and $\alpha _{0},\alpha
_{1}\in C_{\mathrm{loc}}^{\log }$. We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )},\frac{1}{q}:=\frac
1-\theta }{q_{0}}+\frac{\theta }{q_{1}}\text{ and }\alpha (\cdot
):=(1-\theta )\alpha _{0}(\cdot )+\theta \alpha _{1}(\cdot ).
\end{equation*
The
\begin{equation*}
\lbrack f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )},f_{\infty
,q_{1}}^{\alpha _{1}(\cdot )}]_{\theta }=f_{p(\cdot ),q}^{\alpha (\cdot )}
\end{equation*
an
\begin{equation*}
\lbrack F_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )},F_{\infty
,q_{1}}^{\alpha _{1}(\cdot )}]_{\theta }=F_{p(\cdot ),q}^{\alpha (\cdot )}
\end{equation*
holds in the sense of equivalent norms.
\end{theorem}
Using a combination of the arguments used in the proof of Theorems \re
{Inter1} and \ref{Inter2} and the fact that $\frac{\gamma (\cdot )}{\delta
(\cdot )}$\ is a constant function with negative values, we arrive at the
following complex interpolation of variable Triebel-Lizorkin spaces.
\begin{theorem}
\label{calderon-prod11 copy(1)}Let $0<\theta <1$. Let
p_{0},p_{1},q_{0},q_{1}\in C^{\log }$ with $1\leq
p_{0}^{+},q_{0}^{+},p_{1}^{+},q_{1}^{+}<\infty $\ and $\alpha _{0},\alpha
_{1}\in C_{\mathrm{loc}}^{\log }$. We pu
\begin{equation*}
\frac{1}{p(\cdot )}:=\frac{1-\theta }{p_{0}(\cdot )}+\frac{\theta }
p_{1}(\cdot )},\text{ }\frac{1}{q(\cdot )}:=\frac{1-\theta }{q_{0}(\cdot )}
\frac{\theta }{q_{1}(\cdot )},\text{ }\alpha (\cdot ):=(1-\theta )\alpha
_{0}(\cdot )+\theta \alpha _{1}(\cdot )
\end{equation*
an
\begin{equation*}
\gamma (\cdot ):=\frac{p(\cdot )}{p_{0}(\cdot )}-\frac{q(\cdot )}
q_{0}(\cdot )}.
\end{equation*
$\mathrm{(i)}$ We suppose that $\gamma (x)=0$ for any $x\in \mathbb{R}^{n}$.
The
\begin{equation*}
\lbrack f_{p_{0}(\cdot ),q_{0}(\cdot )}^{\alpha _{0}(\cdot )},f_{p_{1}(\cdot
),q_{1}(\cdot )}^{\alpha _{1}(\cdot )}]_{\theta }=f_{p(\cdot ),q(\cdot
)}^{\alpha (\cdot )}
\end{equation*
an
\begin{equation*}
\lbrack F_{p_{0}(\cdot ),q_{0}(\cdot )}^{\alpha _{0}(\cdot )},F_{p_{1}(\cdot
),q_{1}(\cdot )}^{\alpha _{1}(\cdot )}]_{\theta }=F_{p(\cdot ),q(\cdot
)}^{\alpha (\cdot )}
\end{equation*
holds in the sense of equivalent norms.
$\mathrm{(ii)}$ We suppose that $q_{0}$ and $q_{1}$ are constants. In
addition, we assume that $\gamma (x)\neq 0$ for any $x\in \mathbb{R}^{n}$.
The
\begin{equation*}
\lbrack f_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )},f_{p_{1}(\cdot
),q_{1}}^{\alpha _{1}(\cdot )}]_{\theta }=f_{p(\cdot ),q}^{\alpha (\cdot )}
\end{equation*
an
\begin{equation*}
\lbrack F_{p_{0}(\cdot ),q_{0}}^{\alpha _{0}(\cdot )},F_{p_{1}(\cdot
),q_{1}}^{\alpha _{1}(\cdot )}]_{\theta }=F_{p(\cdot ),q}^{\alpha (\cdot )}
\end{equation*
holds in the sense of equivalent norms.
\end{theorem}
\begin{remark}
It should be mentioned that if $\alpha _{0},\alpha _{1},q_{0}$ and $q_{1}$
are constants,\ Theorem \ref{Inter-rest} is more general than what has been
given here.
\end{remark}
\textbf{Acknowledgment}
A great deal of this work has been carried out during the visit of the
author in Jena, Germany. I wish to thank Professor Winfried Sickel for the
hospitality and for many valuable discussions and suggestions\ about this
work.
|
2,877,628,089,459 | arxiv | \section{Introduction}
\label{sec:introduction}
\indent Disk-resolved imaging is a powerful tool to investigate the origin and collisional history
of asteroids. This has been remarkably illustrated by fly-by and rendezvous space missions
\citep{Belton:1992jq, Belton:1996ka, Zuber:2000km,
Fujiwara:2006ca, Sierks:2011il, Russell:2012ep, Russell:2016dj}, as well as observations from the Earth
(e.g., \citealt{Carry:2008ip, Carry:2010je, Merline:2013fs}).
In the late nineties, observations of (4)~Vesta with the Hubble
Space Telescope (HST) led to the discovery
of the now-called ``Rheasilvia basin'' and allowed for establishment of the origin of
the Vestoids and HED meteorites found on Earth \citep{ThomasPC:1997cu, Binzel:1997bj}.
Specifically, it was demonstrated that the basin-forming event on Vesta excavated enough material
to account for the family of small asteroids with spectral properties similar to Vesta.
HST observations thus confirmed the origin of these bodies as fragments from Vesta, as
previously suspected based on spectroscopic measurements \citep{Binzel:1993ju}.
Recently, the Rheasilvia basin was revealed in much greater detail by the Dawn mission, which
unveiled two overlapping giant impact features \citep{Schenk:2012ina}.
\indent In the 2000's, a new generation of ground-based imagers with
high-angular-resolution capability, such as NIRC2
\citep{Wizinowich:2000ua, vanDam:2004jd} on the W. M. Keck II
telescope and NACO \citep{Lenzen:2003iu, Rousset:2003hh} on the
European Southern Observatory (ESO) Very Large Telescope (VLT),
made disk-resolved imaging achievable from the ground for
a larger number of medium-sized ($\sim$100-200-km in diameter) asteroids.
In turn, these observations triggered the development of methods for
modeling the tridimensional shape of these objects by combining the images
with optical light curves
(see, e.g., \citealt{Carry:2010bo, Carry:2012jn, Kaasalainen:2011tk, Viikinkoski:2015jha}).
These models were subsequently validated by in-situ measurements performed by the
ESA Rosetta mission during the fly-by of asteroid (21)~Lutetia
\citep{Sierks:2011il, Carry:2010je, Carry:2012jn, ORourke:2012ch}.
More recently, the newly commissioned VLT/Spectro-Polarimetric High-contrast Exoplanet
Research instrument (SPHERE) and its
very high performance adaptive optics system
\citep{Beuzit:2008gt} demonstrated its ability to reveal in even greater detail
the surface of medium-sized asteroids by resolving their largest (D$>$30km)
craters \citep{Viikinkoski:2015jk, Hanus:2017ed}. This remarkable achievement
opens the prospect of a new era of exploration of the asteroid belt and its collisional history.
\indent Here, we use VLT/SPHERE to investigate the shape and topography of
asteroid (6)~Hebe, a large main-belt asteroid \citep[D$\sim$180-200~km; e.g.,][]
{Tedesco:2004um,Masiero:2011jc}
that has long received particular attention from the community
of asteroid spectroscopists, meteoricists, and dynamicists.
Indeed, Hebe's spectral properties and close proximity to orbital resonances
in the asteroid belt make it a possible
main source of ordinary H~chondrites (i.e., $\sim$34\% of
the meteorite falls, \citealt{Hutchison:2004uk, Farinella:1993fp,
Migliorini:1997hp, Gaffey:1998ck, Bottke:2010vm}).
It was further proposed that Hebe could be the parent
body of an ancient asteroid family \citep{Gaffey:2013vf}. The idea of H chondrites mainly
originating from Hebe, however, was recently
weakened by the discovery of a large number of asteroids
(including several asteroid families) with similar spectral
properties \citep[hence composition,][]{Vernazza:2014uqa}.
Here, we challenge this hypothesis by studying the three-dimensional shape and topography
of Hebe derived from disk-resolved observations.
We observed Hebe throughout its rotation in
order to derive its shape, and to characterize the largest craters at its surface.
When combined with previous adaptive-optics (AO) images and light curves (both from the
literature and from recent optical observations by our team),
these new observations allow us to derive
a reliable shape model and an estimate of Hebe's density based on
its astrometric mass (i.e., the mass derived from the study of planetary ephemeris and orbital deflections).
Finally, we analyse Hebe's topography by means of
an elevation map and discuss the implications for the origin of H chondrites.
\section{Observations and data pre-processing}
\indent We observed (6)~Hebe close to its opposition date while it
was orientated ``equator-on'' (from its spin solution derived
below), that is, with an ideal viewing geometry
exposing its whole surface as it rotated. Observations were acquired at four
different epochs between December 8-12, 2014, such that the variation
of the sub-Earth point longitude was 90$\pm$30$\degr$ between each epoch.
\indent Observations were performed with the recently commissioned
second-generation SPHERE instrument, mounted at the European
Southern Observatory (ESO) Very Large Telescope (VLT)
\citep{Fusco:2006gm,Beuzit:2008gt}, during the science verification of
the instrument\footnote{Observations obtained under ESO programme ID
60.A-9379 (P.I. C. Dumas)}. We used IRDIS broad-band classical
imaging in Y (filter central wavelength=1.043\,$\mu$m, width=0.140\,$\mu$m)
in the pupil-tracking mode, where the pupil remains
fixed while the field orientation varies during the observations,
to achieve the best point-spread function (PSF) stability.
Each observational sequence
consisted in a series of ten images with 2\,s exposure time during
which Hebe was used as a natural guide star for AO
corrections. Observations were performed under average seeing
conditions (0.9-1.1\arcsec) and clear sky transparency, at an airmass of
$\sim$1.1.
\indent Sky backgrounds were acquired along our observations for
data-reduction purposes. At the end of each sequence, we observed the
nearby star HD~26086 under the exact same AO configuration as the
asteroid to estimate the instrument PSF for deconvolution purposes.
Finally, standard
calibrations, which include
detector flat-fields and darks, were acquired in the morning as part of the
instrument calibration plan.
\indent Data pre-processing of the IRDIS data made use of the preliminary
release (v0.14.0-2) of the SPHERE data reduction and handling (DRH)
software \citep{Pavlov:2008di}, as well as additional tools written
in the Interactive Data Language (IDL), in order to perform
background subtraction, flat-fielding and bad-pixel correction. The
pre-processed images were then aligned one with respect to the others
using the IDL ML\_SHIFTFINDER maximum likelihood function,
and averaged to maximise the signal to noise
ratio of the asteroid. Finally, the optimal angular resolution of
each image ($\lambda$/D=0.026", corresponding to a projected distance of 22\,km)
was restored with Mistral, a myopic deconvolution
algorithm optimised for images with sharp boundaries
\citep{Fusco:2002iz, Mugnier:2004kf}, using the stellar PSF acquired
on the same night as our asteroid data.\\
\begin{table*}[!t]
\begin{center}
\caption{
Date, mid-observing time (UTC),
heliocentric distance ($\Delta$) and range to observer ($r$),
phase angle ($\alpha$), apparent size ($\Theta$), longitude ($\lambda$) and latitude ($\beta$)
of the subsolar and subobserver points (SSP, SEP).
PIs of these observations were
$^1$J.-L. Margot,
$^2$W. J. Merline,
$^3$W. M. Keck engineering team,
$^4$F. Marchis,
$^5$B. Carry,
and
$^6$C. Dumas.
\label{tab:obscondition}
}
\begin{tabular}{rcclrrrrrrrr}
\hline\hline
& Date & UTC & \multicolumn{1}{c}{Instrument} & \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$r$} & \multicolumn{1}{c}{$\alpha$} &
\multicolumn{1}{c}{$\Theta$} &
\multicolumn{1}{c}{SEP$_\lambda$} &
\multicolumn{1}{c}{SEP$_\beta$} &
\multicolumn{1}{c}{SSP$_\lambda$} &
\multicolumn{1}{c}{SSP$_\beta$} \\
&&&& \multicolumn{1}{c}{(AU)} & \multicolumn{1}{c}{(AU)} &
\multicolumn{1}{c}{(\degr)} & \multicolumn{1}{c}{(\arcsec)} &
\multicolumn{1}{c}{(\degr)} &\multicolumn{1}{c}{(\degr)} &\multicolumn{1}{c}{(\degr)} &\multicolumn{1}{c}{(\degr)} \\
\hline
1&2002-05-07&14:08:54& Keck/NIRC2$^1$ & 2.52& 1.88& 20.5& 0.131& 66.1& -34.4& 53.3& -17.4 \\
2&2002-05-08&13:55:01& Keck/NIRC2$^1$ & 2.52& 1.86& 20.4& 0.119& 329.8& -34.5& 317.2& -17.5 \\
3&2002-09-27&06:29:51& Keck/NIRC2$^2$ & 2.21& 1.91& 27.0& 0.098& 162.5& -19.4& 187.2& -35.4 \\
4&2007-12-15&14:15:39& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.149& 14.2& 32.9& 356.2& 19.6 \\
5&2007-12-15&14:30:31& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.145& 1.9& 32.9& 343.9& 19.6 \\
6&2007-12-15&14:44:49& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.145& 350.1& 32.9& 332.1& 19.6 \\
7&2007-12-15&15:00:54& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.138& 336.8& 32.9& 318.9& 19.6 \\
8&2007-12-15&15:27:39& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.143& 314.8& 32.9& 296.8& 19.6 \\
9&2007-12-15&16:26:58& Keck/NIRC2$^3$ & 2.47& 1.86& 20.8& 0.151& 265.9& 32.9& 247.9& 19.6 \\
10&2009-06-07&10:52:24& Keck/NIRC2$^2$ & 2.81& 2.01& 15.1& 0.129& 43.1& 8.0& 57.9& 4.9 \\
11&2010-06-28&13:08:00& Keck/NIRC2$^4$ & 2.06& 1.62& 28.9& 0.168& 258.8& -39.3& 221.2& -39.0 \\
12&2010-08-26&12:47:10& Keck/NIRC2$^3$ & 1.98& 1.05& 16.1& 0.260& 48.5& -27.4& 30.6& -31.2 \\
13&2010-08-26&13:04:26& Keck/NIRC2$^3$ & 1.98& 1.05& 16.1& 0.260& 34.3& -27.4& 16.3& -31.2 \\
14&2010-08-26&13:59:47& Keck/NIRC2$^3$ & 1.98& 1.05& 16.1& 0.265& 348.6& -27.4& 330.7& -31.2 \\
15&2010-08-26&14:38:00& Keck/NIRC2$^3$ & 1.98& 1.05& 16.1& 0.270& 317.1& -27.4& 299.2& -31.2 \\
16&2010-11-29&07:10:28& Keck/NIRC2$^4$ & 1.94& 1.39& 28.9& 0.189& 160.9& -22.9& 191.5& -18.5 \\
17&2010-12-13&01:18:16& VLT/NACO$^5$ & 1.94& 1.52& 30.0& 0.153& 28.9& -23.2& 59.6& -15.3 \\
18&2010-12-13&02:40:24& VLT/NACO$^5$ & 1.94& 1.53& 30.0& 0.171& 321.1& -23.2& 351.8& -15.3 \\
19&2010-12-14&00:41:59& VLT/NACO$^5$ & 1.94& 1.53& 30.0& 0.171& 311.5& -23.2& 342.2& -15.1 \\
20&2010-12-14&01:38:22& VLT/NACO$^5$ & 1.94& 1.54& 30.0& 0.158& 265.0& -23.2& 295.7& -15.1 \\
21&2010-12-14&02:14:10& VLT/NACO$^5$ & 1.94& 1.54& 30.0& 0.167& 235.5& -23.2& 266.2& -15.0 \\
22&2014-12-08&00:53:28& VLT/SPHERE$^6$ & 2.03& 1.15& 17.0& 0.216& 208.7& 3.4& 225.7& 2.8 \\
23&2014-12-09&01:04:54& VLT/SPHERE$^6$ & 2.03& 1.16& 17.2& 0.211& 91.6& 3.2& 108.9& 3.0 \\
24&2014-12-10&01:59:38& VLT/SPHERE$^6$ & 2.03& 1.17& 17.5& 0.221& 298.8& 3.0& 316.4& 3.2 \\
25&2014-12-12&04:14:08& VLT/SPHERE$^6$ & 2.04& 1.18& 18.1& 0.221& 332.6& 2.6& 350.7& 3.7 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\FloatBarrier
\section{Additional data}
\subsection{Disk-resolved images}
\indent To reconstruct the
3D shape of (6)~Hebe, we compiled available images
obtained with the earlier-generation AO instruments
NIRC2 \citep{Wizinowich:2000ua, vanDam:2004jd} on the W. M. Keck II telescope
and NACO \citep{Lenzen:2003iu, Rousset:2003hh} on the ESO VLT.
Each of these images, as well as the corresponding
calibration files and stellar PSF, were retrieved from the
Canadian Astronomy Data
Center\footnote{http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/}
\citep{Gwyn:2012gv} or
directly from the observatory's database. Data processing and
Mistral deconvolution of these
images were performed following the
same method as for our SPHERE images.
Only a subset of the \numb{25} different epochs
listed in Table~\ref{tab:obscondition} had been published
\citep{Hanus:2013jh}.
\subsection{Optical light curves}
\indent We used \numb{38} light curves obtained in the years
1953-1993 and available in the Database of Asteroid Models from
Inversion Techniques
\citep[DAMIT\footnote{http://http://astro.troja.mff.cuni.cz/projects/asteroids3D},][]{Durech:2010ey} that were used by
\citet{Torppa:2003ii} to derive
the pole orientation and convex shape of (6)~Hebe from light curve
inversion \citep{Kaasalainen:2001dx, Kaasalainen:2001di}.
We also retrieved \numb{16} light curves observed by the amateurs
F.~Kugel and J.~Caron, from the \textsl{Courbe de Rotation}
group\footnote{http://obswww.unige.ch/{\textasciitilde}behrend/page\_cou.html}, and
84 light curves from the data archive of the SuperWASP survey
\citep{Pollacco:2006gb} for the
period 2006-2009.
This survey aims at finding and characterizing exoplanets by
observation of their transit in front of their host star.
Its large field of view and cadence provides a goldmine
for asteroid light curves \citep{2017-ACM-Grice}.
Finally, four light curves were acquired by our
group during April 2016 with the 60\,cm TRAPPIST telescope
\citep{Jehin:2011vb}.
\subsection{Stellar occultations}
\indent We retrieved the five stellar occultations listed by
\citet{Dunham:2016wz} and publicly available on the Planetary Data
System (PDS)\footnote{http://sbn.psi.edu/pds/resource/occ.html} for
(6)~Hebe. We convert the disappearance and reappearance timings
of the occulted stars into segments (called chords) on the plane
of the sky, using the location of the observers on Earth and the
apparent motion of Hebe following the recipes by \citet{1999-IMCCE-Berthier}.
Of the five events, only two had more than one positive chord (that
is a recorded blink event) and
could be used to constrain the 3D shape (1977-03-05 -- also presented in \citealt{Taylor:1978jw} -- and
2008-02-20).
\subsection{Mid-infrared thermal measurements\label{sec:tm}}
\indent Finally, we compiled available mid-infrared thermal measurements to
1) validate, independently of the AO images, the size of our 3D-shape model and,
2) derive the thermal properties of the surface of Hebe
through thermophysical modeling of the infrared flux.
Specifically, we used a total of 103 thermal data points from
IRAS \citep[12, 25, 60, 100\,$\mu$m,][]{Tedesco:2002fs},
AKARI-IRC \citep[9, 18\,$\mu$m,][]{Usui:2011kh},
ISO-ISOPHOT \citep[25\,$\mu$m,][]{Lagerros:1999ku}, and
Herschel-PACS (70, 100, 160\,$\mu$m, M\"uller et al., in prep).
\section{3D shape, volume, and density\label{sec:3d}}
\indent Recent algorithms such as KOALA \citep{Carry:2010bo, Carry:2012jn,
Kaasalainen:2011tk} and ADAM \citep{Viikinkoski:2015jha} allow
simultaneous derivation of the spin, 3D shape, and size of an asteroid
\citep[see, e.g.,][]{Merline:2013fs, Tanga:2015fx,
Viikinkoski:2015jk, Hanus:2017ed}.
This combined multi-data approach has been validated by comparing
the 3D shape model of (21) Lutetia by \citet{Carry:2010je} with the
images returned by the ESA Rosetta mission during its fly-by of the
asteroid \citep[see][]{Sierks:2011il,Carry:2012jn}.
\indent Here, we reconstruct the spin and shape of
(6)~Hebe with ADAM, which iteratively improves the solution by
minimizing the residuals between the Fourier transformed images
and a projected polyhedral model. This method allows the use of AO images directly
without requiring the extraction of boundary contours. Boundary contours are therefore
used here only as a means to measure the pixel root mean square (RMS) residuals
between the location of the asteroid silhouette on the observed
and modeled images. ADAM offers two different shape supports: subdivision surfaces
and octanoids based on spherical harmonics. Here, we use the subdivision surfaces
parametrisation which offers more local control
on the model than global representations (see \citealt{Viikinkoski:2015jha}).
Two different models depicted in Fig.~\ref{fig:model} were obtained;
the first one using the light curves combined to the full AO sample,
and the second one using the light curves and the SPHERE images only.
Comparison of the SPHERE-based model with our SPHERE images,
earlier AO images, subsets of optical
light curves and stellar occultations are presented in
Figs.~\ref{fig:proj}, \ref{fig:ao}, \ref{fig:lc}, and
\ref{fig:occ}, respectively.
\indent The two models nicely fit all data, the
RMS residuals between the observations and the predictions by the model
being only \numb{0.6} pixels for the location of the asteroid contours,
\numb{0.02} magnitude for the light curves, and \numb{5}\,km for the stellar occultation of 2008
(the occultation of 1977 has very large uncertainties on its timings).
The 3D shape models are close to an oblate spheroid, and have a
volume-equivalent diameter of $196 \pm 6$\,km (all AO) and
$193 \pm 6$\,km (SPHERE-based; Table~\ref{tab:adam}).
Spin-vector coordinates ($\lambda$, $\beta$ in ECJ2000) are close
to earlier estimates
based on light-curve inversion
\citep[(339\degr,+45\degr),][]{Torppa:2003ii}
and on a combination of light curves and AO images \citep[(345\degr,+42\degr),][]{Hanus:2013jh}.
The main difference between the two shape models comes from the presence of
some surface features in the SPHERE-based model that are lacking
in the model obtained using the full dataset of AO images.
This is due to the lower resolution of earlier AO images
that do not address some of the small-scale surface features revealed by the
SPHERE images.
\begin{figure}[h!]
\centering
\includegraphics[angle=0, width=0.7\linewidth, trim=0cm 0cm 0cm 0cm, clip]{xyzModel.png}
\caption[ ]{3D-shape model of (6)~Hebe reconstructed from light curves and all resolved images ({\it left}), and from light curves and SPHERE images only ({\it right}). Viewing directions are two equator-on views rotated by 90$\degr$ and a pole-on view.}
\label{fig:model}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[angle=0, width=\linewidth, trim=0cm 0cm 0cm 0cm, clip]{sphereImgProj3.png}
\caption[ ]{Deconvolved SPHERE images of Hebe obtained between 8 and 12 December 2014 ({\it top}) and corresponding projection of the model ({\it bottom}). Orientation of the four images with respect to the North is 15.2$\degree$, 12.8$\degree$, -5.3$\degree$ and -89.6$\degree$ , respectively.
}
\label{fig:proj}
\end{figure}
\begin{figure*}[!htb]
\includegraphics[angle=0, width=\linewidth, trim=0cm 13.5cm 0cm 0cm, clip]{fig3.pdf}
\caption[ ]{Previous AO images of Hebe obtained with Keck/NIRC2 and VLT/NACO ({\it top of the three rows}) and corresponding projection of the model ({\it bottom}). Each image is 0.8"$\times$0.8" in size.}
\label{fig:ao}
\end{figure*}
\begin{figure}[h!]
\centering
\includegraphics[angle=0, width=\linewidth, trim=0cm 0cm 0cm 0cm, clip]{art-lc-001.eps}
\caption[ ]{Comparison of the synthetic light curves (solid line) from the shape
model with a selection of light curves (gray points). }
\label{fig:lc}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[angle=0, width=\linewidth, trim=0cm 0cm 0cm 0cm, clip]{occ-001.eps}
\caption[ ]{Comparison of the shape model with the chords from the occultation of 1977 and 2008. }
\label{fig:occ}
\end{figure}
\begin{table}[ht]
\begin{center}
\caption{Period, spin (ECJ2000 longitude $\lambda$, latitude $\beta$ and initial Julian date T$_0$), and dimensions
(volume-equivalent diameter $D$, volume $V$, and tri-axial ellipsoid diameters $a$, $b$, $c$ along
principal axes of inertia) of Hebe derived with ADAM.
\label{tab:adam}
}
\begin{tabular}{llllll}
\hline\hline
Parameter & Value & Value & Unc. & Unit \\
& \multicolumn{1}{l}{(all AO)} & \multicolumn{1}{l}{(SPHERE-only)} & & & \\[1mm]
\hline
Period & 7.274467 & 7.274465 & 5.10$^{-5}$ & hour \\
$\lambda$ & 341.7 & 343.2 & 3 & deg. \\
$\beta$ & +49.9 & +46.8 & 4 & deg. \\
T$_0$ & 2434569.00 & 2434569.00 & & \\
\hline
D & 196 & 193 & 6 & km \\
V & 3.95 $\cdot 10^{6}$ & 3.75 $\cdot 10^{6}$ & 1.2 $\cdot 10^{5}$ & km$^3$ \\
a & 218.4 & 213.4 & 6.0 & km \\
b & 206.2 & 200.2 & 6.0 & km \\
c & 172.1 & 172.6 & 6.0 & km \\
a/b & 1.06 & 1.07 & 0.04 & \\
b/c & 1.20 & 1.16 & 0.05 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\indent There are 12 diameter estimates for Hebe in the literature
(Table~\ref{table:diamestimates},
Figure~\ref{fig:hebe-diam}). Rejecting values that do not fall
within one standard deviation of the average value of the full
dataset gives an average equivalent-volume sphere diameter of
$191.5 \pm 8.3$\,km, in very good agreement with the values
of \numb{$193 \pm 6$}\,km and \numb{$196 \pm 6$}\,km derived here
(also supported by the
thermophysical analysis presented in the following section).
In the following, we use the value of the diameter obtained from our
SPHERE-based model, which is more precise
due to the higher angular resolution
of the SPHERE images with respect to the NIRC2 and NACO images.
A main advantage of using a diameter obtained from a full 3D
shape modeling resides in the uncertainty on the derived volume $V$,
which is close to $\delta V / V \approx \delta D / D$, as opposed to
a $\delta V / V \approx 3 \delta D / D$ in the spherical assumption
used in most aforementioned estimates
(see \citealt{Kaasalainen:2012ku} for details).
\indent Combining this diameter with an average mass of
$1.31\pm0.24\times10^{19}$~kg computed from 16 estimates
gathered from the literature (Table~\ref{table:massestimates}, Figure~\ref{fig:hebe-mass}),
provides a bulk density of $3.48\pm0.64$~g.cm$^{-3}$,
in perfect agreement with
the average grain density of ordinary H~chondrites ($
3.42\pm0.18 $~g.cm$^{-3}$; \citealt{Consolmagno:2008cl}).
The derived density therefore suggests a null internal porosity,
consistent with an intact internal structure.
Hebe hence appears to reside in the volumetric and
structural transitional region between the compact and
gravity-shaped dwarf planets, and the medium-sized asteroids
($\sim$10-100 km in diameter) with fractured interior
\citep{Carry:2012cw,2015-AsteroidsIV-Scheeres}.
However, due to the current large mass uncertainty that
dominates the uncertainty of the bulk density,
the possibility of higher internal porosity
cannot be ruled out.
We expect the Gaia mission to trigger higher-precision mass estimates
in the near future \citep{Mignard:2007er, Mouret:2007eg} that will help refine the density
measurement of Hebe.
\section{Thermal parameters and regolith grain size\label{sec:neatm}}
\indent A thermophysical model (TPM; \citealt{Mueller:1998to, Muller:1999vg})
was also used to provide an independent size measurement for Hebe
and to derive its thermal surface properties.
The TPM uses as input our 3D shape model with unscaled diameter.
The procedure is described in detail in Appendix\,\ref{sec:tpm}.
Using absolute magnitude H=5.71 and magnitude slope G=0.27
from the Asteroid Photometric Catalogue \citep{Lagerkvist:2011vg}, the TPM
provides a solution for diameter and albedo of ($D$, p$_{\rm v}$)=(198$\substack{+4 \\ -2}$\,km, 0.24$\pm$0.01),
in good agreement with the size of our 3D-shape model and previous albedo measurements from
IRAS (p$_{\rm v}$=0.27$\pm$0.01; \citealt{Tedesco:2002fs}),
WISE (p$_{\rm v}$=0.24$\pm$0.04; \citealt{Masiero:2014gu}) and
AKARI (p$_{\rm v}$=0.24 0.01; \citealt{Usui:2011kh}).
Best-fitting solutions are found for significant surface
roughness (in agreement with \citealt{Lagerros:1999ku}),
and thermal inertia $\Gamma$ values ranging from 15 to
90~J\,m$^{-2}$\,s$^{-0.5}$K\,$^{-1}$, with a preference for
$\Gamma \approx$\,50~J\,m$^{-2}$\,s$^{-0.5}$K\,$^{-1}$.
Interestingly, we note that the best-fitting solution for $\Gamma$
drops from $\sim$60~J\,m$^{-2}$\,s$^{-0.5}$K\,$^{-1}$
when only considering thermal measurements acquired at r$<$2.1~AU,
to $\sim$40~J\,m$^{-2}$\,s$^{-0.5}$K\,$^{-1}$ for data
taken at r$>$2.6 AU. While this might be indicative of changing thermal inertia with temperature,
this result should be taken with extreme caution, as the TPM probably overfits the data
due to the large error bars on the thermal measurements (see Appendix\,\ref{sec:tpm}).
From the thermal inertia value derived here, one can further derive the grain size of the
surface regolith of Hebe \citep{Gundlach:2013kn}. Assuming values of heat capacity and
material density typical of H5 ordinary chondrites \citep{Opeil:2010hq} and estimated
surface temperature of 230\,K and 180\,K at 1.94 and 2.87\,AU respectively, we find that
the typical grain size of Hebe is about 0.2--0.3\,mm (see Annexe\,\ref{sec:tpm} for more details).
\section{Topography}
\indent Hebe's topography was investigated by generating an elevation
map of its surface with respect to a volume-equivalent ellipsoid
best-fitting our 3D-shape model, following the method by
\citet{Thomas:1999jj}.
This map shown in Figure\,\ref{fig:topo}
allows the identification of several low-topographic and concave
regions possibly created by impacts
(the two shape models depicted in Fig.~\ref{fig:model} produce slightly
different but consistent topographic maps). Specifically, five large
depressions (numbered 1 to 5 on the elevation map) are found at the surface of Hebe, at
(29$\degree$, 43$\degree$), (93$\degree$, -42$\degree$),
(190$\degree$, 35$\degree$), (289$\degree$, -13$\degree$), and near
the south pole. Estimated dimensions (diameter D and maximum depth below the average surface d) are
D$_1$=92--105\,km, d$_1$=13\,km; D$_2$=85--117\,km, d$_2$=12\,km; D$_3$=68--83\,km, d$_3$=11\,km;
D$_4$=75--127\,km, d$_4$=18\,km; and D$_5$=42--52\,km, d$_5$=7\,km, respectively.
\indent Assuming that the volume of a crater relates approximately to the
volume of ejecta produced by the impact -- which is most likely very optimistic
because 1) a significant fraction of impact crater volume comes from
compression \citep{Melosh:1989uq} and, 2) at least a fraction of the ejecta must
have re-accumulated on the surface of the body (e.g., \citealt{Marchi:2015js}),
one can further estimate the
volume of a hypothetical family derived from an impact on Hebe.
The largest depression on Hebe roughly accounts for a volume of
$10^5 $~km$^3$,
corresponding to a body with an
equivalent diameter of $\sim$58\,km.
For comparison, the five known S-type
families spectrally analogous to Hebe (therefore to H chondrites; \citealt{Vernazza:2014uqa})
and located close to the main-belt 3:1 and 5:2 mean-motion resonances,
namely Agnia (located at semi-major axis $a$=2.78~AU and eccentricity
$e$=0.09), Koronis ($a$=2.87~AU, $e$=0.05), Maria ($a$=2.55~AU,
$e$=0.06), Massalia ($a$=2.41~AU, $e$=0.14) and Merxia ($a$=2.75~AU,
$e$=0.13) encompass a total volume of respectively $ \sim 2.4 \times
10^4 $~km$^3$, $ 5.6 \times 10^5 $~km$^3$, $ 3.6 \times 10^5 $~km$^3$,
$ 5.7 \times 10^4 $~km$^3$ and $ 1.8 \times 10^4 $~km$^3$ when the
larger member of each family is removed. Family membership was
determined using \citet{Nesvorny:2015tp}'s Hierarchical Clustering
Method (HCM)-based classification
(http://sbn.psi.edu/pds/resource/nesvornyfam.html) and rejecting
possible interlopers that do not fit the "V-shape" criterion as
defined in \citet{Nesvorny:2015gu}. The diameter of each asteroid
identified as a family member was retrieved from the WISE/NEOWISE
database \citep{Masiero:2011jc} when available, or estimated from
its absolute H~magnitude otherwise, assuming an albedo equal to that
of the largest member of its family (respectively 0.152, 0.213,
0.282, 0.241 and 0.213 for (847)~Agnia, (158)~Koronis, (170)~Maria,
(20)~Massalia and (808)~Merxia; \href{https://mp3c.oca.eu}{https://mp3c.oca.eu}).
We note that these values should be considered as lower limits as those families certainly
include smaller members beyond the detection limit.
\indent We therefore find that the volume of material
corresponding to the largest depression
on Hebe is of the order of some H-chondrite-like S-type
families, and $\sim$4-6 times smaller
than the largest ones.
Therefore, although we cannot firmly exclude Hebe as the main (or unique) source of H chondrites, it appears that such a hypothesis is
not the most likely one.
This is further strengthened by the following two arguments.
First, it seems improbable that the volume excavated from Hebe's largest depression,
which we find to be roughly 10 to 30 times smaller than the volume of the Rheasilvia
basin on Vesta \citep{Schenk:2012in}, would contribute to $\sim$34\% of the meteorite
falls, when HED meteorites only represent $\sim$6\% of the falls \citep{Hutchison:2004uk}.
We note, however, that the low number of HED meteorites may also relate to the relatively old age \citep{Schenk:2012in}
of the Vesta family \citep{Heck:2017jg}.
Second, the current lack of observational evidence for
a Hebe-derived family indicates that such a family,
if it ever existed, must be very ancient and dispersed. Yet, there is growing evidence from laboratory experiments that the current
meteorite flux must be dominated by fragments from recent asteroid breakups \citep{Heck:2017jg}. In the case of H chondrites, this is well supported
by their cosmic ray exposure ages \citep{Marti:1992dv, Eugster:2006vg}. It therefore appears that a recent - yet to be identified -
collision suffered by another H-chondrite-like asteroid is the most
likely source of the vast majority of H chondrites.
\begin{figure*}[h!]
\centering
\includegraphics[angle=0, width=0.6\linewidth, trim=0cm 0cm 0cm 0cm, clip]{map.pdf}
\caption[ ]{Elevation map (in km) of (6)~Hebe, with respect to a volume-equivalent ellipsoid best
fitting our 3D-shape model. The five major depressions are identified by numbers.
}
\label{fig:topo}
\end{figure*}
\section{Conclusion and outlook}
\indent We have reconstructed the spin and tridimensional shape of
(6)~Hebe from combined AO images and optical light curves, and
checked the consistency of the derived model against available
stellar occultations and thermal measurements.
Whereas the irregular shape of Hebe suggests it was moulded by impacts,
its density appears indicative of a compact interior.
Hebe thus seems to reside in the structural regime
in transition between round-shaped dwarf planets shaped by gravity,
and medium-sized asteroids with fractured interiors (i.e.,
significant fractions of macro-porosity; \citealt{Carry:2012cw}).
This however needs to be confirmed by future mass measurements
(e.g., from Gaia high-precision astrometric measurements)
that will help improve the current mass uncertainty that
dominates the uncertainty on density.
The high angular resolution of SPHERE further allowed us to
identify several concave regions at the surface of Hebe possibly
indicative of impact craters.
We find the volume of the largest depression
to be roughly five times smaller than
the volume of the largest S-type H-chondrite-like families located close
to orbital resonances in the asteroid belt.
Furthermore, this volume is more than an order of magnitude smaller than
the volume of the Rheasilvia basin on Vesta \citep{Schenk:2012in}
from which HED meteorites ($\sim$6\% of the falls) originate.
Our results therefore imply that (6) Hebe is not the most likely source
of ordinary H chondrites ($\sim$34\% of the falls).
Finally, this work has demonstrated the potential of SPHERE
to bring important constraints on the origin and collisional history
of the main asteroid belt.
Over the next two years, our team will collect -- via a large
program on VLT/SPHERE (run ID: 199.C-0074, PI: Pierre Vernazza) --
similar volume, shape, and topographic measurements for
a significant number ($\sim$40) of D$\geq$100 km asteroids sampling the
four major compositional classes (S, Ch/Cgh, B/C and P/D).
\begin{acknowledgements}
Based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under programme ID 60.A-9379.
The asteroid diameters and albedos based on NEOWISE observations
were obtained from the Planetary Data System (PDS).
Some of the data presented herein were obtained at the W.M. Keck
Observatory, which is operated as a scientific partnership among the
California Institute of Technology, the University of California and
the National Aeronautics and Space Administration. The Observatory
was made possible by the generous financial support of the W.M. Keck
Foundation.
This research has made use of the Keck Observatory Archive (KOA),
which is operated by the W. M. Keck Observatory and the NASA
Exoplanet Science Institute (NExScI), under contract with the
National Aeronautics and Space Administration.
The authors wish to recognize and acknowledge the very significant
cultural role and reverence that the summit of Mauna Kea has always
had within the indigenous Hawaiian community. We are most fortunate
to have the opportunity to conduct observations from this mountain.
This research used the MP$^{3}$C service developed,
maintained, and hosted at the Lagrange laboratory, Observatoire de la
C{\^o}te d'Azur \citep{2017-ACM-Delbo}.
Photometry of 6 Hebe was identified and extracted from WASP data
with the help of Neil Parley (Open University, now IEA Reading).
The WASP project is currently funded and operated by Warwick University and Keele University, and was originally set up by Queen's University Belfast, the Universities of Keele, St. Andrews and Leicester, the Open University, the Isaac Newton Group, the Instituto de Astrofisica de Canarias, the South African Astronomical Observatory and by STFC.
The WASP project is currently funded and operated by Warwick
University and Keele University, and was originally set up by Queen's
University Belfast, the Universities of Keele, St. Andrews and
Leicester, the Open University, the Isaac Newton Group, the Instituto
de Astrofisica de Canarias, the South African Astronomical Observatory
and by STFC.
TRAPPIST-South is a project funded by the Belgian Funds (National) de la Recherche Scientifique (F.R.S.-FNRS) under grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Foundation (FNS/SNSF).
E.J. and M.G. are F.R.S.-FNRS research associates.
Based on observations with ISO, an ESA project with instruments
funded by ESA Member States and with the participation of ISAS
and NASA.
Herschel is an ESA space observatory with science instruments
provided by European-led Principal Investigator consortia and
with important participation from NASA.
Herschel fluxes of Hebe where extracted by Csaba Kiss (Konkoly Observatory, Research Centre for Astronomy
and Earth Sciences, Hungarian Academy of Sciences, H-1121 Budapest,
Konkoly Thege Mikl{\'o}s {\'u}t 15-17, Hungary).
TM received funding from
the European Union's Horizon 2020 Research and Innovation Programme,
under Grant Agreement no 687378.
\end{acknowledgements}
|
2,877,628,089,460 | arxiv | \section{Introduction}\label{sec:intro}
\emph{Federated Learning} (FL) \citep{ konevcny2015federated, mcmahan2017communication, kairouz2019advances, bonawitz2019towards} is a distributed machine learning paradigm where training data resides
at autonomous client machines and the learning process is facilitated by a central server. The server maintains a shared model and alternates between requesting clients to try and improve it and integrating their suggested improvements back into that shared model.
A few interesting challenges arise from this model. First, the need for communication efficiency, both in terms of the size of data transferred and the number of required messages for reaching convergence. Second, clients are outside of the control of the server and as such may be unreliable, or even malicious. Third, while classical learning models generally assume that data is homogeneous, here privacy and the aforementioned communication concerns force us to deal with the data as it is seen by the clients; that is 1) \emph{non-IID} (identically and independently distributed) -- data may depend on the client it resides at, and 2) \emph{unbalanced} -- different clients may possess different amounts of data.
In previous works \citep{DBLP:journals/corr/abs-1906-06629, NIPS2018_7712, li2019rsa, haddadpour2019convergence, pillutla2019robust}, unbalancedness is either ignored or is represented by a collection of a priori known \emph{client importance weights} that are usually derived from the amount of data each client has. This work investigates aspects that stem from this unbalancedness. Concretely, we focus on the case where unreliable clients declare the amount of data they have and may thus adversely influence their importance weight. We show that without some mitigation, a single malicious client can obstruct convergence in this manner even in the presence of popular FL defense mechanisms. Our experiments consider protections that replace the server step by a robust mean estimator, such as median \citep{10.1145/3154503, pmlr-v80-yin18a, chen2019distributed} and trimmed mean \citep{pmlr-v80-yin18a}.
The rest of this paper is organized as follows. In Section \ref{sec:setup}, we present required definitions and formalize the problem addressed by this work. Section \ref{sec:preprocessing} presents our truncation-based preprocessing method and proves that it can be applied to a randomly-selected sample of client weights . In Section \ref{sec:evaluation}, we report on the results of our empirical evaluation. Conclusions and directions for future work are presented in Section \ref{sec:conclusion}.
\section{Problem setup}\label{sec:setup}
\subsection{Optimization goal}
We are given $K$ clients where each client $k$ has a local collection $Z_k$ of $n_k$ samples taken IID\ from some unknown distribution over sample space $\bm{Z}$. We denote the unified sample collection as $Z = \bigcup_{k \in [K]} Z_k$ and the total number of samples as $n$ (i.e., $n = |Z| = \sum_{k \in [K]} n_k$) .
Our objective is \emph{global} empirical risk minimization (ERM) for some loss function class $ \ell(w; \cdot)\colon \bm{Z} \to \mathbb{R}$, parameterized by $w \in \mathbb{R}^d$ \footnote{We note that some previous FL works specify a more generic finite-sum objective \citep{mcmahan2017communication}. However, this work investigates client-declared sample sizes, whose meaning is clear under the ERM interpretation but seems meaningless in the finite-sum objective setting.}:
\begin{gather} \label{eq:goal}
\min_{w \in \mathbb{R}^d} F(w), \text{ where } F(w) \coloneqq \frac{1}{n} \sum_{z \in Z} \ell(w ; z).
\end{gather}
In the following sections we denote the vector of client sample sizes as $\boldsymbol{N}= (n_1, n_2, \dots, n_K)$ and assume, w.l.o.g., that it is sorted in increasing order.
\subsection{Collaboration model} \label{subs:collab}
We restrict ourselves to the FL paradigm, which leaves the training data distributed on client machines, and learns a shared model by iterating between client updates and server aggregation.
Additionally, a subset of the clients, marked $\mathcal{B}$, can be Byzantine, meaning they can send arbitrary and possibly malicious results on their local updates. Moreover, unlike previous works, we also consider clients' sample sizes to be unreliable because they are reported by possibly Byzantine clients. When the distinction is important, values that are sent by clients are marked with an overdot to signify that they are unreliable (e.g., $\dot{n}_k$), while values that have been preprocessed in some way are marked with a tilde (e.g., $\tilde{n}_k$).
\subsection{Federated learning meta algorithm}
We build upon the baseline federated averaging algorithm ($FedAvg$) described by \citet{mcmahan2017communication}. There, it is suggested that in order to save communication rounds clients perform multiple stochastic gradient descent (SGD) steps while a central server occasionally averages the parameter vectors.
The intuition behind this approach becomes clearer when we mark the $k$\textsuperscript{th}\xspace client's ERM objective function by $F_k(w) \coloneqq \frac{1}{n_k} \sum_{z \in Z_k} \ell(w ; z)$ and observe that the objective function in equation \eqref{eq:goal} can be rewritten as a weighted average of clients' objectives:
\begin{equation} \label{eq:goal-weighted}
F(w) \coloneqq \frac{1}{n} \sum_{k \in [K]} n_k F_k(w).
\end{equation}
Similarly to previous works \citep{pillutla2019robust, chen2019communicationefficient, chen2019distributed}, we capture a large set of algorithms by abstracting $FedAvg$ into a meta-algorithm for FL (Algorithm \ref{algo:meta}). We require three procedures to be specified by any concrete algorithm:
\begin{enumerate}[label=(\alph*)]
\item $Preprocess$ -- receives, possibly byzantine, $\dot{n}_k$'s from clients and produces secure estimates marked as $\tilde{n}_k$'s. To the best of our knowledge, previous works ignore this procedure and assume $n_k$'s are correct.
\item $ClientUpdate$ -- per-client $w_k$ computation. In $FedAvg$, this corresponds to a few local mini-batch SGD rounds. See Algorithm \ref{algo:fedavg-clientupdate} for pseudocode.
\item $Aggregate$ -- the server's strategy for updating $w$. In $FedAvg$, this corresponds to the weighted arithmetic mean, i.e., $w \gets \frac{1}{\dot{n}} \sum_{k \in [K]} \dot{n}_k \dot{w}_k$.
\end{enumerate}
\begin{minipage}{0.45\linewidth}
\vskip -0.9in
\begin{algorithm}[H]
\centering
\caption{$FedAvg$: $ClientUpdate$}
\label{algo:fedavg-clientupdate}
\begin{flushleft}
\textbf{Hyperparameters:} learning rate ($\eta$), number of epochs ($E$), and batch size ($B$).
\end{flushleft}
\begin{algorithmic}[1]
\FOR{$E$ epochs}
\FOR{$B$-sized batch $\mathcal{B} \subseteq Z_k$}
\STATE $w_k \leftarrow w_k - \eta \frac{1}{B} \sum_{z \in \mathcal{B}} \nabla \ell (w_k; z)$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}{0.54\linewidth}
\begin{algorithm}[H]
\centering
\caption{Federated Learning Meta-Algorithm}
\label{algo:meta}
\begin{flushleft}
\textbf{Given procedures:} $Preprocess$, $ClientUpdate$, and $Aggregate$.
\end{flushleft}
\begin{algorithmic}[1]
\STATE $\{\dot{n}_k\}_{k \in [K]} \gets$ collect sample size from each client
\STATE $\{\tilde{n}_k\}_{k \in [K]} \gets Preprocess(\{\dot{n}_k\}_{k \in [K]})$
\STATE $w \gets $ initial guess
\FOR{$t \gets 1$ \textbf{to} $T$}
\STATE $S_t \gets$ a random set of client indices
\FORALL{$k \in S_t$}
\STATE $\dot{w}_k \gets ClientUpdate(\tilde{n}_k, w)$
\ENDFOR
\STATE $w \gets Aggregate(\{\langle \tilde{n}_k, \dot{w}_k \rangle \}_{k \in S_t})$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\end{minipage}
\section{Preprocessing client-declared sample sizes}\label{sec:preprocessing}
\subsection{Preliminaries}
The following assumption is common among works on Byzantine robustness:
\begin{asm}[Bounded Byzantine proportion]\label{asm:1}
The proportion of clients who are Byzantine is bounded by some constant $\alpha$; i.e., $\frac{1}{K}|\mathcal{B}| \le \alpha$.
\end{asm}
The next assumption is a natural generalization when considering unbalancedness:
\begin{asm}[Bounded Byzantine weight proportion]\label{asm:2}
The proportion between the combined weight of Byzantine clients and the total weight is bounded by some constant $\alpha^*$; i.e., $\frac{1}{n}\sum_{b \in \mathcal{B}} n_b \le \alpha^*$.
\end{asm}
Previous works on robust aggregation \citep{DBLP:journals/corr/abs-1906-06629, NIPS2018_7712, li2019rsa, haddadpour2019convergence, zhao2018federated} either used Assumption \ref{asm:1}, without considering the unbalancedness, or implicitly used Assumption \ref{asm:2}. However, we observe that Assumption \ref{asm:2} is unattainable in practice since Byzantine clients can often influence their weight thus rendering their weight proportion unbounded.
We address this gap with the following definition and an appropriate $Preprocess$ procedure.
\begin{definition}[$\mwp$]
Given a proportion $p$, and a vector $\bm{V}=(v_1,...,v_{|\bm{V}|})$ sorted in increasing order, the \emph{maximal weight proportion}, $\mwp(\bm{V}, p)$, is the maximum combined weight for \emph{any} $p$-proportion of the values of $\bm{V}$:
$$
\mwp(\bm{V}, p) \coloneqq \frac{1}{
\sum_{v \in \bm{V}} v
}\sum_{(1 - p)|\bm{V}| < i} v_i.
$$
Note that this is just the weight proportion of the $p |\bm{V}|$ clients with the largest sample sizes.
\end{definition}
In the rest of this work will assume Assumption \ref{asm:1} and design a $Preprocess$ procedure that ensures the following:
\begin{equation} \label{eq:preproc}
\mwp(Preprocess(\bm{N}), \alpha) \leq \alpha^*.
\end{equation}
Observe that this requirement enables the use of weighted robust mean estimators in a realistic setting by ensuring that Assumption \ref{asm:2} holds for the preprocessed client sample sizes. Also note that here, $\alpha$ is our assumption about the proportion of Byzantine clients while $\alpha^*$ relates to an analytical property of the underlying robust algorithm. For example, we may replace the federated average with a weighted median as suggested by \citet{10.1145/3154503}, in which case, $\alpha^*$ must be less than $1/2$.
\subsection{Truncating the values of N}
Our suggested preprocessing procedure uses element-wise truncation of the values of $\bm{N}$ by some value $U$, marked $\trunc(\bm{N}, U) = (\text{min}(n_1, U), \text{min}(n_2, U), \dots, \text{min}(n_K, U))$. Given $\alpha$ and $\alpha^*$, we search for the maximal truncation which satisfies \eqref{eq:preproc}:
\begin{equation} \label{eq:trunc}
U^* \coloneqq \argmax_{U \in \mathbb{N}}\ s.t.\ \mwp(\trunc(\bm{N}, U), \alpha) \leq \alpha^*
\end{equation}
Here $\alpha$ and $U^*$ present a trade-off. Higher $\alpha$ means more Byzantine tolerance but requires smaller truncation value $U^*$, which, may cause slower and less accurate convergence, as we demonstrate empirically and theoretically in Theorem \ref{thm:dist-bound}.
We note that given $\alpha$ and $\alpha^*$, truncating $\bm{N}$ by the solution of \eqref{eq:trunc} is optimal in the sense that any other $Preprocess$ procedure that adheres to \eqref{eq:preproc} has an equal or larger $\bm{L}^1$ distance from the original $\bm{N}$. This follows immediately from the observation that, when truncating the values of $\bm{N}$, the entire distance is due to the truncated elements and if there was another applicable vector closer to $\bm{N}$, we could have redistributed the difference to the largest elements and increase $U^*$ in contradiction to its maximally.
\subsubsection{Finding U given alpha}
If one has an estimate for $\alpha$ it is easy to calculate $U^*$. For example, by going over values in $\bm{N}$ in a decreasing order (i.e., from index $K$ downwards) until finding a value that satisfies the inequality in \eqref{eq:trunc}. Then we mark the index of this value by $u$ and use the fact that in the range $[n_u, n_{u+1}]$ we can express $\mwp(\trunc(\bm{N}, U), \alpha)$ as a simple function of the form $\frac{a + b U}{c + d U}$:
$$
\frac{\sum_{(1-\alpha)K < i \leq u} n_i + |\{n_i : i > \max(u, (1-\alpha)K) \}| U}{\sum_{i \leq u} n_i + |\{n_i : i > u\}| U},
$$
for which we can solve \eqref{eq:trunc} with
\begin{equation} \label{eq:3}
U^* \gets \left\lfloor \frac{a - c \alpha^*}{d \alpha^* -b} \right\rfloor.
\end{equation}
\subsubsection{The alpha-U trade-off}
When we do not know $\alpha$, as a practical procedure, we suggest plotting $U^*$ as a function of $\alpha$. In order to do so we can start with $\alpha \gets \alpha^*$, $U \gets n_1$, and alternate between decreasing $\alpha$ by $1/K$ (one less Byzantine client tolerated) and solving \eqref{eq:trunc}. This procedure can be made efficient by saving intermediate sums and using a specialized data structure for trimmed collections. See Algorithm \ref{algo:report} for pseudocode and Figure \ref{fig:tradeoff} for an example output.
\begin{minipage}{0.49\linewidth}
\captionsetup{type=figure}
\includegraphics[width=\linewidth]{tradeoff}
\captionof{figure}{Example plot of data generated by executing Algorithm \ref{algo:report} on unbalanced vector $\bm{N}$ and $\alpha^* = 50\%$ (this vector corresponds to the partition used in our experiments; See Section \ref{Experimental_Setup} for details).}
\label{fig:tradeoff}
\end{minipage}
\hfill
\begin{minipage}{0.49\linewidth}
\vskip -1.4in
\begin{algorithm}[H]
\centering
\caption{Report ($\alpha$, $U^*$) Pairs}
\label{algo:report}
\begin{algorithmic}
\STATE $\alpha \gets \alpha^*$
\FOR{$u \gets 1$ \textbf{to} $K-1$}
\WHILE{$\mwp(\trunc(\bm{N}, n_{u+1}), \alpha) > \alpha^*$}
\STATE $U^* \gets $ solve \eqref{eq:3} for $U \in [n_u, n_{u+1}]$
\STATE \textbf{report} ($\alpha$, $U^*$)
\STATE $\alpha \gets \alpha - \frac{1}{K}$
\ENDWHILE
\ENDFOR
\end{algorithmic}
\end{algorithm}
\end{minipage}
\subsection{Truncation given a partial view of N}
When K is very large we may want to sample only $k \ll K$ elements IID from $\bm{N}$.
In this case we will need to test that the inequality in \eqref{eq:trunc} holds with high probability.
We consider $k$ discrete random variables taken IID from $\bm{N}$ after truncation by $U$, that is, taken from a distribution over $\{0, 1, \dots, U\}$. We mark these random variables as $X_1, X_2, \dots, X_k$, and their order statistic as $X_{(1)}, X_{(2)}, \dots, X_{(k)}$ where $X_{(1)} \le X_{(2)} \le \dots \le X_{(k)}$.
\begin{theorem}\label{thm:partial-view}
Given parameter $\delta > 0$ and $\varepsilon_1 = \sqrt{\frac{\ln{(3/\delta)}}{2 k}}$, $\varepsilon_2 = U \sqrt{\frac{\ln{\ln{(3/\delta)}}}{2 (k(\alpha - \varepsilon_1) + 1)}}$, $\varepsilon_3 = U \sqrt{\frac{\ln{\ln{(3/\delta)}}}{2 k}}$, we have that $\mwp(\trunc(\bm{N}, U), \alpha) \le \alpha^*$ is true with $1 - \delta$ confidence if the following holds:
\begin{equation}
\begin{split}
\frac{\alpha \big(\frac{\sum^k_{i \gets \lceil(1 - (\alpha - \varepsilon_1))k\rceil} X_{(i)}}{k - \lceil(1 - (\alpha - \varepsilon_1))k\rceil + 1} + \varepsilon_2 \big) }{\big(\frac{1}{k} \sum_{i \in [k]} X_i - \varepsilon_3\big)}
\le \alpha^*
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
See Appendix \ref{prf:partial-view}
\end{proof}
\subsection{Convergence analysis}
After applying our $Preprocess$ procedure we have the truncated number of samples per client, marked $\{\tilde{n}_k\}_{k \in [K]}$. We can trivially ensure that any algorithm instance works as expected by requiring that clients ignore samples that were truncated. That is, even if an honest client $k$ (non-Byzantine) has $n_k$ samples it may use only $\tilde{n}_k$ samples during its $ClienUpdate$.
Although this solution always preserves the semantics of
\emph{any} underlying algorithm, it does hurt convergence guarantees since the total number of samples decreases [\citealt[Tables 5 and 6]{kairouz2019advances}; \citealt{pmlr-v80-yin18a}; \citealt{haddadpour2019convergence}]. Interestingly, \citet[Theorem~3]{li2019convergence} analyze the baseline \emph{FedAvg} and show that its convergence bound decreases with $max\,n_k$ (marked there as $\nu$). This suggests that in some cases truncation itself may mitigate the decrease in total sample size.
Additionally, we note that in practice, the performance of federated averaging based algorithms improves when honest clients use all their original $n_k$ samples. Intuitively, this follows easily from the observation that $Aggregate$ procedures are generally composite mean estimators and $ClientUpdate$ calls are likely to produce more accurate results given more samples.
Lastly, as we have mentioned before, convergence is guaranteed, but we note that the optimization goal itself is inevitably skewed in our Byzantine scenario. The following theorem bounds this difference between original weighted optimization goal \eqref{eq:goal-weighted} and the new goal after truncation.
In order to emphasize the necessity of this bound (in terms of Assumption \ref{asm:2}), we use overdot and tilde to signify unreliable and truncated values, respectively, as previously described in Subsection \ref{subs:collab}.
\begin{theorem}\label{thm:dist-bound}
Given the same setup as in \eqref{eq:goal} and a truncation bound $U$, the following holds for all $w \in \mathbb{R}^d$:
\begin{equation*}
\begin{split}
&
\norm{
\frac{1}{\dot{n}}\sum_{i \in [K]} \dot{n}_i F_i(w) -
\frac{1}{\tilde{n}}\sum_{i \in [K]} \tilde{n}_i F_i(w)} \le \\
&
\norm{
\sum_{i : \dot{n}_i > U} \big(\frac{\dot{n}_i}{\dot{n}} - \frac{1}{K}\big) F_i(w) +
\big(\frac{1}{\dot{n}} - \frac{1}{\tilde{n}} \big) \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)}
\end{split}
\end{equation*}
Where $\mathcal{L}(Z_i)$ is defined as $\sum_{z \in Z_i} \ell(w ; z)$.
\end{theorem}
\begin{proof}
See Appendix \ref{prf:dist-bound}
\end{proof}
From the bound in Theorem \ref{thm:dist-bound} we can clearly see how the coefficients in the left term, $(\dot{n}_i / \dot{n} - 1 / K)$, stem from unbalancedness in the values above the truncation threshold while the coefficient in the right term, $(1 / \dot{n} - 1 / \tilde{n})$, accounts for the increase of relative weight of the values below the truncation threshold. Additionally, note that this formulation demonstrates how a single Byzantine client can increase this difference arbitrarily by increasing its $\dot{n}_i$. Lastly, observe how both terms vanish as $U$ increases, which motivates our selection of $U^*$ as the \emph{maximal} truncation threshold for any given $\alpha$ and $\alpha^*$.
\section{Evaluation}\label{sec:evaluation}
In this section we demonstrate how truncating $\bm{N}$ is a crucial requirement for Byzantine robustness. That is, we show that no matter what is the specific attack or aggregation method, using $\bm{N}$ ``as-is" categorically devoids any robustness guaranties.
\label{p:code}
The code for the experiments is based on the Tensorflow machine learning library \citep{tensorflow2015-whitepaper}. Specifically, the code for the shakespeare experiments is based on the Tensorflow Federated sub-library of Tensorflow. It is given under the Apache license 2.0. Our code can be found in \url{https://github.com/amitport/Towards-Federated-Learning-With-Byzantine-Robust-Client-Weighting}. We perform the experiments using a single NVIDIA GeForce RTX 2080 Ti GPU, but the results are easily reproducible on any device.
\subsection{Experimental setup} \label{Experimental_Setup}
\subsubsection{The machine learning tasks and models}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{partitions}
\caption{Histogram of the sample partitions of the MNIST (left) and Shakespeare (right) datasets.}
\label{fig:partition}
\end{figure*}
\paragraph{Shakespeare: next-character-prediction partitioned by speaker.}
Presented in the original \textit{FedAvg} paper \citep{mcmahan2017communication} and also part of the LEAF benchmark \citep{caldas2019leaf}, the Shakespeare dataset contains 422,615 sentences taken from \emph{The Complete Works of William Shakespeare} \citep{shakespeare1996complete} (freely available public domain texts). The next-character-prediction task with the per-speaker partitioning represents a realistic scenario in the FL domain. Each client trains using an LSTM recurrent model \citep{Hochreiter1997LongSM} with hyperparameters matching those suggested by \citet{reddi2020adaptive} for \emph{FedAvg}.
\paragraph{MNIST: digit recognition with synthetic client partitioning.}
The MNIST database \citep{lecun2010mnist} (available under Creative Commons Attribution-ShareAlike 3.0 license) includes $28{\times}28$ grayscale labeled images of handwritten digits split into a 60,000 training set and a 10,000 test set. We randomly partition the training set among 100 clients. The partition sizes are determined by taking 100 samples from a Lognormal distribution with $\mu\!=\!1.5$, $\sigma\!=\!3.45$, and then interpolating corresponding integers that sum to 60,000. This produces a right-skewed, fat-tailed partition size distribution that emphasizes the importance of correctly weighting aggregation rules and the effects of truncation. Clients train a classifier using a 64-units perceptron with RelU activation and $20\%$ dropout, followed by a softmax layer. Following \citet{pmlr-v80-yin18a}, on every communication round, all clients perform mini-batch SGD with 10\% of their examples.
Note that the Shakespeare and MNIST synthetic tasks were selected because they are relatively simple, unbalanced tasks. Simple, because we want to evaluate
a preprocessing phase and avoid tuning of the underlying algorithms we compare. Unbalanced, since as can be understood from Theorem \ref{thm:dist-bound}, when the client sample sizes are spread mostly evenly, ignoring the client sample size altogether is a viable approach. See Figure \ref{fig:partition} for the histograms of the partitions.
\subsubsection{The server}
We show three $Aggregate$ procedures. Arithmetic mean, as used by the original $FedAvg$, and two additional procedures that replace the arithmetic mean with robust mean estimators. The first of the latter uses the coordinatewise median \citep{10.1145/3154503, pmlr-v80-yin18a}. That is, each server model coordinate is taken as the median of the clients' corresponding coordinates. The second robust aggregation method uses the coordinatewise trimmed mean \citep{pmlr-v80-yin18a} that, for a given hyperparameter $\beta$, first removes $\beta$-proportion lowest and $\beta$-proportion highest values in each coordinate and only then calculates the arithmetic mean of the remaining values.
When preprocessing the client-declared sample size, we compare three options: We either ignore client sample size, truncate according to $\alpha\!=\!10\%$ and $\alpha^*\!=\!50\%$, or just passthrough client sample size as reported.
\subsubsection{The clients and attackers}
For the Shakespeare experiments, we examine a \emph{model negation attack} \citep{NIPS2017_6617}. In this attack, each attacker ``pushes" the model towards zero by always returning a negation of the server's model. When the data distribution is balanced, this attack is easily neutralized since Byzantine clients typically send extreme values. However, in our unbalanced case, we demonstrate that without our preprocessing step, this attack cannot be mitigated even by robust aggregation methods.
For MNIST, in order to provide comparability, we follow the experiment shown by \citet{pmlr-v80-yin18a} in which $10\%$ of the clients use a \emph{label shifting attack}. In this attack, Byzantine clients train normally except for the fact that they replace every training label $y$ with $9\!-\!y$. The values sent by these clients are then incorrect but are relatively moderate in value making their attack somewhat harder to detect. This is in addition to the model negation attacks, already shown in the Shakespeare experiments.
We first execute our experiment without any attacks for every server aggregation and preprocessing combination. Then, for each attack type, we repeat the process two additional times: 1) with a single attacker that declares 10 million samples, and 2) with $10\%$ attackers that declare 1 million samples each.
\subsection{Results}
The Shakespeare experiments without any attackers is shown in Figure \ref{fig:no_attackers} and the executions with attackers are shown in Figure \ref{fig:attack}. The results of the MNIST experiments were almost identical and are deferred to Appendix \ref{sec:mnist} for brevity.
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{shakespeare_no_attackers}
\caption{Accuracy by round without any attackers for the Shakespeare experiments. Curves correspond to preprocessing procedures and columns correspond to different aggregation methods. It can be seen that our method (dashed orange curve) remains comparable to the properly weighted mean estimators (solid blue curve) while ignoring clients' sample sizes (dotted green curve) is sub-optimal. This effect is pronounced when the unweighted median is used, since with our unbalanced partition it is generally very far from the mean. Figure \ref{fig:mnist:no_attackers} in Appendix \ref{sec:mnist} shows similar results for the MNIST experiments.}
\label{fig:no_attackers}
\end{figure*}
The results from the first experiment, running without any attackers (Figure \ref{fig:no_attackers}), demonstrate that ignoring client sample size results in reduced accuracy, especially when median aggregation is used, whereas truncating according to our procedure is significantly better and is on par with properly using all weights. These results highlight the imperativeness of using sample size weights when performing server aggregations.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{shakespeare_model_negation_attack}
\caption{Accuracy by round under Byzantine attacks for the Shakespeare experiments. Curves correspond to preprocessing procedures and columns correspond to different aggregation methods. In the two rows of the experiment the Byzantine clients perform a model negation attack with one and $10\%$ attackers, respectively.
\\\hspace{\linewidth}
We observe that even with a single attacker performing a trivial attack (first row), using the weights directly (solid blue curve) is devastating while when our preprocessing method is used in conjunction with robust mean aggregations (dashed orange curve, two last columns) convergence remains stable even when there are actual $\alpha$ (=$10\%$) attackers (second row). We note that in some cases our method may be slightly less efficient compared with the preprocessing method that ignores sample size altogether (dotted green curve, second row, leftmost column). This is to be expected because we allow Byzantine clients to potentially get close to $\alpha^*$-proportion ($50\%$, in this case) of the weight. However, our method is significantly closer to the optimal solution when there are no or only a few attackers (see Figure \ref{fig:no_attackers}). Moreover, when used in conjunction with robust mean aggregation methods it maintains their robustness properties. Figure \ref{fig:mnist:attack} in Appendix \ref{sec:mnist} shows similar results for the MNIST experiments.}
\label{fig:attack}
\end{figure*}
While Figure \ref{fig:no_attackers} shows that the truncation-based preprocessing performs on par with that of taking all weights into consideration when all clients are honest, Figure \ref{fig:attack} demonstrates that the results are very different when there is an attack. In this case, we see that when even a single attacker reports a highly exaggerated sample size and the server relies on all the values of $\bm{N}$, the performance of all aggregation methods including robust median and trimmed mean quickly degrade.
In contrast, in our experiments robustness is maintained when truncation-based preprocessing is used in conjunction with robust mean aggregations, even when Byzantine clients attain the maximal supported proportion ($\alpha\!=\!10\%$).
\section{Conclusion and future work}\label{sec:conclusion}
Our method is based on truncating the weight values reported by clients in a manner that bounds from above the proportion $\alpha^*$ of weights that can be attributed to Byzantine clients, given an upper bound on the proportion of clients $\alpha$ that may be Byzantine. Different values of parameter $\alpha$ represent different points in the trade-off between model quality and Byzantine-robustness, where higher values increase robustness when attacks do occur but decrease convergence rate even in the lack of attacks.
We evaluated the performance of our truncation method empirically when applied as a preprocessing stage, prior to several aggregation methods. The results of our experiments establish that: 1) in the absence of attacks, model convergence is on par with that of properly using all reported weights, and 2) when attacks do occur, the performance of combining truncation-based preprocessing and robust aggregations incurs almost no penalty in comparison with the performance of using of all weights in the lack of attacks, whereas without preprocessing, even robust aggregation methods collapse to performance worse than that of a random classifier.
When the number of clients is very large, performing server preprocessing and aggregation on the server may become computationally infeasible. We prove that, in this case, truncation-based preprocessing can achieve the same upper bound on $\alpha^*$ w.h.p. based on the weight values reported from a sufficiently large number of the clients selected IID.
As with many Byzantine-robust algorithms, the selection of $\alpha$ has a significant impact on the underlying model and, specifically, on fairness towards clients that hold underrepresented data, which may inadvertently be considered outliers. In future work, we plan to analyze further the trade-off between robustness and the usage of client sample size in rectifying data unbalancedness. We also plan to investigate alternative forms of estimating client importance that may avoid client sample size altogether.
\clearpage
\section{MNIST experiment results}\label{sec:mnist}
This section provides ancillary results on our experiments conducted on the MNIST datased. These results are similar to the results of the Shakespeare experiments.
The experiment without any attackers is shown in Figure \ref{fig:mnist:no_attackers} and the executions with attackers are shown in Figure \ref{fig:mnist:attack}.
\vfill
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{no_attackers_2}
\caption{Accuracy by round without any attackers for the MNIST experiments. Curves correspond to preprocessing procedures and columns correspond to different aggregation methods. It can be seen that our method (dashed orange curve) remains comparable to the properly weighted mean estimators (solid blue curve) while ignoring clients' sample sizes (dotted green curve) is sub-optimal. This effect is pronounced when the unweighted median is used, since with our unbalanced partition it is generally very far from the mean.}
\label{fig:mnist:no_attackers}
\end{figure*}
\vfill
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{attackers_2}
\caption{Accuracy by round under Byzantine attacks for the MNIST experiments. Curves correspond to preprocessing procedures and columns correspond to different aggregation methods. In the first two rows Byzantine clients perform a label shifting attack with one and $10\%$ attackers, respectively. In the last two rows we repeat the experiment with a model negation attack.
\\\hspace{\textwidth}
We observe that even with a single attacker performing a trivial attack (first and third rows), using the weights directly (solid blue curve) is devastating while when our preprocessing method is used in conjunction with robust mean aggregations (dashed orange curve, two last columns) convergence remains stable even when there are actual $\alpha$ (=$10\%$) attackers (second and forth rows). We note that in some cases our method may be slightly less efficient compared with the preprocessing method that ignores sample size altogether (dotted green curve, second row, last column). This is to be expected because we allow Byzantine clients to potentially get close to $\alpha^*$-proportion ($50\%$, in this case) of the weight. However, our method is significantly closer to the optimal solution when there are no or only a few attackers (see Figure \ref{fig:mnist:no_attackers}). Moreover, when used in conjunction with robust mean aggregation methods it maintains their robustness properties.}
\label{fig:mnist:attack}
\end{figure*}
\section{Proofs}\label{proofs}
\subsection{Proof of theorem \ref{thm:partial-view}}\label{prf:partial-view}
First, in the scope of this proof we use a couple of additional notations:
\begin{itemize}
\item $\topa(\bm{V}, p)$: The collection of $p |\bm{V}|$ largest values in $\bm{V}$.
\item $\bm{\sum} \bm{V}$: The sum of all elements in $\bm{V}$.
\end{itemize}
We observe that $\mwp(\trunc(\bm{N}, U), \alpha) \le \alpha^*$ can be rewritten as
\begin{equation}
\begin{split}
&
\mwp(\trunc(\bm{N}, U), \alpha) = \frac{ \bm{\sum} \topa(\trunc(\bm{N}, U), \alpha) }{\bm{\sum} \trunc(\bm{N}, U)}
\label{eq:5}
= \frac{ \alpha \mathbb{E} [ \topa(\trunc(\bm{N}, U), \alpha) ] }{\mathbb{E} [\trunc(\bm{N}, U)]} \le \alpha^*
\end{split}
\end{equation}
Then we note that membership in $\topa(\trunc(\bm{N}, U), \alpha)$ can be viewed as a simple Bernoulli random variable with probability $\alpha$, for which we obtain the following bound using Hoeffding's inequality, $t\geq 0$:
\begin{equation}
\begin{split}
\Pr \big[|\{i \in [k] : X_i \in \topa(\trunc(\bm{N}, U), \alpha)\}| \leq (\alpha - t)k\big]
\leq \mathrm{e}^{-2 t^2 k}
\end{split}
\end{equation}
Therefore with $t = \varepsilon_1$, we have the following with $1 - \frac{\delta}{3}$ confidence:
\begin{equation}
\begin{split}
\bm{\sum} \topa(\{X_i\ |\ i \in [k]\}, \alpha)
\label{eq:6}
\leq
\bm{\sum} \{X_{(i)}\ |\ \lceil(1 - (\alpha - \varepsilon_1))k\rceil \le i \le k\}
\end{split}
\end{equation}
Using Hoeffding's inequality again, we can bound the expectation of $X_{(i)}\ |\ \lceil(1 - (\alpha - \varepsilon_1))k\rceil \le i \le k$
by $\varepsilon_2$ with $1 - \frac{\delta}{3}$ confidence and together with \eqref{eq:6} have that:
\begin{equation}
\begin{split}
\label{eq:7}
\textstyle \mathbb{E} [ \topa(\trunc(\bm{N}, U), \alpha) ]
\leq \frac{\sum^k_{i \gets \lceil(1 - (\alpha - \varepsilon_1))k\rceil} X_{(i)}}{k - \lceil(1 - (\alpha - \varepsilon_1))k\rceil + 1} + \varepsilon_2
\end{split}
\end{equation}
Then, using Hoeffding's inequality for the third time, $\mathbb{E} [\trunc(\bm{N}, U)]$ is bound from below by $\varepsilon_3$ with $1 - \frac{\delta}{3}$ confidence:
\begin{equation}
\begin{split}
\label{eq:8}
\mathbb{E} [\trunc(\bm{N}, U)] \geq \frac{1}{k} \sum_{i \in [k]} X_i - \varepsilon_3
\end{split}
\end{equation}
The proof is concluded by applying (\ref{eq:6}-\ref{eq:8}) to \eqref{eq:5} using the union bound.
\subsection{Proof of theorem \ref{thm:dist-bound}}\label{prf:dist-bound}
Using the fact that $\tilde{n} \leq U K$ we get:
\begin{equation*}
\begin{split}
& \norm{
\frac{1}{\dot{n}}\sum_{i \in [K]} \dot{n}_i F_i(w) -
\frac{1}{\tilde{n}}\sum_{i \in [K]} \tilde{n}_i F_i(w)} = \\ &
\norm{
\sum_{i : \dot{n}_i > U} \frac{\dot{n}_i}{\dot{n}} F_i(w) +
\frac{1}{\dot{n}} \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)
- \sum_{i : \dot{n}_i > U} \frac{U}{\tilde{n}} F_i(w)
- \frac{1}{\tilde{n}} \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)} \leq \\ &
\norm{
\sum_{i : \dot{n}_i > U} \frac{\dot{n}_i}{\dot{n}} F_i(w) +
\frac{1}{\dot{n}} \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)
- \sum_{i : \dot{n}_i > U} \frac{1}{K} F_i(w)
- \frac{1}{\tilde{n}} \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)} = \\ &
\norm{
\sum_{i : \dot{n}_i > U} \big(\frac{\dot{n}_i}{\dot{n}} - \frac{1}{K}\big) F_i(w) +
\big(\frac{1}{\dot{n}} - \frac{1}{\tilde{n}} \big) \sum_{i : \dot{n}_i \le U} \mathcal{L}(Z_i)}
\end{split}
\end{equation*}
\clearpage |
2,877,628,089,461 | arxiv | \section{Introduction}
The local quantum field theory (QFT) is understood as a canonical
Quantum Field Theory in flat space-time
\cite{Ryder}--\cite{Wein-2}. But in what follows it is
demonstrated that a flat geometry of space-time in the processes
of high energy physics is not ensured from the start, being based
on validity of the Einstein's Strong Equivalence Principle (EP).
However, this principle has its applicability boundaries. The
Planck scales present a natural (upper) bound for applicability
because at these scales a natural geometry of space-time,
determined locally by the particular metric
$g_{\mu\nu}(\overline{x})$, disappears due to high fluctuations of
this metric and is replaced by space-time (or quantum) foam.
\\ In \cite{Shalyt-NPCS19},\cite{Shalyt-NPCS20} the author
suggested a hypothesis that actually the real applicability
boundary of EP lies in a domain of the energies $E$ considerably
lower than the Planck energies. The principal objective of this
paper is to demonstrate that the hypothesis is true for some,
quite naturally arising, assumptions. Proceeding from the
afore-said, this condition sets the applicability boundaries for
the canonical QFT. Besides, direct inferences from the obtained
result are considered. Hereinafter, EP is understood as a Strong
Equivalence Principle.
\section{The QFT Applicability Boundaries and Equivalence Principle}
The canonical quantum field theory (QFT) \cite{Ryder}--
\cite{Wein-2} is a local theory considered in continuous
space-time with a plane geometry, i.e with the Minkowskian metric
$\eta_{\mu\nu}(\overline{x})$. In reality, any interaction
introduces some disturbances, introducing an additional local
(little) curvature into the initially flat Minkowskian space
$\mathcal{M}$. Then the metric $\eta_{\mu\nu}(\overline{x})$ is
replaced by the metric
$\eta_{\mu\nu}(\overline{x})+o_{\mu\nu}(\overline{x})$, where the
increment $o_{\mu\nu}(\overline{x})$ is small. But, when it is
assumed that EP is valid, the increment $o_{\mu\nu}(\overline{x})$
in the local theory has no important role and, in a fairly small
neighborhood of the point $\overline{x}$ in virtue EP.
\\The Einstein Equivalence Principle (EP) is a basic principle not
only in the General Relativity (GR) \cite{Einst1}--\cite{Wein-1},
but also in the fundamental physics as a whole. In the standard
formulation it is as follows: (\cite{Wein-1},p.68):
\\<<{\it at every space-time point in an arbitrary
gravitational field it is possible to choose
{\bf a locally inertial coordinate system} such that, within a
sufficiently small region of the point in question, the laws of
nature take the same form as in unaccelerated Cartesian coordinate
systems in the absence of gravitation}>>.
\\Then in (\cite{Wein-1},p.68) <<...There is also a question,
how small is {\bf ''sufficiently small''}. Roughly speaking, we
mean that the region must be small enough so that gravitational
field in sensible constant throughout it...>>.
\\However, the statement {\bf ''sufficiently small''}
is associated with another problem. Indeed, let $\overline{x}$ be
a certain point of the space-time manifold $\mathcal{M}$ (i.e.
$\overline{x}\in\mathcal{M}$) with the geometry given by the
metric $g_{\mu\nu}(\overline{x})$. Next, in accordance with EP,
there is some {\bf sufficiently small} region $\mathcal{V}$ of the
point $\overline{x}$ so that, within $\mathcal{V}$ it is supposed
that space-time has a flat geometry with the Minkowskian metric
$\eta_{\mu\nu}(\overline{x})$.
\\In essence, {\bf sufficiently small} $\mathcal{V}$ means that the
region $\mathcal{V}^{'}$, for which
$\overline{x}\in\mathcal{V}^{'}\subset\mathcal{V}$, satisfies this
condition as well. In this way we can construct the sequence
\begin{eqnarray}\label{Equiv-2}
...\subset\mathcal{V}^{''}\subset\mathcal{V}^{'}\subset\mathcal{V}.
\end{eqnarray}
The problem arises, is there any lower limit for the sequence in formula
(\ref{Equiv-2})?
\\The answer is positive. Currently, there is no doubt that
at very high energies (on the order of Planck energies $E\approx
E_p$), i.e. on Planck scales, $l\approx l_p$ quantum fluctuations
of any metric $g_{\mu\nu}(\overline{x})$ are so high that in this
case the geometry determined by $g_{\mu\nu}(\overline{x})$ is
replaced by the ''geometry'' following from {\bf space-time foam}
that is defined by great quantum fluctuations of
$g_{\mu\nu}(\overline{x})$,i.e. by the characteristic dimensions
of the quantum-gravitational region (for example,
\cite{Wheel}--\cite{Scard1}). The above-mentioned geometry is
drastically differing from the locally smooth geometry of
continuous space-time and EP in it is no longer valid
\cite{Gar1}--\cite{Garay2}.
\\From this it follows that the region $\mathcal{V}_{\overline{r},\overline{t}}$
with the characteristic spatial dimension $\overline{r}\approx
l_p$ (and hence with the temporal dimension $\overline{t}\approx
t_p$) is the lower (approximate) limit for the sequence in
(\ref{Equiv-2}).
\\It is difficult to find the exact lower limit for the sequence in formula
(\ref{Equiv-2})--it seems to be dependent on the processes under
study. Specifically, when the involved particles are considered
to be point, their dimensions may be neglected in a definition of
the EP applicability limit. When the characteristic spatial
dimension of a particle is $\mathbf{r}$, the lower limit of the
sequence from formula (\ref{Equiv-2}) seems to be given by the
region $\mathcal{V}_{\mathbf{r'}}$ containing the above-mentioned
particle with the characteristic dimensions
$\mathbf{r'}>\mathbf{r}$, i.e. the space EP applicability limit
should always be greater than dimensions of the particles
considered in this region. By the present time, it is known that
spatial dimensions of gauge bosons, quarks, and leptons within the
limiting accuracy of the conducted measurements $<10^{-18}m$.
Because of this, the condition $\mathbf{r'}\geq 10^{-18}m$ must be
fulfilled. In addition, the radius of interaction of particles
$\mathbf{r}_{int}$ must be taken into account in quantum theory.
And this fact also imposes a restriction on considering concrete
processes in quantum theory. However, the interactions radii of
all known processes lie in the energy scales $E\ll E_p.$
\\ At the present time there is a series of the results
demonstrating that EP may be violated at the energies $E$
considerably lower than $E_p$(for example, the quantum phenomenon
of neutrino oscillations in a gravitational background
\cite{gasperini} \cite{ahluwalia} \cite{mureika}
and others \cite{Flamb},\cite{Xiao},\cite{Dai}).
\\As QFT is a local theory applicable {\it only} to
space-time with a flat geometry determined by the Minkowskian metric
$\eta_{\mu\nu}(\overline{x})$,the applicability boundary
EP may be considered the applicability boundary of QFT as well.
\\{\bf Main Hypothesis}
\\It is assumed that in the general case EP, and consequently, QFT
is valid for the locally smooth space-time {\it only} if all the
energies $E$ of the particles are satisfied the necessary
condition
\begin{eqnarray}\label{Equiv-3}
E\ll E_p,
\end{eqnarray}
In the following section this hypothesis is proven in the assumption that
space-time foam consists of micro black holes ({\bf mbh})
with the event horizon radius $r\approx l_p$ and mass $m\approx
m_p$.
\\
\\{\bf Remark 2.1}
\\ Why in canonical QFT it is so important never forget
about the fact that space-time has a flat geometry, or the same
possesses the Minkowskian metric $\eta_{\mu\nu}(\overline{x})$?
Simply, in the contrary case we should refuse from some fruitful
methods and from the results obtained by these methods in
canonical QFT, in particular from Wick rotation \cite{Wein-2}. In
fact, in this case the time variable is replaced by $t\mapsto
it\doteq t_E$, and the Minkowskian metric
$\eta_{\mu\nu}(\overline{x})$ is replaced by the four-dimensional
Euclidean metric
\begin{eqnarray}\label{W-1}
ds^{2}=dt^{2}_E+dx^{2}+dy^{2}+dz^{2}.
\end{eqnarray}
Clearly, such replacement is possible only in the case when from
the start space-time (locally) has a flat geometry, i. e.
possesses the Minkowskian metric $\eta_{\mu\nu}(\overline{x})$.
This is another argument supporting the key role of the EP
applicability boundary. Otherwise, when we go beyond this
boundary, Wick rotation becomes invalid. Naturally, some other
methods of canonical QFT will lose their force too.
\section{The Strong Equivalence Principle, Black Holes, and QFT}
It is supposed that a large (i. e, classical)
four-dimensional Schwarzschild black hole is existent with the metric
\begin{equation}\label{bh-1}
ds^{2}=\left( 1-\frac{2MG}{r}\right) dt^{2}-\left(
1-\frac{2MG}{r}\right) ^{-1}dr^{2}-r^{2}d\Omega ^{2},
\end{equation}
where $M$ is the mass of this black hole, and the Schwarzschild
horizon radius $r_{BH}$ is defined by
\begin{equation}
r_{BH}=2MG. \label{bh-2}
\end{equation}
As shown in \cite{Singl-1},\cite{Singl-2}, EP is violated for an
observer distant from the black hole event horizon. Considering
our objective, it seems expedient to give in brief the main
results from \cite{Singl-1},\cite{Singl-2}.
\\ In view of the Unruh effect, an accelerating observer
does detect thermal radiation (so-called Unruh radiation)
with the Unruh temperature given by \cite{unruh}
\begin{eqnarray}\label{Un-1}
T_{Unruh} = \frac{\hbar a}{2 \pi},
\end{eqnarray}
where $a=|{\bf a}|$ is a corresponding acceleration.
\\When an observer is at the {\it fixed} distance,
$r>r_{BH}$, from a Schwarzschild black hole of mass $M$ and event
horizon radius $r_{BH}=2GM$, then, due to the existence of Hawking
radiation \cite{hawking}, the observer will measure radiation with
thermal spectrum and a temperature given by formula \cite{Singl-1}
\begin{eqnarray}\label{Un-2}
T_{H,r}=\frac{\hbar }{8 \pi G M \sqrt{1- \frac{r_{BH}}{r}}}~,
\end{eqnarray}
where $r>r_{BH}$.
\\ In the foregoing formulae and in what follows, we use the normalization
$c=k_B=1$.
\\Besides, in \cite{Singl-1} it is shown that an observer,
positioned at the fixed distance $r>r_{BH}$ from the
above-mentioned black holes and measuring Hawking temperature with
the value $T_{H,r}$, experiences the local acceleration
\begin{eqnarray}\label{Un-3}
a_{BH,r} = \frac{1}{\sqrt{1- \frac{r_{BH}}{r}}}
\left(\frac{r_{BH}}{2r^2} \right) ~.
\end{eqnarray}
Another observer in the Einstein elevator, moving with
acceleration through Minkowskian space-time, will measure the same
acceleration toward the floor of the elevator, thermal radiation
with the Unruh temperature given by formula (\ref{Un-1}). As
shown in \cite{Singl-1}, $a_{BH}$ is coincident with the quantity
$a$ from the formula in (\ref{Un-1}). Then substituting the
acceleration $a=a_{BH}$ from formula (\ref{Un-3}) into formula
(\ref{Un-1}), we can obtain a formula for $T_{Unruh,r}$ in this
case \cite{Singl-2}:
\begin{eqnarray}\label{Un-4}
T_{Unruh,r} = \frac{\hbar}{2 \pi \sqrt{1- \frac{r_{BH}}{r}}}\left(
\frac{r_{BH}}{2r^2}\right) ~.
\end{eqnarray}
If EP is valid, the quantities $T_{Unruh,r}$ from formula
(\ref{Un-4}) and $T_{H,r}$ in (\ref{Un-2}) should be coincident
for $r>r_{BH}$ to a high degree of accuracy. However, we see that
this is not true. In \cite{Singl-2}, e.g. for $r=4GM=2r_{BH}$, we
have $T_{H,r}=4T_{Unruh,r}$.
\\So, far from the event horizon, EP is not the case.
Moreover, violation of EP is the greater the farther it is from
the black hole event horizon. Indeed, for an observer at the
distance $r>r_{BH}$ we can write
$r=\mathcal{\alpha}(r)GM=\frac{1}{2}\mathcal{\alpha}(r)r_{BH},\mathcal{\alpha}(r)>2$.
Then
\begin{eqnarray}\label{Un-4.1}
T_{Unruh,r} = \frac{\hbar}{2\pi \mathcal{\alpha}^{2}(r)G M
\sqrt{1-\frac{2}{\alpha(r)}}}.
\end{eqnarray}
In this way $T_{H}/T_{Unruh,r}=\mathcal{\alpha}^{2}(r)/4$. And the
ratio is the greater, the higher $\mathcal{\alpha}(r)$,i.e. the
farther from horizon the observer is. Next, for compactness, we
denote $T_{Unruh,r}$ in terms of $T_{U,r}$. Of course, in this
case we bear in mind only an observer at a sufficiently great but
finite distance from a black hole, i.e. only when a gravitational
field is thought significant and must be taken into consideration.
So, in the general case $r_{BH}<r\ll\infty$, whereas in the case
of a distant observer we have
\begin{eqnarray}\label{Un-5.1}
r_{BH}\ll r\ll\infty.
\end{eqnarray}
Obviously, this case of violated EP is not directly associated
with the {\bf Main Hypothesis} concerning the boundaries of EP
validity (formula (\ref{Equiv-3})) from the previous section,
because in \cite{Singl-1},\cite{Singl-2} consideration is given
to a large black hole with the event horizon radius $r_{BH}$ much
greater than Planck length $r_{BH}\gg l_p$ at sufficiently low
energies.
\\Really, the resulting distribution of the particles emitted
by a black hole has the form (last formula on p.122 in
\cite{Mukhanov})
\begin{eqnarray}\label{Un-6}
n_E=\Gamma_{gb}[exp(\frac{E}{T_{H}})-1]^{-1},
\end{eqnarray}
where $n_E$ is the number of particles with the energy $E$ and
$\Gamma_{gb}<1$ is the so-called {\it greybody factor}. As the
black hole mass $M$ is large, the temperature $T_{H}$ is low, and
then from the last formula it follows that arbitrary large values
of $n_E$ will be given only by particles with the energies $E$
close to a small value of $T_{H}$.
\\The principal result from the remarkable papers
\cite{Singl-1},\cite{Singl-2} may be summarized as follows:
\\{\bf Comment 3.1}
\\In any point of space-time that is in a field of a large classical Schwarzschild black hole,
and in the cases when this field must be taken into consideration,
it is impossible to remove this field in the vicinity of the point
even locally, i.e. to consider space-time as flat.
\\
\\{\bf Comment 3.2}
\\It is important to refine some formulations from \cite{Singl-1},\cite{Singl-2}.
Specifically, if $r\rightarrow r_{BH}$, then $T_{H,r}\rightarrow
\infty,T_{Unruh,r}\rightarrow\infty$ in formulae (\ref{Un-2}) and
(\ref{Un-4}), respectively. Note that for $r\rightarrow r_{BH}$
these temperatures become infinite
$T_{H,r}=\infty,T_{Unruh,r}=\infty$. Based on this fact, in
\cite{Singl-1},\cite{Singl-2} it is inferred ''that the
equivalence principle is restored on the horizon''. But this
statement is not correct. Restoration of EP is not following from
the fact that the above temperatures take infinite values. We can
only state that temperature on the BH horizon and in its vicinity
cannot be the parameter detecting a deviation from EP. In the
opposite case one can arrive at violation: on the black hole event
horizon, where a gravitational field is very large in value, EP
holds, whereas far from the event horizon, where a gravitational
field is much weaker, this principle is violated.
\\
\\Let us return to high energy physics and to the subject of the previous section.
One of the preferable models for space-time foam is the model
based on the assumption that its unit cells are {\bf mbh}, with
radius and mass on the order of the Planck (for example,
\cite{Scard1},\cite{Scard2}, \cite{Scard3}. Of great importance
for {\bf mbh} are the quantum-gravitational effects and the
corresponding quantum corrections of black hole thermodynamics at
Planck scale (for example, \cite{Nou}).
\\ Then, in line with formula (20) in \cite{Nou},
we have minimal values for radius and mass of a black hole
\begin{equation}\label{min}
r_{min}=\sqrt{\frac{e}{2}}\alpha^{'} l_p,\quad m_{min}\doteq
m_0=\frac{\alpha^{'}\sqrt{e}}{2\sqrt{2}}m_p,
\end{equation}
where the number $\alpha^{'}$ is on the order of 1, and in
\cite{Nou} we take the normalization $\hbar=c=k_B=1$ in which
$l_p=m_{p}^{-1}=T_{p}^{-1}=\sqrt{G}.$
\\From (\ref{min}) it directly follows that the formula
for the event horizon radius $r=2MG$, valid for large classical
black holes, will be valid in the case when we include the
quantum-gravitational effects for {\bf mbh} because
$r_{min}=2m_0G$. Such a black hole of a minimal size is associated
with a maximal temperature (formula (24) in \cite{Nou}):
\begin{equation}\label{tempmax}
T_{H}^{max}=\frac{T_{p}}{2\pi \sqrt{2}\alpha^{'}}.
\end{equation}
A black hole satisfying the formulae
(\ref{min}),(\ref{tempmax}) is termed as {\it minimal} (or {\it Planck}).
\\ Without loss of generality,
it is assumed that for event horizon radii and masses of {\bf mbh}
the following is valid:
\begin{equation}\label{min.2}
r_{mbh}\approx r_{min},m_{mbh}\approx m_0,
\end{equation}
i.e. {\bf mbh} are {\it Planck} black holes.
\\ For the energies $E$ somewhat lower than the Planck energies
(i.e., $E\ll E_p$) involved in the condition (\ref{Equiv-3}), a
semiclassical approximation is valid. This means that, on
substitution of {\bf mbh} with the mass $m_{mbh}$ for a large
(classical) black hole with the mass $M$, in the case under study
the results, substantiated when an observer uses the standard
Unruh-Dewitt detector in radiation measurement for coupled to a
massless scalar field \cite{BD},\cite{akhmedov}, are valid with
the corresponding quantum corrections \cite{Keifer}.
\\ Let us revert to the formulae from \cite{Singl-1},\cite{Singl-2}.
In particular, to formula (\ref{Un-3}) for the real
acceleration measured by an observer who is positioned at the fixed
distance $r\gg r_{BH}$ in the Schwarzschild space-time, given
by (formula (14) in \cite{Singl-1}, formula (9.170) in
\cite{carroll})
\begin{equation}\label{acc-sch}
a_{BH,r}=a_S = \frac{{\sqrt{\nabla _\mu V \nabla ^\mu V}}}{V} =
\frac{MG}{r^2 \sqrt{1-2MG/r}}=\frac{MG}{r^2V},
\end{equation}
where it is supposed that a static observer at the radius $r$ moves
along orbits of the time-like Killing vector
$\textit{K}=\partial_{t}$ and
$V=\sqrt{-\textit{K}_{\mu}\textit{K}^{\mu}}=\sqrt{1-2MG/r}$ is the
red-shift factor for the Schwarzschild space-time (p.413 in
\cite{carroll}).
\\ How changes formula (\ref{acc-sch}) on going to {\bf
mbh}? It is clear that the condition $r\gg r_{BH}$ is replaced by
the condition $r\gg r_{mbh}$ (corresponding to the condition $E\ll
E_p$ and semiclassical approximation),$M$ is replaced by
$m_{mbh}$,the red-shift factor $V$ should be replaced by $V_q$,
where $V_{q}$ is the quantum deformation of $V$ with regard to
quantum corrections in the field {\bf mbh}. Then, for {\bf mbh},
formula (\ref{acc-sch}) is of the form
\begin{equation}\label{acc-sch.q}
a_{S,q} =\frac{m_{mbh}G}{r^2V_q},
\end{equation}
where $a_{S,q}$ is the real acceleration with regard to the
quantum corrections measured by a distant observer in the field
{\bf mbh}.
\\ Clearly, formula (\ref{Un-2}) for $T_{H,r}$, due to formulae for
the red-shift factor $V$, in the general case may be given as
\begin{eqnarray}\label{Un-2.new}
T_{H,r}=\frac{\hbar }{8 \pi GM V}~.
\end{eqnarray}
Then its quantum analog, i.e.
the corresponding formula for temperature in the field {\bf mbh}, for $r\gg r_{mbh}$ is as follows:
\begin{eqnarray}\label{Un-2.new.q}
T_{H,r,q}=\frac{\hbar}{8\pi Gm_{mbh}V_q}.
\end{eqnarray}
In virtue of formula (\ref{acc-sch}), formula
(\ref{Un-4}) takes the form
\begin{eqnarray}\label{Un-4.new1}
T_{U,r} = \frac{\hbar}{2 \pi \sqrt{1- \frac{r_{BH}}{r}}}\left(
\frac{r_{BH}}{2r^2}\right)=\frac{\hbar}{2 \pi V}\left(
\frac{r_{BH}}{2r^2}\right).
\end{eqnarray}
For $r>r_{BH}$ and due to formula (\ref{Un-4.1}), we have
\begin{eqnarray}\label{Un-4.1.new1}
T_{U,r}=\frac{\hbar}{2\pi \mathcal{\alpha}^{2}(r)G M
\sqrt{1-\frac{2}{\alpha(r)}}}= \frac{\hbar}{2\pi
\mathcal{\alpha}^{2}(r)G M V}.
\end{eqnarray}
As indicated above, for $r\gg r_{BH}$ we have $\alpha(r)\gg 1$.
\\ What are the changes on going to {\bf mbh}?
\\ Considering the case $r\gg r_{mbh}$ and semiclassical
picture, we again come to $\alpha(r)\gg 1$, whereas formula
(\ref{Un-4.1.new1}) is replaced by formula
\begin{eqnarray}\label{Un-4.1.new1,q}
T_{U,r,q} = \frac{\hbar}{2\pi \mathcal{\alpha}^{2}(r)G
m_{mbh}V_q}.
\end{eqnarray}
Proceeding from the above, we have \begin{eqnarray}\label{BH-1}
\frac{T_{H,r,q}}{T_{U,r,q}}=\frac{T_{H,r}}{T_{U,r}}=\frac{\mathcal{\alpha}^{2}(r)}{4}
\end{eqnarray}
Formula (\ref{BH-1}) points to the fact that, within the scope of
a semiclassical approximation, relations of a black hole
temperature to the Unruh temperature for a distant observer in the
case of a large (classical) black hole and {\bf mbh} are
coincident because these quantities are dependent on the same
factors:
\\ first, on $1/MV$ and, second, on $1/m_{mbh}V_q$.
\\ Note that in this consideration there
s no need to have an explicit formula for $V_q$ as this quantity
is not involved in the key expression (\ref{BH-1}). Specifically,
to derive an explicit expression for $V_q$, we can use the results
from \cite{Sol} on quantum deformation of the Schwarzschild
solution due to spherically symmetric quantum fluctuations of the
metric and the matter fields. In this case the Schwarzschild
singularity at $r=0$ is shifted to the finite radius
$r_{min}\approx r_{mbh}\propto l_p$, where the scalar curvature is
finite. In this way the results from \cite{Sol} correlate well
with the results from \cite{Nou}.
\\Quantum corrections at Planck scales were obtained in \cite{Nou}
proceeding from validity of the Generalized Uncertainty Principle
(GUP)\cite{GUPg1}--\cite{Tawf}. But the results presented in this
work are independent of this aspect. Actually, during studies of
black hole thermodynamics at Planck scales with the use of other
methods \cite{Corr},\cite{LQG} (differing from those in
\cite{Nou}),in particular, Loop Quantum Gravity (LQG)\cite{LQG},
the obtained results were similar to \cite{Nou}. Because of this,
for {\bf mbh} with all the thermodynamic characteristics
(mass,radius,temperature,...) on the order of the corresponding
Planck quantities, all the calculations in this section are valid.
As noted above, far from horizon of {\bf mbh}, i.e. at the
energies $E\ll E_p$ (\ref{Equiv-3}), the results from
\cite{Singl-1},\cite{Singl-2} remain valid in this case as well.
\\Next, similar to \cite{Scard1}, we assume that in every cell of
space-time foam a micro black hole ({\bf mbh}) with a typical
gravitational radius of $r_{min}\propto l_p$ may be present. Then,
in according with the results in \cite{Singl-1},\cite{Singl-2} and
in virtue of the formula (\ref{BH-1}) we come to violation of the
strong EP for distance $r$, satisfying the condition
\begin{eqnarray}\label{Un-5.2}
l_p\ll r\ll\infty,
\end{eqnarray}
that is equivalent to $\widetilde{E}_r\ll E_p$ for the energies
$\widetilde{E}_r$ associated with the scale of $r$.
\\In the last formula it is implicitly (and purely conditionally)
assumed that a minimum length is equal to $l_p$ and to $r_{min}$.
But, as noted above, in the general case we have $r_{min}\propto
l_p$, i.e., the order is similar to that of $l_p$. Specifically,
in \cite{Rong-1} by natural assumptions it has been demonstrated
that the minimum length may be twice and more as great as the
Planck length. It is obvious that all the above calculations and
derivations of the present work are independent of the specific
value of $r_{min}$.
\\In this way, if the quantum foam structure is determined by {\bf
mbh}, the applicability of QFT is limited to the energies
$E<\widetilde{E}_r\ll E_p$ and the formula (\ref{Equiv-3}) is the
case. This supports the {\bf Main Hypothesis} from Section 2
within the assumption concerning the quantum foam structure made
in this section.
\\{\bf Comment 3.3}
\\ In the case of {\bf mbh} {\bf Comment 3.2} is absolutely clear.
In fact, at a horizon of {\bf mbh},i.e. for $r=r_{min}$,
$T_{H,r}=T_{Unruh,r}=\infty$ similar to large black holes but ,
naturally, without any restoration of EP as the domain
$r=r_{mbh}\approx r_{min}\propto l_p$ is the region of Planck
energies or of quantum foam, where EP in its canonical formulation
becomes invalid. It is obvious that at the event horizon
$r=r_{mbh}$ of {\bf mbh} and in its vicinity a gravitational field
becomes very strong due to quantum effects and nothing could
destroy it.
\section{Some Immediate Consequences}
{\bf 4.1} Based on the above results, all the energies $E$ we can classify into 3 groups:
\\
\\a)low energies $0<E\leq\widetilde{E}\ll E_p$
-- energies, for which the Strong Equivalence Principle is valid
in virtue of formula (\ref{Equiv-3}), and hence this energy
interval sets the QFT applicability boundaries.
\\a1) Since $\widetilde{E}\ll E_p$,
it is natural to assume that $\widetilde{E}\approx 10^{-N}E_p$,
where $N\geq 2$. Obtaining of more accurate estimates for $N$ is a
separate problem;
\\
\\b)intermediate energies $\widetilde{E}<E<E_p$ -- energies,
for which the Strong Equivalence Principle and, consequently QFT,
becomes invalid but the corresponding scales are greater than the
Planck. It can be assumed that QFT in this energy range will be a
theory in a gravitational field that could not be destroyed even
locally. In the case under study it is assumed that this field is
created by {\bf mbh}. Impossibility of destroying this field even
locally is associated with large quantum corrections for the
corresponding quantities which should be taken into consideration
at these energies \cite{Nou},\cite{Corr},\cite{LQG}.
\\ Let us call the energy scale $\widetilde{E}<E< E_p$ as
{\it prequantum gravity phase};
\\
\\c)high (essentially maximal) energies $E\approx E_p$ or
$E>E_p$. This interval is the region of quantum gravity energies.
\\
\\ Next note that, as all the experimentally involved energies $E$ are low,
they satisfy condition a) or b). Specifically, for LHC, maximal
energies are $\approx 10TeV=10^{4}GeV$, that is by 15 orders of
magnitude lower than the Planck energy $\approx 10^{19}GeV$.
Moreover, the characteristic energy scales of all fundamental
interactions also satisfy condition a). Indeed, in the case of
strong interactions this scale is $\Lambda_{QCD}\sim 200MeV$; for
electroweak interactions this scale is determined by the vacuum
average of a Higgs boson and equals $\upsilon\approx 246GeV$;
finally, the scale of the (Grand Unification Theory (GUT))
$M_{GUT}$ lies in the range of $\sim10^{14}GeV--10^{16}GeV$.
\\ It should be noted, however, that on validity of assumption a1)
the energy scale $M_{GUT}$ lies within the applicability region of
the energy group a) and hence of QFT. Provided the EP
applicability boundaries are lying at considerably lower energies,
a study of GUT necessitates a theory with (even locally)
unremovable curvature.
\\At the same time, it is clear that the requirement of the
Lorentz-invariant QFT, due to the action of Lorentz boost (or same
hyperbolic rotations) (for example formula (3) in \cite{Akhm}),
results in however high momenta and energies. But it has been
demonstrated that unlimited growth of the momenta and energies is
impossible because in this case we fall within the energy region,
where the conventional quantum field theory
\cite{Ryder}--\cite{Wein-2} is invalid.
\\Note that at the present time there are experimental indications
that Lorentz-invariance is violated in QFT on passage to higher
energies (for example, \cite{Kost}). Besides, one should note
important recent works associated with EP applicability boundaries
and violation in nuclei and atoms at low energies (for example
\cite{Flamb}). Proceeding from the above, the requirement for
Lorentz-invariance and EP is possible only within the scope of the
condition (\ref{Equiv-3}).
\\
\\{\bf 4.2} Proceeding from the above results,
it is inferred that the well-known QFT
\cite{Ryder}--\cite{Wein-2}, from the start, is a
ultraviolet-finite theory with the natural cutoff parameter
$l_{\widetilde{E}}\propto \hbar/\widetilde{E}.$ Note that the
quantum-gravitational parameter $r_{mbh}\propto l_p$ is beyond the
applicability limits of QFT.
\\
\\{\bf 4.3} In the present approach it is of interest to study
the problem of {\it asymptotic safety} introduced by Steven
Weinberg in \cite{Wein-UV}. We use the definition of this notion
given in (\cite{Keifer--book},p.67): ''A theory is said to be
asymptotically safe if all essential coupling parameters $g_j$
(these are the ones that are invariant under field redefinitions)
approach, for energies $k\rightarrow\infty$, a fixed point where
at least one of them does not vanish.'' If initially it has been
assumed that $r_{min}$ is considered within the scope of
(GUP)\cite{GUPg1}--\cite{Tawf},this definition necessitates
certain refinements. In particular, if (GUP) supposes the
existence of a maximum momentum $p_{max}$, as in \cite{Nozari} or
(Section V in \cite{Tawf}),it is clear that the condition
$k\rightarrow\infty$ can not be fulfilled. So, the condition
$k\rightarrow\infty$ should be replaced by the condition
$k\rightarrow p_{max}.$ Most often, it is assumed that the
momentum $p_{max}$ is on the order of the Planck momentum, i.e.,
we have $p_{max}\propto p_{pl}$. However, in the most general case
the quantity $p_{max}$ may be even trans-Planck.
\\ When $p_{max}$ is inexistent (i.e. $p_{max}=\infty$),
still by this approach of {\it asymptotic safety} the problem
should be reformulated in accordance with the fact (shown above)
that, beginning with the energies $E,E>\widetilde{E}$, a theory
must be considered as QFT in curved space-time in a field created
by {\bf mbh}.
\\In his further works the author is planning to study this problem within the scope of this approach in greater detail.
\\
\\{\bf 4.4} It is possible to correct the estimates
obtained within the scope of the known QFT by means of the
condition (\ref{Equiv-3}). Let us consider a typical example.
\\In his well-known lectures \cite{Wein-cosm} at the Cornell University
Steven Weinberg considered an example of calculating, within the
scope of QFT, the expected value for the vacuum energy density
$<\rho>$ that is proportional to the cosmological term $\lambda$.
To this end, zero-point energies of all normal modes of some field
with the mass $m$ are summed up to the wave number cutoff
$\Lambda\gg m$ for the selected normalization $\hbar=c=1$ (formula
(3.5) in \cite{Wein-cosm}):
\begin{eqnarray}\label{Equiv-4}
<\rho>=\int_0^\Lambda\frac{4\pi
k^{2}dk}{(2\pi)^{3}}\frac{1}{2}\sqrt{k^{2}+m^{2}}\simeq
\frac{\Lambda^{4}}{16\pi^{2}}.
\end{eqnarray}
Assuming, similar to \cite{Wein-cosm}, that GR is valid at all the
energy scales up to the Planck, we have the cutoff $\Lambda\simeq
(8\pi G)^{-1/2}$ and hence (formula (3.6) in \cite{Wein-cosm})
leads to the following result:
\begin{eqnarray}\label{Equiv-5}
<\rho>\approx 2\cdot 10^{71}GeV^{4},
\end{eqnarray}
that by $10^{118}$ orders of magnitude differs from the well-known
experimental value for the vacuum energy density
\begin{eqnarray}\label{Equiv-6}
<\rho_{exp}>\preceq 10^{-29}\mathrm{g}/cm^{3}\approx
10^{-47}GeV^{4}.
\end{eqnarray}
Here $G$ is a gravitational constant.
\\It is clear that in this case the condition (\ref{Equiv-3})
is not fulfilled and this leads to such a monstrous discrepancy
with $<\rho_{exp}>$. Based on the afore-said, the following
estimate for $<\rho>$ is more correct:
\begin{eqnarray}\label{Equiv-4.0}
<\rho_{\widetilde{E}}>\doteq\int_{0}^{\Lambda_{\widetilde{E}}}\frac{4\pi
k^{2}dk}{(2\pi)^{3}}\frac{1}{2}\sqrt{k^{2}+m^{2}}\simeq
\frac{\Lambda_{\widetilde{E}}^{4}}{16\pi^{2}},
\end{eqnarray}
where $\Lambda(\widetilde{E})$--cut-off parameter of the
corresponding energy $\widetilde{E}$ from point a) in {\bf 4.1}.
\\ Of course, the main contribution into the integral
in the right side of (\ref{Equiv-4}) is made by high energies
$\widetilde{E}<E< E_p$ from point b) in {\bf 4.1},which are not
involved in formula (\ref{Equiv-4.0}). Consequently, it seems
possible that $<\rho_{\widetilde{E}}>\ll <\rho>$ and hence
$<\rho_{\widetilde{E}}>$ may be much closer to $<\rho_{exp}>$ than
$<\rho>$.
\\ In the whole, the methods of a quantum field theory may be effectively used to
obtain the vacuum energy density. In his very interesting work
\cite{Rong-2} the author, based on the quantum hoop conjecture,
has obtained a natural cutoff for the vacuum energy of a scalar
field, also giving an estimate for $<\rho>$ much closer to
$<\rho_{exp}>$ than in formulae(\ref{Equiv-4}),(\ref{Equiv-5}).
It is interesting to know how close are the estimates for the
vacuum energy density in \cite{Rong-2} and in formula
(\ref{Equiv-4.0}). To this end, we first need a sufficiently
accurate estimate of the quantity $\widetilde{E}$ in line with the
QFT boundaries applicability. Besides, it is important to find
whether the methods of QFT are enough to obtain $<\rho_{exp}>$ or
some additional assumptions will be required, specifically, the
Holographic Principle applying to the Universe
\cite{Sussk1}--\cite{shalyt-IJMPD} .
\section{Conclusion}
Thus, within the scope of natural assumptions, in this paper it is
demonstrated that the applicability boundary of the well-known QFT
is lying in the region of energies considerably lower than the
Planck energies, i.e. the canonical QFT
\cite{Ryder}--\cite{Wein-2} is an ultraviolet-finite theory.
\\In this paradigm it is important to understand the way to transform
the well-known results for ultraviolet regularization,
renormalization, and so on from QFT within the scope of the
applicability boundary $\widetilde{E}$ of QFT indicated in point
a) of the preceding section. Possibly, this boundary will be
dependent on a nature of the processes under study in high energy
physics.
\begin{center}
{\bf Conflict of Interests}
\end{center}
The author declares that there is no conflict of interests
regarding the publication of this paper.
\\
\begin{center}
{\bf Acknowledgments}
\end{center}
The author would like to express his gratitude to the Reviewers of this work for their valuable remarks and recommendations.
|
2,877,628,089,462 | arxiv | \section{Introduction}
Quantum entanglement has recently been a central topic in theoretical physics.
Many aspects of the dynamics of entanglement have been recently studied, such as
ballistic spreading of the entanglement in integrable \cite{amico,Calabrese:2005,Chiara:2006,Hartman:2013} and nonintegrable \cite{Kim:2013,Liu:2014} systems,
logarithmic spreading in many-body localized systems \cite{Znidaric:2008,Bardarson:2012}, and sub-ballistic spreading due to quantum Griffiths effects \cite{VHA}.
In many of these examples, the entanglement spreads more rapidly than
conserved quantities
that must be transported by currents.
Much of the previous work on the dynamics of entanglement, however, has emphasized far-from-equilibrium regimes,
particularly those following a quantum quench.
Here, we instead explore the entanglement dynamics near equilibrium in nonintegrable, thermalizing spin chains \cite{banuls} of finite length.
For example, if we start in a nonentangled initial pure state, the entanglement entropy grows linearly with time at early time
due to the ``ballistic'' spreading of entanglement \cite{Kim:2013,Liu:2014}, but then
saturates to its ``volume-law'' equilibrium value at long time. The lower two data sets in Fig.~\ref{fig:ent_evolution} illustrate this.
In the limit of a long spin chain, this isolated system is reservoir that thermalizes all of its subsystems. Then the extensive part of the final equilibrium value of the entanglement entropy is equal to the thermal equilibrium entropy
at the corresponding temperature, and that temperature is set by the total energy of the initial state.
We call this process, in which the entanglement entropy approaches the thermal
equilibrium entropy, the ``thermalization of entanglement'' \cite{ETH}.
In this paper we focus on the late time, near equilibrium regime of the
entanglement dynamics, as well as the spontaneous fluctuations of entanglement in pure states sampled from the equilibrium density operator.
In section II, we introduce a nonintegrable, quantum-chaotic model Hamiltonian and its corresponding Floquet operator, where the extensive energy conservation
is removed. In section III, we first examine the distribution of entanglement entropy of eigenstates of the Hamiltonian and the Floquet operators,
finding that the presence of the conservation law affects the distribution. In section IV, we study the dynamics of entanglement entropy
near equilibrium. We study three scenarios: starting from a product state of two random pure states, starting from generalized Bell states with
two different pairing schemes, and the autocorrelation of the spontaneous fluctuations of the entanglement entropy. In all cases, we find the Floquet system
thermalizes entanglement faster than the Hamiltonian system. In section V, we summarize our findings.
\begin{figure}
\includegraphics[width=1.0\linewidth]{Hamiltonian_Floquet_14_compare_linear.pdf}
\centering
\caption{(color online) Time evolution of the entanglement entropy for $L=14$ for: product random (PR) initial states under Floquet dynamics (2) with $\tau=0.8$ (red line with circles) as well as Hamiltonian dynamics (\ref{eqn:Hamiltonian}) (blue line with down triangles); ``generalized Bell'' initial states made from pairs of random pure (RP) states (green line with up triangles) and from ``oppositely paired'' (OP) states (purple line with diamonds) both under Hamiltonian dynamics.
Each case is averaged over 400 initial pure states, and the error estimates are too small to be visible in this figure. See main text for more details.}
\label{fig:ent_evolution}
\end{figure}
\section{Models}
To study a system that is robustly nonintegrable and strongly thermalizing, we choose the
spin-1/2 Ising chain
with both longitudinal and transverse fields. Its Hamiltonian is
\begin{align}
\label{eqn:Hamiltonian}
H=\sum_{i=1}^L g\sigma_i^x + \sum_{i=1}^L h\sigma_i^z + \sum_{i=1}^{L-1}J\sigma_i^z\sigma_{i+1}^z ~,
\end{align}
where $\sigma_i^x$ and $\sigma_i^z$ are Pauli matrices at site $i$.
We use open boundary conditions and set the parameters to $(g,h,J) = (0.9045, 0.8090, 1.0)$,
for which this model has been shown to be robustly nonintegrable and strongly thermalizing for system sizes
readily accessible to exact diagonalization studies \cite{Kim:2013,KIH}.
The only conservation laws that this system is known to have at this parameter choice \cite{E8}
(other than projections on to its exact eigenstates) are
total energy, and parity under spatial reflection of the chain ($i\rightarrow (L+1-i)$).
This system's ``hydrodynamics'' are simply its conserved energy moving diffusively and subject to random local currents due to the system's
quantum-chaotic unitary dynamics.
We set the Planck constant $\hbar$ to unity
so that time and energy have inverse units of each other, and all energies and frequencies are in units of the interaction $J=1$.
To explore the effects of removing the conservation of total energy, we also study a Floquet system that is a modification of (\ref{eqn:Hamiltonian}).
We decompose the Hamiltonian into two parts, $H_z = \sum_i (h \sigma^z_i + \sigma^z_i \sigma^z_{i+1})$
and $H_x = \sum_i g \sigma^x_i$. We periodically drive the system with a time-dependent Hamiltonian that is in turn $H(t)=2H_z$ for
a time interval of $\tau/2$
and then $H(t)=2H_x$ for the next $\tau/2$, and repeat. The time-averaged Hamiltonian is thus unchanged, but the periodic switching changes the energy conservation
from conservation of the extensive total energy to conservation of energy only modulo $(2\pi/\tau)$. This change removes the diffusive transport of energy as a slow
``hydrodynamic'' mode while otherwise changing the model as little as possible.
The Floquet operator that produces the unitary time evolution through one full period is
\begin{align}
U_F(\tau) = \mathrm{e}^{-i H_x \tau} \mathrm{e}^{-i H_z \tau} ~.
\end{align}
We choose time step $\tau = 0.8$, which was found in Ref. \cite{KIH} to produce
a rapid relaxation of the total energy within a few time steps as shown in the Appendix.
The eigenvalues of $U_F(\tau)$ are complex numbers of magnitude one.
Note that time is in a certain sense discrete (integer multiples of $\tau$) for this Floquet system.
The Hamiltonian system, with conserved total energy, is effectively the case $\tau=0$, which we contrast here with the Floquet system
with $\tau=0.8$ where the total energy is not conserved and relaxes very quickly. Of course, there is an interesting crossover between these
two limits \cite{luca}, but we do not explore that crossover in this paper.
Throughout this paper,
we consider the bipartite entanglement entropy of pure states, quantified by the
von Neumann entropy of the reduced density operator of a
half chain: $S=-\mathrm{Tr}\{\rho_L \log_2\rho_L\}=-\mathrm{Tr}\{\rho_R \log_2\rho_R\}$.
We study chains of even length, and $\rho_L$ and $\rho_R$ are the reduced density operators of the left and right half chains, respectively.
Note that we measure the entropy in bits.
\section{Entanglement entropies of eigenstates}
We first look at the entanglement entropy of the eigenstates of the Hamiltonian (1) and of the Floquet operator (2), compared to random pure states of the full chain.
Figure \ref{fig:eigenstates} shows the distributions of these entanglement entropies
for $L = 16$.
We can see that the entanglement of the eigenstates of the Floquet operator is close to
that of random pure states, first derived by Page \cite{Page:1993}:
\begin{equation}
S^R(L)=\frac{L}{2}-\frac{1}{2\ln 2}-\mathcal{O}\left(\frac{1}{2^L}\right) ~.
\label{eq:page}
\end{equation}
This is consistent with previous studies which have shown that a Floquet dynamics thermalizes a subsystem to infinite temperature \cite{luca, lazarides, ponte, KIH}.
The eigenstates of the Hamiltonian, on the other hand, all have entanglement entropies that are a fraction of a bit or more less than random pure states.
What is the source of this difference? It is because the Hamiltonian eigenstates are eigenstates of the extensive conserved total energy, while the random
pure states and the Floquet eigenstates are not constrained by an extensive conserved quantity. This causes the probability distribution of the energy of a
half chain to be narrower for the Hamiltonian system, since if one half chain has a high energy (compared to its share of the eigenenergy) then the other half chain
has to have an energy that is low by the same amount. This suppresses the volume of the possible space
of half-chain states whose energy is either high or low, resulting in a reduced
entropy of the half-chain and thus reduced entanglement entropy, even for the Hamiltonian eigenstates at energies that correspond to infinite temperature. This goes along with the recent observation that the finite-size deviations of the eigenstates of
the Hamiltonian from the Eigenstate Thermalization Hypothesis are larger than those of the eigenstates of the Floquet operator \cite{KIH}: energy conservation
somewhat impedes thermalization.
\begin{figure}
\includegraphics[width=1.0\linewidth]{ham_floquet_ent.pdf}
\centering
\caption{(color online) Normalized histogram of the entanglement entropy for $L=16$ for: (dashed blue line) the eigenstates of the Hamiltonian (\ref{eqn:Hamiltonian});
the eigenstates of the Floquet operator (Eq. (2)) with $\tau = 0.8$ (solid green line); and for random pure states of the full chain (very narrow red distribution), where the histogram is over 2000 randomly generated pure states. }
\label{fig:eigenstates}
\end{figure}
\section{Dynamics of Entanglement near equilibrium}
Now we turn to the dynamics of the entanglement entropy.
The dynamics of a linear operator is set by the matrix elements of the operator between
energy eigenstates (or eigenstates of the Floquet operator) and the eigenenergies. But the entanglement is not a linear operator, so its dynamics cannot
be determined so simply.
We explore the near-equilibrium dynamics of the entanglement in two different ways. First we study the relaxation of the entanglement to its equilibrium value from particular initial states with either low or high entanglement. We then explore the dynamics of the spontaneous
fluctuations of the entanglement entropy during the unitary time evolution of a random pure state of the full spin chain.
From these studies we can clearly show that the entanglement dynamics is slower for the Hamiltonian system, since some of the entanglement entropy is connected to the slow diffusion of energy between the two half-chains. In the Floquet system, on the other hand, near equilibrium the entanglement relaxes to equilibrium with a simple-exponential behavior in time, with a relaxation time that is apparently independent of the system size.
\subsection{Product of Random Pure states}
For initial states with zero entanglement between the two half chains, we use
a product of random (PR in Fig. 1) half-chain pure states $\ket{\psi(t=0)}=\ket{\psi_L}\otimes\ket{\psi_R}$,
where $\ket{\psi_L}$ and $\ket{\psi_R}$ are picked from the ensemble of random pure states of the left and the right half chain, respectively.
On average these states have energy close to 0, so the system is near infinite temperature and starts with zero entanglement entropy.
As random pure states are chosen for the half chains, the expected energy distribution between left and right halves is close to the equilibrium distribution (at infinite temperature), indicating only a small energy transport between two halves is required for thermalization.
Fig.~\ref{fig:ent_evolution} plots the time-dependent entanglement entropy under Hamiltonian and Floquet dynamics for $L=14$. The long-time average $S(\infty)$ is estimated by averaging $S(t)$ from $t=2500\tau$ to $t=2999\tau$.
It is clear from Fig.~\ref{fig:ent_evolution} that the Floquet system has faster relaxation of the entanglement entropy towards its saturation value at long times, even though the initial spreading rate of the entanglement is the same for these two systems.
Since the only significant difference between these two unitary dynamics is whether or not energy conservation and thus energy transport is present,
Fig.~\ref{fig:ent_evolution} suggests that the slow dynamical modes of this system associated with energy transport do also slow down the long-time thermalization of the entanglement.
\subsection{Generalized Bell States}
To explore the thermalization of the entanglement from initial states with {\it higher} entanglement than equilibrium, we use initial states
that maximize the entanglement entropy; we call these ``generalized Bell states''. These states have
Schmidt decomposition
\begin{equation}
\label{eqn:Bell}
\ket{\psi_B} = \frac{1}{\sqrt{2^{L/2}}}\sum_{i=1}^{2^{L/2}}\ket{L_i}\otimes\ket{R_i} ~,
\end{equation}
where the sets $\{\ket{L_i}\}$ and $\{\ket{R_i}\}$ are respectively complete orthonormal bases for left and right half chains. Since these initial states have higher
entanglement entropy than equilibrium, their entropy {\it decreases} as it thermalizes. This is an amusing apparent ``violation'' of the second law of thermodynamics,
but it is actually not thermodynamics, since the decrease is by less than one bit (very close to $1/(2\ln 2)$ by Eq. \eqref{eq:page})
, and thus far from extensive.
The random pure (RP) Bell states are made by independently choosing a random orthonormal basis for each half-chain.
To make initial Bell states that also have very large energy differences between the two half-chains, we make the opposite paired (OP) states that
can be written as
\begin{equation}
\ket{\psi(t=0)}=\frac{1}{\sqrt{2^{L/2}}}\sum_{i=1}^{2^{L/2}}\mathrm{e}^{i\theta_i}\ket{E_i^{h}}_{left}\otimes\ket{E_{2^{L/2}+1-i}^{h}}_{right} ~,
\end{equation}
where the $\ket{E_i^{h}}$ are the eigenstates of the half-chain Hamiltonian (Hamiltonian (\ref{eqn:Hamiltonian}) with $L/2$ sites), with their eigenenergies ordered according to $E_i^{h}\leq E_{i+1}^{h}$.
Therefore, by construction, many Schmidt pairs in these opposite paired Bell states have large energy imbalance between the two half-chains, unlike the random pure Bell states where the energy imbalance between the two halves is small. The contrast between them shows how the slow diffusive relaxation of the energy imbalance affects entanglement thermalization.
The ensemble of OP states that we average over is obtained by choosing random phases $\{\theta_i\}$.
The time evolution of the entanglement entropy for a $L=14$ spin chain starting from generalized Bell states of pairs of random pure (RP) states as well as generalized Bell states with opposite pairing (OP) under Hamiltonian dynamics are shown in Fig.~\ref{fig:ent_evolution}, with the estimated long time average subtracted.
For the opposite paired (OP) initial states, the initial large energy differences between the two half-chains in many of the Schmidt pairs make the excess
entanglement long-lived, since the relaxation of these energy differences requires diffusion of the energy over the full length of the chain.
For the RP initial states, on the other hand, the half-chain states are random so do not show nonequilibrium energy correlations, and the excess entanglement
relaxes to equilibrium much
more rapidly than it does for the OP states.
This observation hence provides additional evidence of the coupling between entanglement entropy relaxation and energy transport under Hamiltonian dynamics.
Fig.~\ref{fig:excess entropy} gives a more detailed view of the thermalization of the excess entanglement entropy starting from these generalized Bell initial
states. Here RP initial states under Floquet dynamics are also shown; since the Floquet system does not have conserved energy we cannot construct an OP
initial state for it. This figure again shows the clear importance of energy transport for entanglement thermalization.
The excess entropy of the RP initial states decays away faster for the Floquet system as compared to the Hamiltonian system, since the thermalization of the Floquet system is not constrained by an extensive conserved energy.
The strong initial anticorrelation between the energies of
the two half-chains greatly slows down the thermalization of the entanglement for the OP initial states under Hamiltonian dynamics.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{entropy_mean_subtracted_14_v2.pdf}
\caption{(color online) Thermalization of
entanglement entropy in three cases for $L=14$ (log-log scale): from random pure (RP) generalized Bell states under Hamiltonian dynamics (circular markers, green);
from RP generalized Bell states under Floquet dynamics (square markers, blue); and from oppositely paired (OP) generalized Bell states under Hamiltonian dynamics
(triangular markers, red). Within Hamiltonian dynamics, the larger initial energy imbalance for the OP initial states dictates a slower thermalization of the
entanglement, while the absence
of energy conservation for the Floquet system allows the fastest thermalization of entanglement entropy among all cases considered.
See main text for description of initial states.}
\label{fig:excess entropy}
\end{center}
\end{figure}
\subsection{Autocorrelation of entanglement}
Next we examine the dynamics of the spontaneous fluctuations of the entanglement entropy at equilibrium at infinite temperature, where
all pure states are equally likely.
Therefore, we simply pick many random pure states of the full chain and calculate the unitary time evolution of each initial state over many time steps.
We measure the autocorrelation of the entropy for each realization (indexed by $i$) as
\begin{equation}
\label{eqn:auto run}
R_i(t)=\frac{1}{M}\sum_{m=1}^{M}\left[S_i(t_m)-\bar{S}_i\right]\left[S_i(t_m+t)-\bar{S}_i\right] ~,
\end{equation}
where each run has in total 30000 time points $t_m$, equally spaced in time by $\Delta t$, and $S_i(T)$ is the entropy at time $T$.
Thus we measure the autocorrelation at integer multiples of the time step: $t = n\Delta t$.
$M$ is the maximum number of pairs that can be extracted from the time series.
Each random initial state
gives slightly different time-averaged entropies $\bar{S}_i$,
and thus for each run
we subtract its average in Eq. \eqref{eqn:auto run}. Then we average over runs and normalize the autocorrelation to be one at time difference $t=0$. The statistical errors are estimated from this averaging over runs.
\begin{figure}
\includegraphics[width=1.0\linewidth]{entropy_autocorrelaton_compare_v4.pdf}
\centering
\caption{(color online) Autocorrelation of the entanglement entropy vs. time for random pure states under Hamiltonian (solid lines) as well as Floquet dynamics (dashed lines) with different system sizes $L$ in log-linear scale. For each case, the autocorrelation is normalized to be one at time zero. Under Floquet dynamics, the autocorrelation decays as a simple exponential function of time, and faster than under Hamiltonian dynamics. Only weak size dependence of this normalized autocorrelation is observed under either dynamics.}
\label{fig:autocorrelation}
\end{figure}
Fig.~\ref{fig:autocorrelation} plots the autocorrelation under Hamiltonian and Floquet dynamics with systems of different sizes: $L=10$, $L=12$ and $L=14$.
For $L=10$ the number of independent runs in each case is $N=400$, while for $L=12$ and $L=14$ we chose $N=100$.
With $\tau=0.8$ as before, our time points are spaced by
$\Delta t=3\tau$ for Hamiltonian dynamics and $\Delta t=2\tau$ for Floquet dynamics.
It can be easily seen from Fig.~\ref{fig:autocorrelation} that
the relaxation of autocorrelations under Floquet dynamics is systematically faster.
Particularly, the autocorrelation in the Floquet system assumes a simple exponential decay.
This observation indicates that under the Floquet dynamics a random state ``relaxes'' to equilibrium by independent and unconstrained local relaxation.
In the Hamiltonian system, on the other hand, a spontaneous fluctuation that rearranges the energy density on a long length scale is
necessarily slow, due to the slow energy diffusion. Thus any influence of such fluctuations on the entanglement must relax slowly.
Clearly we are seeing such an influence that is causing the slower long-time relaxation of the entanglement autocorrelation in the Hamiltonian system.
Fig.~\ref{fig:Ham_auto} suggests that the autocorrelation under Hamiltonian dynamics decays exponentially in square root of time, as curves become roughly straight when plotted against $\sqrt{t}$ in a semi-log plot, and the straightness increases as system size increases. This scaling may be understood as fluctuation of entanglement entropy coupled to operators on the spin chain.
At time $t$ the fluctuation couples to the $O(4^l)$ operators on a size of $l\sim\sqrt{t}$ by diffusion, of which only $O(1)$ operators are slow, so only $O(1/4^l)\sim\exp(-c\sqrt{t})$ fraction of the information about the initial fluctuation is left at time $t$, where $c$ is some constant, resulting in an exponential decay of autocorrelation in $\sqrt{t}$.
The same reasoning may also be applied to the Floquet system, where the slowest modes instead have \cite{Kim: 2014} $l\sim t$, thus leading to a the observed simple exponential decay as is shown in Figure \ref{fig:autocorrelation}.
\begin{figure}
\includegraphics[width=1.0\linewidth]{Ham_ent_auto_compare_sqrt.pdf}
\centering
\caption{(color online) Autocorrelation of the entanglement entropy under Hamiltonian dynamics vs. square root of time for random pure states with different system sizes $L$. On this semi-log scale, all three curves roughly follow a straight line and the tail becomes more straight as system size increases.}
\label{fig:Ham_auto}
\end{figure}
One may also note here that under either dynamics, the relaxation of these autocorrelations has little dependence on system size.
This indicates that the fluctuations that are contributing here are on length scales smaller than the $L=10$ systems.
For the Floquet system this is consistent with the relaxation being simply local, so any longer length scale slow operators \cite{Kim: 2014}
apparently do not couple substantially to the entanglement fluctuations.
For the Hamiltonian systems this absence of size dependence suggests that over the time range probed here, the energy fluctuations that couple to the
entanglement are on length scales smaller than the length of the smaller $L=10$ system. But the substantially slower relaxation as compared to the
Floquet system suggests that energy transport over a few lattice spacings does couple to the entanglement fluctuations.
\section{Conclusion}
In conclusion, we have investigated the thermalization of the entanglement entropy by comparing state evolution of spin chains under Hamiltonian dynamics and Floquet dynamics, with the two systems having the same time-averaged Hamiltonian.
Eigenstates of these two dynamics have quite different distributions of the entanglement entropy. The Floquet eigenstates all have entanglement close to that of random pure states, while the Hamiltonian eigenstates all have significantly less entanglement due to the constraint of total energy conservation.
We show that the entanglement entropy relaxes to equilibrium more slowly under Hamiltonian dynamics, both for initial states well away from equilibrium and for the
spontaneous fluctuations of the entanglement entropy at equilibrium. The Hamiltonian system has slow diffusive energy transport, while the Floquet system does not.
This slow diffusive relaxation of the energy distribution in the Hamiltonian system results in slow relaxation near equilibrium of the entanglement entropy.
\section{Acknowledgement}
H.K. is supported by NSF DMR-1308141.
|
2,877,628,089,463 | arxiv | \section{Introduction}
Having computers able to recognize text from images is an old problem that has many practical applications, such as automatic content search on scanned documents. Transcribing \textit{printed} text is now a reliable technology. However, automatically recognizing \textit{handwritten} text is still a hard and open problem. Unlike printed text, cursive handwriting cannot be segmented into individual characters, since their boundaries are ill-defined. Graves et al. \cite{Graves:06icml} introduced the Connectionist Temporal Classification (CTC), a loss enabling to train neural networks to recognize sequences without explicit segmentation. Today, in order to deal with such a complex problem, state-of-the-art solutions \cite{Bluche2017,8270157} are all based on deep neural networks and the CTC loss.
The supervised training of these neural networks requires large amounts of annotated data; in our case, images of handwritten text with corresponding transcripts. However, annotating images of text is a costly, time-consuming task. We therefore propose a system to reverse the annotation process: starting from a given word, we generate a corresponding image of cursive text. We first tackle the challenge of generating realistic data, and then address the question of using such synthetic data to train neural networks in order to improve the performance of handwritten text recognition.
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\columnwidth]{1_principle_sansserif.png}
\caption{Adversarial generation of an image of text, conditioned on the textual content (``Dimanche''). The differences with a standard GAN are shown in blue.}
\label{fig:1_principle}
\end{figure}
The problem of generating images of handwritten text has already been addressed in the past. Many techniques \cite{Elanwar13} are based on a collection of templates of a few characters, either human-written or built using Bezier curves. These templates are possibly perturbed and finally concatenated. However, this class of solutions, that simply concatenates character models, cannot faithfully reproduce the distribution of real-world images. It is also complex to have templates that are generic enough to result in truly cursive text. Alternatively, following the online approach, we can consider the handwriting as a trajectory, typically recorded with pen computing. In this setting, the model aims at producing a sequence of pen positions to generate a given word. Graves et al. \cite{Graves13} use a Long Short-Term Memory (LSTM) \cite{Hochreiter,Gers2000} recurrent neural network to predict such a sequence, and let the network condition its prediction on the target string to synthesize handwriting. However, this method does not allow to deal with some features useful for offline recognition, such as background texture or line thickness variations.
Generative Adversarial Networks (GAN) \cite{NIPS2014_5423} offer a powerful framework for generative modeling. This architecture enables the generation of highly realistic and diverse images.
The original GAN does not allow any control over the generated images, but many works \cite{Mirza2014,Odena2016,Brock18} proposed a modified GAN for class-conditional image generation. However, we want to condition our generation on the sequence of characters to render, not on a single class. Closer to our goal, Reed et al. \cite{reed2016} conditions the generation on a textual description of the image to be produced. In addition to a random vector, their generator receives an embedding of the description text, and their discriminator is trained to classify as fake a real image with a non-matching description, to enforce the generator to produce description-matching images.
To the best of our knowledge, there is only one work \cite{chang2018generating} on a GAN for text image synthesis. While our generation process is directly conditioned on a sequence of characters, this method follows a style transfer approach, resorting to a CycleGAN \cite{Zhu17} to render images of isolated handwritten Chinese characters from a printed font.
Data augmentation techniques based on distortion and additive noise do not allow to enlarge the textual contents of the training data. Moreover, having control of the generated text enables the creation of training material that covers even rare sequences of characters, which can be expected to improve the recognition performance. This, combined with the intrinsic diversity, provides a strong motivation to use a conditional GAN for the generation of cursive text images.
In this paper, we make the following contributions:
\begin{itemize}
\item We propose an adversarial architecture, schematically represented in Fig. \ref{fig:1_principle}, to generate realistic images of handwritten words.
\begin{itemize}
\item We use bidirectional LSTM recurrent layers to encode the sequence of characters to be produced.
\item We introduce an auxiliary network for text recognition, in order to control the textual content of the generated images.
\end{itemize}
\item We obtain realistic images on both French and Arabic datasets.
\item Finally, we slightly improve text recognition performance on the RIMES dataset \cite{grosicki2009icdar}, using a neural network trained on a dataset extended with synthetic images.
\end{itemize}
\section{Proposed adversarial model}
\label{sec:model}
We introduce here our adversarial model for handwritten word generation. Section \ref{subsec:argan} gives the general idea and defines the training objectives of the different parts. We detail the network architectures in Section \ref{subsec:archi} and describe our optimization settings in Section \ref{subsec:optim}.
\subsection{Auxiliary Recognizer Generative Adversarial Networks}
\label{subsec:argan}
A standard GAN \cite{NIPS2014_5423} comprises a generator ($G$) and a discriminator ($D$) network, shown in gray in Fig. \ref{fig:1_principle}. $G$ maps a random noise $\boldsymbol{z}$ to a sample in the image space. $D$ is trained to discriminate between real and generated (fake) images. Adversarially, $G$ is trained to produce images that $D$ fails to discriminate correctly. These networks hence have competing objectives.
\begin{center}
\centering
\includegraphics[width=0.9\columnwidth]{2_model_sansserif.png}
\captionof{figure}{Architecture of the networks $\varphi$, $G$, $D$ and $R$. For ease of reading, not all layers are represented (refer to the text for exact details). $G$ receives a chunk of noise and $\varphi(\boldsymbol{s})$ in each ResBlock (details in Fig. \ref{fig:2_resblock}). Both $G$ and $D$ include a self-attention layer. $R$ follows the architecture of \cite{Bluche2017} and is trained with only real data using the CTC loss. We resort to the hinge version of the adversarial loss for $D$. When training $G$, we balance the gradients coming from $D$ and $R$ (details in Section \ref{subsec:optim}).}
\label{fig:2_model}
\end{center}
In order to control the textual content of the generated images, we modify the standard GAN as follows. First, we use a recurrent network ($\varphi$) to encode $\boldsymbol{s}$, the sequence of characters to be rendered in an image. $G$ takes this embedding $\varphi(\boldsymbol{s})$ as a second input. Then, in the vein of \cite{Odena2016}, the generator is asked to carry out a secondary task. To this end, we introduce an auxiliary network for text recognition ($R$). We then train $G$ to produce images that $R$ is able to recognize correctly, thereby completing its original adversarial objective with a ``collaboration'' constraint with $R$. We use the hinge version of the adversarial loss \cite{Lim2017} and the CTC loss \cite{Graves:06icml} to train this system. Formally, $D$, $R$, $G$ and $\varphi$ are trained to minimize the following objectives:
\begin{align*}
L_D = & - \mathbb{E}_{(\boldsymbol{x}, \boldsymbol{s}) \sim p_{data}} \Big[ \min\big(0, -1 + D(\boldsymbol{x}) \big) \Big] \\
& - \mathbb{E}_{\boldsymbol{z} \sim p_z, \boldsymbol{s} \sim p_{w}} \Big[ \min\Big(0, - 1 - D\big(G(\boldsymbol{z}, \varphi(\boldsymbol{s}))\big) \Big) \Big] \\
L_R = & + \mathbb{E}_{(\boldsymbol{x}, \boldsymbol{s}) \sim p_{data}} \Big[ \mathrm{CTC}(\boldsymbol{s}, R(\boldsymbol{x})) \Big] \\
L_{(G, \varphi)} = & - \mathbb{E}_{\boldsymbol{z} \sim p_z, \boldsymbol{s} \sim p_{w}} \Big[ D\big( G(\boldsymbol{z}, \varphi(\boldsymbol{s})) \big) \Big] \\
& + \mathbb{E}_{\boldsymbol{z} \sim p_z, \boldsymbol{s} \sim p_{w}} \Big[ \mathrm{CTC}\Big(\boldsymbol{s}, R\big( G(\boldsymbol{z}, \varphi(\boldsymbol{s})) \big) \Big) \Big]
\end{align*}
with $p_{data}$ the joint distribution of real \texttt{[image, word]} pairs, $p_{z}$ a prior distribution on input noise and $p_{w}$ a prior distribution of words, potentially different from the word distribution of the real dataset.
\subsection{Networks architecture}
\label{subsec:archi}
Fig. \ref{fig:2_model} and the text below describe the architecture of networks $\varphi$, $G$, $D$ and $R$. The residual blocks (ResBlocks) we used are detailed in Fig. \ref{fig:2_resblock}.
The network $\varphi$ first embeds each character of the sequence $\boldsymbol{s}$ in $\mathbb{R}^{128}$, then encodes it with a four-layer bidirectional LSTM \cite{Hochreiter,Gers2000} recurrent network (with a hidden state of size 128). $\varphi(\boldsymbol{s})$ is the output of the last bidirectional LSTM layer.
The network $G$ is derived from \cite{Brock18}. The input noise, of dimension 128, is split into eight equal-sized chunks. The first one is passed to a fully connected layer of dimension 1024, whose output is reshaped to $256 \times 1 \times 4$ (with the convention $\mathrm{depth} \times \mathrm{height} \times \mathrm{width}$). Each of the seven remaining chunks is concatenated with the embedding $\varphi(\boldsymbol{s})$, and fed to an up-sampling ResBlock through Conditional Batch Normalization (CBN) \cite{de2017modulating} layers (see Fig. \ref{fig:2_resblock}). The consecutive ResBlocks have the following number of filters: 256, 128, 128, 64, 32, 16, 16. A self-attention layer \cite{Zhang2018} is used between the fourth and the fifth ResBlocks. We add a final convolutional layer and a tanh activation in order to obtain a $1 \times 128 \times 512$ image.
The network $D$ is made up of seven down-sampling ResBlocks (with the following number of filters: 16, 16, 32, 64, 128, 128, 256), a self-attention layer between the third and the fourth ResBlocks, and a normal ResBlock (with 256 filters). We then sum the output along horizontal and vertical dimensions and project it on $\mathbb{R}$.
The auxiliary network $R$ is a Gated Convolutional Network, introduced in \cite{Bluche2017} (we used the ``big architecture''). This network consists in an encoder of five convolutional layers, with Tanh activations and convolutional gates, followed by a max pooling on the vertical dimension and a decoder made up of two stacked bidirectional LSTM layers.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{2_resblock_sansserif.png}
\caption{Detail of a ResBlock. The base components are shown in gray. In the ResBlocks of $G$, we concatenate a noise chunk with $\varphi(\boldsymbol{s})$ and feed it to CBN \cite{de2017modulating} layers (red). The unique hidden layer in CBN has 512 units. We also perform up-sampling (blue) with nearest neighbor interpolation. The ResBlocks of $D$ resort to standard Batch Normalization \cite{ioffe} and operate down-sampling (green) with an average pooling. The activation is ReLU \cite{NairH10} in $G$ and LeakyReLU \cite{maas2013rectifier} in $D$. The 1x1 convolution is only used when input (in) and output (out) numbers of channels are different. In the two 3x3 convolutions, the padding and stride are set to 1.}
\label{fig:2_resblock}
\end{figure}
\subsection{Optimization settings}
\label{subsec:optim}
We used spectral normalization \cite{MiyatoSpectral18} in $G$ and $D$, following recent works \cite{Zhang2018,Brock18,MiyatoSpectral18} that found that it stabilizes the training. We optimized our system with the Adam algorithm \cite{kingma2014adam} (for all networks: $lr = 2\times10^{-4}, \beta_1 = 0, \beta_2 = 0.999$) and we used gradient clipping in $D$ and $R$. We trained our model for several hundred thousand iterations with mini-batches of 64 images of the same type, either real or generated.
While $D$ processes one real batch and one generated batch per training step, $R$ is trained with real data only, to prevent it from learning how to recognize generated (and potentially false) images of text. To train the networks $G$ and $\varphi$, we first produce a batch of ``fake'' images $\boldsymbol{x_{\mathrm{fake}}} := G(\boldsymbol{z}, \varphi(\boldsymbol{s}))$, and then pass it through $D$ and $R$. $(G, \varphi)$ learn from the gradients $\boldsymbol{\nabla_D} := -\frac{\partial D(\boldsymbol{x_{\mathrm{fake}}})}{\partial \boldsymbol{x_{\mathrm{fake}}}}$ and $\boldsymbol{\nabla_R} := \frac{\partial\mathrm{CTC}(\boldsymbol{s}, R(\boldsymbol{x_{\mathrm{fake}}}))}{\partial \boldsymbol{x_{\mathrm{fake}}}}$ coming from these two networks. Since $R$ and $D$ have different architectures and losses, the norms of $\boldsymbol{\nabla_D}$ and $\boldsymbol{\nabla_R}$ can differ by several orders of magnitudes (we observed that $||\boldsymbol{\nabla_R}||_2$ is typically $10^2$ to $10^3$ times greater than $||\boldsymbol{\nabla_D}||_2$). To have $(G, \varphi)$ learn from both $D$ and $R$, we found it useful to balance the two gradients before propagating them to $G$. Therefore, we apply the following affine transformation to $\boldsymbol{\nabla_R}$:
\[
\boldsymbol{\nabla_R} \leftarrow \alpha \times \big( \frac{\sigma_D}{\sigma_R} (\boldsymbol{\nabla_R} - \mu_R ) + \mu_D \big)
\]
\noindent With $\mu_\bullet$ and $\sigma_\bullet$ being the mean and the standard deviation of $\boldsymbol{\nabla_\bullet}, \bullet \in \{D, R\}$. $\alpha$ controls the relative importance of $R$ with respect to $D$ and is set to $1$ in our model. The concrete impact of this transformation is discussed in Section \ref{subsubsec:gb}.
\section{Results}
\label{sec:results}
\subsection{Experimental setup}
\label{subsec:setup}
In our experiments, we use $128 \times 512$ images of handwritten words obtained with the following preprocessing: we isometrically resize the images to a height of 128 pixels, then remove the images of width greater than 512 pixels and finally, pad them with white to reach a width of 512 pixels for all the images (right-padding for French, left-padding for Arabic). Table \ref{tab:3_data} summarizes the meaningful characteristics of the two datasets we work with, namely the RIMES \cite{grosicki2009icdar} and the OpenHaRT \cite{openhart2010} datasets, while Fig. \ref{fig:3_data} shows some images from these two datasets.
\begin{table}[h!]
\centering
\caption{Characteristics of the subsets of RIMES and OpenHaRT.}
\label{tab:3_data}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Dataset & Language & Images & Words & Characters \\
\hline
RIMES & French & 129414 & 6780 & 86 \\
OpenHaRT & Arabic & 710892 & 65575 & 77 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{3_data.png}
\caption{Images after preprocessing. First line: RIMES. Second line: OpenHaRT.}
\label{fig:3_data}
\end{figure}
To reflect the distribution found in natural language, the words to be generated are sampled from a large list of words (French Wikipedia for French, OpenHaRT for Arabic). For the text recognition experiments on the RIMES dataset (Section \ref{subsec:reco}), we use a separate validation dataset of 6060 images
We evaluate the performance with Fr\'{e}chet Inception Distance (FID) \cite{FID2017} and Geometry Score (GS) \cite{Khrulkov2018}. FID is widely used and gives a distance between real and generated data. GS compares the topology of the underlying real and generated manifolds and provides a way to measure the mode collapse. For these two indicators, lower is better. In general, we observed that FID correlates with visual impression better than GS. For each experiment, we computed FID (with 25k real and 25k generated images) and GS (with 5k real and 5k generated images, 100 repetitions and default settings for the other parameters) every 10000 iterations and trained the system with different random seeds. We then chose independently the best FID and the best GS among the different runs. To verify the textual content, we relied on visual inspection. To measure the impact of data augmentation on the text recognition performance, we used Levenshtein distance at the character level (Edit Distance) and Word Error Rate.
\subsection{Ablation study}
\label{subsec:ablation}
For all the experiments in this section, we used the RIMES database described in Section \ref{subsec:setup}.
\subsubsection{Gradient balancing}
\label{subsubsec:gb}
When training the networks $(G, \varphi)$, the norms of the gradients coming from $D$ and $R$ may differ by several orders of magnitudes. As mentioned in Section \ref{sec:model}, we found it useful to balance these two gradients. Table \ref{tab:gb} reports FID, GS and a generated image for different gradient balancing settings.
\begin{table}[h!]
\centering
\caption{FID, GS and a generated image of the word ``r\'{e}parer'', for four settings: no gradient balancing, $\alpha=0.1$, $\alpha=1$ (our model) and $\alpha=10$.}
\label{tab:gb}
\begin{tabular}{|c||c|c|c|}
\hline
$\alpha$ & FID & GS & Images \\
\hline
None & 141.35 & $2.44 \times 10^{-3}$ & \begin{minipage}{3cm}
\centering
\includegraphics[width=\linewidth]{3_samples_nogb}
\end{minipage} \\
0.1 & 72.93 & $4.23 \times 10^{-2}$ & \begin{minipage}{3cm}
\centering
\includegraphics[width=\linewidth]{3_samples_01gb}
\end{minipage} \\
10 & 222.47 & $2.92 \times 10^{-3}$ & \begin{minipage}{3cm}
\centering
\includegraphics[width=\linewidth]{3_samples_10gb}
\end{minipage} \\
\textbf{1} & \textbf{23.94} & $\mathbf{8.58 \times 10^{-4}}$ & \begin{minipage}{3cm}
\centering
\includegraphics[width=\linewidth]{3_samples_1gb}
\end{minipage} \\
\hline
\end{tabular}
\end{table}
Without gradient balancing, we observed that $||\boldsymbol{\nabla_R}||_2$ was typically $10^2$ to $10^3$ times greater than $||\boldsymbol{\nabla_D}||_2$, meaning that the learning signal for $(G, \varphi)$ is biased toward satisfying $R$. As a result, the word ``r\'{e}parer'' is clearly readable, but the FID is high (141.35) and the generated image is not realistic (the background is noisy, the letters are too far apart).
With $\alpha=0.1$, $||\boldsymbol{\nabla_R}||_2$ is much smaller than $||\boldsymbol{\nabla_D}||_2$, meaning that $G$ and $\varphi$ take little account of the auxiliary recognition task. As illustrated by the second image in Table \ref{tab:gb}, we lose control of the textual content of the generated image. FID is better than before, but still high (72.93). In a way, the generated image is quite realistic, since the background is whiter and the writing more cursive.
On the contrary, when setting $\alpha$ to 10, $G$ and $\varphi$ mostly learn from the feedback of $R$ and the generation is thus successfully conditioned on the textual content. In fact, we can distinguish the letters of ``r\'{e}parer'' in the third generated image in Table \ref{tab:gb}. However, as we are focusing on optimizing the generation process to have a minimal CTC cost, we observe strong visual artifacts that remind of the one obtained by Deep Dream generators \cite{mordvintsev2015inceptionism}. FID is much higher (222.47) and the resulting images are very noisy, as demonstrated by the third image in Table \ref{tab:gb}.
The best compromise corresponds to $\alpha=1$. We obtain the best FID of 23.94 and GS of \num{8.58e-4}, while the generated image is both readable and realistic. For all other experiments, we set $\alpha$ to 1.
\subsubsection{Adversarial loss}
Using the network architecture described in Section \ref{sec:model}, we test three different adversarial training procedures: the ``vanilla'' GAN \cite{NIPS2014_5423} (GAN), the Least Squares GAN \cite{mao2016least} (LSGAN) and the Geometric GAN \cite{Lim2017, Zhang2018, Brock18}, used in our model. FID and GS are reported in Table \ref{tab:3_losses}.
\begin{table}[h]
\centering
\caption{FID and GS for different adversarial losses.}
\label{tab:3_losses}
\begin{tabular}{|c||c|c|c|c|}
\hline
Adversarial Loss & FID & GS \\
\hline
GAN & 36.32 & $5.29 \times 10^{-3}$ \\
LSGAN & 116.09 & $3.78 \times 10^{-3}$ \\
\textbf{Geometric GAN} & \textbf{23.94}& $\mathbf{8.58 \times 10^{-4}}$ \\
\hline
\end{tabular}
\end{table}
As shown in Table \ref{tab:3_losses}, Geometric GAN leads to the best performance in terms of FID and GS. LSGAN fails to produce text-like outputs in three out of five trials. The low FID for vanilla GAN indicates that it produces realistic images. The high GS in Table \ref{tab:3_losses} shows that both GAN and LSGAN suffer from a style collapse, and we observed that the textual content was not controlled. The trends given by FID and GS have been successfully confirmed by visual inspection of the generated samples.
\subsubsection{Self-attention}
We use a self-attention layer \cite{Zhang2018}, in both the generator and the discriminator, as it may help to keep coherence across the full image. We trained our model with and without this module to measure its impact.
\begin{table}[h]
\centering
\caption{Impact of self-attention.}
\label{tab:3_sa}
\begin{tabular}{|c||c|c|c|c|}
\hline
& FID & GS \\
\hline
Without self-attention & 67.86 & $4.51 \times 10^{-3}$ \\
\textbf{With self-attention} & \textbf{23.94} & $\mathbf{8.58 \times 10^{-4}}$ \\
\hline
\end{tabular}
\end{table}
Without self-attention, we still obtain realistic samples with correct textual content, but using self-attention improves performance both in terms of FID and GS, as shown in Table \ref{tab:3_sa}.
\subsubsection{Conditional Batch Normalization}
As described in Section \ref{sec:model}, $G$ is provided a noise chunk and $\varphi(s)$ through each CBN layer. Another reasonable option, closer to \cite{Odena2016}, is to concatenate the whole noise $\boldsymbol{z}$ with $\varphi(\boldsymbol{s})$, and feed it to the first linear layer of $G$ (in this scenario, CBN is replaced with standard Batch Normalization). Table \ref{tab:concat} reports FID and GS for these two solutions.
\begin{table}[h]
\centering
\caption{Generator input via the first linear layer or via CBN layers.}
\label{tab:concat}
\begin{tabular}{|c||c|c|}
\hline
& FID & GS \\
\hline
First linear layer & 42.23 & $1.81 \times 10^{-3}$ \\
\textbf{CBN layers} & \textbf{23.94} & $\mathbf{8.58 \times 10^{-4}}$\\
\hline
\end{tabular}
\end{table}
FID and GS in Table \ref{tab:concat} indicates that feeding the generator inputs through CBN layers improves realism and reduces mode collapse. The visual inspection of the generated samples confirmed these trends and showed that the other solution prevents from correctly conditioning on the textual content. \\
\subsection{Generation of handwritten text images}
We trained the model detailed in Section \ref{sec:model} on the two datasets described in Section \ref{subsec:setup}, RIMES and OpenHaRT.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{3_french.png}
\caption{Images generated with our system trained on RIMES. Targets: \textit{olibrius}, \textit{Dimanche}, \textit{inventif}, \textit{r\'{e}parer}, \textit{bonjour}, \textit{famille}, \textit{ionique}, \textit{gorille}, \textit{malade}, \textit{certes}, \textit{golf}, \textit{des}, \textit{ski}, \textit{le}.}
\label{fig:3_french}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{3_arabic.png}
\caption{Images generated with our system trained on OpenHaRT. Targets: \novocalize \<st_tny>, \<b^srA'>, \<trtkb>, \<.hdy_tA>, \<bA^srt>, \<Alrd`>, \<AtxA_d>, \<tbnt>, \<.hbAn>, \<Al.hs>, \<ny^gr>, \<kyAn>. }
\label{fig:3_arabic}
\end{figure}
Fig. \ref{fig:3_french} and Fig. \ref{fig:3_arabic} display some randomly generated (not cherry-picked) samples in French and Arabic respectively. For these two languages, we observe that our model is able to produce images of cursive handwriting, successfully conditioned on variable-length words (even if some words remain barely readable, e.g. \textit{le} and \textit{olibrius} in Fig. \ref{fig:3_french}). The typography of the individual characters is varied, but we can detect a slight collapse of writing style among the images. For French, as we trained the generator to produce words from all Wikipedia, we are able to successfully synthesize words that are not present in the training dataset. In Fig. \ref{fig:3_french} for instance, the words \textit{olibrius}, \textit{inventif}, \textit{ionique}, \textit{gorille} and \textit{ski} are not in RIMES, while \textit{Dimanche}, \textit{bonjour}, \textit{malade} and \textit{golf} appear in the corpus but with a different case.
\subsection{Data augmentation for handwritten text recognition}
\label{subsec:reco}
We aim at evaluating the benefits of generated data to train a model for handwritten text recognition. To this end, we trained from scratch a Gated Convolutional Network \cite{Bluche2017} (identical to the network $R$ described in Section \ref{subsec:archi}) with the CTC loss, RMSprop optimizer \cite{rmsprop} and a learning rate of $10^{-4}$. We used the validation data described in \ref{subsec:setup} for early stopping.
\begin{table}[h]
\centering
\caption{Extending the RIMES dataset with 100k generated images. Impact on the text recognition performance in terms of Edit Distance (ED) and Word Error Rate (WER) on the validation set.}
\label{tab:data}
\begin{tabular}{|l|c|c|}
\hline
Data & ED & WER \\
\hline
RIMES only & 4.34 & 12.1 \\
RIMES + 100k & 4.03 & 11.9 \\
\hline
\end{tabular}
\end{table}
Table \ref{tab:data} shows that extending the RIMES dataset with data generated with our adversarial model brings a slight improvement in terms of Edit Distance and Word Error Rate. Note that using only GAN-made synthetic images for training the text recognition model does not yield competitive results.
\section{Conclusion}
We presented an adversarial model to produce synthetic images of handwritten word images, conditioned on the sequence of characters to render. Beyond the classical use of a generator and a discriminator to create plausible images, we employ recurrent layers to embed the word to condition on, and add an auxiliary recognition network in order to generate an image with legible text. Another crucial component of our model lies in balancing the gradients coming from the discriminator and from the recognizer when training the generator.
We obtained realistic word images in both French and Arabic.
Our experiments showed a slight reduction in error rate for the French model trained on combined data.
An immediate continuation of our experiments would be to train the described model on more challenging datasets, with textured background for instance. Furthermore, deeper investigation to reduce the observed phenomenon of style collapse would be a significant improvement. Another important line of work is to extend this system to the generation of line images of varying size.
\section{Introduction}
|
2,877,628,089,464 | arxiv | \section{Introduction\label{section:introduction}}
Since the 125~GeV/$c^2$ Higgs boson was discovered at the Large Hadron Collider (LHC) in 2012~\cite{Chatrchyan:2012xdj,Aad:2012tfa}, its properties have been tested more and more precisely~\cite{Sirunyan:2018koj,Tao:2018zeu,Magnan:2018spp}.
Even though no new physics
beyond Standard Model (SM) has been confirmed so far,
it is still necessary and meaningful to
search for new physics.
In this paper we study the anomalous $HZZ$ couplings.
The new physics beyond the SM in the SM effective field theory (SMEFT)
is shown as higher-dimensional operators in the Lagrangian which later supply non-SM interactions.
In this analysis we note these non-SM $HVV$ ($V$ represents $Z,W,\gamma$)
interactions from
six-dimensional operators as anomalous $HVV$ couplings, and consider
them separately from SM loop contributions.
To scrutinize the Lorentz structures from several anomalous couplings, we calculate
the scattering amplitudes in the spinor helicity method,
and the analytic formulas are shown symmetrically and elegantly
in the spinor notations.
$HVV$ couplings can be probed at the LHC through
processes including $V^\ast\to VH$ or $H\to VV$ decays.
Among these processes,
the $gg\to H\to ZZ \to 4\ell$ process,
which is called the golden channel, is the most precise
and has been studied extensively
in both theoretical studies
~\cite{Gao:2010qx,Bolognesi:2012mm,Anderson:2013afp,
Chen:2012jy,Chen:2013ejz,Chen:2014pia,Chen:2014hqs,Nelson:1986ki,Soni:1993jc,
Chang:1993jy,Arens:1994wd,Choi:2002jk,Buszello:2002uu,
Godbole:2007cn,Kovalchuk:2008zz,Cao:2009ah,DeRujula:2010ys,
Gainer:2011xz,Coleppa:2012eh,Stolarski:2012ps,Boughezal:2012tz,
Avery:2012um,Campbell:2012ct,Campbell:2012cz,Modak:2013sb,Sun:2013yra,
Gainer:2013rxa,Buchalla:2013mpa,
Chen:2013waa,Kauer:2012hd,
Beneke:2014sba,Falkowski:2014ffa,Modak:2014zca,Gonzalez-Alonso:2014rla,
Belyaev:2015xwa,
Gainer:2018qjm}
and experiments at LHC
~\cite{Chatrchyan:2012jja,Khachatryan:2014kca,
Khachatryan:2014iha,
deFlorian:2016spz,Sirunyan:2017tqd,Sirunyan:2019twz,
Aad:2015xua,Aaboud:2018wps,
Aaboud:2018puo}.
Thus, we also choose this golden channel to study anomalous $HVV$ couplings.
To reach a more precise result, both on-shell
and off-shell Higgs regions can be exploited.
At the same time, the interference effects between this process and the SM processes
should be included. Especially in the off-shell Higgs region, the
interference between this process and the continuum process $gg\to ZZ \to 4\ell$
should not be ignored~\cite{Campbell:2013una,Campbell:2011cu}.
Based on a modified $\texttt{MCFM}$~\cite{Campbell:2013una, Ellis:2014yca} package with anomalous $HZZ$ couplings, we study the interference effects quantitatively.
Furthermore, we estimate the constraints on the anomalous coupling using CMS
experimental data at LHC.
The rest of the paper is organized as follows. In Section~\ref{section:theory_calc}, the spinor helicity amplitudes with anomalous couplings are calculated. In Section~\ref{section:numerical result}, the analytic formulas are embedded into the \texttt{MCFM8.0} package and the cross sections for proton - proton collision, especially the interference effects, are shown numerically.
In Section ~\ref{section:constraints}, the constraints on the $HZZ$ anomalous couplings are estimated.
Section~\ref{section:conclusion} is the conclusion and discussion.
\section{theoretical calculation\label{section:theory_calc}}
In this section firstly we introduce the $HZZ$ anomalous couplings, and then we calculate the spinor helicity amplitudes.
\subsection{$HZZ$ anomalous couplings}
In the SM effective field theory~\cite{Buchmuller:1985jz, Grzadkowski:2010es}
the complete form of higher-dimensional operators can be written as
\begin{equation}
\mathcal{L}=\mathcal{L}_{SM}+\frac{1}{\Lambda}\sum_k C_k^{5}\mathcal{O}_k^{5}
+\frac{1}{\Lambda^2}\sum_k C_k^{6}\mathcal{O}_k^{6}+\mathcal{O}(\frac{1}{\Lambda^3})~,
\label{complete_form}
\end{equation}
where $\Lambda$ is the new physics energy scale, and $C_k^{i}$ with $i=5,6$
are Wilson loop coefficients.
As the dimension-five operators $\mathcal{O}_k^{5}$ have no contribution
to anomalous $HZZ$ couplings, the dimension-six operators $\mathcal{O}_k^{6}$ have leading contributions.
The relative dimension-six operators in the Warsaw basis~\cite{Grzadkowski:2010es} are
\begin{eqnarray}
&&\mathcal{O}^6_{\Phi D} =(\Phi^{\dagger}D^{\mu}\Phi)^{\ast}(\Phi^{\dagger}D^{\mu}\Phi), \nonumber \\
&&\mathcal{O}^6_{\Phi W}=
\Phi^\dagger \Phi W^{I}_{\mu\nu}W^{I\mu\nu},~~
\mathcal{O}^6_{\Phi B}= \Phi^\dagger\Phi B_{\mu\nu}B^{\mu\nu},~~
\mathcal{O}^6_{\Phi WB}= \Phi^\dagger \tau^I \Phi W^{I}_{\mu\nu}B^{\mu\nu},
\nonumber \\
&& \mathcal{O}^6_{\Phi \tilde{W}}= \Phi^\dagger\Phi \tilde{W}^{I}_{\mu\nu}W^{I\mu\nu},
~~\mathcal{O}^6_{\Phi \tilde{B}}= \Phi^\dagger\Phi \tilde{B}_{\mu\nu}B^{\mu\nu},
~~\mathcal{O}^6_{\Phi \tilde{W}B}= \Phi^\dagger \tau^I \Phi \tilde{W}^{I}_{\mu\nu}B^{\mu\nu},
\label{operatorphiw}
\end{eqnarray}
where $\Phi$ is a doublet representation under the $SU(2)_L$ group and the aforementioned Higgs field $H$ is one of its four components;
$D_\mu=\partial_\mu-i g W^{I}_{\mu}T^{I}-ig^{\prime}YB_\mu$, where $g$ and $g^\prime$ are coupling constants, $T^{I}=\tau^{I}/2$, where $\tau^{I}$ are Pauli matrices, $Y$ is the $U(1)_Y$ generator;
$W^{I}_{\mu\nu}=\partial_\mu W^{I}_{\nu}-\partial_\nu W^{I}_\mu-g\epsilon^{IJK}W^{J}_\mu W^{K}_\nu$,
$B_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu$,
$\tilde{W}^{I}_{\mu\nu}= \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}W^
{I\rho\sigma}$,
$\tilde{B}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}B^
{\rho\sigma}$.
For the $H\to 4\ell$ process that we are going to take to constrain the
anomalous $HZZ$ couplings numerically, there are dimension-six operators include $HZ\ell\ell$ contact interaction ~\cite{Barklow:2017awn,Cohen:2016bsd} that can also contribute non-SM effects,
which are
\begin{equation}
\mathcal{O}^6_{\Phi L}= (\Phi^\dagger \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm} D$}_\mu\Phi)( \bar{L}\gamma_\mu L),
~~\mathcal{O}^6_{\Phi LT}= (\Phi^\dagger T^I \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm} D$}_\mu \Phi)( \bar{L}\gamma_\mu T^I L),
~~\mathcal{O}^6_{\Phi e}= (\Phi^\dagger \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm} D$}_\mu\Phi)(\bar{e}\gamma_\mu e),
\label{operatorphil}
\end{equation}
where
$\Phi^\dagger \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm} D$}_\mu\Phi=\Phi^\dagger D_\mu\Phi-D_\mu\Phi^\dagger\Phi$,
~~$\Phi^\dagger T^I\mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm} D$}_\mu\Phi=\Phi^\dagger T^I D_\mu\Phi-D_\mu\Phi^\dagger T^I \Phi$,
$L$,$e$ represent left- and right-handed charged leptons.
One may worry about the pollution caused by the $HZ\ell\ell$ contact interaction from these operators to the $4\ell$ final state when probing $HZZ$ couplings.
Nevertheless, we can use certain additional methods to distinguish them. In the off-shell Higgs region, the on-shell $Z$ boson selection
cut can reduce much of the $HZ\ell\ell$ background. In on-shell Higgs region,
the non-leptonic $Z$ decay channel can also be adopted in constraining $HZZ$ couplings. These discussions are not the focus of the current paper and we are not going to examine them in detail here.
After spontaneous symmetry breaking, we get the anomalous $HZZ$ interactions
\begin{equation}
\mathcal{L}_{a}=\frac{a_1}{v}M_Z^2 HZ^{\mu}Z_{\mu}-\frac{a_2}{v}HZ^{\mu\nu}Z_{\mu\nu}
-\frac{a_3}{v}HZ^{\mu\nu}\tilde{Z}_{\mu\nu}~,
\label{lagrangian}
\end{equation}
with
\begin{eqnarray}
a_1&=& \frac{v^2}{\Lambda^2}C^6_{\Phi D}, \nonumber \\
a_2&=&-\frac{v^2}{\Lambda^2}(C^6_{\Phi W}c^2+C^6_{\Phi B}s^2+C^6_{\Phi WB}cs), \nonumber \\
a_3&=&-\frac{v^2}{\Lambda^2}(C^6_{\Phi \tilde{W}}c^2+C^6_{\Phi \tilde{B}}s^2
+C^6_{\Phi \tilde{W}B}cs),
\label{a1a2a3}
\end{eqnarray}
where $c$ and $s$ stand for the cosine and sine of the weak mixing angle respectively,
$a_1,a_2,a_3$ are dimensionless complex numbers and $v = 246$~GeV is the electroweak vacuum expectation value.
Notice that the signs before $a_2$ and $a_3$ are same as in \cite{Gao:2010qx,Khachatryan:2014kca,Sirunyan:2019twz}, but have an additional minus sign from the definition in \cite{Chen:2013ejz}.
$Z_{\mu}$ is $Z$ boson field, $Z_{\mu\nu}=\partial_{\mu} Z_{\nu}-\partial_{\nu} Z_{\mu}$ is the field strength tensor of the $Z$ boson and $\tilde{Z}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}Z^
{\rho\sigma}$ represents its dual field strength.
The loop corrections in SM can contribute similarly as
the $a_2$ and $a_3$ terms.
Quantitatively, the one-loop correction can contribute to $a_2$ term with small contributions
$\mathcal{O}(10^{-2}-10^{-3})$,
while the $a_3$ term appear in SM only at a three-loop level and thus has
a even smaller contribution~\cite{Khachatryan:2014kca}. Therefore, only if the contributions from the $a_2$ and $a_3$ terms are larger than these loop contributions can we consider them as from new physics.
The $HZZ$ interaction vertex from Eq.~\eqref{lagrangian} is
\begin{equation}
\Gamma_a^{\mu\nu}(k,k^{\prime})=i\frac{2}{v}\sum_{i=1}^3a_i\Gamma_{a,i}^{\mu\nu}(k,k^{\prime})=i\frac{2}{v}[a_1M_Z^2g^{\mu\nu}-2a_2(k^{\nu} k^{\prime\mu}-k\cdot k^{\prime}g^{\mu\nu})-2a_3\epsilon^{\mu\nu\rho\sigma} k_{\rho}k^{\prime}_{\sigma} ]~,
\label{eqn:gamma}
\end{equation}
where $k$,$k^{\prime}$ are the momenta of the two $Z$ bosons.
It is worthy to notice that the $HZZ$ vertices in the SM are
\begin{equation}
\Gamma_\text{SM}^{\mu\nu}(k,k^{\prime})=i\frac{2}{v}M_Z^2g^{\mu\nu}~,
\end{equation}
so the Lorentz structure of the $a_1$ term is same as the SM case.
While the $a_2$ and $a_3$ terms have different Lorentz structures, which
represent non-SM $CP$-even and $CP$-odd cases respectively.
\subsection{Helicity amplitude of the process $gg\rightarrow H\rightarrow ZZ\to2e2\mu$ }
\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.5\textwidth]{1.pdf} }
\captionsetup{singlelinecheck = false, justification=justified}
\caption{ Feynman diagram of the Higgs-mediated process $gg \rightarrow H \rightarrow ZZ\to2e2\mu$. The black dot represents an effective $ggH$ coupling from loop contributions.}
\label{ggHZZ}
\end{figure}
The total helicity amplitude for the process $gg\rightarrow H\rightarrow ZZ\to2e2\mu$ in Fig.~{\ref{ggHZZ}} is composed of three individual amplitudes $A^H_{\rm SM}, A^H_{CP-\rm{even}}$ and $A^H_{CP-\rm{odd}}$,
which have the same production process but different Higgs decay modes according to the three kinds of $HZZ$ vertices in Eq.~\eqref{eqn:gamma}. The specific formulas are
\begin{eqnarray}
&&~~~~\mathcal{A}^{gg\to H\rightarrow ZZ\to2e2\mu}
(1_g^{h_1},2_g^{h_2},3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},6^{h_6}_{\mu^+})\\ \label{eqn:a3}
&&=[a_1\mathcal{A}^{H}_{\rm SM}+a_2\mathcal{A}^{H}_{CP-\rm{even}}+a_3\mathcal{A}^{H}_{CP-\rm{odd}}]
(1_g^{h_1},2_g^{h_2},3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},6^{h_6}_{\mu^+})~, \\ \label{eqn:aphd}
&&=\mathcal{A}^{gg\rightarrow H}(1_g^{h_1},2_g^{h_2})\times
\frac{P_H(s_{12})}{s_{12}}\times \sum_{i=1}^3 a_i\mathcal{A}_i^{H\rightarrow ZZ\to2e2\mu}(3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},
6^{h_6}_{\mu^+})~,
\end{eqnarray}
where $h_i$ $(i=1\cdots 6)$ are helicity indices of external particles, $s_{ij}=(k_i+k_j)^2$ and $P_H(s)=\frac{s}{s-M_H^2+iM_H\Gamma_H}$ is the Higgs propagator.
The production part $\mathcal{A}^{gg\to H}(1^{h_1}_g,2^{h_2}_g)$ is the helicity amplitude of gluon-gluon
fusion to Higgs process, in which $h_1, h_2$ represent the helicities of gluons with outgoing momenta.
For all the other helicity amplitudes in this paper, we also keep the convention that the momentum of each external particle is outgoing.
When writing the helicity amplitudes, we adopt the conventions used in~\cite{Dixon:1996wi, Campbell:2013una}:
\begin{eqnarray}
&&\langle ij \rangle = \bar{u}_-(p_i) u_+(p_j), \qquad ~~{[ ij ]} = \bar{u}_+(p_i) u_-(p_j)~,\nonumber\\
&&\langle ij \rangle[ ji ] = 2 p_i \cdot p_j, \qquad ~~s_{ij} = (p_i+p_j)^2,
\end{eqnarray}
and we have
\begin{eqnarray}
\mathcal{A}^{gg\to H}(1^{+}_g,2^{+}_g)&=&\frac{2c_g}{v}[12]^2~, \nonumber\\
\mathcal{A}^{gg\to H}(1^{-}_g,2^{-}_g)&=&\frac{2c_g}{v}\langle12\rangle^2~.
\label{eqn:ggh}
\end{eqnarray}
To keep the $ggH$ coupling consistent with SM, we make
\begin{equation}
\frac{c_g}{v}=\frac{1}{2}\sum_f\frac{\delta^{a b}}{2}\frac{i}{16\pi^2}g^2_s4e
\frac{m_f^2}{2M_W s_W}\frac{1}{s_{12}}[2+s_{12}(1-\tau_H)C^{\gamma\gamma}_0(m_f^2)]~,
\label{eqn::FggH}
\end{equation}
with
\begin{equation}
C^{\gamma \gamma}_0(m^2) = 2\tau_H f(\tau_H)/4m^2~, \tau_H=4m^2/M^2_{H},
\end{equation}
\begin{equation}
f(\tau) = \left\{ \begin{array}{ll}
{\rm arcsin}^2 \sqrt{1/\tau} & \tau \geq 1 \\
-\frac{1}{4} \left[ \log \frac{1 + \sqrt{1-\tau } }
{1 - \sqrt{1-\tau} } - i \pi \right]^2 \ \ \ & \tau <1~
\end{array} \right.~,
\label{eqn:c0gammagamma}
\end{equation}
where $a, b=1,...,8$ are $SU(3)_c$ adjoint representation indices for the gluons,
the index $f$ represents quark flavor and $C^{\gamma\gamma}_{0}(m^2)$ is the Passarino-Veltman three-point scalar function \cite{Passarino:1978jh,Chen:2017plj}.
The decay part $\mathcal{A}^{H\rightarrow ZZ\to2e2\mu}(3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},6^{h_6}_{\mu^+})$ is the helicity amplitude of
the process $H\rightarrow ZZ\to e^-e^+\mu^-\mu^+$, which have three sources according to the three types of vertices as written in Eq.~\eqref{eqn:gamma}.
Correspondingly we write it as
\begin{equation}
\mathcal{A}^{H\rightarrow ZZ\to2e2\mu}(3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},6^{h_6}_{\mu^+})=\sum_{i=1}^3 a_i\mathcal{A}_i^{H\rightarrow ZZ\to2e2\mu}(3_{e^-}^{h_3},4^{h_4}_{e^+},5_{\mu^-}^{h_5},6^{h_6}_{\mu^+})
\end{equation}
with
\begin{eqnarray}
\label{eqn:amp1}
&& \mathcal{A}_1^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f \times l^2_e\frac{M_W^2}{\cos^2\theta_W}\langle35\rangle[46] , \\ \nonumber \label{eqn:amp2}
&& \mathcal{A}_2^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times l^2_e\times \\
&&
\Big[2k\cdot k^{\prime}\langle35\rangle[46]
+\big(\langle35\rangle[45]+
\langle36\rangle[46]\big)\big(\langle35\rangle[36]+\langle45\rangle[46]\big) \Big], \\ \nonumber
&& \mathcal{A}_3^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times l^2_e\times (-i) \times \\ \nonumber
&& \Big[2\big(k\cdot k^{\prime}+\langle46\rangle[46]\big)\langle35\rangle[46]
+\langle35\rangle[45]\big(\langle35\rangle[36]+\langle45\rangle[46]\big) \\
&&+\langle36\rangle[46]\big(\langle35\rangle[36]-\langle45\rangle[46]\big)\Big]~\label{eqn:amp3}.
\end{eqnarray}
and
\begin{equation}
f=-2ie^3\frac{1}{M_W \sin\theta_W}\frac{P_Z(s_{34})}{s_{34}}\frac{P_Z(s_{56})}{s_{56}}~,
\end{equation}
where $P_Z(s)=\frac{s}{s-M_Z^2+iM_Z\Gamma_Z}$ is the $Z$ boson propagator, $M_Z$,$M_W$ are the masses of the $Z$,$W$ bosons, $\theta_W$ is the Weinberg angle, $l_e$ and $r_e$ ( will appear for other helicity combinations) are the coupling factors of the $Z$ boson to left-handed and right-handed leptons:
\begin{equation}
l_e=\frac{-1+2\sin^2\theta_W}{\sin(2\theta_W)}~, r_e=\frac{2\sin^2\theta_W}{\sin(2\theta_W)}~.
\end{equation}
In Eq.s~\eqref{eqn:amp1}\eqref{eqn:amp2}\eqref{eqn:amp3}, we only show the case in which
the helicities of the four leptons
($h_3,h_4,h_5,h_6$) are equal to ($-,+,-,+$). As for the other three non-zero helicity combinations
($-,+,+,-$), ($+,-,-,+$), ($+,-,+,-$), their helicity amplitudes are similar to Eq.s~\eqref{eqn:amp1}\eqref{eqn:amp2}\eqref{eqn:amp3}, but with some exchanges such as
\begin{equation}
l_e \leftrightarrow r_e ~,~ 4 \leftrightarrow 6~,~ 3 \leftrightarrow 5~,~ []\leftrightarrow \langle\rangle~.
\end{equation}
Their specific formulas are shown in Appendix~\ref{app:amp}.
\subsection{Helicity amplitude of the box process $gg\rightarrow ZZ\to2e2\mu$}
\begin{figure}[htbp]
\centering
\centerline{ \includegraphics[width=0.5\textwidth]{2.pdf} }
\caption{ Feynman diagram of the box process $gg \rightarrow ZZ\to2e2\mu$~.}
\label{image3}
\end{figure}
The box process $gg\rightarrow ZZ\to2e2\mu$ is a continuum background of the Higgs-mediated $gg\to H\to2e2\mu$ process. The interference between these two kinds of processes can have nonnegligible contribution in the off-shell Higgs region.
The Feynman diagram of the process $gg\rightarrow ZZ\to2e2\mu$ is a box diagram
which is induced by fermion loops (see Fig.~\ref{image3}).
The helicity amplitude $A^{gg\rightarrow ZZ\to2e2\mu}_\text{box}$
has been calculated analytically
and coded in \texttt{MCFM8.0} package.
Another similar calculation that using a different method can be found in
\texttt{gg2VV} code~\cite{Binoth:2008pr}.
\subsection{Helicity amplitude of the process $gg\rightarrow H\rightarrow ZZ \to 4\ell$}
The process $gg\rightarrow H\rightarrow ZZ \to 4\ell$ with identical $4e$ or $4\mu$ final states can also be used to probe the anomalous $HZZ$ couplings.
In SM the differential cross sections of the $4\ell$ (include both $4e$ and $4\mu$ ) and $2e2\mu$ processes are nearly the same in both on-shell and off-shell Higgs regions~\cite{Ellis:2014yca},
which indicates adding the $4e/4\mu$ process can almost double experimental statistics.
This situation can probably be similar for the anomalous Higgs-mediated processes.
The $4e/4\mu$ Feynman diagrams consist of two different topology structures as shown in Fig.~\ref{image4}.
Fig.~\ref{image4}(b) is different from Fig.~\ref{image4}(a) just by swapping the positive charged leptons (4$\leftrightarrow$6). The helicity amplitude of each diagram is similar to the former $2e2\mu$ cases but need to be multiplied by a symmetry factor $\frac{1}{2}$.
While calculating the total cross section the interference term between Fig.~\ref{image4}(a) and (b)
need an extra factor of -1 comparing to the self-conjugated terms
because it connects all of the decayed leptons in one fermion loop while each self-conjugated term
has two fermion loops.
After considering these details, the summed cross section of $4e$ and $4\mu$ processes
is comparable to the $2e2\mu$ process. More details are shown in the following numerical results.
\begin{figure}[htbp]
\centering
\begin{minipage}[c]{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{3.pdf}\\
(a)
\end{minipage}%
\begin{minipage}[c]{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{4.pdf}\\
(b)
\end{minipage}
\caption{ The Feynman diagrams of the process $gg \rightarrow H\rightarrow ZZ\to 4\ell$, where $4\ell=4e\ \mbox{or}\ 4\mu$. Note that diagram (b)
is obtained by swapping the two positive charged leptons (4$\leftrightarrow$6) in diagram (a). }
\label{image4}
\end{figure}
\section{Numerical result\label{section:numerical result}}
In this section we present the integrated cross sections and differential distributions in both
on-shell and off-shell Higgs regions, especially
the interference between anomalous Higgs-mediated processes and SM processes.
\subsection{The cross sections}
To compare theoretical calculation with experimental observation at LHC,
we need to further calculate the cross sections at hadron level.
From helicity amplitude to the cross section,
there need two more steps. Firstly we should
sum and square the amplitudes to
get the differential cross section at
parton level, then integrate phase space and parton distribution function (PDF)
to get the cross section at hadron level.
As following we show these two steps conceptually.
The squared amplitude in the differential cross section at parton level $d\hat{\sigma}(s_{12})$ is
\begin{eqnarray}
& & \left|\mathcal{A}_\text{box}^{gg\to ZZ\to 4\ell}+\mathcal{A}^{gg\to H \to ZZ\to 4\ell}\right|^2 \\
&= & \left|\mathcal{A}_\text{box}^{gg\to ZZ\to 4\ell}+\mathcal{A}^{H}_{\rm SM}+
a_1\mathcal{A}^{H}_{\rm SM}+a_2\mathcal{A}^{H}_{CP-\rm{even}}+a_3\mathcal{A}^{H}_{CP-\rm{odd}}
\right|^2~.
\label{eqn:dsigma}
\end{eqnarray}
After expanding it, there left self-conjugated terms
and interference terms that have different amplitude sources.
As in the next step the integral of phase space and PDF are same for each term,
we note the integrated cross sections separately by the amplitude sources, which are
\begin{equation}
\sigma_{k,l}\sim
\left\{
\begin{array}{ll}
|\mathcal{A}_k|^2, \quad\quad & k=l ;\\
2\text{Re}(\mathcal{A}^\ast_k\mathcal{A}_l), \quad\quad& k\ne l ,\\
\end{array} \right.
\label{eq:sigmabox}
\end{equation}
where $k,l=$~\{box, SM, $CP$-even, $CP$-odd\}.
The superscripts of $\mathcal{A}$ are omitted for brevity.
\subsection{Numerical results for $gg\to 2e 2\mu$ process}
We make the integral of phase space and the PDF in the \texttt{MCFM}~8.0 package~\cite{Campbell:2015qma,Boughezal:2016wmq}.
The simulation is performed for the proton-proton collision at the
center-of-mass energy $\sqrt{s}=13$~TeV.
The Higgs mass is set to be $M_H=125 \rm~GeV$.
The renormalization $\mu_r$ and factorization scale $\mu_f$ are
set as the dynamic scale $m_{4\ell}/2$.
For PDF we choose the leading-order MSTW 2008 PDFs MSTW08LO~\cite{Martin:2009iq}.
Some basic phase space cuts are exerted as follows, which are similar to
the event selection cuts used in CMS experiment~\cite{CMS-PAS-HIG-13-002}.
\begin{equation}
\begin{aligned}
&P_{T,\mu}>5\rm~GeV, \ |\eta_{\mu}|<2.4~,\\
&P_{T,e}>7\rm~GeV, \ |\eta_{e}|<2.5~,\\
&\ m_{\ell\ell}>4 \rm~GeV, \ m_{4\ell}>100\rm~GeV~.\\
\end{aligned}
\end{equation}
Besides, for the $2e2\mu$ channel,
the hardest (second-hardest) lepton should satisfy $P_T> 20~(10)\rm~GeV$;
one pair of leptons with the
same flavour and opposite charge is required to have $40 \rm~GeV<m_{\ell^+\ell^-}<120 \rm~GeV$ and the other pair needs to fulfill $12 \rm~GeV <m_{\ell^+\ell^-}<120 \rm~GeV$.
For the $4e$ or $4\mu$ channel, four oppositely charge lepton pairs exist as $Z$ boson candidates. The
selection strategy is to first choose one pair nearest to the $Z$ boson mass as one $Z$ boson,
then consider the left two leptons as the other $Z$ boson. The other requirements are similar to the
$2e2\mu$ channel.
\begin{table*}[!htp]
\begin{floatrow}
\begin{minipage}{0.5\linewidth}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{13~\text{TeV}, $m_{2e2\mu}<130\rm~GeV$, on-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.024 & 0 & 0 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & 0 & 0.503 & 0.558 & 0\\
\cline{2-6}
& \footnotesize $CP$-even & 0 & 0.558 & 0.202 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.075 \\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{13~\text{TeV}, $m_{2e2\mu}>220\rm~GeV$, off-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 1.283 & -0.174 & -0.571 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & -0.174 & 0.100 & 0.137 & 0\\
\cline{2-6}
& \footnotesize $CP$-even & -0.571 & 0.137 & 0.720 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.716 \\
\hline
\end{tabular}
\end{minipage}
\caption{ The cross sections of $gg\to2e2\mu$ processes in proton-proton collision at
center-of-mass energy $\sqrt{s}=13$~TeV with $a_1=0, a_2=a_3=1$ in Eq.~\eqref{eqn:gamma}.
}
\label{table:13tevsigma}
\end{floatrow}
\end{table*}
Table~\ref{table:13tevsigma} shows the cross sections $\sigma_{k,l}$ with
$k,l=$~\{box, SM, $CP$-even, $CP$-odd\}
while $a_1,a_2,a_3$ are all set to 1 for convinience. The cross section values
can be converted easily by multiplying a scale factor for small $a_i$s.
In the left and right panels, the integral regions
of $m_{4\ell}$ are separately set as $m_{4\ell}<130~\text{GeV}$ and $m_{4\ell}>220~\text{GeV}$,
which correspond to the on-shell and off-shell Higgs regions, respectively.
Next we focus on two kinds of interference effects: the interference
between each Higgs-mediated process and box continuum background, denoted as $\sigma_{\text{box},l}$ (or $\sigma_{l,\text{box}}$) with $l\ne\text{box}$; and the interference between different Higgs-mediated processes, denoted as $\sigma_{k,l}$ with $k,l\ne {\rm box}$.
The interference terms
between Higgs-mediated processes
and the continuum background $\sigma_{\text{box},l}$ are all zeros in on-shell Higgs region, but relatively sizeble in the off-shell regions
except for the cases with the $CP$-odd Higgs-mediated process as shown in
Table~\ref{table:13tevsigma}.
There is an interesting reason for it. As from Eq.~\eqref{eqn:a3}\eqref{eqn:aphd}\eqref{eq:sigmabox},
\begin{eqnarray}
\nonumber
\sigma_{\text{box},l}&\sim & 2\text{Re}(\mathcal{A}^\ast_\text{box}\mathcal{A}_l)~, \\
\nonumber
&\sim & 2\text{Re}\big(\mathcal{A}^\ast_{\rm box} \mathcal{A}^{gg\to H}P_H(s_{12})\mathcal{A}_i\big)~,\\
&\sim & 2\frac{(s_{12}-M^2_H)\text{Re}\big(\mathcal{A}^\ast_{\rm box}\mathcal{A}^{gg\to H}\mathcal{A}_i\big)+M_H\Gamma_H\text{Im}\big(\mathcal{A}^\ast_{\rm box}\mathcal{A}^{gg\to H}\mathcal{A}_i\big)}
{(s_{12}-M^2_H)^2+M^2_H\Gamma^2_H}~,
\label{eqn:sigmaint}
\end{eqnarray}
which means the integrand of $\sigma_{\text{box},l}$ consists of two parts, one is antisymmetric
around $M^2_H$, the other is proportional to $M_H\Gamma_H\text{Im}\big(\mathcal{A}^\ast_{\rm box}\mathcal{A}^{gg\to H}\mathcal{A}_i\big)$.
The first part can be largely suppressed almost to zero
in the integral with an integral region symmetric around $M_H$.
The second part is also suppressed not only by the small factor
of $\Gamma_H/M_H$
but also by a small value of $\text{Im}\big(\mathcal{A}^\ast_{\rm box}\mathcal{A}^{gg\to H}\mathcal{A}_i\big)$ in the on-shell Higgs region.
By contrary, in the off-shell Higgs region
the integral regions are not symmetric around $M_H$ but in one side larger than $M_H$, which makes the first term have some non-zero contribution.
Both the first and the second terms can also be enhanced
when $\sqrt{s_{12}}$ is a little larger than twice of the top quark mass.
That is because the $gg\to H$ process is induced mainly by
top quark loop,
both the real part and the imaginary part of the amplitude
(Re$\mathcal{A}^{gg\to H}$ and Im$\mathcal{A}^{gg\to H}$)
can be enhanced when $\sqrt{s_{12}}$ is just larger than the $2M_t$ threshold (see Eq.~\eqref{eqn::FggH}).
Then $\text{Im}\big(\mathcal{A}^\ast_{\rm box}\mathcal{A}^{gg\to H}\mathcal{A}_i\big)$ can have a larger value, even though
the relative contribution from the second term can be still suppressed by the smallness of the factor $\Gamma_H/M_H$.
In conclusion, mainly due to the nonsymmetric integral region
and some enhancement of $\mathcal{A}^{gg\to H}$,
the interferece contribution
in the off-shell Higgs region becomes comparable with the self-conjugated
contributions.
It is also worthwhile to point out there is no cross section contribution from the
interference between the $CP$-odd Higgs-mediated process and other three processes, which include the continuumm background process, SM Higgs-mediated process and anomalous $CP$-even Higgs-mediated process. It is because there is an antisymmetric tensor $\epsilon^{\mu\nu\rho\sigma}$ in the $CP$-odd $HZZ$ interaction vertex (see last term in Eq.~\eqref{eqn:gamma}), while in the other three processes, the two indices are symmetrically paired
and so the contract of the indices makes the interference term zero.
Nevertheless, these $CP$-odd interference term
can show angular distributions, include polar angle
distribution of $\ell$ in $Z$ boson rest frame and azimuthal angular distribution
between two $z$ decay planes~\cite{Buchalla:2013mpa,Beneke:2014sba}, even though its contribution to the total cross scetion is still zero.
The interference between $CP$-even Higgs-mediated process
and SM Higgs-mediated process
is nonnegligible both in on-shell and off-shell Higgs regions.
In on-shell Higgs region, the contribution from interfernce terms
is larger than that from the self-conjugated terms. Furthermore,
for $a_1=0, a_2=-1$
choice(as in \cite{Chen:2013ejz}), the interference terms would have a minus sign, comparing to
the relative values in Table~\ref{table:13tevsigma}, which makes the total contribution of
$CP$-even Higgs-mediated process beyond SM a destructive effect.
In the off-shell region, the $CP$-even Higgs-mediated process
have two interference terms, separately between SM Higgs-mediated process
and the box process. These two interference terms have opposite sign,
which means they cancel each other partly. Even though,
the summed interfernce effect is still comparable to the self-conjugated contribution.
\begin{figure}[!htbp]
\centering
\centerline{\includegraphics[width=0.8\textwidth]{m4l_13.pdf} }
\caption{ Differential cross sections of the $gg\to2e2\mu$ processes and $q\bar{q}\to2e2\mu$ process in proton-proton collision at $\sqrt{s}=13$~TeV with $a_2=1, a_1=a_3=0$ in Eq.~\eqref{eqn:gamma}.
}
\label{fig:dsigma_int}
\end{figure}
Fig.~\ref{fig:dsigma_int} shows the differential cross sections.
The black histogram is from its main background process $q\bar{q}\to 2e 2\mu$,
which is a huge background but still controllable. The red dashed histogram
is from the SM $gg\to 2e 2\mu$
processes including contributions from both
the box and SM Higgs-mediated amplitudes.
The blue dotted histogram adds contribution from the $CP$-even Higgs-mediated amplitude to
the SM signal and background amplitudes. Therefore three kinds of interference terms are included.
For comparison,
we also show the green dashed-dotted histogram without interference terms
from $CP$-even Higgs amplitudes with others,
so the interference contribution can be calculated by the difference
between blue and green histograms.
In the on-shell region we can see the $CP$-even Higgs-mediated
process have a total positive contribution (blue histogram) compare to the SM process(red histogram), while the green histogram shows
the main positive contribution is from the interference term.
In the off-shell region, the interference contribution
is obvious in $200~\rm GeV < m_{4\ell} < 600~\rm GeV$ region.
There is a bump in blue and green histograms when $m_{4\ell}\approx350$~GeV,
which is caused by the total cross section of the $CP$-even Higgs-mediated
process increase suddenly
beyond the $2M_t$ (twice of the top quark mass) threshold.
The differential cross section for the $CP$-odd Higgs-mediated
process is similar to the green histogram in off-shell region since it has no interference
contribution after the angular distributions being integrated.
The numerical results at center-of-mass energy $\sqrt{s}=8$~TeV
are shown in Table~\ref{table:8tevsigma} in Appendix \ref{table8tev}. By comparing them to
the results at $\sqrt{s}=13$~TeV in Table~\ref{table:13tevsigma},
we can find that
each cross section is decreased by about one or two times
and their relative ratios have some minor changes. That can be caused by
both PDF functions and kinematic distributions.
\subsection{Numerical results for $gg\to 4e/4\mu$ processes}
\begin{table*}[!htp]
\begin{floatrow}
\begin{minipage}{0.5\linewidth}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{13~\text{TeV},~$m_{4e/4\mu}<130\rm~GeV$, on-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.045 & 0 & 0 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & 0 & 0.540 & 0.568 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & 0 & 0.568 & 0.186 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.060 \\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{13~\text{TeV},~$m_{4e/4\mu}>220\rm~GeV$, off-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 1.303 & -0.176 & -0.575 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & -0.176 & 0.101 & 0.137 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & -0.575 & 0.137 & 0.740 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.708 \\
\hline
\end{tabular}
\end{minipage}
\caption{ Cross sections of $gg\to4e/4\mu$ processes in proton-proton collisions at
center-of-mass energy $\sqrt{s}=13$~TeV with $a_1=0, a_2=a_3=1$ in Eq.~\eqref{eqn:gamma}.
}
\label{table:13tevsigma_4l}
\end{floatrow}
\end{table*}
The cross sections of $gg\to4e/4\mu$ processes are listed in
Table~\ref{table:13tevsigma_4l} (Table~\ref{table:8tevsigma_4l} in Appendix~\ref{table8tev})
for comparison and next use.
Here $gg\to4e/4\mu$ represents the sum of $gg\to4e$ and $gg\to 4\mu$.
Comparing Table~\ref{table:13tevsigma_4l} with
Table~\ref{table:13tevsigma},
the numbers in the right panels are similar, while
the numbers in the left panels have relatively large differences.
That is mainly because the different selection cuts~\cite{Ellis:2014yca}.
If apply the $4e/4\mu$ selection cuts to the $gg\to 2e2\mu$ process,
$\sigma_{\rm box, box}$ in the left panels can become similar.
\section{ Constraints: a naive estimation \label{section:constraints}}
In this section we show a naive estimation to constrain $a_1$, $a_2$ and $a_3$
by using the data in both the on-shell and off-shell Higgs regions.
First, we estimate the expected number of events $N^{\rm exp}(a_1, a_2,a_3)$
in the off-shell Higgs region, which is defined as the contribution from the processes with anomalous couplings after excluding the pure SM contributions.
A theoretical observed total number of events should be
\begin{equation}
N^{\rm theo}(a_1,a_2,a_3)= \sigma_{\rm tot}\times \mathcal{L}\times
k\times\epsilon~,
\end{equation}
where $\sigma_{\rm tot}$ is the total cross section,
$\mathcal{L}$ is the integrated luminosity,
$k$ represents the $k$-factor and $\epsilon$ is the total efficiency.
The simulation in the CMS experiment~\cite{Sirunyan:2019twz}
with an integrated luminosity of
$\mathcal{L}\sim 80$~fb$^{-1}$ at $\sqrt{s}=13$~TeV
shows that
for the $gg\to4\ell$ process, the expected
numbers of events in the off-shell Higgs region ($m_{4\ell}>220\rm~GeV$) can be divided into two categories:
$N_{gg~~ {\rm signal}}=20.3$ and $N_{gg~~{\rm interference}}=-34.4$,
where the subscript ``$gg$ signal'' represents the SM Higgs-mediated signal term,
``$gg$ interference'' represents the interference term between
SM Higgs-mediated process and the box process.
For high-order corrections that may change the $k$-factor, some existing studies~\cite{Caola:2015psa,Melnikov:2015laa,Campbell:2016ivq,Caola:2016trd} show that the loop corrections on the box diagram~(Fig.\ref{image3}) and the Higgs-mediated diagram are different. Therefore, we also group the expected event number contributed from the anomalous couplings into two categories.
\begin{eqnarray}
\nonumber
&&N^{\rm exp}(a_1,a_2,a_3)\\
\nonumber
&=&
\frac{N_{gg~~{\rm signal}}}{\sigma^{H}_{\text{SM}}}
\times[(a_1+1)^2\sigma_{\text{SM}}^H-\sigma_{\text {SM}}^H+a^2_2\sigma^H_{CP-{\text{even}}}
+a^2_3\sigma^H_{CP-{\text{odd}}}+(a_1+1)a_2\sigma_{CP-{\text{even}},{\text {SM}}}^{\rm int}]\\
&&+
\frac{N_{gg~~{\rm interference}}}{\sigma^{\rm int}_{\text{SM}}}
\times[a_1\sigma_{{\rm SM},{\rm box}}^{\rm int}+a_2\sigma^{\rm int}_{CP-{\text{even},{\rm box}}}],
\label{Nexp}
\end{eqnarray}
where $N^{\rm exp}(a_1,a_2,a_3)$ represents the expected number of events
from anomalous $CP$-even and $CP$-odd processes,
$\sigma_k^H$ is the self-conjugate Higgs-mediated cross section, and $\sigma_{k,l}^{\rm int}$ is the interference cross section with $k,l=$~\{box, SM, $CP$-even, $CP$-odd\}.
The first term on the right-hand side of the equation is the contribution from the s-channel processes, and the second part is the contribution from the interference between the s-channel processes and the box diagram. For each category with the same topological Feynman diagrams, it is assumed to have the same $k$-factor and total efficiency $\epsilon$, which are equal to the corresponding values for the SM process. These coefficients are extracted from experimental measurements, which are similar as the treatment in the experiments~\cite{Sirunyan:2019twz,Ellis:2014yca}.
The cross section of $4\ell$ final states is the sum of
the cross sections of $2e2\mu$, $4e$ and $4\mu$ final states.
$N^{\rm exp}(a_1,a_2,a_3)$ can be obtained by combining the corresponding cross sections from both Table.~\ref{table:13tevsigma}
and Table.~\ref{table:13tevsigma_4l}.
The experimental observed number $N^{\rm obs}(a_1,a_2,a_3)$ that corresponds to $N^{\rm exp}(a_1,a_2,a_3)$ is defined as
$N^{\rm obs}(a_1,a_2,a_3)=N_{\rm total~~observed }-N^{\rm SM}_{\rm total~~expected }=38.7$ in the CMS experiment\cite{Sirunyan:2019twz}. Its fluctuation is estimated as the $\delta_{\rm off-shell}=\sqrt{N_{\text{total observed}}}=\sqrt{1325}$~(including both signal and background).
Second, the observed signal strength of the $gg\to H\to 4l$ process measured by CMS~\cite{CMS-PAS-HIG-19-001} is $\mu_{ggH}^{\rm obs}=0.97^{+0.09}_{-0.09}\mbox{(stat.)}^{+0.09}_{-0.07}\mbox{(syst.)}$. Its fluctuation is $\delta_{\rm on-shell}=0.127 $ after a combination of both statistical and systematic errors. Theoretically, the signal strength with anomalous couplings can be estimated as
\begin{equation}
\mu_{ggH}^{\rm{exp}}(a_1,a_2,a_3)=\frac{1}{\sigma_{\rm SM}^H}[(a_1+1)^2\sigma_{\rm SM}^H+a_2^2\sigma_{CP-{\rm even}}^H+a_3^2\sigma_{CP-{\rm odd}}^H+(a_1+1)a_2\sigma_{CP-{\rm even},{\rm SM}}^{\rm int}],
\label{mu}
\end{equation}
where $\sigma_k^H$ and $\sigma_{k,l}^{\rm int}$ are same as in Eq.(\ref{Nexp}) except in the on-shell region.
Equation~(\ref{mu}) is shorter than Eq.(\ref{Nexp}) because in the on-shell Higgs region
the interference term with box diagram $\sigma_{\rm SM, box}$ and $\sigma_{\rm CP-even, box}$ are zero.
The survival parameter regions of $a_1,a_2$ and $a_3$ can be obtained by a global $\chi^2$ fit, which can be constructed as
\begin{equation}
\chi^2=\left(\frac{N^{\rm exp}-N^{\rm obs}}{\delta_{\rm off-shell}}\right)^2+\left(\frac{\mu_{ggH}^{\rm exp}-\mu_{ggH}^{\rm{obs}}}{\delta_{\rm on-shell}}\right)^2.
\label{chi2}
\end{equation}
The adoption of the $\chi^2$ fit here can be controversial, as we only have two input data points (on-shell and off-shell) and have to find parameter regions for three variables ($a_1,a_2$ and $a_3$). We claim that the result here is just for a complete analysis including both theoretical calculation and experimental constraints and it is very preliminary. The situation can be improved if experimental collaborations can collect sufficient statistics in the future. Nevertheless, the $\chi^2$ fit can also provide some interesting results.
\begin{figure}[htbp]
\centering
\centerline{\includegraphics[height=5cm,width=5cm]{contour_a2a3_20200625.pdf}
\includegraphics[height=5cm,width=5cm]{contour_a2a3_20200625big.pdf}
}
\centerline{\includegraphics[height=5cm,width=5cm]{contour_a2a1_20200625.pdf}
\includegraphics[height=5cm,width=5cm]{contour_a2a1_20200625big.pdf}}
\centerline{\includegraphics[height=5cm,width=5cm]{contour_a3a1_20200625.pdf}
\includegraphics[height=5cm,width=5cm]{contour_a3a1_20200625big.pdf}}
\caption{Two-dimensional constraints on the new physics coefficients $a_1$,$a_2$ and $a_3$ from $\chi^2$ fits. To illustrate the constraints from different energy regions, three $1\sigma$ regions (green concentric circles, blue concentric circles, and red region) from three individual $\chi^2$ fits (on-shell, off-shell, and both) are drawn here. CMS $2\sigma$ constraints~($95\%$ confidence level)~\cite{Sirunyan:2019twz} are drawn as the lines~(magenta for $a_2$ when $a_1=a_3=0$ and grey for $a_3$ when $a_1=a_2=0$) in the right zoom in plots.}
\label{contour}
\end{figure}
Fig.\ref{contour} shows the two dimensional contour diagram of the anomalous couplings.
There are three colored regions~(green, blue, and red) in each small plot and the red areas are the final $1\sigma$ survival parameter regions from the global $\chi^2$ fit. In the actual two dimensional fitting procedure, we take two anomalous couplings to be free and fix the third one to be zero. Three individual $\chi^2$ fits are operated, constraint only from the off-region (first part in Eq.~\eqref{chi2}), constraint from the on-shell region~(second part in Eq.~\eqref{chi2}) and both of the two. The purpose is to show how the irregular overlap red regions come from. As discussed in above sections, we have equal number of experimental data points and free parameters here and the $\chi^2$ fit degenerates to an equation solving problem. Survival parameter regions from either on-shell or off-shell constraint come to be concentric circles and the global fitting results are almost the overlap region between them.
In the recently updated CMS experiment~\cite{Sirunyan:2019twz},
it use both on-shell and off-shell data, construct
kinematic discriminants, and get the limit (at 95\% confidence level) of the parameters $a_2\subset [-0.09,0.19]$,~$a_3\subset [-0.21,0.18]$~(there is no corresponding constraint on $a_1$). This experimental analysis is based on one free parameter-fitting schedule so we draw them as the line segments in the right plots of Fig.~\ref{contour}~(magenta for $a_2$ and grey for $a_3$). Our global fit results is roughly consistent with the CMS's, although within a first glance the two seems have some tension(Pay attention that we draw $1\sigma$ contour while CMS's results are the limit at 95\% confidence level which corresponds to $2\sigma$ intervals in the hypothesis of Gaussian distribution). The CMS's results seems to be more stringent than ours. This maybe caused by more kinematic information in detail they used in their analysis. Besides, we have some parameter regions with $a_1\sim -2$ or $a_2$ approaching 1. These regions show the correlations of each pairs of parameters. There is cancellation on the cross sections when the parameters coexist. In principle, the anomalous couplings should be much smaller than 1 to validate the operator expansion. Therefore, these parameter regions should be ruled out. Nevertheless, our global fit provides a complementary perspective of how the final anomalous coupling parameters contour regions are obtained from the individual on-shell/off-shell energy region constraints. These preliminary fitting results can be optimized in the case of more statistics in the future.
\section{conclusion and discussion\label{section:conclusion}}
When considering the anomalous $HZZ$ couplings, we calculate
the cross sections induced by these new couplings,
and special attention is focused on the interference effects.
In principle, there are three kinds of interference:
1. the interference between anomalous $CP$-even Higgs-mediated process and the continuum background box process $\sigma_{CP\text{-even},\text{box}}$;
2. the interference between anomalous $CP$-even Higgs-mediated process and SM Higgs-mediated process $\sigma_{CP\text{-even},\text{SM}}$;
and 3. the interference between the anomalous $CP$-odd Higgs-mediated process
and all other processes $\sigma_{CP\text{-odd},k}$ with $k = {\text{box}, \text{SM},
CP\text{-even}}$.
The numerical results of the integrated cross sections show that
the first kind of interference can be neglected in the on-shell Higgs region but is nonnegligible in the off-shell Higgs region,
the second kind of interference
is important in both the on-shell and off-shell Higgs regions,
and the third kind of interference has zero contribution for the total cross section in both regions.
By using the theoretical calculation together with both on-shell and off-shell Higgs experimental data,
we estimate the constraints on the anomalous $HZZ$ couplings.
The correlations of the different kinds of anomalous couplings
are shown in contour plots, which illustrate how the
anomalous contributions cancel each other out and the extra parameter
regions survive when they coexist.
In this research we only use the numerical results of integrated
cross sections, whereas in fact more information can be fetched from the
differential cross sections (kinematic distributions).
Furthermore, the $k$-factors and total efficiencies
should also be estimated separately according to different sources.
We leave them for our future work.
\begin{acknowledgements}
We thank John M. Campbell for his helpful explanation of the code in the \texttt{MCFM} package. The work is supported by the National Natural Science Foundation of China under Grant No.11847168, the Fundamental Research Funds for the Central Universities of China under Grant No. GK201803019, GK202003018, 1301031995, and the Natural Science Foundation of Shannxi Province, China (2019JM-431, 2019JQ-739).
\end{acknowledgements}
\begin{appendix}
\section{Helicity amplitudes for the process $H\rightarrow ZZ \to e^-e^+\mu^-\mu^+$\label{app:amp}}
The helicity amplitudes $\mathcal{A}_1$, $\mathcal{A}_2$ and $\mathcal{A}_3$ are shown separately.
The common factor $f$ is defined as
\begin{equation}
f=-2ie^3\frac{1}{M_W \sin\theta_W}\frac{P_Z(s_{34})}{s_{34}}\frac{P_Z(s_{56})}{s_{56}}~. \nonumber
\end{equation}
\begin{equation}
\begin{aligned}
& \mathcal{A}_1^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f \times l^2_e\frac{M_W^2}{\cos^2\theta_W}\langle35\rangle[46] , \\
& \mathcal{A}_1^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f \times l_er_e\frac{M_W^2}{\cos^2\theta_W}\langle36\rangle[45] , \\
& \mathcal{A}_1^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f \times l_er_e\frac{M_W^2}{\cos^2\theta_W}\langle45\rangle[36] , \\
& \mathcal{A}_1^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f \times r^2_e\frac{M_W^2}{\cos^2\theta_W}\langle46\rangle[35] ~.
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&
\mathcal{A}_2^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times l^2_e\times \\
&
\Big[2k\cdot k^{\prime}\langle35\rangle[46]
+\big(\langle35\rangle[45]+
\langle36\rangle[46]\big)\big(\langle35\rangle[36]+\langle45\rangle[46]\big) \Big], \\
&
\mathcal{A}_2^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f\times l_er_e\times \\
&
\Big[2k\cdot k^{\prime}\langle36\rangle[45]
+\big(\langle35\rangle[45]+
\langle36\rangle[46]\big)\big(\langle36\rangle[35]+\langle46\rangle[45]\big) \Big], \\
&
\mathcal{A}_2^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times r_el_e\times \\
&
\Big[2k\cdot k^{\prime}\langle45\rangle[36]
+\big(\langle45\rangle[35]+
\langle46\rangle[36]\big)\big(\langle35\rangle[36]+\langle45\rangle[46]\big) \Big], \\
&
\mathcal{A}_2^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f\times r^2_e\times \\
&
\Big[2k\cdot k^{\prime}\langle46\rangle[35]
+\big(\langle45\rangle[35]+
\langle46\rangle[36]\big)\big(\langle36\rangle[35]+\langle46\rangle[45]\big) \Big].
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&
\mathcal{A}_3^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times l^2_e\times (-i) \times \\
& \Big[2\big(k\cdot k^{\prime}+\langle46\rangle[46]\big)\langle35\rangle[46]
+\langle35\rangle[45]\big(\langle35\rangle[36]+\langle45\rangle[46]\big) \\
&+\langle36\rangle[46]\big(\langle35\rangle[36]-\langle45\rangle[46]\big)\Big]~, \\
&
\mathcal{A}_3^{H\rightarrow ZZ\to2e2\mu}(3^-_{e^-},4^+_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f\times l_er_e\times (-i) \times \\
& \Big[2\big(k\cdot k^{\prime}+\langle45\rangle[45]\big)\langle36\rangle[45]
+\langle36\rangle[46]\big(\langle36\rangle[35]+\langle46\rangle[45]\big) \\
&+\langle35\rangle[45]\big(\langle36\rangle[35]-\langle46\rangle[45]\big)\Big], \\
&
\mathcal{A}_3^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^-_{\mu^-},6^+_{\mu^+})=
f\times r_el_e\times (-i) \times \\
& \Big[2\big(k\cdot k^{\prime}+\langle36\rangle[36]\big)\langle45\rangle[36]
+\langle45\rangle[35]\big(\langle45\rangle[46]+\langle35\rangle[36]\big) \\
&+\langle46\rangle[36]\big(\langle45\rangle[46]-\langle35\rangle[36]\big)\Big], \\
&
\mathcal{A}_3^{H\rightarrow ZZ\to2e2\mu}(3^+_{e^-},4^-_{e^+},5^+_{\mu^-},6^-_{\mu^+})=
f\times r^2_e\times (-i) \times \\
& \Big[2\big(k\cdot k^{\prime}+\langle35\rangle[35]\big)\langle46\rangle[35]
+\langle46\rangle[36]\big(\langle46\rangle[45]+\langle36\rangle[35]\big) \\
&+\langle45\rangle[35]\big(\langle46\rangle[45]-\langle36\rangle[35]\big)\Big].
\end{aligned}
\end{equation}
\section{The cross sections at $\sqrt{s}=8$~TeV
\label{table8tev}}
\begin{table*}[!htp]
\begin{floatrow}
\begin{minipage}{0.5\linewidth}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{8~\text{TeV},~$m_{2e2\mu}<130\rm~GeV$, on-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.011 & 0 & 0 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & 0 & 0.232 & 0.257 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & 0 & 0.257 & 0.093 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.035 \\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{8~\text{TeV}~,~$m_{2e2\mu}>220\rm~GeV$, off-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.479 & -0.056 & -0.198 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & -0.056 & 0.031 & 0.047 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & -0.198 & 0.047 & 0.228 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.219 \\
\hline
\end{tabular}
\end{minipage}
\caption{ Cross sections of $gg\to2e2\mu$ process in proton-proton collision at $\sqrt{s}=8$~TeV with $a_1=0,a_2=a_3=1$ in Eq.~\eqref{eqn:gamma}.}
\label{table:8tevsigma}
\end{floatrow}
\end{table*}
\begin{table*}[!htp]
\begin{floatrow}
\begin{minipage}{0.5\linewidth}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{8~\text{TeV}~,~$m_{4e/4\mu}<130\rm~GeV$, on-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.021 & 0 & 0 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & 0 & 0.248 & 0.261 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & 0 & 0.261 & 0.086 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.028 \\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{8~\text{TeV}~,~$m_{4e/4\mu}>220\rm~GeV$, off-shell} \\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$\sigma_{k,l}$(fb)}} & \multirow{2}{*}{box} &
\multicolumn{3}{c|}{\footnotesize Higgs-med.}\\
\cline{4-6}
\multicolumn{2}{|c|}{} & & {\footnotesize SM} & {\footnotesize $CP$-even} & {\footnotesize $CP$-odd} \\
\hline
\multicolumn{2}{|c|}{box} & 0.485 & -0.056 & -0.199 & 0 \\
\hline
\multirow{3}{*}{ \begin{tabular}{c} \rotatebox{90}{\footnotesize Higgs-med.} \end{tabular}} &
{\footnotesize SM} & -0.056 & 0.031 & 0.047 & 0 \\
\cline{2-6}
& \footnotesize $CP$-even & -0.199 & 0.047 & 0.229 & 0 \\
\cline{2-6}
& \footnotesize $CP$-odd & 0 & 0 & 0 & 0.215 \\
\hline
\end{tabular}
\end{minipage}
\caption{ The cross sections of $gg\to4e/4\mu$ processes in proton-proton collision at
center-of-mass energy $\sqrt{s}=8$~TeV with $a_1=0, a_2=a_3=1$ in Eq.~\eqref{eqn:gamma}.
}
\label{table:8tevsigma_4l}
\end{floatrow}
\end{table*}
\end{appendix}
\bibliographystyle{utphys}
|
2,877,628,089,465 | arxiv | \section{Introduction}\label{sect1}
Variable selection in regression models is one of the big challenges in the era of high-dimensional data where the number of explanatory variables might largely exceed the sample size. During the last two decades, many classical variable selection algorithms have been proposed which are often based on finding the solution to an appropriate optimization problem. As the most famous example, the Lasso \citep{tibshirani1996} relies on an \(\ell_1\)-type relaxation of the original \(\ell_0\)-type optimization problem.
Convex methods like the Lasso are computationally very efficient and are therefore routinely used in high-dimensional statistical applications. However, such classical methods mainly focus on point estimation and do not provide a measure of uncertainty concerning the best model, per se, although recent works aim at addressing these issues as well (see e.g. \citealp{wasserman2009}, \citealp{meinshausen2010} and \citealp{lee2016}). On the other hand, a major advantage of a fully Bayesian approach is that it automatically accounts for model uncertainty. In particular, Bayesian model averaging \citep{raftery1997} and the median probability model \citep{barbieri2004} can be used for predictive inference. Furthermore, posterior inclusion probabilities can be interpreted as natural ``importance'' measures of the individual explanatory variables.
Important \(\ell_0\)-type criteria like the Bayesian Information Criterion (BIC, \citealp{schwarz1978}) and the Extended Bayesian Information Criterion (EBIC, \citealp{chen2008}) can be derived as asymptotic approximations to a fully Bayesian approach (compare e.g. \citealp{liang2013}).
It has been argued that \(\ell_0\)-type methods posses favourable statistical properties in comparison to convex \(\ell_1\)-type methods with respect to variable selection and prediction (see e.g. \citealp{raskutti2011} and \citealp{narisetty2014}). Since solving the associated, generally NP-hard, discrete optimization problems by an exhaustive search is computationally prohibitive, there have been recent attempts in providing more efficient methods for resolving such issues, as for example, mixed integer optimization methods \citep{bertsimas2016} and Adaptive Subspace (AdaSub) methods (\citealp{staerk2018}; \citealp{staerk2021}).
The challenging practical issue of a fully Bayesian approach is similar to that of optimizing \(\ell_0\)-type information criteria: computing (approximate) posterior model probabilities for all possible models is not feasible if the number of explanatory variables \(p\) is very large, since there are in general \(2^p\) possible models which have to be considered.
Often, Markov Chain Monte Carlo (MCMC) methods based on Metropolis-Hastings steps (e.g. \citealp{madigan1995}), Gibbs samplers (e.g. \citealp{george1993}; \citealp{dellaportas2002}) and ``reversible jump'' updates (e.g. \citealp{green1995}) are used in order to obtain a representative sample from the posterior model distribution.
However, the effectiveness of MCMC methods depends heavily on a sensible choice of the proposal distributions being used. Therefore, such methods may suffer from bad mixing resulting in a slow exploration of the model space, especially when the number of covariates is large. Moreover, tuning of the proposal distribution is often only feasible after manual ``pilot'' runs of the algorithm.
Adaptive MCMC methods aim to address these issues by updating the proposal parameters ``on the fly'' during a single run of the algorithm so that the proposal distribution automatically adjusts according to the currently available information. Recently, a number of different adaptive MCMC algorithms have been proposed in the Bayesian variable selection context, see e.g. \citet{nott2005}, \citet{lamnisos2013}, \citet{ji2013}, \citet{griffin2014}, \citet{griffin2018} and \cite{wan2021}.
In this work we propose an alternative, simple and efficient adaptive independent Metropolis-Hastings algorithm for Bayesian variable selection, called the Metropolized Adaptive Subspace (MAdaSub) algorithm, and compare it to existing adaptive MCMC algorithms. In MAdaSub the individual proposal probabilities of the explanatory variables are sequentially adapted after each iteration. The employed updating scheme is inspired by the AdaSub method introduced in \citet{staerk2021} and can itself be motivated in a Bayesian way, such that the individual proposal probabilities finally converge against the true respective posterior inclusion probabilities. In the limit, the algorithm can be viewed as a simple Metropolis-Hastings sampler using a product of independent Bernoulli proposals which is the closest to the unknown target distribution in terms of Kullback-Leibler divergence (among the distributions in the family of independent Bernoulli form).
The paper is structured as follows. The considered setting of Bayesian variable selection in generalized linear models (GLMs) is briefly described in Section~\ref{sec:setting}. The MAdaSub algorithm is motivated and introduced in Section~\ref{sec:MAdaSub}. By making use of general results obtained by \citet{roberts2007}, it is shown that the MAdaSub algorithm is ergodic despite its continuing adaptation (see Theorem~\ref{thm:MAdaSub}). Alternative adaptive approaches are also briefly discussed and conceptually compared to the newly proposed algorithm. In Section~\ref{sec:parallel}, a parallel version of MAdaSub is presented where the proposal probabilities can be adapted using the information from all available chains, without affecting the ergodicity of the algorithm (see Theorem~\ref{thm:parallel}).
Detailed proofs of the theoretical results of Sections~\ref{sec:MAdaSub} and~\ref{sec:parallel} can be found in the supplement to this paper.
The adaptive behaviour of MAdaSub and the choice of its tuning parameters are illustrated via simulated data applications in Section~\ref{sec:sim}. In Section~\ref{sec:realdata} various real data applications demonstrate that MAdaSub provides an efficient and stable way for sampling from high-dimensional posterior model distributions. The paper concludes with a discussion in Section~\ref{sec:discussion}. An R-implementation of MAdaSub is available at \url{https://github.com/chstaerk/MAdaSub}.
\section{The setting}\label{sec:setting}
In this work we consider variable selection in univariate generalized linear models (GLMs), where the response variable \(Y\) is modelled in terms of \(p\) possible explanatory variables \(X_1,\dots,X_p\). More precisely, for a sample of size \(n\), the components of the response vector \(\boldsymbol Y=(Y_1,\dots,Y_n)^T\) are assumed to be independent with each of them having a distribution from a fixed exponential dispersion family with
\begin{equation} g\big( E(Y_i \,|\, \boldsymbol X_{i,*})\big) = \mu + \sum_{j=1}^p \beta_j X_{i,j},~~~i=1,\dots, n \, , \label{eq:GLM}\end{equation}
where \(g\) is a (fixed) link function, \(\mu\in\mathbb{R}\) is the intercept and \(\boldsymbol \beta=(\beta_1,\dots,\beta_p)^T\in\mathbb{R}^{p}\) is the vector of regression coefficients.
Here, \(\boldsymbol X=(X_{i,j})\in\mathbb{R}^{n \times p}\) is the design matrix; it's \(i\)-th row \(\boldsymbol X_{i,*}\) corresponds to the \(i\)-th observation and it's \(j\)-th column \(\boldsymbol X_{*,j}\equiv \boldsymbol X_j\) corresponds to the values of the \(j\)-th predictor.
For a subset \(S\subseteq\{1,\dots,p\}\), the model induced by \(S\) is defined by a GLM of the form~(\ref{eq:GLM}) but with design matrix \(\boldsymbol X_S\in\mathbb{R}^{n\times |S|}\) in place of \(\boldsymbol X\in\mathbb{R}^{n\times p}\) and corresponding vector of coefficients \(\boldsymbol\beta_S\in\mathbb{R}^{|S|}\), where \(\boldsymbol X_S\) denotes the submatrix of the original design matrix \(\boldsymbol X\) containing only the columns with indices in \(S\). For brevity, we often simply refer to the model \(S\). Without further notice, we assume that we always include an intercept \(\mu\) in the corresponding GLM with design matrix \(\boldsymbol X_{S}\).
We denote the set of labelled explanatory variables by \(\mathcal{P}=\{1,\dots,p\}\) and the full model space by \( \mathcal{M}=\{S;\, S\subseteq\mathcal{P}\} \).
In a fully Bayesian approach we assign prior probabilities \(\pi(S)\) to each of the considered models \(S\in\mathcal{M}\) as well as priors \(\pi(\mu_S,\psi_S,\boldsymbol\beta_S \,|\, S)\) for the parameters of each model \(S\in\mathcal{M}\), where \(\psi_S\) denotes a possibly present dispersion parameter (e.g.\ the variance in a normal linear model). After observing some data \(\mathcal{D}=(\boldsymbol X,\boldsymbol y)\), with \(\boldsymbol X\in\mathbb{R}^{n\times p}\) and \(\boldsymbol y\in\mathbb{R}^n\), the posterior model probabilities are proportional to
\begin{equation}
\pi( S \,|\, \mathcal{D} ) \propto \pi( \boldsymbol y \,|\,\boldsymbol X, S) \, \pi( S ) \, , ~ S\in\mathcal{M} \,,
\end{equation}
where
\begin{equation}
\pi( \boldsymbol y \,|\,\boldsymbol X, S) = \int\int\int f(\boldsymbol y \,|\,\boldsymbol X, S, \mu_S,\psi_S, \boldsymbol\beta_S) \, \pi(\mu_S,\psi_S, \boldsymbol\beta_S \,|\, S) \, d\mu_S \, d\psi_S \, d\boldsymbol\beta_S \,
\end{equation}
is the marginal likelihood of the data \(\boldsymbol y\) under model \(S\), while \(f(\boldsymbol y \,|\,\boldsymbol X, S, \mu_S,\psi_S, \boldsymbol\beta_S) \) denotes the likelihood of the data \(\boldsymbol y\) under model \(S\) given the parameter values \(\mu_S,\psi_S,\boldsymbol\beta_S\) and the values of the explanatory variables \(\boldsymbol X\).
Note that the marginal likelihood \(\pi( \boldsymbol y \,|\,\boldsymbol X, S) \) is generally only available in closed form when conjugate priors are used.
\begin{remark}\label{remark:conjugate} A prominent example in normal linear models is a conjugate prior structure, where the prior on the variance \(\psi = \sigma^2\) is given by Jeffreys prior (independent of the model~\(S\)) and the prior on the vector of coefficients \(\boldsymbol\beta_S\) in model \(S\in\mathcal{M}\) is given by a multivariate normal distribution, i.e.
\begin{equation}\label{eq:prior1}
\boldsymbol \beta_S \, | \, S , \sigma^2 \sim \mathcal{N}_{|S|} (\boldsymbol\theta_S , \sigma^2 g \, {\boldsymbol V}_{\!\! S} ) , ~~~ \pi(\sigma^2) \propto \frac{1}{\sigma^2} \,,
\end{equation}
where \( \boldsymbol\theta_S\in\mathbb{R}^{|S|}\), \(g>0\) and \({\boldsymbol V}_{\!\! S}\in \mathbb{R}^{|S|\times |S|}\)
are hyperparameters. After centering each of the covariates \(\boldsymbol X_j\), \(j\in\mathcal{P}\), the improper prior \(\pi(\mu)\propto 1\) is a common choice for the intercept~\(\mu\) (again, independent of the model \(S\)). With no specific prior information, the prior mean of~\(\boldsymbol\beta_S\) can be set to the zero vector (\(\boldsymbol\theta_S= \boldsymbol 0\)). The
matrix \(\boldsymbol {\boldsymbol V}_{\!\! S}\) is often chosen to be the identity matrix of dimension \(|S|\)
or to be \( {\boldsymbol V}_{\!\! S} = (\boldsymbol X_S^T \boldsymbol X_S)^{-1} \) yielding Zellner's g-prior \citep{zellner1986}.
The first choice corresponds to Ridge Regression and implies prior independence of the regression coefficients, while the second choice with \(g=n\) corresponds to a unit information prior. In case no specific prior information is available about the possible regressors, a natural choice for the model prior is an independent Bernoulli prior of the form
\begin{equation}\label{eq:modelprior}
\pi(S\,|\,\omega) = \omega^{|S|} (1-\omega)^{p-|S|} ,\, S\in\mathcal{M} \,,
\end{equation}
where \(\omega=\pi(j\in S)\) is the prior probability that variable \(X_j\) is included in the model, for all \(j\in\mathcal{P}\). One can either set the prior inclusion probability \(\omega\) to some fixed value or consider an additional hyperprior for \(\omega\), with the latter option yielding more flexibility. A convenient choice is the (conjugate) beta prior
\(\omega \sim \mathcal{B}e(a_\omega,b_\omega)\), where \(a_\omega>0\) and \(b_\omega>0\) can be chosen in order to reflect the prior expectation and prior variance of the model size \(s=|S|\), \(S\in\mathcal{M}\) (see \citealp{kohn2001} for details).
\end{remark}
In the general non-conjugate case the marginal likelihood is not readily computable and numerical methods may be used for deriving an approximation to the marginal likelihood. Laplace's method yields an asymptotic analytic approximation to the marginal likelihood \citep{kass1995}. Similarly, different information criteria like the Bayesian Information Criterion (BIC, \citealp{schwarz1978}) or the Extended Bayesian Information Criterion (EBIC, \citealp{chen2008}) can be used directly as asymptotic approximations to fully Bayesian posterior model probabilities under suitable choices of model priors. Under a uniform model prior, i.e. \(\pi(S)=\frac{1}{2^p}\) for all \(S\in\mathcal{M}\), the BIC can be derived as an approximation to \(-2\log(\text{BF}(S)) = -2\log(\text{PO}(S))\), where \(\text{BF}(S) = \pi( \boldsymbol y \,|\,\boldsymbol X , S)/ \pi( \boldsymbol y \,|\,\boldsymbol X , \emptyset) \) denotes the Bayes factor of model \(S\in\mathcal{M}\) versus the null model \(\emptyset\in\mathcal{M}\) and \(\text{PO}(S)\)
denotes the corresponding posterior odds (\citealp{schwarz1978}; \citealp{kass1995b}).
In a high-dimensional but sparse situation, in which only a few of the many possible predictors contribute substantially to the response, a uniform prior on the model space is a naive choice since it induces severe overfitting. Therefore, \citet{chen2008} propose the prior
\begin{equation}\pi(S)\propto {\binom{p}{|S|}}^{-\gamma}\,,\label{eq:EBICprior}\end{equation}
where \(\gamma\in[0,1]\) is an additional parameter.
If~\(\gamma=1\), then \(\pi(S)=\frac{1}{p+1}{ \binom{p}{|S|}}^{-1}\), so the prior gives equal probability to each model size, and to each model of the same size; note that this prior does also coincide with the beta-binomial model prior discussed above when setting \(a_\omega=b_\omega=1\), providing automatic multiplicity correction \citep{scott2010}. If~\(\gamma=0\), then we obtain the uniform prior used in the original BIC. Similar to the derivation of the BIC one asymptotically obtains the EBIC with parameter \(\gamma\in[0,1]\) as
\begin{equation}\text{EBIC}_\gamma(S) = -2\log\left( f(\boldsymbol y \,|\,\boldsymbol X, S, \hat{\mu}_S,\hat{\psi}_S, \hat{\boldsymbol\beta}_S) \right)+ \Big(\log(n)+2\gamma\log(p)\Big) |S| \,,\label{def:EBIC2}\end{equation}
where \(f(\boldsymbol y \,|\,\boldsymbol X, S, \hat{\mu}_S,\hat{\psi}_S, \hat{\boldsymbol\beta}_S) \) denotes the maximized likelihood under the model \(S\in\mathcal{M}\) (compare \citealp{chen2012}). Under the model prior~(\ref{eq:EBICprior}) and a unit-information prior on the regression coefficients for each model \(S\in\mathcal{M}\), one can asymptotically approximate the model posterior by
\begin{equation} \pi(S\,|\,\mathcal{D}) \approx \frac{\exp\left(-\frac{1}{2}\times \text{EBIC}_\gamma(S)\right)}{\sum_{S'\in\mathcal{M}} \exp\left(-\frac{1}{2}\times \text{EBIC}_\gamma(S')\right)} \,, ~ S\in\mathcal{M} \, . \label{eq:EBICkernel}\end{equation}
In this work we consider situations where the marginal likelihood \(\pi( \boldsymbol y \,|\,\boldsymbol X , S)\) is available in closed form due to the use of conjugate priors (see Remark~\ref{remark:conjugate}) or where an approximation to the posterior~\(\pi(S\,|\,\mathcal{D})\) is used (e.g. via equation~(\ref{eq:EBICkernel})).
This assumption allows one to focus on the essential part of efficient sampling in very large model spaces, since additional sampling from the respective model parameters is not necessary.
\section{The MAdaSub algorithm}\label{sec:MAdaSub}
A simple way to sample from a given target distribution is to use an independent Metropolis-Hastings algorithm. Clearly, the efficiency of such an MCMC algorithm depends on the choice of the proposal distribution, which is in general not an easy task (see e.g.\ \citealp{rosenthal2011}).
In the ideal situation, the proposal distribution for an independence sampler should be the same as the target distribution \(\pi(S\,|\,\mathcal{D})\), leading to an independent sample from the target distribution with corresponding acceptance probability of one. Adaptive MCMC algorithms aim to sequentially update the proposal distribution during the algorithm based on the previous samples such that, in case of the independence sampler, the proposal becomes closer and closer to the target distribution as the MCMC sample grows (see e.g.\ \citealp{holden2009}, \citealp{giordani2010}). However, especially in high-dimensional situations, it is crucial that the adaptation of the proposal as well as sampling from the proposal can be carried out efficiently. For this reason, we restrict ourselves to proposal distributions which have an independent Bernoulli form:\ if \(S\in\mathcal{M}\) is the current model, then we propose model \(V\in\mathcal{M}\) with probability
\begin{equation}\label{eq:inde}
q(V\,|\,S;\boldsymbol r) \equiv q(V;\boldsymbol r) = \prod_{j\in V} r_j \prod_{j\in\mathcal{P}\setminus V} (1-r_j) \,,
\end{equation}
for some vector \(\boldsymbol r=(r_1,\dots,r_p)\in (0,1)^p\) of individual proposal probabilities.
\begin{algorithm}[]
\caption{Metropolized Adaptive Subspace (MAdaSub) algorithm}\label{algo:MCMC}
\begin{flushleft} \textbf{Input:} \end{flushleft}
\vspace{-7mm}
\begin{itemize}
\item Data $\mathcal{D}=(\boldsymbol X,\boldsymbol y)$.
\vspace{-2mm}
\item (Approximate) kernel of posterior \(\pi(S\,|\,\mathcal{D})\propto \pi( \boldsymbol y \,|\,\boldsymbol X, S) \, \pi( S ) \) for \(S\in\mathcal{M}\).
\vspace{-2mm}
\item Vector of initial proposal probabilities \(\boldsymbol r^{(0)} = \left(r_1^{(0)},\dots,r_p^{(0)}\right)^T\in(0,1)^p\).
\vspace{-2mm}
\item Parameters $L_j>0$ for \(j\in\mathcal{P}\), controlling the adaptation rate of the algorithm (e.g.\ $L_j=L=p$).
\vspace{-2mm}
\item Constant \(\epsilon\in(0,0.5)\) (chosen to be small, e.g.\ \(\epsilon\leq\frac{1}{p}\)).
\vspace{-2mm}
\item Number of iterations $T\in\mathbb{N}$.
\vspace{-2mm}
\item Starting point \(S^{(0)}\in\mathcal{M}\) (optional).
\end{itemize}
\vspace{-5mm}
\begin{flushleft} \textbf{Algorithm: } \end{flushleft}
\vspace{-5mm}
\begin{enumerate}
\vspace{-2mm}
\item[(1)]
If starting point \(S^{(0)}\) not specified:
\vspace{-1mm}
\par
\begingroup
\leftskip=0.5cm
\noindent Sample $b_j^{(0)}\sim\text{Bernoulli}\left(r_j^{(0)}\right)$ independently for $j\in\mathcal{P}$. \\
Set \(S^{(0)} = \{j\in\mathcal{P};~b_j^{(0)}=1\}\).
\par
\endgroup
\item[(2)] For $t=1,\dots,T$:
\vspace{-2mm}
\begin{enumerate}
\item[(a)] Truncate vector of proposal probabilities to
\(\tilde{\boldsymbol r}^{(t-1)} = \left(\tilde{r}_1^{(t-1)},\dots,\tilde{r}_p^{(t-1)}\right)^T\), i.e. for \(j\in\mathcal{P}\) set
\vspace{-1mm}
\begin{align*} \tilde{r}_j^{(t-1)} = \begin{cases} r_j^{(t-1)} &\mbox{, if } r_j^{(t-1)}\in[\epsilon,1-\epsilon] \,, \\
\epsilon & \mbox{, if } r_j^{(t-1)}<\epsilon \,,\\
1-\epsilon & \mbox{, if } r_j^{(t-1)}>1-\epsilon \,. \\ \end{cases} \end{align*}
\vspace{-3mm}
\item[(b)] Draw $b_j^{(t)}\sim\text{Bernoulli}\left(\tilde{r}_j^{(t-1)}\right)$ independently for $j\in\mathcal{P}$.
\vspace{-1mm}
\item[(c)] Set $V^{(t)}=\{j\in\mathcal{P};~b_j^{(t)}=1\}$.
\vspace{-1mm}
\item[(d)] Compute acceptance probability \[ \alpha^{(t)}
= \min\left\{ \frac{\pi( \boldsymbol y \,|\,\boldsymbol X, V^{(t)}) \, \pi( V^{(t)} ) \, q(S^{(t-1)}; \tilde{\boldsymbol r}^{(t-1)})} { \pi( \boldsymbol y \,|\,\boldsymbol X, S^{(t-1)} ) \, \pi( S^{(t-1)} ) \, q(V^{(t)}; \tilde{\boldsymbol r}^{(t-1)} )} ,\, 1\right\} \,.\]
\vspace{-3mm}
\item[(e)] Set \(S^{(t)} = \begin{cases} V^{(t)} &\mbox{, with probability } \alpha^{(t)}, \\
S^{(t-1)} & \mbox{, with probability } 1-\alpha^{(t)}. \\ \end{cases} \)
\vspace{-1mm}
\item[(f)] Update vector of proposal probabilities \(\boldsymbol r^{(t)} = \left(r_1^{(t)},\dots,r_p^{(t)}\right)^T\) via
\vspace{-1mm}
\[r_j^{(t)}= \frac{L_j r_j^{(0)}+\sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j)}{L_j+t} \,,~~ j\in\mathcal{P} \, . \]
\end{enumerate}
\end{enumerate}
\vspace{-5mm}
\begin{flushleft} \textbf{Output:} \end{flushleft}
\vspace{-7mm}
\begin{itemize}
\item Approximate sample \(S^{(b+1)},\dots,S^{(T)}\) from posterior distribution \(\pi(\cdot\,|\,\mathcal{D})\), after burn-in period of length \(b\).
\end{itemize}
\end{algorithm}
\subsection{Serial version of the MAdaSub algorithm}
The fundamental idea of the newly proposed MAdaSub algorithm (given below as Algorithm~\ref{algo:MCMC}) is to sequentially update the individual proposal probabilities according to the currently ``estimated'' posterior inclusion probabilities. In more detail, after initializing the vector of proposal probabilities \(\boldsymbol r^{(0)} = \left(r_1^{(0)},\dots,r_p^{(0)}\right)\in(0,1)^p\), the individual proposal probabilities \(r_j^{(t)}\) of variables \(X_j\) are updated after each iteration \(t\) of the algorithm, such that \(r_j^{(t)}\) finally converges to the actual posterior inclusion probability \(\pi_j = \pi(j\in S\,|\,\mathcal{D})\), as \(t\rightarrow\infty\) (see Corollary~\ref{cor:MAdaSub} below). Therefore, in the limit, we make use of the proposal
\begin{equation} \label{eq:limitingprop}
q(V;\boldsymbol r^*) = \prod_{j\in V} \pi_j \prod_{j\in\mathcal{P}\setminus V} (1-\pi_j) ,~~ V\in\mathcal{M} \,, ~~ \text{with } \boldsymbol r^* = (\pi_1,\dots,\pi_p) \, ,
\end{equation}
which is the closest distribution (in terms of Kullback-Leibler divergence) to the actual target \(\pi(S\,|\,\mathcal{D})\), among all distributions of independent Bernoulli form~(\ref{eq:inde}) (see \citealp{clyde2011}). Note that the median probability model \citep{barbieri2004, barbieri2021}, defined by \(S_{\text{MPM}}=\{j\in\mathcal{P}:\pi_j\geq 0.5\}\), has the largest probability in the limiting proposal~(\ref{eq:limitingprop}) of MAdaSub, i.e.\ \(\mathsf{arg \ max}_{V\in\mathcal{M}} q(V;\boldsymbol r^*) = S_{\text{MPM}}\). Thus, MAdaSub can be interpreted as an adaptive algorithm which aims to adjust the proposal so that models in the region of the median probability model are proposed with increasing probability.
For \(j\in\mathcal{P}\), the concrete update of \(r_j^{(t)}\) after iteration \(t\in\mathbb{N}\) is given by
\begin{equation} \label{eq:update}
r_j^{(t)} \,=\, \frac{L_j r_j^{(0)}+\sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j)}{L_j+t} \,=\, \left(1-\frac{1}{L_j+t}\right)r_j^{(t-1)} + \frac{\mathbbm{1}_{S^{(t)}}(j)}{L_j+t} \,,
\end{equation}
where, for \(j\in\mathcal{P}\), \(L_j>0\) are additional parameters controlling the adaptation rate of the algorithm and \(\mathbbm{1}_{S^{(i)}}\) denotes the indicator function of the set \(S^{(i)}\). If \(j\in S^{(t)}\) (i.e.\ \(\mathbbm{1}_{S^{(t)}}(j)=1\)), then variable~\(X_j\) is included in the sampled model in iteration \(t\) of the algorithm and the proposal probability \(r_j^{(t)}\) of \(X_j\) increases in the next iteration~\(t+1\); similarly, if \(j\notin S^{(t)}\) (i.e.\ \(\mathbbm{1}_{S^{(t)}}(j)=0\)), then the proposal probability decreases. The additional ``truncation'' step 2 (a) in the MAdaSub algorithm
ensures that the truncated individual proposal probabilities \(\tilde{r}_j^{(t)}\), \(j\in\mathcal{P}\), are always included in the compact interval \(\mathcal{I}=[\epsilon,1-\epsilon]\), where \(\epsilon\in(0,0.5)\) is a pre-specified ``precision'' parameter.
This adjustment simplifies the proof of the ergodicity of MAdaSub.
In practice, note that the mean size of the proposed model~\(V\) from the proposal density \(q(V;\tilde{\boldsymbol r})\) in equation~(\ref{eq:inde}) with \(\tilde{\boldsymbol r}\in[\epsilon,1-\epsilon]^p\) is at least \(E|V|\geq\epsilon \times p\); thus, in practice we recommended to set \(\epsilon\leq\frac{1}{p}\), so that models of small size including the null model can be proposed with sufficiently large probability. On the other hand, if \(\epsilon\) is chosen to be very small, then the MAdaSub algorithm may take a longer time to convergence in case proposal probabilities of informative variables are close to \(\epsilon\approx 0\) during the initial burn-in period of the algorithm. Simulations and real data applications show that the choice \(\epsilon=\frac{1}{p}\) works well in all considered situations (see Sections~\ref{sec:sim} and~\ref{sec:realdata}).
The updating scheme of the individual proposal probabilities is inspired by the AdaSub method proposed in \citet{staerk2018} and \citet{staerk2021} and can itself be motivated in a Bayesian way: since we do not know the true posterior inclusion probability \(\pi_j\) of variable \(X_j\) for \(j\in\mathcal{P}\), we place a beta prior on \(\pi_j\) with the following parametrization
\begin{equation}
\pi_j \sim \mathcal{B}e\left(L_j r_j^{(0)},\, L_j \left(1- r_j^{(0)}\right)\right) \, ,
\end{equation}
where \(r_j^{(0)}=E[\pi_j]\) is the prior expectation of \(\pi_j\) and \(L_j>0\) controls the variance of \(\pi_j\) via
\begin{equation}
\text{Var}(\pi_j) = \frac{1}{L_j+1} \times r_j^{(0)}\, \left(1-r_j^{(0)}\right) \,.
\end{equation}
If \(L_j\rightarrow 0\), then \(\text{Var}(\pi_j) \rightarrow r_j^{(0)} \, (1-r_j^{(0)}) \), which is the variance of a Bernoulli random variable with mean \(r_j^{(0)}\). If \(L_j\rightarrow \infty\), then \(\text{Var}(\pi_j) \rightarrow 0\).
Now, one might view the samples \(S^{(1)},\dots,S^{(t)}\) obtained after \(t\) iterations of MAdaSub as ``new'' data and interpret the information learned about \(\pi_j\) as \(t\) approximately independent Bernoulli trials, where \(j\in S^{(i)}\) corresponds to ``success'' and \(j\notin S^{(i)}\) corresponds to ``failure''. Then the (pseudo) posterior of \(\pi_j\) after iteration \(t\) of the algorithm is given by
\begin{equation}
\pi_j \,|\,S^{(1)},\dots,S^{(t)} \sim \mathcal{B}e\left( L_j r_j^{(0)} + \sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j),\, L_j (1- r_j^{(0)}) + \sum_{i=1}^t \mathbbm{1}_{\mathcal{P}\setminus S^{(i)}}(j) \right) \, ,
\label{eq:posterior}
\end{equation}
with posterior expectation
\begin{equation}\label{eq:update_again}
E(\pi_j \,|\, S^{(1)},\dots,S^{(t)}) = \frac{L_j r_j^{(0)}+\sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j)}{L_j+t} = r_j^{(t)} \,
\end{equation}
and posterior variance
\begin{equation}\label{eq:updated_var}
\text{Var}(\pi_j\,|\, S^{(1)},\dots,S^{(t)}) = \frac{1}{L_j+t+1} \times r_j^{(t)}\, \left(1-r_j^{(t)}\right) \,.
\end{equation}
The interpretation of~\(r_j^{(0)}\) as the prior expectation for the posterior inclusion probability~\(\pi_j\) motivates the choice of \(r_j^{(0)}= \pi(j\in S)\) as the actual prior inclusion probability of variable~\(X_j\).
If no particular prior information about specific variables is available, but the prior expected model size is equal to $q\in(0,p)$, then we recommend to set \(r_j^{(0)}=\frac{q}{p}\) and \(L=L_j=p\) for all \(j\in\mathcal{P}\). In this particular situation, equation~(\ref{eq:update}) reduces to
\begin{equation}\label{eq:update_no}
E(\pi_j \,|\, S^{(1)},\dots,S^{(t)}) = \frac{q+\sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j)}{p+t} = r_j^{(t)} \,.
\end{equation}
Equation~(\ref{eq:update_no}) is closely related to the AdaSub update in \citet[equation (3.6)]{staerk2021}. A crucial difference between AdaSub and the proposed MAdaSub algorithm is that in AdaSub we only learn something about variable \(X_j\) if it is included in the currently sampled subspace in which the optimization is carried out, while in MAdaSub we obtain ``new knowledge'' about variable \(X_j\) in each iteration.
This may also explain the empirical observation that it is beneficial to choose a larger adaptation rate in the optimization method AdaSub in comparison to the adaptive MCMC algorithm MAdaSub. Even though it seems natural to choose the parameters \(r_j^{(0)}\) and \(L_j\) of MAdaSub as the respective prior quantities, this choice is not imperative. In fact, simulations indicate that choosing \(r_j^{(0)}=\frac{q}{p}\) with \(q\in[2,10]\) and \(L_j=p\) for all \(j\in\mathcal{P}\) yields a stable algorithm with good mixing in most sparse high-dimensional set-ups irrespective of the actual prior (see also Sections~\ref{sec:sim} and~\ref{sec:realdata}). Furthermore, if one has already run and stopped the MAdaSub algorithm after a certain number of iterations \(T\), then one can simply restart the algorithm with the already updated parameters \(r_j^{(T)}\) and \(L_j+T\) (compare equation~(\ref{eq:updated_var})) as new starting values for the corresponding parameters.
Using general results for adaptive MCMC algorithms by \citet{roberts2007}, we show that the MAdaSub algorithm is ergodic despite its continuing adaptation, i.e.\ that ``in the limit'' it samples from the targeted posterior model distribution.
\begin{theorem}\label{thm:MAdaSub}
The MAdaSub algorithm (Algorithm~\ref{algo:MCMC}) is ergodic for all choices of \(\boldsymbol r^{(0)}\in(0,1)^p\), \(L_j>0\) and \(\epsilon\in(0,0.5)\) and fulfils the weak law of large numbers.
\end{theorem}
The proof of Theorem~\ref{thm:MAdaSub} can be found in the supplement to this paper, where it is shown that MAdaSub satisfies both the simultaneous uniform ergodicity condition and the diminishing adaptation condition (cf.\ \citealp{roberts2007}). As an immediate consequence of Theorem~\ref{thm:MAdaSub} we obtain the following important result.
\begin{corollary}\label{cor:MAdaSub}
For all choices of \(\boldsymbol r^{(0)}\in(0,1)^p\), \(L_j>0\) and \(\epsilon\in(0,0.5)\), the proposal probabilities \(r_j^{(t)}\) of the explanatory variables \(X_j\) in MAdaSub converge (in probability) to the respective posterior inclusion probabilities \(\pi_j=\pi(j\in S\,|\,\mathcal{D})\), i.e. for all \(j\in\mathcal{P}\) it holds that
\( r_j^{(t)} \overset{\text{P}}{\rightarrow} \pi_j \) as \(t\rightarrow\infty\).
\end{corollary}
\subsection{Comparison to related adaptive approaches}\label{sec:comp}
In this section we conceptually compare the proposed MAdaSub algorithm (Algorithm~\ref{algo:MCMC}) with other approaches for high-dimensional Bayesian variable selection, focusing on adaptive MCMC algorithms that are most closely related to the new algorithm.
In a pioneering work, \citet{nott2005} propose an adaptive sampling algorithm for Bayesian variable selection based on a Metropolized Gibbs sampler, showing empirically that the adaptive algorithm outperforms different non-adaptive algorithms in terms of efficiency per iteration.
However, since their approach requires the computation of inverses of estimated covariance matrices, it does not scale well to very high-dimensional settings.
\citet{clyde2011} propose a Bayesian Adaptive Sampling (BAS) algorithm which is based on sampling without replacement from the posterior model distribution, where the individual sampling probabilities of the variables are adapted during the algorithm in such a manner that they converge against the posterior inclusion probabilities. By construction, if the number of iterations is equal to the number of possible models, the BAS algorithm enumerates all possible models. However, since BAS samples without replacement, it has to be ensured that no model is sampled twice and therefore, after each iteration of the algorithm, the sampling probabilities of some of the remaining models have to be renormalized.
Additionally, BAS differs from the other methods discussed since it is not an MCMC algorithm and may yield biased estimates of posterior inclusion probabilities after a limited number of iterations.
Another related adaptive method for Bayesian variable selection has been proposed by \citet{ji2013}. They consider an adaptive independence Metropolis-Hastings algorithm for sampling directly from the posterior distribution of the regression coefficients \(\boldsymbol\beta=(\beta_1,\dots,\beta_p)^T\), assuming that the prior of \(\beta_j\) for \(j\in\mathcal{P}\) is given by a mixture of a point-mass at zero (indicating that the corresponding variable~\(X_j\) is not included in the model) and a continuous normal distribution (indicating that variable~\(X_j\) is ``relevant''). Mixtures of normal distributions are used as proposals in the Metropolis-Hastings step, which are adapted during the algorithm to minimize the Kullback-Leibler divergence from the target distribution. The considered family of mixture distributions should ideally have sufficiently many mixture components to be able to approximate the
multimodal posterior distribution of~\(\boldsymbol\beta\). In comparison, MAdaSub focuses on sampling from the discrete model distribution and makes use of independent Bernoulli distributions as approximations to the targeted posterior model distribution, while the updating scheme is motivated in a Bayesian way. Further, it is not clear how the adaptive mixture approach of \citet{ji2013} scales to very high-dimensional problems.
\citet{schafer2013} develop sequential Monte Carlo algorithms (cf.\ \citealp{south2019})
using model proposals which
can take correlations between different explanatory variables into account. In contrast, MAdaSub is an adaptive MCMC algorithm which is based on independent Bernoulli proposals. While similar extensions of MAdaSub (i.e.\ including proposal distributions which allow for dependencies between variables) might be desirable in order to better approximate the target posterior model distribution, this may come at the price of a larger computational cost for updating and sampling from the proposal. Empirical results for MAdaSub (see Sections~\ref{sec:sim} and~\ref{sec:realdata}) indicate that even the simple independent Bernoulli proposals yield good mixing and sufficiently large acceptance rates in various settings.
Recently, several variants and extensions of the original adaptive MCMC sampler of \citet{nott2005} have been developed, including an adaptive Metropolis-Hastings algorithm by \citet{lamnisos2013}, where the expected number of variables to be changed by the proposal
is adapted during the algorithm. \citet{zanella2019} propose a tempered Gibbs sampling algorithm with adaptive choices of components to be updated in each iteration.
Furthermore, different individual adaptation algorithms have been developed in \citet{griffin2014} as well as in the follow-up works of \citet{griffin2018} and \citet{wan2021}.
These strategies are based on adaptive Metropolis-Hastings algorithms with the adaptation of the proposal distributions aiming at achieving a desirable pre-specified acceptance rate \(\tau\in(0,1)\). The employed proposal distributions are of the following form: if \(S\in\mathcal{M}\) is the current model, then the probability of proposing the model \(V\in\mathcal{M}\) is given by
\begin{equation} \label{eq:propalt}
\tilde{q}(V\,|\,S;\boldsymbol \eta) = \prod_{j\in V\setminus S} A_j \prod_{j\in S\setminus V} D_j
\prod_{j\in \mathcal{P}\setminus( S\cup V)} (1-A_j) \prod_{j\in S\cap V} (1-D_j) \, ,
\end{equation}
where \(\boldsymbol\eta=(A_1,\dots,A_p,D_1,\dots,D_p)^T\in(0,1)^{2p}\) is a vector of tuning parameters with the following interpretation: For \(j\in\mathcal{P}\), \(A_j\) is the probability of adding variable~\(X_j\) if it is not included in the current model \(S\) and \(D_j\) is the probability of deleting variable \(X_j\) if it is included in the current model \(S\). An important difference between the adaptive algorithms proposed in \citet{griffin2018} and MAdaSub is the different target for updating the proposal parameters. The adaptation strategies in \citet{griffin2018} aim at achieving a prespecified acceptance rate of the proposal~(\ref{eq:propalt}), while MAdaSub aims at obtaining a global independent proposal with the largest possible acceptance rate, focusing on regions close to the median probability model. Furthermore, the adaptation of the individual proposal probabilities in MAdaSub can be motivated in a Bayesian way, leading to a natural parallel implementation of the algorithm with an efficient joint updating scheme for the shared adaptive parameters (see Section~\ref{sec:parallel}).
\comment{
One of the strategies proposed in \citet{griffin2018} is called ``adaptive scale individual adaptation'' and adapts a single tuning parameter \(\zeta\in[0,1]\) after iteration \(t\) via
\begin{equation}
\text{logit}_\epsilon(\zeta^{(t)}) = \text{logit}_\epsilon(\zeta^{(t-1)}) + \phi^{(t-1)}
\left( \tilde{\alpha}^{(t)} - \tau \right) \,,
\end{equation}
where \(\phi^{(t-1)}\) controls the ``adaptation rate'' and \(\tau\in(0,1)\) is the targeted acceptance rate of the sampler.
The corresponding proposal in iteration \(t+1\) is given by equation~(\ref{eq:propalt}) with \(\boldsymbol\eta^{(t)}= (A_1^{(t)},\dots,A_p^{(t)},D_1^{(t)},\dots,D_p^{(t)})^T\) defined by
\begin{equation} \label{eq:update2}
A_j^{(t)} = \zeta^{(t)} \min\left\{ \frac{\hat{\pi}_j^{(t)}}{1-\hat{\pi}_j^{(t)}},1\right\} , ~~
D_j^{(t)} = \zeta^{(t)} \min\left\{ \frac{1-\hat{\pi}_j^{(t)}}{\hat{\pi}_j^{(t)}},1\right\} , ~~ j\in\mathcal{P} \, ,
\end{equation}
where \(\hat{\pi}_j^{(t)}\) is a Rao-Blackwellized estimate (see \citealp{ghosh2011}) of the posterior inclusion probability of variable \(X_j\) after iteration \(t\).
\citet{griffin2018} motivate their choice of the proposal parameters \(A_j^{(t)}\) and \(D_j^{(t)}\) by considering the idealized situation of a target distribution which itself has an independent Bernoulli form of~(\ref{eq:inde}) and showing that, in such a situation, the expected squared jumping distance (compare \citealp{pasarica2010}) of the resulting (non-adaptive) MCMC algorithm is maximized when the true marginal probabilities \(\pi_j\) are known and used as (fixed) estimates \(\hat{\pi}_j^{(t)}=\pi_j\) for \(j\in\mathcal{P}\) in~(\ref{eq:update2}), as well as \(\zeta^{(t)}=1\). Furthermore, they argue that this proposal leads to a small asymptotic variance when estimating linear transformations of marginal quantities (like posterior inclusion probabilities). They thus conclude that the proposal~(\ref{eq:propalt}) with \(A_j^{(t)}\) and \(D_j^{(t)}\) as in~(\ref{eq:update2})
should be preferred over an independence sampler of the form~(\ref{eq:inde}) (which is used in the MAdaSub algorithm).
This argument may not be applicable in general, since the definition~(\ref{eq:update2}) of \(A_j^{(t)}\) and \(D_j^{(t)}\) can lead to large negative autocorrelations between succeeding samples of the resulting chain: If \(\hat{\pi}_j^{(t)}\approx 0.5\) for some \(j\in\mathcal{P}\) and \(\zeta^{(t)}\approx1\), then \(A_j^{(t)}\approx D_j^{(t)} \approx 1\), i.e.\ with large probability the chain will ``oscillate'' between models including the variable \(X_j\) and excluding the variable \(X_j\).
Of course, this behaviour of the algorithm is beneficial for estimating the marginal inclusion probabilities \(\pi_j\) of the target, but only given that the estimated inclusion probabilities are already close to the true ones (\(\hat{\pi}_j^{(t)}\approx \pi_j\)) and, even more importantly, given that the target has an independent Bernoulli form. However, posterior model distributions will rarely have an exact independent Bernoulli form.
We want to illustrate the behaviour of the different proposals in such situations via a simple toy example.
\begin{example}
{\itshape
Let \(\mathcal{P}=\{1,2\}\) and suppose that the target \(\pi\) is given by \(\pi(\emptyset) = 0.5-2\delta\), \(\pi(\{1\}) = \pi(\{2\}) = \delta\) and \(\pi(\{1,2\}) = 0.5\) for some small constant \(\delta\in(0,0.25)\). Then the true marginal inclusion probabilities are \(\pi_1=0.5+\delta\) and \(\pi_2=0.5+\delta\). Suppose that we use the non-adaptive MCMC algorithm based on the proposal~(\ref{eq:propalt}) with \(A_j^{(t)}\) and \(D_j^{(t)}\) as in (\ref{eq:update2}) and \(\hat{\pi}_j^{(t)}=\pi_j\) for \(j=1,2\) and \(t\in\mathbb{N}\). Then, for \(\delta\approx 0\) and \(\zeta^{(t)}\approx1\), we have \(A_j^{(t)}\approx1\) and \(D_j^{(t)}\approx1\). Now if we start the chain with \(S^{(0)}=\{1\}\), then with large probability we will oscillate between the two models with only a single variable (i.e.\ \(S^{(1)}=\{2\}\), \(S^{(2)}=\{1\}\), \(S^{(3)}=\{2\}\), \(\dots\)) and for a long time we will not be able to visit the other two models which have the largest probability. On the other hand, the independent Bernoulli proposal (\ref{eq:inde}) (used in MAdaSub) with \(r_j=\pi_j\approx0.5\) for \(j=1,2\), has probability \((1-r_1)(1-r_2)\approx0.25\) of proposing the ``null'' model \(V=\emptyset\) and probability \(r_1r_2\approx0.25\) of proposing the ``full'' model \(V=\{1,2\}\). Thus, we avoid being stuck at models with low target probability.
}
\end{example}
}
\section{Parallelization of the MAdaSub algorithm}\label{sec:parallel}
In this section we present a parallel version of the MadaSub algorithm which aims at increasing the computational efficiency and accelerating the convergence of the chains. The simplest approach to parallelization would be to independently run the MAdaSub algorithm in parallel on each of \(K\in\mathbb{N}\) different workers, yielding \(K\) individual chains which, in the limit, sample from the posterior model distribution (see Theorem~\ref{thm:MAdaSub}). However, it is desirable that the information learned about the adaptive parameters can be shared efficiently between the different chains, so that the convergence of the adaptive parameters to their optimal values can be accelerated, leading to a faster convergence of the chains to their common limiting distribution.
\begin{algorithm}[]
\caption{Parallel version of MAdaSub}\label{algo:parallel}
\begin{flushleft} \textbf{Input:} \end{flushleft}
\vspace{-7mm}
\begin{itemize}
\item Number of workers \(K\in\mathbb{N}\).
\vspace{-2mm}
\item Number of rounds \(R\in\mathbb{N}\).
\vspace{-2mm}
\item Number of iterations per round \(T\in\mathbb{N}\).
\vspace{-2mm}
\item Data $\mathcal{D}=(\boldsymbol X,\boldsymbol y)$.
\vspace{-2mm}
\item (Approximate) kernel of posterior \(\pi(S\,|\,\mathcal{D})\propto \pi( \boldsymbol y \,|\,\boldsymbol X, S) \, \pi( S ) \) for \(S\in\mathcal{M}\).
\vspace{-2mm}
\item Vector of initial proposal probabilities \(\boldsymbol r^{(k,0)} = \left(r_1^{(k,0)},\dots,r_p^{(k,0)}\right)^T\in(0,1)^p\) for each worker \(k=1,\dots,K\).
\vspace{-2mm}
\item Adaptation parameters $L^{(k)}_j>0$ for \(j\in\mathcal{P}\) and each worker \(k=1,\dots,K\).
\vspace{-2mm}
\item Constant \(\epsilon\in(0,0.5)\) (chosen to be small, e.g.\ \(\epsilon\leq\frac{1}{p}\)).
\vspace{-2mm}
\item Starting points \(S^{(k,0)}\in\mathcal{M}\) for \(k=1,\dots,K\) (optional).
\end{itemize}
\vspace{-5mm}
\begin{flushleft} \textbf{Algorithm: } \end{flushleft}
\vspace{-5mm}
\begin{enumerate}
\vspace{-2mm}
\item[(1)]
Set \(\bar{\boldsymbol r}^{(k,0)} = \boldsymbol r^{(k,0)}\) for \(k=1,\dots,K\). \\[1mm]
For \(k=1,\dots,K\): If starting point \(S^{(k,0)}\) not specified:
\vspace{-1mm}
\par
\begingroup
\leftskip=0.5cm
\noindent Sample $b_j^{(k,0)}\sim\text{Bernoulli}\left(r_j^{(k,0)}\right)$ independently for $j\in\mathcal{P}$. \\
Set \(S^{(k,0)} = \{j\in\mathcal{P};~b_j^{(k,0)}=1\}\).
\par
\endgroup
\item[(2)] For $m=1,\dots,R$: (for each round)
\vspace{-1mm}
\begin{enumerate}
\item[(a)] For \(k=1,\dots,K\): (for each worker in parallel)
\vspace{1mm}
\par
\begingroup
\leftskip=0.25cm
\noindent Run MAdaSub (Algorithm~\ref{algo:MCMC}) on worker \(k\) for \(T\) iterations with
\begin{itemize}
\item starting point \(S^{(k,(m-1)T)}\),
\item initial proposal probabilities \(\bar{\boldsymbol r}^{(k,m-1)}\),
\item initial adaptation parameters \(L^{(k)}_j + (m-1)TK\), for \(j\in\mathcal{P}\).
\end{itemize}
Output: Sampled models \(S^{(k,(m-1)T + t)}\) for \(t=1,\dots,T\).
\par
\endgroup
\item[(b)] Exchange information between workers: \\
For \(k=1,\dots,K\) compute \(\bar{\boldsymbol r}^{(k,m)} = \left(\bar{r}_1^{(k,m)},\dots,\bar{r}_p^{(k,m)}\right)^T\) with
\[\bar{ r}_j^{(k,m)} = \frac{ L_j^{(k)} r_j^{(k,0)} + \sum_{t=1}^{mT} \sum_{l=1}^K \mathbbm{1}_{S^{(l,t)}}(j) }{ L_j^{(k)} + mTK },~~ j\in\mathcal{P} \,.\]
\end{enumerate}
\end{enumerate}
\vspace{-5mm}
\begin{flushleft} \textbf{Output:} \end{flushleft}
\vspace{-7mm}
\begin{itemize}
\item For each worker \(k=1,\dots,K\) approximate sample \(S^{(k,b+1)},\dots,S^{(k,RT)}\) from posterior distribution \(\pi(\cdot\,|\,\mathcal{D})\), after burn-in period of length \(b\).
\end{itemize}
\end{algorithm}
A parallel version of MAdaSub is given as Algorithm~\ref{algo:parallel}, where the workers sample individual MAdaSub chains in parallel, but the acquired information is exchanged periodically between the chains and the adaptive proposal probabilities are updated together. More specifically, let \(S^{(k,1)},\dots,S^{(k,T)}\) denote the models sampled by MAdaSub (see Algorithm~\ref{algo:MCMC}) for the first \(T\) iterations on worker \(k\), for \(k\in\{1,\dots,K\}\). Then, for each worker \(k\in\{1,\dots,K\}\), we define the jointly updated proposal probabilities after the first round~(\(m=1\)) of \(T\) iterations by
\begin{equation} \bar{ r}_j^{(k,1)} = \frac{ L_j^{(k,0)} r_j^{(k,0)} + \sum_{t=1}^{T} \sum_{l=1}^K \mathbbm{1}_{S^{(l,t)}}(j) }{ L_j^{(k,0)} + TK } ,~~j\in\mathcal{P} \,,\end{equation}
where \(r_j^{(k,0)}\) denotes the initial proposal probability for variable \(X_j\) and \(L_j^{(k,0)}\) the corresponding adaptation parameter (both can be different across the chains).
After the joint update, each MAdaSub chain is resumed (with \(\bar{ r}_j^{(k,1)}\) as initial proposal probabilities and \(L^{(k)}_j + TK\) as initial prior variance parameters for \(j\in\mathcal{P}\)) and is run independently on each of the workers for \(T\) additional iterations in a second round (\(m=2\)); then the proposal probabilities are updated jointly again to \(\bar{ r}_j^{(k,2)}\), and so on (up to \(m=R\) rounds in Algorithm~\ref{algo:parallel}). The joint updates of the proposal probabilities after \(m\in\mathbb{N}\) rounds of \(T\) iterations are given by
\begin{equation} \bar{ r}_j^{(k,m)} = \frac{ L_j^{(k,0)} r_j^{(k,0)} + \sum_{t=1}^{mT} \sum_{l=1}^{K} \mathbbm{1}_{S^{(l,t)}}(j) }{ L_j^{(k,0)} + mTK } ,~~k\in\{1,\dots,K\}, ~ j\in\mathcal{P} . \label{eq:jointupdate}\end{equation}
Similarly to the serial version of MAdaSub, the adaptive learning of its parallel version can be naturally motivated in a Bayesian way:
each worker \(k=1,\dots,K\) can be thought of as an individual subject continuously updating its prior belief about the true posterior inclusion probability \(\pi_j\) of variable \(X_j\) through new information from its individual chain; additionally, after a period of \(T\) iterations the subject updates its prior belief also by obtaining new information from the \(K-1\) other subjects. If the (possibly different) priors of subjects \(k=1,\dots,K\) on \(\pi_j\) are
\begin{equation}
\pi_j \sim \mathcal{B}e\left(L_j^{(k)} r_j^{(k,0)},\, L_j^{(k)} \left(1- r_j^{(k,0)}\right)\right), \, j\in\mathcal{P}\,,
\end{equation}
where \(r_j^{(k,0)}=E[\pi_j]\) is the prior expectation of subject \(k\) about \(\pi_j\) and \(L_j^{(k)}>0\) controls its prior variance, then the (pseudo) posterior of subject \(k\) about \(\pi_j\) after \(m\) rounds of \(T\) iterations of the parallel MAdaSub algorithm
is given by (compare to equation~(\ref{eq:posterior}))
\begin{align}
\pi_j \,\big|\,S^{(1,1)},\dots,S^{(k,mT)} \sim \mathcal{B}e\bigg( & L_j^{(k)} r_j^{(k,0)} + \sum_{i=1}^{mT} \sum_{l=1}^K \mathbbm{1}_{S^{(l,i)}}(j) ,\, \nonumber \\
& L_j^{(k)} (1- r_j^{(k,0)}) + \sum_{i=1}^{mT} \sum_{l=1}^K \mathbbm{1}_{\mathcal{P}\setminus S^{(l,i)}}(j) \bigg)
\end{align}
with posterior expectation (compare to equation~(\ref{eq:update}))
\begin{equation}
E(\pi_j \,|\, S^{(1,1)},\dots,S^{(k,mT)}) = \bar{r}_j^{(k,m)} \,,
\end{equation}
corresponding to the joint update in equation~(\ref{eq:jointupdate}).
Although the individual chains in the parallel MAdaSub algorithm make use of the information from all the other chains in order to update the proposal parameters, the ergodicity of the chains is not affected.
\begin{theorem}\label{thm:parallel}
Consider the parallel version of MAdaSub (Algorithm~\ref{algo:parallel}). Then, for each worker \(k\in\{1,\dots,K\}\) and all choices of \(\boldsymbol r^{(k,0)}\in(0,1)^p\), \(L^{(k)}_j>0\), \(j\in\mathcal{P}\) and \(\epsilon\in(0,0.5)\), each induced chain \(S^{(k,0)},S^{(k,1)},\dots\) of the workers \(k=1,\dots,K\) is ergodic and fulfils the weak law of large numbers.
\end{theorem}
\begin{corollary}\label{cor:Mparallel}
For each worker \(k\in\{1,\dots,K\}\) and all choices of \(\boldsymbol r^{(k,0)}\in(0,1)^p\), \(L^{(k)}_j>0\), \(j\in\mathcal{P}\) and \(\epsilon\in(0,0.5)\), the proposal probabilities \(\bar{r}_j^{(k,m)}\) of the explanatory variables \(X_j\) converge (in probability) to the respective posterior inclusion probabilities \(\pi_j=\pi(j\in S\,|\,\mathcal{D})\), i.e. for all \(j\in\mathcal{P}\) and \(k=1,\dots,K\) it holds that
\( \bar{r}_j^{(k,m)} \overset{\text{P}}{\rightarrow} \pi_j \) as \(m\rightarrow\infty\).
\end{corollary}
Thus, the same convergence results hold for the parallel version (Algorithm~\ref{algo:parallel}) as for the serial version (Algorithm~\ref{algo:MCMC}) of MAdaSub. The benefit of the parallel algorithm is that the convergence of the proposal probabilities against the posterior inclusion probabilities can be accelerated via the exchange of information between the parallel chains, so that the MCMC chains can converge faster against the full posterior distribution. There is a practical trade-off between the effectiveness regarding the joint update for the proposal probabilities and the efficiency regarding the communication between the different chains. If the number of rounds~\(R\) is chosen to be small with a large number of iterations~\(T\) per round, the available information from the multiple chains is not fully utilized during the algorithm; however, if the number of rounds~\(R\) is chosen to be large with a small number of iterations~\(T\) per round, then the computational cost of communication between the chains increases and may outweigh the benefit of the accelerated convergence of the proposal probabilities.
If~\(T_{\text{max}}\) denotes the maximum number of iterations, we observe that choosing the number of rounds \(R\in[10,100]\) with \(T=T_{\text{max}}/R\) iterations per round works well in practice (see Section~\ref{sec:realdata}).
\section{Simulated data applications}\label{sec:sim}
\subsection{Illustrative example}
We first illustrate the adaptive behaviour of the serial MAdaSub algorithm (Algorithm~\ref{algo:MCMC}) in a relatively low-dimensional setting. By this way, it is computationally feasible to derive the exact posterior model distribution via a full model enumeration and to compare it with the estimates from MAdaSub.
In particular, we consider an illustrative simulated dataset \(\mathcal{D}=(\boldsymbol X,\boldsymbol y)\) with sample size \(n=60\) and \(p=20\) explanatory variables, by generating \( \boldsymbol X=(X_{i,j})\in\mathbb{R}^{n\times p}\) with \(i\)-th row \(\boldsymbol X_{i,*}\sim\mathcal{N}_p(\boldsymbol 0,\boldsymbol \Sigma)\), where \(\boldsymbol \Sigma=(\Sigma_{i,j})\in\mathbb{R}^{p \times p}\) is the covariance matrix with entries \(\Sigma_{k,l}=\rho^{|k-l|}\), \(k,l\in\{1,\dots,p\}\), corresponding to a Toeplitz-correlation structure with \(\rho=0.9\). The true vector of regression coefficients is considered to be \[\boldsymbol \beta_0=(0.4,0.8,1.2,1.6,2.0,0,\dots,0)^T\in\mathbb{R}^p\,,\] with active set \(S_0=\{1,\dots,5\}\). The response \(\boldsymbol y=(y_1,\dots,y_n)^T\) is then simulated from the normal linear model via \(y_i\overset{\text{ind.}}{\sim} N(\boldsymbol X_{i,*} \boldsymbol \beta_0, 1)\), \(i=1,\dots,n\).
We employ the g-prior with \(g=n\) and an independent Bernoulli model prior with inclusion probability \(\omega=0.5\), resulting in a uniform prior over the model space (see Remark~\ref{remark:conjugate}). In the MAdaSub algorithm we set \(r_j^{(0)}=\frac{1}{2}\) for \(j\in\mathcal{P}\), i.e.\ we use the prior inclusion probabilities as initial proposal probabilities.
We consider the choice \(L_j=p\) (for \(j\in\mathcal{P}\)) for the variance parameters of MAdaSub, corresponding to equation~(\ref{eq:update_no}). Furthermore, we set \(\epsilon=\frac{1}{p}\) and run the MAdaSub algorithm for \(T=50{,}000\) iterations. To compare the results of MAdaSub with the true posterior model distribution, we have also conducted a full model enumeration using the Bayesian Adaptive Sampling (BAS) algorithm, which is implemented in the R-package \texttt{BAS} \citep{clyde2017}. By this, we have obtained the exact posterior inclusion probabilities \(\pi(j\in S\,|\,\mathcal{D})\) for each \(j\in\mathcal{P}\).
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/MAdaSub_lowdim_c09_p20_gprior_sizes_12_10_eps1p.pdf}
\caption{\label{fig:lowdim_sizes}Illustrative example with g-prior. \small Evolution of the sizes \(|V^{(t)}|\) of the proposed models (black) and of the sizes \(|S^{(t)}|\) of the sampled models (red) along the first 5,000 iterations (\(t\)) for non-adaptive sampler with prior marginals as proposal probabilitie
, for MAdaSub (with \(L_j=p\)), for non-adaptive sampler with posterior marginals as proposal probabilities and for local \(\text{MC}^3\) sampler (from top to bottom).}
\end{figure}
To illustrate the efficient adaptation of MAdaSub, we present comparisons with independent Metropolis-Hastings algorithms where the individual proposal probabilities are \textit{not} adapted during the algorithm, i.e.\ we set \(r_j^{(t)}=r_j^{(0)}\) for all \(t\in\mathbb{N}\) and \(j\in\mathcal{P}\). In particular, we consider the choice \(r_j^{(t)}=r_j^{(0)}=0.5\), corresponding to the initial proposal distribution in MAdaSub, and the choice \(r_j^{(t)}=r_j^{(0)}=\pi(j\in S\,|\,\mathcal{D})\), corresponding to the targeted proposal distribution, which is, as stated above, the closest independent Bernoulli proposal to the target \(\pi(\cdot\,|\,\mathcal{D})\) in terms of Kullback-Leibler divergence (\citealp{clyde2011}). Note that the non-adaptive independence sampler with posterior inclusion probabilities as proposal probabilities (\(r_j^{(t)}=\pi(j\in S\,|\,\mathcal{D})\)) is only considered as a benchmark and cannot be used in practice, since the true posterior probabilities are initially unknown and are to be estimated by the MCMC algorithms. Furthermore, we also present comparisons with a standard local ``Markov chain Monte Carlo model composition'' (\(\text{MC}^3\)) algorithm \citep{madigan1995}, which in each iteration proposes to delete or add a single variable to the current model.
\begin{figure}[!t]\centering
\includegraphics[width=0.9\textwidth]{Plots/MAdaSub/MAdaSub_lowdim_c09_p20_gprior_acc3_12_6_eps1p.pdf}
\caption{\label{fig:lowdim_acc} Illustrative example with g-prior. \small Evolution of acceptance rates along the iterations for non-adaptive independence sampler with prior marginals (blue) and posterior marginals (red) as proposal probabilities, for \(\text{MC}^3\) sampler (gray), as well as for MAdaSub with \(L_j=p\) (black), \(L_j=p/n\) (orange) and \(L_j=100p\) (purple) for \(j\in\mathcal{P}\).}
\end{figure}
Figure~\ref{fig:lowdim_sizes} depicts the sizes \(|V^{(t)}|\) of the proposed models and the sizes \(|S^{(t)}|\) of the sampled models, while Figure~\ref{fig:lowdim_acc} shows the evolution of the acceptance rates along the iterations \(t\) of the different MCMC algorithms.
As might have been expected, the non-adaptive sampler with prior marginals as proposal probabilities performs poorly with a very slow exploration of the model space and a small acceptance rate which remains close to zero. On the other hand, the non-adaptive sampler with posterior marginals as proposal probabilities leads to fast mixing with corresponding acceptance rate of approximately \(0.54\). Even though the MAdaSub algorithm starts with exactly the same ``initial configuration'' as the non-adaptive sampler with prior marginals, it quickly adjusts the proposal probabilities accordingly, so that the resulting acceptance rate approaches the target value of \(0.54\) from the non-adaptive sampler with posterior marginals. In particular, when inspecting the evolution of the sampled model sizes in Figure~\ref{fig:lowdim_sizes}, the MAdaSub algorithm is very difficult to distinguish from the sampler with posterior marginals after a very short burn-in period (see also Figure~\ref{fig:lowdim_values} of the supplement).
In order to investigate the behaviour of the MAdaSub algorithm with respect to the variance parameters~\(L_j\), additionally to the choice \(L_j=p\) we examine two further runs of MAdaSub with the same specifications as before, but with \(L_j=p/n\)
and with \(L_j=100p\), respectively. Figure~\ref{fig:lowdim_acc} indicates that the original choice \(L_j=p\) is favourable, yielding a fast and ``sustainable'' increase of the acceptance rate. On the other hand, for \(L_j=100p\) the proposal probabilities in MAdaSub are slowly adapted, while for \(L_j=p/n\) the proposal probabilities are adapted very quickly, resulting in initially large acceptance rates; however, this increase is only due to a premature focus of the proposal on certain parts of the model space and thus the acceptance rate decreases at some point when the algorithm identifies other areas of high posterior probability that have not been covered by the proposal.
Figure~\ref{fig:lowdim_probabilities} shows the evolution of the proposal probabilities~\(r_j^{(t)}\) corresponding to variables~\(X_j\), with \(j=1,\dots,9\), along the iterations~\(t\) of MAdaSub for the three different choices of the variance parameter~\(L_j\). For~\(L_j=p\), the proposal probabilities quickly converge to the correct posterior inclusion probabilities; for \(L_j=100p\) and \(L_j=p/n\), the proposal probabilities also ``converge correctly'' but at a considerably slower pace.
This illustrative example shows that --- despite the ergodicity of the MAdaSub algorithm for all choices of its tuning parameters (Theorem~\ref{thm:MAdaSub}) --- the speed of convergence against the target distribution crucially depends on an appropriate choice of these parameters. Regarding the variance parameters we observe that the choice~\(L_j=p\) for~\(j\in\mathcal{P}\) works generally well in practice (see also results below).
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/MAdaSub_lowdim_c09_p20_gprior_probabilities3_12_6_eps1p.pdf}
\caption{\label{fig:lowdim_probabilities} Illustrative example with g-prior. \small Evolution of the proposal probabilities~\(r_j^{(t)}\), for \(j=1,\dots,9\), along the iterations~(\(t\)) of MAdaSub with \(L_j=p\) (black), \(L_j=p/n\) (orange) and \(L_j=100p\) (purple) for \(j\in\mathcal{P}\). The red horizontal lines indicate the true posterior inclusion probabilities.}
\end{figure}
The adaptive nature of MAdaSub entails the possibility for an automatic check of convergence of the algorithm: as the proposal probabilities~\(r_j^{(t)}\) are continuously adjusted towards the current empirical inclusion frequencies \(f_j^{(t)}=\frac{1}{t}\sum_{i=1}^t \mathbbm{1}_{S^{(i)}}(j)\) (see equation~(\ref{eq:update})), the MAdaSub algorithm may be stopped as soon as the individual proposal probabilities and empirical inclusion frequencies are within a prespecified distance \(\delta\in(0,1)\) (e.g.~\(\delta=0.005\), see~Figure~\ref{fig:lowdim_convergence_check} of the supplement), i.e.\ the algorithm is stopped at iteration~\(t_c\) if \(\max_{j\in\mathcal{P}}|f_j^{(t_c)}-r_j^{(t_c)}|\leq \delta\). Even when automatic stopping may be applied, we additionally recommend to investigate the convergence of the MAdaSub algorithm via the diagnostic plots presented in this section.
\subsection{Simulation study}
In this simulation study we further investigate the performance of the serial MAdaSub algorithm in relation to a local non-adaptive \(\text{MC}^3\) algorithm. In particular, we analyse how the algorithms are affected by high correlations between the covariates.
We consider a similar setting as in the illustrative data application with \(p=20\) covariates and sample size \(n=60\). To evaluate the performance in a variety of different data situations, for each simulated dataset the number \(s_0\) of informative variables is randomly drawn from \(\{0,1,\dots,10\}\) and the true active set \(S_0\subseteq\mathcal{P}\) of size \(|S_0|=s_0\) is randomly selected from the full set of covariates \(\mathcal{P}=\{1,\dots,p\}\); then, for each \(j\in S_0\), the \(j\)-th component \(\beta_{0,j}\) of the true coefficient vector \(\boldsymbol\beta_0\in\mathbb{R}^p\) is simulated from a uniform distribution \(\beta_{0,j}\sim U(-2,2)\). As before, the covariates are simulated using a Toeplitz correlation structure, while the response is simulated from a normal linear model with error variance \(\sigma^2=1\). We consider three different correlation settings by varying the correlation \(\rho\) between adjacent covariates in the Toeplitz structure:\ a low-correlated setting with \(\rho=0.3\), a highly-correlated setting with \(\rho=0.9\) and a very highly-correlated setting with \(\rho=0.99\). For each of the three settings, 100 different datasets are simulated as described above; in each case, we employ a g-prior with \(g=n\) on the regression coefficients and a uniform prior on the model space.
For each simulated dataset we apply MAdaSub with 20,000 iterations, using \(L_j=p\) for \(j\in\mathcal{P}\) and \(\epsilon=\frac{1}{p}\). In order to investigate the influence of the initial proposal probabilities \(r_j^{(0)}\) in MAdaSub, two different choices for \(r_j^{(0)}\) are considered: choice~(a)~based on prior inclusion probabilities \(r_j^{(0)}=\frac{1}{2}\) and choice~(b)~based on (approximated) marginal posterior odds
\begin{equation} \pi_j^{\text{marg}} = \frac{\text{PO}_j}{1+\text{PO}_j} ~~~ \text{with } ~~ \text{PO}_j = \frac{P( S=\{j\}\,|\,\mathcal{D})}{P(S=\emptyset\,|\,\mathcal{D})},~ j\in\mathcal{P} , \label{eq:margodds} \end{equation}
and setting \(r_j^{(0)} = \min \{ \max\{ \pi_j^{\text{marg}}, \frac{1}{p} \} , 0.9 \} \) to prevent the premature focus of the algorithm on some covariates (if \(\pi_j^{\text{marg}}\approx 1\)) or the avoidance of other covariates (if \(\pi^{\text{marg}}_j\approx 0\)). Here, the marginal posterior odds \(\text{PO}_j\) are crude approximations to the true posterior odds, derived under the assumption of posterior independence of variable inclusion. The local \(\text{MC}^3\) algorithm \citep{madigan1995} is applied as before with 20,000 iterations. Exact posterior inclusion probabilities are again derived using the BAS algorithm \citep{clyde2017}. The different MCMC algorithms are evaluated based on final acceptance rates and the number of iterations for convergence of the posterior inclusion probabilities (PIP), where PIP convergence is defined to occur at the smallest iteration~\(t_c\) for which \(\max_{j\in\mathcal{P}}|r_j^{(t_c)}-\pi_j|\leq 0.05\); if \(t_c\geq20{,}000\), then the number of iterations for convergence is displayed as 20,000 in Figure~\ref{fig:simulations_corr}.
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/Simulations_Correlation_12_6_epsilon.pdf}
\caption{\label{fig:simulations_corr}\small Simulation study with varying correlation \(\rho\in\{0.3,0.9,0.99\}\) in Toeplitz structure. Performance of MAdaSub --- with (a) \(r_j^{(0)}=\frac{q}{p}\) and (b) \(r_j^{(0)}\) based on marginal posterior odds~(\ref{eq:margodds}) --- and \(\text{MC}^3\) sampler in terms of acceptance rates and numbers of iterations for convergence of posterior inclusion probabilities (PIP).}
\end{figure}
Figure~\ref{fig:simulations_corr} shows that the acceptance rates of the MAdaSub samplers tend to be substantially larger in comparison to the local \(\text{MC}^3\) algorithm in all correlation settings. Nevertheless, for the MAdaSub samplers a decreasing trend of acceptance rates can be observed with increasing correlations. This observation reflects that for low-correlated situations the resulting posterior distribution is often closer to an independent Bernoulli form than for highly-correlated cases, and thus can be better approximated by the proposal distributions of MAdaSub, leading to larger acceptance rates. In the low-correlated setting (\(\rho=0.3\)), the choice~(b) for the initial proposal probabilities in MAdaSub based on marginal posterior odds leads to slightly larger acceptance rates and a faster PIP convergence compared to the MAdaSub sampler~(a) based on the prior inclusion probabilities. However, in cases of high correlations among some of the covariates (\(\rho=0.9\) and \(\rho=0.99\)), the prior choice~(a) is clearly favorable yielding larger acceptance rates and a faster PIP convergence compared to the MAdaSub sampler~(b) and the \(\text{MC}^3\) algorithm. Thus, while in low-correlated settings the marginal posterior odds yield reasonable first approximations to the true posterior odds, the prior inclusion probabilities are more robust and to be preferred as initial proposal probabilities in MAdaSub in situations with high correlations. In fact, the results show that the MAdaSub sampler~(a) yields a well-mixing algorithm even in very highly-correlated settings.
\section{Real data applications}\label{sec:realdata}
\subsection{Tecator data}\label{sec:tecator}
We first examine the Tecator dataset which has already been investigated in \citet{griffin2010}, \citet{lamnisos2013} and \citet{griffin2018}. The data has been recorded by \citet{borggaard1992} on a Tecator Infratec Food Analyzer and consists of \(n=172\) meat samples and their near-infrared absorbance spectra, represented by \(p=100\) channels in the wavelength range 850-1050nm (compare \citealp{griffin2010}). The fat content of the samples is considered as the response variable.
Here we analyse the efficiency of MAdaSub under the same setting as in \citet{lamnisos2013}, where several adaptive and non-adaptive MCMC algorithms are compared using normal linear models for the Tecator data. In particular, \citet{lamnisos2013} consider a classical \(\text{MC}^3\) algorithm (\citealp{madigan1995}), the adaptive Gibbs sampler of \citet{nott2005} and adaptive and non-adaptive Metropolis-Hastings algorithms based on the tunable model proposal of \citet{lamnisos2009}. In order to compare their results with MAdaSub, we choose the same conjugate prior set-up as in \citet{lamnisos2013}, i.e.\ we use the prior given in equation~(\ref{eq:prior1}) with \(g=5\), \(\boldsymbol V_S = \boldsymbol I_{|S|}\) for \(S\in\mathcal{M}\) (where \(\boldsymbol I_{|S|}\) denotes the identity matrix of dimension \(|S|\))
and we employ the independent Bernoulli model prior given in equation~(\ref{eq:modelprior}) with (fixed) prior inclusion probability \(\omega=\frac{5}{100}\), so that the mean prior model size is equal to five.
In the comparative study of \citet{lamnisos2013} each algorithm is run for 2,000,000 iterations, including an initial burn-in period of 100,000 iterations. Furthermore, thinning is applied using only every 10th iteration, so that the finally obtained MCMC sample has size 190,000. For comparison reasons, after a burn-in period of 100,000 iterations, we run the serial MAdaSub algorithm for 190,000 iterations, so that the considered MCMC sample has the same size as in \citet{lamnisos2013}. In the serial MAdaSub algorithm we first set \(r_j^{(0)}=\frac{5}{100}\) for \(j\in\mathcal{P}\), i.e.\ we use the prior inclusion probabilities as the initial proposal probabilities in MAdaSub; further, we set \(L_j=p\) for \(j\in\mathcal{P}\) and \(\epsilon=\frac{1}{p}\). Since the acceptance rate of MAdaSub is already sufficiently large in the considered setting yielding a well-mixing algorithm, we do not consider additional thinning of the resulting chain. In fact, the acceptance rate of the serial MAdaSub chain is approximately 0.38 for the 190,000 iterations (excluding the burn-in period). We note that in this example the relatively large number of 100,000 burn-in iterations is not necessarily required for MAdaSub and is only used for comparison reasons. \citet{lamnisos2013} report estimated median effective sample sizes of the different samplers
for the evolution of the indicators \(\big(\gamma_j^{(t)}\big)_{t=1}^T\) for \(j\in\mathcal{P}\), where \(\gamma_j^{(t)}=\mathbbm{1}_{S^{(t)}}(j)\) indicates whether variable \(X_j\) is included in the sampled model \(S^{(t)}\) in iteration \(t\). The estimated median effective sample size for the 190,000 iterations of the serial MAdaSub algorithm is approximately 38,012 (using the R-package \texttt{coda}), which is slightly larger than the values for the competing algorithms reported in \citealp{lamnisos2013} (the largest one is 37,581 for the ``optimally'' tuned Metropolis-Hastings algorithm). Note that when using 1,900,000 iterations with thinning (every 10th iteration after 100,000 burn-in iterations) as in the other algorithms, the estimated median effective sample size for MAdaSub is much larger (178,334), yielding almost independent samples of size 190,000.
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/Tecator_parallel_random_epsilon_12_6.pdf}
\caption{\label{fig:Tecator_parallel}Tecator data application. \small Results of 25 independent serial MAdaSub chains (Algorithm~\ref{algo:MCMC}) and of 25 parallel MAdaSub chains exchanging information after every 5,000 iterations (Algorithm~\ref{algo:parallel}) in terms of empirical variable inclusion frequencies~\(f_j\) for \(j\in\{1,\dots,100\}\).}
\end{figure}
To investigate the stability of the results for different choices of the tuning parameters of MAdaSub, we run 25 independent serial MAdaSub chains (Algorithm~\ref{algo:MCMC}) with the same set-up as described above but with random initializations of the sampling probabilities \(r_j^{(k,0)}=q^{(k)}/p\), \(j\in\mathcal{P}\), with \(q^{(k)}\sim U(2,10)\) and of the variance parameters \(L_j^{(k)}=L^{(k)}\), \(j\in\mathcal{P}\), with \(L^{(k)}\sim U(p/2,2p)\), for each chain \(k=1,\dots,25\). Furthermore, we run 25 additional parallel MAdaSub chains (Algorithm~\ref{algo:parallel}) with the described random initializations, exchanging the information after each of \(R=58\) rounds of \(T=5{,}000\) iterations (yielding in total again 290,000 iterations for each of the additional 25 parallel chains).
Figure~\ref{fig:Tecator_parallel} shows the resulting empirical variable inclusion frequencies (as estimates of posterior inclusion probabilities) for the 25 serial and 25 parallel MAdaSub chains. From left to right, the first three plots of Figure~\ref{fig:Tecator_parallel} depict the development of the empirical inclusion frequencies for the first three rounds of 5,000 iterations each, while the rightmost plots depict the final empirical inclusion frequencies after 290,000 iterations (disregarding the burn-in period of 100,000 iterations). After the first 5,000 iterations, the empirical inclusion frequencies show a similar variability for the serial and parallel chains, as no communication between the parallel chains has yet occurred. After the second round of 5,000 further iterations, the benefit of the communication between the 25 parallel chains is apparent, leading to less variable estimates due to a faster convergence of the proposal probabilities against the posterior inclusion probabilities. Nevertheless, also the serial MAdaSub chains (with different initial tuning parameters) provide quite accurate estimates after only 10,000 iterations.
After 290,000 iterations, all of the serial and parallel MAdaSub chains yield very stable estimates of posterior inclusion probabilities, reproducing the results shown in Figure~1 of \citet{lamnisos2013}. As the covariates represent 100 channels of the near-infrared absorbance spectrum, it is not surprising that adjacent channels have similar posterior inclusion probabilities. If one is interested in selecting a final single model, the median probability model (which includes all variables with posterior inclusion probability greater than 0.5, see \citealp{barbieri2004}) might not be the best choice in this particular situation, since then only variables corresponding to the ``global mode'' and no variables from the two other ``local modes'' in Figure~\ref{fig:Tecator_parallel} are selected. Alternatively, one may choose one or two variables from each of the three ``local modes'' or make use of Bayesian model averaging \citep{raftery1997} for predictive inference.
The computational time for each of the 5000 iterations of the serial MAdaSub algorithm is approximately 3.5 seconds (using an R implementation of MAdaSub on an Intel(R) Core(TM) i7-7700K, 4.2 GHz processor); thus, even without parallelization, one obtains accurate posterior estimates with the serial MAdaSub algorithm within seconds using a usual desktop computer (e.g.\ after 10,000 or 15,000 iterations, see Figure~\ref{fig:Tecator_parallel}). \citet{lamnisos2013} report that the computational times for each of the other considered MCMC methods were in the order of 25,000 seconds for the total number of 2,000,000 iterations (using a MATLAB implementation). Although the computational times are not directly comparable, these results indicate that the serial MAdaSub algorithm is already very efficient. When using a computer cluster with 50 CPUs, the overall computational time for all considered 50 MAdaSub chains (each with 290,000 iterations) is 568 seconds, while the computational time for a single chain is 231 seconds on the same system. This shows that, even though 25 of the 50 MAdaSub chains communicate with each other after every 5,000 iterations, the parallelization yields a substantial speed-up in comparison to a serial application of 50 independent chains.
\subsection{PCR and Leukemia data}\label{sec:PCR}
We illustrate the effectiveness of MAdaSub for two further high-dimensional datasets. In particular, we consider the polymerase chain reaction (PCR) dataset of \citet{lan2006} with \(p=22{,}575\) explanatory variables (expression levels of genes), sample size \(n=60\) (mice) and continuous response data (the dataset is available in JRSS(B) Datasets Vol. 77(5), \citealp{Song2015}). Furthermore, we consider the leukemia dataset of \citet{golub1999} with \(6817\) gene expression measurements of \(n=72\) patients and binary response data (the dataset can be loaded via the R-package \texttt{golubEsets}, \citealp{golub2017}). For the PCR dataset we face the problem of variable selection in a linear regression framework, while for the leukemia dataset we consider variable selection in a logistic regression framework. We have preprocessed the leukemia dataset as described in \citet{dudoit2002} (compare also \citealp{ai2009}), resulting in a restricted design matrix with \(p=3571\) columns (genes). Furthermore, in both datasets we have mean-centered the columns of the design matrix after the initial preprocessing.
Here we adopt the posterior approximation induced by \(\text{EBIC}_\gamma\) with \(\gamma=1\) (see equation~(\ref{eq:EBICkernel})), corresponding to a beta-binomial model prior with \(a_\omega=b_\omega=1\) as parameters in the beta distribution (see Section~\ref{sec:setting}).
For both datasets we run 25 independent serial MAdaSub chains (Algorithm~\ref{algo:MCMC}) with 2,500,000 iterations and 25 parallel MAdaSub chains (Algorithm~\ref{algo:parallel}) exchanging information after each of \(R=50\) rounds of \(T=50{,}000\) iterations (yielding also 2,500,000 iterations for each parallel chain). For each serial and parallel chain \(k=1,\dots,50\), we set~\(\epsilon=\frac{1}{p}\) and randomly initialize the sampling probabilities \(r_j^{(k,0)}=q^{(k)}/p\), \(j\in\mathcal{P}\), with \(q^{(k)}\sim U(2,5)\) and the variance parameters \(L_j^{(k)}=L^{(k)}\), \(j\in\mathcal{P}\), with \(L^{(k)}\sim U(p/2,2p)\). For the leukemia dataset we make use of a fast C\texttt{++} implementation for ML-estimation in logistic regression models via a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, which is available in the R-package \texttt{RcppNumerical} \citep{qiu2016}. For both datasets, the 50 MAdaSub chains are run in parallel on a computer cluster with 50 CPUs, yielding overall computational times of 7,710 seconds for the PCR data (6,502 seconds for a single chain) and 5,886 seconds for the Leukemia data (2,509 seconds for a single chain).
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/PCR_evolution_14_7_annotated.pdf}
\caption{\label{fig:PCR_evolution}PCR data application. \small
Evolution of empirical variable inclusion frequencies
for 25 serial MAdaSub chains (Algorithm~\ref{algo:MCMC}, top) and 25 parallel MAdaSub chains exchanging information after every round of 50,000 iterations
(Algorithm~\ref{algo:parallel}, bottom). Bold lines represent median frequencies with 5\%- and 95\%-quantiles (shaded area) over the chains within each round, for most informative variables \(X_j\) (with final estimate \(f_j\geq 0.05\) for at least one chain).
}
\end{figure}
Figures~\ref{fig:PCR_evolution} and~\ref{fig:Leukemia_evolution} show the evolution of empirical inclusion frequencies of the most informative variables along the number of rounds of MAdaSub for the PCR and leukemia data, respectively. Despite the high-dimensional model spaces and the different initializations of each chain, it is apparent that for both datasets the parallel MAdaSub algorithm provides stable estimates of posterior inclusion probabilities after a small number of rounds.
In particular, the estimates from the parallel MAdaSub algorithm stabilize after only three rounds of 50,000 iterations (see also Figures~\ref{fig:PCR_parallel} and~\ref{fig:Leukemia_parallel} of the supplement). For the PCR data, all serial and parallel MAdaSub chains yield congruent and very accurate estimates of posterior inclusion probabilities after 2,500,000 iterations (Figures~\ref{fig:PCR_evolution}, \ref{fig:highdimreal_scatter} and~\ref{fig:PCR_parallel}).
The final acceptance rates of MAdaSub for the PCR dataset are between 0.20 and 0.24, while the acceptance rates for the leukemia dataset are between 0.03 and 0.14. The smaller acceptance rates for the leukemia dataset indicate that this corresponds to a more challenging scenario (i.e.~the targeted posterior model distribution seems to be ``further away'' from an independent Bernoulli form). This observation is also reflected in the larger variability of the estimates from the MAdaSub chains without parallel updating (Figures~\ref{fig:Leukemia_evolution}, \ref{fig:highdimreal_scatter} and~\ref{fig:Leukemia_parallel}). The leukemia data application particularly illustrates the benefits of the parallel version of MAdaSub, where multiple chains with different initializations sequentially explore different regions of the model space, but exchange the information after each round of 50,000 iterations, increasing the speed of convergence of the proposal probabilities to the posterior inclusion probabilities.
Note that in very high-dimensional settings such as for the PCR data (with \(p=22{,}575\)), the classical \(\text{MC}^3\) algorithm \citep{madigan1995} does not yield stable estimates due to slow mixing (cf.\ \citealp{griffin2018}), while the BAS algorithm \citep{clyde2017} using sampling without replacement is computationally intractable. Further results in \cite{griffin2018} show that several competing adaptive algorithms --- including sequential Monte Carlo algorithms of \cite{schafer2013} and tempered Gibbs sampling algorithms of \cite{zanella2019} --- do not provide reliable estimates of posterior inclusion probabilities for the PCR data; only the adaptively scaled individual adaptation algorithm of \cite{griffin2018} with proposals of the form (\ref{eq:propalt}) yields stable results for the PCR data similarly to MAdaSub with a slightly different prior set-up (see Figures~10 and~11 of the supplement of \citealp{griffin2018}).
\begin{figure}[!t]\centering
\includegraphics[width=\textwidth]{Plots/MAdaSub/Golub_evolution_12_6_annotated.pdf}
\caption{\label{fig:Leukemia_evolution}Leukemia data application. \small Evolution of empirical variable inclusion frequencies for 25 serial MAdaSub chains (Algorithm~\ref{algo:MCMC}, top) and 25 parallel MAdaSub chains exchanging information after each round of 50,000 iterations (Algorithm~\ref{algo:parallel}, bottom) for most informative variables \(X_j\) (with final estimate \(f_j\geq 0.1\) for at least one chain), cf.\ Figure~\ref{fig:PCR_evolution}.}
\end{figure}
Due to the very large model spaces in both considered examples, posterior probabilities of individual models are generally small and corresponding MCMC estimates will typically not be very reliable. Therefore, as in similar studies (see \citealp{griffin2018}), we have focused on the estimation of posterior inclusion probabilities. For the PCR data two variables (genes) stand out with respect to their final estimates of posterior inclusion probabilities, namely the gene 1438937\_x\_at (covariate index \(j=7640\)) with estimated probability between 0.52 and 0.59, and the gene 1438936\_s\_at (\(j=7639\)) with estimated probability between 0.33 and 0.37.
Similarly, for the leukemia data two genes stand out, namely the genes M23197\_at (\(j=956\)) with estimated probability between 0.35 and 0.49
and X95735\_at (\(j=2481\)) with estimated probability between between 0.17 and 0.27 (considering final estimates from the 25 parallel chains only). We note that these two genes are also among the four top scoring genes in a Bayesian probit regression analysis of the leukemia dataset in \citet{ai2009}.
\section{Discussion}\label{sec:discussion}
We introduced a new adaptive MCMC algorithm, the Metropolized Adaptive Subspace (MAdaSub) algorithm, for sampling from high-dimensional posterior model distributions in situations where conjugate priors or approximations to the posterior are employed. We further developed an efficient parallel version of MAdaSub, where the information regarding the adaptive proposal probabilities of the variables can be shared periodically between the different chains, leading to a faster convergence of the individual chains in practice.
Simulated and real data applications illustrated that MAdaSub can efficiently sample from multimodal posterior model distributions, yielding stable estimates of posterior inclusion probabilities even for ten thousands of possible covariates.
The reliable estimation of posterior inclusion probabilities is particularly important for Bayesian inference, since the median probability model (MPM) --- including all variables with posterior inclusion probability larger than 0.5 --- has been shown to yield optimal predictions for uncorrelated covariates \citep{barbieri2004} and also a favourable performance for correlated designs \citep{barbieri2021}, e.g.\ compared to the largest posterior probability model. MAdaSub provides a natural adaptive MCMC algorithm which focuses on the sequential adaptation of currently estimated inclusion probabilities, with the aim of driving the sampler quickly into regions near to the MPM; in the limit, the MPM itself is the model which receives the largest probability under the independent Bernoulli proposal of MAdaSub.
Despite the continuing adaptation of the proposals, we have shown that MAdaSub constitutes a valid MCMC algorithm which samples from the full posterior model distribution.
While the serial and parallel versions of MAdaSub are ergodic for all choices of their tuning parameters (see Theorem~\ref{thm:MAdaSub} and Theorem~\ref{thm:parallel}), we have also emphasized that the ``speed of convergence'' against the targeted posterior model distribution depends crucially on a proper choice of their tuning parameters (see Section~\ref{sec:sim}). Deriving theoretical results regarding the mixing time of the proposed algorithms is an important but challenging issue for further research.
Since MAdaSub is based on adaptive independent proposal distributions, in each iteration of the algorithm the proposed model is (almost) independent of the current model, so that ``distant'' moves in the model space are encouraged. This can be advantageous in comparison to Gibbs samplers and Metropolis-Hastings algorithms based on local proposal distributions, which may yield larger acceptance rates but are more prone to be stuck in local modes of the posterior model distribution. In future work one may also consider combinations of the adaptive independent proposals in MAdaSub with adaptive local proposals as for example in \citet{lamnisos2013} and \citet{zanella2019}.
Furthermore, the extension of MAdaSub to settings with non-conjugate priors is interesting to be investigated by using similar ideas as in \citet{ji2013} or by incorporating additional reversible-jump moves \citep{green1995}.
\bibliographystyle{Chicago}
{\footnotesize
\setlength{\bibsep}{0.5pt}
|
2,877,628,089,466 | arxiv | \section{Introduction}\label{sec:introduction}
In the last two decades, owing to the advent of powerful new space and ground-based telescopes for imaging and spectroscopic observations,
many new strong gravitational lensing (SGL) systems have been discovered. The sample size of available SGL systems has grown to be large enough for statistical analysis to study lens properties and to constrain cosmological parameters. Since the number of observed galaxy-scale SGL systems
is much more than that of galaxy cluster-scale SGL systems, most statistical analyses have utilized the galaxy-scale SGL sample.
In practice, several different quantities can be adopted as statistical quantities with galaxy-scale SGL sample, including the distribution of image angular separations (see, e.g., Turner et al. 1984; Dyer 1984; Chiba \& Yoshii 1999; Dev et al 2004; Cao \& Zhu 2012), the distribution of lens redshifts (see, e.g., Turner et al. 1984; Kochanek 1992; Ofek et al. 2003; Mitchell et al. 2005; Cao et al. 2012a), and the velocity dispersion ($\sigma$) of lenses (see, e.g., Futamase \& Yoshida 2001; Biesiada 2006; Grillo et al. 2008; Schwab et al. 2010; Cao et al. 2017).
The major disadvantage of using the distributions of image angular separations and lens redshifts as statistical quantities is that the
theoretically predicted values of these two are dependent not only on the lens mass model but
also on the lens luminosity function. While the theoretical prediction of $\sigma$ is dependent only on the lens mass model but not on the luminosity function. Besides, the gravitational lens time-delay ($\Delta\tau$) method is another cosmological application of SGL systems
(see, e.g., Refsdal 1964; Treu \& Marshall 2016; Bonvin et al. 2017; Birrer et al. 2019), which is different
from the three methods mentioned above, since the time-delay analysis is done for one system at a time rather than performing on a sample of lens galaxies simultaneously.
The methods of using $\Delta\tau$ and $\sigma$ as observed quantities for the SGL systems are both popular at present.
The theoretical analysis shows that $\Delta\tau$ is more sensitive to the cosmological
parameters than $\sigma$ (Paraficz \& Hjorth 2009; Wei \& Wu 2017). The fact also proves that the measurements of $\Delta\tau$ are very powerful at constraints on the cosmological parameters and especially sensitive to the Hubble constant $H_0$ (Bonvin et al. 2017; Suyu et al. 2017; Liao et al. 2017; Birrer et al. 2019). The measurements of $\sigma$ are weak at confining the cosmological parameters (see, e.g., Biesiada 2006; Cao et al. 2012b; Wang \& Xu 2013; Chen et al. 2015; Cao et al. 2015; An et al. 2016; Xia et al. 2017; Cui et al. 2017; Li et al. 2018a), but they are useful for investigating the lens mass models if the priors on cosmological parameters are given (see, e.g., Koopmans et al. 2009; Sonnenfeld et al. 2013a; Cao et al. 2016; Holanda et al. 2017).
Additionally, a combination of time delay and velocity dispersion (i.e., $\Delta\tau/\sigma^2$) is proved to be more sensitive to
the cosmological parameters (see., e.g., Paraficz \& Hjorth 2009; Jee et al. 2015, 2016; Wei \& Wu 2017; Shajib et al. 2018) than using $\Delta\tau$ and $\sigma$ separately.
By combining the observations of SGL and stellar dynamics in elliptical
galaxies, one can use the lens velocity dispersion (VD) as statistical quantity to put constrains on both the cosmological parameters and the density profiles
of galaxies. The core idea of this method is that the gravitational mass $M_{\textrm{grl}}^E$ and the dynamical mass $M_{\textrm{dyn}}^E$
enclosed within the disk defined by the so-called Einstein ring should be equivalent, namely, $M_{\textrm{grl}}^E=M_{\textrm{dyn}}^E$.
Further, $M_{\textrm{grl}}^E$ inferring from the strong lensing data depends on cosmological distances, and $M_{\textrm{dyn}}^E$ inferring from the stellar VD depends on both the lens mass model and the cosmological distance, so one can relate the VD with the model parameters including cosmological and lens mass model parameters. This method can be traced back to Futamase \& Yoshida (2001), but at that time there were no available observational data of lens VD. Grillo et al. (2008) first applied this method to constrain cosmological parameters with observational data, wherein the sample included 20 SGL systems from the Lens Structure and Dynamics (LSD) survey (Koopmans \& Treu 2002, 2003; Treu \& Koopmans 2002, 2004) and the Sloan Lens ACS (SLACS) survey (Bolton et al. 2006a; Treu et al. 2006; Koopmans et al.2006).
In the literature, a recent compiled sample which can be used in this method includes 118 galaxy-scale
SGL systems (Cao et al. 2015, hereafter C15) from the SLACS survey, the Baryon Oscillation Spectroscopic Survey (BOSS) emission-line lens survey (BELLS; see, Brownstein et al. 2012), the LSD survey,
and the Strong Lensing Legacy Survey (SL2S; see, Gavazzi et al. 2012; Ruff et al. 2011; Sonnenfeld et al. 2013a,b, 2015). In this paper, we update the sample with definite criteria by taking advantage of new observational data, and then explore the effect of lens mass model on constraining cosmological parameters, as well as evaluate several different lens mass models.
The rest of the paper is organized as follows. In Section 2, we demonstrate the methodology of using the lens
velocity dispersion as statistical quantity to constrain model parameters. Then,
in Section 3 the SGL data sample used in our analysis is introduced. In Section 4, we first investigate the sensitivity of the sample under consideration to cosmological parameters, and diagnose whether the lens mass density profile is universal for the entire sample via the qualitative and semi-quantitative analysis; and then carry out observational constraints on parameters of cosmology and lens mass models.
In the last section, the main conclusions are summarized.
\section{Methodology}\label{sec:method}
As discussed in the last section, the method of using the galaxy lens VD as statistical quantity has some special merits.
However, in this method, besides the imaging data of the SGL systems, one also has to possess the spectroscopic data
of the systems and measure the central velocity dispersion of the lens galaxies from the spectroscopy.
On the basis of various recent lensing surveys which have carried out both imaging and spectroscopic observations,
this method has become feasible.
In this method, the main idea is that the projected gravitational mass $M_{\textrm{grl}}^E$ and the projected dynamical
mass $M_{\textrm{dyn}}^E$ within the Einstein radius should be equivalent, i.e.,
\begin{equation}
\label{eq:Mass_eq}
M_{\textrm{grl}}^E= M_{\textrm{dyn}}^E.
\end{equation}
From the theory of gravitational lensing,
the projected gravitational mass within the Einstein radius is $M_{\textrm{grl}}^E=\Sigma_{cr}\pi R_E^2$. The Einstein
radius $R_E$ is determined by $R_E=\theta_E D_l$, wherein $\theta_E$ is the Einstein angle, and $D_l$ is the angular
diameter distance between observer and lens. The critical surface mass density $\Sigma_{cr}$ is defined
by $\Sigma_{cr} = \frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}}$, where $D_{ls}$ is the angular diameter distance
between lens and source, and $D_s$ is that between observer and source. Thus, one can further figure out
\begin{equation}
\label{eq:Mgrl}
M_{\textrm{grl}}^E= \frac{c^2}{4G}\frac{D_sD_l}{D_{ls}}\theta_E^2,
\end{equation}
wherein the distances $D_s$, $D_l$ and $D_{ls}$ are dependent on the cosmological model.
To estimate the projected dynamical mass $M_{\textrm{dyn}}^E$ from the lens galaxy VD,
one must first suppose the mass distribution model for the lens galaxy.
Here we choose a general mass model (Koopmans 2006) for the lens galaxies in our sample, which are early-type galaxies (ETGs) with E/S0 morphologies:
\begin{eqnarray}
\label{eq:profile}
\left\{
\begin{array}{lll}
\rho(r)&=& \rho_0\; (r/r_0)^{-\gamma}\\
\nu(r)&=& \nu_0\; (r/r_0)^{-\delta} \\
\beta(r)&=&1-\sigma_{\theta}^2/ \sigma_r^2
\end{array}
\right.
\end{eqnarray}
where $\rho(r)$ is the total (i.e. luminous plus dark-matter) mass density distribution, and $\nu(r)$ is the luminosity density of stars.
The parameter $\beta(r)$ denotes the anisotropy of the stellar velocity dispersion, and is also called as the stellar orbital anisotropy, where $\sigma_\theta$ and $\sigma_r$ are
the tangential and radial velocity dispersions, respectively.
Based on the assumption that the relationship between stellar number density $n(r)$ and stellar luminosity density $\nu(r)$ is
spatially constant, an assumption unlikely to be violated appreciably within the effective radius of the
early-type lens galaxies under consideration, the radial Jeans equation in Spherical Coordinate can be written as
\begin{equation}
\frac{\emph{d}}{\emph{d} r} [\nu(r)\sigma^2_r]+\frac{2\beta}{r}\nu(r)\sigma^2_r =-\nu(r)\frac{\emph{d} \Phi}{\emph{d} r},
\label{eq:Radial_JeansEq}
\end{equation}
where
\begin{equation}
\frac{\emph{d}\Phi}{\emph{d}r}=\frac{GM(r)}{r^2},
\label{eq:dPhi_dr}
\end{equation}
and $M(r)$ is the total mass inside a sphere with radius $r$.
By substituting Eq. (\ref{eq:dPhi_dr}) into Eq. (\ref{eq:Radial_JeansEq}), one can get the expression for $\sigma_r^2$,
\begin{equation}
\sigma_r^2(r) = \frac{G\int^{\infty}_{r}\emph{d}r^{\prime}r^{\prime 2\beta-2}\nu(r^{\prime})M(r^{\prime})}{r^{2\beta}\nu(r)},
\label{eq:sigma2r}
\end{equation}
By defining $r$ to be the spherical radial coordinate from the lens center, $Z$ to be the axis along the line of sight (LOS) , and $R$ to be the cylindrical radius which is perpendicular to the LOS, then one has $r^2=R^2+Z^2$. The projected dynamical
mass $M_{\textrm{dyn}}$ contained within a cylinder of radius equal to the Einstein radius $R_E$ can be calculated with
\begin{equation}
M_{\textrm{dyn}}^E = \int_{0}^{R_E}\emph{d}R 2\pi R'\Sigma(R'),
\label{eq:ME}
\end{equation}
where
\begin{eqnarray}
\Sigma(R)&=&\int_{-\infty}^{\infty}\rho(r)dZ\nonumber\\
&=& \int_{-\infty}^{\infty}\emph{d}Z \frac{\rho_0}{r_0^{-\gamma}}(Z^2+R^2)^{-\gamma/2}\nonumber\\
&=&\sqrt(\pi)R^{1-\gamma}\frac{\Gamma\left(\frac{\gamma-1}{2}\right)}{\Gamma(\gamma/2)}\frac{\rho_0}{r_0^{-\gamma}}
\label{eq:Sigma_R}
\end{eqnarray}
By substituting Eq.(\ref{eq:Sigma_R}) into Eq.(\ref{eq:ME}), one can have
\begin{equation}
M_{\textrm{dyn}}^E = 2\pi^{3/2}\frac{R_E^{3-\gamma}}{3-\gamma}\frac{\Gamma\left(\frac{\gamma-1}{2}\right)}{\Gamma(\gamma/2)}\frac{\rho_0}{r_0^{-\gamma}}.
\label{eq:ME2}
\end{equation}
The total mass contained within a sphere with radius $r$ is
\begin{equation}
M(r)=\int_0^r dr' 4\pi r'^2\rho(r')=4\pi\frac{\rho_0}{r_0^{-\gamma}}\frac{r^{3-\gamma}}{3-\gamma}.
\label{eq:Mr}
\end{equation}
By combining Eqs. (\ref{eq:ME2}) and (\ref{eq:Mr}), one can further have
\begin{equation}
M(r)=\frac{2}{\sqrt{\pi}}\frac{\Gamma(\gamma/2)}{\Gamma(\frac{\gamma-1}{2})}\left(\frac{r}{R_E}\right)^{3-\gamma}M_{\textrm{dyn}}^E.
\label{eq:Mr2}
\end{equation}
By substituting Eqs. (\ref{eq:Mr2}) and (\ref{eq:profile}) into Eq. (\ref{eq:sigma2r}), one reads
\begin{equation}
\sigma_r^2(r)
=
\frac{2}{\sqrt{\pi}}\frac{GM_{\textrm{dyn}}^E}{R_E}\frac{1}{\xi-2\beta}\frac{\Gamma(\gamma/2)}
{\Gamma(\frac{\gamma-1}{2})}\left(\frac{r}{R_E}\right)^{2-\gamma},
\label{eq:sigma_2r_2}
\end{equation}
where $\xi=\gamma+\delta-2$, and $\beta$ is assumed to be independent of the radius $r$.
The actual velocity dispersion of the lens galaxy measured by the observation is the component of luminosity-weighted average along the LOS and over the effective spectroscopic aperture $R_A$, that can be expressed mathematically
\begin{equation}
\sigma^2_{\parallel}(\leq R_A) = \frac{\int_{0}^{R_A}\emph{d}R\, 2\pi R\int_{-\infty}^{\infty}\emph{d}Z\, \sigma^2_{\textrm{los}}\nu(r)}{\int_{0}^{R_A}\emph{d}R\, 2\pi R\int_{-\infty}^{\infty}\emph{d}Z\,\nu(r)}
\label{eq:sigma2ll_1}
\end{equation}
where $\sigma^2_{\textrm{los}}$ is the LOS velocity dispersion, which is a combination of the radial ($\sigma_r^2$) and tangential ($\sigma_t^2$) velocity dispersions. Using $\theta$ to indicate the angle between the LOS (Z-axis) and the radial direction (r-axis), then
one reads
\begin{eqnarray}
\sigma^2_{los}&=&(\sigma_r \cos\theta)^2+(\sigma_t \sin\theta)^2\nonumber\\
&=&\sigma_r^2\frac{r^2-R^2}{r^2}+\sigma_t^2\frac{R^2}{r^2}\nonumber\\
&=&\sigma_r^2(1-\frac{R^2}{r^2})+(1-\beta)\sigma_r^2\frac{R^2}{r^2}\nonumber\\
&=&\sigma_r^2(1-\beta\frac{R^2}{r^2})
\label{eq:sigma_los}
\end{eqnarray}
By substituting Eq.(\ref{eq:sigma_los}) into Eq.(\ref{eq:sigma2ll_1}), one can read
\begin{equation}
\sigma^2_{\parallel}(\leq R_A) = \frac{\int_{0}^{R_A}\emph{d}R\, 2\pi R\int_{-\infty}^{\infty}\emph{d}Z\, \sigma^2_r(r)(1-\beta\frac{R^2}{r^2})\nu(r)}{\int_{0}^{R_A}\emph{d}R\, 2\pi R\int_{-\infty}^{\infty}\emph{d}Z\,\nu(r)}
\label{eq:sigma2ll_2}
\end{equation}
Further, by substituting Eq.(\ref{eq:sigma_2r_2}) and (\ref{eq:profile}) into Eq.(\ref{eq:sigma2ll_2}), one obtains
\begin{eqnarray}
\begin{array}{rr}
\sigma^2_{\parallel}(\leq R_A) = \frac{2}{\sqrt{\pi}}\frac{GM_{\textrm{dyn}}^E}{R_E}\frac{3-\delta}{(\xi-2\beta)(3-\xi)}\left[\frac{\Gamma
\left[(\xi-1)/2\right]}{\Gamma(\xi/2)}-\beta\frac{\Gamma\left[(\xi+1)/2\right]}{\Gamma\left[(\xi+2)/2
\right]}\right]\\
\frac{\Gamma(\gamma/2)\Gamma(\delta/2)}{\Gamma\left[(\gamma-1)/2\right]\Gamma\left[(\delta-1)/2\right]}
\left(\frac{R_A}{R_E}\right)^{2-\gamma}.
\end{array}
\end{eqnarray}
Finally, with the relation expressed in Eq.(\ref{eq:Mass_eq}), the above formula can be rewritten as
\begin{eqnarray}
\label{eq:sigma_thetaA}
\begin{array}{rr}
\sigma^2_{\parallel}(\leq R_A) = \frac{c^2}{2\sqrt{\pi}}\frac{D_s}{D_{ls}}\theta_E\frac{3-\delta}{(\xi-2\beta)(3-\xi)}\left[\frac{\Gamma
\left[(\xi-1)/2\right]}{\Gamma(\xi/2)}-\beta\frac{\Gamma\left[(\xi+1)/2\right]}{\Gamma\left[(\xi+2)/2
\right]}\right]\\
\frac{\Gamma(\gamma/2)\Gamma(\delta/2)}{\Gamma\left[(\gamma-1)/2\right]\Gamma\left[(\delta-1)/2\right]}
\left(\frac{\theta_A}{\theta_E}\right)^{2-\gamma},
\end{array}
\end{eqnarray}
where $R_A=\theta_A D_l$.
From the spectroscopic data, one can measure the velocity dispersion $\sigma_{\textrm{ap}}$ inside the circular aperture with
the angular radius $\theta_{\textrm{ap}}$. In practice, if the $\sigma_{\textrm{ap}}$ are measured within rectangular apertures,
one usually derives the equivalent circular apertures with
the angular radii $\theta_{\textrm{ap}}$ following J{\o}rgensen et al. (1995),
\begin{equation}
\label{eq:theta_ap_eff}
\theta_{\textrm{\textsf{ap}}}\approx 1.025\times \sqrt{(\theta_x\theta_y/\pi)},
\end{equation}
where $\theta_x$ and $\theta_y$ are the angular sizes of
width and length of the rectangular aperture.
More precisely, $\sigma_{\textrm{ap}}$ is the luminosity weighted average of the line-of-sight velocity dispersion of the lensing galaxy inside $\theta_{\textrm{ap}}$.
For a fair comparison and in consideration of the effect of the aperture size on the measurements of velocity dispersions, all
velocity dispersions $\sigma_{\textrm{ap}}$ measured within apertures of arbitrary sizes,
are normalized to a typical physical aperture, $\sigma_{\textrm{e2}}$, with the
radius $R_{\textrm{eff}}/2$, where $R_{\textrm{eff}}$ is the half-light radius of the lens galaxy. The radius $R_{\textrm{eff}}/2$ is chosen because it is well-matched to the typical
Einstein radius, therefore just a small error is brought in when the relation satisfied in the Einstein radius (e.g., Eq.(\ref{eq:Mass_eq}) ) is extrapolated to the radius $R_{\textrm{eff}}/2$ (Auger et al. 2010).
Following the prescription, one can use the aperture correction formula,
\begin{eqnarray}
\label{eq:sigma_obs}
\sigma^{\textrm{obs}}_{\parallel} \equiv \sigma_{\textrm{e2}} = \sigma_{\textrm{ap}}[\theta_{\textrm{eff}}/(2\theta_{\textrm{ap}})]^{\eta},
\end{eqnarray}
where $\theta_{\textrm{eff}} = R_{\textrm{eff}}/D_l$.
The best-fitting values of the
correction factor $\eta$ are different when using different observational samples. For example, the best-fitting values of $\eta$ are $-0.04$, $-0.06$ and $-0.066\pm 0.035$ found by J{\o}rgensen et al. (1995), Mehlert et al. (2003) and Cappellari et al. (2006), respectively, where the third value is consistent with the former two at the 1$\sigma$ level.
In this work, we adopt the value $\eta = -0.066\pm 0.035$ from Cappellari et al. (2006). Then, the total uncertainty of $\sigma_{\textrm{e2}}$, i.e., $\Delta\sigma_{\textrm{e2}}^{\textrm{tot}}$, satisfies
\begin{equation}
\label{eq:err_sigma_e2}
(\Delta\sigma_{\textrm{e2}}^{\textrm{tot}})^2 = (\Delta\sigma_{\textrm{e2}}^{\textrm{stat}})^2+(\Delta\sigma_{\textrm{e2}}^{\textrm{AC}})^2+
(\Delta\sigma_{\textrm{e2}}^{\textrm{sys}})^2.
\end{equation}
The the statistical error, $\Delta\sigma_{\textrm{e2}}^{\textrm{stat}}$, is propagated from the measurement error of $\sigma_{\textrm{ap}}$. The error due to the aperture correction, $\Delta\sigma_{\textrm{e2}}^{\textrm{AC}}$, is propagated from the uncertainty of $\eta$.
In addition to the measurement errors, we should also consider the systematic error $\Delta\sigma_{\textrm{e2}}^{\textrm{sys}}$. The essential assumption of the method is that the projected mass within the Einstein radius, $M^E$, can be uniformly estimated from both the gravitational and dynamical masses, i.e., $M^E = M^E_\textrm{grl} = M^E_\textrm{dyn} $. In practice, the model-predicted value of $M^E = M^E_
\textrm{dyn}$ from Eq.(\ref{eq:ME2}) only includes the contribution from the lens galaxy, while the value of $M^E = M^E_\textrm{grl}$ from Eq.(\ref{eq:Mgrl}) includes the extra contribution from other matters (outside of the lens galaxy) along the line of sight. The extra mass from the lensing data
can be treated as a systematic error, which contributes uncertainty of $\sim$3\% to the model-predicted value of the velocity dispersion (Jiang \& Kochanek 2007).
In order to compare the observational values of the VD with the corresponding model-predicted ones, one needs to
calculate the theoretical value of the VD within the radius $R_{\textrm{eff}}/2$ from Eq. (\ref{eq:sigma_thetaA}) (Koopmans 2006),
\begin{equation}
\label{eq:sigma_th}
\sigma_{\parallel(\leq\theta_{\textrm{eff}}/2)}=\sqrt{\frac{c^2}{2\sqrt{\pi}}\frac{D_s}{D_{ls}}\theta_E\frac{3-\delta}{(\xi-2\beta)(3-\xi)}F(\gamma,\delta, \beta)\left(\frac{\theta_{\textrm{eff}}}{2 \theta_\textrm{E}}\right)^{(2-\gamma)}},
\end{equation}
where
\begin{equation}
F=\left[\frac{\Gamma
\left[(\xi-1)/2\right]}{\Gamma(\xi/2)}-\beta\frac{\Gamma\left[(\xi+1)/2\right]}{\Gamma\left[(\xi+2)/2
\right]}\right]
\frac{\Gamma(\gamma/2)\Gamma(\delta/2)}{\Gamma\left[(\gamma-1)/2\right]\Gamma\left[(\delta-1)/2\right]}.
\end{equation}
In the case of $\gamma = \delta =2$ and $\beta= 0$, the mass model is reduced to the well-known Singular Isothermal Sphere(SIS) model,
and the predicted value of the VD is recovered to
\begin{equation}
\label{eq:sigma_th_SIS}
\sigma_{\textrm{SIS}}= \sqrt{\frac{c^2}{4\pi}\frac{D_{s}}{D_{ls}}\theta_E}.
\end{equation}
In our analysis, the likelihood is assumed to be
\begin{equation}
\label{eq:likelihood}
\mathcal{L}\propto e^{-\chi^2/2}.
\end{equation}
$\chi^2$ is constructed as
\begin{equation}
\label{eq:chi2}
\chi^2=\sum^{N}_{i=1}\left(\frac{\sigma^{\textrm{th}}_{\parallel ,i}-\sigma^{\textrm{obs}}_{\parallel ,i}}
{\Delta\sigma^{\textrm{tot}}_{\parallel ,i}}\right)^2,
\end{equation}
where $N$ is the number of the data points, $\Delta\sigma^{\textrm{tot}}_{\parallel ,i}$ is
the uncertainty of $\sigma^{\textrm{obs}}_{\parallel ,i}$, which is calculated with Eq. (\ref{eq:err_sigma_e2}). One can obtain $\sigma^{\textrm{obs}}_{\parallel ,i}$ and $\sigma^{\textrm{th}}_{\parallel ,i}$
from Eqs. (\ref{eq:sigma_obs}) and (\ref{eq:sigma_th}), respectively.
In the following analyses, we derive the posterior probability distributions of model parameters through an affine--invariant Markov chain Monte Carlo (MCMC) Ensemble sampler (emcee; Foreman-Mackey et al. 2013), where the likelihood is computed with Eqs. (\ref{eq:likelihood}) and (\ref{eq:chi2}).
For the purpose of the analysis in this work, it would suffice to assume that $\beta$ is independent of $r$ (see, e.g., koopmans et al. 2006; Treu et al. 2010). Because we cannot
independently measure $\beta$ for individual lensing systems, We then treat $\beta$
as a nuisance parameter and marginalize over it using a Gaussian prior with $\beta=0.18\pm0.13$, that is an independent constraint on $\beta$ from a well-studied sample of nearby elliptical galaxies
(see, e.g.,Gerhard et al. 2001), and adopted in the previous works (see, e.g., Bolton et al. 2006; Schwab et al. 2010; Cao et al. 2017). Thus, throughout this paper a Gaussian prior on $\beta$ with $\beta=0.18\pm0.13$ is used over the range of $[\bar{\beta}-2\sigma_{\beta}, \bar{\beta}+2\sigma_{\beta}]$ where $\bar{\beta}=0.18$ and $\sigma_{\beta}=0.13$, unless some special instructions are made.
In addition, we assume a flat prior for each remaining parameter over a range of interest.
\section{Data sample}\label{sec:data}
According to the analysis in the last section, one can learn that the method under consideration requires the following information
from observations, including the lens redshift $z_l$, the source redshift $z_s$, the Einstein angle $\theta_E$, the central VD of
the lens galaxy $\sigma_{\textrm{ap}}$, the spectroscopic aperture angular radius $\theta_{\textrm{ap}}$, and the half-light angular radius of the
lens galaxy $\theta_{\textrm{eff}}$. Additionally, to ensure the validity of the assumption of spherical symmetry on the lens galaxy,
the selected lens galaxies should satisfy the following
conditions:(i) the lens galaxy should be ETGs with E/S0 morphologies; and (ii) the lens galaxy should
not have significant substructure or close massive companion.
Some lens galaxies from C15's sample do not satisfy the
above conditions. Here we assemble a sample including 161 galaxy-scale SGL systems which meet all the requirements mentioned above,
where 5 systems from the LSD survey\footnote{http://web.physics.ucsb.edu/$\sim$tt/LSD/}(Koopmans \& Treu 2002, 2003; Treu \& Koopmans 2002, 2004), 26 from the SL2S (Ruff et al. 2011; Sonnenfeld et al. 2013a,b; Sonnenfeld et al. 2015), 57 from the SLACS (Bolton et al. 2008; Auger et al. 2009, 2010), 38 from the an extension of the SLACS survey known
as ``SLACS for the Masses'' (hereafter S4TM, Shu et al. 2015; Shu et al. 2017), 21 from the BELLS (Brownstein et al. 2012, hereafter B12), and 14 from
the BELLS for GALaxy-Ly$\alpha$ EmitteR sYstemsGALLERY (hereafter BELLS GALLERY, Shu et al. 2016a,b). The useful information of these 161 systems is listed in Appendix (i.e., Table \ref{tab:sample}). The SLACS lenses used in this work are selected from the full SLACS sample (Bolton et al. 2008, hereafter B08) with high-fidelity observations carried out using the Advanced Camera for Surveys (ACS) on Hubble Space Telescope (HST), where the data of $\theta_{E}$ and $\theta_{\textrm{eff}}$ are taken from Tables 4 and 5 of B08, and the data of $z_l$, $z_s$ and $\sigma_{\textrm{ap}}$ from the Table 3 of Auger et al. (2009). All the observational data of BELLS lenses are taken from B12. In B12, the effective radii $\theta_{\textrm{eff}}$ of the lenses are measured from both the BOSS and HST-ACS imaging data, and the measurements from the later are much more precise than those from the former. So, We choose to use the $\theta_{\textrm{eff}}$ data from the HST-ACS observations, which are listed in Table 3 of B12. In addition, the lenses from S4TM and BELLS GALLERY surveys are not included in C15's sample.
The velocity dispersions of the lenses from LSD and SL2S surveys, which are measured within rectangular slits, are transformed into velocity dispersions $\sigma_{e2}$, within a circular aperture with radius $R_{\textrm{eff}}/2$ based on Eqs.(\ref{eq:theta_ap_eff}) and (\ref{eq:sigma_obs}). The SLACS and S4TM surveys select candidates from Sloan Digital Sky Survey I (SDSS-I, Eisenstein et al. 2001; Strauss et al. 2002) data, in which the velocity dispersions of the lenses are measured within the
$1.5^{\prime\prime}$--radius fibers. The lens candidates of the BELLS and BELLS GALLERY surveys are spectroscopically selected from the BOSS (Dawson et al. 2013) of the Sloan Digital Sky Survey-III (SDSS-III, Eisenstein et al.
2011), in which the VD of the lenses are measured within the
$1^{\prime\prime}$--radius fibers. These velocity dispersions measured with fibers are corrected to $\sigma_{e2}$ based on Eq.(\ref{eq:sigma_obs}). The distribution of the whole SGL sample is shown in Figure \ref{fig:distributions}.
From the upper panels of Figure \ref{fig:distributions}, one can see that $\sim$30\% of the lenses are located at $z_l \sim 0.2$, and only $\sim$5\% located at $z_l>0.75$. The lower panels of Figure \ref{fig:distributions} show that $\sim$80\% of the lenses possess the velocity dispersions $180\, \textrm{km}\,\textrm{s}^{-1} <\sigma_{e2}<300\,\textrm{km}\,\textrm{s}^{-1}$.
\section{Analysis and Results}\label{sec:analysis}
\subsection{Qualitative analysis}
From Eq. (\ref{eq:sigma_th}) one can see that the cosmological model enters into the theoretical observable $\sigma^{\textrm{th}}_{\parallel}$ not through a distance measure directly, but rather through a distance ratio
\begin{equation}
\label{eq:Dth}
\frac{D_s}{D_{ls}}=\frac{\int_0^{z_s} \frac{dz}{E(z;\textbf{p})}}{\int_{z_l}^{z_s} \frac{dz}{E(z;\textbf{p})}},
\end{equation}
where in the framework of the flat FLRW metric the theoretical values of $D_s$ and $D_{ls}$ can be obtained by
\begin{equation}
\label{eq:Dsth} D_s(z_s; \textbf{p}, H_0) = \frac{c}{H_0(1+z_s)} \int_0^{z_s} \frac{dz}{E(z;\textbf{p})},
\end{equation}
and
\begin{equation}
\label{eq:Ddsth} D_{ls}(z_l, z_s; \textbf{p}, H_0) = \frac{c}{H_0(1+z_s)} \int_{z_l}^{z_s} \frac{dz}{E(z;\textbf{p})},
\end{equation}
respectively, where $\textbf{p}$ denotes the parameter space of the considered cosmological model, and $E=H/H_0$ is the dimensionless Hubble parameter, and $c$ is the velocity of light. The theoretical prediction of the observable is independent of the Hubble constant $H_0$
which gets canceled in the distance ratio. On the other side, the distance ratio $D_s/D_{ls}$ is a ratio of two integrals which have the same integrand (i.e., $1/E(z;\textbf{p})$) and differ only by the limits of integration, so the theoretical observable $\sigma^{\textrm{th}}_{\parallel}\propto \sqrt{D_s/D_{ls}}$ is insensitive to the cosmological parameters involved in the integrand (Biesiada et al. 2010). In Figure \ref{fig:sen_om}, we show the impact of the matter-density parameter $\Omega_m$ on the distance ratio by taking a spatially flat $\Lambda$CDM model with $\Omega_m = 0.3$ as a fiducial cosmological model. The three panels of Figure \ref{fig:sen_om} display the evolution of $D_s/D_{ls}$ with respect to the source redshift $z_s$ along with variety of $\Omega_m$, corresponding to the cases of the lens redshift $z_l = 0.1, 0.5, \textrm{and} 1$ from left to right. The general trend is that the sensitivity of $D_s/D_{ls}$ to $\Omega_m$ increases with the increase of $z_l$.
In Figure \ref{fig:sen_om}, the shadows denote the cases that the relative uncertainties of $D_s/D_{ls}$ are 10\% and 20\%, respectively, with
respect to the fiducial value. One can see that an individual lens with $z_l = 0.1$ cannot put any constraint on $\Omega_m$
even when $D_s/D_{ls}$ only has 10\% uncertainty. An individual lens with $z_l = 0.5$ can bound on $\Omega_m$ with
$\sim$80\%--160\% relative uncertainty when $D_s/D_{ls}$ only has 10\% uncertainty, but cannot put any constraint on $\Omega_m$
when the uncertainty of $D_s/D_{ls}$ increases to 20\%. Unfortunately, with regard to the SGL sample under consideration,
the typical values of the relative uncertainties of $D_s/D_{ls}$
are approximately 10\% and 20\% at $z_l \approx 0.1$ and $0.5$, respectively\footnote{The uncertainty on $D_s/D_{ls}$ is mainly propagated from that on $\sigma_{e2}$. The relative uncertainty on
$D_s/D_{ls}$ is about 2 times of that on $\sigma_{e2}$ because of $D_s/D_{ls}\propto \sigma_{e2}^2$.}. It means that most individual lenses with $z_l < 0.5$ in our sample do not contribute to the limit on $\Omega_m$. An
individual lens with $z_l = 1$ can put a limit on $\Omega_m$ with $\sim$ 50\% -- 100\% ($\sim$ 80\% -- 200\%) relative uncertainty,
corresponding to $D_s/D_{ls}$ with 10\%(20\%) uncertainty. In our sample, there is only one system with $z_l > 1$, that is MG2016$+$112 with
$z_l = 1.004$ from LSD survey. In general, one is not able to make a high-precision estimate on $\Omega_m$ with the sample under consideration.
After repeating similar analyses for other cosmological parameters (i.e., the equation of state parameter of dark energy, and the curvature parameter), we find out that the current sample is really weak at confining these cosmological parameters. Besides, the distance ratio $D_s/D_{ls}$ is more sensitive to $\Omega_m$ than to the equation of state parameter of dark energy (Sereno 2002).
\subsection{Observational constraints} \label{sec:Obs_Cons}
We assume a kind of spherically symmetric mass distributions (i.e. Eq. (\ref{eq:profile})) for the lens galaxies in the kinematic analysis. As discussed above, the dependence of $\gamma$ on the properties of lens galaxies should be taken into account. In the previous works, the dependence of the total mass density slope $\gamma$ on the redshift has been widely studied (see, e.g., Ruff et al. 2011; Bolton et al 2012; Cao et al. 2015; Cao et al. 2016; Cui et al. 2017; Holanda et al. 2017). Besides, Auger et al. (2010) also found a significant correlation between $\gamma$ and total mass surface density, that has also been confirmed by Dutton \& Treu (2014) and Sonnenfeld et al. (2013a). In the light of these works, we specifically consider three parameterizations for $\gamma$, namely:
\begin{itemize}
\item $P_1:$ $\gamma =\gamma_0,$
\item $P_2:$ $\gamma=\gamma_0+\gamma_z*z_l,$
\item $P_3:$ $\gamma=\gamma_0+\gamma_z*z_l+\gamma_s*\log\tilde{\Sigma},$
\end{itemize}
where $\gamma$ is treated as an arbitrary constant in case $P_1$, and its dependence on the lens redshift $z_l$ is considered in
case $P_2$. Besides, the dependence on both the redshift and the surface mass density is taken into account in case $P_3$. According to the virial theorem, the projected dynamical mass within the radius $R_{\textrm{eff}}/2$ satisfies $M_{\textrm{e2}}^{\textrm{dyn}}\propto\sigma_{\textrm{e2}}^2R_{\textrm{eff}}$ (see, e.g., Auger et al. 2010), so the corresponding surface mass density is $\Sigma \propto \sigma_{\textrm{e2}}^2/R_{\textrm{eff}}$.
Here, we use $\tilde{\Sigma}$ to denote the normalized surface mass density of the lens galaxy, which is expressed as
\begin{equation}
\tilde{\Sigma}=\frac{(\sigma_{\textrm{e2}}/100 \textrm{km}\, \textrm{s}^{-1})^2}{R_{\textrm{eff}}/10h^{-1}\textrm{kpc}},
\end{equation}
where the usual convention of writing the Hubble constant as $H_0 = 100 h$ $\textrm{km} \textrm{s}^{-1} \textrm{Mpc}^{-1}$ is adopted.
As mentioned above, the sample under consideration is quite weak at constraining cosmological parameters, so constraining too many cosmological parameters simultaneously would only distort the results. Thus, we only attempt to fit $\Omega_m$ in the framework of flat $\Lambda$CDM model, where $\Omega_m$ is the only free parameter of cosmology. We then conduct observational constraints on $\Omega_m$ and lens mass model parameters in the scenarios of ``$P_1$'', ``$P_2$'' and ``$P_3$'', respectively. As discussed at the end of Sec. {\ref{sec:method}}, the orbit anisotropy
parameter $\beta$ is treated as a nuisance parameter and marginalize over it using a Gaussian
prior, and the flat priors are assumed for other free parameters. In the following, we consider two different treatment schemes for the slope ($\delta$) of
the luminosity density profile.
\subsubsection{The case of treating $\delta$ as a universal parameter for all lens galaxies in the sample}
We first consider the case of treating the luminosity density slope $\delta$ as a universal parameter for all lens galaxies in the entire sample. In other words, the intrinsic scatter of $\delta$ among the lens galaxies is assumed to be ignorable.
This treatment scheme to $\delta$ is the same as that adopted in Cao et al. (2016, 2017) and Xia et al. (2017).
In the previous studies, another treatment to $\delta$ is setting $\delta = \gamma$ (see., e.g., C15; An et al. 2016; Cui et al. 2017), that is
not adopted in this work.
The results corresponding to three parameterizations for $\gamma$ are displayed in Figure {\ref{fig:om_1d}}. The limits on $\Omega_m$ at 68\% (95\%) confidence level are $\Omega_m < 0.067$($\Omega_m < 0.158$), $\Omega_m < 0.227$($\Omega_m < 0.731$) and $\Omega_m>0.832$ ($\Omega_m>0.616$) in the scenarios of parameterizing $\gamma$ with ``$P_1$'', ``$P_2$'' and ``$P_3$'', respectively. The main tendencies can be summarized to three aspects.
First, the limits on $\Omega_m$ are significantly dependent on the lens mass model. The allowed range of $\Omega_m$ in the third scenario is inconsistent with those obtained in the former two at 68\% confidence level.
Second, the constraints on $\Omega_m$ are weak. In the first two scenarios, the lower limits on $\Omega_m$ are unavailable. Conversely , the upper limit on $\Omega_m$ is absent in the last scenario. It is consistent with the qualitative analysis mentioned previously, which reveals that the sample under consideration is insensitivity to $\Omega_m$. Third, the estimations on $\Omega_m$ are biased. The mean value of $\Omega_m$ constrained from the standard cosmological probes is around 0.3 (see, e.g. Huterer \& Shafer 2018; Scolnic et al. 2018; Alam et al. 2017), such as $\Omega_m = 0.315\pm0.007$ in the framework of flat $\Lambda$CDM model obtained from the recent Planck 2018 result (Planck Collaboration: Aghanim et al. 2018). In the first scenario, the limit on $\Omega_m$ is inconsistent with that from the Placnck result at 95\% confidence level, and the allowed values of $\Omega_m$ are especially low. In the second scenario, it is consistent with the Placnk result at 95\% confidence level, but the mean value of $\Omega_m$ is much lower. In the last scenario, it is inconsistent with the Placnk result at 95\% confidence level, and the mean value is much higher.
This is the first time to constrain the cosmological parameter in the scenario of considering the dependence of $\gamma$ on both redshift and surface mass density. In C15, they constrained the equation of state (EoS) of dark energy (with other cosmological parameters fixed) from their sample with 118 systems in the scenarios of $P_1$ and $P_2$. From the results listed in Table 2 of C15, one can see that the constraints on the EoS of dark energy are also quite weak, the uncertainties are bigger than 30\%.
\subsubsection{The case of treating $\delta$ as an observable for each lens galaxy}
In the previous analyses, the most troublesome issue is the bias in the estimation of $\Omega_m$, which may be due to some unconsidered systematic errors. The sample dependence of $\delta$ has been ignored in the above analysis, that may be a potential source of bias in the estimation of $\Omega_m$. To verify this conjecture, we choose to consider the intrinsic scatter of $\delta$ among the lens galaxies by treating $\delta$ as an observable for each lens galaxy.
We obtain $\delta$ values by fitting the two-dimensional power-law luminosity profile convolved with the instrumental point-spread function (PSF)
to HST F814W or F606W imaging data over a circle of radius $\theta_{\textrm{eff}}/2$ centered on the lens galaxies\footnote{The observational and model-predicted values of the velocity dispersion (i.e., Eqs. (\ref{eq:sigma_obs}) and (\ref{eq:sigma_th})) used in the analysis are the components of luminosity-weighted average over the aperture with radius $\theta_{\textrm{eff}}/2$, so here we do the luminosity fitting inside $\theta_{\textrm{eff}}/2$ for each lens galaxy.}, where the projected two-dimensional profile, $I(R)\propto R^{-\delta+1}$, is derived from the corresponding three-dimensional profile, i.e., $\nu(r)\propto r^{-\delta}$ (Bolton et al. 2006b and Schwab et al. 2010).
Obviously, the high-resolution HST imaging data must be available for the selected lens galaxies. In view of this requirement,
a truncated sample with 130 SGL systems is used here, where the chosen systems come from the SLACS, S4TM, BELLS and BELLS GALLERY surveys.
According to the actually available imaging data, we use the HST F814W data for the SLACS, S4TM, and BELLS lenses, and HST F606W data for BELLS GALLERY lenses. It is worth pointing out that the region of interest in our method is the innermost area for each lens galaxy, wherein
the luminosity density profile can be well fitted with a power-law distribution. The Figure \ref{fig:LuminosityFitting} shows the luminosity fitting results, where SLACSJ0008$-$0004 from SLACS and SDSSJ1215$+$0047 from BELLS are
taken as examples. It turns out that the statistical error on the measured $\delta$
for each lens galaxy is smaller than 1\%, which is ignorable.
Besides, the measured values of $\delta$ for the 130 lenses have a mean of $<\delta> = 2.173$ and a standard deviation $\sigma_{\delta}=0.085$.
The observational constraints on the free parameters are presented in Figure \ref{fig:combine_130_delta_ob} and Table \ref{tab:results}. It turns out that the estimation of $\Omega_m$ is still obviously biased in the scenarios of parameterizing $\gamma$ with $P_1$ and $P_2$. On the contrary,
the unbiased estimate of $\Omega_m$ is obtained in the scenario of parameterizing $\gamma$ with $P_3$,
though the uncertainty on $\Omega_m$ is about 50\%.
To make a comparison, the constraints on $\Omega_m$ from the truncated sample in the case of treating $\delta$ as a universal parameter are also displayed in Figure \ref{fig:om_1d}, which show that the estimations on $\Omega_m$ are still biased like those from the entire sample. It implies that the unbiased estimate on $\Omega_m$
in the scenario of parameterizing $\gamma$ with $P_3$ results from treating $\delta$ as an observable for each lens rather than reducing the sample from 161 to 130 systems.
Hence, in order to get the unbiased estimate for $\Omega_m$, one should properly consider
both the dependence of $\gamma$ on the lens properties and the intrinsic scatter of $\delta$ among the lenses.
Moreover, when replacing the previously adopted prior value on $\beta$ with $\beta = 0$, the estimation of $\Omega_m$ is shifted
from $\Omega_m=0.381^{+0.185}_{-0.154}$ to $\Omega_m = 0.176^{+0.134}_{-0.101}$ at 68\% confidence level in the scenario of $P_3$, these two estimations are consistent at 68\% confidence level, but the relative change in the mean value is $\sim$50\%. The effect of the prior on $\beta$ is noticeable, so a precise prior on $\beta$ is very important.
The analyses above reveal that the limit on the cosmological parameter is quite dependent on the lens mass model. So it is necessary to compare the lens mass models and select the most compatible one, that can supply helpful reference for future studies on selecting the lens mass model. First of all, it is easy to imagine that the lens model which can result in an unbiased estimate of $\Omega_m$ should be preferred. In order to ensure the rigorousness of the consequence, we employ the Bayesian information criterion (BIC) to compare the lens models. The BIC (Schwarz 1978) is defined as
\begin{equation}
\label{eq:BIC}
\textrm{BIC}=-2\ln \mathcal{L}_{max}+k\ln N,
\end{equation}
where $\mathcal{L}_{max}$ is the maximum likelihood (satisfying $-2\ln \mathcal{L}_{max}=\chi^2_{min}$ under the Gaussian assumption), $k$ is the number of the parameters of the considered model, and $N$ is the number of data points used in the fitting. The BIC is widely used in a cosmological context(see, e.g., Liddle 2004; God{\l}owski \& Szyd{\l}owski 2005;
Magueijo \& Sorkin 2007; Mukherjee et al. 2006; Biesiada 2007; Davis et al. 2007; Li et al. 2013; Wen et al. 2018; Birrer et al. 2019). This statistic prefers
models that give a good fit with fewer parameters. The favorite model is the one with the minimum BIC value.
The BIC values for the scenarios of parameterizing $\gamma$ with $P_1$, $P_2$ and $P_3$ are 334.7, 332.6 and 207.5, respectively, which are also listed in the last column of Table \ref{tab:results}.
So, the most compatible lens model is the third scenario, which is exactly the one that results in an unbiased estimate of $\Omega_m$.
What's more, in the framework of the third scenario, $\gamma_z=0$ is ruled out at $\sim$$2\sigma$ level, and $\gamma_s = 0$ is ruled out at $\sim$$10\sigma$ level, wherein $\gamma_z = -0.218^{+0.089}_{-0.087}$
and $\gamma_s = 0.661^{+0.054}_{-0.055}$ at 68\% confidence level.
By fixing the cosmological parameters at the fiducial values, Sonnenfeld et al. (2013a) found the dependence of $\gamma$ on redshift and surface stellar mass density ($\Sigma_{\ast}$) at $\sim$$3.1\sigma$ and $\sim$$5.4\sigma$ levels from the SL2S, SLACS and LSD lenses, wherein $\partial \gamma/\partial z_l=-0.31^{+0.09}_{-0.10}$ and
$\partial \gamma/\partial \log\Sigma_{\ast}=0.38\pm0.07$ at 68\% confidence level.
Based on the previous analysis, we conclude that dependencies of $\gamma$ on both the redshift and the surface mass density are significant. Besides, $\gamma$ has a positive correlation with the surface mass density,
and a negative correlation with the redshift.
\section{Summary and conclusions}\label{sec:Conclusions}
We have compiled a galaxy-scale strong gravitational lensing sample including 161 systems with the gravitational lensing and stellar velocity dispersion measurements, which are selected with strict criteria to satisfy the assumption of spherical symmetry on the lens mass model. Actually, the selected lenses are all early-type galaxies with E/S0 morphologies. A kind of spherically symmetric mass distributions expressed with Eq.(\ref{eq:profile}) is assumed for the lens galaxies throughout this paper.
After carrying out the qualitative and semi-quantitative analysis, we find that the current sample is weak at confining cosmological parameters.
Besides, the slope of the total mass density profile, i.e., $\gamma$, presents a significant sample-dependent signal. On the other side,
the sample dependence of the slope of the luminosity density, i.e., $\delta$, is much weaker than that of $\gamma$, but stronger than that of the orbit anisotropy parameter $\beta$.
Given this, we specifically consider three parameterizations for the slope $\gamma$.
The slope $\gamma$ is treated as an arbitrary constant without considering any dependency in the first scenario (namely ``P1''). And its dependence on the lens redshift is considered in the second scenario (namely ``P2''). Further, its dependencies on both the redshift and the surface mass density of the lens are taken into account in the last scenario (namely ``P3''). Moreover, $\beta$ is treated as a nuisance parameter and has been marginalized over with a Gaussian prior $\beta = 0.18\pm0.13$ from the independent constraint based on the observations of nearby elliptical galaxies.
Regarding to the parameter $\delta$, we treat it in two different ways.
In the first case, we treat $\delta$ as a universal parameter for all lens galaxies in the entire sample.
It turns out that the limit on the cosmological parameter, $\Omega_m$, is quite weak and biased, as well as quite dependent on the
parametrization of $\gamma$.
In the second case, we turn to consider the sample-dependence of $\delta$ by treating $\delta$ as an observable for each lens.
Then, the observational constraints show that the unbiased estimate of $\Omega_m$ can be obtained in the scenario of parameterizing $\gamma$
with $P_3$, although the estimates are still biased in the scenarios of $P_1$ and $P_2$.
The dependencies of $\gamma$ on the redshift and on the surface mass density are observed at $\sim$$2\sigma$ and $\sim$$10\sigma$ levels, respectively.
Consequently, both the dependence of $\gamma$ on the lens properties and the intrinsic scatter of $\delta$ among
the lenses should be properly taken into account to get the unbiased estimate for the cosmological parameter in the method under consideration. Besides, the effect of the prior on $\beta$ is also noticeable, so a empirically-motivated prior on $\beta$ is very essential for our study.
It also shows that the slope $\gamma$ has a positive correlation with the surface mass density, and a negative correlation with the redshift. The overall trends show that, at a given redshift, the galaxies with high density also have steeper slopes; and, at fixed surface mass density, the galaxies at a lower redshift have steeper slopes. These trends are consistent with those obtained in the previous studies (e.g., Auger et al. 2010; Ruff et al. 2011; Bolton et al 2012; Holanda et al. 2017; Sonnenfeld et al. 2013a; Li et al. 2018b).
It is worth noting that what's measured here is how the mean density slope
for the population of ETGs considered changes in the $(z_l, \tilde{\Sigma})$ space, and not how $\gamma$ changes along the
lifetime of an individual galaxy. In order to infer
the latter quantity one needs to evaluate the variation of $\gamma$
along the evolutionary track of a galaxy as it moves
in the $(z_l, \tilde{\Sigma})$ space. This requires to know how
both mass and size of a galaxy change with time (Sonnenfeld et al. 2013a), since
the slope depends on these parameters, however, that is impractical in the actual observations. Hence, the numerical simulations are usually needed in order to obtain the evolutionary track of an individual galaxy.
Finally, we point out that besides the dependence of $\gamma$ on redshift and surface mass density considered in this work, other important dependencies may also be found in future, that can lead to a more accurate phenomenological model for lens galaxies.
In addition, although the measurements of the velocity dispersions ($\sigma$) of lens galaxies alone are weak at constraining the cosmological parameters, measurements of time delays ($\Delta \tau$) and the joint measurements of the former two ($\Delta \tau/\sigma^2$) are both proved to be more sensitive to the cosmological parameters (Paraficz \& Hjorth 2009; Wei \& Wu 2017; Jee et al. 2015, 2016; Shajib et al. 2018). At present, the cosmological implementation is limited by the uncertainty in the lens modeling.
One can anticipate a dramatic increase in the number of SGL systems in view of the forthcoming optical imaging surveys (see, e.g., Oguri \& Marshall 2010; Collett 2015; Shu et al. 2018). The high-quality imaging and spectroscopic observations on SGL systems will certainly be very helpful in improving the lens mass modelling (Barnab\`e et al. 2013; Suyu et al. 2017; Treu et al. 2018). In addition, the mass and surface-brightness structures of early-type lensing galaxies may be better quantified with the aid of hydrodynamic simulations of galaxy formation n (Xu et al. 2017; Mukherjee et al. 2018; Wang et al. 2018).
\section*{Acknowledgments}
We would like to thank the anonymous referee for her/his comments that helped to greatly improve and clarify several points of the paper, and thank Adam S. Bolton for the helpful explanation on how to fit the luminosity distribution with a single power-law profile,
and thank Shuo Cao, Marek Biesiada, Kai Liao, Xuheng Ding, Lixin Xu, Guojian Wang and Jingzhao Qi for other helpful discussions.
YC has been supported by the National Natural Science Foundation of China (Nos. 11703034 and 11573031), and the NAOC Nebula Talents Program.
RL has been supported by the National Key Program for Science and Technology Research and Development of China (2017YFB0203300), the National Natural Science Foundation of China (Nos. 11773032 and 118513), and the NAOC Nebula Talents Program. YS has been supported by the Royal Society -- K.C. Wong International Fellowship (NF170995).
|
2,877,628,089,467 | arxiv | \chapter*{Introduction}
The classification of all irreducible unitary representations of a semisimple Lie group is a natural problem in representation theory which is still unsolved for most groups. A guiding principle for the classification is the orbit philosophy which proposes a tight relation between the unitary dual of a semisimple group $G$ and the set of coadjoint orbits in the dual $\mathfrak{g}^*$ of the Lie algebra $\mathfrak{g}$ of $G$ by \emph{quantization}. For elliptic and hyperbolic coadjoint orbits, cohomological and parabolic induction provide explicit constructions of the corresponding unitary representations and the resulting representations make up a large part of the unitary dual. For nilpotent orbits, however, it is not clear how to apply the orbit philosophy to construct unitary representations. Among the (finitely many) nilpotent coadjoint orbits, there are one or two of minimal dimension (depending on whether the group is of Hermitian type or not). Irreducible unitary representations corresponding to a minimal nilpotent coadjoint orbit are called \emph{minimal representations}.
Minimal representations are not only of importance in representation theory, but are for instance related to Fourier coefficients of automorphic forms \cite{BS18,GGKPS18,GGKPS19,KPW02}, and their realizations often establish connections to families of special functions \cite{HKMM11,Kob14} as well as interesting analytic and geometric questions \cite{KO03a,KO03b,KO03c}. It is therefore desirable to have convenient models for these representations.
For reductive groups $G$ possessing a parabolic subgroup $P=MAN$ with abelian nilradical $N$, minimal representations can often be constructed as subrepresentations of the corresponding degenerate principal series $\Ind_P^G(\chi)$ for a certain character $\chi$. The subrepresentations are given as the kernel of a system of second order differential operators. Realizing the degenerate principal series in the non-compact picture on a space of functions on the opposite nilradical $\overline{N}$ and identifying $\overline{N}$ with its Lie algebra $\overline{\mathfrak{n}}$, these differential operators become constant coefficient differential operators on $\overline{\mathfrak{n}}$. Taking the Euclidean Fourier transform $\mathcal{S}'(\overline{\mathfrak{n}})\to\mathcal{S}'(\mathfrak{n})$ turns the systems of differential operators into a system of multiplication operators and hence the differential equations turn into a support condition on $\mathfrak{n}$. In this way, a realization of the minimal representation on a space of functions on the smallest non-trivial $\Ad(MA)$-orbit $\mathcal{O}\subseteq\mathfrak{n}$ is obtained. The relevant Hilbert space turns out to be $L^2(\mathcal{O})$ with respect to a certain $\Ad(MA)$-equivariant measure on $\mathcal{O}$. This was first observed by Vergne and Rossi~\cite{VR76} in the case of Hermitian groups and later generalized by Dvorsky and Sahi~\cite{DS99}, Kobayashi and {\O}rsted~\cite{KO03b} and M\"{o}llers and Schwarz~\cite{MS17} to cover all cases (see also Goncharov~\cite{Gon82} for the underlying Lie algebra representation).
In the case of parabolic subgroups $P=MAN$ with Heisenberg nilradical $N$, analogous \emph{conformally invariant systems of differential operators} on $\overline{\mathfrak{n}}$ have been constructed by Barchini, Kable and Zierau~\cite{BKZ08}, but so far their kernels have only been studied in a few examples, mostly algebraically (see e.g. \cite{Kab11,Kab12a,Kab12b,KO19}). In particular, an analysis of the corresponding representations is missing due to the fact that the Euclidean Fourier transform has to be replaced by the Heisenberg group Fourier transform which is considerably more complicated.
In this work we systematically study the kernels of conformally invariant systems of differential operators and their Fourier transforms in the case of non-Hermitian groups. This leads to $L^2$-models for the minimal representations of the split groups $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, the quaternionic groups $G_{2(2)}$, $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$, the groups $\SL(n,\mathbb{R})$ and $\widetilde{SL}(3,\mathbb{R})$ as well as the groups $\widetilde{\SO}(p,q)$ with either $p\geq q=3$ or $p,q\geq4$ with $p+q$ even. For some of these groups, the relevant $L^2$-models have been found earlier by different methods, and our approach gives a uniform construction for all these cases and provides an explanation for the corresponding formulas that occur in the literature. Moreover, our models seem to be new for the quaternionic groups $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$ and for the indefinite orthogonal groups $\widetilde{\SO}(p,q)$, except for $\widetilde{\SO}(4,3)$ where there seems to be a relation to the realization constructed by Sabourin~\cite{Sab96}.
We now describe our results in detail.
\section*{Conformally invariant systems}
Let $G$ be a connected non-compact simple real Lie group with finite center and denote by $\mathfrak{g}$ its Lie algebra. We assume that $G$ has a parabolic subgroup $P=MAN$ whose nilradical $N$ is a Heisenberg group and write $\mathfrak{m}$, $\mathfrak{a}$ and $\mathfrak{n}$ for the Lie algebras of $M$, $A$ and $N$. There exists a unique element $H\in\mathfrak{a}$ such that $\ad(H)$ has eigenvalues $+1$ and $+2$ on $\mathfrak{n}$ and $-1$ and $-2$ on the opposite nilradical $\overline{\mathfrak{n}}$. We decompose
$$ \mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2 $$
into eigenspaces for $\ad(H)$ where $\mathfrak{g}_0=\mathfrak{m}\oplus\mathfrak{a}$. In all cases but $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ the parabolic subgroup $P$ is maximal and hence $\mathfrak{a}=\mathbb{R} H$. To simplify notation we therefore put
$$ \mathfrak{a}=\mathbb{R} H \qquad \mbox{and} \qquad \mathfrak{m}=\{X\in\mathfrak{g}_0:\ad(X)|_{\mathfrak{g}_2}=0\} $$
also in the case $\mathfrak{g}\simeq\sl(n,\mathbb{R})$. Further, we write $\rho=\frac{1}{2}\ad|_{\mathfrak{n}}\in\mathfrak{a}^*$ as usual.
For an irreducible smooth admissible representation $(\zeta,V_\zeta)$ of $M$ and $\nu\in\mathfrak{a}_\mathbb{C}^*$ we form the degenerate principal series (smooth normalized parabolic induction)
$$ \pi_{\zeta,\nu} = \Ind_P^G(\zeta\otimes e^\nu\otimes\1) $$
and realize it on a subspace $I(\zeta,\nu)\subseteq C^\infty(\overline{\mathfrak{n}})\otimeshat V_\zeta$ of functions on the opposite nilradical $\overline{\mathfrak{n}}\simeq\overline{N}$ which is a Heisenberg Lie algebra (the \emph{non-compact picture}). The representation $\pi_{\zeta,\nu}$ is irreducible for generic $\nu$, but may contain irreducible subrepresentations for singular parameters.
Extending $H$ to an $\sl_2$-triple by $E\in\mathfrak{g}_2$ and $F\in\mathfrak{g}_{-2}$, we can endow $V=\mathfrak{g}_{-1}$ with a symplectic form $\omega$ characterized by
$$ [x,y]=\omega(x,y)F \qquad \mbox{for }x,y\in V. $$
The identity component $M_0$ of $M$ acts symplectically on $(V,\omega)$ and the $5$-grading of $\mathfrak{g}$ gives rise to three additional symplectic invariants:
\begin{align*}
\mu:V\to\mathfrak{m}, &\quad \mu(x)=\frac{1}{2!}\ad(x)^2E && \mbox{(the moment map)}\\
\Psi:V\to V, &\quad \Psi(x)=\frac{1}{3!}\ad(x)^3E && \mbox{(the cubic map)}\\
Q:V\to\mathbb{R}, &\quad Q(x)=\frac{1}{4!}\ad(x)^4E && \mbox{(the quartic)}
\end{align*}
which are all $M$-equivariant polynomials. In \cite{BKZ08} Barchini, Kable and Zierau constructed for each of the invariants $\omega$, $\mu$, $\Psi$ and $Q$ a system of differential operators on $\overline{\mathfrak{n}}$ which is \emph{conformally invariant}. These systems can be seen as quantizations of the symplectic invariants. The joint kernel of each system gives rise to a subrepresentation of a degenerate principal series representation. For instance, for the conformally invariant system $\Omega_\mu(T)$ ($T\in\mathfrak{m}$) corresponding to the moment map, the conformal invariance implies that for every simple or one-dimensional abelian ideal $\mathfrak{m}'\subseteq\mathfrak{m}$ there exists a parameter $\nu=\nu(\mathfrak{m}')\in\mathfrak{a}^*$ such that the joint kernel
$$ I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')} = \{u\in I(\zeta,\nu):\Omega_\mu(T)u=0\,\forall\,T\in\mathfrak{m}'\} $$
is a subrepresentation of $I(\zeta,\nu)$ whenever $\zeta$ is trivial on the connected component $M_0$ of $M$. Note that $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')}$ could be trivial.
\section*{The Heisenberg group Fourier transform}
The infinite-dimensional irreducible unitary representations $(\sigma_\lambda,\mathcal{H}_\lambda)$ of $\overline{N}$ are parameterized by their central character $\lambda\in\mathbb{R}^\times$ in the sense that $\sigma_\lambda(e^{tF})=e^{i\lambda t}\id$. They give rise to operator-valued maps
$$ \sigma_\lambda:L^1(\overline{N}) \to \End(\mathcal{H}_\lambda), \quad \sigma_\lambda(u)=\int_{\overline{N}}u(\overline{n})\sigma_\lambda(\overline{n})\,d\overline{n}. $$
The Heisenberg group Fourier transform is the collection of all $\sigma_\lambda$ and it extends to a unitary isomorphism
$$ \mathcal{F}:L^2(\overline{N})\to L^2(\mathbb{R}^\times,{\textup{HS}}(\mathcal{H});|\lambda|^{\frac{\dim V}{2}}\,d\lambda)\simeq L^2(\mathbb{R}^\times;|\lambda|^{\frac{\dim V}{2}}\,d\lambda)\otimeshat{\textup{HS}}(\mathcal{H}), \quad \mathcal{F} u(\lambda)=\sigma_\lambda(u), $$
where $\mathcal{H}=\mathcal{H}_\lambda$ is a Hilbert space which realizes all representations $\sigma_\lambda$ and ${\textup{HS}}(\mathcal{H})$ denotes the Hilbert space of all Hilbert--Schmidt operators on $\mathcal{H}$. It is a non-trivial problem to extend $\mathcal{F}$ to tempered distributions (see e.g. \cite{Dah19,Fab91} on this issue). For our purpose it is enough to show that $\mathcal{F}$ extends for $\Re\nu>-\rho$ to an injective linear map (see Corollary \ref{cor:FTinjectiveOnPS})
$$ \mathcal{F}:I(\zeta,\nu)\to\mathcal{D}'(\mathbb{R}^\times)\otimeshat\Hom(\mathcal{H}^\infty,\mathcal{H}^{-\infty}), $$
where $\mathcal{H}^\infty$ denotes the space of smooth vectors in $\mathcal{H}=\mathcal{H}_\lambda$ and $\mathcal{H}^{-\infty}=(\mathcal{H}^\infty)'$ its dual space, the space of distribution vectors (both spaces are independent of $\lambda$).
For $\Re\nu>-\rho$ we call the realization
$$ \widehat{\pi}_{\zeta,\nu}(g)=\mathcal{F}\circ\pi_{\zeta,\nu}(g)\circ\mathcal{F}^{-1} \qquad (g\in G) $$
on the subspace $\widehat{I}(\zeta,\nu):=\mathcal{F}(I(\zeta,\nu))\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\Hom(\mathcal{H}^\infty,\mathcal{H}^{-\infty})$ the \emph{Fourier transformed picture}. In order to understand the Fourier transformed picture of the subrepresentations $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')}$ we study the Fourier transform of the conformally invariant system $\Omega_\mu$. For this denote by $d\omega_{\met,\lambda}$ the metaplectic representation of $\sp(V,\omega)$ which is uniquely defined by
$$ d\sigma_\lambda([T,X]) = [d\omega_{\met,\lambda}(T),d\sigma_\lambda(X)] \qquad (T\in\sp(V,\omega),X\in\overline{\mathfrak{n}}). $$
Note that the adjoint representation $\ad:\mathfrak{m}\to\gl(V),\,T\mapsto\ad(T)|_V$ identifies $\mathfrak{m}$ with a subalgebra of $\sp(V,\omega)$ so that we can restrict $d\omega_{\met,\lambda}$ to $\mathfrak{m}$.
\begin{thmalph}
For $\lambda\in\mathbb{R}^\times$, $T\in\mathfrak{m}$ and $u\in I(\zeta,\nu)$ we have
$$ \sigma_\lambda(\Omega_\mu(T)u) = 2i\lambda\, \sigma_\lambda(u)\circ d\omega_{\met,\lambda}(T). $$
\end{thmalph}
This implies that the Fourier transform of $u\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')}$ satisfies
$$ \mathcal{F} u(\lambda)\circ d\omega_{\met,\lambda}(T) = 0 \qquad \mbox{for all }T\in\mathfrak{m}'. $$
We also obtain formulas for the Fourier transform of the conformally invariant systems $\Omega_\omega$, $\Omega_\Psi$ and $\Omega_Q$ associated to $\omega$, $\Psi$ and $Q$ in Sections~\ref{sec:FTOmegaOmega}, \ref{sec:FTOmegaPsi} and \ref{sec:FTOmegaQ}.
\section*{The Fourier transformed picture of the minimal representation}
In this work we restrict our attention to the case of non-Hermitian $G$. More details about the differences between Hermitian and non-Hermitian $G$ can be found in Section~\ref{sec:HermVsNonHerm}. We hope to return to the Hermitian case in a future work.
Since $G$ is non-Hermitian, one can use the structure theory developed in \cite{SS} to obtain a bigrading on $\mathfrak{g}$ (see Section~\ref{sec:Bigrading} for details). This results in a particular choice of a Lagrangian subspace $\Lambda\subseteq V$ with decomposition $\Lambda=\mathbb{R} A\oplus\mathcal{J}$, where in most cases $\mathcal{J}$ is a semisimple Jordan algebra of degree $3$ with norm function $n(z)$ defined by $\Psi(z)=n(z)A$. (In fact, all semisimple Jordan algebras of rank three arise in this way, cf. Table~\ref{tab:Classification}). The two exceptions are $\mathfrak{g}\simeq\mathfrak{g}_{2(2)}$ where $\mathcal{J}\simeq\mathbb{R}$ and $n(z)=z^3$ and $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ where $n(z)=0$ for all $z\in\mathcal{J}$. We write $(a,z)$ for $aA+z\in\mathbb{R} A\oplus\mathcal{J}=\Lambda$.
We realize the representations $\sigma_\lambda$ on the common Hilbert space $\mathcal{H}=L^2(\Lambda)$, the Schr\"{o}dinger model of $\sigma_\lambda$. In this realization we have $\mathcal{H}^\infty=\mathcal{S}(\Lambda)$, the space of Schwartz functions on $\Lambda$, and $\mathcal{H}^{-\infty}=\mathcal{S}'(\Lambda)$, the space of tempered distributions, so that the Schwartz Kernel Theorem implies
$$ \Hom(\mathcal{H}^\infty,\mathcal{H}^{-\infty}) \simeq \mathcal{S}'(\Lambda\times\Lambda) \simeq \mathcal{S}'(\Lambda)\otimeshat\mathcal{S}'(\Lambda). $$
From here on we assume that the parameter $\nu=\nu(\mathfrak{m}')\in\mathfrak{a}^*$, for which the joint kernel of $\Omega_\mu(\mathfrak{m}')$ is a subrepresentation of $I(\zeta,\nu)$, is the same for all factors of $\mathfrak{m}$. This is in particular the case when $\mathfrak{m}$ is simple, but also for $\mathfrak{g}\simeq\sl(3,\mathbb{R})$ where $\mathfrak{m}\simeq\gl(1,\mathbb{R})$ and for $\mathfrak{g}\simeq\so(4,4)$ where $\mathfrak{m}\simeq\sl(2,\mathbb{R})\oplus\sl(2,\mathbb{R})\oplus\sl(2,\mathbb{R})$. The reason for this assumption is that we need invariance under the full Lie algebra $\mathfrak{m}$ in the following result:
\begin{thmalph}\label{thm:IntroThmB}
For every $\lambda\in\mathbb{R}^\times$ the space $L^2(\Lambda)^{-\infty,\mathfrak{m}}$ of $\mathfrak{m}$-invariant distribution vectors in $d\omega_{\met,\lambda}$ is two-dimensional and spanned by $\xi_{\lambda,\varepsilon}$ ($\varepsilon\in\mathbb{Z}/2\mathbb{Z}$), where
$$ \xi_{\lambda,\varepsilon}(a,z) = \sgn(a)^\varepsilon|a|^{s_\min}e^{-i\lambda\frac{n(z)}{a}}, \qquad (a,z)\in\mathbb{R}\times\mathcal{J}, $$
with $s_\min=-\frac{1}{6}(\dim\Lambda+2)$.
\end{thmalph}
We remark that these distributions also occur in the classification \cite{EKP02} of certain generalized functions whose Euclidean Fourier transform is of the same type.
Theorem \ref{thm:IntroThmB} implies that the Fourier transform $\mathcal{F} u\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)\otimeshat\mathcal{S}'(\Lambda)$ of a function $u\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ in the kernel of $\Omega_\mu(\mathfrak{m})$ can be written as
$$ \mathcal{F} u(\lambda,x,y) = \xi_{-\lambda,\varepsilon}(x)\widetilde{u}(\lambda,y) $$
for some $\widetilde{u}(\lambda,\cdot)\in\mathcal{S}'(\Lambda)$, where $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ is determined in Corollary \ref{cor:PSEmbedding}. The map
$$ I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})} \to \mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda), \quad u\mapsto\widetilde{u} $$
is injective and provides a new realization $\rho_\min$ of the subrepresentation $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ on $J_\min\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$. Note that still $J_\min$ could be trivial. To show that there exists a representation $\zeta$ of $M$ such that $J_\min\neq\{0\}$, we compute the Lie algebra action in the new realization and find $K$-finite vectors.
We remark that this construction only excludes the non-Hermitian Lie algebras $\mathfrak{g}=\sl(n,\mathbb{R})$ ($n>3$) and $\mathfrak{g}=\so(p,q)$ ($p,q\geq3$, $(p,q)\neq(4,4)$). In Section~\ref{sec:FTpictureMinRepSLn} we explain how for $\mathfrak{g}=\sl(n,\mathbb{R})$ a generalization of the first order system $\Omega_\omega$ (a quantization of the symplectic form $\omega$) to the case of vector-valued principal series induced from characters $\zeta=\zeta_r$ of $M$ ($r\in\mathbb{C}$) yields a one-parameter family $d\rho_{\min,r}$ of subrepresentations of $I(\zeta_r,\nu)$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$. And for $\mathfrak{g}=\so(p,q)$ we combine in Section~\ref{sec:FTpictureMinRepSOpq} generalizations of both $\Omega_\omega$ and $\Omega_\mu$ to the vector-valued degenerate principal series induced from representations of $\SL(2,\mathbb{R})\subseteq M$ to find the analogous representation $d\rho_\min$ of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$.
\section*{The Lie algebra action and lowest $K$-types}
Heuristic arguments involving the standard Knapp--Stein intertwining operators (see Remark~\ref{rem:MotivationL2}) show that $\widetilde{\pi}_\min$ should be unitary on $L^2(\mathbb{R}^\times\times\Lambda,|\lambda|^{\dim\Lambda-2s_\min}d\lambda\,dy)$. To obtain a unitary representation on $L^2(\mathbb{R}^\times\times\Lambda)$, we twist the representation with the isomorphism
$$ \Phi_\delta:\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)\to\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda), \quad \Phi_\delta u(\lambda,x) = \sgn(\lambda)^\delta|\lambda|^{-s_\min}u(\lambda,\tfrac{x}{\lambda}) $$
which restricts to an isometry $L^2(\mathbb{R}^\times\times\Lambda,|\lambda|^{\dim\Lambda-2s_\min}d\lambda\,dy)\to L^2(\mathbb{R}^\times\times\Lambda)$ (see Corollary \ref{cor:PSEmbedding} for the choice of $\delta\in\mathbb{Z}/2\mathbb{Z}$). Let
$$ I_\min := \Phi_\delta(J_\min), \qquad \pi_\min(g) := \Phi_\delta\circ\rho_\min(g)\circ\Phi_\delta^{-1}. $$
\begin{thmalph}
The Lie algebra action $d\pi_\min$ is by algebraic differential operators of degree $\leq3$ on $\mathbb{R}^\times\times\Lambda$ and is explicitly computed in Proposition~\ref{prop:dpimin}. It extends naturally to $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ and is infinitesimally unitary on $L^2(\mathbb{R}^\times\times\Lambda)$.
\end{thmalph}
Comparing this action with formulas in the literature shows that our representation agrees with the one for $\mathfrak{g}=\so(n,n),\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ in \cite{KPW02,KS90}, for $\mathfrak{g}=\mathfrak{g}_{2(2)}$ in \cite{Gel80, Sav93}, and for $\mathfrak{g}=\sl(3,\mathbb{R})$ in \cite{Tor83} (see Section \ref{sec:LAactionLiterature} for details). There also seems to be a relation to the formulas for $\mathfrak{g}=\so(4,3)$ in Sabourin \cite{Sab96}. Similar formulas also appear in \cite{GP05,GP06} but without addressing the question of unitarizability. In this sense, our computations give a new explanation of the formulas in the literature and provide an explicit (degenerate) principal series embedding of the representations as well as generalize them to a larger class of groups.
In order to show that the Lie algebra representation $d\pi_\min$ on $I_\min\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ integrates to an irreducible unitary representation on $L^2(\mathbb{R}^\times\times\Lambda)$ for some representation $\zeta$, we find the lowest $K$-type in the representation.
\begin{thmalph}
\begin{enumerate}[(1)]
\item For $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ there exists a $\mathfrak{k}$-subrepresentation $W\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, explicitly given in Theorem~\ref{thm:LKTQuat}, which is isomorphic to the representation $S^{2,4,8}(\mathbb{C}^2)\boxtimes\mathbb{C}$ of $\mathfrak{k}\simeq\su(2)\oplus\mathfrak{k}''$.
\item For $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ there exists a $\mathfrak{k}$-subrepresentation $W\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, explicitly given in Theorem~\ref{thm:LKTSplit}, which is isomorphic to the trivial representation.
\item For $\mathfrak{g}=\mathfrak{g}_{2(2)}$ there exists a $\mathfrak{k}$-subrepresentation $W\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, explicitly given in Theorem~\ref{thm:LKTG2}, which is isomorphic to the representation $S^2(\mathbb{C}^2)\boxtimes\mathbb{C}$ of $\mathfrak{k}\simeq\mathfrak{k}'\oplus\su(2)$.
\item For $\mathfrak{g}=\sl(n,\mathbb{R})$ there exist for every $r\in\mathbb{C}$ two $\mathfrak{k}$-subrepresentations $W_{\varepsilon,r}\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ ($\varepsilon\in\mathbb{Z}/2\mathbb{Z}$) of the representation $d\pi_{\min,r}$ of $\mathfrak{g}$, explicitly given in Theorem~\ref{thm:LKTSLn}. The representation $W_{0,r}$ is isomorphic to the trivial representation of $\mathfrak{k}$ and $W_{1,r}$ is isomorphic to the standard representation $\mathbb{C}^n$ of $\mathfrak{k}\simeq\so(n)$.
\item For $\mathfrak{g}=\sl(3,\mathbb{R})$ and $r=0$ there exists a third $\mathfrak{k}$-subrepresentation $W_{\frac{1}{2}}\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ of $d\pi_{\min,r}$, explicitly given in Theorem~\ref{thm:LKTSL3}, which is isomorphic to the representation $\mathbb{C}^2$ of $\mathfrak{k}\simeq\su(2)$.
\item For $\mathfrak{g}=\so(p,q)$, $p\geq q\geq4$ with $p+q$ even, there exists a $\mathfrak{k}$-subrepresentation $W\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, explicitly given in Theorem~\ref{thm:LKTSOpq}, which is isomorphic to the representation $\mathbb{C}\boxtimes\mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^q)$ of $\mathfrak{k}\simeq\so(p)\oplus\so(q)$.
\item For $\mathfrak{g}=\so(p,3)$, $p\geq3$, there exists a $\mathfrak{k}$-subrepresentation $W\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, explicitly given in Theorem~\ref{thm:LKTSOp3}, which is isomorphic to the representation $\mathbb{C}\boxtimes S^{p-3}(\mathbb{C}^2)$ of $\mathfrak{k}\simeq\so(p)\oplus\su(2)$.
\end{enumerate}
\end{thmalph}
The explicit form for the spherical vector for $\mathfrak{g}=\so(n,n),\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ has previously been found by Kazhdan, Pioline and Waldron~\cite{KPW02} and a similar formula for a $K$-finite vector in the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$ can be found in \cite{GNPP08}. Further, for $\sl(3,\mathbb{R})$ the lowest $K$-types $W_{0,r}$, $W_{1,r}$ and $W_{\frac{1}{2}}$ were obtained by Torasso~\cite{Tor83}. We believe that the other formulas are new.
\section*{The minimal representation}
A careful study of the action of $\mathfrak{g}$ on the lowest $K$-type $W$ shows:
\begin{thmalph}\label{thmintro:IntegrationMinRep}
The $K$-type $W$ generates an irreducible $(\mathfrak{g},K)$-module $\overline{W}=d\pi_\min(U(\mathfrak{g}))W$ with lowest $K$-type $W$. This $(\mathfrak{g},K)$-module integrates to an irreducible unitary representation of $G$ on $L^2(\mathbb{R}^\times\times\Lambda)$ which is minimal in the sense that its annihilator is a completely prime ideal with associated variety equal to the minimal nilpotent coadjoint orbit. For $\mathfrak{g}$ not of type $A$, the annihilator is the Joseph ideal.
\end{thmalph}
For the split groups $G=\SO(n,n),E_{6(6)},E_{7(7)},E_{8(8)}$ the same realization has been constructed by Kazhdan and Savin~\cite{KS90} and for $G=G_{2(2)}$ by Gelfand~\cite{Gel80} (see also Savin~\cite{Sav93}). For $G=\widetilde{\SL}(3,\mathbb{R})$ the representations were studied in detail by Torasso~\cite{Tor83}, and for $G=\widetilde{\SO}(4,3)$ the realization is similar to the one constructed by Sabourin~\cite{Sab96}. We believe that the realization is new for the quaternionic groups $G=E_{6(2)},E_{7(-5)},E_{8(-24)}$ in which case the representations can also be obtained as continuation of the quaternionic discrete series by algebraic methods (see Gross and Wallach~\cite{GW94,GW96}) and also for the groups $G=\widetilde{\SO}(p,q)$.
In view of the classification of minimal representations in \cite{Tam19}, the construction in Theorem \ref{thmintro:IntegrationMinRep} together with the constructions in \cite{HKM14}, \cite{MS17} and \cite{Sav93} yield $L^2$-models for all minimal representations except for the ones of $F_{4(4)}$ and the complex groups $E_8(\mathbb{C})$ and $F_4(\mathbb{C})$. We believe that the case of $F_{4(4)}$ can be treated with a slight generalization of our methods to the vector-valued case (see Remark \ref{rem:LKTF4}). It is further feasible that a construction similar to the one in \cite[Section 7]{Sav93} may construct an $L^2$-model for the complex groups $E_8(\mathbb{C})$ and $F_4(\mathbb{C})$ from the one for $E_{8(8)}$ and $F_{4(4)}$. This would give $L^2$-models for all minimal representations of simple Lie groups.
\section*{The action of a non-trivial Weyl group element}
The group $G$ is generated by its maximal parabolic subgroup $\overline{P}$ and a non-trivial Weyl group element $w_1\in K$. While the action of $\overline{P}$ in $\pi_\min$ is relatively simple (see Lemma \ref{lem:RepOfPbar}), it is non-trivial to find explicit formulas for other group elements. In \cite{KS90,Sav93} the authors were able to construct the above $L^2$-models for split groups by extending the action of $\overline{P}$ to $G$ in terms of $\pi_\min(w_1)$ (see also \cite{Rum97} for the case of $p$-adic groups). Here, one has to verify that the definition for the operator $\pi_\min(w_1)$ satisfies several relations, and the methods does not seem to generalize in a straightforward way.
Having constructed the representations $\pi_\min$ by different methods, we are able to find the action $\pi_\min(w_1)$ of the Weyl group element $w_1$ in all cases explicitly. The action depends on the eigenvalues of a certain Lie algebra element on the lowest $K$-type. Those can be integers of half-integers, and we refer to these two cases as the \emph{integer case} and the \emph{half-integer case} (see Section~\ref{sec:ActionWeylGroupElts} for details).
\begin{thmalph}[see Theorem~\ref{thm:ActionW1}]
The element $w_1$ acts in the $L^2$-model of the minimal representation by
\begin{equation*}
\pi_\min(w_1)f(\lambda,a,x) = e^{-i\frac{n(x)}{\lambda a}}f(\sqrt{2}a,-\tfrac{\lambda}{\sqrt{2}},x)\times\begin{cases}1&\mbox{in the integer case,}\\\varepsilon(a\lambda)&\mbox{in the half-integer case,}\end{cases}\label{eq:ActionW1}
\end{equation*}
where
$$ \varepsilon(x) = \begin{cases}1&\mbox{for $x>0$,}\\i&\mbox{for $x<0$.}\end{cases} $$
\end{thmalph}
This gives a complete description of $\pi_\min$ on the generators $\overline{P}$ and $w_1$ and generalizes the formulas for $\pi_\min(w_1)$ in \cite{KS90,Sav93} . We remark that this viewpoint was also advocated in \cite{KM11} where the action of a non-trivial Weyl group element was obtained in a different $L^2$-model for the minimal representation of $\upO(p,q)$.
\section*{Outlook}
Minimal representations have shown to be of importance in the theory of automorphic representations, for instance in the construction of exceptional theta series (see e.g. \cite{KPW02}), the study of Fourier coefficients of automorphic forms (see e.g. \cite{GGKPS18,GGKPS19}) or the study of local components of global automorphic representations (see e.g. \cite{BS18,KS15}). Some of these works use $L^2$-realizations of minimal representations. We hope that our new $L^2$-models might help to generalize some of their results.
Another possible application concerns branching laws for unitary representations, i.e. the restriction of representations to subgroups. $L^2$-models have proven to be useful in the decomposition of restricted representations since here classical spectral theory of differential operators can be applied (see e.g. \cite{Dvo07,KO03b,MO15}). We expect our new $L^2$-models to be useful for the decomposition of restrictions of minimal representations.
It has further been observed that explicit realizations of small representations have fruitful connections to geometry, analysis and special functions (see e.g. \cite{FL12,HKMM11,HSS12,Kob14,KM11,KO03a,KO03c,KoMo11}). In our $L^2$-models the explicit $K$-finite vectors exhibited in Chapter~\ref{ch:LKT} are for instance expressed in terms of $K$-Bessel functions. We believe that there are many additional connections between our new realizations and other branches of mathematics. For instance, it would be interesting to relate the $L^2$-model for the minimal representation of $G=\SO(p,q)$ constructed in this paper to the one obtained in \cite{KM11,KO03c} by an explicit integral transformation.
A more direct further line of research is the investigation of the missing case $G=F_{4(4)}$ for which we expect a similar, possibly vector-valued, $L^2$-model. Also the case of Hermitian groups, which is missing in this work, is a possible further research question (see Section~\ref{sec:HermVsNonHerm} for some structural results in this situation).
\section*{Acknowledgments}
We thank Marcus Slupinski and Robert Stanton for sharing early versions of their manuscript \cite{SS} with us. We are particularly indebted to Robert Stanton for numerous discussions about the structure of Heisenberg graded Lie algebras and for his help with Lemma \ref{lem:BezoutianSum}.
\chapter[Structure theory]{Structure theory for Heisenberg parabolic subgroups}
In this preliminary section we study the structure of Heisenberg graded real Lie algebras. For much of this we follow \cite{SS}, the statements in Sections \ref{sec:BruhatDecomposition}, \ref{sec:MaxCptSubgroups} and \ref{sec:HermVsNonHerm} are new.
\section{Heisenberg parabolic subgroups}
Let $G$\index{G1@$G$} be a connected non-compact simple real Lie group with finite center. We assume that $G$ has a parabolic subgroup $P$\index{P1@$P$} whose nilradical is a Heisenberg group, i.e. two-step nilpotent with one-dimensional center. Then $P$ is maximal parabolic except in the case where $G$ is locally isomorphic to $\SL(n,\mathbb{R})$. Let $P=MAN$\index{M1@$M$}\index{A1@$A$}\index{N1@$N$} be the Langlands decomposition of $P$ and denote by $\mathfrak{g}$\index{g3@$\mathfrak{g}$}, $\mathfrak{p}$\index{p3@$\mathfrak{p}$}, $\mathfrak{m}$\index{m3@$\mathfrak{m}$}, $\mathfrak{a}$\index{a3@$\mathfrak{a}$} and $\mathfrak{n}$\index{n3@$\mathfrak{n}$} the corresponding Lie algebras of $G$, $P$, $M$, $A$ and $N$. See Table~\ref{tab:Classification} for a classification due to Cheng~\cite{Che87}.
There is a unique grading element $H\in\mathfrak{a}$\index{H1@$H$} such that $\ad(H)$ has eigenvalues $1$ and $2$ on $\mathfrak{n}$. Write
$$ \mathfrak{g} = \mathfrak{g}_{-2}+\mathfrak{g}_{-1}+\mathfrak{g}_0+\mathfrak{g}_1+\mathfrak{g}_2\index{g3i@$\mathfrak{g}_i$} $$
for the decomposition of $\mathfrak{g}$ into eigenspaces of $\ad(H)$, so that $\mathfrak{m}\oplus\mathfrak{a}=\mathfrak{g}_0$ and $\mathfrak{n}=\mathfrak{g}_1+\mathfrak{g}_2$. Denote by $\overline{\mathfrak{p}}=\mathfrak{g}_{-2}+\mathfrak{g}_{-1}+\mathfrak{g}_0$\index{p3@$\overline{\mathfrak{p}}$} the opposite parabolic subalgebra with nilradical $\overline{\mathfrak{n}}=\mathfrak{g}_{-2}+\mathfrak{g}_{-1}$\index{n3@$\overline{\mathfrak{n}}$} and let $\overline{P}$\index{P1@$\overline{P}$} and $\overline{N}$\index{N1@$\overline{N}$} be the corresponding groups. In all cases but $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ we then have
$$ \mathfrak{a} = \mathbb{R} H, \qquad \mathfrak{m} = \{X\in\mathfrak{g}_0:\ad(X)|_{\mathfrak{g}_2}=0\}. $$
To simplify notation, we use this as a definition for $\mathfrak{a}$ and $\mathfrak{m}$ in the case $\mathfrak{g}=\sl(n,\mathbb{R})$.
Since $M$ commutes with $A=\exp(\mathbb{R} H)$, $M$ preserves the $5$-grading of $\mathfrak{g}$. In particular, $M$ acts on the subspaces $\mathfrak{g}_{\pm2}$ by the adjoint representation. These subspaces are one-dimensional since they are the center of the Heisenberg algebra $\mathfrak{n}$ resp. $\overline{\mathfrak{n}}$, so there exists a character $\chi:M\to\{\pm1\}$\index{1xchi@$\chi$} such that
$$ \Ad(m)|_{\mathfrak{g}_{\pm2}}=\chi(m)\cdot\id_{\mathfrak{g}_{\pm2}}. $$
\section{The $\sl_2$-triple and the Weyl group element $w_0$}\label{sec:W0}
We can choose $E\in\mathfrak{g}_2$\index{E@$E$} and $F\in\mathfrak{g}_{-2}$\index{F@$F$} such that $[E,F]=H$. Then $\{E,H,F\}$ forms an $\sl_2$-triple. Put
$$ w_0 := \exp\left(\frac{\pi}{2}(E-F)\right),\index{w20@$w_0$} $$
then $\Ad(w_0):\mathfrak{g}\to\mathfrak{g}$ is of order four. On the $\sl_2$-triple it is given by
$$ \Ad(w_0)E = -F, \qquad \Ad(w_0)H = -H, \qquad \Ad(w_0)F = -E, $$
and it acts trivially on $\mathfrak{m}$. Further, $\Ad(w_0)$ restricts to isomorphisms $\mathfrak{g}_1\to\mathfrak{g}_{-1}$ and $\mathfrak{g}_{-1}\to\mathfrak{g}_1$ which compose to $-1$ times the identity. We put
\begin{align*}
\overline{X} &:= \ad(X)F = \Ad(w_0)X = -\Ad(w_0^{-1})X, && X\in\mathfrak{g}_1,\\
\overline{Y} &:= \ad(Y)E = -\Ad(w_0)Y = \Ad(w_0^{-1})Y, && Y\in\mathfrak{g}_{-1}.\index{1ABar@$\overline{(\cdot)}$}
\end{align*}
Note that these maps are mutually inverse and
$$ \overline{\Ad(m)X} = \chi(m)\Ad(m)\overline{X} \qquad \forall\,m\in M,X\in\mathfrak{g}_{\pm1}. $$
We further remark that $P$ and $\overline{P}$ are conjugate via $w_0$:
$$ \overline{P} = w_0Pw_0^{-1} = w_0^{-1}Pw_0. $$
\section{Minimal adjoint orbits}
The adjoint orbit
$$ \mathcal{O}_{\textup{min}} = \Ad(G)E\subseteq\mathfrak{g}\index{Omin@$\mathcal{O}_{\textup{min}}$} $$
is a minimal nilpotent orbit. If $G$ is non-Hermitian then $\mathcal{O}_{\textup{min}}=-\mathcal{O}_{\textup{min}}$ is the unique minimal nilpotent orbit. If $G$ is Hermitian then $\mathcal{O}_{\textup{min}}$ and $-\mathcal{O}_{\textup{min}}$ are the two distinct minimal nilpotent orbits.
\begin{lemma}
The stabilizer of $E$ in $G$ is given by $M_1N$ where $M_1=\{m\in M:\Ad(m)E=E\}=\{m\in M:\chi(m)=1\}\subseteq M$\index{M11@$M_1$} is a subgroup of index at most $2$.
\end{lemma}
\begin{proof}
Let $g\in G$ such that $\Ad(g)E=E$. We claim that $g\in N_G(\mathfrak{p})=P$. In fact, for $x\in\mathfrak{g}_1$ and $T\in\mathfrak{m}$ we have
$$ [\Ad(g)x,E] = \Ad(g)[x,E] = 0 \qquad \mbox{and} \qquad [\Ad(g)T,E] = \Ad(g)[T,E] = 0 $$
and hence $\Ad(g)x,\Ad(g)T\in Z_\mathfrak{g}(E)=\mathfrak{m}+\mathfrak{g}_1+\mathfrak{g}_2$. Further,
$$ [\Ad(g)H,E] = \Ad(g)[H,E] = 2\Ad(g)E = 2E $$
and hence $\Ad(g)H\in H+\mathfrak{m}+\mathfrak{g}_1+\mathfrak{g}_2$. This shows that $\Ad(g)\mathfrak{p}\subseteq\mathfrak{p}$ and therefore $g\in N_G(\mathfrak{p})=P=MAN$. Write $g=man$ with $m\in M$, $a=\exp(rH)\in A$ and $n\in N$, then $\Ad(g)E=\chi(m)e^{2r}E$. This shows that $r=0$ and hence $a=1$, and $m\in M_1=\{g\in M:\chi(g)=1\}$.
\end{proof}
\section{The symplectic invariants}
Put $V:=\mathfrak{g}_{-1}$\index{V@$V$}. The bilinear form $\omega:V\times V\to\mathbb{R}$\index{1zomegaxy@$\omega(x,y)$} given by
$$ [X,Y] = \omega(X,Y)F, \qquad X,Y\in V, $$
turns $V$ into a symplectic vector space. We note that for $X,Y\in\mathfrak{g}_1$ we have
$$ [X,Y] = -\omega(\overline{X},\overline{Y})E. $$
We now define three maps on $V$ related to the symplectic structure.
\subsection{The moment map}
Put
$$ \mu:V\to\mathfrak{g}_0, \quad x\mapsto\frac{1}{2!}\ad(x)^2E.\index{1mux@$\mu(x)$} $$
Then $\mu$ actually maps into $\mathfrak{m}$ and is $\Ad(M)$-equivariant, i.e.
$$ \mu(\Ad(m)x) = \chi(m)\Ad(m)\mu(x), \qquad m\in M. $$
\subsection{The cubic map}
Put
$$ \Psi:V\to V, \quad x\mapsto\frac{1}{3!}\ad(x)^3E.\index{1Psix@$\Psi(x)$} $$
Then $\Psi$ is $\Ad(M)$-equivariant, i.e.
$$ \Psi(\Ad(m)x) = \chi(m)\Ad(m)\Psi(x), \qquad m\in M. $$
\subsection{The quartic}
Define $Q:V\to\mathbb{R}$ by
$$ Q(x)F = \frac{1}{4!}\ad(x)^4E.\index{Qx@$Q(x)$} $$
Then $Q$ transforms under the action $\Ad(M)$ by the character $\chi$, i.e.
$$ Q(\Ad(m)x) = \chi(m)Q(x), \qquad m\in M. $$
\subsection{Symplectic formulas}
We state some formulas for the symplectic covariants $\mu$, $\Psi$ and $Q$ proved in \cite{SS,SS15}. (Note that $\mu$, $\Psi$ and $Q_n$ in their notation are $-2\mu$, $6\Psi$ and $36Q$ in our notation.)
\begin{lemma}[{\cite[Proposition 3.36]{SS}}]\label{lem:SymplecticFormulas}
For $x\in V$ the following identities hold:
\begin{enumerate}[(1)]
\item $\mu(ax+b\Psi(x))=(a^2-b^2Q(x))\mu(x)$,
\item $\Psi(ax+b\Psi(x))=(a^2-b^2Q(x))(bQ(x)x+a\Psi(x))$,
\item $Q(ax+b\Psi(x))=(a^2-b^2Q(x))^2Q(x)$,
\item $\mu(x)\Psi(x)=-3Q(x)x$.
\end{enumerate}
\end{lemma}
Let $B_\mu$\index{Bmuxy@$B_\mu(x,y)$}, $B_\Psi$\index{BPsixyz@$B_\Psi(x,y,z)$} and $B_Q$\index{BQxyzw@$B_Q(x,y,z,w)$} denote the symmetrizations of $\mu$, $\Psi$ and $Q$, respectively. Further, let
$$ \tau(x)y=\omega(x,y)x \qquad \mbox{and} \qquad B_\tau(x,y)z=\tfrac{1}{2}(\omega(x,z)y+\omega(y,z)x).\index{1tauxy@$\tau(x,y)$}\index{Bmuxy@$B_\tau(x,y)$} $$
\begin{lemma}[{\cite[Proposition 2.2]{SS15}}]\label{lem:SymmetrizationsOfSymplecticCovariants}
Let $x,y,z,w\in V$.
\begin{enumerate}[(1)]
\item $B_\mu(x,y) = \tfrac{1}{4}([x,[y,E]]+[y,[x,E]])$,
\item $B_\Psi(x,y,z) = -\tfrac{1}{3}[B_\mu(x,y),z]-\tfrac{1}{6}B_\tau(x,y)z$,
\item $B_Q(x,y,z,w) = \tfrac{1}{4}\omega(x,B_\Psi(y,z,w))$.
\end{enumerate}
\end{lemma}
Note that this implies in particular that
$$ \omega(B_\mu(x,y)z,w) = \omega(B_\mu(z,w)x,y) \qquad \mbox{for all }x,y,z,w\in V. $$
\begin{lemma}[{\cite[Proposition 3.9]{SS}}]\label{lem:RewriteBmu}
For $x,y,z\in V$
$$ 4[B_\mu(x,y),z]-4[B_\mu(x,z),y] = \omega(x,y)z-\omega(x,z)y-2\omega(y,z)x. $$
\end{lemma}
\subsection{The Lie bracket}
We express the Lie bracket $[\mathfrak{g}_1,\mathfrak{g}_{-1}]$ in terms of the moment map and the symplectic form.
\begin{lemma}\label{lem:G1bracketG-1}
For $v\in\mathfrak{g}_1$ and $w\in\mathfrak{g}_{-1}$ the decomposition of $[v,w]\in\mathfrak{g}_0$ in terms of the decomposition $\mathfrak{g}_0=\mathfrak{m}+\mathfrak{a}$ is given by
$$ [v,w] = -2B_\mu(\overline{v},w) - \tfrac{1}{2}\omega(\overline{v},w)H. $$
\end{lemma}
\begin{proof}
We have
\begin{align*}
[v,w] &= [\overline{\overline{v}},w] = [[\overline{v},E],w] = -[w,[\overline{v},E]]\\
&= -\tfrac{1}{2}\left([\overline{v},[w,E]]+[w,[\overline{v},E]]\right) +\tfrac{1}{2}[\ad(\overline{v}),\ad(w)]E\\
&= -2B_\mu(\overline{v},w)+\tfrac{1}{2}\ad([\overline{v},w])E = -2B_\mu(\overline{v},w)+\tfrac{1}{2}\omega(\overline{v},w)\ad(F)E\\
&= -2B_\mu(\overline{v},w)-\tfrac{1}{2}\omega(\overline{v},w)H.\qedhere
\end{align*}
\end{proof}
\subsection{Simple factors of $\mathfrak{m}$}
We note that $\mathfrak{m}$ is reductive with at most one-dimensional center. The proof of the following result was communicated to us by R. Stanton:
\begin{lemma}\label{lem:BezoutianSum}
Let $(e_\alpha)$ be a basis of $V$ and $(\widehat{e}_\alpha)$ its dual basis with respect to the symplectic form $\omega$, i.e. $\omega(e_\alpha,\widehat{e}_\beta)=\delta_{\alpha\beta}$. Then the map
$$ \mathfrak{m}\to\mathfrak{m}, \quad T\mapsto\sum_\alpha B_\mu(Te_\alpha,\widehat{e}_\alpha) $$
is a scalar multiple $\mathcal{C}(\mathfrak{m}')\cdot\id_{\mathfrak{m}'}$\index{Cmprime@$\mathcal{C}(\mathfrak{m}')$} of the identity on the center and on each simple factor $\mathfrak{m}'$ of $\mathfrak{m}$. In particular, if $\mathfrak{m}$ is simple the map is a scalar multiple of the identity.
\end{lemma}
\begin{proof}
The expression is obviously independent of the chosen basis, so that we may assume $\{e_\alpha\}$ to be a symplectic basis of the form $\{e_i,f_j\}$ with $\omega(e_i,f_j)=\delta_{ij}$ and $\omega(e_i,e_j)=\omega(f_i,f_j)=0$. Then $\{\widehat{e}_\alpha\}=\{f_i,-e_j\}$ and the sum becomes
$$ \sum_i\Big(B_\mu(Te_i,f_i)+B_\mu(Tf_i,-e_i)\Big) = \sum_i\Big(B_\mu(Te_i,f_i)-B_\mu(e_i,Tf_i)\Big). $$
For $x,y\in V$ we define
$$ \Bz(x,y):\mathfrak{m}\to\mathfrak{m}, \quad \Bz(x,y)T=B_\mu(Tx,y)-B_\mu(x,Ty). $$
$\Bz$ is called the \textit{Bezoutian} and it is easy to see that $\Bz:\Lambda^2V\to\End(\mathfrak{m})$ is $\mathfrak{m}$-equivariant. Now, the trivial representation of $\mathfrak{m}$ is contained in $\Lambda^2V$ as the symplectic form, and therefore its image in $\End(\mathfrak{m})$ under $\Bz$ has to be a copy of the trivial representation. Since $\mathfrak{m}$ is reductive with at most one-dimensional center, the copies of the trivial representation in $\End(\mathfrak{m})$ are given by the identity on the center and on each simple factor of $\mathfrak{m}$ . This shows the claim.
\end{proof}
In \cite{BKZ08} the numbers $\mathcal{C}(\mathfrak{m}')$ are computed for all simple factors $\mathfrak{m}'$ of $\mathfrak{m}$ and we have included them in Table~\ref{tab:Classification}.
\begin{corollary}
For every factor $\mathfrak{m}'$ of $\mathfrak{m}$ we have
$$ \tr(T\mu(x)) = \mathcal{C}(\mathfrak{m}')\omega(Tx,x) \qquad \forall\,x\in V,T\in\mathfrak{m}'. $$
\end{corollary}
\begin{proof}
Let $(e_\alpha)$ be a basis of $V$ and $\widehat{e}_\alpha$ the dual basis with respect to the symplectic form, i.e. $\omega(e_\alpha,\widehat{e}_\beta)=\delta_{\alpha\beta}$. Then
\begin{align*}
\tr(T\mu(x)) &= \sum_\alpha\omega(T\mu(x)e_\alpha,\widehat{e}_\alpha) = \sum_\alpha\omega(T\widehat{e}_\alpha,\mu(x)e_\alpha)\\
&= \sum_\alpha\omega(T\widehat{e}_\alpha,-3B_\Psi(x,x,e_\alpha)-\frac{1}{2}\tau(x)e_\alpha)\\
&= -12\sum_\alpha B_Q(x,x,e_\alpha,T\widehat{e}_\alpha)+\frac{1}{2}\omega(Tx,x)\\
&= -3\sum_\alpha\omega(x,B_\Psi(e_\alpha,T\widehat{e}_\alpha,x))+\frac{1}{2}\omega(Tx,x)\\
&= \sum_\alpha\omega(x,B_\mu(e_\alpha,T\widehat{e}_\alpha)x)+\frac{1}{2}\omega(x,B_\tau(e_\alpha,T\widehat{e}_\alpha)x)+\frac{1}{2}\omega(Tx,x)\\
&= \mathcal{C}(\mathfrak{m}')\omega(Tx,x).\qedhere
\end{align*}
\end{proof}
\section{The Killing form}
We compute the Killing form $\kappa(X,Y)=\tr(\ad(X)\circ\ad(Y))$\index{1kappaXY@$\kappa(X,Y)$} on $\mathfrak{g}$. For this let
$$ p := \dim\mathfrak{g}_1+4.\index{p2@$p$} $$
\begin{lemma}\label{lem:KillingForm}
Let $X\in\mathfrak{g}_i$ and $Y\in\mathfrak{g}_j$, then $\kappa(X,Y)=0$ unless $i+j=0$. Further,
\begin{align*}
\kappa(E,F) &= p,\\
\kappa(v,w) &= -p\omega(\overline{v},w), && v\in\mathfrak{g}_1,w\in\mathfrak{g}_{-1},\\
\kappa(S+aH,T+bH) &= \kappa_\mathfrak{m}(S,T) + 2\tr(\ad(S)\circ\ad(T))|_{\mathfrak{g}_1} + 2p ab, && S,T\in\mathfrak{m},a,b\in\mathbb{R}.
\end{align*}
\end{lemma}
\begin{proof}
It is clear that $\kappa(\mathfrak{g}_i,\mathfrak{g}_j)=\{0\}$ unless $i+j=0$. The formulas for $\kappa(E,F)$ and $\kappa(H,H)$ are proven in \cite[Proposition 2.2]{SS} and the formula for $\kappa(S,T)$ is clear since $\mathfrak{m}$ acts trivially on $E$ and $F$. It therefore remains to show the formula for $\kappa(v,w)$, $v\in\mathfrak{g}_1$ and $w\in\mathfrak{g}_{-1}$. Using $\ad$-invariance of the Killing form we have
\begin{equation*}
\kappa(v,w) = \kappa([\overline{v},E],w) = -\kappa(E,[\overline{v},w]) = -\kappa(E,\omega(\overline{v},w)F) = -p\omega(\overline{v},w).\qedhere
\end{equation*}
\end{proof}
\section{The Heisenberg nilradical}
The unipotent subgroup $\overline{N}$ is a Heisenberg group and hence diffeomorphic to its Lie algebra $\overline{\mathfrak{n}}$. We identify $\overline{N}\simeq\mathfrak{g}_{-1}\oplus\mathfrak{g}_{-2}\simeq V\times\mathbb{R}$ via
$$ V\times\mathbb{R}\stackrel{\sim}{\to}\overline{N}, \quad (x,s)\mapsto\overline{n}_{(x,s)}:= \exp(x+sF). $$
The group multiplication in $\overline{N}$ is given by
$$ \overline{n}_{(x,s)}\cdot\overline{n}_{(y,t)} = \overline{n}_{(x+y,s+t+\frac{1}{2}\omega(x,y))}, \qquad x,y\in V,s,t\in\mathbb{R}. $$
Hence, the map $V\times\mathbb{R}\to\overline{N},\,(x,s)\mapsto\overline{n}_{(x,s)}$ turns into a group isomorphism if we equip $V\times\mathbb{R}$ with the product
$$ (x,s)\cdot(y,t) := (x+y,s+t+\tfrac{1}{2}\omega(x,y)). $$
\section{Bruhat decomposition}\label{sec:BruhatDecomposition}
The natural multiplication map
$$ \overline{N}\times M\times A\times N\to G $$
is a diffeomorphism onto an open dense subset of $G$, the open dense Bruhat cell. Hence, every $g\in\overline{N}MAN\subseteq G$ decomposes uniquely into
$$ g = \overline{n}(g)m(g)a(g)n.\index{n2g@$\overline{n}(g)$}\index{m2g@$m(g)$}\index{a2g@$a(g)$} $$
We identify $\mathfrak{a}_\mathbb{C}^*\simeq\mathbb{C}$ by $\nu\mapsto\nu(H)$. For $\lambda\in\mathfrak{a}_\mathbb{C}^*$ we write $a^\lambda=e^{\lambda(X)}$ where $a=e^X\in A$ with $X\in\mathfrak{a}$.
\begin{lemma}\label{lem:AProjectionOnw0Nbar}
For $(x,s)\in V\times\mathbb{R}$ we have $w_0^{-1}\overline{n}_{(x,s)}\in\overline{N}MAN$ if and only if $s^2-Q(x)\neq0$. In this case
$$ a(w_0^{-1}\overline{n}_{(x,s)})^\lambda = |s^2-Q(x)|^{\lambda/2} \qquad \mbox{and} \qquad \log \overline{n}(w_0^{-1}\overline{n}_{(x,s)}) = \frac{1}{s^2-Q(x)}(\Psi(x)-sx,-s). $$
Moreover, the quartic $Q$ is non-positive if and only if the character $\chi:M\to\{\pm1\}$ is trivial.
\end{lemma}
\begin{proof}
Assume $w_0^{-1}\overline{n}_{(x,s)}=\overline{n}_{(y,t)}m\exp(rH)n\in\overline{N}MAN$. We let both sides act on $E$ by the adjoint representation and then compare the results. Let us first compute $\Ad(w_0^{-1}\overline{n}_{(x,s)})E$. We have
\begin{align*}
\ad(x+sF)E &= \overline{x}-sH,\\
\ad(x+sF)^2E &= 2(\mu(x)-sx-s^2F),\\
\ad(x+sF)^3E &= 6\Psi(x),\\
\ad(x+sF)^4E &= 24Q(x)F,\\
\ad(x+sF)^5E &= 0,
\end{align*}
and hence
\begin{equation}
\Ad(\overline{n}_{(x,s)})E = E+\overline{x}+\mu(x)-sH+\Psi(x)-sx+(Q(x)-s^2)F.\label{eq:AdNbarOnE}
\end{equation}
Applying $\Ad(w_0^{-1})=\Ad(w_0)^{-1}$ yields
\begin{equation}
\Ad(w_0^{-1}\overline{n}_{(x,s)})E = (s^2-Q(x))E+\overline{\Psi(x)-sx}+\mu(x)+sH-x-F.\label{eq:Adw0NbarOnE}
\end{equation}
Now let us compute $\Ad(\overline{n}_{(y,t)}m\exp(rH)n)E$. First, $N$ acts trivially on $E$. Further, $M$ acts on $E$ by the character $\chi:M\to\{\pm1\}$. Therefore,
\begin{multline}
\Ad(\overline{n}_{(y,t)}m\exp(rH)n)E = \chi(m)e^{2r}\Ad(\overline{n}_{(y,t)})E\\
= \chi(m)e^{2r}\left(E+\overline{y}+\mu(y)-tH+\Psi(y)-ty+(Q(y)-t^2)F\right).\label{eq:BruhatAppliedToE}
\end{multline}
Comparing with \eqref{eq:Adw0NbarOnE} shows that
$$ s^2-Q(x) = \chi(m)e^{2r}, \quad \Psi(x)-sx = \chi(m)e^{2r}y, \quad \mu(x) = \chi(m)e^{2r}\mu(y)$$
$$ s = -\chi(m)e^{2r}t, \qquad x = -\chi(m)e^{2r}(\Psi(y)-ty), \qquad 1 = \chi(m)e^{2r}(t^2-Q(y)). $$
The first identity shows that if $\{x\in\overline{V}:Q(x)>0\}\neq\emptyset$ then there exists $m\in M$ such that $\chi(m)=-1$, because otherwise the non-empty open set of all $w_0^{-1}\overline{n}_{(x,t)}man$ with $t^2-Q(x)<0$ and $m\in M$, $a\in A$, $n\in N$, would have trivial intersection with the open dense Bruhat cell $\overline{N}MAN$. Further, the first, second and fourth identities show that
$$ s^2-Q(x)=\pm e^{2r}\neq0 \qquad \mbox{and} \qquad (y,t)=\frac{1}{s^2-Q(x)}(\Psi(x)-sx,-s). $$
Conversely, if $s^2-Q(x)\neq0$ then let
$$ r:=\frac{1}{2}\log|s^2-Q(x)| \qquad \mbox{and} \qquad (y,t)=\frac{1}{s^2-Q(x)}(\Psi(x)-sx,-s), $$
and choose $m\in M$ such that $\chi(m)=\sgn(s^2-Q(x))$. Using the above computation as well as Lemma~\ref{lem:SymplecticFormulas} one can show that
$$ \Ad(w_0^{-1}\overline{n}_{(x,s)})E = \Ad(\overline{n}_{(y,t)}me^{rH})E. $$
Since the stabilizer of $E$ in $G$ is equal to $M_1N$ there exist $m'\in M_1$ and $n\in N$ such that $w_0^{-1}\overline{n}_{(x,s)}=\overline{n}_{(y,t)}mm'e^{rH}n\in\overline{N}MAN$. This show the claim.
\end{proof}
\begin{lemma}\label{lem:BruhatDecompForNNbar}
For $(x,t)\in V\times\mathbb{R}$ and $s\in\mathbb{R}$ sufficiently close to $0$ we have
\begin{align*}
\log\overline{n}(e^{-sE}\overline{n}_{(x,t)}) &= \left(\frac{x+s(\Psi(x)-tx)}{1-2st-s^2(Q(x)-t^2)},\frac{t+s(Q(x)-t^2)}{1-2st-s^2(Q(x)-t^2)}\right),\\
\left.\frac{d}{ds}\right|_{s=0}m(e^{-sE}\overline{n}_{(x,t)}) &= -\mu(x),\\
a(e^{-sE}\overline{n}_{(x,t)})^\lambda &= (1-2st-s^2(Q(x)-t^2))^{\frac{\lambda}{2}}.
\end{align*}
\end{lemma}
\begin{proof}
Write
\begin{equation}
e^{-sE}\overline{n}_{(x,t)} = \exp(y+uF)m_se^{rH}\exp(z+vE)\label{eq:BruhatDecompForNNbar}
\end{equation}
for $y,z\in V$, $r,u,v\in\mathbb{R}$ and $m_s\in M$. We first act with both sides of \eqref{eq:BruhatDecompForNNbar} on $E$ by the adjoint action. By \eqref{eq:AdNbarOnE} we have
$$ \Ad(\overline{n}_{(x,t)})E = E+\overline{x}+\mu(x)-tH+(\Psi(x)-tx)+(Q(x)-t^2)F. $$
Now, $\Ad(e^{-sE})=e^{-s\ad(E)}$ and
\begin{align*}
\ad(E)\Ad(\overline{n}_{(x,t)})E &= 2tE+\overline{tx-\Psi(x)}+(Q(x)-t^2)H,\\
\ad(E)^2\Ad(\overline{n}_{(x,t)})E &= -2(Q(x)-t^2)E,\\
\ad(E)^3\Ad(\overline{n}_{(x,t)})E &= 0,
\end{align*}
hence
\begin{multline*}
\Ad(e^{-sE}\overline{n}_{(x,t)})E = (1-2st-s^2(Q(x)-t^2))E + \overline{x+s(\Psi(x)-tx)}+\mu(x)\\
-(t+s(Q(x)-t^2))H+(\Psi(x)-tx)+(Q(x)-t^2)F.
\end{multline*}
On the other hand, by \eqref{eq:BruhatAppliedToE}:
\begin{multline*}
\Ad\big(\exp(y+uF)m_se^{rH}\exp(z+vE)\big)E\\
= \chi(m_s)e^{2r}\left(E+\overline{y}+\mu(y)-uH+\Psi(y)-uy+(Q(y)-u^2)F\right).
\end{multline*}
Comparing the two expressions shows the formulas for $\overline{n}(e^{-sE}\overline{n}_{(x,t)})$ and $a(e^{-sE}\overline{n}_{(x,t)})$.
To find $\left.\frac{d}{ds}\right|_{s=0}m_s$ we let both sides of \eqref{eq:BruhatDecompForNNbar} act on $\overline{a}\in\mathfrak{g}_1$. By similar computations we arrive at
\begin{multline*}
\Ad\big(e^{-sE}\overline{n}_{(x,t)}\big)\overline{a} = \big(-s\omega(x,a)+s^2\omega(tx+\Psi(x),a)\big)E+\overline{(a-s(ta+\mu(x)a+\omega(x,a)x))}\\
+2B_\mu(x,a)+\big(-\tfrac{1}{2}\omega(x,a)+s\omega(tx+\Psi(x),a)\big)H\\
-\big(ta+\mu(x)a+\omega(x,a)x\big)-\omega(tx+\Psi(x),a)F
\end{multline*}
and
\begin{multline*}
\Ad\big(\exp(y+uF)m_se^{rH}\exp(z+vE)\big)\overline{a} \\= e^r\Big(\overline{m_sa}+2B_\mu(y,m_s)-\tfrac{1}{2}\omega(y,m_sa)H-\big(um_sa+\mu(y)m_sa+\omega(y,m_sa)y\big)-\omega(uy+\Psi(y),m_sa)F\Big)\\
+\omega(a,z)e^{2r}\Big(E+\overline{y}+\mu(y)-uH+\big(\Psi(y)-uy\big)+(Q(y)-u^2)F\Big).
\end{multline*}
Comparing the coefficients of $E$ shows
$$ z = \frac{sx-s^2(tx+\Psi(x))}{1-2st-s^2(Q(x)-t^2)}. $$
Next, comparing the terms in $\mathfrak{g}_1$ and using the previously obtained formula for $y$ yields
$$ e^rm_sa = a-s(ta+\mu(x)a+\omega(x,a)x) + \omega(z,a)(x-s(tx-\Psi(x))). $$
Note that $\left.\frac{d}{ds}\right|_{s=0}e^r=-t$ and $\left.\frac{d}{ds}\right|_{s=0}z=x$, hence differentiating the above identity gives
$$ -ta+\left.\frac{d}{ds}\right|_{s=0}m_sa = -(ta+\mu(x)a+\omega(x,a)x) + \omega(x,a)x = -ta-\mu(x)a $$
and the claim follows.
\end{proof}
\begin{lemma}\label{lem:GenJosephIdeal}
Assume that $\mathfrak{g}_\mathbb{C}$ is not of type A. Then the elements
\begin{align*}
& 2\cdot F\cdot T-\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\,e_\alpha\cdot e_\beta && T\in\mathfrak{m},\\
& 6(F\overline{v}+\overline{v}F)+3(H\overline{v}+\overline{v}H)+4\sum_\alpha\big(e_\alpha B_\mu(v,\widehat{e}_\alpha)+B_\mu(v,\widehat{e}_\alpha)e_\alpha\big), && v\in V,\\
& 4(EF+FE)+2H^2+\sum_\alpha\big(e_\alpha\overline{\widehat{e}_\alpha}+\overline{\widehat{e}_\alpha}e_\alpha\big)
\end{align*}
generate an ideal in $U(\mathfrak{g}_\mathbb{C})$ whose associated variety is the minimal nilpotent coadjoint orbit in $\mathfrak{g}_\mathbb{C}$.
\end{lemma}
\begin{proof}
The minimal nilpotent adjoint orbit is given by $\mathcal{O}_{\textup{min}}=\Ad(G)E$. Since $\overline{N}MAN$ is open dense in $G$, the set $\Ad(\overline{N}MAN)E\subseteq$ is dense in $\mathcal{O}_{\textup{min}}$. We have $\Ad(MAN)E=\pm\mathbb{R}_+E$, the sign depending on whether $M$ acts trivially on $E$ or not. It follows that $\Ad(\overline{N}MAN)E=\pm\mathbb{R}_+\Ad(\overline{N})E$ and hence
$$ \overline{\mathcal{O}_{\min,\mathbb{C}}} = \mathbb{C}\Ad(\overline{N}_\mathbb{C})E. $$
From \eqref{eq:AdNbarOnE} it is easy to see that this affine variety is the zero set of the second order (possibly vector-valued) polynomials
$$ aS-\mu(x), \quad 3ay+Sx-3tx, \quad ab-\frac{1}{4}\omega(x,y)+t^2 $$
in the variable $X=aE+\overline{x}+S+tH+y+bF$. Carefully applying the identification $\mathfrak{g}_\mathbb{C}\simeq\mathfrak{g}_\mathbb{C}^*$ given by the Killing form and the natural isomorphism $\mathbb{C}[\mathfrak{g}_\mathbb{C}^*]\simeq S(\mathfrak{g}_\mathbb{C})\simeq\gr U(\mathfrak{g}_\mathbb{C})$ produces the claimed generators.
\end{proof}
\section{Maximal compact subgroups}\label{sec:MaxCptSubgroups}
Let $\theta$\index{1htheta@$\theta$} be a Cartan involution of $G$ and denote by $\theta$ also the corresponding involution on $\mathfrak{g}$. We may choose $\theta$ such that $\theta\mathfrak{g}_i=\mathfrak{g}_{-i}$, $i\in\{-2,-1,0,1,2\}$ and also $\theta\mathfrak{m}=\mathfrak{m}$ and $\theta H=-H$. Possibly rescaling $E$ and $F$ we may further assume that $\theta E=-F$ and $\theta F=-E$. Define $J\in\End(V)$\index{J1@$J$} by
$$ Jv := \overline{\theta x} = -\theta\overline{x}, \qquad x\in V. $$
Then $J^2=-\1$ and the bilinear form on $V$ given by.
$$ (x|y) = \tfrac{1}{4}\omega(Jx,y), \qquad x,y\in V,\index{1AIP@$(\cdot\vert\cdot)$} $$
is positive definite. Write $|x|^2=(x|x)=\tfrac{1}{4}\omega(Jx,x)$\index{1ANorm@$\vert\cdot\vert$} for the corresponding norm on $V$. Further note that
$$ \Ad(\theta(T))|_V = J\circ\Ad(T)|_V\circ J^{-1} \qquad \forall\,T\in\mathfrak{m}. $$
It is easy to see that
$$ \omega(Jx,Jy)=\omega(x,y) \qquad \forall\,x,y\in V, $$
so that $J\in\Sp(V,\omega)$. For the symplectic covariants we further have for $x\in V$:
\begin{equation}
\mu(Jx)=J\circ\mu(x)\circ J^{-1}, \qquad \Psi(Jx) = J\Psi(x), \qquad Q(Jx) = Q(x).\label{eq:JonSymplCov}
\end{equation}
The following result is a converse to the construction of $J$ from $\theta$:
\begin{lemma}\label{lem:CartanInvFromJ}
Let $J\in\End(V)$ and define $\theta:\mathfrak{g}\to\mathfrak{g}$ by
$$ \theta E=-F, \qquad \theta H=-H, \qquad \theta F=-E, $$
and for $x\in\mathfrak{g}_1$, $y\in\mathfrak{g}_{-1}$ and $T\in\mathfrak{m}$ by
$$ \theta x=-J\overline{x}, \qquad \theta T=JTJ^{-1}, \qquad \theta y=\overline{Jy}. $$
Then $\theta$ is a Cartan involution of $\mathfrak{g}$ if and only if the following conditions are satisfied:
\begin{enumerate}[(1)]
\item\label{lem:CartanInvFromJ1} $J^2=-\1$,
\item\label{lem:CartanInvFromJ2} $\omega(Jx,x)\geq 0$ for all $x\in V$,
\item\label{lem:CartanInvFromJ3} $\omega(Jx,Jy)=\omega(x,y)$ for all $x,y\in V$,
\item\label{lem:CartanInvFromJ4} $\mu(Jx)=J\mu(x)J^{-1}$ for all $x\in V$.
\end{enumerate}
\end{lemma}
\begin{proof}
It is easy to see that conditions (3) and (4) imply \eqref{eq:JonSymplCov}. Then a short computation shows that $\theta$ is indeed a Lie algebra automorphism. That $\kappa(\theta X,X)\geq0$ for all $X\in\mathfrak{g}$ now follows from (2) and Lemma~\ref{lem:KillingForm}, and that $\theta^2=\1$ is a consequence of (1).
\end{proof}
Let $K=G^\theta$\index{K1@$K$} be the subgroup of $\theta$-fixed points in $G$. Then $K$ is maximal compact in $G$ and its Lie algebra $\mathfrak{k}$\index{k3@$\mathfrak{k}$} has the form
$$ \mathfrak{k} = \mathbb{R}(E-F)\oplus\{x+\theta(x):x\in\mathfrak{g}_1\}\oplus\mathfrak{k}_\mathfrak{m}, $$
where $\mathfrak{k}_\mathfrak{m}=\mathfrak{k}\cap\mathfrak{m}\subseteq\mathfrak{m}$\index{k3m@$\mathfrak{k}_\mathfrak{m}$} is maximal compact in $\mathfrak{m}$. Write $K_M=K\cap M\subseteq M$ for the corresponding group. Using the decomposition $G=KMAN$ we define a map $H:G\to\mathfrak{a}$ by
$$ g\in KMe^{H(g)}N.\index{H1g@$H(g)$} $$
We now compute $H$ on $\overline{N}$.
\begin{lemma}\label{lem:IwasawaAProjectionOnNbar}
For $(x,t)\in V\times\mathbb{R}$ we have
\begin{equation*}
e^{\lambda(H(\overline{n}_{(x,t)}))} = \Big(1+4|x|^2-\frac{1}{2}\omega(\mu(Jx)x,x)+2t^2+4|tx-\Psi(x)|^2+(t^2-Q(x))^2\Big)^{\lambda/4}.
\end{equation*}
\end{lemma}
We remark that for some special cases a similar formula was obtained in \cite[equation (5.5)]{GP09} and \cite[equation (5.4)]{GP10}.
\begin{remark}\label{rem:SphVectorPositive}
Since
\begin{align*}
\kappa(\mu(x),T) &= \frac{1}{2}\kappa(\ad(x)^2E,T) = \frac{1}{2}\kappa(E,\ad(x)^2T)\\
&= \frac{1}{2}\kappa(E,[[T,x],x]) = \frac{1}{2}\omega(Tx,x)\kappa(E,F) = \frac{p}{2}\omega(Tx,x)
\end{align*}
we have
$$ \omega(\mu(Jx)x,x) = \frac{2}{p}\kappa(\mu(x),\mu(Jx)) = \frac{2}{p}\kappa(\mu(x),\theta\mu(x)) \leq 0. $$
\end{remark}
\begin{proof}
Let $(x,t)\in V\times\mathbb{R}$ and write $\overline{n}_{(x,t)}=kman$, so that $a=e^{H(\overline{n}_{(x,t)})}$. Then
$$ \theta(\overline{n}_{(x,t)})^{-1}\overline{n}_{(x,t)} = \theta(n)^{-1}(\theta(m)^{-1}m)a^2n\in\overline{N}MAN. $$
Hence $a=a(\theta(\overline{n}_{(x,t)})^{-1}\overline{n}_{(x,t)})^{1/2}$ which we compute by letting $\theta(\overline{n}_{(x,t)})^{-1}\overline{n}_{(x,t)}$ act on $E$ via the adjoint action. First, by \eqref{eq:AdNbarOnE} we have
\begin{equation}
\Ad(\overline{n}_{(x,t)})E = E+\overline{x}+\mu(x)-tH+\Psi(x)-tx+(Q(x)-t^2)F.\label{eq:ActionNbarOnE}
\end{equation}
Next, $\theta(\overline{n}_{(x,t)})^{-1}=\exp(-\overline{Jx}+tE)$. We let this act on each of the summands of \eqref{eq:ActionNbarOnE}. First, it acts trivially on $E$:
$$ \Ad(\exp(-\overline{Jx}+tE))E = E. $$
The action on the $\mathfrak{g}_1$-part $\overline{x}$ is given by
$$ \Ad(\exp(-\overline{Jx}+tE))\overline{x} = \overline{x}+\ad(-\overline{Jx}+tE)\overline{x} = \overline{x}+\omega(Jx,x)E = \overline{x}+4|x|^2E. $$
Next, on the $\mathfrak{m}$-part $\mu(x)$ we have
\begin{align*}
& \Ad(\exp(-\overline{Jx}+tE))\mu(x)\\
={}& \mu(x) + \ad(-\overline{Jx}+tE)\mu(x) + \tfrac{1}{2}\ad(-\overline{Jx}+tE)^2\mu(x)\\
={}& \mu(x) + [\mu(x),\overline{Jx}] + \tfrac{1}{2}\omega(Jx,[\mu(x),Jx])E\\
={}& \mu(x) + [\mu(x),\overline{Jx}] - \Big(\tfrac{3}{2}\omega(Jx,B_\Psi(x,x,Jx))+\tfrac{1}{4}\omega(Jx,\omega(x,Jx)x\Big)E\\
={}& \mu(x) + [\mu(x),\overline{Jx}] - (6B_Q(x,x,Jx,Jx)-4|x|^4)E.
\end{align*}
For the action on $H$ we find
$$ \Ad(\exp(-\overline{Jx}+tE))H = H + \ad(-\overline{Jx}+tE)H = H + \overline{Jx} -2tE. $$
Next, the action on the $\mathfrak{g}_{-1}$-part $\Psi(x)-tx$ is given by
\begin{align*}
& \Ad(\exp(-\overline{Jx}+tE))(\Psi(x)-tx)\\
={}& (\Psi(x)-tx)+\ad(-\overline{Jx}+tE)(\Psi(x)-tx)+\tfrac{1}{2}\ad(-\overline{Jx}+tE)^2(\Psi(x)-tx)\\
& +\tfrac{1}{6}\ad(-\overline{Jx}+tE)^3(\Psi(x)-tx)\\
={}& (\Psi(x)-tx) + \Big(2B_\mu(Jx,\Psi(x)-tx)+\tfrac{1}{2}\omega(Jx,\Psi(x)-tx)H-t\overline{(\Psi(x)-tx)}\Big)\\
& +\tfrac{1}{2}\Big(2[B_\mu(Jx,\Psi(x)-tx),\overline{Jx}]+\tfrac{1}{2}\omega(Jx,\Psi(x)-tx)\overline{Jx}-2t\omega(Jx,\Psi(x)-tx)E\Big)\\
& +\tfrac{1}{6}\Big(2\omega(Jx,[B_\mu(Jx,\Psi(x)-tx),Jx])E\Big)\\
={}& \Big(-t\omega(Jx,\Psi(x)-tx)+\tfrac{1}{3}\omega(Jx,[B_\mu(Jx,\Psi(x)-tx),Jx])\Big)E + \cdots
\end{align*}
And finally, for the action on $F$ we have
\begin{align*}
\Ad(\exp(-\overline{Jx}+tE))F ={}& F + \ad(-\overline{Jx}+tE)F + \tfrac{1}{2}\ad(-\overline{Jx}+tE)^2F + \tfrac{1}{6}\ad(-\overline{Jx}+tE)^3F\\
& + \tfrac{1}{24}\ad(-\overline{Jx}+tE)^4F\\
={}& F + \Big(-Jx+tH\Big) + \tfrac{1}{2}\Big(-2\mu(Jx)+2t\overline{Jx}-2t^2E\Big)\\
& +\tfrac{1}{6}\Big(6\overline{\Psi(Jx)}\Big)+\tfrac{1}{24}\Big(24Q(Jx)E\Big).
\end{align*}
Altogether we obtain
\begin{multline*}
\Ad(\theta(\overline{n}_{(x,t)})^{-1}\overline{n}_{(x,t)})E = \Big(1+4|x|^2-6B_Q(x,x,Jx,Jx)+4|x|^4+2t^2-t\omega(Jx,\Psi(x)-tx)\\
+\tfrac{1}{3}\omega(Jx,[B_\mu(Jx,\Psi(x)-tx),Jx])+(Q(Jx)-t^2)(Q(x)-t^2)\Big)E + \cdots
\end{multline*}
This expression can be simplified. In fact,
\begin{align*}
\omega(Jx,\Psi(x)) &= 4B_Q(Jx,x,x,x),\\
\omega(Jx,[B_\mu(Jx,x),Jx]) &= -12B_Q(Jx,Jx,Jx,x),\\
\omega(Jx,[B_\mu(Jx,\Psi(x)),Jx]) &= -12B_Q(Jx,Jx,Jx,\Psi(x)).
\end{align*}
Hence
\begin{multline*}
1+4|x|^2-6B_Q(Jx,Jx,x,x)+4|x|^4+2t^2-4tB_Q(Jx,x,x,x)+4t^2|x|^2\\
-4B_Q(Jx,Jx,Jx,\Psi(x))+4tB_Q(Jx,Jx,Jx,x)+(Q(Jx)-t^2)(Q(x)-t^2).
\end{multline*}
Finally, using \eqref{eq:JonSymplCov} we find that
\begin{align*}
B_Q(Jx,x,x,x) &= (x|\Psi(x)), & B_Q(Jx,Jx,Jx,\Psi(x)) &= -\|\Psi(x)\|^2,\\
B_Q(Jx,Jx,Jx,x) &= -(x|\Psi(x)), & B_Q(Jx,Jx,x,x) &= \frac{1}{12}\omega(\mu(Jx)x,x)+\frac{2}{3}|x|^4
\end{align*}
and $Q(Jx)=Q(x)$ and the claimed formula follows.
\end{proof}
\section{Hermitian vs. non-Hermitian}\label{sec:HermVsNonHerm}
We derive several equivalent properties characterizing the Hermitian Lie algebras among all Heisenberg graded Lie algebras.
\begin{theorem}\label{thm:CharacterizationHermitian}
The following are equivalent:
\begin{enumerate}[(1)]
\item The group $G$ is of Hermitian type,
\item There exists $J\in\mathfrak{m}$ such that $\ad(J)^2|_{\mathfrak{g}_{\pm1}}=-1$ and $(X,Y)\mapsto\omega(\ad(J)X,Y)$ is positive definite on $V$,
\item The minimal adjoint orbits $\mathcal{O}_{\textup{min}}$ and $-\mathcal{O}_{\textup{min}}$ are distinct,
\item The quartic $Q$ is non-positive,
\item The character $\chi$ of $M$ is trivial.
\end{enumerate}
\end{theorem}
\begin{proof}
We first show (1)$\Leftrightarrow$(2). If $G$ is of Hermitian type then the center of $\mathfrak{k}$ is non-trivial, i.e. there exists $0\neq X\in\mathfrak{k}$ such that $[X,Y]=0$ for any $Y\in\mathfrak{k}$. Write $X=s(E-F)+(x+\theta(x))+T$ with $s\in\mathbb{R}$, $x\in\mathfrak{g}_1$ and $T\in\mathfrak{k}_\mathfrak{m}$. Then
$$ 0 = [X,E-F] = -\overline{x}+\overline{\theta(x)} $$
where $\overline{x}\in\mathfrak{g}_{-1}$ and $\overline{\theta(x)}\in\mathfrak{g}_1$. Hence $x=0$. Further, for every $v\in\mathfrak{g}_1$ we have
$$ 0 = [X,v+\theta(v)] = s(\overline{\theta(v)}-\overline{v}) + [T,v]+[T,\theta(v)], $$
and therefore $[T,v]+s\overline{\theta(v)}=0$ and $[T,\theta(v)]-s\overline{v}=0$. This implies
$$ \ad(T)v = -s\overline{\theta(v)} \qquad \mbox{and} \qquad \ad(T)w = s\overline{\theta(w)} $$
for $v\in\mathfrak{g}_1$ and $w\in\mathfrak{g}_{-1}$. Now, if $s=0$ then $\ad(T)=0$ on $\mathfrak{g}_{\pm1}$, and trivially also on $\mathfrak{g}_{\pm2}$, hence on $\mathfrak{g}$ which is only possible if $T=0$, because $\mathfrak{g}$ is assumed to be simple. So $s\neq0$ and therefore $\ad(T)^2|_{\mathfrak{g}_{\pm1}}=-s^2$. Then $J:=s^{-1}T\in\mathfrak{m}$ satisfies $\ad(J)^2|_{\mathfrak{g}_{\pm1}}=-1$. Further, for $X,Y\in V=\mathfrak{g}_{-1}$ we have
$$ 0\leq -B(X,\theta Y) = -B(\theta X,Y) = p\cdot\omega(\ad(J)X,Y) $$
and hence $\omega(\ad(J)X,Y)$ is positive definite.\\
Conversely, let $J\in\mathfrak{m}$ with $\ad(J)^2|_{\mathfrak{g}_{\pm1}}=-1$ and $\omega(\ad(J)X,Y)$ positive definite on $V$. Note that $\widetilde{J}:=\exp(\frac{\pi}{2}J)\in M$ satisfies $\Ad(\widetilde{J}^2)=-1$ and $\Ad(\widetilde{J})|_{\mathfrak{g}_{\pm1}}=\ad(J)|_{\mathfrak{g}_{\pm1}}$. Then one can define an involution $\theta$ on $\mathfrak{g}$ by
$$ \theta(E):=-F, \qquad \theta(F):=-E, \qquad \theta(H):=-H, $$
and for $v\in\mathfrak{g}_1$, $w\in\mathfrak{g}_{-1}$ and $T\in\mathfrak{m}$ by
$$ \theta(v) := -\overline{\ad(J)v}, \qquad \theta(w) := \overline{\ad(J)w}, \qquad \theta(T) = \Ad(\widetilde{J})T. $$
It is immediate that $\theta^2=1$ and hence $\theta$ is in fact an involution. We now show that $\theta$ is a Cartan involution. Then, by the same computations as above, the center of the corresponding maximal compact subalgebra $\mathfrak{k}=\mathfrak{g}^\theta$ is spanned by $X=E-F+J$ and hence $G$ is of Hermitian type. To show that $\theta$ is a Cartan involution we compute $B(X,\theta(X))$ for $X\in\mathfrak{g}_i$, $i=-2,-1,0,1,2$.
\begin{enumerate}[(1)]
\item For $X=E\in\mathfrak{g}_2$ we have $B(X,\theta(X))=-p<0$.
\item For $X\in\mathfrak{g}_1$ we have $B(X,\theta(X))=p\cdot\omega(\overline{X},\overline{\ad(J)X})=-p\cdot\omega(\ad(J)\overline{X},\overline{X})$ which is strictly negative for $X\neq0$.
\item For $X\in\mathfrak{m}$ we have $\theta(X)=\widetilde{J}X\widetilde{J}^{-1}$. Since $\mathfrak{m}\subseteq\sp(V,\omega)$ via $X\mapsto\ad(X)|_{\mathfrak{g}_{-1}}$ the Killing form $B_\mathfrak{m}$ has to be a scalar multiple of the Killing form of $\sp(V,\omega)$ on each simple factor of $\mathfrak{m}$. It is well-known that the involution $\theta(X)=\widetilde{J}X\widetilde{J}^{-1}$ extends to a Cartan involution of $\sp(V,\omega)$ and hence $B(X,\theta(X))<0$ for all $X\neq0$.
\item For $X=H\in\mathfrak{a}$ we have $B(X,\theta(X))=-2p<0$.
\item For $X\in\mathfrak{g}_{-1}$ we have $B(X,\theta(X))=-p\cdot\omega(\ad(J)X,X)$ and hence strictly negative for $X\neq0$.
\item For $X=F\in\mathfrak{g}_{-2}$ we have $B(X,\theta(X))=-p<0$.
\end{enumerate}
Next, the equivalence (1)$\Leftrightarrow$(3) follows from \cite[Theorem 1.4]{Oku15}. (Note that $\mathfrak{g}$ cannot have a complex structure, because in this case $\mathfrak{g}_2$ as the highest root space would have a complex structure and have real dimension $\geq2$.)
Let us show (3)$\Leftrightarrow$(4). If the orbits $\mathcal{O}_{\textup{min}}$ and $-\mathcal{O}_{\textup{min}}$ are distinct then $\Ad(M)\cdot E=\{E\}$. Hence, by Lemma~\ref{lem:AProjectionOnw0Nbar} the quartic $Q$ is non-positive. If conversely $Q\leq0$ then by Lemma~\ref{lem:AProjectionOnw0Nbar} we have $\Ad(m)E=E$ for all $m\in M$. We obtain $\Ad(MAN)E=\{e^{2r}E:r\in\mathbb{R}\}$ and further
\begin{multline*}
\Ad(\overline{N}MAN)E = \{e^{2r}\left(E+\overline{x}+\mu(x)-sH+(\Psi(x)-sx)+(Q(x)-s^2)F\right):\\
r\in\mathbb{R},(x,s)\in V\times\mathbb{R}\}.
\end{multline*}
In particular, the coefficient of $E$ of every element in $\Ad(\overline{N}MAN)E$ is positive. Since the set $\Ad(\overline{N}MAN)E$ is dense in $\mathcal{O}_{\textup{min}}$, the element $-E$ cannot be in $\mathcal{O}_{\textup{min}}$ and therefore, $\mathcal{O}_{\textup{min}}$ and $-\mathcal{O}_{\textup{min}}$ are distinct.
Finally, (4)$\Leftrightarrow$(5) follows from Lemma~\ref{lem:AProjectionOnw0Nbar}.
\end{proof}
\begin{corollary}
If $G$ is Hermitian, then the formula in Lemma~\ref{lem:IwasawaAProjectionOnNbar} simplifies to
$$ e^{\lambda(H(\overline{n}_{(x,t)}))} = (1+2|x|^2-Q(x)+t^2)^{\lambda/2}. $$
\end{corollary}
\begin{proof}
If $G$ is Hermitian, $J\in\mathfrak{m}$ and hence
$$ J\Psi(x)=3B_\Psi(Jx,x,x) \qquad \mbox{and} \qquad B_Q(Jx,x,x,x)=0, $$
so that
\begin{align*}
\omega(\mu(Jx)x,x) &= 12B_Q(Jx,Jx,x,x)-8|x|^4 = 3\omega(Jx,B_\Psi(Jx,x,x))-8|x|^4\\
&= \omega(Jx,J\Psi(x)) - 8|x|^4 = 4Q(x) -8|x|^4,\\
(x|\Psi(x)) &= \frac{1}{4}\omega(Jx,\Psi(x)) = B_Q(Jx,x,x,x) = 0.
\end{align*}
Further, polarizing Lemma~\ref{lem:SymplecticFormulas}~(2) gives $B_\Psi(\Psi(x),x,x)=\frac{1}{3}Q(x)x$ and hence
\begin{align*}
|\Psi(x)|^2 &= \frac{1}{4}\omega(J\Psi(x),\Psi(x)) = B_Q(J\Psi(x),x,x,x) = -3B_Q(\Psi(x),x,x,Jx)\\
&= -\frac{3}{4}\omega(Jx,B_\Psi(\Psi(x),x,x)) = -Q(x)|x|^2.
\end{align*}
Inserting this into the formula in Lemma~\ref{lem:IwasawaAProjectionOnNbar} and rearranging shows the claim.
\end{proof}
\begin{remark}
Note that if $\mathfrak{g}=\su(n+1,1)$ then $V=\mathbb{C}^n$ and $Q(x)=-|x|^4$, so that the above expression becomes
$$ e^{\lambda(H(\overline{n}_{(x,t)}))}=((1+|x|^2)^2+t^2). $$
This formula is well-known (see e.g. \cite[Theorem IX.3.8]{Hel78}).
\end{remark}
In the case where $G$ is Hermitian we further show that the pair $(\sp(V,\omega),\mathfrak{m})$ is of \emph{holomorphic type} in the sense of Kobayashi~\cite[Definition 1.4]{Kob08}. Recall that for reductive Hermitian Lie algebras $\mathfrak{h}\subseteq\mathfrak{g}$, the pair $(\mathfrak{g},\mathfrak{h})$ is said to be of holomorphic type if there exists a Cartan involution $\theta$ of $\mathfrak{g}$ which leaves $\mathfrak{h}$ invariant and an element $z\in\mathfrak{k}_\mathfrak{h}=\mathfrak{h}^\theta$ such that $\ad(z)=0$ on $\mathfrak{k}=\mathfrak{g}^\theta$ and $\ad(z)^2=-1$ on $\mathfrak{p}=\mathfrak{g}^{-\theta}$. In this case the natural embedding $H/(K\cap H)\subseteq G/K$ of Hermitian symmetric spaces is holomorphic.
\begin{corollary}
If $G$ is Hermitian then $\mathfrak{m}$ is also Hermitian and the pair $(\sp(V,\omega),\mathfrak{m})$ is holomorphic.
\end{corollary}
\begin{proof}
We can choose the element $z$ to be $z=\tfrac{1}{2}J\in\mathfrak{m}$. A Cartan involution of $\sp(V,\omega)$ is given by $\theta(T)=JTJ^{-1}$, hence $z\in\sp(V,\omega)^\theta$. Further $\ad(z)=0$ on $\sp(V,\omega)^\theta=\{T\in\sp(V,\omega):TJ=JT\}$, and for $T\in\sp(V,\omega)^{-\theta}=\{T\in\sp(V,\omega):TJ=-JT\}$ we have $\ad(z)T=JT$ and $\ad(z)^2T=\tfrac{1}{2}[J,JT]=J^2T=-T$.
\end{proof}
\chapter[Principal series representations]{Principal series representations and intertwining operators}
We study the degenerate principal series representations induced from the parabolic subgroup $P$ and intertwining operators between them.
\section{Degenerate principal series representations}
For a smooth admissible representation $(\zeta,V_\zeta)$\index{1fzeta@$\zeta$}\index{Vzeta@$V_\zeta$} of $M$ and $\nu\in\mathfrak{a}_\mathbb{C}^*$ we let $(\widetilde{\pi}_{\zeta,\nu},\widetilde{I}(\zeta,\nu))$ be the induced representation $\Ind_P^G(\zeta\otimes e^\nu\otimes\1)$, acting by left-translation on
$$ \widetilde{I}(\zeta,\nu) = \{f\in C^\infty(G,V_\zeta):f(gman)=a^{-\nu-\rho}\zeta(m)^{-1}f(g)\,\forall\,man\in MAN\}. $$
Here $\rho\in\mathfrak{a}^*$\index{1rho@$\rho$} denotes as usual the half sum of all positive roots.
\section{The non-compact picture}
Since $\overline{N}MAN\subseteq G$ is open dense, functions in $\widetilde{I}(\zeta,\nu)$ are uniquely determined by their restriction to $\overline{N}$. Therefore, we define for any $f\in\widetilde{I}(\zeta,\nu)$ a function $f_{\overline{\mathfrak{n}}}$ on $V\times\mathbb{R}$ by
$$ f_{\overline{\mathfrak{n}}}(x,s) := f(\overline{n}_{(x,s)}), \qquad (x,s)\in V\times\mathbb{R}\index{fn@$f_{\overline{\mathfrak{n}}}$} $$
and let
$$ I(\zeta,\nu) := \{f_{\overline{\mathfrak{n}}}:f\in\widetilde{I}(\zeta,\nu)\}.\index{Izetanu@$I(\zeta,\nu)$} $$
The representation $\widetilde{\pi}_{\zeta,\nu}$ on $\widetilde{I}(\zeta,\nu)$ defines an equivalent representation $\pi_{\zeta,\nu}$\index{1pi1zetanu@$\pi_{\zeta,\nu}$} on $I(\zeta,\nu)$ by
$$ \pi_{\zeta,\nu}(g)f_{\overline{\mathfrak{n}}} = (\widetilde{\pi}_{\zeta,\nu}(g)f)_{\overline{\mathfrak{n}}}, \qquad g\in G,f\in\widetilde{I}(\zeta,\nu). $$
The realization $(I(\zeta,\nu),\pi_{\zeta,\nu})$ is called the \textit{non-compact picture} of the degenerate principal series. Note that
$$ \mathcal{S}(V\times\mathbb{R})\otimeshat V_\zeta\subseteq I(\zeta,\nu)\subseteq C^\infty_{\temp}(V\times\mathbb{R})\otimeshat V_\zeta. $$
We compute the action of $MA\overline{N}$ and $w_0$ in this realization:
\begin{proposition}\label{prop:GroupActionNonCptPicture}
For $f\in I(\zeta,\nu)$ and $(x,s)\in V\times\mathbb{R}$ we have
\begin{align*}
\pi_{\zeta,\nu}(\overline{n}_{(y,t)})f(x,s) &= f(x-y,s-t+\tfrac{1}{2}\omega(x,y)), && \overline{n}_{(y,t)}\in\overline{N},\\
\pi_{\zeta,\nu}(m)f(x,s) &= \zeta(m)f(m^{-1}x,\chi(m)^{-1}s), && m\in M,\\
\pi_{\zeta,\nu}(e^{rH})f(x,s) &= e^{(\nu+\rho)r}f(e^rx,e^{2r}s), && e^{rH}\in A.
\end{align*}
Moreover, for $\zeta=\1$ the trivial representation of $M$ we have
$$ \pi_{\1,\nu}(w_0^{\pm1})f(x,s) = |s^2-Q(x)|^{-\frac{\nu+\rho}{2}}f\left(\pm\frac{\Psi(x)-sx}{s^2-Q(x)},-\frac{s}{s^2-Q(x)}\right). $$
\end{proposition}
\begin{proof}
The formulas for $M$, $A$ and $\overline{N}$ are obvious. For $\pi_{\1,\nu}(w_0)$ we use Lemma \ref{lem:AProjectionOnw0Nbar}.
\end{proof}
This yields the differentiated action $d\pi_{\zeta,\nu}(X)=\left.\frac{d}{dt}\right|_{t=0}\pi_{\zeta,\nu}(\exp(tX))$\index{dpi1zetanu@$d\pi_{\zeta,\nu}$} of the Lie algebra $\mathfrak{g}$ on $I(\zeta,\nu)$. To simplify the formulas we $\mathbb{E}$ denote the weighted Euler operator on $V\times\mathbb{R}$, i.e.
$$ \mathbb{E} = \sum_\alpha x_\alpha\frac{\partial}{\partial x_\alpha} + 2s\frac{\partial}{\partial s},\index{E@$\mathbb{E}$} $$
where $x=\sum_\alpha x_\alpha e_\alpha$ for any basis $(e_\alpha)$ of $V$.
\begin{corollary}\label{cor:LieAlgActionNonCptPicture}
The Lie algebra representation $d\pi_{\zeta,\nu}$ of $\mathfrak{g}$ is given by
\begin{align*}
d\pi_{\zeta,\nu}(F) &= -\partial_s,\\
d\pi_{\zeta,\nu}(v) &= -\partial_v+\tfrac{1}{2}\omega(x,v)\partial_s, && v\in\mathfrak{g}_{-1}\\
d\pi_{\zeta,\nu}(T) &= -\partial_{Tx}+d\zeta(T), && T\in\mathfrak{m},\\
d\pi_{\zeta,\nu}(H) &= \mathbb{E}+(\nu+\rho),\\
d\pi_{\zeta,\nu}(w) &= \partial_{\mu(x)\overline{w}+\omega(x,\overline{w})x-s\overline{w}}+\tfrac{1}{2}\omega(sx+\psi(x),\overline{w})\partial_s+\tfrac{\nu+\rho}{2}\omega(x,\overline{w})-2d\zeta(B_\mu(x,\overline{w})), && w\in\mathfrak{g}_1,\\
d\pi_{\zeta,\nu}(E) &= \partial_{sx+\Psi(x)}+(s^2+Q(x))\partial_s+(\nu+\rho)s+d\zeta(\mu(x)).
\end{align*}
\end{corollary}
\begin{proof}
The formulas for $d\pi_\nu(F)$, $d\pi_\nu(v)$, $d\pi_\nu(T)$ and $d\pi_\nu(H)$ follow easily by differentiating the corresponding group actions in Proposition~\ref{prop:GroupActionNonCptPicture}. For $d\pi_\nu(E)$ we use Lemma~\ref{lem:BruhatDecompForNNbar} and $d\pi_\nu(w)$ can be obtained from $d\pi_\nu(E)$ and $d\pi_{\zeta,\nu}(\overline{w})$ by $d\pi_{\zeta,\nu}(w)=[d\pi_{\zeta,\nu}(\overline{w}),d\pi_{\zeta,\nu}(E)]$.
\end{proof}
\section{Intertwining operators}
For $\Re\nu\gg0$ the standard Knapp--Stein intertwining operator $\widetilde{A}(\zeta,\nu):\widetilde{I}(\zeta,\nu)\to\widetilde{I}(w_0\zeta,-\nu)$ (with $w_0\zeta(m)=\zeta(w_0^{-1}mw_0)$) is given by the convergent integral
$$ \widetilde{A}(\zeta,\nu)f(g) = \int_{\overline{N}} f(gw_0\overline{n}) \,d\overline{n}. $$
It is well-known that $\widetilde{A}(\zeta,\nu)$ extends meromorphically in $\nu\in\mathbb{C}$. We consider the Knapp--Stein operator $A(\zeta,\nu)$ in the non-compact picture:
$$ A(\zeta,\nu)f_{\overline{\mathfrak{n}}} := (\widetilde{A}(\zeta,\nu)f)_{\overline{\mathfrak{n}}} \qquad (f\in\widetilde{I}(\zeta,\nu)).\index{A1zetanu@$A(\zeta,\nu)$} $$
\begin{proposition}\label{prop:KnappSteinIntegralFormula}
Assume $\zeta=\1$ is the trivial representation, then the operator $A(\zeta,\nu)$ is the convolution operator
$$ A(\1,\nu)f(x,s) = \int_{V\times\mathbb{R}} |t^2-Q(y)|^{\frac{\nu-\rho}{2}} f((x,s)\cdot(y,t)) \,d(y,t). $$
\end{proposition}
\begin{proof}
In \cite[Chapter VII, \S7]{Kna86} it is shown that
$$ \widetilde{A}(\zeta,\nu)f(g) = \int_{\overline{N}} a(w_0^{-1}\overline{n})^{\nu-\rho}\zeta(m(w_0^{-1}\overline{n}))f(g\overline{n}) \,d\overline{n}. $$
Then Lemma~\ref{lem:AProjectionOnw0Nbar} immediately yields the claim.
\end{proof}
\begin{remark}
Of course one can also write down a formula for $A(\zeta,\nu)$ for general $\zeta$, but we will not need this in what follows.
\end{remark}
\section{The Fourier transform on the Heisenberg group}\label{sec:HeisFT}
The infinite-dimensional irreducible unitary representations of the Heisenberg group $\overline{N}$ are parameterized by their central character $i\lambda\in i\mathbb{R}^\times$. More precisely, for each $\lambda\in\mathbb{R}^\times$ there exists a unique (up to equivalence) infinite-dimensional unitary representation $(\sigma_\lambda,\mathcal{H}_\lambda)$\index{1sigmalambda@$\sigma_\lambda$}\index{H2lambda@$\mathcal{H}_\lambda$} of $\overline{N}$ such that $d\sigma_\lambda(0,t)=i\lambda t$ for $(0,t)\in\overline{\mathfrak{n}}\simeq V\times\mathbb{R}$. There are two standard realizations of $\sigma_\lambda$, the Schrödinger model and the Fock model. The Schrödinger model is realized on the space $\mathcal{H}_\lambda=L^2(\Lambda)$ for a Lagrangian subspace $\Lambda\subseteq V$, whereas the Fock model is realized on the Fock space $\mathcal{H}_\lambda=\mathcal{F}(V)$ consisting of holomorphic functions on $V$ (with respect to a certain complex structure) which are square-integrable with respect to a Gaussian measure on $V$.
Since $M$ acts on $\overline{N}$ by automorphisms, the map $\overline{n}\mapsto\sigma_\lambda(\Ad(m)\overline{n})$ defines an irreducible unitary representation of $\overline{N}$ with central character $i\chi(m)\lambda$ and hence there exists a projective unitary representation $\omega_{\met,\lambda}$\index{1zomegametlambda@$\omega_{\met,\lambda}$} of $M$ on the same representation space such that
\begin{equation}
\sigma_\lambda(\Ad(m)\overline{n}) = \omega_{\met,\lambda}(m)\circ\sigma_{\chi(m)\lambda}(\overline{n})\circ\omega_{\met,\lambda}(m)^{-1} \qquad \forall\,m\in M,\overline{n}\in\overline{N}.\label{eq:DefMetaplecticRep}
\end{equation}
Since $M_1=\{m\in M:\chi(m)=1\}$ acts symplectically on $V$, the representation $\omega_{\met,\lambda}|_{M_1}$ is simply the restriction of the metaplectic representation of $\operatorname{Sp}(V,\omega)$ to $M_1$.
For $f\in L^1(\overline{N})$ we form
$$ \sigma_\lambda(f) = \int_{\overline{N}} f(\overline{n})\sigma_\lambda(\overline{n})\,d\overline{n}, $$
where $d\overline{n}$ is a fixed Haar measure on $\overline{N}$. If we identify $X\in\overline{\mathfrak{n}}$ with the corresponding left-invariant vector field on $\overline{N}$, then
$$ \sigma_\lambda(Xf) = -\sigma_\lambda(f)\circ d\sigma_\lambda(X). $$
Further, if $f*g$ denotes the convolution of $f,g:\overline{N}\to\mathbb{C}$ given by
$$ f*g(x) = \int_{\overline{N}}f(y)g(xy^{-1})\,dy,\index{1AStar@$*$} $$
then
$$ \sigma_\lambda(f*g) = \sigma_\lambda(g)\circ\sigma_\lambda(f). $$
We have the following Plancherel Formula (after appropriate normalization of the measures involved):
$$ \|f\|_{L^2(\overline{N})}^2 = \int_{\mathbb{R}^\times}\|\sigma_\lambda(f)\|_{\textup{HS}}^2 |\lambda|^{\frac{\dim\mathfrak{g}_1}{2}}\,d\lambda. $$
Here $\|T\|_{\textup{HS}}^2=\tr(TT^*)$\index{1@$\|\cdot\|_{\textup{HS}}$} denotes the Hilbert--Schmidt norm of a Hilbert--Schmidt operator on a Hilbert space. Realizing all infinite-dimensional irreducible unitary representations of $\overline{N}$ on the same Hilbert space $\mathcal{H}_\lambda=\mathcal{H}$ and writing ${\textup{HS}}(\mathcal{H})$\index{H1SH@${\textup{HS}}(\mathcal{H})$} for the Hilbert space of Hilbert--Schmidt operators on $\mathcal{H}$, the Fourier transform can be viewed as an isometric isomorphism
$$ \mathcal{F}: L^2(\overline{N})\to L^2(\mathbb{R}^\times,{\textup{HS}}(\mathcal{H});|\lambda|^{\frac{\dim\mathfrak{g}_1}{2}}\,d\lambda), \quad \mathcal{F} f(\lambda) = \sigma_\lambda(f). $$
\section[The Heisenberg Fourier transform]{The Schr\"{o}dinger model and the Fourier transform of distributions}
For the Schr\"{o}dinger model of the infinite-dimensional irreducible unitary representations of $\overline{N}\simeq V\times\mathbb{R}$ one has to choose a Lagrangian subspace $\Lambda\subseteq V$\index{1Lambda@$\Lambda$} and a Lagrangian complement $\Lambda^*\subseteq V$\index{1LambdaStar@$\Lambda^*$}. Then the Schr\"{o}dinger model is a realization of $\sigma_\lambda$ on $\mathcal{H}=L^2(\Lambda)$ given by
\begin{equation}
\sigma_\lambda(z,t)\varphi(x) = e^{i\lambda t}e^{i\lambda(\omega(z'',x)+\frac{1}{2}\omega(z',z''))}\varphi(x-z'), \qquad (x\in\Lambda)\label{eq:DefSchroedingerModel}
\end{equation}
for $z=(z',z'')\in\Lambda\oplus\Lambda^*=V$ and $t\in\mathbb{R}$. The corresponding differentiated representation of $\overline{\mathfrak{n}}\simeq V\times\mathbb{R}$ is given by
$$ d\sigma_\lambda(z,t) = -\partial_{z'}+i\lambda\omega(z'',x)+i\lambda t, \qquad z=(z',z'')\in\Lambda\oplus\Lambda^*=V, t\in\mathbb{R}. $$
For $\varphi\in\mathcal{S}(\overline{N})$ we have
\begin{align*}
\sigma_\lambda(u)\varphi(y) &= \int_{\overline{N}} u(z,t)\sigma_\lambda(z,t)\varphi(y)\,dz\,dt\\
&= \int_{\Lambda}\int_{\Lambda^*}\int_\mathbb{R} u(z',z'',t)e^{i\lambda t}e^{i\lambda(\omega(z'',y)+\frac{1}{2}\omega(z',z''))}\varphi(y-z')\,dt\,dz''\,dz'\\
&= \int_\Lambda\left(\int_{\Lambda^*}\int_\mathbb{R} u(y-x,z'',t)e^{i\lambda t}e^{-\frac{i\lambda}{2}(\omega(x+y,z''))}\,dt\,dz''\right)\varphi(x)\,dx\\
&= \int_{\Lambda}\widehat{u}(\lambda,x,y)\varphi(x)\,dx,
\end{align*}
where
$$ \widehat{u}(\lambda,x,y) = \int_{\Lambda^*}\int_\mathbb{R} u(y-x,z'',t)e^{i\lambda t}e^{-\frac{i\lambda}{2}\omega(x+y,z'')}\,dt\,dz'' = \mathcal{F}_2\mathcal{F}_3u(y-x,\tfrac{\lambda}{2}(x+y),-\lambda).\index{uhatlambdaxy@$\widehat{u}(\lambda,x,y)$} $$
Here $\mathcal{F}_2$ denotes the symplectic Fourier transform with respect to $\omega$ in the second variable, and $\mathcal{F}_3$ denotes the Euclidean Fourier transform with respect to the third variable.
\begin{proposition}\label{prop:KernelFT}
The linear map
$$ \mathcal{S}'(\overline{N})\to\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda\times\Lambda)\simeq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\Hom(\mathcal{S}(\Lambda),\mathcal{S}'(\Lambda)), \quad u\mapsto\widehat{u}, $$
is defined and continuous. Its kernel is given by those distributions which are polynomial in $t$.
\end{proposition}
\begin{proof}
Clearly $\mathcal{F}_2\mathcal{F}_3$ is a topological isomorphism $\mathcal{S}'(\overline{N})\to\mathcal{S}'(\Lambda\times\Lambda\times\mathbb{R})$. Restricting the last coordinate to $\mathbb{R}^\times$ defines a continuous linear map $\mathcal{S}'(\Lambda\times\Lambda\times\mathbb{R})\simeq\mathcal{S}'(\Lambda\times\Lambda)\otimeshat\mathcal{S}'(\mathbb{R})\to\mathcal{S}'(\Lambda\times\Lambda)\otimeshat\mathcal{D}'(\mathbb{R}^\times)$. Finally, the change of coordinates $(x,y,\lambda)\mapsto(y-x,\frac{\lambda}{2}(x+y),-\lambda)$ induces a continuous linear isomorphism on $\mathcal{S}'(\Lambda\times\Lambda)\otimeshat\mathcal{D}'(\mathbb{R}^\times)$. Composing these three maps shows continuity of the map $u\mapsto\widehat{u}$. To determine its kernel we observe that the only non-bijective map in this three-fold composition is the restriction to $\mathbb{R}^\times$. Its kernel is given by all distributions $v\in\mathcal{S}'(\Lambda\times\Lambda\times\mathbb{R})$ with $\supp v\subseteq\Lambda\times\Lambda\times\{0\}$. Such distributions are necessarily of the form
$$ v(x,y,\lambda) = \sum_{k=0}^m v_k(x,y)\delta^{(k)}(\lambda) $$
for some distributions $v_k\in\mathcal{S}'(\Lambda\times\Lambda)$. Taking the inverse Fourier transforms $\mathcal{F}_2^{-1}\circ\mathcal{F}_3^{-1}$ shows the claim.
\end{proof}
\begin{remark}
The map $u\mapsto\widehat{u}$ is essentially the group Fourier transform of $\overline{N}$, but only evaluated at the infinite-dimensional unitary representations $\sigma_\lambda$, $\lambda\in\mathbb{R}^\times$. Therefore, it has a kernel which can be treated using the finite-dimensional unitary representations of $\overline{N}$.
\end{remark}
\begin{corollary}\label{cor:FTinjectiveOnPS}
For $\Re\nu>-\rho$ the Fourier transform
$$ \mathcal{F}:I(\zeta,\nu)\subseteq\mathcal{S}'(V\times\mathbb{R}) \to \mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda\times\Lambda)\index{F@$\mathcal{F}$} $$
is injective.
\end{corollary}
\begin{proof}
Assume $u=f_{\overline{\mathfrak{n}}}\in I(\zeta,\nu)$ is in the kernel of the Fourier transform. By Proposition~\ref{prop:KernelFT} $u$ is polynomial in $t$. On the other hand, we can write $\overline{n}_{(z,t)}=kme^{H(\overline{n}_{(z,t)})}n\in KMAN$ and hence
$$ u(z,t) = f(k)\zeta(m)^{-1}e^{-(\nu+\rho)H(\overline{n}_{(z,t)})}. $$
Since $K$ is compact, the first factor $f(k)$ is bounded. Further, since $\zeta$ is a unitary character, we have $|\zeta(m)|=1$. By Lemma~\ref{lem:IwasawaAProjectionOnNbar} the third factor $e^{-(\nu+\rho)H(\overline{n}_{(z,t)})}$ is a polynomial in $t$ lifted to the power $-\frac{1}{4}(\nu+\rho)$ which decays as $|t|\to\infty$ since $\Re(\nu+\rho)>0$. The only polynomial $u(z,t)$ in $t$ with this property is the zero polynomial, so the kernel of the Fourier transform restricted to $I(\zeta,\nu)$ is trivial.
\end{proof}
\begin{remark}
We note that
$$ \mathcal{S}'(\Lambda\times\Lambda)\simeq\Hom(\mathcal{S}(\Lambda),\mathcal{S}'(\Lambda))=\Hom(\mathcal{H}^\infty,\mathcal{H}^{-\infty}) $$
since the space $\mathcal{H}^\infty$\index{H2infty@$\mathcal{H}^\infty$} of smooth vectors in $\mathcal{H}=L^2(\Lambda)$ is given by the space $\mathcal{S}(\Lambda)$ of Schwarz functions.
\end{remark}
The previous observation allows us to define a representation $\widehat{\pi}_{\zeta,\nu}$ of $G$ on $\widehat{I}(\zeta,\nu)=\mathcal{F}(I(\zeta,\nu))\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda\times\Lambda)$ by
$$ \widehat{\pi}_{\zeta,\nu}(g) = \mathcal{F}\circ\pi_{\zeta,\nu}(g)\circ\mathcal{F}^{-1}, \qquad g\in G.\index{1pi2hatzetanu@$\widehat{\pi}_{\zeta,\nu}$} $$
We call this realization the \emph{Fourier transformed picture}. In this picture, the action of the opposite parabolic subgroup $\overline{P}$ is easily expressed in terms of the representation $\sigma_\lambda$ and the metaplectic representation $\omega_{\met,\lambda}$.
\begin{proposition}\label{prop:ActionFTpicture}
For $f\in\widehat{I}(\zeta,\nu)\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\Hom(\mathcal{H}^\infty,\mathcal{H}^{-\infty})$ we have
\begin{align*}
\widehat{\pi}_{\zeta,\nu}(\overline{n}_{(z,t)})f(\lambda) &= \sigma_\lambda(z,t)\circ f(\lambda), && \overline{n}_{(z,t)}\in\overline{N},\\
\widehat{\pi}_{\zeta,\nu}(m)f(\lambda) &= \zeta(m)\cdot\omega_{\met,\lambda}(m)\circ f(\chi(m)\lambda)\circ\omega_{\met,\lambda}(m)^{-1}, && m\in M,\\
\widehat{\pi}_{\zeta,\nu}(e^{tH})f(\lambda) &= e^{(\nu-\rho)t}\delta_{e^t}\circ f(e^{-2t}\lambda)\circ\delta_{e^{-t}}, && e^{tH}\in A,
\end{align*}
where $\delta_s\varphi(x)=\varphi(sx)$\index{1deltas@$\delta_s$} ($s>0$). Alternatively, viewing $f$ as a distribution in $(\lambda,x,y)\in\mathbb{R}^\times\times\Lambda\times\Lambda$, we have
\begin{align*}
\widehat{\pi}_{\zeta,\nu}(\overline{n}_{(z,t)})f(\lambda,x,y) &= e^{i\lambda t}e^{i\lambda(\omega(z'',y)+\frac{1}{2}\omega(z',z''))}f(\lambda,x,y-z'), && \overline{n}_{(z,t)}\in\overline{N},\\
\widehat{\pi}_{\zeta,\nu}(m)f(\lambda,x,y) &= \zeta(m)(\id_{\mathbb{R}^\times}^*\otimes\omega_{\met,-\lambda}(m)\otimes\omega_{\met,\lambda}(m))f(\chi(m)\lambda,x,y), && m\in M,\\
\widehat{\pi}_{\zeta,\nu}(e^{tH})f(\lambda,x,y) &= e^{(\nu-1)t}f(e^{-2t}\lambda,e^tx,e^ty), && e^{tH}\in A,
\end{align*}
\end{proposition}
\begin{proof}
Use Proposition \ref{prop:GroupActionNonCptPicture} and note that
$$ \sigma_\lambda(\overline{n})^\top=\sigma_{-\lambda}(\overline{n}^{-1}) \qquad \mbox{and} \qquad \omega_{\met,\lambda}(m)^\top=\omega_{\met,-\lambda}(m^{-1}) $$
as operators $\mathcal{S}(\Lambda)\to\mathcal{S}(\Lambda)$, $\mathcal{S}(\Lambda)\to\mathcal{S}'(\Lambda)$ or $\mathcal{S}'(\Lambda)\to\mathcal{S}'(\Lambda)$.
\end{proof}
It seems difficult to express the action of $N$ or $w_0$ in the Fourier transformed picture. More accessible is the action of the Lie algebra $\mathfrak{g}$ in the differentiated representation $d\widehat{\pi}_{\zeta,\nu}$ which can be obtained using Corollary \ref{cor:LieAlgActionNonCptPicture} and the formulas in the following lemma. We will not carry out the computation of the Lie algebra action on the whole principal series representation, but rather restrict to a certain subrepresentation in Section \ref{sec:FTpictureMinRep}.
\begin{lemma}\label{lem:FTMultDiff}
Let $u\in\mathcal{S}'(V\times\mathbb{R})$.
\begin{enumerate}[(1)]
\item For $v\in\Lambda$ we have
\begin{align*}
\widehat{\omega(v,z)u}(\lambda,x,y) &= -\frac{1}{i\lambda}(\partial_{v,x}+\partial_{v,y})\widehat{u}(\lambda,x,y),\\
\widehat{\partial_vu}(\lambda,x,y) &= -\frac{1}{2}(\partial_{v,x}-\partial_{v,y})\widehat{u}(\lambda,x,y).
\end{align*}
\item For $w\in\Lambda^*$ we have
\begin{align*}
\widehat{\omega(z,w)u}(\lambda,x,y) &= \omega(y-x,w)\widehat{u}(\lambda,x,y),\\
\widehat{\partial_wu}(\lambda,x,y) &= \frac{i\lambda}{2}\omega(x+y,w)\widehat{u}(\lambda,x,y).
\end{align*}
\item For differentiation and multiplication with respect to the central variable we have
\begin{align*}
\widehat{\partial_tu}(\lambda,x,y) &= -i\lambda \widehat{u}(\lambda,x,y),\\
\widehat{tu}(\lambda,x,y) &= -i\partial_\lambda \widehat{u}(\lambda,x,y) - \frac{1}{2i\lambda}(\partial_{x+y,x}+\partial_{x+y,y})\widehat{u}(\lambda,x,y).
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
We only show the last formula, the rest is standard. For this let $(e_\alpha)\subseteq\Lambda$ be a basis of $\Lambda$ with dual basis $(\widehat{e}_\alpha)\subseteq\Lambda^*$. Then using (1) we find
\begin{align*}
\widehat{tu}(\lambda,x,y) &= -i\int_{\Lambda^*}\int_\mathbb{R} u(y-x,z'',t)\partial_\lambda\left[e^{i\lambda t}\right]e^{-\frac{i\lambda}{2}\omega(x+y,z'')}\,dt\,dz''\\
&= -i\partial_\lambda \widehat{u}(x,y,\lambda) + \frac{1}{2}\int_{\Lambda^*}\int_\mathbb{R}\omega(x+y,z'')u(y-x,z'',t)e^{i\lambda t}e^{-\frac{i\lambda}{2}\omega(x+y,z'')}\,dt\,dz''\\
&= -i\partial_\lambda \widehat{u}(\lambda,x,y) + \frac{1}{2}\sum_\alpha\omega(x+y,\widehat{e}_\alpha)\widehat{\omega(e_\alpha,z'')u}(\lambda,x,y)\\
&= -i\partial_\lambda \widehat{u}(\lambda,x,y) - \frac{1}{2i\lambda}\sum_\alpha\omega(x+y,\widehat{e}_\alpha)(\partial_{e_\alpha,x}+\partial_{e_\alpha,y})\widehat{u}(\lambda,x,y)
\end{align*}
and the claimed formula follows.
\end{proof}
\chapter[Conformally invariant systems]{Conformally invariant systems and their Fourier transform}
We recall the construction of conformally invariant systems on Heisenberg nilradicals due to \cite{BKZ08} and compute their action in the Fourier transformed picture.
\section{Quantization of the symplectic covariants}
In \cite{BKZ08} four conformally invariant systems of differential operators on $\overline{N}$ are constructed. We briefly recall their construction and properties. For this let $(e_\alpha)\subseteq V$ be a basis and $\widehat{e}_\alpha$ be the dual basis with respect to the symplectic form, i.e. $\omega(e_\alpha,\widehat{e}_\beta)=\delta_{\alpha\beta}$. Denote by $X_\alpha$ the left-invariant differential operator on $\overline{N}\simeq V\times\mathbb{R}$ corresponding to $e_\alpha$, i.e.
$$ X_\alpha f(\overline{n}) = \left.\frac{d}{dt}\right|_{t=0}f(\overline{n}e^{te_\alpha}). $$
In the coordinates $(x,t)\in V\times\mathbb{R}$ on $\overline{N}$ this operator takes the form
\begin{equation}
X_\alpha = \partial_\alpha+\tfrac{1}{2}\omega(x,e_\alpha)\partial_t,\label{eq:LeftInvVectorFields}
\end{equation}
where $\partial_\alpha=\partial_{e_\alpha}$.
\subsection{Quantization of $\omega$}
For $v\in V$ we let
$$ \Omega_\omega(v) := \sum_\alpha\omega(v,\widehat{e}_\alpha)X_\alpha = \partial_v+\frac{1}{2}\omega(x,v)\partial_t.\index{1ZOmega1omegav@$\Omega_\omega(v)$} $$
\subsection{Quantization of $\mu$}
For $T\in\mathfrak{m}$ we let
$$ \Omega_\mu(T) = \sum_{\alpha,\beta} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)X_\alpha X_\beta.\index{1ZOmega2muT@$\Omega_\mu(T)$} $$
Using the explicit expression \eqref{eq:LeftInvVectorFields} of $X_\alpha$ in the coordinates $(x,t)\in V\times\mathbb{R}$ we find
\begin{align}
\Omega_\mu(T) ={}& \tfrac{1}{2}\sum_{\alpha,\beta} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\Big[(\partial_\alpha+\tfrac{1}{2}\omega(x,e_\alpha)\partial_t)(\partial_\beta+\tfrac{1}{2}\omega(x,e_\beta)\partial_t)\notag\\
& \hspace{4cm}+(\partial_\beta+\tfrac{1}{2}\omega(x,e_\beta)\partial_t)(\partial_\alpha+\tfrac{1}{2}\omega(x,e_\alpha)\partial_t)\Big]\notag\\
={}& \sum_{\alpha,\beta} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\Big[\partial_\alpha\partial_\beta+\tfrac{1}{2}\omega(x,e_\alpha)\partial_\beta\partial_t+\tfrac{1}{2}\omega(x,e_\beta)\partial_\alpha\partial_t+\tfrac{1}{4}\omega(x,e_\alpha)\omega(x,e_\beta)\partial_t^2\Big]\notag\\
={}& \sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta-\partial_{Tx}\partial_t+\tfrac{1}{4}\omega(Tx,x)\partial_t^2.\label{eq:ExplicitDmuT}
\end{align}
\subsection{Quantization of $\Psi$ and $Q$}
For $X\in V$ we let
\begin{align*}
\Omega_\Psi(X) &:= \sum_{\alpha,\beta,\gamma} \omega(X,B_\Psi(\widehat{e}_\alpha,\widehat{e}_\beta,\widehat{e}_\gamma))X_\alpha X_\beta X_\gamma,\index{1ZOmega3PsiX@$\Omega_\Psi(X)$}\\
\Omega_Q &:= \sum_{\alpha,\beta,\gamma,\delta} B_Q(\widehat{e}_\alpha,\widehat{e}_\beta,\widehat{e}_\gamma,\widehat{e}_\delta)X_\alpha X_\beta X_\gamma X_\delta.\index{1ZOmega4Q@$\Omega_Q$}
\end{align*}
\section{Conformal invariance}
In \cite{BKZ08} it is shown that all four systems are conformally invariant for $d\zeta=0$ and certain special parameters $\nu$. Since we also need to involve non-trivial representations $d\zeta$ in Sections~\ref{sec:FTpictureMinRepSLn} and \ref{sec:FTpictureMinRepSOpq}, we give a self-contained proof of conformal invariance for the systems $\Omega_\omega$ and $\Omega_\mu$ to have all relevant formulas available.
\begin{theorem}\label{thm:ConfInvOmegaOmega}
For every $v\in V$ we have
\begin{align*}
[\Omega_\omega(v),d\pi_{\zeta,\nu}(X)] &= 0 && (X\in\overline{\mathfrak{n}}),\\
[\Omega_\omega(v),d\pi_{\zeta,\nu}(H)] &= \Omega_\omega(v),\\
[\Omega_\omega(v),d\pi_{\zeta,\nu}(S)] &= -\Omega_\omega(Sv) && (S\in\mathfrak{m}).
\end{align*}
Moreover,
$$ [\Omega_\omega(v),d\pi_{\zeta,\nu}(E)] = t\Omega_\omega(v)-\Omega_\omega(\mu(x)v)+\frac{\nu+\rho}{2}\omega(x,v)+2d\zeta(B_\mu(x,v)). $$
In particular, for $d\zeta=0$ and $\nu=-\rho$ the space
$$ I(\zeta,\nu)^{\Omega_\omega(V)} = \{f\in I(\zeta,\nu):\Omega_\omega(v)f=0\,\forall\,v\in V\}\index{IzetanuOmegaomegaV@$I(\zeta,\nu)^{\Omega_\omega(V)}$} $$
is a subrepresentation of $(\pi_{\zeta,\nu},I(\zeta,\nu))$.
\end{theorem}
\begin{proof}
This is a straightforward verification using the formulas in Corollary~\ref{cor:LieAlgActionNonCptPicture}.
\end{proof}
\begin{remark}
It is easy to see that $I(\zeta,\nu)^{\Omega_\omega(V)}$ only consists of constant functions and is therefore the trivial representation. To obtain a non-trivial representation we try to reduce the conformally invariant system $\Omega_\omega(V)$ to $\Omega_\omega(W)$ for a subspace $W\subseteq V$. However, in all cases except $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ the Lie algebra $\mathfrak{m}$ acts irreducibly on $V$ so that $[\Omega_\omega(w),d\pi_{\zeta,\nu}(S)]f=0$ for some $w\in V$ implies $\Omega_\omega(v)f=0$ for all $v\in V$. For $\mathfrak{g}=\sl(n,\mathbb{R})$ the adjoint representation of $\mathfrak{m}$ on $V$ splits into two irreducible subspaces. In Section~\ref{sec:FTpictureMinRepSLn} we show that restricting $\Omega_\omega$ to one of those subspaces yields a conformally invariant subsystem for certain parameters $(\zeta,\nu)$.
\end{remark}
\begin{theorem}[{\cite[Theorem 5.2]{BKZ08}}]\label{thm:ConfInvOmegaMu}
For every $T\in\mathfrak{m}$ we have
\begin{align*}
[\Omega_\mu(T),d\pi_{\zeta,\nu}(X)] &= 0 && (X\in\overline{\mathfrak{n}}),\\
[\Omega_\mu(T),d\pi_{\zeta,\nu}(H)] &= 2\Omega_\mu(T),\\
[\Omega_\mu(T),d\pi_{\zeta,\nu}(S)] &= \Omega_\mu([T,S]) && (S\in\mathfrak{m}).
\end{align*}
Further, if $T\in\mathfrak{m}'$ where $\mathfrak{m}'$ is any simple or abelian factor of $\mathfrak{m}$, then
\begin{multline*}
[\Omega_\mu(T),d\pi_{\zeta,\nu}(E)] = 2s\Omega_\mu(T)+\Omega_\mu([T,\mu(x)])+(2\,\mathcal{C}(\mathfrak{m}')-2-(\nu+\rho))\Omega_\omega(Tx)\\
+4\sum_\alpha d\zeta(B_\mu(x,e_\alpha))\Omega_\omega(T\widehat{e}_\alpha)-2\,\mathcal{C}(\mathfrak{m}')d\zeta(T).
\end{multline*}
In particular, for any simple or abelian factor $\mathfrak{m}'$ of $\mathfrak{m}$, $d\zeta=0$ and $\nu=2\,\mathcal{C}(\mathfrak{m}')-\rho-2$ the space
$$ I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')} = \{f\in I(\zeta,\nu):\Omega_\mu(T)f=0\,\forall\,T\in\mathfrak{m}'\} $$
is a subrepresentation of $(\pi_{\zeta,\nu},I(\zeta,\nu))$.\index{IzetanuOmegamumprime@$I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')}$}
\end{theorem}
\begin{proof}
The first three identities are easy to verify. To calculate $[\Omega_\mu(T),d\pi_{\zeta,\nu}(E)]$ we compute all commutators between the different summands of
$$ \Omega_\mu(T) = \sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta-\partial_{Tx}\partial_s+\tfrac{1}{4}\omega(Tx,x)\partial_s^2 $$
and
$$ d\pi_{\zeta,\nu}(E) = \partial_{sx}+\partial_{\Psi(x)}+(s^2+Q(x))\partial_s+(\nu+\rho)s+d\zeta(\mu(x)) $$
separately. First,
$$ \Big[\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta,\partial_{sx}\Big] = 2s\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta. $$
Next, using Lemma~\ref{lem:SymmetrizationsOfSymplecticCovariants}~(2) we have
\begin{align*}
\Big[\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta,\partial_{\Psi(x)}\Big] ={}& 6\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{B_\Psi(e_\alpha,x,x)}\partial_\beta+6\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{B_\Psi(e_\alpha,e_\beta,x)}\\
={}& -\sum_{\alpha,\beta}\Big(2\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{\mu(x)e_\alpha}\partial_\beta+\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\omega(x,e_\alpha)\partial_x\partial_\beta\Big)\\
& -\sum_{\alpha,\beta}\Big(2\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{B_\mu(e_\alpha,e_\beta)x}+\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{\omega(e_\alpha,x)e_\beta}\Big)\\
={}& \sum_{\alpha,\beta}\omega([T,\mu(x)]\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta+\sum_{\alpha,\beta}\omega(x,\widehat{e}_\alpha)\omega(Tx,\widehat{e}_\beta)\partial_\alpha\partial_\beta\\
& -\sum_{\alpha,\beta}2\partial_{B_\mu(e_\alpha,T\widehat{e}_\alpha)x}-\partial_{Tx}.
\intertext{By Lemma~\ref{lem:BezoutianSum} the sum in the third summand evaluates to $\sum_\alpha B_\mu(e_\alpha,T\widehat{e}_\alpha)=-\mathcal{C}(\mathfrak{m}')T$ and together we obtain}
={}& \sum_{\alpha,\beta}\omega([T,\mu(x)]\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta+\sum_{\alpha,\beta}\omega(x,\widehat{e}_\alpha)\omega(Tx,\widehat{e}_\beta)\partial_\alpha\partial_\beta\\
& +(2\,\mathcal{C}(\mathfrak{m}')-1)\partial_{Tx}.
\end{align*}
Next, by Lemma~\ref{lem:SymmetrizationsOfSymplecticCovariants}~(2) and (3)
\begin{align*}
& \Big[\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta,(s^2+Q(x))\partial_s\Big]\\
={}& 8\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)B_Q(e_\alpha,x,x,x)\partial_\beta\partial_s+12\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)B_Q(e_\alpha,e_\beta,x,x)\partial_s\\
={}& 2\partial_{T\Psi(x)}\partial_s-\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\Big(\omega(x,B_\mu(e_\alpha,e_\beta)x)+\frac{1}{2}\omega(x,e_\beta)\omega(e_\alpha,x)\Big)\partial_s\\
={}& -\frac{2}{3}\partial_{T\mu(x)x}\partial_s-\sum_\alpha\omega(x,B_\mu(e_\alpha,T\widehat{e}_\alpha)x)\partial_s+\frac{1}{2}\omega(Tx,x)\partial_s
\intertext{which is, again by Lemma~\ref{lem:BezoutianSum}, equal to}
={}& -\frac{2}{3}\partial_{T\mu(x)x}\partial_s+(\frac{1}{2}-\mathcal{C}(\mathfrak{m}'))\omega(Tx,x)\partial_s.
\end{align*}
Finally, the last commutator of this type is
$$ \Big[\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta,(\nu+\rho)s\Big] = 0. $$
Next, we have
$$ [\partial_{Tx}\partial_s,\partial_{sx}] = \partial_{Tx} + \sum_{\alpha,\beta}\omega(x,\widehat{e}_\alpha)\omega(Tx,\widehat{e}_\beta)\partial_\alpha\partial_\beta. $$
Further, again by Lemma~\ref{lem:SymmetrizationsOfSymplecticCovariants}
\begin{align*}
[\partial_{Tx}\partial_s,\partial_{\Psi(x)}] &= \sum_\alpha\Big(3\omega(B_\Psi(Tx,x,x),\widehat{e}_\alpha)-\omega(T\Psi(x),\widehat{e}_\alpha)\Big)\partial_\alpha\partial_s\\
&= -\partial_{\mu(x)Tx}\partial_s+\frac{1}{2}\omega(Tx,x)\partial_x\partial_s+\frac{1}{3}\partial_{T\mu(x)x}\partial_s.
\end{align*}
Next,
\begin{align*}
[\partial_{Tx}\partial_s,(s^2+Q(x))\partial_s] &= 2s\partial_{Tx}\partial_s + 4B_Q(Tx,x,x,x)\partial_s^2\\
&= 2s\partial_{Tx}\partial_s + \omega(Tx,\Psi(x))\partial_s^2\\
&= 2s\partial_{Tx}\partial_s - \frac{1}{6}\omega([T,\mu(x)]x,x)\partial_s^2.
\end{align*}
And the last commutator of this type is
\begin{align*}
[\partial_{Tx}\partial_s,(\nu+\rho)s] &= (\nu+\rho)\partial_{Tx}.
\end{align*}
Next, we have
\begin{align*}
[\omega(Tx,x)\partial_s^2,\partial_{sx}] &= 2\omega(Tx,x)\partial_x\partial_s-2s\omega(Tx,x)\partial_s^2.
\end{align*}
Further,
\begin{align*}
[\omega(Tx,x)\partial_s^2,\partial_{\Psi(x)}] &= -\Big(\omega(T\Psi(x),x)+\omega(Tx,\Psi(x))\Big)\partial_s^2 = \frac{1}{3}\omega([T,\mu(x)]x,x)\partial_s^2.
\end{align*}
Next
\begin{align*}
[\omega(Tx,x)\partial_s^2,(s^2+Q(x))\partial_s] &= 2\omega(Tx,x)\partial_s+4s\omega(Tx,x)\partial_s^2
\end{align*}
and
\begin{align*}
[\omega(Tx,x)\partial_s^2,(\nu+\rho)s] &= 2(\nu+\rho)\omega(Tx,x)\partial_s.
\end{align*}
Finally,
\begin{align*}
[\Omega_\mu(T),d\zeta(\mu(x))] &= 2\sum_\alpha d\zeta(B_\mu(e_\alpha,T\widehat{e}_\alpha))+4\sum_\alpha d\zeta(B_\mu(x,e_\alpha))\partial_{T\widehat{e}_\alpha}-2d\zeta(B_\mu(Tx,x))\partial_t\\
&= -2\,\mathcal{C}(\mathfrak{m}')d\zeta(T)+4\sum_\alpha d\zeta(B_\mu(x,e_\alpha))\Omega_\omega(T\widehat{e}_\alpha)
\end{align*}
Collecting all terms shows the claimed formula.
\end{proof}
\section{The Fourier transform of $\Omega_\omega$}\label{sec:FTOmegaOmega}
Since $\Omega_\omega(v)$ is acting by the left-invariant vector field corresponding to $v\in V$, it follows immediately that
\begin{equation}
\sigma_\lambda(\Omega_\omega(v)u) = -\sigma_\lambda(u)d\sigma_\lambda(v).\label{eq:FTofOmegaOmega}
\end{equation}
\section{The Fourier transform of $\Omega_\mu$}\label{sec:FTOmegaMu}
We show that in the Fourier transformed picture the conformally invariant system $\Omega_\mu(T)$, $T\in\mathfrak{m}$, is nothing else but the metaplectic representation of $\sp(V,\omega)$ restricted to $\mathfrak{m}$ as defined in Section \ref{sec:HeisFT}. Differentiating \eqref{eq:DefMetaplecticRep} we obtain that $d\omega_{\met,\lambda}$ is the unique representation of $\mathfrak{m}$ on $\mathcal{S}(\Lambda)$ by differential operators such that
$$ d\sigma_\lambda([T,X]) = [d\omega_{\met,\lambda}(T),d\sigma_\lambda(X)] \qquad \forall\,T\in\mathfrak{m},X\in\overline{\mathfrak{n}}. $$
\begin{theorem}\label{thm:FTofOmegaMu}
For every $\lambda\in\mathbb{R}^\times$ and $T\in\mathfrak{m}$ we have
$$ d\sigma_\lambda(\Omega_\mu(T)) = 2i\lambda d\omega_{\met,\lambda}(T). $$
\end{theorem}
\begin{proof}
It suffices to show that
$$ [d\sigma_\lambda(\Omega_\mu(T)),d\sigma_\lambda(X)] = 2i\lambda d\sigma_\lambda([T,X]) \qquad \forall\,X\in V. $$
We compute
\begin{align*}
& [d\sigma_\lambda(\Omega_\mu(T)),d\sigma_\lambda(X)] = \sum_{\alpha,\beta}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)[d\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta),d\sigma_\lambda(X)]\\
={}& \sum_{\alpha,\beta}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)\Big(d\sigma_\lambda(X_\alpha)[d\sigma_\lambda(X_\beta),d\sigma_\lambda(X)]+[d\sigma_\lambda(X_\alpha),d\sigma_\lambda(X)]d\sigma_\lambda(X_\beta)\Big)\\
={}& \sum_{\alpha,\beta}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)\Big(d\sigma_\lambda(X_\alpha)d\sigma_\lambda([X_\beta,X])+d\sigma_\lambda([X_\alpha,X])d\sigma_\lambda(X_\beta)\Big)\\
={}& i\lambda\sum_{\alpha,\beta}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)\left(\omega(X_\beta,X)d\sigma_\lambda(X_\alpha)+\omega(X_\alpha,X)d\sigma_\lambda(X_\beta)\right)\\
={}& 2i\lambda\sum_\beta\omega([T,X],\widehat{e}_\beta)d\sigma_\lambda(X_\beta) = 2i\lambda d\sigma_\lambda([T,X]).\qedhere
\end{align*}
\end{proof}
\section{The Fourier transform of $\Omega_\Psi$}\label{sec:FTOmegaPsi}
\begin{corollary}\label{cor:FTofOmegaPsi}
For every $\lambda\in\mathbb{R}^\times$ and $X\in V$ we have
\begin{align*}
d\sigma_\lambda(\Omega_\Psi(X)) &= \frac{2i\lambda}{3}\sum_\alpha\sigma_\lambda(e_\alpha)d\omega_{\met,\lambda}(B_\mu(X,\widehat{e}_\alpha)) + \frac{i\lambda}{12}(\dim V+1)\sigma_\lambda(X)\\
&= \frac{i\lambda p}{3}\sum_i\sigma_\lambda(T_i'X)d\omega_{\met,\lambda}(T_i) + \frac{i\lambda}{12}(\dim V+1)\sigma_\lambda(X),
\end{align*}
where $(e_\alpha)$ is a basis of $V$, $(\widehat{e}_\alpha)$ the dual basis with respect the symplectic form $\omega$, $(T_i)$ a basis of $\mathfrak{m}$ and $(T_i')$ the dual basis with respect to the Killing form $\kappa$ of $\mathfrak{g}$.
\end{corollary}
\begin{proof}
By Lemma \ref{lem:SymmetrizationsOfSymplecticCovariants} and Theorem \ref{thm:FTofOmegaMu} we have
\begin{align*}
\sigma_\lambda(\Omega_\Psi(X)) ={}& \sum_{\alpha,\beta,\gamma}\omega(\widehat{e}_\beta,B_\Psi(X,\widehat{e}_\alpha,\widehat{e}_\gamma))\sigma_\lambda(e_\alpha)\sigma_\lambda(e_\beta)\sigma_\lambda(e_\gamma)\\
={}& -\frac{1}{3}\sum_{\alpha,\beta,\gamma}\omega(\widehat{e}_\beta,B_\mu(X,\widehat{e}_\alpha)\widehat{e}_\gamma)\sigma_\lambda(e_\alpha)\sigma_\lambda(e_\beta)\sigma_\lambda(e_\gamma)\\
& \qquad -\frac{1}{12}\sum_{\alpha,\beta,\gamma}\omega(\widehat{e}_\beta,\omega(X,\widehat{e}_\gamma)\widehat{e}_\alpha+\omega(\widehat{e}_\alpha,\widehat{e}_\gamma)X)\sigma_\lambda(e_\alpha)\sigma_\lambda(e_\beta)\sigma_\lambda(e_\gamma)\\
={}& \frac{1}{3}\sum_\alpha\sigma_\lambda(e_\alpha)\sigma_\lambda(\Omega_\mu(B_\mu(X,\widehat{e}_\alpha))\\
& \qquad + \frac{1}{12}\sum_\alpha\Big(\sigma_\lambda(e_\alpha)\sigma_\lambda(\widehat{e}_\alpha)\sigma_\lambda(X) + \sigma_\lambda(e_\alpha)\sigma_\lambda(X)\sigma_\lambda(\widehat{e}_\alpha)\Big)\\
={}& \frac{2i\lambda}{3}\sum_\alpha\sigma_\lambda(e_\alpha)d\omega_{\met,\lambda}(B_\mu(X,\widehat{e}_\alpha))\\
& \qquad + \frac{1}{6}\sum_\alpha\sigma_\lambda(e_\alpha)\sigma_\lambda(\widehat{e}_\alpha)\sigma_\lambda(X) + \frac{1}{12}\sum_\alpha\sigma_\lambda(e_\alpha)\sigma_\lambda([X,\widehat{e}_\alpha]).
\end{align*}
Using the independence of the chosen basis we find
\begin{equation}
\sum_\alpha\sigma_\lambda(e_\alpha)\sigma_\lambda(\widehat{e}_\alpha) = \frac{1}{2}\sum_\alpha\big(\sigma_\lambda(e_\alpha)\sigma_\lambda(\widehat{e}_\alpha)-\sigma_\lambda(\widehat{e}_\alpha)\sigma_\lambda(e_\alpha)\big) = \frac{1}{2}\sum_\alpha\sigma_\lambda([e_\alpha,\widehat{e}_\alpha]) = \frac{i\lambda}{2}\dim V\label{eq:SumSigmaSquared}
\end{equation}
and $\sigma_\lambda([X,\widehat{e}_\alpha])=i\lambda\omega(X,\widehat{e}_\alpha)$. This shows
$$ \sigma_\lambda(\Omega_\Psi(X)) = \frac{2i\lambda}{3}\sum_\alpha\sigma_\lambda(e_\alpha)d\omega_{\met,\lambda}(B_\mu(X,\widehat{e}_\alpha)) + \frac{i\lambda}{12}(\dim V+1)\sigma_\lambda(X). $$
The second identity follows by expanding $B_\mu(X,\widehat{e}_\alpha)=\sum_i\kappa(B_\mu(X,\widehat{e}_\alpha),T_i')T_i$ and using $\kappa(B_\mu(x,y),T)=\frac{p}{2}\omega(Tx,y)$ (see Remark \ref{rem:SphVectorPositive}).
\end{proof}
\section{The Fourier transform of $\Omega_Q$}\label{sec:FTOmegaQ}
The Fourier transform of the conformally invariant differential operator $\Omega_Q$ is essentially the Casimir element in the restriction of the metaplectic representation of $\sp(V,\omega)$ to $\mathfrak{m}$.
\begin{corollary}\label{cor:FTofOmegaQ}
For every $\lambda\in\mathbb{R}^\times$ we have
$$ \sigma_\lambda(\Omega_Q) = \frac{p}{3}\lambda^2 d\omega_{\met,\lambda}(\Cas_\mathfrak{m}) - \frac{(\dim V)^2}{96}\lambda^2, $$
where $\Cas_\mathfrak{m}\in U(\mathfrak{m})$\index{Casm@$\Cas_\mathfrak{m}$} denotes the Casimir element of $\mathfrak{m}$ with respect to the Killing form $\kappa$.
\end{corollary}
\begin{proof}
Using Lemma \ref{lem:SymmetrizationsOfSymplecticCovariants} and we find
\begin{align*}
\sigma_\lambda(\Omega_Q) ={}& -\frac{1}{12}\sum_{\alpha,\beta,\gamma,\delta}\omega(\widehat{e}_\gamma,B_\mu(\widehat{e}_\alpha,\widehat{e}_\beta)\widehat{e}_\delta)\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta)\sigma_\lambda(X_\gamma)\sigma_\lambda(X_\delta)\\
& \qquad -\frac{1}{24}\sum_{\alpha,\beta,\gamma,\delta}\omega(\widehat{e}_\gamma,B_\tau(\widehat{e}_\alpha,\widehat{e}_\beta)\widehat{e}_\delta)\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta)\sigma_\lambda(X_\gamma)\sigma_\lambda(X_\delta)\\
={}& \frac{1}{12}\sum_{\alpha,\beta}\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta)\sigma_\lambda(\Omega_\mu(B_\mu(\widehat{e}_\alpha,\widehat{e}_\beta))\\
& \qquad +\frac{1}{24}\sum_{\alpha,\beta}\sigma_\lambda(e_\alpha)\sigma_\lambda(e_\beta)\sigma_\lambda(\widehat{e}_\beta)\sigma_\lambda(\widehat{e}_\alpha).
\end{align*}
With \eqref{eq:SumSigmaSquared} the latter sum becomes $-\frac{1}{4}\lambda^2(\dim V)^2$. For the first sum let $(T_i)$ be a basis of $\mathfrak{m}$ and $(T_i')$ its dual basis with respect to the Killing form $\kappa$ of $\mathfrak{g}$. Then
\begin{align*}
\sum_{\alpha,\beta}\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta)\sigma_\lambda(\Omega_\mu(B_\mu(\widehat{e}_\alpha,\widehat{e}_\beta)) &= \sum_i\sum_{\alpha,\beta}\kappa(B_\mu(\widehat{e}_\alpha,\widehat{e}_\beta),T_i')\sigma_\lambda(X_\alpha)\sigma_\lambda(X_\beta)\sigma_\lambda(\Omega_\mu(T_i))\\
&= -p\sum_i\sigma_\lambda(\Omega_\mu(T_i'))\sigma_\lambda(\Omega_\mu(T_i))\\
&= 4p\lambda^2\sum_i d\omega_{\met,\lambda}(T_i')d\omega_{\met,\lambda}(T_i)\\
&= 4p\lambda^2 d\omega_{\met,\lambda}(\Cas_\mathfrak{m}),
\end{align*}
where $\Cas_\mathfrak{m}=\sum_i T_i'T_i\in U(\mathfrak{m})$ is the Casimir element of $\mathfrak{m}$.
\end{proof}
\chapter{Analysis of the Fourier transform of $\Omega_\mu$}\label{ch:AnalysisFTofOmegaMu}
We study the subrepresentation $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ and its image in $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda\times\Lambda)$ under the Fourier transform in the case where $G$ is non-Hermitian. For this we first develop some more structure theory following \cite{SS}.
\section{The Lagrangian decomposition}
By Theorem~\ref{thm:CharacterizationHermitian} the group $G$ is non-Hermitian if and only if there exists $O\in V$\index{O@$O$} such that $Q(O)>0$. We renormalize $O$ such that $Q(O)=1$. Any such $O\in V$ has by \cite[Theorem 6.35]{SS} a Lagrangian decomposition
$$ O=A+B\index{A1@$A$}\index{B@$B$} $$
where $A,B\in Z=\mu^{-1}(0)$\index{Z@$Z$} and $\omega(A,B)=2$. This decomposition is unique and $A$ and $B$ are given by
$$ A = \tfrac{1}{2}(O-\Psi(O)), \qquad B = \tfrac{1}{2}(O+\Psi(O)). $$
Further, the tangent spaces
$$ \Lambda := T_AZ \qquad \mbox{and} \qquad \Lambda^* := T_BZ\index{1Lambda@$\Lambda$}\index{1LambdaStar@$\Lambda^*$}$$
are complementary Lagrangian subspaces. We use this particular Lagrangian decomposition for the Schr\"{o}dinger model of the representation $\sigma_\lambda$ of $\overline{N}$.
Note that we use the same letter $A$ for the element $A\in V$ and the one-dimensional subgroup $A=\exp(\mathbb{R} H)\subseteq G$. It should be clear from the context which object is meant.
\section{The bigrading}\label{sec:Bigrading}
Let
$$ H_\alpha = \tfrac{1}{2}(H+\mu(O))=\tfrac{1}{2}(H+2B_\mu(A,B)), \qquad H_\beta = \tfrac{1}{2}(H-\mu(O))=\tfrac{1}{2}(H-2B_\mu(A,B)).\index{H1alpha@$H_\alpha$}\index{H1beta@$H_\beta$} $$
Then $[H_\alpha,H_\beta]=0$ and $\ad(H_\alpha)$ and $\ad(H_\beta)$ are simultaneously diagonalizable. Write
$$ \mathfrak{g}_{(i,j)} := \{X\in\mathfrak{g}:\ad(H_\alpha)X=iX,\ad(H_\beta)X=jX\},\index{g3ij@$\mathfrak{g}_{(i,j)}$} $$
then we have the bigrading
$$ \mathfrak{g} = \bigoplus_{i,j}\mathfrak{g}_{(i,j)}. $$
More precisely
\begin{align*}
\mathfrak{g}_2 &= \mathfrak{g}_{(1,1)},\\
\mathfrak{g}_1 &= \mathfrak{g}_{(2,-1)}+\mathfrak{g}_{(1,0)}+\mathfrak{g}_{(0,1)}+\mathfrak{g}_{(-1,2)},\\
\mathfrak{g}_0 &= \mathfrak{g}_{(1,-1)}+\mathfrak{g}_{(0,0)}+\mathfrak{g}_{(-1,1)},\\
\mathfrak{g}_{-1} &= \mathfrak{g}_{(1,-2)}+\mathfrak{g}_{(0,-1)}+\mathfrak{g}_{(-1,0)}+\mathfrak{g}_{(-2,1)},\\
\mathfrak{g}_{-2} &= \mathfrak{g}_{(-1,-1)}.
\end{align*}
Note that $\Ad(w_0)\mathfrak{g}_{(i,j)}=\mathfrak{g}_{(-j,-i)}$, i.e. $w_0$ flips the star diagram along the axis $i+j=0$.
$$
\begin{xy}
\xymatrix{
& & & \mathfrak{g}_{(1,1)} \ar@{-}[ddl] \ar@{-}[ddr]\\ \\
\mathfrak{g}_{(2,-1)} \ar@{-}[ddr] \ar@{-}[rr] & & \mathfrak{g}_{(1,0)} \ar@{-}[ddl] \ar@{-}[ddr] \ar@{-}[rr] & & \mathfrak{g}_{(0,1)} \ar@{-}[ddl] \ar@{-}[ddr] \ar@{-}[rr] & & \mathfrak{g}_{(-1,2)} \ar@{-}[ddl]\\ \\
& \mathfrak{g}_{(1,-1)} \ar@{-}[ddr] \ar@{-}[ddl] \ar@{-}[rr] & & \mathfrak{g}_{(0,0)} \ar@{-}[ddr] \ar@{-}[ddl] \ar@{-}[rr] & & \mathfrak{g}_{(-1,1)} \ar@{-}[ddr] \ar@{-}[ddl]\\ \\
\mathfrak{g}_{(1,-2)} \ar@{-}[rr] & & \mathfrak{g}_{(0,-1)} \ar@{-}[ddr] \ar@{-}[rr] & & \mathfrak{g}_{(-1,0)} \ar@{-}[ddl] \ar@{-}[rr] & & \mathfrak{g}_{(-2,1)}\\ \\
& & & \mathfrak{g}_{(-1,-1)}
}
\end{xy}
$$
Here $\mathfrak{m}=\mathfrak{g}_{(1,-1)}\oplus(\mathfrak{m}\cap\mathfrak{g}_{(0,0)})\oplus\mathfrak{g}_{(-1,1)}$ and $\mathfrak{g}_{(0,0)}=\mathbb{R} H\oplus(\mathfrak{m}\cap\mathfrak{g}_{(0,0)})$. Further, $\mathfrak{m}\cap\mathfrak{g}_{(0,0)}=\mathbb{R} B_\mu(A,B)\oplus\mathfrak{m}^O$, where
$$ \mathfrak{m}^O=\{T\in\mathfrak{m}:[T,O]=0\}=\{T\in\mathfrak{m}:[T,A]=[T,B]=0\}.\index{m3O@$\mathfrak{m}^O$} $$
Further,
$$ \mathfrak{g}_{(1,-2)} = \mathbb{R} A \qquad \mbox{and} \qquad \mathfrak{g}_{(-2,1)} = \mathbb{R} B, $$
and the Lagrangians $\Lambda$ and $\Lambda^*$ are given by
$$ \Lambda = \mathbb{R} A+\mathfrak{g}_{(0,-1)}, \qquad \mbox{and} \qquad \Lambda^* = \mathbb{R} B+\mathfrak{g}_{(-1,0)}. $$
Note that $\omega(A,\mathfrak{g}_{(-1,0)})=0$ and $\omega(B,\mathfrak{g}_{(0,-1)})=0$. Moreover, the maps
$$ \mathfrak{g}_{(0,-1)}\to\mathfrak{g}_{(-1,1)}, \, X\mapsto B_\mu(X,B) \qquad \mbox{and} \qquad \mathfrak{g}_{(-1,0)}\to\mathfrak{g}_{(1,-1)}, \, X\mapsto B_\mu(A,X) $$
are $\mathfrak{g}_{(0,0)}$-equivariant isomorphisms. Note that $B_\mu(A,B)$ acts on $\mathfrak{g}_{-1}$ by
\begin{equation}
B_\mu(A,B) = \begin{cases}\frac{3}{2} & \mbox{on $\mathfrak{g}_{(1,-2)}$,}\\\frac{1}{2} & \mbox{on $\mathfrak{g}_{(0,-1)}$,}\\-\frac{1}{2} & \mbox{on $\mathfrak{g}_{(-1,0)}$,}\\-\frac{3}{2} & \mbox{on $\mathfrak{g}_{(-2,1)}$.}\end{cases}\label{eq:BmuABonG-1}
\end{equation}
Further, note that
$$ \mathfrak{g}_{(2,-1)} = \mathbb{R}\overline{A}, \qquad \mathfrak{g}_{(-1,2)} = \mathbb{R}\overline{B}, $$
and
$$ [\overline{A},B] = -2H_\alpha, \qquad [\overline{B},A] = 2H_\beta. $$
The choice of $O\in V$ further determines a subalgebra of $\mathfrak{g}$ isomorphic to $\sl(3,\mathbb{R})$ given by
$$ \mathfrak{g}_{(1,1)}+\mathfrak{g}_{(2,-1)}+\mathfrak{g}_{(-1,2)}+\mathbb{R} H_\alpha+\mathbb{R} H_\beta+\mathfrak{g}_{(-2,1)}+\mathfrak{g}_{(1,-2)}+\mathfrak{g}_{(-1,-1)}. $$
For $z\in\mathfrak{g}_{(0,-1)}$ we have $\mu(z)\in\mathfrak{g}_{(1,-1)}$ and hence $\Psi(z)\in\mathfrak{g}_{(1,-2)}=\mathbb{C} A$. Write
$$ \Psi(z) = n(z)A, \qquad z\in\mathfrak{g}_{(0,-1)}.\index{n2z@$n(z)$} $$
Then the function $n(z)$ is a polynomial of degree $3$ which vanishes identically if and only if $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ (see \cite[Proposition 7.9]{SS}). In all other cases,
$$ \mathcal{J}=\mathfrak{g}_{(0,-1)}\index{J3@$\mathcal{J}$} $$
carries the structure of a rank $3$ Jordan algebra with Jordan determinant $n(z)$. Note that $\Psi^{-1}(0)\cap\mathcal{J}$ resp. $\mu^{-1}(0)\cap\mathcal{J}$ is the subvariety of elements of rank $\leq2$ resp. $\leq1$. We write $\mathcal{J}^*=\mathfrak{g}_{(-1,0)}$\index{J3@$\mathcal{J}^*$} which can be identified with the dual of $\mathcal{J}$ using the symplectic form.
\begin{remark}
A slightly different and more natural point of view is to endow $(V^+,V^-)=(\mathcal{J},\mathcal{J}^*)$ with the structure of a \emph{Jordan pair}. This structure consists of trilinear maps
$$ \{\cdot,\cdot,\cdot\}_\pm:V^\pm\times V^\mp\times V^\pm\to V^\pm, $$
such that
\begin{enumerate}[(1)]
\item $\{u,v,w\}_\pm=\{w,v,u\}_\pm$ for all $u,w\in V^\pm$ and $v\in V^\mp$,
\item $\{x,y,\{u,v,w\}_\pm\}_\pm=\{\{x,y,u\}_\pm,v,w\}_\pm-\{u,\{v,x,y\}_\mp,w\}_\pm+\{u,v,\{x,y,w\}_\pm\}_\pm$ for all $x,u,w\in V^\pm$ and $y,v\in V^\mp$.
\end{enumerate}
If we define
$$ \{u,v,w\}_\pm := \pm B_\Psi(u,v,w), $$
property (1) follows immediately from the symmetry of $B_\Psi$ and with Lemma~\ref{lem:SymmetrizationsOfSymplecticCovariants} it is easy to see that property (2) is equivalent to the $\mathfrak{m}$-equivariance of $B_\mu$.
\end{remark}
In addition to $w_0$ we define the additional Weyl group elements
$$ w_1 = \exp\left(\frac{\pi}{2\sqrt{2}}(A-\overline{B})\right), \qquad w_2 = \exp\left(\frac{\pi}{2\sqrt{2}}(B+\overline{A})\right).\index{w21@$w_1$}\index{w22@$w_2$} $$
\begin{lemma}\label{lem:W1W2}
\begin{enumerate}[(1)]
\item\label{lem:W1W2-1} The elements $w_1$ and $w_2$ have the following mapping properties:
$$ \Ad(w_1)\mathfrak{g}_{(i,j)} = \mathfrak{g}_{(i+j,-j)}, \qquad \Ad(w_2)\mathfrak{g}_{(i,j)} = \mathfrak{g}_{(-i,i+j)}. $$
\item\label{lem:W1W2-2} For $v\in\mathcal{J}$, $w\in\mathcal{J}^*$ and $T\in\mathfrak{m}^O$ we have
\begin{align*}
\Ad(w_1)F &= -\frac{1}{\sqrt{2}}B, & \Ad(w_1)E &= \frac{1}{\sqrt{2}}\overline{A},\\
\Ad(w_1)A &= -\overline{B}, & \Ad(w_1)\overline{A} &= -\sqrt{2}E,\\
\Ad(w_1)v &= \sqrt{2}B_\mu(v,B), & \Ad(w_1)\overline{v} &= \overline{v},\\
\Ad(w_1)w &= w, & \Ad(w_1)\overline{w} &= \sqrt{2}B_\mu(A,w),\\
\Ad(w_1)B &= \sqrt{2}F, & \Ad(w_1)\overline{B} &= -A,\\
\Ad(w_1)B_\mu(v,B) &= -\frac{1}{\sqrt{2}}v, & \Ad(w_1)B_\mu(A,w) &= -\frac{1}{\sqrt{2}}\overline{w},\\
\Ad(w_1)H &= B_\mu(A,B)+\frac{1}{2}H, & \Ad(w_1)B_\mu(A,B) &= -\frac{1}{2}B_\mu(A,B)+\frac{3}{4}H,\\
\Ad(w_1)T &= T,
\end{align*}
and similar for $w_2$ by substituting $(A,B)\mapsto(B,-A)$.
\item\label{lem:W1W2-3} We have $w_1^2,w_2^2\in M$ with $\chi(w_1^2)=\chi(w_2^2)=-1$ and
\begin{equation*}
\begin{split}
\Ad(w_1^2)(aA+v+w+bB) &= aA-v+w-bB\\
\Ad(w_2^2)(aA+v+w+bB) &= -aA+v-w+bB
\end{split}
\qquad(a,b\in\mathbb{R},v\in\mathcal{J},w\in\mathcal{J}^*).
\end{equation*}
\item\label{lem:W1W2-4} The following relations hold:
\begin{align*}
w_0w_1w_0^{-1} &= w_2^{-1}, & w_1w_0w_1^{-1} &= w_2, & w_2w_0w_2^{-1} &= w_1^{-1}\\
w_0w_2w_0^{-1} &= w_1, & w_1w_2w_1^{-1} &= w_0^{-1}, & w_2w_1w_2^{-1} &= w_0.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{lem:W1W2-2} is an easy computation using the definitions in Section~\ref{sec:W0} as well as Lemma~\ref{lem:G1bracketG-1}. The formulas for $\Ad(w_1)$ and $\Ad(w_2)$ then imply \eqref{lem:W1W2-1} and \eqref{lem:W1W2-3}. Finally, \eqref{lem:W1W2-4} follows with the identity
$$ w\exp(X)w^{-1} = \exp(\Ad(w)X) $$
and the definitions in Section~\ref{sec:W0}.
\end{proof}
\section{Identities for the moment map $\mu$ in the bigrading}
We show several identities for the moment map $\mu$ and its symmetrization $B_\mu$ acting on different parts of the decomposition
$$ V = \mathbb{R} A\oplus\mathcal{J}\oplus\mathcal{J}^*\oplus\mathbb{R} B. $$
\begin{lemma}\label{lem:TraceOnG0-1}
Assume $\mathfrak{g}\not\simeq\sl(n,\mathbb{R}),\so(p,q)$. For $z\in\mathcal{J}$ and $w\in\mathcal{J}^*$ we have
$$ \tr(B_\mu(A,w)\circ B_\mu(z,B)|_\mathcal{J}) = \left(\frac{1}{2}+\frac{1}{6}\dim\mathcal{J}\right)\omega(z,w). $$
\end{lemma}
We remark that the formula also holds for $\mathfrak{g}=\sl(3,\mathbb{R})$ and $\so(4,4)$, but we do not need this.
\begin{proof}
First note that since $B_\mu$ is $\mathfrak{m}$-equivariant
$$ [B_\mu(A,w),B_\mu(z,B)] = B_\mu(B_\mu(A,w)z,B)+B_\mu(z,B_\mu(A,w)B). $$
By Lemma~\ref{lem:RewriteBmu} we have $B_\mu(A,w)z=\frac{1}{2}\omega(z,w)A$ and $B_\mu(A,w)B=-w$ and hence
$$ [B_\mu(A,w),B_\mu(z,B)] = \frac{1}{2}\omega(z,w)B_\mu(A,B)-B_\mu(z,w). $$
Choose a basis $(e_\alpha)$ of $\mathfrak{g}_{(0,-1)}$ and let $(\widehat{e}_\alpha)$ be the basis of $\mathfrak{g}_{(-1,0)}$ such that $\omega(e_\alpha,\widehat{e}_\beta)=\delta_{\alpha\beta}$. Then
\begin{align*}
& \tr(B_\mu(A,w)\circ B_\mu(z,B)|_{\mathfrak{g}_{(0,-1)}}) = \sum_\alpha \omega(B_\mu(A,w)B_\mu(z,B)e_\alpha,\widehat{e}_\alpha)\\
={}& \sum_\alpha \Big(\omega(B_\mu(z,B)B_\mu(A,w)e_\alpha,\widehat{e}_\alpha)+\frac{1}{2}\omega(z,w)\omega(B_\mu(A,B)e_\alpha,\widehat{e}_\alpha)-\omega(B_\mu(z,w)e_\alpha,\widehat{e}_\alpha)\Big).
\end{align*}
Again by Lemma~\ref{lem:RewriteBmu} we find that $B_\mu(A,w)e_\alpha=\frac{1}{2}\omega(e_\alpha,w)A$ and $B_\mu(z,B)A=z$, and further $B_\mu(A,B)e_\alpha=\frac{1}{2}e_\alpha$ so that
$$ \tr(B_\mu(A,w)\circ B_\mu(z,B)|_{\mathfrak{g}_{(0,-1)}}) = (\frac{1}{2}+\frac{1}{4}\dim\mathfrak{g}_{(0,-1)})\omega(z,w)-\tr(B_\mu(z,w)|_{\mathfrak{g}_{(0,-1)}}). $$
We have $B_\mu(z,w)\in\mathfrak{g}_{(0,0)}=\mathbb{R} H_\alpha\oplus\mathbb{R} H_\beta\oplus\mathfrak{m}^O$, where $\mathfrak{m}^O=\{T\in\mathfrak{m}:[T,O]=0\}$. Write $B_\mu(z,w)=aH_\alpha+bH_\beta+T$ with $T\in\mathfrak{m}^O$. Then $\tr(T|_{\mathfrak{g}_{(0,-1)}})=0$ (since $\mathfrak{g}\not\simeq\sl(n,\mathbb{R}),\so(p,q)$ and hence $\mathfrak{m}^O$ is semisimple) and $\tr(H_\alpha|_{\mathfrak{g}_{(0,-1)}})=0$, $\tr(H_\beta|_{\mathfrak{g}_{(0,-1)}})=-\dim\mathfrak{g}_{(0,-1)}$. We determine $a$ and $b$ which will complete the proof. Since $[T,A]=[T,B]=0$ we have
$$ B_\mu(z,w)A = (a-2b)A \qquad \mbox{and} \qquad B_\mu(z,w)B = (b-2a)B. $$
On the other hand, by Lemma~\ref{lem:RewriteBmu} we find
\begin{align*}
B_\mu(z,w)A &= \frac{1}{2}\omega(B_\mu(z,w)A,B)A = \frac{1}{2}\omega(B_\mu(A,B)z,w)A = \frac{1}{4}\omega(z,w)A,\\
B_\mu(z,w)B &= \frac{1}{2}\omega(A,B_\mu(z,w)B)B = \frac{1}{2}\omega(z,B_\mu(A,B)w)B = -\frac{1}{4}\omega(z,w)B.
\end{align*}
Thus, $a=-b=\frac{1}{12}\omega(z,w)$ and the claimed formula follows.
\end{proof}
\begin{lemma}\label{lem:MuSquared}
For $z\in\mathcal{J}$ we have
$$ \mu(z)^2B = -4n(z)z. $$
\end{lemma}
\begin{proof}
Using Lemma~\ref{lem:RewriteBmu} and $\mu(z)z=-3\Psi(z)=-3n(z)A$ we find
\begin{align*}
\mu(z)^2B &= \mu(z)B_\mu(z,z)B = \mu(z)B_\mu(z,B)z = [\mu(z),B_\mu(z,B)]z + B_\mu(z,B)\mu(z)z\\
&= [\mu(z),B_\mu(z,B)]z - 3n(z)B_\mu(z,B)A = [\mu(z),B_\mu(z,B)]z - 3n(z)z.
\end{align*}
Now by the $\mathfrak{m}$-equivariance of $B_\mu$:
\begin{align*}
[\mu(z),B_\mu(z,B)]z &= B_\mu(\mu(z)z,B)z + B_\mu(z,\mu(z)B)z\\
&= -3n(z)B_\mu(A,B)z + B_\mu(z,B_\mu(z,B)z)z\\
&= -\frac{3}{2}n(z)z + \frac{1}{2}[B_\mu(z,B),\mu(z)]z
\end{align*}
and hence $[\mu(z),B_\mu(z,B)]z=-n(z)z$ and the claim follows.
\end{proof}
\begin{lemma}
If $\mathfrak{g}$ is non-Hermitian and $\mathfrak{m}$ is simple then the number $\mathcal{C}(\mathfrak{m})$ in Lemma \ref{lem:BezoutianSum} is given by
$$ \mathcal{C}(\mathfrak{m}) = \frac{3}{2}+\frac{\dim\mathcal{J}}{6}.\index{Cmprime@$\mathcal{C}(\mathfrak{m}')$} $$
\end{lemma}
\begin{proof}
Let $(e_\alpha)$ be a basis of $\mathfrak{g}_{-1}$ and $(\widehat{e}_\alpha)$ its dual basis with respect to $\omega$, then by Lemma \ref{lem:BezoutianSum}
\begin{equation}
\sum_\alpha B_\mu(Te_\alpha,\widehat{e}_\alpha) = \mathcal{C}(\mathfrak{m})\cdot T \qquad \forall\,T\in\mathfrak{m}.\label{eq:CofM}
\end{equation}
We may choose $e_\alpha\in\{A,B\}\cup\mathfrak{g}_{(0,-1)}\cup\mathfrak{g}_{(-1,0)}$, then $\widehat{e}_\alpha\in\{-\frac{1}{2}A,\frac{1}{2}B\}\cup\mathfrak{g}_{(0,-1)}\cup\mathfrak{g}_{(-1,0)}$. Now put $T=B_\mu(A,B)\in\mathfrak{m}$, then the left hand side of \eqref{eq:CofM} becomes
$$ \frac{1}{2}B_\mu(TA,B) - \frac{1}{2}B_\mu(TB,A) + \sum_{e_\alpha\in\mathfrak{g}_{(0,-1)}} B_\mu(Te_\alpha,\widehat{e}_\alpha) + \sum_{e_\alpha\in\mathfrak{g}_{(-1,0)}} B_\mu(Te_\alpha,\widehat{e}_\alpha) $$
and by \eqref{eq:BmuABonG-1} this is
$$ \frac{3}{2}B_\mu(A,B) + \sum_{e_\alpha\in\mathfrak{g}_{(0,-1)}} B_\mu(e_\alpha,\widehat{e}_\alpha). $$
Hence
$$ \sum_{e_\alpha\in\mathfrak{g}_{(0,-1)}} B_\mu(e_\alpha,\widehat{e}_\alpha) = \left(\mathcal{C}(\mathfrak{m})-\frac{3}{2}\right) B_\mu(A,B). $$
We apply both sides to $A$ and find
$$ \sum_{e_\alpha\in\mathfrak{g}_{(0,-1)}}B_\mu(e_\alpha,\widehat{e}_\alpha)A = \left(\mathcal{C}(\mathfrak{m})-\frac{3}{2}\right)B_\mu(A,B)A = \frac{3}{2}\left(\mathcal{C}(\mathfrak{m})-\frac{3}{2}\right)A. $$
As in the proof of Lemma~\ref{lem:TraceOnG0-1} we have $B_\mu(e_\alpha,\widehat{e}_\alpha)A=\frac{1}{4}\omega(e_\alpha,\widehat{e}_\alpha)A=\frac{1}{4}A$ and the claim follows.
\end{proof}
\begin{lemma}\label{lem:DecompBigradingMuPsiQ}
Let $x=aA+z+w+bB$ with $a,b\in\mathbb{R}$, $z\in\mathfrak{g}_{(0,-1)}$ and $w\in\mathfrak{g}_{(-1,0)}$. Then
\begin{align*}
\mu(x) ={}& \underbrace{(\mu(z)+2aB_\mu(A,w))}_{\in\mathfrak{g}_{(1,-1)}} + \underbrace{(2abB_\mu(A,B)+2B_\mu(z,w))}_{\in\mathfrak{g}_{(0,0)}} + \underbrace{(\mu(w)+2bB_\mu(z,B))}_{\in\mathfrak{g}_{(-1,1)}},\\
\Psi(x) ={}& \underbrace{(-a^2b-\tfrac{1}{2}a\omega(z,w)+n(z))A}_{\in\mathfrak{g}_{(1,-2)}} + \underbrace{\Big[(-ab-\tfrac{1}{2}\omega(z,w))z-a\mu(w)A-\mu(z)w\Big]}_{\in\mathfrak{g}_{(0,-1)}}\\
&\hspace{1.5cm}+\underbrace{\Big[(ab+\tfrac{1}{2}\omega(z,w))w-b\mu(z)B-\mu(w)z\Big]}_{\in\mathfrak{g}_{(-1,0)}} + \underbrace{(ab^2+\tfrac{1}{2}b\omega(z,w)+n(w))B}_{\in\mathfrak{g}_{(-2,1)}},\\
Q(x) ={}& a^2b^2+ab\omega(z,w)-2bn(z)+2an(w) + \frac{1}{4}\omega(z,w)^2+\frac{1}{2}\omega(\mu(z)w,w),
\end{align*}
where $\Psi(z)=n(z)A$ and $\Psi(w)=n(w)B$. Moreover, we have $ \mu(z) = -B_\mu(A,\mu(z)B)$, $\mu(w) = B_\mu(\mu(w)A,B) $ and
$$ B_\mu(z,w) \in \tfrac{1}{6}\omega(z,w)B_\mu(A,B) + \mathfrak{m}^O. $$
\end{lemma}
\section{Invariant distribution vectors in $\mathcal{S}'(\Lambda)$}
Using Theorem \ref{thm:FTofOmegaMu} we compute $d\omega_{\met,\lambda}(T)$ explicitly for $T\in\mathfrak{m}$. In view of the decomposition $\Lambda=\mathbb{R} A+\mathcal{J}$ we write $x\in\Lambda$ as $x=aA+z$ with $a\in\mathbb{R}$ and $z\in\mathcal{J}$.
\begin{proposition}
For every $\lambda\in\mathbb{R}^\times$ the representation $d\omega_{\met,\lambda}$ of $\mathfrak{m}$ is given by
$$ d\omega_{\met,\lambda}(T) = \begin{cases}\frac{1}{2i\lambda}\sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta-\frac{1}{2}\omega(TB,z)\partial_A & T\in\mathfrak{g}_{(1,-1)},\\-\frac{1}{2}\omega(TA,B)a\partial_A-\partial_{Tz}-\frac{1}{2}\tr(T|_\Lambda) & T\in\mathfrak{g}_{(0,0)}\cap\mathfrak{m},\\-a\partial_{TA}+\frac{1}{2}i\lambda\omega(Tz,z) & T\in\mathfrak{g}_{(-1,1)}.\end{cases} $$
\end{proposition}
\begin{proof}
By Theorem \ref{thm:FTofOmegaMu}:
$$ d\omega_{\met,\lambda}(T) = \frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)d\sigma_\lambda(X_\alpha)d\sigma_\lambda(X_\beta). $$
Since this expression is independent of the choice of the basis $(e_\alpha)$ we may choose $e_\alpha\in\{A,B\}\cup\mathfrak{g}_{(0,-1)}\cup\mathfrak{g}_{(-1,0)}$. Then $\widehat{e}_\alpha\in\{-\tfrac{1}{2}A,\tfrac{1}{2}B\}\cup\mathfrak{g}_{(0,-1)}\cup\mathfrak{g}_{(-1,0)}$. Further, the representation $d\sigma_\lambda$ is in the coordinates $(a,z)$ given by
$$ d\sigma_\lambda(A) = -\partial_A, \qquad d\sigma_\lambda(X) = -\partial_X, \quad X\in\mathfrak{g}_{(0,-1)} $$
and
$$ d\sigma_\lambda(B) = -2i\lambda a, \qquad d\sigma_\lambda(X) = i\lambda\omega(X,z), \quad X\in\mathfrak{g}_{(-1,0)}. $$
First let $T\in\mathfrak{g}_{(1,-1)}$, then
\begin{align*}
d\omega_{\met,\lambda}(T) ={}& \frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)d\sigma_\lambda(X_\alpha)d\sigma_\lambda(X_\beta)\\
={}& \frac{1}{2i\lambda}\Big(-\frac{1}{2}i\lambda\sum_{e_\beta\in\mathfrak{g}_{(-1,0)}}\omega(TB,\widehat{e}_\beta)\omega(e_\beta,z)\partial_A+\sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta\\
& -\frac{1}{2}i\lambda\sum_{e_\alpha\in\mathfrak{g}_{(-1,0)}}\omega(T\widehat{e}_\alpha,B)\omega(e_\alpha,z)\partial_A\Big)\\
={}& \frac{1}{2i\lambda}\sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta-\frac{1}{2}\omega(TB,z)\partial_A.
\end{align*}
Next let $T\in\mathfrak{g}_{(0,0)}$, then $\ad(T)$ preserves each $\mathfrak{g}_{(i,j)}$ and we find
\begin{align*}
d\omega_{\met,\lambda}(T) ={}& \frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)d\sigma_\lambda(X_\alpha)d\sigma_\lambda(X_\beta)\\
={}& \frac{1}{2i\lambda}\Big(-\frac{1}{2}i\lambda\omega(TB,A)\partial_Aa-i\lambda\sum_{\substack{e_\alpha\in\mathfrak{g}_{(0,-1)}\\e_\beta\in\mathfrak{g}_{(-1,0)}}}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)\partial_\alpha\omega(e_\beta,z)\\
& -i\lambda\sum_{\substack{e_\alpha\in\mathfrak{g}_{(-1,0)}\\e_\beta\in\mathfrak{g}_{(0,-1)}}}\omega([T,\widehat{e}_\alpha],\widehat{e}_\beta)\omega(e_\alpha,z)\partial_\beta-\frac{1}{2}i\lambda\omega(TA,B)a\partial_A\Big)\\
={}& -\frac{1}{2}\Big(\omega(TA,B)a\partial_A+2\partial_{Tz}+\tr(T|_\Lambda)\Big).
\end{align*}
Finally let $T\in\mathfrak{g}_{(-1,1)}$, then
\begin{align*}
d\omega_{\met,\lambda}(T) ={}& \frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)d\sigma_\lambda(X_\alpha)d\sigma_\lambda(X_\beta)\\
={}& \frac{1}{2i\lambda}\Big(-i\lambda\sum_{e_\alpha\in\mathfrak{g}_{(0,-1)}}\omega(T\widehat{e}_\alpha,A)a\partial_\alpha-\lambda^2\sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(-1,0)}}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\omega(e_\alpha,z)\omega(e_\beta,z)\\
& -i\lambda\sum_{e_\beta\in\mathfrak{g}_{(0,-1)}}\omega(TA,\widehat{e}_\beta)a\partial_\beta\Big)\\
={}& -a\partial_{TA}+\frac{1}{2}i\lambda\omega(Tz,z).\qedhere
\end{align*}
\end{proof}
Recall that $\mathcal{J}=\mathfrak{g}_{(0,-1)}$ is a rank $3$ Jordan algebra with norm function $n(z)=\frac{1}{2}\omega(\Psi(z),B)$, except in the case $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ where $n(z)=0$. We note that for any $w\in\mathcal{J}$ we have by Lemma~\ref{lem:SymmetrizationsOfSymplecticCovariants}
\begin{equation}
\partial_wn(z) = \frac{1}{2}\omega(3B_\Psi(z,z,w),B) = -\frac{1}{2}\omega(\mu(z)w+\frac{1}{2}\tau(z)w,B) = -\frac{1}{2}\omega(\mu(z)w,B).\label{eq:DerivativeOfN}
\end{equation}
\begin{theorem}\label{thm:InvDistributionVector}
Assume $\mathfrak{g}\not\simeq\sl(n,\mathbb{R}),\so(p,q)$. For every $\lambda\in\mathbb{R}^\times$ the space $L^2(\Lambda)^{-\infty,\mathfrak{m}}=\mathcal{S}'(\Lambda)^{\mathfrak{m}}$ of $\mathfrak{m}$-invariant distribution vectors in $\omega_{\met,\lambda}$ is two-dimensional. More precisely, $\mathcal{S}'(\Lambda)^{\mathfrak{m}}=\mathbb{C}\xi_{\lambda,0}\oplus\mathbb{C}\xi_{\lambda,1}$, where
$$ \xi_{\lambda,\varepsilon}(a,z) = \sgn(a)^\varepsilon|a|^{s_\min}e^{-i\lambda\frac{n(z)}{a}}, \qquad s_\min = -\tfrac{1}{6}(\dim\Lambda+2),\varepsilon\in\mathbb{Z}/2\mathbb{Z}.\index{1oxilambdaepsilon@$\xi_{\lambda,\varepsilon}$}\index{smin@$s_\min$} $$
\end{theorem}
\begin{remark}
For $s_\min\leq-1$ the definition of $\xi_{\lambda,\varepsilon}(a,z)$ does not make sense as a function on $\Lambda$, but has to be interpreted as a distribution. In Appendix~\ref{app:MeromFamily} we show that $\xi_{\lambda,\varepsilon}$ is indeed the special value of a meromorphic family of distributions on $\Lambda$ at a regular point.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:InvDistributionVector}]
Let $\xi\in\mathcal{S}'(\Lambda)$ such that $d\omega_{\met,\lambda}(T)\xi=0$ for all $T\in\mathfrak{m}_0$. By definition of $\mathfrak{m}_0$ this is equivalent to $d\omega_{\met,\lambda}(T)\xi=0$ for $T\in\mathfrak{g}_{(1,-1)}$ and $T\in\mathfrak{g}_{(-1,1)}$. First, let $T=B_\mu(w,B)\in\mathfrak{g}_{(-1,1)}$, $w\in\mathfrak{g}_{(0,-1)}$, then by Lemma~\ref{lem:RewriteBmu}
$$ TA = B_\mu(B,w)A = B_\mu(B,A)w +\frac{1}{4}\omega(B,w)A-\frac{1}{4}\omega(B,A)w-\frac{1}{2}\omega(w,A)B = w $$
and for $z\in\mathcal{J}$:
$$ \omega(Tz,z) = \omega(B_\mu(w,B)z,z) = \omega(B_\mu(w,z)B,z) = \omega(B_\mu(z,w)z,B) = \omega(\mu(z)w,B). $$
Hence $d\omega_{\met,\lambda}(T)\xi=0$ implies
$$ a\partial_w\xi = \frac{1}{2}i\lambda\omega(\mu(z)w,B)\xi = -i\lambda(\partial_wn(z))\xi $$
which is equivalent to
$$ a\partial_w(\xi\cdot e^{i\lambda\frac{n(z)}{a}}) = 0. $$
Since $w\in\mathfrak{g}_{(0,-1)}$ was arbitrary we have
$$ \xi(a,z) = \xi_0(a)e^{-i\lambda\frac{n(z)}{a}} + \xi_1(a,z), $$
where $\xi_0(a)$ is independent of $z$ and $\xi_1(a,z)$ has support on $\{a=0\}$. Next let $T=B_\mu(A,w)\in\mathfrak{g}_{(1,-1)}$, $w\in\mathfrak{g}_{(-1,0)}$, then again by Lemma~\ref{lem:RewriteBmu}
$$ TB = B_\mu(A,w)B = B_\mu(A,B)w+\frac{1}{4}\omega(A,w)B-\frac{1}{4}\omega(A,B)w-\frac{1}{2}\omega(w,B)A = -w. $$
Therefore, $d\omega_{\met,\lambda}(T)\xi=0$ implies
$$ \sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta\xi = i\lambda\omega(z,w)\partial_A\xi. $$
Let us first assume $a\neq0$, then $\xi(a,z)=\xi_0(a)e^{-i\lambda\frac{n(z)}{a}}$ and hence
\begin{align*}
\partial_A\xi(a,z) &= \xi_0'(a)e^{i\lambda\frac{n(z)}{a}}+i\lambda a^{-2}n(z)\xi(a,z) \\
\partial_\alpha\xi(a,z) &= \frac{1}{2}i\lambda a^{-1}\omega(\mu(z)e_\alpha,B)\xi(a,z),\\
\partial_\alpha\partial_\beta\xi(a,z) &= i\lambda a^{-1}\omega(B_\mu(z,e_\beta)e_\alpha,B)\xi(a,z)-\frac{1}{4}\lambda^2a^{-2}\omega(\mu(z)e_\alpha,B)\omega(\mu(z)e_\beta,B)\xi(a,z).
\end{align*}
We sum the two terms for $\partial_\alpha\partial_\beta\xi$ over $\alpha$ and $\beta$ separately. For the first term we obtain, using Lemma~\ref{lem:RewriteBmu} and \ref{lem:TraceOnG0-1}:
\begin{align*}
& \sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\omega(B_\mu(z,e_\beta)e_\alpha,B) = \sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\beta,\widehat{e}_\alpha)\omega(B_\mu(z,e_\beta)B,e_\alpha)\\
={}& \sum_{e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(B_\mu(z,e_\beta)B,T\widehat{e}_\beta) = -\sum_{e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(TB_\mu(z,B)e_\beta,\widehat{e}_\beta)\\
={}& -\tr(B_\mu(A,w)\circ B_\mu(z,B)|_{\mathfrak{g}_{(0,-1)}}) = -\left(\frac{1}{2}+\frac{1}{6}\dim\mathfrak{g}_{(0,-1)}\right)\omega(z,w).
\end{align*}
For the second term we have
$$ \sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(z)e_\alpha,B)\omega(\mu(z)e_\beta,B) = -\omega(\mu(z)T\mu(z)B,B). $$
But $[T,\mu(z)]=2B_\mu(Tz,z)=0$ since $T\in\mathfrak{g}_{(1,-1)}$ implies $Tz\in\mathfrak{g}_{(1,-2)}$ and $B_\mu(Tz,z)\in\mathfrak{g}_{(2,-2)}=0$. Therefore
$$ \omega(\mu(z)T\mu(z)B,B) = -\omega(\mu(z)^2B,TB) = -4n(z)\omega(z,w) . $$
by Lemma~\ref{lem:MuSquared}. This implies
$$ \sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta\xi = -i\lambda a^{-1}\left(\frac{1}{2}+\frac{1}{6}\dim\mathfrak{g}_{(0,-1)}\right)\omega(z,w)\xi - \lambda^2a^{-2}n(z)\omega(z,w)\xi $$
and hence $d\omega_{\met,\lambda}(T)\xi=0$ becomes
$$ a\xi_0'(a) = s_\min \xi_0(a). $$
Hence $\xi_0(a)=c_1|a|^{s_\min}+c_2\sgn(a)|a|^{s_\min}$. Now assume $\xi_1(a,z)$ has support in $\{a=0\}$ and solves $d\omega_{\met,\lambda}(T)\xi_1=0$. Then there exists $m\in\mathbb{N}$ and $\xi_k\in\mathcal{S}'(\mathcal{J})$, $k=0,\ldots,m$, such that $\xi_1(a,z)=\sum_{k=0}^m\xi_k(z)\delta^{(k)}(a)$, where $\delta^{(k)}(a)$ denotes the $k$-th derivative of the Dirac distribution $\delta(a)$. The differential equation $d\omega_{\met,\lambda}(T)\xi_1=0$ then reads
$$ \sum_{k=0}^m\sum_{e_\alpha,e_\beta\in\mathfrak{g}_{(0,-1)}} \omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta\xi_k(z)\delta^{(k)}(a) = -i\lambda\omega(z,w)\sum_{k=0}^m\xi_k(z)\delta^{(k+1)}(a). $$
Comparing coefficients of $\delta_a^{(k)}$ we find inductively that $\xi_k=0$ so that $\xi_1=0$. This finishes the proof.
\end{proof}
\section{The Fourier transformed picture}\label{sec:FTpictureMinRep}
Assume $\mathfrak{g}\not\simeq\sl(n,\mathbb{R}),\so(p,q)$. Then $\mathfrak{m}$ is simple and $\mathcal{C}:=\mathcal{C}(\mathfrak{m})=\frac{3}{2}+\frac{\dim\mathcal{J}}{6}$. It follows from Theorem~\ref{thm:ConfInvOmegaMu} that for $\nu=2\,\mathcal{C}-\rho-2=-\frac{2}{3}\dim\mathcal{J}-1$ the space
$$ I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})} = \{u\in I(\zeta,\nu):\Omega_\mu(T)u=0\,\forall\,T\in\mathfrak{m}\} \subseteq I(\zeta,\nu)\index{IzetanuOmegamumprime@$I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m}')}$} $$
is a subrepresentation of $(\pi_{\zeta,\nu},I(\zeta,\nu))$. We study this subrepresentation in the Fourier transformed picture. Let $u\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$, then for every $\lambda\in\mathbb{R}^\times$:
$$ 0 = \sigma_\lambda(\Omega_\mu(T)u) = \sigma_\lambda(u)\circ d\sigma_\lambda(\Omega_\mu(T)) \qquad \forall\,T\in\mathfrak{m}. $$
\begin{corollary}\label{cor:FTImageOfSubrep}
Let $A:\mathcal{S}(\Lambda)\to\mathcal{S}'(\Lambda)$ be a continuous linear operator such that $A\circ d\sigma_\lambda(\Omega_\mu(T))=0$ for all $T\in\mathfrak{m}$. Then there exist $u_0,u_1\in\mathcal{S}'(\Lambda)$ such that
$$ A\varphi = \langle\varphi,\xi_{-\lambda,0}\rangle u_0 + \langle\varphi,\xi_{-\lambda,1}\rangle u_1 \qquad \forall\,\varphi\in\mathcal{S}(\Lambda). $$
\end{corollary}
\begin{proof}
Since $d\sigma_\lambda(\Omega_\mu(T))=2i\lambda d\omega_{\met,\lambda}(T)$ we have $A\circ d\omega_{\met,\lambda}(T)=0$ for all $T\in\mathfrak{m}$. Let $A^\top:\mathcal{S}(\Lambda)\to\mathcal{S}'(\Lambda)$ denote the transpose of $A$ and note that $d\omega_{\met,\lambda}(T)^\top=-d\omega_{\met,-\lambda}(T)$ for $T\in\mathfrak{m}$. Hence $d\omega_{\met,-\lambda}(T)\circ A^\top=0$ for all $T\in\mathfrak{m}$. This implies that the image of $A^\top$ is contained in $\mathcal{S}'(\Lambda)^\mathfrak{m}=\mathbb{C}\xi_{-\lambda,0}\oplus\mathbb{C}\xi_{-\lambda,1}$, so there exist unique $u_0,u_1\in\mathcal{S}'(\Lambda)$ such that $A^\top\varphi=\langle\varphi,u_0\rangle\xi_{-\lambda,0}+\langle\varphi,u_1\rangle\xi_{-\lambda,1}$ for $\varphi\in\mathcal{S}(\Lambda)$. Passing to the transposed operator once more shows the claimed formula for $A$.
\end{proof}
By Corollary~\ref{cor:FTImageOfSubrep} we can write
$$ \sigma_\lambda(u)\varphi(y) = \langle\varphi,\xi_{-\lambda,0}\rangle u_0(\lambda,y) + \langle\varphi,\xi_{-\lambda,1}\rangle u_1(\lambda,y) $$
for unique $u_0,u_1\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$\index{u0lambday@$u_0(\lambda,y)$}\index{u1lambday@$u_1(\lambda,y)$}. In terms of the integral kernel $\widehat{u}(\lambda,x,y)$ of $\sigma_\lambda(u)$ this can be written as
\begin{equation}
\widehat{u}(x,y,\lambda) = \xi_{-\lambda,0}(x)u_0(\lambda,y) + \xi_{-\lambda,1}(x)u_1(\lambda,y).\label{eq:IntegralFormulaIntertwiner}
\end{equation}
The map
$$ I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})} \to (\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))\oplus(\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)), \quad u\mapsto(u_0,u_1) $$
is injective and we denote its image by $J_\min$\index{J1min@$J_\min$}. Let $\rho_\min$\index{1rhomin@$\rho_\min$} denote the representation of $G$ on $J_\min$ which turns the map $u\mapsto(u_0,u_1)$ into an isomorphism of $G$-representations.
We remark that $J_\min$ could be zero, which is equivalent to $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}=\{0\}$. In order to show that there exists some representation $\zeta$ of $M$ such that $J_\min\neq\{0\}$ and to extend $\rho_\min$ to an irreducible unitary representation of $G$, we compute the Lie algebra action $d\rho_\min(\mathfrak{g})$. For this we first state the action of the identity component of $\overline{P}$ which is easily derived from Proposition~\ref{prop:ActionFTpicture}.
\begin{proposition}\label{prop:FTActionPbar}
The representation $\rho_\min$ is for $g\in\overline{P}_0$ given by
$$ \rho_\min(g)(u_0,u_1)=(\rho_{\min,0}(g)u_0,\rho_{\min,1}u_1), $$
where $\rho_{\min,\varepsilon}$ is the representation of $\overline{P}_0$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ given by
\begin{align*}
\rho_{\min,\varepsilon}(\overline{n}_{(z,t)})f(\lambda,y) &= e^{i\lambda t}e^{i\lambda(\omega(z'',y)+\frac{1}{2}\omega(z',z''))}f(\lambda,y-z') && \overline{n}_{(z,t)}\in\overline{N},\\
\rho_{\min,\varepsilon}(m)f(\lambda,y) &= (\id_{\mathbb{R}^\times}^*\otimes\,\omega_{\met,\lambda}(m))f(\lambda,y) && m\in M_0,\\
\rho_{\min,\varepsilon}(e^{tH})f(\lambda,y) &= e^{(\nu+s_\min-1)t}f(e^{-2t}\lambda,e^ty) && e^{tH}\in A.
\end{align*}
Since $\rho_{\min,\varepsilon}$ is independent of $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$, we abuse notation and write $\rho_\min=\rho_{\min,0}=\rho_{\min,1}$.
\end{proposition}
To state the Lie algebra action we write $y\in\Lambda$ as $y=aA+y'$.
\begin{proposition}\label{prop:drhomin}
The Lie algebra representation $d\rho_\min$\index{drhomin@$d\rho_\min$} of $\mathfrak{g}$ is given by
$$ d\rho_\min(X)(u_0,u_1) = (d\rho_{\min,0}(X)u_0,d\rho_{\min,1}(X)u_1)), $$
where $d\rho_{\min,\varepsilon}$ is the representation of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ given by
\begin{align*}
d\rho_{\min,\varepsilon}(F) ={}& i\lambda\\
d\rho_{\min,\varepsilon}(v) ={}& -\partial_v && (v\in\Lambda)\\
d\rho_{\min,\varepsilon}(w) ={}& -i\lambda\omega(y,w) && (w\in\Lambda^*)\\
d\rho_{\min,\varepsilon}(T) ={}& d\omega_{\met,\lambda}(T) && (T\in\mathfrak{m})\\
d\rho_{\min,\varepsilon}(H) ={}& \partial_y-2\lambda\partial_\lambda+2s_\min-\frac{\dim\Lambda}{2}-1\\
d\rho_{\min,\varepsilon}(\overline{A}) ={}& i\partial_\lambda\partial_A+\frac{s_\min-\frac{\dim\Lambda}{2}}{i\lambda}\partial_A-\frac{2}{\lambda^2}n(\partial')\\
d\rho_{\min,\varepsilon}(\overline{v}) ={}& i\partial_\lambda\partial_v+\frac{3s_\min+1}{i\lambda}\partial_v+\frac{1}{2}\omega(\mu(y')v,B)\partial_A\\
& -\frac{1}{i\lambda}\sum_{\alpha,\beta}\omega(B_\mu(y',v)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta} && (v\in\mathfrak{g}_{(0,-1)})\\
d\rho_{\min,\varepsilon}(\overline{w}) ={}& -\omega(y,w)\lambda\partial_\lambda+\omega(y,w)\partial_y+2s_\min\omega(y,w)\\
& +\partial_{\mu(y')w}-\frac{1}{2i\lambda}\omega(y,B)\sum_{\alpha,\beta}\omega(B_\mu(A,w)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta} && (w\in\mathfrak{g}_{(-1,0)})\\
d\rho_{\min,\varepsilon}(\overline{B}) ={}& -\omega(y,B)\lambda\partial_\lambda+\omega(y,B)\partial_y+s_\min\omega(y,B)+2i\lambda n(y')\\
d\rho_{\min,\varepsilon}(E) ={}& i\lambda\partial_\lambda^2-ia\partial_\lambda\partial_A-i\partial_\lambda\partial_{y'}-5is_\min\partial_\lambda-\frac{4s_\min+1}{i\lambda}a\partial_A+n(y')\partial_A\\
& +\frac{2}{\lambda^2}an(\partial')+\frac{s_\min}{i\lambda}\dim\Lambda-\frac{3s_\min+1}{i\lambda}\partial_{y'}\\
& +\frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(\mu(y')\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta},
\end{align*}
where $(e_\alpha)$ is a basis of $\mathcal{J}=\mathfrak{g}_{(0,-1)}$, $(\widehat{e}_\alpha)$ the corresponding dual basis of $\mathfrak{g}_{(-1,0)}$ with respect to the symplectic form, and
$$ n(\partial')=\frac{1}{2}\sum_{\alpha,\beta,\gamma}\omega(A,B_\Psi(\widehat{e}_\alpha,\widehat{e}_\beta,\widehat{e}_\gamma))\partial_{e_\alpha}\partial_{e_\beta}\partial_{e_\gamma}. $$\index{n2zz@$n(\partial')$}
Since $d\rho_{\min,\varepsilon}$ is independent of $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$, we abuse notation and write $d\rho_\min=d\rho_{\min,0}=d\rho_{\min,1}$.
\end{proposition}
\begin{proof}
The formulas for $\mathfrak{m}$, $\mathfrak{a}$ and $\overline{\mathfrak{n}}$ follow easily by differentiating the formulas in Proposition~\ref{prop:FTActionPbar}. We next compute $d\rho_\min(\overline{B})$. Writing $z=aA+v+w+bB$ with $a,b\in\mathbb{R}$ and $v\in\mathcal{J}$, $w\in\mathcal{J}^*$ we have by Lemma~\ref{lem:DecompBigradingMuPsiQ}:
\begin{multline*}
d\pi_{\zeta,\nu}(\overline{B}) = 2a^2\partial_A+2a\partial_v+\partial_{\mu(v)B}-(ab+\tfrac{1}{2}\omega(v,w)+t)\partial_B\\
+(at+n(v)-\tfrac{1}{2}a\omega(v,w)-a^2b)\partial_t+(\nu+\rho)a.
\end{multline*}
A careful application of Lemma~\ref{lem:FTMultDiff} yields
\begin{align}
\begin{split}
d\widehat{\pi}_{\zeta,\nu}(\overline{B}) ={}& -\omega(y,B)\lambda\partial_\lambda+(\omega(y,B)-\tfrac{1}{2}\omega(x,B))\omega(x,B)\partial_{A,x}+\omega(y-x,B)\partial_{x',x}\\
&+\omega(x,B)\partial_{y',x}+\tfrac{1}{2}\omega(y,B)^2\partial_{A,y}+\omega(y,B)\partial_{y',y}\\
&+4i\lambda n(x')+i\lambda\omega(\mu(x')y',B)+2i\lambda n(y')+\tfrac{\nu+\rho}{2}\omega(y-x,B)
\end{split}\label{eq:FTpiBbar}
\end{align}
where $\partial_{v,x}$ resp. $\partial_{w,y}$ means differentiation in the direction $v$ resp. $w$ with respect to the variable $x$ resp. $y$ of $\widehat{u}(\lambda,x,y)$, and $x\in\mathbb{R} A+x'$, $y\in\mathbb{R} A+y'$. In view of \eqref{eq:IntegralFormulaIntertwiner} we compute with $x=aA+x'$:
\begin{align*}
\lambda\partial_\lambda\xi_{-\lambda,\varepsilon}(x) &= \tfrac{i\lambda n(x')}{a}\xi_{-\lambda,\varepsilon}(a,x'),\\
\omega(x,B)\partial_{A,x}\xi_{-\lambda,\varepsilon}(x) &= 2a\partial_A\xi_{-\lambda,\varepsilon}(a,x') = 2s_\min\xi_{-\lambda,\varepsilon}(a,x')-\tfrac{2i\lambda n(x')}{a}\xi_{-\lambda,\varepsilon}(a,x'),\\
\partial_{x',x}\xi_{-\lambda,\varepsilon}(x) &= \tfrac{3i\lambda n(x')}{a}\xi_{-\lambda,\varepsilon}(a,x'),\\
\partial_{y',x}\xi_{-\lambda,\varepsilon}(x) &= -\frac{i\lambda}{2a}\omega(\mu(x')y',B)\xi_{-\lambda,\varepsilon}(a,x').
\end{align*}
A short computation using $\frac{\nu+\rho}{2}=-s_\min$ then shows
\begin{multline*}
\widehat{(d\pi_{\zeta,\nu}(\overline{B})u)}(\lambda,x,y) = \sum_{\varepsilon\in\mathbb{Z}/2\mathbb{Z}}\xi_{-\lambda,\varepsilon}(x)\Big[-\omega(y,B)\lambda\partial_\lambda+\tfrac{1}{2}\omega(y,B)^2\partial_A+\omega(y,B)\partial_{y'}\\
+s_\min\omega(y,B)+2i\lambda n(y')\Big]u_\varepsilon(\lambda,y).
\end{multline*}
This shows the claimed formula for $d\rho_{\min,\varepsilon}(\overline{B})$. Since $\mathfrak{g}$ is generated by $\mathfrak{m}$, $\mathfrak{a}$, $\overline{\mathfrak{n}}$ and $\overline{B}$, the remaining formulas can be obtained as commutators, more precisely we use that for $v\in\mathcal{J}$, $w\in\mathcal{J}^*$ and $T\in\mathfrak{g}_{(1,-1)}$ (so that $Tw\in\mathcal{J}^*$):
\begin{equation*}
[\overline{B},B_\mu(A,w)] = \overline{w}, \qquad [T,\overline{w}] = \overline{Tw}, \qquad [B_\mu(A,w),\overline{v}] = \tfrac{1}{2}\omega(v,w)\overline{A}, \qquad [\overline{B},\overline{A}]=2E.\qedhere
\end{equation*}
\end{proof}
\begin{remark}
The representation $d\rho_\min$ obviously extends to a representation of $\mathfrak{g}$ on the space $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, and we will use the same notation for this extension.
\end{remark}
\begin{remark}\label{rem:MotivationL2}
We give a formal heuristic argument why the representation $\rho_\min$ should extend to a unitary representation on $L^2(\mathbb{R}^\times\times\Lambda,|\lambda|^{\dim\Lambda-2s_\min}d\lambda)$. Note that several steps of the argument need a certain regularization to make sense. However, since we prove that $\rho_\min$ extends to a unitary representation by different means, we do not justify all steps.\\
By Proposition~\ref{prop:KnappSteinIntegralFormula} the invariant Hermitian form on $I(\zeta,\nu)$ is given by a regularization of
\begin{align*}
\langle u,\overline{\Delta^{\frac{\nu-\rho}{2}}*v}\rangle &= \int_{\mathbb{R}^\times} \tr\left(\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}}*v)^*\sigma_\lambda(u)\right) |\lambda|^{\dim\Lambda}\,d\lambda\\
&= \int_{\mathbb{R}^\times} \tr\left(\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})\sigma_\lambda(v)^*\sigma_\lambda(u)\right) |\lambda|^{\dim\Lambda}\,d\lambda,
\end{align*}
where $\Delta(z,t)=t^2-Q(z)$\index{1Deltazt@\par\indexspace$\Delta(z,t)$}. Assume $u,v\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ satisfy $\sigma_\lambda(u)\varphi=\langle\xi_{-\lambda,\varepsilon},\varphi\rangle u_\varepsilon(\lambda)$ and $\sigma_\lambda(v)^*\psi=\langle\overline{v_\varepsilon(\lambda)},\psi\rangle\overline{\xi_{-\lambda,\varepsilon}}$ and hence
$$ \sigma_\lambda(v)^*\sigma_\lambda(u)\varphi = \langle\xi_{-\lambda,\varepsilon},\varphi\rangle\langle u_\varepsilon(\lambda),\overline{v_\varepsilon(\lambda)}\rangle\overline{\xi_{-\lambda,\varepsilon}}, $$
so that
$$ \tr\left(\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})\sigma_\lambda(v)^*\sigma_\lambda(u)\right) = \langle u_\varepsilon(\lambda),\overline{v_\varepsilon(\lambda)}\rangle\langle\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})\overline{\xi_{-\lambda,\varepsilon}},\xi_{-\lambda,\varepsilon}\rangle. $$
Now, $\Delta(-z,-t)=\Delta(z,t)$ and $\sigma_\lambda(z,t)^\top=\sigma_{-\lambda}(-z,-t)$ so that
$$ \langle\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})\overline{\xi_{-\lambda,\varepsilon}},\xi_{-\lambda,\varepsilon}\rangle = \langle\sigma_{-\lambda}(\Delta^{\frac{\nu-\rho}{2}})\xi_{-\lambda,\varepsilon},\overline{\xi_{-\lambda,\varepsilon}}\rangle. $$
A short computation shows that
$$ \sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})=|\lambda|^{-\nu}\delta_{|\lambda|^{\frac{1}{2}}}\circ\sigma_{\sgn\lambda}(\Delta^{\frac{\nu-\rho}{2}})\circ\delta_{|\lambda|^{-\frac{1}{2}}} \qquad \mbox{and} \qquad \xi_{\lambda,\varepsilon} = |\lambda|^{-\frac{s_\min}{2}}\delta_{|\lambda|^{\frac{1}{2}}}\xi_{\sgn\lambda,\varepsilon}, $$
where $\delta_s\varphi(x)=\varphi(sx)$\index{1deltas@$\delta_s$} ($s>0$), and hence
\begin{align*}
\langle\sigma_{-\lambda}(\Delta^{\frac{\nu-\rho}{2}})\xi_{-\lambda,\varepsilon},\overline{\xi_{-\lambda,\varepsilon}}\rangle &= |\lambda|^{-\nu-s_\min}\langle\delta_{|\lambda|^{\frac{1}{2}}}\circ\sigma_{-\sgn\lambda}(\Delta^{\frac{\nu-\rho}{2}})\xi_{-\sgn\lambda,\varepsilon},\delta_{|\lambda|^{\frac{1}{2}}}\overline{\xi_{-\sgn\lambda,\varepsilon}}\rangle\\
&= |\lambda|^{-\nu-s_\min-\frac{1}{2}\dim\Lambda}\langle\sigma_{-\sgn\lambda}(\Delta^{\frac{\nu-\rho}{2}})\xi_{-\sgn\lambda,\varepsilon},\overline{\xi_{-\sgn\lambda,\varepsilon}}\rangle.
\end{align*}
Since $\Delta(z,t)$ is $M_0$-invariant, the operator $\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})$ is $\omega_{\met,\lambda}(M_0)$-invariant. The subspace of $M_0$-invariant vectors in $\omega_{\met,\lambda}$ is spanned $\xi_{\lambda,0}$ and $\xi_{\lambda,1}$, so $\sigma_\lambda(\Delta^{\frac{\nu-\rho}{2}})\xi_{\lambda,\varepsilon}$ is a linear combination of $\xi_{\lambda,0}$ and $\xi_{\lambda,1}$. This shows that
\begin{align*}
\langle u,\overline{\Delta^{\frac{\nu-\rho}{2}}*v}\rangle &= \const\cdot\int_{\mathbb{R}^\times} \langle u_\varepsilon(\lambda),\overline{v_\varepsilon(\lambda)}\rangle |\lambda|^{\frac{1}{2}\dim\Lambda-\nu-s_\min}\,d\lambda\\
&= \const\cdot\int_{\mathbb{R}^\times}\int_\Lambda u_\varepsilon(\lambda,x)\overline{v_\varepsilon(\lambda,x)} |\lambda|^{\dim\Lambda-2s_\min}\,dx\,d\lambda.
\end{align*}
Hence, the representation $\rho_\min$ is unitary on $L^2(\mathbb{R}^\times\times\Lambda,|\lambda|^{\dim\Lambda-2s_\min}\,d\lambda\,dx)$.
\end{remark}
We renormalize $\rho_\min$ to obtain a unitary representation on $L^2(\mathbb{R}^\times\times\Lambda)$. For $\delta\in\mathbb{Z}/2\mathbb{Z}$ let
\begin{equation}
\Phi_\delta:\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)\to\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda), \quad \Phi_\delta u(\lambda,x) = \sgn(\lambda)^\delta|\lambda|^{-s_\min}u(\lambda,\tfrac{x}{\lambda}),\label{eq:DefPhi}\index{1Phidelta@$\Phi_\delta$}
\end{equation}
then $\Phi_\delta$ restricts to an isometric isomorphism
$$ L^2(\mathbb{R}^\times\times\Lambda,|\lambda|^{\dim\Lambda-2s_\min}\,d\lambda\,dy) \to L^2(\mathbb{R}^\times\times\Lambda). $$
We define
$$ \pi_\min(g) := \Phi_\delta\circ\rho_\min(g)\circ\Phi_\delta^{-1},\index{1pimin@$\pi_\min$} $$
then
\begin{align*}
\Phi_\delta\circ\partial_v\circ\Phi_\delta^{-1} &= \lambda\partial_v, &
\Phi_\delta\circ\partial_\lambda\circ\Phi_\delta^{-1} &= (\partial_\lambda+\lambda^{-1}\partial_x+s\lambda^{-1}),\\
\Phi_\delta\circ\omega(y,w)\circ\Phi_\delta^{-1} &= \lambda^{-1}\omega(x,w), & \Phi_\delta\circ\lambda\circ\Phi_\delta^{-1} &= \lambda,
\end{align*}
and hence:
\begin{proposition}\label{prop:dpimin}
The representation $d\pi_\min$\index{dpimin@$d\pi_\min$} of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is given by
\begin{align*}
d\pi_\min(F) ={}& i\lambda\\
d\pi_\min(v) ={}& -\lambda\partial_v && (v\in\Lambda)\\
d\pi_\min(w) ={}& -i\omega(x,w) && (w\in\Lambda^*)\\
d\pi_\min(T) ={}& -\frac{i\lambda}{2}\sum_{\alpha,\beta}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta-\frac{1}{2}\omega(TB,x')\partial_A && (T\in\mathfrak{g}_{(1,-1)})\\
d\pi_\min(T) ={}& -\partial_{Tx}-\frac{1}{2}\tr(T|_\Lambda) && (T\in\mathfrak{g}_{(0,0)}\cap\mathfrak{m})\\
d\pi_\min(T) ={}& -a\partial_{TA}-\frac{1}{2i\lambda}\omega(Tx',x') && (T\in\mathfrak{g}_{(-1,1)})\\
d\pi_\min(H) ={}& -\partial_x-2\lambda\partial_\lambda-\frac{\dim\Lambda+2}{2}\\
d\pi_\min(\overline{A}) ={}& i\lambda\partial_\lambda\partial_A+i\partial_x\partial_A+i\frac{\dim\Lambda+2}{2}\partial_A-2\lambda n(\partial')\\
d\pi_\min(\overline{v}) ={}& i\lambda\partial_\lambda\partial_v+i\partial_x\partial_v-2is_\min\partial_v+\frac{1}{2}\lambda^{-1}\omega(\mu(x')v,B)\partial_A\\
& -\frac{1}{i}\sum_{\alpha,\beta}\omega(B_\mu(x',v)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta} && (v\in\mathfrak{g}_{(0,-1)})\\
d\pi_\min(\overline{w}) ={}& -\omega(x,w)\partial_\lambda+s_\min\lambda^{-1}\omega(x,w)\\
& +\lambda^{-1}\partial_{\mu(x')w}-\frac{1}{2i}\omega(x,B)\sum_{\alpha,\beta}\omega(B_\mu(A,w)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta} && (w\in\mathfrak{g}_{(-1,0)})\\
d\pi_\min(\overline{B}) ={}& -\omega(x,B)\partial_\lambda+2i\lambda^{-2}n(x')\\
d\pi_\min(E) ={}& i\lambda\partial_\lambda^2 + i\partial_\lambda\partial_x-3is_\min\partial_\lambda+\lambda^{-2}n(x')\partial_A+2an(\partial')\\
& -\frac{is_\min}{3\lambda}(\dim\Lambda-1)+\frac{s_\min}{i\lambda}\partial_{x'} +\frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(\mu(x')\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}.
\end{align*}\index{n2zz@$n(\partial')$}
\end{proposition}
\begin{proposition}\label{prop:AnnihilatorJosephIdeal}
The annihilator of $d\pi_\min$ in $U(\mathfrak{g}_\mathbb{C})$ is a completely prime ideal whose associated variety is equal to the closure of the minimal nilpotent coadjoint orbit. In particular, for $\mathfrak{g}_\mathbb{C}$ not of type $A$ the annihilator is the Joseph ideal.
\end{proposition}
\begin{proof}
It is easily verified that $d\pi_\min$ maps the generators of Lemma \ref{lem:GenJosephIdeal} to zero. Therefore, the annihilator ideal of $d\pi_\min$ has associated variety contained in the closure of the minimal nilpotent coadjoint orbit. Since this orbit is minimal, the associated variety is actually equal to the orbit's closure. By the same argument as in \cite[Theorem 2.18]{HKM14} the annihilator ideal is completely prime since it is given by the kernel of an algebra homomorphism into the ring of regular differential operators on the irreducible variety $\mathbb{R}^\times\times\Lambda$ which does not have zero divisors. The second claim now follows from the uniqueness result for the Joseph ideal \cite[Theorem 3.1]{GS04}.
\end{proof}
\section{The case $\mathfrak{g}=\sl(n,\mathbb{R})$}\label{sec:FTpictureMinRepSLn}
For $\mathfrak{g}=\sl(n,\mathbb{R})$ the previous arguments cannot be applied in the same way since here $\mathfrak{m}$ is not simple and the value $\nu\in\mathfrak{a}_\mathbb{C}^*$ for which $\Omega_\mu(\mathfrak{m}')$ is conformally invariant is different for the two factors $\mathfrak{m}'$ of $\mathfrak{m}$. We discuss how to use the first order system $\Omega_\omega$ instead to obtain a subrepresentation of $I(\zeta,\nu)$ for some $\nu$ which has a Fourier transformed picture similar to the other cases.
The subalgebra $\mathfrak{m}$ decomposes into the direct sum of two ideals
$$ \mathfrak{m} = \mathfrak{m}_0\oplus\mathfrak{m}_1 $$
with $\mathfrak{m}_0=\mathbb{R} T_0$\index{T0@$T_0$} and $\mathfrak{m}_1\simeq\sl(n-2,\mathbb{R})$. We can normalize $T_0$ such that it has eigenvalues $\pm1$ on $V$. Then the eigenspaces $\Lambda=\ker(T_0-\id_V)$ and $\Lambda^*=\ker(T_0+\id_V)$ are dual Lagrangian subspaces with $V=\Lambda\oplus\Lambda^*$. Note that both $\Lambda$ and $\Lambda^*$ are invariant under $\mathfrak{m}_1$. The following lemma is easily verified using the explicit realization of $\mathfrak{g}$ given in Appendix \ref{app:SLn}:
\begin{lemma}\label{lem:SLnIdentities}
\begin{enumerate}[(1)]
\item\label{lem:SLnIdentities1} For $v\in\Lambda$ and $w\in\Lambda^*$ we have $\mu(v)=\mu(w)=0$.
\item\label{lem:SLnIdentities3} $B_\mu(x,y)\equiv\frac{n}{4(n-2)}\omega(T_0x,y)T_0\mod\mathfrak{m}_1$ for all $x,y\in V$.
\end{enumerate}
\end{lemma}
For $r\in\mathbb{C}$ let $\zeta_r$\index{1fzetar@$\zeta_r$} denote a character of $M$ for which $d\zeta_r(\mathfrak{m}_1)=0$ and $d\zeta_r(T_0)=\frac{n-2}{2}+\frac{n-2}{n}r$.
\begin{theorem}
For any $r\in\mathbb{C}$ and $\zeta=\zeta_r$, the system of differential operators $\Omega_\omega(v)$ ($v\in\Lambda$) is conformally invariant for $\pi_{\zeta,\nu}$ with $\nu+\rho=\frac{n}{2}+r$.
\end{theorem}
\begin{proof}
Since $[\mathfrak{m},\Lambda]\subseteq\Lambda$, Theorem \ref{thm:ConfInvOmegaOmega} implies that $\Omega_\omega(v)$ ($v\in\Lambda$) is conformally invariant if and only if
$$ \frac{\nu+\rho}{2}\omega(x,v)+2d\zeta(B_\mu(x,v)) = 0 $$
for all $v\in\Lambda$. This follows from Lemma \ref{lem:SLnIdentities} \eqref{lem:SLnIdentities3}.
\end{proof}
\begin{remark}
$\Omega_\omega(v)u=0$ for all $v\in\Lambda$ implies $\Omega_\mu(T)u=0$ for all $T\in\mathfrak{m}_1$. In fact, since $T\Lambda\subseteq\Lambda$ and $T\Lambda^*\subseteq\Lambda^*$ we have
\begin{align*}
\Omega_\mu(T) &= \sum_{e_\alpha\in\Lambda,e_\beta\in\Lambda^*}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)(X_\alpha X_\beta+X_\beta X_\alpha)\\
&= \sum_{e_\alpha\in\Lambda,e_\beta\in\Lambda^*}\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)([X_\alpha,X_\beta]+2X_\beta X_\alpha)
\end{align*}
for a basis $(e_\alpha)$ of $V$ with $e_\alpha\in\Lambda\cup\Lambda^*$. If $\Omega_\omega(v)u=0$ for all $v\in\Lambda$, then $X_\alpha u=0$ for $e_\alpha\in\Lambda$. Further, $[X_\alpha,X_\beta]=\omega(e_\alpha,e_\beta)\partial_t$ so that $\sum\omega(T\widehat{e}_\alpha,\widehat{e}_\beta)[X_\alpha,X_\beta]=\tr(T|_\Lambda)\partial_t$ which vanishes for $T\in\mathfrak{m}_1\simeq\sl(n-2,\mathbb{R})$.
\end{remark}
By \eqref{eq:FTofOmegaOmega} the Fourier transform of $\Omega_\omega(v)$ is given by composition with $\sigma_\lambda(v)$. In terms of the distribution kernel $\widehat{u}(\lambda,x,y)$ of $\sigma_\lambda(u)$ this means
$$ \widehat{\Omega_\omega(v)u}(\lambda,x,y) = \sigma_{-\lambda}(v)_x\widehat{u}(\lambda,x,y). $$
This implies that for every $u\in I(\zeta_r,\nu)^{\Omega_\omega(\Lambda)}$, the distribution $\widehat{u}(\lambda,x,y)$ is in the $x$-variable a distribution vector in $L^2(\Lambda)^{-\infty}=\mathcal{S}'(\Lambda)$ which is invariant under $\sigma_{-\lambda}(v)=-\partial_v$ for all $v\in\Lambda$. These are obviously only the constant functions:
\begin{proposition}
For every $\lambda\in\mathbb{R}^\times$ the space $L^2(\Lambda)^{-\infty,\Lambda}=\mathcal{S}'(\Lambda)^{\Lambda}$ of $\Lambda$-invariant distribution vectors in $\sigma_\lambda$ is one-dimensional and spanned by the constant function
$$ \xi_\lambda(x) = 1.\index{1oxilambda@$\xi_\lambda$} $$
\end{proposition}
It follows that for $u\in I(\zeta_r,\nu)^{\Omega_\omega(\Lambda)}$ we can write
$$ \widehat{u}(\lambda,x,y) = \xi_{-\lambda}(x)u_0(\lambda,y) = u_0(\lambda,y)\index{u0lambday@$u_0(\lambda,y)$} $$
for some $u_0\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$. Let $J_{\min,r}\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$\index{J1minr@$J_{\min,r}$} denote the image of the map
$$ I(\zeta_r,\nu)^{\Omega_\omega(\Lambda)} \to \mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda), \quad u\mapsto u_0 $$
and write $\rho_{\min,r}$\index{1rhominr@$\rho_{\min,r}$} for the representation of $G$ on $J_{\min,r}$ which makes this map $G$-equivariant.
\begin{proposition}\label{prop:LAactionMinRepSLn}
The representation $d\rho_{\min,r}$\index{drhominr@$d\rho_{\min,r}$} of $\mathfrak{g}$ on $J_{\min,r}\subseteq\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is given by
\begin{align*}
d\rho_{\min,r}(F) ={}& i\lambda\\
d\rho_{\min,r}(v) ={}& -\partial_v && (v\in\Lambda)\\
d\rho_{\min,r}(w) ={}& -i\lambda\omega(y,w) && (w\in\Lambda^*)\\
d\rho_{\min,r}(T) ={}& -\partial_{Ty}-\tfrac{n-2r}{2n}\tr(T|_\Lambda) && (T\in\mathfrak{m})\\
d\rho_{\min,r}(H) ={}& \partial_y-2\lambda\partial_\lambda-\tfrac{n-2r}{2}\\
d\rho_{\min,r}(\overline{v}) ={}& i(\partial_\lambda+\tfrac{n-2r-2}{2\lambda})\partial_v\\
d\rho_{\min,r}(\overline{w}) ={}& \omega(y,w)(\partial_y-\lambda\partial_\lambda),\\
d\rho_{\min,r}(E) ={}& i(\lambda\partial_\lambda^2-\partial_\lambda\partial_y-\tfrac{n-2r-2}{2\lambda}\partial_y+\tfrac{n-2r}{2}\partial_\lambda).
\end{align*}
\end{proposition}
\begin{proof}
We proceed as in the proof of Proposition~\ref{prop:drhomin}. The formulas for $\mathfrak{m}$, $\mathfrak{a}$ and $\overline{\mathfrak{n}}$ follow easily from Proposition~\ref{prop:ActionFTpicture}, and for $w\in\Lambda^*$ we find, using Lemma~\ref{lem:FTMultDiff} and Lemma~\ref{lem:SLnIdentities}
\begin{multline*}
d\widehat{\pi}_{\zeta,\nu}(\overline{w}) = \omega(x,w)\left[\partial_{y-x,x}-\frac{\nu+\rho}{2}+\frac{n}{2(n-2)}d\zeta(T_0)\right]\\
+\omega(y,w)\left[\partial_{x,x}+\partial_{y,y}-\lambda\partial_\lambda+\frac{\nu+\rho}{2}-\frac{n}{2(n-2)}d\zeta(T_0)\right].
\end{multline*}
Since $\nu+\rho=\frac{n}{n-2}d\zeta_r(T_0)$ and $\partial_{v,x}\xi_\lambda(x)=0$ for all $v\in\Lambda$ it follows that for $u(\lambda,x,y)=\xi_{-\lambda}(x)u_0(\lambda,y)$:
\begin{equation*}
d\widehat{\pi}_{\zeta,\nu}(\overline{w})u(\lambda,x,y) = \xi_{-\lambda}(x)\cdot\omega(y,w)(\partial_y-\lambda\partial_\lambda)u_0(\lambda,y).
\end{equation*}
The formula for $d\rho_{\min,r}(\overline{v})$ is obtained by a similar computation, and for $d\rho_{\min,r}(E)$ we use that $[\overline{v},\overline{w}]=-\omega(v,w)E$.
\end{proof}
The change of coordinates $x=\lambda y$ finally yields a representation $d\pi_{\min,r}$\index{dpiminr@$d\pi_{\min,r}$} of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ given by
\begin{align*}
d\pi_{\min,r}(F) ={}& i\lambda\\
d\pi_{\min,r}(v) ={}& -\lambda\partial_v && (v\in\Lambda)\\
d\pi_{\min,r}(w) ={}& -i\omega(x,w) && (w\in\Lambda^*)\\
d\pi_{\min,r}(T) ={}& -\partial_{Tx}-\tfrac{n-2r}{2n}\tr(T|_\Lambda) && (T\in\mathfrak{m})\\
d\pi_{\min,r}(H) ={}& -\partial_x-2\lambda\partial_\lambda-\tfrac{n-2r}{2}\\
d\pi_{\min,r}(\overline{v}) ={}& i(\lambda\partial_\lambda+\partial_x+\tfrac{n-2r}{2})\partial_v\\
d\pi_{\min,r}(\overline{w}) ={}& -\omega(x,w)\partial_\lambda,\\
d\pi_{\min,r}(E) ={}& i(\lambda\partial_\lambda+\partial_x+\tfrac{n-2r}{2})\partial_\lambda.
\end{align*}
\begin{remark}
It can be shown that for $\zeta=\zeta_r$, $r\in\mathbb{C}$, and $\nu+\rho=n-2$ the second order differential operator
$$ \Omega_\mu^\zeta(T_0) = \Omega_\mu(T_0)+\frac{n}{n-2}d\zeta(T_0)\partial_t $$
is conformally invariant for $\pi_{\zeta,\nu}$. For $n>3$ the single equation $\Omega_\mu^\zeta(T_0)u=0$ is not sufficient to give a small representation similar to the previous cases; only for $n=3$ this is the case, since here $\mathfrak{m}=\mathfrak{m}_0=\mathbb{R} T_0$. In fact, for $n=3$ the same arguments as before identify $I(\zeta,\nu)^{\Omega_\mu^\zeta(T_0)}$ with a subspace of $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ and the corresponding Lie algebra action agrees with the one obtained in Proposition~\ref{prop:LAactionMinRepSLn}. This is due to the fact that for $n=3$ the parameter families $(\nu,r)=(-\frac{1}{2}+r,r)$ and $(\nu,r)=(-1,r)$ are related by the Weyl group element
$$ w(\diag(H_1,H_2,H_3)) = \diag(H_1,H_3,H_2). $$
It is likely that the corresponding standard intertwining operator identifies the two subrepresentations $I(\zeta,\nu)^{\Omega_\omega(\Lambda)}$ and $I(\zeta,\nu)^{\Omega_\mu^\zeta(T_0)}$.
\end{remark}
\section{The case $\mathfrak{g}=\so(p,q)$}\label{sec:FTpictureMinRepSOpq}
We also treat the case $\mathfrak{g}=\so(p,q)$ separately since here $\mathfrak{m}$ is not simple either and the value $\nu\in\mathfrak{a}_\mathbb{C}^*$ for which $\Omega_\mu(\mathfrak{m}')$ is conformally invariant is different for the two factors $\mathfrak{m}'$ of $\mathfrak{m}$. Instead, we combine variations of the first order system $\Omega_\omega$ and the second order system $\Omega_\mu$ to the case of vector-valued principal series in order to obtain a subrepresentation of $I(\zeta,\nu)$ which has a Fourier transformed picture similar to the other cases.
For $\mathfrak{g}=\so(p,q)$ the lack of simplicity of $\mathfrak{m}$ stems from the fact that $\mathcal{J}=\mathfrak{g}_{(0,-1)}$ is not a simple Jordan algebra but the sum of two simple Jordan algebras, the one-dimensional Jordan algebra which is of rank one and a $(p+q-6)$-dimensional Jordan algebra of rank two. Write $\mathcal{J}=\mathcal{J}_0\oplus\overline{\mathcal{J}}$\index{J3@$\mathcal{J}$} with $\mathcal{J}_0=\mathbb{R} P$\index{J30@$\mathcal{J}_0$}\index{P1@$P$} and $\overline{\mathcal{J}}\simeq\mathbb{R}^{p-3,q-3}$\index{J3@$\overline{\mathcal{J}}$}\index{Rpq@$\mathbb{R}^{p-3,q-3}$} and similarly $\mathcal{J}^*=\mathcal{J}_0^*\oplus\overline{\mathcal{J}}^*$\index{J3@$\mathcal{J}^*$}\index{J30star@$\mathcal{J}^*_0$}\index{J3@$\overline{\mathcal{J}}^*$} with $\mathcal{J}_0^*=\mathbb{R} Q$\index{Q@$Q$} such that $\omega(P,Q)=1$ and $\omega(\mathcal{J}_0,\overline{\mathcal{J}}^*)=0=\omega(\overline{\mathcal{J}},\mathcal{J}_0^*)$. We decompose $v\in\mathcal{J}$ into $v=v_0+\overline{v}$ with $v_0\in\mathcal{J}_0$ and $\overline{v}\in\overline{\mathcal{J}}$ and similar for $w\in\mathcal{J}^*$.
Note that we use the same letter $P$ for the element $P\in\mathcal{J}$ and the parabolic subgroup $P=MAN\subseteq G$. It should be clear from the context which object is meant.
The following statement is the analog of Lemma~\ref{lem:TraceOnG0-1}:
\begin{lemma}\label{lem:SOpqTrace}
For $v\in\mathcal{J}$ and $w\in\mathcal{J}^*$ we have
$$ \tr(B_\mu(A,w)\circ B_\mu(v,B)|_\mathcal{J}) = \frac{p+q-6}{2}\omega(v_0,w_0)+\omega(\overline{v},\overline{w}). $$
\end{lemma}
\begin{proof}
This is a straightforward computation using Appendix~\ref{app:SOpq}.
\end{proof}
According to the decomposition $\mathcal{J}=\mathcal{J}_0\oplus\overline{\mathcal{J}}$ the Lie algebra $\mathfrak{m}$ splits into
$$ \mathfrak{m}=\mathfrak{m}_0\oplus\overline{\mathfrak{m}} \qquad \mbox{with} \qquad \mathfrak{m}_0\simeq\sl(2,\mathbb{R}) \quad \mbox{and} \quad \overline{\mathfrak{m}}\simeq\so(p-2,q-2).\index{m30@$\mathfrak{m}_0$}\index{m3@$\overline{\mathfrak{m}}$} $$
We collect a few more identities related to the bigrading, all which can be verified by direct computations using the explicit realization of $\mathfrak{g}$ given in Appendix~\ref{app:SOpq}.
\begin{lemma}\label{lem:SOpqIdentities}
\begin{enumerate}[(1)]
\item $\mathfrak{m}_0\simeq\sl(2,\mathbb{R})$ is spanned by the $\sl(2)$-triple
$$ (e,f,h)=(\sqrt{2}B_\mu(A,Q),\sqrt{2}B_\mu(P,B),B_\mu(A,B)-2B_\mu(P,Q)).\index{e@$e$}\index{f@$f$}\index{h3@$h$} $$
\item $B_\mu(A,Q):\overline{\mathcal{J}}\to\overline{\mathcal{J}}^*$ and $B_\mu(P,B):\overline{\mathcal{J}}^*\to\overline{\mathcal{J}}$ are isomorphisms satisfying
$$ B_\mu(A,Q)\circ B_\mu(P,B)|_{\overline{\mathcal{J}}^*}=\frac{1}{2}\id_{\overline{\mathcal{J}}^*} \qquad \mbox{and} \qquad B_\mu(P,B)\circ B_\mu(A,Q)|_{\overline{\mathcal{J}}}=\frac{1}{2}\id_{\overline{\mathcal{J}}}. $$
\item $B_\mu(v,B)\in\overline{\mathfrak{m}}$ if and only if $v\in\overline{\mathcal{J}}$.
\item $\mu(P)=\mu(Q)=0$ and $\mu(\overline{\mathcal{J}})\in\mathbb{R} B_\mu(A,Q)$
\item For $v\in\mathcal{J}$:
\begin{align*}
\mu(v)Q &= -\omega(v_0,Q)\overline{v},\\
\mu(v)w &= -\omega(\overline{v},w)\omega(v,Q)P+\omega(\mu(v)P,B)B_\mu(A,Q)w && (w\in\overline{\mathcal{J}}^*),\\
\mu(v)B &= -\omega(\mu(\overline{v})P,B)Q+2\omega(v_0,Q)B_\mu(P,B)\overline{v}.
\end{align*}
\item $B_\mu(P,Q)$ acts on $V$ as follows:
\begin{align*}
B_\mu(P,Q)A &= \frac{1}{4}A, & B_\mu(P,Q)P &= \frac{3}{4}P, & B_\mu(P,Q)v &= -\frac{1}{4}v &&(v\in\overline{\mathcal{J}}),\\
B_\mu(P,Q)B &= -\frac{1}{4}B, & B_\mu(P,Q)Q &= -\frac{3}{4}Q, & B_\mu(P,Q)w &= \frac{1}{4}w &&(v\in\overline{\mathcal{J}}^*).
\end{align*}
\end{enumerate}
\end{lemma}
Let $\zeta$ be a representation of $M$ such that $d\zeta$ is trivial on $\mathfrak{m}_1$.
\begin{proposition}
The system of differential operators
$$ \Omega_\omega^\zeta(v) = \sum_\alpha \Big((\nu+\rho-2)\omega(v,\widehat{e}_\alpha)+4d\zeta(B_\mu(v,\widehat{e}_\alpha))\Big)\Omega_\omega(e_\alpha)\index{1ZOmega1omegazetav@$\Omega_\omega^\zeta(v)$} $$
is conformally invariant for $\pi_{\zeta,\nu}$ if and only if $d\zeta(\Cas_{\mathfrak{m}_0})=(\nu+\rho)(\nu+\rho-2)$, where $\Cas_{\mathfrak{m}_0}=h^2+2ef+2fe$\index{Casm0@$\Cas_{\mathfrak{m}_0}$}. In this case, the joint kernel
$$ I(\zeta,\nu)^{\Omega_\omega^\zeta(\Lambda)} = \{u\in I(\zeta,\nu):\Omega_\omega^\zeta(v)u=0\,\forall\,v\in\Lambda\}\index{IzetanuOmegaomegazetaLambda@$I(\zeta,\nu)^{\Omega_\omega^\zeta(\Lambda)}$} $$
is a subrepresentation of $I(\zeta,\nu)$.
\end{proposition}
\begin{proof}
Using Theorem~\ref{thm:ConfInvOmegaMu} we find
\begin{align*}
[\Omega_\omega^\zeta(v),d\pi_{\zeta,\nu}(X)] &= 0, && (X\in\overline{\mathfrak{n}}),\\
[\Omega_\omega^\zeta(v),d\pi_{\zeta,\nu}(H)] &= \Omega_\omega^\zeta(v),\\
[\Omega_\omega^\zeta(v),d\pi_{\zeta,\nu}(S)] &= -\Omega_\omega^\zeta(Sv) && (S\in\mathfrak{m}).
\end{align*}
Moreover, we have
\begin{align*}
& [\Omega_\omega^\zeta(v),d\pi_{\zeta,\nu}(E)]\\
&= \sum_\alpha\Big((\nu+\rho-2)\omega(v,\widehat{e}_\alpha)+4d\zeta(B_\mu(v,\widehat{e}_\alpha))\Big)[\Omega_\omega(e_\alpha),d\pi_{\zeta,\nu}(E)]\\
&\hspace{9cm}+ 4\sum_\alpha d\zeta([B_\mu(v,\widehat{e}_\alpha),\mu(x)])\Omega_\omega(e_\alpha)\\
&= \sum_\alpha\Big((\nu+\rho-2)\omega(v,\widehat{e}_\alpha)+4d\zeta(B_\mu(v,\widehat{e}_\alpha))\Big)\Big(t\Omega_\omega(e_\alpha)-\Omega_\omega(\mu(x)e_\alpha)\\
&\hspace{9.5cm}+\tfrac{\nu+\rho}{2}\omega(x,e_\alpha)+2d\zeta(B_\mu(x,e_\alpha))\Big)\\
&\qquad\qquad- 4\sum_\alpha\Big(d\zeta(B_\mu(\mu(x)v,\widehat{e}_\alpha)+d\zeta(B_\mu(v,\mu(x)\widehat{e}_\alpha)])\Big)\Omega_\omega(e_\alpha)\\
&= t\Omega_\omega^\zeta(v)-\Omega_\omega^\zeta(\mu(x)v)+\tfrac{(\nu+\rho)(\nu+\rho-2)}{2}\omega(x,v)-4d\zeta(B_\mu(x,v))\\
&\hspace{9.2cm}+8\sum_\alpha d\zeta(B_\mu(v,\widehat{e}_\alpha))d\zeta(B_\mu(x,e_\alpha)).
\end{align*}
The last term is evaluated in the next lemma and the claim follows.
\end{proof}
\begin{lemma}
For $v,w\in V$ we have
$$ \sum_\alpha B_\mu(v,\widehat{e}_\alpha)B_\mu(e_\alpha,w) \equiv \frac{1}{16}\omega(v,w)(h^2+2ef+2fe)+\frac{1}{2}B_\mu(v,w) \mod \overline{\mathfrak{m}}\,\mathcal{U}(\mathfrak{m}_0) $$
\end{lemma}
\begin{proof}
We first note that
$$ B_\mu(v,w) \equiv \frac{1}{4}\omega(hv,w)h+\frac{1}{2}\omega(fv,w)e+\frac{1}{2}\omega(ev,w)f \mod\overline{\mathfrak{m}}. $$
This can be shown by applying both sides to $A$, $B$, $P$ and $Q$ and pairing with another element in this list with respect to the symplectic form. Plugging this into the sum and using the following identities on $V$ (which is a direct sum of $p+q-4$ copies of the standard representation $\mathbb{R}^2$ of $\mathfrak{m}_0\simeq\sl(2,\mathbb{R})$)
\begin{equation}\label{eq:SL2FormulasSOpq}
\begin{aligned}
\ad(h)^2 &= 1, & \ad(e)^2 &= \ad(f)^2=0,\\
\ad(e)\ad(f) &= \frac{1}{2}(1+\ad(h)), & \ad(f)\ad(e) &= \frac{1}{2}(1-\ad(h)),\\
\ad(h)\ad(e) &= -\ad(e)\ad(h)=\ad(e), & \ad(h)\ad(f) &= -\ad(f)\ad(h)=-\ad(f),
\end{aligned}
\end{equation}
shows the desired formula.
\end{proof}
Now let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}=\so(p,q)$ such that the analytic subgroup $\langle\exp\mathfrak{m}_0\rangle$ of $M$ corresponding to $\mathfrak{m}_0\simeq\sl(2,\mathbb{R})$ is the non-trivial double cover of $SL(2,\mathbb{R})$. For $k\in\mathbb{Z}/4\mathbb{Z}$ and $s\in\mathbb{C}$ there exists a principal series representation $(\zeta_{k,s},V_{k,s})$\index{1fzetaks@$\zeta_{k,s}$}\index{Vks@$V_{k,s}$} of $\langle\exp\mathfrak{m}_0\rangle$ with $K$-types $v_n$, $n\in2\mathbb{Z}+\frac{k}{2}$, on which the basis
$$ \kappa = f-e, \qquad x_\pm = h\mp i(e+f)\index{1kappa@$\kappa$}\index{xpm@$x_\pm$} $$
of $\mathfrak{m}_0$ acts by
\begin{equation}
d\zeta_{k,s}(\kappa)v_n = inv_n, \qquad d\zeta_{k,s}(x_\pm)v_n = (s\pm n+1)v_{n\pm2}.\label{eq:ActionSL2ZetaSOpq}
\end{equation}
Note that for $k=0,2$ these representations factor through $SL(2,\mathbb{R})$ and become the usual even and odd principal series for $\SL(2,\mathbb{R})$ while for $k=1,3$ the representations are genuine. We also write $(\zeta,V_\zeta)=(\zeta_{k,s},V_{k,s})$ for short and denote by $\zeta$ any extension to $M$ which is trivial on $\mathfrak{m}_1$. Note that for $\zeta=\zeta_{k,s}$ we have $d\zeta(\Cas_{\mathfrak{m}_0})=s^2-1$, so that $\Omega_\omega^\zeta$ is conformally invariant for $\pi_{\zeta,\nu}$ if and only if $s=\pm(\nu+\rho-1)$. We therefore let $s=-(\nu+\rho-1)$.
The Fourier transform of the equation $\Omega_\omega^\zeta(v)u=0$ is by \eqref{eq:FTofOmegaOmega}:
$$ 0 = \widehat{\Omega_\omega^\zeta(v)u}(\lambda,x,y) = \Big((\nu+\rho-2)d\sigma_{-\lambda}(v)+4\sum_\alpha d\zeta(B_\mu(v,\widehat{e}_\alpha))d\sigma_{-\lambda}(e_\alpha)\Big)_x\widehat{u}(\lambda,x,y) $$
This motivates the following:
\begin{proposition}\label{prop:InvDistVectSOpq1}
For every $\lambda\in\mathbb{R}^\times$ the space of all $\xi\in\mathcal{S}'(\Lambda)\otimes V_\zeta$ satisfying
$$ \Big((\nu+\rho-2)d\sigma_\lambda(v)+4\sum_\alpha d\zeta(B_\mu(v,\widehat{e}_\alpha))d\sigma_\lambda(e_\alpha)\Big)\xi=0 $$
consists of all distributions of the form $\xi(a,z)=\xi_0(a,p)e^{-i\lambda\frac{n(z)}{a}}$ with
$$ \xi_0=\sum_{n\equiv\frac{k}{2}\mod2}\xi_{0,n}\otimes v_n\in\mathcal{S}'(\mathbb{R}^2)\otimes V_\zeta $$
and each $\xi_{0,n}\in\mathcal{S}'(\mathbb{R}^2)$ homogeneous of degree $-1$ satisfying the recurrence relation
$$ (s+n+1)(p-i\sqrt{2}a)\xi_{0,n} = (s-n-1)(p+i\sqrt{a})\xi_{0,n+2}, $$
where $p=\omega(x,Q)$.
\end{proposition}
\begin{proof}
A short computation shows that the above equation is equivalent to
\begin{align}
& ((s+1)-d\zeta(h))\partial_A\xi-2\sqrt{2}d\zeta(e)\partial_P\xi = 0,\label{eq:InvDistVectSOpq1-1}\\
& ((s+1)+d\zeta(h))a\xi+\sqrt{2}d\zeta(f)p\xi = 0,\label{eq:InvDistVectSOpq1-2}\\
& ((s+1)+d\zeta(h))\partial_P\xi-\sqrt{2}d\zeta(f)\partial_A\xi = 0,\label{eq:InvDistVectSOpq1-3}\\
& ((s+1)-d\zeta(h))p\xi+2\sqrt{2}d\zeta(e)a\xi = 0,\label{eq:InvDistVectSOpq1-4}
\end{align}
and for $v\in\overline{\mathcal{J}}$
\begin{align}
& ((s+1)-d\zeta(h))\partial_v\xi+2\sqrt{2}i\lambda d\zeta(e)\omega(B_\mu(x,v)P,B)\xi = 0,\label{eq:InvDistVectSOpq1-5}\\
& i\lambda((s+1)+d\zeta(h))\omega(B_\mu(x,v)P,B)\xi+\sqrt{2}d\zeta(f)\partial_v\xi = 0.\label{eq:InvDistVectSOpq1-6}
\end{align}
Combining \eqref{eq:InvDistVectSOpq1-4} and \eqref{eq:InvDistVectSOpq1-5} resp. \eqref{eq:InvDistVectSOpq1-2} and \eqref{eq:InvDistVectSOpq1-6} we find
$$ d\zeta(e)\big(a\partial_v-i\lambda p\omega(B_\mu(x,v)P,B)\big)\xi = 0 = d\zeta(f)\big(a\partial_v-i\lambda p\omega(B_\mu(x,v)P,B)\big)\xi, $$
hence $(a\partial_v-i\lambda p\omega(B_\mu(x,v)P,B))\xi=0$. Note that
$$ \partial_v n(x) = -\frac{1}{2}\omega(\mu(x)v,B) = -p\omega(B_\mu(x,v)P,B), $$
so that this equation is equivalent to $\partial_v(\xi\cdot e^{i\lambda\frac{n(x)}{a}})=0$. It follows that $\xi(a,x)=\xi_0(a,p)e^{-i\lambda\frac{n(x)}{a}}$. Combining \eqref{eq:InvDistVectSOpq1-1} and \eqref{eq:InvDistVectSOpq1-4} resp. \eqref{eq:InvDistVectSOpq1-2} and \eqref{eq:InvDistVectSOpq1-3} yields
$$ d\zeta(e)\big(a\partial_A+p\partial_P+1\big)\xi_0 = 0 = d\zeta(f)\big(a\partial_A+p\partial_P+1\big)\xi_0, $$
hence $(a\partial_A+p\partial_P+1)\xi_0=0$ and $\xi_0$ is homogeneous of degree $-1$. Write $\xi_0=\sum_n\xi_{0,n}\otimes v_n$, then using \eqref{eq:ActionSL2ZetaSOpq} we find that \eqref{eq:InvDistVectSOpq1-1} and \eqref{eq:InvDistVectSOpq1-3} are equivalent to
$$ (s+n+1)(\partial_A+i\sqrt{2}\partial_P)\xi_{0,n} = (s-n-1)(\partial_A-i\sqrt{2}\partial_P)\xi_{0,n+2} $$
and that \eqref{eq:InvDistVectSOpq1-2} and \eqref{eq:InvDistVectSOpq1-4} are equivalent to
$$ (s+n+1)(p-i\sqrt{2}a)\xi_{0,n} = (s-n-1)(p+i\sqrt{2}a)\xi_{0,n+2}. $$
It is easy to see that the latter identity implies the first one whenever $\xi_{0,n}$ and $\xi_{0,n+2}$ are homogeneous of degree $-1$.
\end{proof}
In Proposition~\ref{prop:InvDistVectSOpq1} the space of invariant distribution vectors $\xi$ is still infinite-dimensional. This indicates that the kernel of the system $\Omega_\omega^\zeta(v)$, $v\in\Lambda$, is not small enough to yield a representation in the same way as in Section~\ref{sec:FTpictureMinRep}. We therefore also use construct a vector-valued version of the second order system $\Omega_\mu$:
\begin{proposition}
For $\nu=-\frac{p+q-2}{2}$ the system of differential operators
$$ \Omega_\mu^\zeta(T) = \Omega_\mu(T) + 2d\zeta(T)\partial_t \qquad (T\in\mathfrak{m})\index{1ZOmega2muzetaT@$\Omega_\mu^\zeta(T)$} $$
is conformally invariant on the kernel of the system $\Omega_\omega^\zeta$, i.e. the joint kernel
$$ I(\zeta,\nu)^{\Omega_\omega^\zeta(\Lambda),\Omega_\mu^\zeta(\mathfrak{m})} = \{u\in I(\zeta,\nu)^{\Omega_\omega^\zeta(\Lambda)}:\Omega_\mu^\zeta(T)u=0\,\forall\,T\in\mathfrak{m}\}\index{IzetanuOmegaomegazetaLambdaOmegamuzetam@$I(\zeta,\nu)^{\Omega_\omega^\zeta(\Lambda),\Omega_\mu^\zeta(\mathfrak{m})}$} $$
is a subrepresentation of $I(\zeta,\nu)$.
\end{proposition}
\begin{proof}
Using Theorem~\ref{thm:ConfInvOmegaMu} we find that
\begin{align*}
[\Omega_\mu^\zeta(T),d\pi_{\zeta,\nu}(\overline{\mathfrak{n}})] &= 0,\\
[\Omega_\mu^\zeta(T),d\pi_{\zeta,\nu}(H)] &= 2\Omega_\mu^\zeta(T),\\
[\Omega_\mu^\zeta(T),d\pi_{\zeta,\nu}(S)] &= \Omega_\mu^\zeta([T,S]) && (S\in\mathfrak{m}).
\end{align*}
We further show that $[\Omega_\mu^\zeta(T),d\pi_{\zeta,\nu}(E)]$ can be expressed as a $C^\infty(\overline{\mathfrak{n}})$-linear combination of operators in $\Omega_\mu^\zeta(\mathfrak{m})$ and $\Omega_\omega^\zeta(\Lambda)$. First note that
$$ [d\zeta(T)\partial_t,d\pi_{\zeta,\nu}(E)] = d\zeta(T)(\partial_x+2t\partial_t)+(\nu+\rho)d\zeta(T)+d\zeta([T,\mu(x)])\partial_t. $$
Together with the formula for $[\Omega_\mu(T),d\pi_{\zeta,\nu}(E)]$ in Theorem~\ref{thm:ConfInvOmegaMu} this yields
\begin{multline*}
[\Omega_\mu^\zeta(T),d\pi_{\zeta,\nu}(E)] = 2t\Omega_\mu^\zeta(T)+\Omega_\mu^\zeta([T,\mu(x)])+(2\,\mathcal{C}(\mathfrak{m}')-2-(\nu+\rho))\Omega_\omega(Tx)\\
+4\sum_\alpha d\zeta(B_\mu(x,T\widehat{e}_\alpha))\Omega_\omega(e_\alpha)+2d\zeta(T)\partial_x+2(\nu+\rho-\mathcal{C}(\mathfrak{m}'))d\zeta(T).
\end{multline*}
Note that $\mathcal{C}(\mathfrak{m}_0)=\frac{p+q-4}{2}$ and $\mathcal{C}(\overline{\mathfrak{m}})=2$ and $\nu+\rho=\frac{p+q-4}{2}$. Let us first assume that $\mathfrak{m}'=\overline{\mathfrak{m}}$, then $d\zeta(T)=0$. Further, since $[T,B_\mu(x,\widehat{e}_\alpha)]\in\overline{\mathfrak{m}}$ we have $d\zeta([T,B_\mu(x,\widehat{e}_\alpha)])=0$ and hence
\begin{multline*}
(2\,\mathcal{C}(\mathfrak{m}')-2-(\nu+\rho))\Omega_\omega(Tx) + 4\sum_\alpha d\zeta(B_\mu(x,T\widehat{e}_\alpha))\Omega_\omega(e_\alpha)\\
= -(\nu+\rho-2)\Omega_\omega(Tx)-4\sum_\alpha d\zeta(B_\mu(Tx,\widehat{e}_\alpha))\Omega_\omega(e_\alpha) = -\Omega_\omega^\zeta(Tx).
\end{multline*}
Now let $\mathfrak{m}'=\mathfrak{m}_0$, then $\nu+\rho-\mathcal{C}(\mathfrak{m}')=0$, so the last term vanishes. Using $\Omega_\omega^\zeta(Tx)=0$ the first two terms combine to
$$ 4\sum_\alpha d\zeta\big(B_\mu(x,T\widehat{e}_\alpha)-B_\mu(Tx,\widehat{e}_\alpha)\big)\Omega_\omega(e_\alpha). $$
which, by the following lemma, equals
\begin{equation*}
-2d\zeta(T)\Omega_\omega(x) = -2d\zeta(T)\partial_x.\qedhere
\end{equation*}
\end{proof}
\begin{lemma}
For $T\in\mathfrak{m}_0$ and $x,y\in V$ we have
$$ B_\mu(Tx,y)-B_\mu(x,Ty) \equiv \frac{1}{2}\omega(x,y)T \mod\overline{\mathfrak{m}}. $$
\end{lemma}
\begin{proof}
Modulo $\overline{\mathfrak{m}}$ we have
\begin{align*}
B_\mu(Tx,y)-B_\mu(x,Ty) \equiv{}& \frac{1}{4}\big(\omega(hTx,y)-\omega(hx,Ty)\big)h + \frac{1}{2}\big(\omega(fTx,y)-\omega(fx,Ty)\big)e\\
& \hspace{5cm}+ \frac{1}{2}\big(\omega(eTx,y)-\omega(ex,Ty)\big)f\\
={}& \frac{1}{4}\omega((hT+Th)x,y)h + \frac{1}{2}\omega((fT+Tf)x,y)e + \frac{1}{2}\omega((eT+Te)x,y)f.
\end{align*}
Using \eqref{eq:SL2FormulasSOpq} one shows that for $T=T_hh+T_ee+T_ff$ we have
$$ (hT+Th)x=2T_hx, \qquad (eT+Te)x=T_fx, \qquad (fT+Tf)x=T_ex, $$
so the result follows.
\end{proof}
We finally fix $\nu=-\frac{p+q-2}{2}$, $k=p+q$ and $s=-(\nu+\rho-1)=-\frac{p+q-6}{2}$. Note that $\zeta=\zeta_{k,s}$ is reducible and has a unique irreducible subrepresentation. This subrepresentation is finite-dimensional for $p+q$ even and spanned by
$$ v_{\frac{p+q-8}{2}},v_{\frac{p+q-8}{2}-2},\ldots,v_{-\frac{p+q-8}{2}}, $$
and it is infinite-dimensional for $p+q$ odd and spanned by
$$ v_{\frac{p+q-8}{2}},v_{\frac{p+q-8}{2}-2},\ldots $$
By Theorem~\ref{thm:FTofOmegaMu} and Lemma~\ref{lem:FTMultDiff} the Fourier transform of $\Omega_\mu^\zeta(T)$ takes the form
$$ \widehat{\Omega_\mu^\zeta(T)u}(\lambda,x,y) = -2i\lambda\left( d\omega_{\met,-\lambda}(T)_x+d\zeta(T)\right)\widehat{u}(\lambda,x,y). $$
We therefore study $\mathfrak{m}$-invariant distribution vectors in $L^2(\Lambda)^{-\infty}\otimes V_\zeta=\mathcal{S}'(\Lambda)\otimes V_\zeta$.
\begin{proposition}\label{prop:InvDistVectSOpq2}
For every $\lambda\in\mathbb{R}^\times$ the space $(\mathcal{S}'(\Lambda)\otimes V_\zeta)^{\mathfrak{m}}$ of $\mathfrak{m}$-invariant distribution vectors in $\omega_{\met,-\lambda}\otimes\zeta$ is two-dimensional and spanned by the distributions
$$ \xi_{\lambda,\varepsilon}(a,z)=\xi_{0,\varepsilon}(a,p)e^{-i\lambda\frac{n(z)}{a}} \qquad \mbox{with} \qquad \xi_{0,\varepsilon} = \sum_{n\equiv\frac{k}{2}\mod2}\xi_{0,\varepsilon,n}\otimes v_n\index{1oxilambdaepsilon@$\xi_{\lambda,\varepsilon}$} $$
and
$$ \xi_{0,\varepsilon,n}(a,p) = c_n\sgn(a)^\varepsilon|a|^{-\frac{p+q-6}{2}}(\sqrt{2}|a|+i\sgn(a)p)^n(2a^2+p^2)^{\frac{p+q-2n-8}{4}} $$
with $(c_n)_n$ satisfying
$$ (n+2+\tfrac{p+q-8}{2})c_{n+2} = (n-\tfrac{p+q-8}{2})c_n. $$
\end{proposition}
\begin{proof}
We first study invariance under $T\in\mathfrak{m}_1$ in the same way as in Theorem~\ref{thm:InvDistributionVector}. For $v\in\mathcal{J}$ we have
\begin{equation}
d\omega_{\met,\lambda}(B_\mu(v,B))\xi = -a\partial_v(\xi\cdot e^{i\lambda\frac{n(z)}{a}})\cdot e^{-i\lambda\frac{n(z)}{a}}\label{eq:InvDistribVectSOpq1}
\end{equation}
so that invariance under $B_\mu(\mathcal{J}_1,B)\subseteq\mathfrak{m}_1$ implies $\xi(a,z)=\xi_0(a,p)e^{-i\lambda\frac{n(z)}{a}}$ with $\xi_0\in\mathcal{S}'(\mathbb{R}^2)\otimes V_\zeta$ and $p=\omega(z,Q)$. For $w\in\mathcal{J}^*$ we further have
$$ d\omega_{\met,\lambda}(B_\mu(A,w))\xi = \frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(B_\mu(A,w)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta\xi-\frac{1}{2}\omega(z,w)\partial_A\xi. $$
For $\xi$ as above we find
\begin{align*}
\partial_A\xi ={}& \partial_A\xi_0\cdot e^{-i\lambda\frac{n(z)}{a}}+\frac{i\lambda n(z)}{a^2}\xi,\\
\partial_\alpha\xi ={}& \omega(e_\alpha,Q)\partial_P\xi_0\cdot e^{-i\lambda\frac{n(z)}{a}}+\frac{i\lambda}{2a}\omega(\mu(z)e_\alpha,B)\xi,\\
\partial_\alpha\partial_\beta\xi ={}& \omega(e_\alpha,Q)\omega(e_\beta,Q)\partial_P^2\xi_0\cdot e^{-i\lambda\frac{n(z)}{a}}+\frac{i\lambda}{a}\omega(e_\alpha,Q)\omega(\mu(z)e_\beta,B)\partial_P\xi_0\cdot e^{-i\lambda\frac{n(z)}{a}},\\
& -\frac{\lambda^2}{4a^2}\omega(\mu(z)e_\alpha,B)\omega(\mu(z)e_\beta,B)\xi + \frac{i\lambda}{a}\omega(B_\mu(z,e_\alpha)e_\beta,B)\xi.
\end{align*}
Combined with Lemma~\ref{lem:SOpqTrace} and \ref{lem:SOpqIdentities} this gives
\begin{multline}
d\omega_{\met,\lambda}(B_\mu(A,w))\xi = -\frac{\omega(z_0,w_0)}{2a}\Big[a\partial_A\xi_0+\frac{p+q-6}{2}\xi_0\Big]e^{-i\lambda\frac{n(z)}{a}}\\
-\frac{\omega(z_1,w_1)}{2a}\Big[a\partial_A\xi_0+p\partial_P\xi_0+\xi_0\Big]e^{-i\lambda\frac{n(z)}{a}},\label{eq:InvDistribVectSOpq2}
\end{multline}
so that invariance under $B_\mu(A,\mathcal{J}_1^*)$ implies that $\xi_0$ is homogeneous of degree $-1$. Next we consider the action of $\mathfrak{m}_0$ on $\xi$ of this form. By \eqref{eq:InvDistribVectSOpq1} and \eqref{eq:InvDistribVectSOpq2} a distribution $\xi(a,z)=\xi(a,p)e^{-i\lambda\frac{n(z)}{a}}$ is invariant under $\mathfrak{m}_0$ if and only if
$$ -\frac{p}{a}\left(a\partial_A+\frac{p+q-6}{2}\right)\xi_0+\sqrt{2}d\zeta(e)\xi_0 = 0 \qquad \mbox{and} \qquad -2a\partial_P\xi_0+\sqrt{2}d\zeta(f)\xi_0 = 0. $$
Writing $\xi_0=\sum_n\xi_{0,n}\otimes v_n$ and using \eqref{eq:ActionSL2ZetaSOpq} shows that this is equivalent to
\begin{align*}
\left(2a\partial_P-p\partial_A-\frac{p+q-6}{2}\frac{p}{a}\right)\xi_{0,n} &= in\sqrt{2}\xi_{0,n},\\
\left(2a\partial_P+p\partial_A+\frac{p+q-6}{2}\frac{p}{a}\right)\xi_{0,n} &= i\frac{\sqrt{2}}{2}\big((s+n-1)\xi_{0,n-2}-(s-n-1)\xi_{0,n+2}\big).
\end{align*}
The first equation has the solutions
$$ \xi_{0,n}(a,p)=c_n\sgn(a)^\varepsilon|a|^{-\frac{p+q-6}{2}}(\sqrt{2}|a|+i\sgn(a)p)^n(2a^2+p^2)^{\frac{p+q-2n-8}{4}} \qquad (\varepsilon\in\mathbb{Z}/2\mathbb{Z}), $$
and for this choice of $\xi_{0,n}$ the second equation is equivalent to
$$ (n-s+1)c_{n+2} = (n+s+1)c_n. $$
It follows that $c_n=0$ for $n>-(s+1)=\frac{p+q-8}{2}$ and for $n\leq\frac{p+q-8}{2}$ the sequence $(c_n)$ is uniquely determined by $c_{\frac{p+q-8}{2}}$. The result follows.
\end{proof}
\begin{remark}
Comparing the invariant distribution vectors in Proposition~\ref{prop:InvDistVectSOpq1} and Proposition~\ref{prop:InvDistVectSOpq2} suggests that $\Omega_\mu^\zeta(\mathfrak{m})u=0$ implies $\Omega_\omega^\zeta(V)u=0$. However, we were not able to show this only using the differential operators $\Omega_\mu^\zeta(T)$ and $\Omega_\omega^\zeta(v)$.
\end{remark}
By the same arguments as in the other cases, $u\in I(\zeta,\nu)^{\Omega_\omega^\zeta(V),\Omega_\mu^\zeta(\mathfrak{m})}$ implies
$$ \widehat{u}(\lambda,x,y) = \xi_{-\lambda,0}(x)u_0(\lambda,y)+\xi_{-\lambda,1}(x)u_1(\lambda,y)\index{u0lambday@$u_0(\lambda,y)$}\index{u1lambday@$u_1(\lambda,y)$} $$
and we obtain a representation $\rho_\min=(\rho_{\min,0},\rho_{\min,1})$\index{1rhomin@$\rho_\min$} of $G$ on a subspace $J_\min\subseteq(\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))\oplus(\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$\index{J1min@$J_\min$} which makes the map $u\mapsto(u_0,u_1)$ equivariant. Also here, $d\rho_{\min,\varepsilon}$\index{drhomin@$d\rho_\min$} is independent of $\varepsilon$ and we simply write $d\rho_\min=d\rho_{\min,0}=d\rho_{\min,1}$ and extend $d\rho_\min$ to $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$.
\begin{proposition}
The representation $d\rho_\min$ of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is given by the same formulas as in Proposition~\ref{prop:drhomin} with $s_\min=-1$, except for the following:
\begin{align*}
d\rho_\min(\overline{v}) ={}& i\partial_\lambda\partial_v+\frac{1}{2}\omega(\mu(y')v,B)\partial_A-\frac{1}{i\lambda}\sum_{\alpha,\beta}\omega(B_\mu(y',v)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}\\
& \hspace{7.35cm}+\begin{cases}(s_\min-1)\frac{1}{i\lambda}\partial_v&(v\in\mathcal{J}_0)\\(s_\min-\frac{\dim\Lambda}{2}+1)\frac{1}{i\lambda}\partial_v&(v\in\mathcal{J}_1)\end{cases}\\
d\rho_\min(\overline{w}) ={}& -\omega(y,w)\lambda\partial_\lambda+\omega(y,w)\partial_y+\partial_{\mu(y')w}-\frac{1}{2i\lambda}\omega(y,B)\sum_{\alpha,\beta}\omega(B_\mu(A,w)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}\\
& \hspace{6.7cm}+\begin{cases}(s_\min-\frac{\dim\Lambda}{2}+1)\omega(y,w)&(w\in\mathcal{J}_0^*)\\(s_\min-1)\omega(y,w)&(w\in\mathcal{J}_1^*)\end{cases}\\
d\rho_\min(E) ={}& i\lambda\partial_\lambda^2-ia\partial_\lambda\partial_A-i\partial_\lambda\partial_{y'}-i(2s-\tfrac{\dim\Lambda}{2}-1)\partial_\lambda-\frac{s-\frac{\dim\Lambda}{2}}{i\lambda}a\partial_A+n(y')\partial_A\\
& +\frac{2}{\lambda^2}an(\partial')-\frac{(s-1)(s-\frac{\dim\Lambda}{2}+1)}{i\lambda}-\frac{s-1}{i\lambda}\partial_{y_0'}-\frac{s-\frac{\dim\Lambda}{2}+1}{i\lambda}\partial_{y_1'}\\
& +\frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(\mu(y')\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}.
\end{align*}
\end{proposition}
\begin{proof}
We proceed as in the proof of Proposition~\ref{prop:drhomin} and first compute $d\rho_\min(\overline{B})$ by taking the Fourier transform of $d\pi_{\zeta,\nu}(\overline{B})u$. Since
$$ d\pi_{\zeta,\nu}(\overline{B})=d\pi_{\1,\nu}(\overline{B})-2d\zeta(B_\mu(x,B)) $$
we can consider the two terms $d\pi_{\1,\nu}(\overline{B})$ and $d\zeta(B_\mu(x,B))$ separately. For the first term the Fourier transform was computed in \eqref{eq:FTpiBbar} and for the second term we use
$$ B_\mu(x,B)=\frac{1}{2}\omega(x,B)B_\mu(A,B)+\omega(x,Q)B_\mu(P,B), $$
so that
$$ \widehat{d\zeta(B_\mu(x,B))u} = \frac{1}{2}\omega(y-x,B)d\zeta(B_\mu(A,B))\widehat{u}+\omega(y-x,Q)d\zeta(B_\mu(P,B)). $$
Applying the result to $$\widehat{u}(\lambda,x,y)=\sum_\varepsilon\xi_{-\lambda,\varepsilon}(x)u_\varepsilon(\lambda,y)$$ using
\begin{align*}
\lambda\partial_\lambda\xi_{-\lambda,\varepsilon}(x) &= \tfrac{i\lambda n(x')}{a}\xi_{-\lambda,\varepsilon}(a,x'),\\
(a\partial_A+p\partial_P)\xi_{-\lambda,\varepsilon}(x) &= -\xi_{-\lambda,\varepsilon}(x)\\
\partial_{\overline{x}}\xi_{-\lambda,\varepsilon}(x) &= \tfrac{2i\lambda n(x')}{a}\xi_{-\lambda,\varepsilon}(a,x'),\\
\partial_{\overline{y}}\xi_{-\lambda,\varepsilon}(x) &= -\frac{i\lambda}{2a}\omega(\mu(x')\overline{y},B)\xi_{-\lambda,\varepsilon}(a,x').
\end{align*}
gives
\begin{multline*}
d\widehat{\pi}_{\zeta,\nu}(\overline{B})\widehat{u}(\lambda,x,y) = \sum_\varepsilon\xi_{-\lambda,\varepsilon}(x)\Bigg[-\omega(y,B)\lambda\partial_\lambda+\omega(y,B)\partial_y-\omega(y,B)+2i\lambda n(y')\Bigg]u_\varepsilon(\lambda,y)\\
+\omega(y-x,B)\sum_\varepsilon u_\varepsilon(\lambda,y)\left[\frac{1}{2}(a\partial_A-p\partial_P+\partial_{\overline{x}})+\frac{\nu+\rho-1}{2}-d\zeta(B_\mu(A,B))\right]\xi_{-\lambda,\varepsilon}(x)\\
+\omega(y-x,Q)\sum_\varepsilon u_\varepsilon(\lambda,y)\left[2a\partial_P+i\lambda\omega(\mu(z)P,B)-2d\zeta(B_\mu(P,B))\right]\xi_{-\lambda,\varepsilon}(x).
\end{multline*}
The $\mathfrak{m}$-invariance of $\xi_{-\lambda,\varepsilon}$ further implies that
\begin{align*}
d\zeta(B_\mu(P,B))\xi_{-\lambda,\varepsilon} &= \left(a\partial_P+\frac{1}{2}i\lambda\omega(\mu(z)P,B)\right)\xi_{-\lambda,\varepsilon}\\
d\zeta(B_\mu(A,B))\xi_{-\lambda,\varepsilon} &= \frac{1}{2}\left(a\partial_A-p\partial_P+\partial_{\overline{x}}+\frac{p+q-6}{2}\right)\xi_{-\lambda,\varepsilon},
\end{align*}
so that the claimed formula follows.
\end{proof}
As before, we change coordinates using the map $\Phi_\delta$ in \eqref{eq:DefPhi} with $s_\min=-1$ and obtain a representation $d\pi_\min$\index{dpimin@$d\pi_\min$} of $\mathfrak{g}$ which is given by the same formulas as in Proposition~\ref{prop:dpimin} except for
\begin{align*}
d\rho_\min(\overline{v}) ={}& i\lambda\partial_\lambda\partial_v+i\partial_x\partial_v+\frac{1}{2\lambda}\omega(\mu(x')v,B)\partial_A+i\sum_{\alpha,\beta}\omega(B_\mu(x',v)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}\\
& \hspace{8.2cm}+\begin{cases}2i\partial_v&(v\in\mathcal{J}_0)\\i\frac{p+q-4}{2}\partial_v&(v\in\mathcal{J}_1)\end{cases}\\
d\rho_\min(\overline{w}) ={}& -\omega(x,w)\partial_\lambda +\frac{1}{\lambda}\partial_{\mu(x')w}+\frac{1}{2}i\omega(x,B)\sum_{\alpha,\beta}\omega(B_\mu(A,w)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}\\
& \hspace{7.35cm}-\begin{cases}\frac{p+q-6}{2\lambda}\omega(x,w)&(w\in\mathcal{J}_0^*)\\\frac{1}{\lambda}\omega(x,w)&(w\in\mathcal{J}_1^*)\end{cases}\\
d\rho_\min(E) ={}& i\lambda\partial_\lambda^2+i\partial_\lambda\partial_x+i\frac{p+q-2}{2}\partial_\lambda+\frac{n(x')}{\lambda^2}\partial_A+2an(\partial')\\
& -\frac{p+q-6}{2i\lambda}-\frac{p+q-6}{2i\lambda}\partial_{x_0'}-\frac{1}{i\lambda}\partial_{x_1'}+\frac{1}{2i\lambda}\sum_{\alpha,\beta}\omega(\mu(x')\widehat{e}_\alpha,\widehat{e}_\beta)\partial_{e_\alpha}\partial_{e_\beta}.
\end{align*}
\section{Matching the Lie algebra action with the literature}\label{sec:LAactionLiterature}
For some cases, the Lie algebra representation $d\pi_\min$ can be found in the existing literature.
\subsection{The split cases $\mathfrak{g}=\so(n,n),\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$}
For the split cases $\mathfrak{g}=\so(n,n)$, $\mathfrak{e}_{6(6)}$, $\mathfrak{e}_{7(7)}$ and $\mathfrak{e}_{8(8)}$ our formulas for the representation $d\pi_\min$ agree with the formulas in \cite[Appendix]{KPW02}.
\subsection{The case $\mathfrak{g}=\mathfrak{g}_{2(2)}$}
Let $\mathfrak{g}=\mathfrak{g}_{2(2)}$, the split real form of $\mathfrak{g}_2(\mathbb{C})$. Then the subspace $\mathfrak{h}=\mathbb{R} H_\alpha\oplus\mathbb{R} H_\beta$ is a Cartan subalgebra of $\mathfrak{g}$. The roots $\lambda=\alpha-2\beta$ and $\mu=-\alpha+\beta$ form a system of simple roots with $\lambda$ a long root and $\mu$ a short root. A Chevalley basis of $\mathfrak{g}$ is given by
\begin{align*}
X_\mu &= -2B_\mu(B,C), & X_{-\mu} &= -2B_\mu(A,D), & X_{2\lambda+3\mu} &= F, & X_{-2\lambda-3\mu} &= E,\\
X_\lambda &= -\frac{1}{\sqrt{2}}A, & X_{\lambda+\mu} &= -\sqrt{2}C, & X_{\lambda+2\mu} &= -\sqrt{2}D, & X_{\lambda+3\mu} &= -\frac{1}{\sqrt{2}}B,\\
X_{-\lambda} &= -\frac{1}{\sqrt{2}}\overline{B}, & X_{-\lambda-\mu} &= -\sqrt{2}\overline{D}, & X_{-\lambda-2\mu} &= \sqrt{2}\overline{C}, & X_{-\lambda-3\mu} &= \frac{1}{\sqrt{2}}\overline{A}.
\end{align*}
Using the the coordinates
$$ (\lambda,x) = \left(z,\frac{x}{\sqrt{2}}A+\sqrt{2}yC\right), $$
the Lie algebra action $d\pi_\min$ equals the one given in \cite[pages 124--125]{Sav93} which is due to Gelfand \cite{Gel80}. (In \cite{Sav93} the simple roots are denoted by $\alpha$ and $\beta$ instead of $\lambda$ and $\mu$. Further note that in \cite{Sav93} the term $-\frac{iz}{27}D_y^3$ in the formula for $T(X_{-\alpha-3\beta})$ has to be replaced by $-\frac{z}{27}D_y^3$, cf. \cite{Gel80}.)
\subsection{The case $\mathfrak{g}=\sl(n,\mathbb{R})$}
For $\mathfrak{g}=\sl(n,\mathbb{R})$ the Lie algebra action $d\pi_{\min,r}$ agrees with the action of $\mathfrak{g}$ on the Fourier transformed picture of a different degenerate principal series, namely one corresponding to a maximal parabolic subgroup. Let $Q=L_QN_Q\subseteq G$ be a parabolic subgroup with $L_Q\simeq\GL(n-1,\mathbb{R})$. The characters $\chi_{r,\varepsilon}$ of $L_Q$ are parameterized by $r\in\mathbb{C}$ and $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ and we form the degenerate principal series $\Ind_Q^G(\chi_{r,\varepsilon})$. In \cite[Proposition 4.2]{MS17} this representation is realized in the non-compact picture on $\overline{N}_Q\simeq\mathbb{R}^{n-1}$ and the Euclidean Fourier transform on $\mathbb{R}^{n-1}$ is applied. Surprisingly, this results in the same formulas as the ones obtained for $d\pi_{\min,r}$ in Section~\ref{sec:FTpictureMinRepSLn}, if we identify the tuple $(\lambda,x)\in\mathbb{R}^\times\times\Lambda$ with a vector in $\mathbb{R}^{n-1}\simeq\overline{N}_Q$.
In Section~\ref{sec:Int(g,K)Module} we integrate $d\pi_{\min,r}$ to irreducible unitary representations of $\SL(n,\mathbb{R})$ which are equivalent to the unitary degenerate principal series $\Ind_Q^G(\chi_{r,\varepsilon})$ with $r\in i\mathbb{R}$ and $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$.
\subsection{The case $\mathfrak{g}=\so(4,3)$}
For $\mathfrak{g}=\so(4,3)$ Sabourin~\cite{Sab96} constructed an explicit $L^2$-realization of the minimal representation. His formulas have a lot in common with our realization, but the major difference is that in his model the Lie algebra acts by differential operators of order $\leq2$ while we need order $3$ as well. We believe that Sabourin's realization can be obtained with our methods by choosing a different Lagrangian subspace $\Lambda\subseteq V$. More precisely, for $\mathfrak{g}=\so(p,q)$ the Lie algebra $\mathfrak{m}\simeq\sl(2,\mathbb{R})\otimes\so(p-2,q-2)$ acts on $V\simeq\mathbb{R}^2\otimes\mathbb{R}^{p+q-4}$ by the tensor product of the two standard representations. We believe that choosing $\Lambda$ and $\Lambda^*$ to be $\so(p-2,q-2)$-invariant one obtains Sabourin's formulas.
\chapter{Lowest $K$-types}\label{ch:LKT}
To show that for some representation $\zeta$ of $M$, the subrepresentation $I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ is non-trivial, we find the lowest $K$-type in the Fourier transformed picture explicitly. For this we first discuss maximal compact subalgebras of $\mathfrak{g}$.
\section{Cartan involutions}\label{sec:CartanInvolutions}
We study Cartan involutions in the case where $\mathfrak{m}$ is simple. This excludes the cases $\mathfrak{g}=\sl(n,\mathbb{R})$ and $\mathfrak{g}=\so(p,q)$ for which we discuss Cartan involutions in Section~\ref{sec:LKTsln} and \ref{sec:LKTSOpq}, respectively.
To construct a Cartan involution on $\mathfrak{g}$, we first choose a unit in the Jordan algebra $\mathcal{J}$. Let $C\in\mathcal{J}=\mathfrak{g}_{(0,-1)}$\index{C@$C$} with $\Psi(C)\neq0$ and put $D:=\mu(C)B\in\mathfrak{g}_{(-1,0)}$\index{D@$D$}. Then by Lemma~\ref{lem:DecompBigradingMuPsiQ} we have $\mu(C)=-B_\mu(A,\mu(C)B)=-B_\mu(A,D)$ and $\mu(D)=B_\mu(\mu(D)A,B)$. On the other hand, by the $\mathfrak{m}$-equivariance of $B_\mu$ and Lemma~\ref{lem:MuSquared}:
\begin{align*}
\mu(D) &= B_\mu(\mu(C)B,\mu(C)B) = [\mu(C),\underbrace{B_\mu(B,\mu(C)B)}_{\in\mathfrak{g}_{(-2,2)}=0}] - B_\mu(B,\mu(C)^2B) = 4n(C)B_\mu(C,B).
\end{align*}
It follows that $\mu(D)A=4n(C)C$ and hence $\mu(D)=4n(C)B_\mu(C,B)$. We therefore renormalize $C$ such that $n(C)=\frac{1}{4}$, then
$$ \mu(C) = -B_\mu(A,D) \qquad \mbox{and} \qquad \mu(D) = B_\mu(C,B). $$
as well as
$$ \mu(C)D = -C \qquad \mbox{and} \qquad \mu(D)C = D. $$
It further follows that $\Psi(C)=\frac{1}{4}A$ and
\begin{align*}
\Psi(D) &= \frac{1}{2}\omega(A,\Psi(D))B = -\frac{1}{6}\omega(A,\mu(D)D)B = \frac{1}{6}\omega(\mu(D)A,D)B = \frac{1}{6}\omega(C,D)B\\
&= \frac{1}{6}\omega(C,\mu(C)B)B = -\frac{1}{6}\omega(\mu(C)C,B)B = \frac{1}{2}\omega(\Psi(C),B)B = \frac{1}{4}B.
\end{align*}
Note that this computation also shows that $\omega(C,D)=\frac{3}{2}$.
\begin{lemma}\label{lem:BmuCD}
$B_\mu(C,D)=\frac{1}{4}B_\mu(A,B)$.
\end{lemma}
\begin{proof}
We have
$$ [\mu(C),\mu(D)] = 2B_\mu(\mu(C)D,D) = -2B_\mu(C,D). $$
On the other hand, $\mu(D)=B_\mu(C,B)$, so that
$$ [\mu(C),\mu(D)] = B_\mu(\mu(C)C,B)+B_\mu(C,\mu(C)B) = -\frac{3}{4}B_\mu(A,B)+B_\mu(C,D).\qedhere $$
\end{proof}
\begin{lemma}
The elements $\{2\mu(C),2B_\mu(A,B),-2\mu(D)\}$ form an $\sl(2)$-triple.
\end{lemma}
\begin{proof}
By the proof of Lemma~\ref{lem:BmuCD} we have $[\mu(C),\mu(D)]=-\frac{1}{2}B_\mu(A,B)$. Further,
\begin{align*}
[B_\mu(A,B),\mu(C)] &= 2B_\mu(B_\mu(A,B)C,C) = \mu(C),\\
[B_\mu(A,B),\mu(D)] &= 2B_\mu(B_\mu(A,B)D,D) = -\mu(D).\qedhere
\end{align*}
\end{proof}
Let $\mathcal{J}_0=\{v\in\mathcal{J}:\omega(v,D)=0\}$\index{J30@$\mathcal{J}_0$}.
\begin{lemma}\label{lem:muCmuD}
$\mu(C)\mu(D)|_{\mathcal{J}_0}=-\frac{1}{4}\id_{\mathcal{J}_0}$
\end{lemma}
\begin{proof}
Let $v\in\mathcal{J}_0$, then
\begin{align*}
\mu(C)\mu(D)v &= [\mu(C),\mu(D)]v + \mu(D)\mu(C)v = -\frac{1}{4}v + \mu(D)\mu(C)v.
\end{align*}
Since
$$ \mu(C)v = \frac{1}{2}\omega(\mu(C)v,B) = -\frac{1}{2}\omega(v,\mu(C)B) = -\frac{1}{2}\omega(v,D) = 0 $$
the claim follows.
\end{proof}
In \cite[Theorem 7.32]{SS} it is shown that $\mathcal{J}$ can be endowed with a natural Jordan algebra structure with unit element $C$ and norm function $N(v)=4n(v)$. The corresponding trace form is given by
$$ T(u,v) = \partial_uN(c)\partial_vN(c)-\partial_u\partial_vN(c) = 4\omega(u,D)\omega(v,D)+4\omega(\mu(D)u,v), \qquad u,v\in\mathcal{J}.\index{Tuv@$T(u,v)$} $$
Note that since $T(C,v)=2\omega(v,D)$, the $T$-orthogonal complement of $C$ in $\mathcal{J}$ equals $\mathcal{J}_0$. Write $u=u_0C+u'$ and $v=v_0C+v'$ with $u',v'\in\mathcal{J}_0$, then
\begin{equation}
T(u,v) = 3u_0v_0 + 4\omega(\mu(D)u',v').\label{eq:FormulaTraceForm}
\end{equation}
\begin{definition}
A \emph{Cartan involution} of a Jordan algebra $\mathcal{J}$ with trace form $T$ is an involutive algebra automorphism $\vartheta$\index{1htheta@$\vartheta$} of $\mathcal{J}$ such that $T(\vartheta v,v)>0$ for all $v\in\mathcal{J}\setminus\{0\}$.
\end{definition}
\begin{proposition}\label{prop:JFromJordanCartanInv}
For every Cartan involution $\vartheta$ of the Jordan algebra $\mathcal{J}$ the map $J\in\End(V)$\index{J1@$J$} given by
\begin{align*}
JA &= -B, & JC &= -D, & Jv &= 2\mu(D)\vartheta v && (x\in\mathcal{J}_0),\\
JB &= A, & JD &= C, & Jw &= 2\vartheta\mu(C)w && (w\in\mathfrak{g}_{(-1,0)},\omega(C,w)=0),
\end{align*}
satisfies the conditions of Lemma~\ref{lem:CartanInvFromJ}.
\end{proposition}
\begin{proof}
Condition \eqref{lem:CartanInvFromJ1} follows from $\vartheta^2=\id$ and Lemma~\ref{lem:muCmuD}. Condition \eqref{lem:CartanInvFromJ2} follows from \eqref{eq:FormulaTraceForm} and Lemma~\ref{lem:muCmuD}. The only non-trivial computation for condition \eqref{lem:CartanInvFromJ3} is
\begin{align*}
\omega(Jv,Jw) &= 4\omega(\mu(D)\vartheta v,\vartheta\mu(C)w) = T(\vartheta v,\vartheta\mu(C)w) = T(v,\mu(C)w)\\
&= 4\omega(\mu(D)v,\mu(C)w) = -4\omega(\mu(C)\mu(D)v,w) = \omega(v,w).
\end{align*}
by Lemma~\ref{lem:muCmuD} and the fact that $\vartheta$ is a Jordan algebra automorphism and therefore leaves the trace form invariant. Condition \eqref{lem:CartanInvFromJ4} is equivalent to
$$ JB_\mu(x,y)J^{-1}=B_\mu(Jx,Jy) \qquad \mbox{for all $x,y\in V$,} $$
which is checked by a lengthy case-by-case computation.
\end{proof}
We fix a Cartan involution $\vartheta$ of $\mathcal{J}$, let $J$ be the map constructed in Proposition~\ref{prop:JFromJordanCartanInv} and $\theta$ the corresponding Cartan involution of $\mathfrak{g}$ constructed in Lemma~\ref{lem:CartanInvFromJ}. Write $\mathfrak{k}=\mathfrak{g}^\theta$ for the corresponding maximal compact subalgebra. We let
$$ T_0 = B_\mu(B,C)-B_\mu(A,D) = \mu(C)+\mu(D) \in \mathfrak{k}\cap\mathfrak{m}.\index{T0@$T_0$} $$
\begin{proposition}\label{prop:SU2Ideal}
If $\mathcal{J}$ is a Euclidean Jordan algebra (i.e. $T$ is positive definite and $\vartheta=\id_\mathcal{J}$), then the elements
$$ T_1=2T_0-(E-F), \quad T_2=A-2D+\theta(A-2D), \quad T_3=B+2C+\theta(B+2C)\index{T1@$T_1$}\index{T2@$T_2$}\index{T3@$T_3$} $$
span an ideal $\mathfrak{k}_1\subseteq\mathfrak{k}$\index{k31@$\mathfrak{k}_1$} which is isomorphic to $\su(2)$.
\end{proposition}
\begin{proof}
We first note the following commutator formulas in $\mathfrak{k}$:
\begin{align*}
[E-F,x+\theta(x)] &= Jx+\theta(Jx) && \forall\,x\in V,\\
[S,x+\theta(x)] &= Sx+\theta(Sx) && \forall\,S\in\mathfrak{k}\cap\mathfrak{m},x\in V,\\
[E-F,S] &= 0 && \forall\,S\in\mathfrak{k}\cap\mathfrak{m},\\
[x+\theta(x),y+\theta(y)] &= -\omega(x,y)(E-F)-2(B_\mu(Jx,y)-B_\mu(x,Jy)) && \forall\,x,y\in V.
\end{align*}
Further, we have
$$ T_0A = C, \quad T_0B=D, \quad T_0C=-\frac{3}{4}A+D, \quad T_0D=-\frac{3}{4}B-C. $$
We first show that $\mathfrak{k}_1$ is a subalgebra. For this we compute
\begin{align*}
[T_1,T_2] ={}& (2T_0-J)(A-2D)+\theta((2T_0-J)(A-2D)) = 4T_3,\\
[T_1,T_3] ={}& (2T_0-J)(B+2C)+\theta((2T_0-J)(B+2C)) = -4T_2,\\
[T_2,T_3] ={}& -\omega(A-2D,B+2C)(E-F)\\
& -2(B_\mu(J(A-2D),B+2C)-B_\mu(A-2D,J(B+2C)))\\
={}& 8T_1.
\end{align*}
It remains to show that $\mathfrak{k}_1$ is an ideal, i.e. $[\mathfrak{k},\mathfrak{k}_1]\subseteq\mathfrak{k}_1$. First, by similar computations as above, $T_1$, $T_2$ and $T_3$ commute with
$$ 2T_0+3(E-F), \quad 3A+2D+\theta(3A+2D) \quad \mbox{and} \quad 3B-2C+\theta(3B-2C). $$
Finally, similar computations show that $T_1$, $T_2$ and $T_3$ commute with
\begin{itemize}
\item $v+\theta v$, $v\in\mathcal{J}_0$ or $v\in\overline{\mathcal{J}}_0$,
\item $B_\mu(v,B)+B_\mu(A,Jv)$, $v\in\mathcal{J}_0$,
\item $S\in\mathfrak{g}_{(0,0)}\cap\mathfrak{k}$.\qedhere
\end{itemize}
\end{proof}
\begin{remark}\label{rem:SU2Ideal}
The renormalized generators
$$ \widetilde{T}_1 = \frac{1}{2}T_1, \quad \widetilde{T}_2 = \frac{1}{2\sqrt{2}}T_2, \quad \widetilde{T}_3 = \frac{1}{2\sqrt{2}}T_3 $$
satisfy the standard $\su(2)$-relations
$$ [\widetilde{T}_1,\widetilde{T}_2]=2\widetilde{T}_3, \quad [\widetilde{T}_2,\widetilde{T}_3]=2\widetilde{T}_1, \quad [\widetilde{T}_3,\widetilde{T}_1]=2\widetilde{T}_2. $$
\end{remark}
\section{The quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$}
Assume that the Jordan algebra $\mathcal{J}$ is simple and Euclidean, i.e. the identity $\vartheta=\id_{\mathcal{J}}$ is a Cartan involution. Then by Proposition \ref{prop:SU2Ideal} the group $G$ is of quaternionic type. By the classification, we have $\mathfrak{g}\simeq\mathfrak{f}_{4(4)},\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ with $s_\min=-\frac{3}{2},-2,-3,-5$ respectively. We decompose $\mathfrak{k}$ into simple ideals:
$$ \mathfrak{k}=\mathfrak{k}_1\oplus\mathfrak{k}_2\index{k31@$\mathfrak{k}_1$}\index{k32@$\mathfrak{k}_2$} $$
with $\mathfrak{k}_1=\mathbb{R} T_1\oplus\mathbb{R} T_2\oplus\mathbb{R} T_3\simeq\su(2)$. We further abbreviate $n=-s_\min-1\in\{\frac{1}{2},1,2,4\}$\index{n2@$n$}.
\begin{theorem}\label{thm:LKTQuat}
For $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ the space $W=\bigoplus_{k=-n}^n\mathbb{C} f_k$\index{W1@$W$} with
$$ f_k(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k(\lambda^2+2a^2)^{\frac{s_\min-k}{2}}\exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+2a^2)}\right) \sum_{m=-n}^nh_{k,m}K_m(r)e^{im\theta},\index{fk@$f_k$} $$
where
$$ (r\cos\theta,r\sin\theta) = \left(\frac{2(\lambda^2+2a^2)I_1-I_3}{\sqrt{2}(\lambda^2+2a^2)},\frac{I_2+\lambda^2+2a^2}{(\lambda^2+2a^2)^{\frac{1}{2}}}\right) $$
with
$$ I_1=\omega(x,D), \qquad I_2=\omega(\mu(x)C,B), \qquad I_3=\omega(\Psi(x),B)\index{I1@$I_1$}\index{I2@$I_2$}\index{I3@$I_3$} $$
and $(h_{k,m})_{m=-n,\ldots,n}$\index{h3km@$h_{k,m}$} is given by \eqref{eq:QuatRecurrenceSolution}, is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the representation $S^{2n}(\mathbb{C}^2)\boxtimes\mathbb{C}$ of $\mathfrak{k}\simeq\su(2)\oplus\mathfrak{k}_2$.
\end{theorem}
\begin{remark}\label{rem:LKTF4}
We excluded the case $\mathfrak{g}=\mathfrak{f}_{4(4)}$ in the theorem, because here $n=\frac{1}{2}$ and therefore the summation would have to be over $m=\pm\frac{1}{2}$. This is not immediately possible since it would require the use of $e^{\pm\frac{i\theta}{2}}=(\cos\theta+i\sin\theta)^{\frac{1}{2}}$ which cannot be defined as a smooth function on $\mathbb{R}_+\times\Lambda$ or $\mathbb{R}_-\times\Lambda$, because the image of both $2(\lambda^2+2a^2)I_1-I_3$ and $I_2+\lambda^2+2a^2$ is $\mathbb{R}$. However, it might be possible to find the lowest $K$-type $S^1(\mathbb{C}^2)\boxtimes\mathbb{C}$ of the minimal representation in a space of vector-valued functions as a subrepresentation of a vector-valued degenerate principal series (cf. \cite[Section 12]{GW96}).
\end{remark}
We prove this result in several steps. For this we decompose $\mathfrak{k}_2$ as follows:
\begin{align*}
\mathfrak{k}_2 ={}& \mathbb{R}(2T_0+3(E-F))\oplus\mathbb{R}(3A+2D+\theta(3A+2D))\oplus\mathbb{R}(3B-2C+\theta(3B-2C))\\
&\oplus\{v+\theta v:v\in\mathcal{J}_0\}\oplus\{\overline{v}+\theta\overline{v}:v\in\mathcal{J}_0\}\\
&\oplus\{B_\mu(v,B)+B_\mu(A,Jv):v\in\mathcal{J}_0\} \oplus(\mathfrak{g}_{(0,0)}\cap\mathfrak{k}).
\end{align*}
\begin{lemma}\label{lem:QuatStep0}
The Lie algebra $\mathfrak{k}_2$ is generated by
$$ \mathfrak{g}_{(0,0)}\cap\mathfrak{k}\oplus\{v+\theta v:v\in\mathcal{J}_0\}\oplus\{B_\mu(v,B)+B_\mu(A,Jv):v\in\mathcal{J}_0\}. $$
\end{lemma}
\begin{proof}
Let $\mathfrak{h}\subseteq\mathfrak{k}$ denote the subalgebra generated by the above elements. For $v,w\in\mathcal{J}_0$ we have
\begin{align*}
[B_\mu(v,B)+B_\mu(A,Jv),w+\theta w] &= B_\mu(v,B)w+B_\mu(A,Jv)w+\theta(B_\mu(v,B)w+B_\mu(A,Jv)w)\\
&= -\frac{1}{6}\omega(Jv,w)(3A+2D+\theta(3A+2D)) + u+\theta(u)
\end{align*}
with $u=B_\mu(v,w)B+\frac{1}{3}\omega(Jv,w)D\in\overline{\mathcal{J}}_0$. Note that since the Jordan algebra $\mathcal{J}$ is simple, we can always find $v,w\in\mathcal{J}_0$ such that $u\neq0$. If we further act by $S\in\mathfrak{g}_{(0,0)}\cap\mathfrak{k}$, using $SA=SD=0$, we obtain
$$ [S,[B_\mu(v,B)+B_\mu(A,Jv),w+\theta w]] = Su+\theta(Su). $$
Now $\mathfrak{g}_{(0,0)}\cap\mathfrak{k}$ is the Lie algebra of the automorphism group of the Jordan algebra $\mathcal{J}$ which acts irreducibly on $\mathcal{J}_0$. It follows that $\{u+\theta u:u\in\overline{\mathcal{J}}_0\}\subseteq\mathfrak{h}$. Further, by choosing $v,w\in\mathcal{J}_0$ above such that $\omega(Jv,w)\neq0$ we obtain $3A+2D+\theta(3A+2D)$. A similar argument with
$$ [B_\mu(v,B)+B_\mu(A,Jv),\overline{w}+\theta(\overline{w})] $$
shows $3B-2C+\theta(3B-2C)\in\mathfrak{h}$. Finally,
\begin{equation*}
[3A+2D+\theta(3A+2D),3B-2C+\theta(3B-2C)] = -8(2T_0+3(E-F))\in\mathfrak{h}.\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lem:QuatStep1}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is $(\mathfrak{g}_{(0,0)}\cap\mathfrak{k})$-invariant, it is of the form
$$ f(\lambda,a,x) = f_1(\lambda,a,I_1,I_2,I_3), $$
where
$$ I_1=\omega(x,D), \qquad I_2=\omega(\mu(x)C,B), \qquad I_3=\omega(\Psi(x),B). $$
\end{lemma}
\begin{proof}
The Lie algebra $\mathfrak{g}_{(0,0)}$ decomposes as $\mathbb{R} H\oplus(\mathfrak{m}\cap\mathfrak{g}_{(0,0)})$ and $\mathfrak{m}\cap\mathfrak{g}_{(0,0)}$ is the Lie algebra of the structure group of the Jordan algebra $\mathcal{J}$ (see \cite{FK94} for details on Jordan algebras). Since $\mathcal{J}$ is Euclidean, the maximal compact subalgebra $\mathfrak{k}\cap\mathfrak{g}_{(0,0)}$ is therefore the Lie algebra of the automorphism group of $\mathcal{J}$. Its invariants are the coefficients $a_1(x),a_2(x),a_3(x)$ of the minimal polynomial $X^3-a_1(x)X^2+a_2(x)X-a_3(x)$ of a generic element $x\in\mathcal{J}$ (see . These are given by $a_1(x)=\tr(x)=T(x,C)=2\omega(x,D)$, $a_3(x)=\det(x)=4n(x)=2\omega(\Psi(x),B)$ and
\begin{equation*}
a_2(x)=\partial_C\det(x)=6\omega(B_\Psi(x,x,C),B)=-2\omega(\mu(x)B,C).\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lem:QuatStep2}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally an eigenfunction of $d\pi_\min(A-\overline{B})$ to the eigenvalue $ik\sqrt{2}$, it is of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k\exp\left(-\frac{iaI_3}{\lambda R}\right)f_2(R,I_1,I_2,I_3), $$
where
$$ R=\lambda^2+2a^2.\index{R@$R$} $$
\end{lemma}
\begin{proof}
The method of characteristics applied to the first order equation
$$ d\pi_\min(A-\overline{B})f = \left(-\lambda\partial_A+2a\partial_\lambda-\frac{iI_3}{\lambda^2}\right)f=ik\sqrt{2}f $$
shows the claim.
\end{proof}
\begin{lemma}\label{lem:QuatStep3}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $\{\lambda\,d\pi_\min(v+\theta v)+2a\,d\pi_\min(B_\mu(v,B)+B_\mu(A,Jv)):v\in\mathcal{J}_0\}$, it is of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k(\lambda^2+2a^2)^{\frac{s_\min-k}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right) f_3(S,T) $$
with
$$ S=\frac{2RI_1-I_3}{\sqrt{2}R}, \quad T=\frac{I_2+R}{R^{\frac{1}{2}}}.\index{S@$S$}\index{T@$T$} $$
\end{lemma}
\begin{proof}
Applying
\begin{multline*}
\lambda\,d\pi_\min(v+\theta v)+2a\,d\pi_\min(B_\mu(v,B)+B_\mu(A,Jv)) = -\omega(x,Jv)(\lambda\partial_\lambda+a\partial_A-s_\min)\\
-(2a^2+\lambda^2)\partial_v+\partial_{\mu(x)Jv}+i\frac{a}{\lambda}\omega(\mu(x)v,B)
\end{multline*}
to $f(\lambda,a,x)=(\lambda-i\sqrt{2}a)^k\exp\left(-\frac{iaI_3}{\lambda R}\right)f_2(R,I_1,I_2,I_3)$ gives
\begin{multline*}
(\lambda-i\sqrt{2}a)^k\exp\left(-\frac{iaI_3}{\lambda R}\right)\Bigg[\omega(x,Jv)\Big(-2R\partial_R+(R-I_2)\partial_2-2I_3\partial_3+s_\min-k\Big)f_2\\
+\omega(\mu(x)v,B)\Big(R\partial_3+\frac{1}{2}\partial_1\Big)f_2\Bigg] = 0.
\end{multline*}
Here we have used
\begin{align*}
\omega(B_\mu(x,v)C,B) &= -\frac{1}{2}\omega(x,Jv), & \omega(B_\mu(x,\mu(x)Jv)C,B) &= -\frac{1}{2}I_2\omega(x,Jv),\\
\omega(\mu(x)Jv,D) &= \frac{1}{2}\omega(\mu(x)v,B), & \omega(B_\Psi(x,x,\mu(x)Jv),B) &= -\frac{2}{3}I_3\omega(x,Jv).
\end{align*}
This yields two first order partial differential equations:
\begin{align}
& -2R\partial_Rf_2+(R-I_2)\partial_2f_2-2I_3\partial_3f_2 = (k-s_\min)f_2,\label{eq:QuatStep1Eq1}\\
& R\partial_3f_2+\frac{1}{2}\partial_1f_2 = 0.\label{eq:QuatStep1Eq2}
\end{align}
Solving \eqref{eq:QuatStep1Eq2} using the method of characteristics gives
$$ f_2(R,I_1,I_2,I_3) = \widetilde{f_2}(R,I_2,U) \qquad \mbox{with }U=2RI_1-I_3. $$
Applying \eqref{eq:QuatStep1Eq1} to this expression and using again the method of characteristics yields
\begin{equation*}
\widetilde{f_2}(\lambda,a,I_2,U) = R^{\frac{s_\min-k}{2}} f_3(S,T).\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lem:QuatStep4}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $\{B_\mu(v,B)+B_\mu(A,Jv):v\in\mathcal{J}_0\}$, it is of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k(\lambda^2+2a^2)^{\frac{s_\min-k}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right) \sum_{m\in\mathbb{Z}}h_{k,m}K_m(r)e^{im\theta}, $$
where $(S,T)=(r\cos\theta,r\sin\theta)$ and $(h_{k,m})_{m\in\mathbb{Z}}$ is a sequence satisfying
\begin{equation}
\frac{m+s_\min}{2}h_{k,m-1}-\frac{m-s_\min}{2}h_{k,m+1}+kh_{k,m} = 0.\qedhere\label{eq:QuatStep2Recurrence}
\end{equation}
\end{lemma}
\begin{proof}
Using the identities
\begin{align*}
\sum_{\alpha,\beta} \omega(e_\alpha,D)\omega(e_\beta,D)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= 0,\\
\sum_{\alpha,\beta} \omega(e_\alpha,D)\omega(B_\mu(x,e_\beta)C,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= -\frac{1}{4}\omega(x,Jv),\\
\sum_{\alpha,\beta} \omega(e_\alpha,D)\omega(\mu(x)e_\beta,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= \frac{1}{2}\omega(\mu(x)v,B),\\
\sum_{\alpha,\beta} \omega(B_\mu(e_\alpha,e_\beta)x,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= s_\min\omega(x,Jv),\\
\sum_{\alpha,\beta} \omega(\mu(x)e_\alpha,B)\omega(\mu(x)e_\beta,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= 2I_3\omega(x,Jv),\\
\sum_{\alpha,\beta} \omega(\mu(x)e_\alpha,B)\omega(B_\mu(x,e_\beta)C,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= -\frac{1}{2}I_2\omega(x,Jv),\\
\sum_{\alpha,\beta} \omega(B_\mu(x,e_\alpha)C,B)\omega(B_\mu(x,e_\beta)C,B)\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta) &= \frac{1}{2}I_1\omega(x,Jv) + \frac{1}{4}\omega(\mu(x)v,B)
\end{align*}
the equation $d\pi_\min(B_\mu(v,B)+B_\mu(A,Jv))f=0$ for $f$ as in Lemma \ref{lem:QuatStep3} becomes
\begin{multline*}
\omega(x,Jv)\Big(\frac{I_3}{R}\partial_S^2-\sqrt{2}T\partial_S\partial_T+2I_1\partial_T^2+\sqrt{2}s_\min\partial_S-\frac{I_3}{R}-k\sqrt{2}\Big)f_3\\
+\omega(\mu(x)v,B)\Big(\partial_S^2+\partial_T^2-1\Big)f_3 = 0.
\end{multline*}
Again, this gives rise to two partial differential equations, this time of second order:
\begin{align}
& \frac{I_3}{R}\partial_S^2f_3-\sqrt{2}T\partial_S\partial_Tf_3+2I_1\partial_T^2f_3+\sqrt{2}s_\min\partial_Sf_3-\frac{I_3}{R}f_3-k\sqrt{2}f_3 = 0,\label{eq:QuatStep2Eq1}\\
& \partial_S^2f_3+\partial_T^2f_3 = f_3.\label{eq:QuatStep2Eq2}
\end{align}
While \eqref{eq:QuatStep2Eq2} only contains the variables $S$ and $T$, \eqref{eq:QuatStep2Eq1} also contains $I_1$ and $I_3$. We therefore subtract $2I_1$ times \eqref{eq:QuatStep2Eq2} from \eqref{eq:QuatStep2Eq1} to obtain the equivalent equation
\begin{equation}
(S\partial_S+T\partial_T-s_\min)\partial_Sf_3-Sf_3+kf_3 = 0.\label{eq:QuatStep2Eq3}
\end{equation}
Using polar coordinates $(S,T)=(r\cos\theta,r\sin\theta)$ we expand $f_3$ into a Fourier series
$$ f_3(S,T) = \sum_{m\in\mathbb{Z}} g_m(r)e^{im\theta}. $$
Then \eqref{eq:QuatStep2Eq2} becomes
$$ \partial_r^2g_m+\frac{1}{r}\partial_rg_m-\Big(1+\frac{m^2}{r^2}\Big)g_m = 0. $$
The two solutions to this ordinary differential equation are the Bessel functions $I_m(r)$ and $K_m(r)$. The $I$-Bessel function grows exponentially as $r\to\infty$ while the $K$-Bessel function decays exponentially. Since we are only interested in tempered distributions (in fact, only $L^2$-functions) we write
$$ g_m(r) = h_{k,m}K_m(r) $$
for some scalars $h_{k,m}\in\mathbb{C}$. Applying \eqref{eq:QuatStep2Eq3} to the Fourier expansion finally yields the relation \eqref{eq:QuatStep2Recurrence}.
\end{proof}
\begin{lemma}\label{lem:QuatStep5}
For $-n\leq k\leq n$ there is a unique (up to scalar multiples) sequence $(h_{k,m})_{m\in\mathbb{Z}}$ satisfying \eqref{eq:QuatStep2Recurrence}. It satisfies $h_{k,m}=0$ for $|m|>n$ and, normalizing $h_{k,m}=1$ for $m=-n$, is given by
\begin{equation}\label{eq:QuatRecurrenceSolution}
h_{k,m} = \sum_{j=0}^{m+n}(-1)^{m+n-j}{n+k\choose j}{n-k\choose m+n-j}
\end{equation}
for $m=-n,\ldots,n$.
\end{lemma}
\begin{proof}
Let $(h_{k,m})_{m\in\mathbb{Z}}$ be a sequence satisfying \eqref{eq:QuatStep2Recurrence} and form the generating function
$$ h(t) = \sum_{m\in\mathbb{Z}}h_{k,m}t^m. $$
Then \eqref{eq:QuatStep2Recurrence} becomes the differential equation
$$ h'(t) = \frac{-nt^2+2kt-n}{t(1-t^2)}h(t). $$
Writing
$$ \frac{-nt^2+2kt-n}{t(1-t^2)} = -\frac{n}{t} + \frac{k-n}{1-t} + \frac{k+n}{1+t} $$
reveals the solution
$$ h(t) = t^{-n}(1-t)^{n-k}(1+t)^{n+k}. $$
Expanding both $(1-t)^{n-k}$ and $(1+t)^{n+k}$ into Taylor series around $t=0$ shows the claim.
\end{proof}
Recall the elements $\widetilde{T}_1,\widetilde{T}_2,\widetilde{T}_3\in\mathfrak{k}$ from Proposition \ref{prop:SU2Ideal} and Remark \ref{rem:SU2Ideal} which span the ideal $\mathfrak{k}_1\simeq\su(2)$. The element $\widetilde{T}_2$ spans a maximal torus in $\mathfrak{k}_1$ and $\ad(\widetilde{T}_2)$ has eigenvalues $0,\pm2i$. By the representation theory of $\su(2)$, the action of $\widetilde{T}_2$ in every finite-dimensional representation of $\mathfrak{k}_1$ is diagonalizable with eigenvalues $2ik$, $k\in\mathbb{Z}\cup(\mathbb{Z}+\frac{1}{2})$, and the elements $\widetilde{T}_3\pm i\widetilde{T}_1$ step between the subsequent eigenspaces. Lemma~\ref{lem:QuatStep2} shows that $f_k$ is in fact an eigenfunction of $d\pi_\min(\widetilde{T}_2)$ to the eigenvalue $2ik$. We therefore compute the action of $d\pi_\min(\widetilde{T}_3\pm i\widetilde{T}_1)$ on $f_k$.
\begin{lemma}\label{lem:QuatStep6}
For $-n\leq k\leq n$ we have
$$ d\pi_\min(2T_0\pm i\sqrt{2}(C-\overline{D}))f_k = -3(k\mp n)f_{k\pm1}. $$
\end{lemma}
\begin{proof}
For $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k(\lambda^2+2a^2)^{\frac{s_\min-k}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right) f_3(S,T) $$
with $f_3$ satisfying \eqref{eq:QuatStep2Eq2} and \eqref{eq:QuatStep2Eq3} a lengthy computation shows that
\begin{multline*}
d\pi_\min(2T_0\pm i\sqrt{2}(C-\overline{D}))f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^{k\pm1}(\lambda^2+2a^2)^{\frac{s_\min-(k\pm1)}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\\
\times3i\Big[T\partial_S^2-S\partial_S\partial_T\mp T\partial_S\pm S\partial_T+s_\min\partial_T\Big]f_3(S,T).
\end{multline*}
If now $f_3(S,T)=\sum_m h_{k,m}K_m(r)e^{im\theta}$ with $(S,T)=(r\cos\theta,r\sin\theta)$ it can further be shown that
\begin{multline*}
\Big[T\partial_S^2-S\partial_S\partial_T\mp T\partial_S\pm S\partial_T+s_\min\partial_T\Big]f_3(S,T)\\
= i\sum_m\left[\frac{m+s_\min}{2}h_{k,m-1}+\frac{m-s_\min}{2}h_{k,m+1}\pm mh_{k,m}\right]K_m(r)e^{im\theta}.
\end{multline*}
Finally, it can be shown using Lemma~\ref{lem:QuatStep5} that
\begin{equation*}
\frac{m+s_\min}{2}h_{k,m-1}+\frac{m-s_\min}{2}h_{k,m+1}\pm mh_{k,m} = (k\pm(s_\min+1))h_{k+1,m}.\qedhere
\end{equation*}
\end{proof}
Combining the various lemmas, we obtain Theorem~\ref{thm:LKTQuat}.
\begin{corollary}\label{cor:ActionWeylSquaresLKTquat}
The elements $w_0^2,w_1^2,w_2^2\in K$ act on $W$ in the following way:
\begin{align*}
\pi_\min(w_0^2)f_k &= (-1)^{n-k}f_{-k}, & \pi_\min(w_1^2)f_k &= (-1)^kf_k, & \pi_\min(w_2^2)f_k &= (-1)^nf_{-k}.
\end{align*}
\end{corollary}
\begin{proof}
From Lemma~\ref{lem:QuatStep2} and \ref{lem:QuatStep6} it follows that the $\su(2)$-triple $\widetilde{T}_1,\widetilde{T}_2,\widetilde{T}_3$ acts on $W$ by
\begin{equation}
d\pi_\min(\widetilde{T}_2)f_k = 2ikf_k, \qquad d\pi_\min(\widetilde{T}_3\pm i\widetilde{T}_1)f_k = 2i(s_\min+1\mp k)f_{k\mp1}.\label{eq:SU2TripleActionQuat}
\end{equation}
Since $\mathfrak{k}_2$ acts trivially on $W$, we find
\begin{align*}
\pi_\min(w_0^2)f_k &= \pi_\min(\exp(-\tfrac{\pi}{2}\widetilde{T}_1))f_k,\\
\pi_\min(w_1^2)f_k &= \pi_\min(\exp(\tfrac{\pi}{2}\widetilde{T}_2))f_k,\\
\pi_\min(w_2^2)f_k &= \pi_\min(\exp(\tfrac{\pi}{2}\widetilde{T}_3))f_k.
\end{align*}
The formulas now follow from the representation theory of $\SU(2)$.
\end{proof}
\section{The split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$}
Assume that the Jordan algebra $\mathcal{J}$ is simple, non-Euclidean and split. Then the group $G$ is split. By the classification we have $\mathfrak{g}\simeq\mathfrak{e}_{6(6)}$, $\mathfrak{e}_{7(7)}$ or $\mathfrak{e}_{8(8)}$.
The lowest $K$-type in this case turns out to be the trivial representation. It is spanned by a vector which is most easily described using a renormalization $\overline{K}_\alpha(x)=x^{-\frac{\alpha}{2}}K_\alpha(\sqrt{x})$ of the $K$-Bessel function (see Appendix~\ref{app:KBessel} for details).
\begin{theorem}\label{thm:LKTSplit}
The space $W=\mathbb{C} f_0$\index{W1@$W$} with
$$ f_0(\lambda,a,x) = (\lambda^2+2a^2)^{\frac{s_\min}{2}}\exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+2a^2)}\right)\overline{K}_{-\frac{s_\min+1}{2}}\Bigg(\frac{2R^2I_2+I_3^2-RI_4+2R^3}{2R^2}\Bigg),\index{fk@$f_k$} $$
where
$$ R=\lambda^2+2a^2, \qquad I_2=\omega(Jx,x), \qquad I_3=\omega(\Psi(x),B), \qquad I_4=\omega(\mu(x)Jx,Jx),\index{R@$R$}\index{I2@$I_2$}\index{I3@$I_3$}\index{I4@$I_4$} $$
is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the trivial representation.
\end{theorem}
\begin{remark}
The spherical vector $f_0$ has previously been found in \cite{KPW02} using case-by-case computations. Note that $-\frac{s_\min+1}{2}$ equals $\frac{1}{2},1,2$ for $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$.
\end{remark}
We prove this result in several steps. The following lemma is proven in a similar way as Lemma~\ref{lem:QuatStep0}:
\begin{lemma}\label{lem:SplitStep0}
The Lie algebra $\mathfrak{k}$ is generated by $\mathfrak{g}_0\cap\mathfrak{k}$ and $\{v+\theta v:v\in\Lambda\}$.
\end{lemma}
\begin{lemma}\label{lem:SplitStep1}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is $(\mathfrak{g}_{(0,0)}\cap\mathfrak{k})$-invariant, it is of the form
$$ f(\lambda,a,x) = f_1(\lambda,a,I_2,I_3,I_4), $$
where
$$ I_2=\omega(Jx,x), \qquad I_3=\omega(\Psi(x),B), \qquad I_4=\omega(\mu(x)Jx,Jx). $$
\end{lemma}
\begin{proof}
In a non-Euclidean split Jordan algebra $\mathcal{J}$ of degree $3$ the polynomials invariant under a maximal compact subgroup of the structure group are $\tr(\vartheta(x)x)$, $\det(x)$ and $\tr(\vartheta(x)x\vartheta(x)x)$. These are essentially $I_2$, $I_3$ and $I_4$.
\end{proof}
\begin{lemma}\label{lem:SplitStep2}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $A-\overline{B}$, it is of the form
$$ f(\lambda,a,x) = \exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+2a^2)}\right)f_2(R,I_2,I_3,I_4) $$
with
$$ R = \lambda^2+2a^2. $$
\end{lemma}
\begin{proof}
We have
$$ d\pi_\min(A-\overline{B}) = -\lambda\partial_A+2a\partial_\lambda-\frac{iI_3}{\lambda^2}. $$
As in the quaternionic case, the method of characteristics shows the claim.
\end{proof}
\begin{lemma}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $\{\lambda\,d\pi_\min(v+\theta v)+2a\,d\pi_\min(B_\mu(v,B)+B_\mu(A,Jv)):v\in\mathcal{J}\}$, it is of the form
$$ f(\lambda,a,x) = (\lambda^2+2a^2)^{\frac{s_\min}{2}}\exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+2a^2)}\right)f_3(Z) $$
with
$$ Z=\frac{2R^2I_2+I_3^2-RI_4+2R^3}{2R^2}. $$
\end{lemma}
\begin{proof}
We have
\begin{multline*}
\lambda\,d\pi_\min(v+\theta v)+2a\,d\pi_\min(B_\mu(v,B)+B_\mu(A,Jv))\\
= -\omega(x,Jv)(\lambda\partial_\lambda+a\partial_A-s_\min)-(\lambda^2+2a^2)\partial_v+\partial_{\mu(x)Jv}+\frac{ia}{\lambda}\omega(\mu(x)v,B).
\end{multline*}
Using
\begin{align*}
\omega(\mu(x)Jx,J\mu(x)Jv) &= \frac{1}{2}I_3\omega(\mu(x)v,B)-\frac{1}{2}I_4\omega(x,Jv),\\
\omega(\mu(x)\mu(x)Jv,B) &= 2I_3\omega(x,Jv),\\
\omega(x,J\mu(x)Jv) &= \omega(\mu(x)Jx,Jv)
\end{align*}
we find that this equals
\begin{align*}
\exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+2a^2)}\right)\Bigg[
&\omega(x,Jv)\Big(-2R\partial_Rf_2+2R\partial_2f_2-2I_3\partial_3f_2-2I_4\partial_4f_2\Big)\\
&+\omega(\mu(x)v,B)\Big(R\partial_3f_2+2I_3\partial_4f_2\Big)\\
&+\omega(\mu(x)Jx,Jv)\Big(-4R\partial_4f_2-2\partial_2f_2\Big)\Bigg],
\end{align*}
resulting in three first order differential equations for $f_2$. Solving all three using the method of characteristics yields
\begin{equation*}
f_2(R,I_2,I_3,I_4) = R^{\frac{s_\min}{2}}f_3(Z).\qedhere
\end{equation*}
\end{proof}
\begin{lemma}
The distribution $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $\{d\pi_\min(v+\theta v):v\in\mathcal{J}\}$ if and only if the function $f_3(Z)$ solves the differential equation
$$ Zf_3''(Z)+\frac{1-s_\min}{2}f_3'(Z)-\frac{1}{4}f_3(Z)=0. $$
\end{lemma}
\begin{proof}
We have
\begin{multline*}
\lambda d\pi_\min(v+\theta v) = -\lambda^2\partial_v-\omega(x,Jv)\lambda\partial_\lambda+s_\min\omega(x,Jv)+\partial_{\mu(x)Jv}\\
+ia\lambda\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\partial_\alpha\partial_\beta.
\end{multline*}
Using
\begin{align*}
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(B_\mu(e_\alpha,e_\beta)x,B) ={}& s_\min\omega(x,Jv),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(x)e_\alpha,B)\omega(\mu(x)e_\beta,B) ={}& 2I_3\omega(x,Jv),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(x)e_\alpha,B)\omega(x,Je_\beta) ={}& \omega(\mu(x)Jx,Jv),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(x)e_\alpha,B)\omega(\mu(x)Jx,Je_\beta) ={}& \frac{1}{2}I_3\omega(\mu(x)v,B)-\frac{1}{2}I_4\omega(x,Jv),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(e_\beta,Je_\alpha) ={}& 0,\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(x,Je_\alpha)\omega(x,Je_\beta) ={}& \omega(\mu(x)v,B),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(x,Je_\alpha)\omega(\mu(x)Jx,Je_\beta) ={}& \frac{1}{2}I_2\omega(\mu(x)v,B)+\frac{1}{2}I_3\omega(x,Jv),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(x)Je_\alpha,Je_\beta) ={}& s_\min\omega(\mu(x)v,B),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(B_\mu(x,e_\beta)Jx,Je_\alpha) ={}& -\frac{1}{2}\omega(\mu(x)v,B),\\
\sum_{\alpha,\beta}\omega(B_\mu(A,Jv)\widehat{e}_\alpha,\widehat{e}_\beta)\omega(\mu(x)Jx,Je_\alpha)\omega(\mu(x)Jx,Je_\beta) ={}& I_2I_3\omega(x,Jv) - \frac{1}{2}I_4\omega(\mu(x)v,B)\\
& - I_3\omega(\mu(x)Jx,Jv)
\end{align*}
we find that
\begin{equation*}
\lambda d\pi_\min(v+\theta v)f = \frac{ia\lambda}{R^2}\Big(I_3\omega(x,Jv)+R\omega(\mu(x)v,B)\Big) \Big(4Z\partial_Z^2f_3+2(1-s_\min)\partial_Zf_3-f_3\Big).\qedhere
\end{equation*}
\end{proof}
By Appendix~\ref{app:KBessel}, the unique tempered solution to this ordinary differential equation is the renormalized $K$-Bessel function $\overline{K}_\alpha(Z)$ with $\alpha=-\frac{s_\min+1}{2}$. This shows Theorem~\ref{thm:LKTSplit}.
\begin{corollary}\label{cor:ActionWeylSquaresLKTsplit}
The elements $w_0^2,w_1^2,w_2^2\in K$ act on $W$ in the following way:
$$ \pi_\min(w_0^2)f_0 = \pi_\min(w_1^2)f_0 = \pi_\min(w_2^2)f_0 = f_0. $$
\end{corollary}
\section{The case $\mathfrak{g}=\mathfrak{g}_{2(2)}$}
Let $\mathfrak{g}=\mathfrak{g}_{2(2)}$, then $\mathcal{J}=\mathbb{R} C$ is one-dimensional and therefore, strictly speaking, not of rank three. We treat this case separately. For simplicity we write $x\in\mathcal{J}$ as $x=cC$ and $f(\lambda,a,x)=f(\lambda,a,c)$.
\begin{theorem}\label{thm:LKTG2}
The space $W=\mathbb{R} f_{-1}\oplus\mathbb{R} f_0\oplus\mathbb{R} f_1$\index{W1@$W$} with
\begin{align*}
f_0(\lambda,a,c) ={}& (\lambda^2+2a^2)^{-\frac{1}{6}}\exp\left(-\frac{iac^3}{2\lambda(\lambda^2+2a^2)}\right)\overline{K}_{-\frac{1}{3}}(S),\\
f_{\pm1}(\lambda,a,c) ={}& (\lambda\mp i\sqrt{2}a)(\lambda^2+2a^2)^{-\frac{7}{6}}\exp\left(-\frac{iac^3}{2\lambda(\lambda^2+2a^2)}\right)\\
& \hspace{5cm}\times\left[c\overline{K}_{-\frac{1}{3}}(S)\mp\frac{\sqrt{2}(2\lambda^2+4a^2+c^2)^2}{4(\lambda^2+2a^2)}\overline{K}_{\frac{2}{3}}(S)\right],\index{fk@$f_k$}
\end{align*}
where
$$ S = \frac{(2\lambda^2+4a^2+c^2)^3}{8(\lambda^2+2a^2)^2},\index{S@$S$} $$
is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimes\mathcal{S}'(\Lambda))$ isomorphic to the representation $\mathbb{C}\boxtimes S^2(\mathbb{C}^2)$ of $\mathfrak{k}\simeq\su(2)\oplus\su(2)$.
\end{theorem}
\begin{remark}
In \cite{GNPP08} one vector in the $K$-type $W$ is found explicitly, but the formula differs slightly from ours. In comparison to the other cases, our formula looks more natural than the one in \cite{GNPP08} which contains an additional transcendental function.
\end{remark}
We prove this result in several steps. First note that the Lie algebra $\mathfrak{k}$ splits into the direct sum of two ideals
$$ \mathfrak{k} = \mathfrak{k}_1\oplus\mathfrak{k}_2 $$
with $\mathfrak{k}_1,\mathfrak{k}_2\simeq\su(2)$ given by
$$ \mathfrak{k}_1 = \mathbb{R} T_1\oplus\mathbb{R} T_2\oplus\mathbb{R} T_3, \qquad \mathfrak{k}_2 = \mathbb{R} S_1\oplus\mathbb{R} S_2\oplus\mathbb{R} S_3,\index{k31@$\mathfrak{k}_1$}\index{k32@$\mathfrak{k}_2$} $$
where $T_1,T_2,T_3$ are as in Proposition~\ref{prop:SU2Ideal} and
$$ S_1 = 2T_0+3(E-F), \qquad S_2 = 3A+2D+\theta(3A+2D), \qquad S_3 = 3B-2C+\theta(3B-2C).\index{S1@$S_1$}\index{S2@$S_2$}\index{S3@$S_3$} $$
We expect to find the $K$-type $\mathbb{C}\boxtimes S^2(\mathbb{C}^2)$ on which $\mathfrak{k}_1$ acts trivially and $\mathfrak{k}_2\simeq\su(2)$ acts by the three-dimensional representation $S^2(\mathbb{C}^2)$. Writing $S^2(\mathbb{C}^2)$ as the direct sum of weight spaces relative to the maximal torus $\mathbb{R} S_2\subseteq\mathfrak{k}_2$, we expect to find a vector of weight $0$, i.e. an $S_2$-invariant vector.
\begin{lemma}\label{lem:G2Step1}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is $d\pi_\min(S_2+T_2)$-invariant, it is of the form
$$ f(\lambda,a,x) = \exp\left(-\frac{iac^3}{2\lambda(\lambda^2+2a^2)}\right)f_1(R,c), $$
where $R=\lambda^2+2a^2$.
\end{lemma}
\begin{proof}
We have $S_2+T_2=4(A-\overline{B})$ and
$$ d\pi_\min(A-\overline{B}) = -\lambda\partial_A+2a\partial_\lambda-\frac{iI_3}{\lambda^2}, $$
where $I_3=2n(x)=\frac{1}{2}c^3$. Applying the method of characteristics shows the claim.
\end{proof}
\begin{lemma}\label{lem:G2Step2}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ is additionally invariant under $d\pi_\min(S_2-3T_2)$ and $\lambda d\pi_\min(T_1)-a\,d\pi_\min(T_3)$, the function $f_1(R,c)$ satisfies
$$ \left[2R\partial_R\partial_C+\frac{1}{3}c\partial_C^2+\frac{4}{3}\partial_C+\frac{3c}{8R^2}(c^2+2R)(c^2-2R)\right]f_1 = 0 $$
and
$$ \left[\frac{2R}{3}\partial_C^2-4R^2\partial_R^2-6R\partial_R+\frac{2}{3}c\partial_C-\frac{2}{9}+\frac{8R^3-24c^2R^2-6c^4R+4c^6}{8R^2}\right]f_1 = 0. $$
\end{lemma}
\begin{proof}
This is an elementary computation.
\end{proof}
Inspired by the previous cases we make the Ansatz $f_1(R,c)=R^{-\frac{1}{6}}f_2(S)$ with
$$ S=\frac{(2R+c^2)^3}{8R^2}. $$
\begin{lemma}\label{lem:G2Step3}
A function of the form $f_1(R,c)=R^{-\frac{1}{6}}f_2(S)$ satisfies the differential equations in Lemma \ref{lem:G2Step2} if and only if $f_2$ satisfies
$$ Tf_2''(S) +\frac{2}{3}f_2'(S)-\frac{1}{4}f_2(S) = 0. $$
\end{lemma}
\begin{proof}
Another elementary computation.
\end{proof}
From Appendix~\ref{app:KBessel} we know that $f_2(S)=\overline{K}_{-\frac{1}{3}}(S)$ is the unique tempered solution to the above differential equation. This leads to the function $f_0(\lambda,a,x)$. We now apply $S_3\pm i\sqrt{2}S_1$ to $f_0$.
\begin{lemma}\label{lem:G2Step4}
Let
$$ f(\lambda,a,x) = (\lambda^2+2a^2)^{-\frac{1}{6}}\exp\left(-\frac{iac^3}{2\lambda(\lambda^2+2a^2)}\right)f_2(S) $$
be invariant under $T_1,T_2,T_3$ and $S_2$. Then
\begin{multline*}
d\pi_\min(S_3\pm i\sqrt{2}S_1)f = -8(\lambda\mp i\sqrt{2}a)(\lambda^2+2a^2)^{-\frac{7}{6}}\exp\left(-\frac{iac^3}{2\lambda(\lambda^2+2a^2)}\right)\\
\times\left(\frac{c}{2}f_2(S)\pm\frac{\sqrt{2}(2R+c^2)^2}{4R}f_2'(S)\right).
\end{multline*}
\end{lemma}
\begin{proof}
Since $f$ is invariant under $T_1$ and $T_3$ we find that
$$ d\pi_\min(S_1)f = 8\,d\pi_\min(T)f \qquad \mbox{and} \qquad d\pi_\min(S_3)f = -8\,d\pi_\min(C-\overline{D}). $$
The rest is a simple computation.
\end{proof}
Since $\overline{K}_{-\frac{1}{3}}'(x)=-\frac{1}{2}\overline{K}_{\frac{2}{3}}(x)$ by \eqref{eq:BesselDerivative1} this leads to the functions $f_1$ and $f_{-1}$.
\begin{proposition}\label{prop:G2Step5}
The functions $f_{-1}$, $f_0$ and $f_1$ are $\mathfrak{k}_1$-invariant and transform in the following way under the action of $\mathfrak{k}_2$:
\begin{align*}
d\pi_\min(S_2)f_k &= 4\sqrt{2}ikf_k & d\pi_\min(S_3\pm i\sqrt{2}S_1)f_0 &= -4f_{\pm1},\\
d\pi_\min(S_3\pm i\sqrt{2}S_1)f_{\pm1} &= 0, & d\pi_\min(S_3\mp i\sqrt{2}S_1)f_{\pm1} &= 16f_0.
\end{align*}
\end{proposition}
\begin{proof}
The first formula follows as in the proof of Lemma~\ref{lem:G2Step1}. The second formula is essentially Lemma~\ref{lem:G2Step4}. The third and the fourth formula are easily computed using
$$ S_3\pm i\sqrt{2}S_1 \equiv 8(-C+\overline{D}\pm i\sqrt{2}T_0) \mod \mathfrak{k}_1 $$
and \eqref{eq:BesselDerivative1} and \eqref{eq:BesselDerivative2}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:LKTG2}]
Note that the elements
$$ \widetilde{S}_1 = \frac{1}{2}S_1, \qquad \widetilde{S}_2 = \frac{1}{2\sqrt{2}}S_2 \qquad \mbox{and} \qquad \widetilde{S}_3 = -\frac{1}{2\sqrt{2}}S_3 $$
form an $\su(2)$-triple, then the statement follows from Proposition~\ref{prop:G2Step5}.
\end{proof}
\begin{corollary}\label{cor:ActionWeylSquaresLKTG2}
The elements $w_0^2,w_1^2,w_2^2\in K$ act on $W$ in the following way:
\begin{align*}
\pi_\min(w_0^2)f_k &= (-1)^{k-1}f_{-k}, & \pi_\min(w_1^2)f_k &= (-1)^kf_k, & \pi_\min(w_2^2)f_k &= -f_{-k}.
\end{align*}
\end{corollary}
\begin{proof}
Since $\mathfrak{k}_1$ acts trivially on $W$, we find
\begin{align*}
\pi_\min(w_0^2)f_k &= \pi_\min(\exp(\tfrac{\pi}{2}\widetilde{S}_1))f_k,\\
\pi_\min(w_1^2)f_k &= \pi_\min(\exp(\tfrac{\pi}{2}\widetilde{S}_2))f_k,\\
\pi_\min(w_2^2)f_k &= \pi_\min(\exp(-\tfrac{\pi}{2}\widetilde{S}_3))f_k.
\end{align*}
The rest is an $\SU(2)$ computation.
\end{proof}
\section{The case $\mathfrak{g}=\sl(n,\mathbb{R})$}\label{sec:LKTsln}
Let $\mathfrak{g}=\sl(n,\mathbb{R})$, then $\mathcal{J}\simeq\mathbb{R}^{n-3}$ does not carry the structure of a Jordan algebra. In this case, it is not necessary to use the decomposition $\mathfrak{g}_{-1}=\mathbb{R} A\oplus\mathcal{J}\oplus\mathcal{J}^*\oplus\mathbb{R} B$ at all. We choose any Cartan involution $\theta$ on $\mathfrak{g}$ associated to a map $J:V\to V$ as in Lemma~\ref{lem:CartanInvFromJ} and let $\mathfrak{k}=\mathfrak{g}^\theta\simeq\so(n)$ denote the maximal compact subalgebra.
In contrast to the previous cases, we have a continuous family $(d\pi_{\min,r},\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ ($r\in\mathbb{C}$) of representations, and for each $r\in\mathbb{C}$ we find two non-equivalent $(\mathfrak{g},K)$-submodules of $d\pi_{\min,r}$ which integrate to non-equivalent irreducible unitary representations of $L^2(\mathbb{R}^\times\times\Lambda)$. In this section we determine their lowest $K$-types.
\begin{theorem}\label{thm:LKTSLn}
Let $r\in\mathbb{C}$.
\begin{enumerate}[(1)]
\item The space $W_{0,r}=\mathbb{C} f_{0,r}$\index{W10r@$W_{0,r}$} with
$$ f_{0,r}(\lambda,x) = \overline{K}_{\frac{n-2r-2}{4}}(\lambda^2+4|x|^2)\index{f0r@$f_{0,r}$} $$
is a $\mathfrak{k}$-subrepresentation of $(d\pi_{\min,r},\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the trivial representation.
\item The space $W_{1,r}=\mathbb{C} g_{0,r}\oplus\mathbb{C} g_{1,r}\oplus\{g_{w,r}:w\in\Lambda^*\}$\index{W11r@$W_{1,r}$} with
\begin{align*}
g_{0,r}(\lambda,x) &= \lambda\overline{K}_{\frac{n-2r}{4}}(\lambda^2+4|x|^2)\index{g20r@$g_{0,r}$}\\
g_{1,r}(\lambda,x) &= \overline{K}_{\frac{n-2r-4}{4}}(\lambda^2+4|x|^2)\index{g21r@$g_{1,r}$}\\
g_{w,r}(\lambda,x) &= \omega(x,w)\overline{K}_{\frac{n-2r}{4}}(\lambda^2+4|x|^2) && (w\in\Lambda^*)\index{g2wr@$g_{w,r}$}
\end{align*}
is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the standard representation of $\mathfrak{k}\simeq\so(n)$ on $\mathbb{C}^n$.
\end{enumerate}
\end{theorem}
\begin{proof}
We first find the spherical vector $f_{0,r}$. If $f_{0,r}$ is invariant under $\mathfrak{k}\cap\mathfrak{g}_0=\so(n-2)$, it has to be of the form
$$ f_{0,r}(\lambda,x) = f_1(\lambda,I_2) \qquad \mbox{with} \qquad I_2=|x|^2=\frac{1}{4}\omega(Jx,x). $$
For such $f_{0,r}$ the equation $d\pi_{\min,r}(v+\theta v)f_{0,r}=0$ takes the form
$$ \omega(x,Jv)\left(\frac{1}{2}\lambda\partial_2-\partial_\lambda\right)f_1 = 0, $$
so that $f_1(\lambda,I_2)=f_2(\lambda^2+4I_2)$. Finally, the equation $d\pi_{\min,r}(w+\theta w)=0$ reduces to
$$ 4zf_2''+(n-2r+2)f_2'-f_2=0 $$
which has by Appendix~\ref{app:KBessel} the unique tempered solution $f_2(z)=\overline{K}_{\frac{n-2r-2}{4}}(z)$.\\
Now let us show that $g_{0,r}$, $g_{1,r}$ and the $g_{w,r}$'s span a finite-dimensional $\mathfrak{k}$-representation. Similar to (1) one verifies for $T\in\mathfrak{k}\cap\mathfrak{g}_0\simeq\so(p-2)$:
$$ d\pi_\min(T)g_{w,r} = g_{Tw,r}, \qquad d\pi_\min(T)g_{0,r} = d\pi_\min(T)g_{1,r}=0, $$
for $v\in\Lambda$:
$$ d\pi_\min(v+\theta v)g_{0,r} = -g_{Jv,r}, \quad d\pi_\min(v+\theta v)g_{1,r} = 0, \quad d\pi_\min(v+\theta v)g_{w,r} = -\omega(v,w)g_{0,r}, $$
and for $w\in\Lambda^*$:
$$ d\pi_\min(w+\theta w)g_{0,r} = 0, \quad d\pi_\min(w+\theta w)g_{1,r} = -ig_{w,r}, \quad d\pi_\min(w+\theta w)g_{w',r} = i\omega(w',Jw)g_{1,r}, $$
where we have used \eqref{eq:BesselDerivative1} and \eqref{eq:BesselDerivative2}.
\end{proof}
\section{The case $\mathfrak{g}=\sl(3,\mathbb{R})$}
For $\mathfrak{g}=\sl(3,\mathbb{R})$ there is an additional $(\mathfrak{g},K)$-module contained in $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ which gives rise to a genuine irreducible unitary representation of the double cover $\widetilde{SL}(3,\mathbb{R})$ of $\SL(3,\mathbb{R})$. For this we choose $A,B\in V$ such that $JA=-B$ and $JB=A$ and use coordinates $x=aA\in\Lambda$.
\begin{theorem}\label{thm:LKTSL3}
The space $W_{\frac{1}{2}}=\mathbb{C} h_{-\frac{1}{2}}\oplus\mathbb{C} h_{\frac{1}{2}}$\index{W12@$W_{\frac{1}{2}}$} with
\begin{align*}
h_{\frac{1}{2}}(\lambda,a) &= (|\lambda|-i\sqrt{2}\sgn(\lambda)a)^{\frac{1}{2}}\overline{K}_{\frac{1}{2}}(\lambda^2+2a^2),\\
h_{-\frac{1}{2}}(\lambda,a) &= \sgn(\lambda)(|\lambda|+i\sqrt{2}\sgn(\lambda)a)^{\frac{1}{2}}\overline{K}_{\frac{1}{2}}(\lambda^2+2a^2)\index{h312@$h_{\pm\frac{1}{2}}$}
\end{align*}
is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the representation $S^1(\mathbb{C}^2)$ of $\mathfrak{k}=\so(3)\simeq\su(2)$.
\end{theorem}
\begin{proof}
The elements $U_1=\sqrt{2}(A-\overline{B})$, $U_2=\sqrt{2}(B+\overline{A})$ and $U_3=-2(E-F)$ form an $\su(2)$-triple, i.e. $[U_1,U_2]=2U_3$, $[U_2,U_3]=2U_1$ and $[U_3,U_1]=2U_2$. With respect to the maximal torus $\mathbb{R} U_1$ in $\mathfrak{k}$ the vectors $U_2\mp iU_3$ are root vectors:
$$ [U_1,U_2\mp iU_3] = \pm2i(U_2\mp iU_3). $$
The highest weight vector of a $\mathfrak{k}$-type isomorphic to $S^1(\mathbb{C}^2)$ solves the weight equation
$$ d\pi_\min(U_1)h = \sqrt{2}(2a\partial_\lambda-\lambda\partial_A)h = ih. $$
Using the method of characteristics we find that
$$ h(\lambda,a) = (|\lambda|-i\sqrt{2}\sgn(\lambda)a)^{\frac{1}{2}}\cdot u(\lambda^2+2a^2) $$
is a solution. The highest weight equation
$$ d\pi_\min(U_2-iU_3)h = 0 $$
then gives
$$ 4zu''(z)+6u'(z)-u(z) = 0. $$
By Appendix~\ref{app:KBessel} the unique tempered solution is $u(z)=\overline{K}_{\frac{1}{2}}(z)$ which leads to the highest weight vector $h_{\frac{1}{2}}$. A straightforward computation using $\overline{K}_{\frac{1}{2}}(x)=\sqrt{\frac{\pi}{2}}x^{-\frac{1}{2}}e^{-\sqrt{x}}$ (see Appendix~\ref{app:KBessel}) shows that
\begin{equation*}
d\pi_\min(U_2+iU_3)h_{\frac{1}{2}} = -2h_{-\frac{1}{2}} \qquad \mbox{and} \qquad d\pi_\min(U_2+iU_3)h_{-\frac{1}{2}}=0.\qedhere
\end{equation*}
\end{proof}
\begin{remark}
We observe that these functions together with the ones from Theorem~\ref{thm:LKTSLn} in the case $n=3$ agree (up to a change of coordinates) with the ones found by Torasso \cite[Proposition 14, 15 \& 16]{Tor83}.
\end{remark}
\section{The case $\mathfrak{g}=\so(p,q)$}\label{sec:LKTSOpq}
The construction of a Cartan involution in Section~\ref{sec:CartanInvolutions} does not apply in the case $\mathfrak{g}\simeq\so(p,q)$ since here $\mathcal{J}$ is not simple but the direct sum of a rank one Jordan algebra $\mathcal{J}_0\simeq\mathbb{R}$ and a rank two Jordan algebra $\overline{\mathcal{J}}\simeq\mathbb{R}^{p-3,q-3}$. We therefore give a separate construction.
According to the decomposition $\mathcal{J}=\mathcal{J}_0\oplus\overline{\mathcal{J}}$, the norm function decomposes into
$$ n(x) = -\frac{1}{2}\omega(x_0,Q)\omega(\mu(\overline{x})P,B) $$
where $-\omega(\mu(\overline{x})P,B)$ is a quadratic form on $\overline{\mathcal{J}}$ of signature $(p-3,q-3)$, the norm function of the quadratic Jordan algebra $\overline{\mathcal{J}}\simeq\mathbb{R}^{p-3,q-3}$. The following result can be proven using the explicit decompositions in Appendix~\ref{app:SOpq}:
\begin{proposition}\label{prop:CartanInvSOpq}
Let $\vartheta:\overline{\mathcal{J}}\to\overline{\mathcal{J}}$\index{1htheta@$\vartheta$} be a Jordan algebra automorphism such that the symmetric bilinear form $(v_1,v_2)\mapsto-\omega(B_\mu(v_1,\vartheta v_2)P,B)$ is positive definite. Then the map $J:V\to V$ given by
\begin{align*}
JA &= -B, & JP &= -Q, & Jv &= -\sqrt{2}B_\mu(P,B)\vartheta v && (v\in\overline{\mathcal{J}}),\\
JB &= A, & JQ &= P, & Jw &= \sqrt{2}\vartheta B_\mu(A,Q)w && (w\in\overline{\mathcal{J}}^*),
\end{align*}
satisfies the conditions of Lemma~\ref{lem:CartanInvFromJ}.
\end{proposition}
We fix $\vartheta$ as in the proposition and let $\mathcal{J}_1$\index{J31@$\mathcal{J}_1$} denote the $+1$ eigenspace and $\mathcal{J}_2$\index{J32@$\mathcal{J}_2$} the $-1$ eigenspace of $\vartheta$ in $\overline{\mathcal{J}}$. Then $\mathcal{J}=\mathcal{J}_0\oplus\mathcal{J}_1\oplus\mathcal{J}_2$, and using the symplectic form we obtain a dual decomposition $\mathcal{J}^*=\mathcal{J}_0^*\oplus\mathcal{J}_1^*\oplus\mathcal{J}_2^*$\index{J31star@$\mathcal{J}_1^*$}\index{J32star@$\mathcal{J}_2^*$}.
\begin{lemma}
\begin{enumerate}[(1)]
\item $B_\mu(A,Q)$ and $B_\mu(P,B)$ map $\mathcal{J}_i$ and $\mathcal{J}_i^*$ to each other ($i=1,2$).
\item $B_\mu(\mathcal{J}_1,\mathcal{J}_2)=0$ and $\mu(\mathcal{J}_1),\mu(\mathcal{J}_2)\in\mathbb{R} B_\mu(A,Q)$.
\end{enumerate}
\end{lemma}
Denote by $\theta$ the corresponding Cartan involution of $\mathfrak{g}$ and by $\mathfrak{k}=\mathfrak{g}^\theta$ the corresponding maximal compact subalgebra of $\mathfrak{g}$. Then
$$ T_0=B_\mu(A,Q)-B_\mu(P,B) \in \mathfrak{k}.\index{T0@$T_0$} $$
\begin{proposition}\label{prop:SOpqKstructure}
The Lie algebra $\mathfrak{k}$ decomposes into the sum of two ideals $\mathfrak{k}=\mathfrak{k}_1\oplus\mathfrak{k}_2$ with $\mathfrak{k}_1\simeq\so(p)$ and $\mathfrak{k}_2\simeq\so(q)$ given by
\begin{multline*}
\mathfrak{k}_1 = \mathbb{R}(2T_0+\sqrt{2}(E-F))\oplus\mathbb{R}(A-\sqrt{2}Q+\theta(A-\sqrt{2}Q))+\mathbb{R}(B+\sqrt{2}P+\theta(B+\sqrt{2}P))\\
\oplus\{B_\mu(v,B)+B_\mu(A,Jv):v\in\mathcal{J}_1\}\oplus\{x+\theta x:x\in\mathcal{J}_1\oplus\mathcal{J}_1^*\}\oplus\so(p-3)\index{k31@$\mathfrak{k}_1$}
\end{multline*}
and
\begin{multline*}
\mathfrak{k}_2 = \mathbb{R}(2T_0-\sqrt{2}(E-F))\oplus\mathbb{R}(A+\sqrt{2}Q+\theta(A+\sqrt{2}Q))+\mathbb{R}(B-\sqrt{2}P+\theta(B-\sqrt{2}P))\\
\oplus\{B_\mu(v,B)+B_\mu(A,Jv):v\in\mathcal{J}_2\}\oplus\{x+\theta x:x\in\mathcal{J}_2\oplus\mathcal{J}_2^*\}\oplus\so(q-3),\index{k32@$\mathfrak{k}_2$}
\end{multline*}
where $\so(p-3)$ resp. $\so(q-3)$ denotes the ideal of $\mathfrak{k}\cap\mathfrak{g}_{(0,0)}\simeq\so(p-3)\oplus\so(q-3)$ which acts trivially on $\mathcal{J}_2\simeq\mathbb{R}^{q-3}$ resp. $\mathcal{J}_1\simeq\mathbb{R}^{p-3}$.
\end{proposition}
\begin{proof}
Note that
$$ T_0A=-P, \qquad T_0P=\frac{1}{2}A, \qquad T_0Q=\frac{1}{2}B, \qquad T_0B=-Q. $$
The rest is along the same lines as the proof of Proposition~\ref{prop:SU2Ideal}.
\end{proof}
To state explicit formulas for $K$-finite vectors we first need the following result:
\begin{lemma}\label{lem:LKTSOpqUniquePolys}
For $j\in\mathbb{N}$ and $k\in\mathbb{Z}$ there exists a unique family of polynomials $(p_{j,k,m})_{m=0,\ldots,j}\subseteq\mathbb{C}[S,T]$\index{p2jkmST@$p_{j,k,m}(S,T)$} satisfying
\begin{enumerate}[(1)]
\item\label{lem:LKTSOpqUniquePolys1} $(2\partial_S\partial_T+k\sqrt{2})p_{j,k,m}-T\partial_Sp_{j,k,m-1}-2S\partial_Tp_{j,k,m+1}=0$,
\item\label{lem:LKTSOpqUniquePolys2} $\partial_T^2p_{j,k,m+1}+(m-T\partial_T)p_{j,k,m}=0$,
\item\label{lem:LKTSOpqUniquePolys3} $p_{j,k,0}(S,T)=S^j$.
\end{enumerate}
For the family of polynomials $(p_{j,k,m})_{j,k,m}$ the following identities hold:
\begin{enumerate}[(a)]
\item\label{lem:LKTSOpqUniquePolysA} $\partial_Sp_{j,k,m}\mp\sqrt{2}\partial_Tp_{j,k,m+1}=(j\mp k)p_{j-1,k\pm1,m}$,
\item\label{lem:LKTSOpqUniquePolysB} $\pm\frac{\sqrt{2}}{2}Tp_{j,k,m-1}+Sp_{j,k,m}\mp\sqrt{2}\partial_Tp_{j,k,m}=p_{j+1,k\pm1,m}$,
\item\label{lem:LKTSOpqUniquePolysC} $(S\partial_S-T\partial_T)p_{j,k,m}=(j-2m)p_{j,k,m}$,
\item\label{lem:LKTSOpqUniquePolysD} $-\frac{T^2}{2}p_{j,k,m-2}+(2m-1)p_{j,k,m-1}+S^2p_{j,k,m}=p_{j+2,k,m}$,
\item\label{lem:LKTSOpqUniquePolysE} $p_{j,k,m}(-S,T)=(-1)^jp_{j,-k,m}(S,T)$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first show uniqueness. For this we write $p_m=p_{j,k,m}$ for short. Every polynomial $p_m$ can be written as the sum of homogeneous polynomials
$$ p_m = \sum_{\alpha\geq0} p_m^\alpha $$
with $p_m^\alpha$ homogeneous of degree $\alpha$. Let $\alpha$ be maximal with $p_m^\alpha\neq0$ for some $m$. We claim that $\alpha=j$. By \eqref{lem:LKTSOpqUniquePolys3} we have $\alpha\geq j$, so we assume $\alpha>j$. Then \eqref{lem:LKTSOpqUniquePolys2} would imply $p_m^\alpha=c_m^\alpha S^{\alpha-m}T^m$ and \eqref{lem:LKTSOpqUniquePolys1} would imply
$$ k\sqrt{2}c_m^\alpha-(\alpha-m+1)c_{m-1}^\alpha-2(m+1)c_{m+1}^\alpha=0. $$
From \eqref{lem:LKTSOpqUniquePolys3} we know that $c_0^\alpha=0$ and recursively we find $c_m^\alpha=0$ for all $m$ which is a contradiction, so $\alpha=j$ is maximal with the property that $p_m^\alpha\neq0$ for some $m$. The previous argument also shows that $p_m^j=c_m^jS^{j-m}T^m$ with $c_m^j$ uniquely determined by $c_0^j=1$. For the lower order terms we observe that $p_0^\alpha=0$ for $\alpha<j$. For fixed $\alpha<j$ equation \eqref{lem:LKTSOpqUniquePolys1} determines $\partial_Tp_{m+1}^\alpha$ from $p_{m-1}^\alpha$, $p_m^\alpha$ and $p_m^{\alpha+2}$, so $p_{m+1}^\alpha$ is unique modulo polynomials in $S$ independent of $T$. This disambiguity is removed by \eqref{lem:LKTSOpqUniquePolys2} for $m>0$.\\
Now let us prove existence by induction on $j$. For $j=0$ we also have $k=m=0$ and $p_{0,0,0}=1$ by \eqref{lem:LKTSOpqUniquePolys3} which satisfies also \eqref{lem:LKTSOpqUniquePolys1} and \eqref{lem:LKTSOpqUniquePolys2}. Next we note that the left hand side of \eqref{lem:LKTSOpqUniquePolysB} satisfies \eqref{lem:LKTSOpqUniquePolys1}, \eqref{lem:LKTSOpqUniquePolys2} and \eqref{lem:LKTSOpqUniquePolys3} for $j$ replaced by $j+1$ and $k$ replaced by $k\pm1$. Therefore, \eqref{lem:LKTSOpqUniquePolysB} can be used to recursively define the family $(p_{j+1,k,m})_{k,m}$ using $(p_{j,k,m})_{k,m}$ which establishes the existence part of the proof.\\
Finally, the identities \eqref{lem:LKTSOpqUniquePolysA}, \eqref{lem:LKTSOpqUniquePolysB}, \eqref{lem:LKTSOpqUniquePolysC}, \eqref{lem:LKTSOpqUniquePolysD} and \eqref{lem:LKTSOpqUniquePolysE} are proven by showing that the left hand side satisfies \eqref{lem:LKTSOpqUniquePolys1} and \eqref{lem:LKTSOpqUniquePolys2} for certain values of $j$ and $k$ (for \eqref{lem:LKTSOpqUniquePolysC} the term $2mp_{j,k,m}$ has to be moved to the left hand side first) and then using the previously established uniqueness result.
\end{proof}
\begin{remark}
It is easy to see that
$$ p_{j,k,m}(S,T) = \const\times S^{j-m}T^m + \const\times S^{j-m-1}T^{m-1} + \cdots. $$
For $k=\pm j$ it is possible to find the coefficient of $S^{j-m}T^m$:
$$ p_{j,\pm j,m}(S,T) = (\pm1)^m2^{-\frac{m}{2}}{j\choose m}S^{j-m}T^m + \mbox{lower order terms}. $$
However, we were not able to find a closed formula in general.
\end{remark}
We assume from now on that $p\geq q\geq3$. The lowest $K$-type turns out to be isomorphic to $\mathbb{C}\boxtimes\mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^q)$ as a representation of $\mathfrak{k}\simeq\so(p)\oplus\so(q)$, where $\mathcal{H}^\alpha(\mathbb{R}^n)$\index{H2alphaRn@$\mathcal{H}^\alpha(\mathbb{R}^n)$} denotes the space of homogeneous polynomials on $\mathbb{R}^n$ of degree $\alpha$ which are harmonic. Since only the subalgebra $\mathfrak{k}_2\cap\mathfrak{g}_{(0,0)}\simeq\so(q-3)$ acts geometrically in the representation $d\pi_\min$, it is helpful to use the following multiplicity-free branching rule:
$$ \mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^q)|_{\so(2)\oplus\so(q-2)} \simeq \bigoplus_{j=0}^{\frac{p-q}{2}}\bigoplus_{\substack{k=-j\\k\equiv j\mod2}}^j\mathbb{C}_k\boxtimes\mathcal{H}^{\frac{p-q}{2}-j}(\mathbb{R}^{q-2}), $$
where $\mathbb{C}_k$\index{Ck@$\mathbb{C}_k$} denotes the obvious character of $\so(2)$, and further decompose
$$ \mathcal{H}^{\frac{p-q}{2}-j}(\mathbb{R}^{q-2})|_{\so(q-3)} \simeq \bigoplus_{\ell=0}^{\frac{p-q}{2}-j}\mathcal{H}^\ell(\mathbb{R}^{q-3}). $$
Together we find that
\begin{equation*}
\mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^q)|_{\so(2)\oplus\so(q-3)} \simeq \bigoplus_{j=0}^{\frac{p-q}{2}}\Bigg(\bigoplus_{\substack{k=-j\\k\equiv j\mod2}}^j\mathbb{C}_k\Bigg)\boxtimes\Bigg(\bigoplus_{\ell=0}^{\frac{p-q}{2}-j}\mathcal{H}^\ell(\mathbb{R}^{q-3})\Bigg).
\end{equation*}
For a distribution $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ we write
$f\otimes\mathcal{H}^\ell(\mathbb{R}^{q-3})$ for the space of distributions $f\otimes\varphi$, $\varphi\in\mathcal{H}^\ell(\mathbb{R}^{q-3})$, given by
$$ (f\otimes\varphi)(\lambda,a,x) = f(\lambda,a,x)\varphi(x_2). $$
Here we define spherical harmonics with respect to the positive definite quadratic form $x_2\mapsto\omega(\mu(x_2)P,B)$ on $\mathcal{J}_2\simeq\mathbb{R}^{q-3}$.
We further recall the renormalized $K$-Bessel function $\overline{K}_\alpha(z)$ from Appendix~\ref{app:KBessel}.
\begin{theorem}\label{thm:LKTSOpq}
Let $\mathfrak{g}=\so(p,q)$ with $p\geq q\geq3$ and $p+q$ even. Then
$$ W = \bigoplus_{j=0}^{\frac{p-q}{2}}\bigoplus_{\substack{k=-j\\k\equiv j\mod2}}^j\bigoplus_{\ell=0}^{\frac{p-q}{2}-j}f_{j,k,\ell}\otimes\mathcal{H}^\ell(\mathbb{R}^{q-3})\index{W1@$W$} $$
with
$$ f_{j,k,\ell}(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k(\lambda^2+2a^2)^{-\frac{k+1}{2}}\exp\left(-\frac{2ian(x)}{\lambda(\lambda^2+a^2)}\right)h_{j,k,\ell}(S,T,U)\index{fjkl@$f_{j,k,\ell}$} $$
and
$$ h_{j,k,\ell}(S,T,U) = \sum_{m=0}^jp_{j,k,m}(S,T)(1+S^2)^{\frac{q+2\ell+2m-4}{2}}\overline{K}_{\frac{q+2\ell+2m-4}{2}}(U)\index{h3jkl@$h_{j,k,\ell}$} $$
with
$$ S = \frac{I_1}{\sqrt{\lambda^2+2a^2}} \qquad \mbox{and} \qquad T = \frac{I_2^p+I_2^q-\sqrt{2}(\lambda^2+2a^2)}{\sqrt{\lambda^2+2a^2}}\index{S@$S$}\index{T@$T$} $$
and
$$ I_1 = \omega(x_0,Q), \qquad I_2^p = -2\sqrt{2}|x_1|^2, \qquad I_2^q = 2\sqrt{2}|x_2|^2\index{I1@$I_1$}\index{I2p@$I_2^p$}\index{I2q@$I_2^q$} $$
is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the representation $\mathbb{C}\boxtimes\mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^q)$ of $\mathfrak{k}\simeq\so(p)\oplus\so(q)$.
\end{theorem}
We begin with the action of $\mathfrak{k}\cap\mathfrak{g}_{(0,0)}\simeq\so(p-3)\oplus\so(q-3)$.
\begin{lemma}\label{lem:LKTSOpqStep1}
If $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ generates under the action of $\mathfrak{k}\cap\mathfrak{g}_{(0,0)}\simeq\so(p-3)\oplus\so(q-3)$ a subrepresentation isomorphic to $\mathbb{C}\boxtimes\mathcal{H}^\ell(\mathbb{R}^{q-3})$, it has to be a linear combination of distributions of the form
$$ f(\lambda,a,x) = f_1(\lambda,a,I_1,I_2^p,I_2^q)\varphi(x_2), $$
where
$$ I_1 = \omega(x,Q), \qquad I_2^p = \omega(\mu(x_1)P,B)=-2\sqrt{2}|x_1|^2, \qquad I_2^q = \omega(\mu(x_2)P,B)=2\sqrt{2}|x_2|^2 $$
and $\varphi\in\mathcal{H}^\ell(\mathbb{R}^{q-3})$.
\end{lemma}
\begin{proof}
The subalgebra $\mathfrak{k}\cap\mathfrak{g}_{(0,0)}\simeq\so(p-3)\oplus\so(q-3)$ acts on $\mathcal{J}_1\oplus\mathcal{J}_2\simeq\mathbb{R}^{p-3}\oplus\mathbb{R}^{q-3}$ by the direct sum of the standard representations of $\so(p-3)$ and $\so(q-3)$. An $\so(p-3)$-invariant distribution on $\mathcal{J}_1\simeq\mathbb{R}^{p-3}$ only depends on $I_2^p=-2\sqrt{2}|x_1|^2$, and a distribution on $\mathcal{J}_2\simeq\mathbb{R}^{q-3}$ that belongs to the isotypic component of $\mathcal{H}^\ell(\mathbb{R}^{q-3})$ has to be a linear combination of products of $\varphi\in\mathcal{H}^\ell(\mathbb{R}^{q-3})$ and distributions only depending on $I_2^q=2\sqrt{2}|x_2|^2$.
\end{proof}
We note the following identities for derivatives of the invariants $I_1$, $I_2^p$, $I_2^q$ in the directions $P$, $v\in\mathcal{J}_1$ and $w\in\mathcal{J}_2$:
\begin{align*}
\partial_PI_1 &= 1, & \partial_PI_2^p &= 0, & \partial_PI_2^q &= 0,\\
\partial_vI_1 &= 0, & \partial_vI_2^p &= \sqrt{2}\omega(x,Jv), & \partial_vI_2^q &= 0,\\
\partial_wI_1 &= 0, & \partial_wI_2^p &= 0, & \partial_wI_2^q &= -\sqrt{2}\omega(x,Jv).
\end{align*}
Further, we have
$$ I_3 = \omega(\Psi(x),B) = 2n(x) = -I_1(I_2^p+I_2^q).\index{I3@$I_3$} $$
\begin{lemma}\label{lem:LKTSOpqStep2}
If $f$ is additionally an eigenfunction of $d\pi_\min(A-\overline{B})$ to the eigenvalue $ik\sqrt{2}$, it is of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^k\exp\left(-\frac{iaI_3}{\lambda R}\right)f_2(R,I_1,I_2^p,I_2^q)\varphi(x_2), $$
where
$$ R = \lambda^2+2a^2. $$
\end{lemma}
\begin{proof}
The method of characteristics applied to the first order equation
$$ d\pi_\min(A-\overline{B})f = \left(-\lambda\partial_A+2a\partial_\lambda-\frac{2in(x)}{\lambda^2}\right)f = ik\sqrt{2}f $$
shows the claim.
\end{proof}
\begin{lemma}\label{lem:LKTSOpqStep3}
If $f$ is additionally invariant under $\{\lambda d\pi_\min(v+\theta v)+2a(B_\mu(v,B)+B_\mu(A,Jv)):v\in\mathcal{J}_1\}\subseteq\mathfrak{k}_1$, it has to be of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)f_3\left(S,T,I_2^q\right)\varphi(x_2), $$
where
$$ S = \frac{I_1}{R^{\frac{1}{2}}} \qquad \mbox{and} \qquad T = \frac{I_2^p+I_2^q-\sqrt{2}R}{R^{\frac{1}{2}}}. $$
\end{lemma}
\begin{proof}
Applying $\lambda d\pi_\min(v+\theta v)+2a(B_\mu(v,B)+B_\mu(A,Jv))$ to $f(\lambda,a,x)$ as in Lemma~\ref{lem:LKTSOpqStep2} leads to the differential equation
$$ 2R\partial_R+I_1\partial_1+(I_2^p+I_2^q+\sqrt{2}R)\partial_{2p})f_2=-(k+1)f_2 $$
which can be solved using the method of characteristics.
\end{proof}
\begin{lemma}\label{lem:LKTSOpqStep4}
If $f$ is additionally invariant under $\{d\pi_\min(Jv-\overline{v}):v\in\mathcal{J}_1\}\subseteq\mathfrak{k}_1$, the function $f_3(S,T,I_2^q)$ solves the following two partial differential equations:
\begin{align}
\Big(-2\partial_T^2-\sqrt{2}I_2^q\partial_{2q}^2-\tfrac{\sqrt{2}}{2}(2\ell+q-3)\partial_{2q}+(1+S^2)\Big)f_3 &= 0,\label{eq:LKTSOpqPDE1}\\
\Big(2\partial_S\partial_T-ST+k\sqrt{2}\Big)f_3 &= 0.\label{eq:LKTSOpqPDE2}
\end{align}
\end{lemma}
\begin{proof}
A lengthy computation involving Lemma~\ref{lem:SOpqTrace} and Lemma~\ref{lem:SOpqIdentities} shows that
\begin{multline*}
id\pi_\min(Jv-\overline{v})f = \omega(x,Jv)(\lambda-i\sqrt{2}a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\varphi(x_2)\\
\times\Bigg[\Big(-2\partial_T^2-\sqrt{2}I_2^q\partial_{2q}^2-\tfrac{\sqrt{2}}{2}(2\ell+q-3)\partial_{2q}+(1+S^2)\Big)f_3\\
+R^{-\frac{1}{2}}\Big(-\sqrt{2}S\partial_S\partial_T+\tfrac{\sqrt{2}}{2}S^2T-kS\Big)f_3\Bigg].
\end{multline*}
Since $R^{-\frac{1}{2}}$ is independent of $S$, $T$ and $I_2^q$ this implies two equations.
\end{proof}
We remark at this point that
$$ \frac{1}{2}T^2+2\sqrt{2}I_2^q = \frac{1}{2R}(I_2^p+I_2^q)^2-\sqrt{2}(I_2^p-I_2^q)+R > 0. $$
Together with a deeper analysis of the equations in Lemma~\ref{lem:LKTSOpqStep4} this leads us to introducing a new variable
$$ U = (1+S^2)\Big(\frac{1}{2}T^2+2\sqrt{2}I_2^q\Big). $$
and making the Ansatz
$$ f_3(S,T,I_2^q) = f_4(U)g_4(S). $$
Equation \eqref{eq:LKTSOpqPDE1} applied to this gives
$$ Uf_4''(U)+\tfrac{q+2\ell-2}{2}f_4'(U)-\tfrac{1}{4}f_4(U) = 0 $$
which has the solution $f_4(U)=\overline{K}_{\frac{q+2\ell-4}{2}}(U)$. Plugging this into \eqref{eq:LKTSOpqPDE2} gives
$$ 2Tf_4'(U)\Big((1+S^2)g_4'(S)-(q+2\ell-4)Sg_4(S)\Big)+k\sqrt{2}f_4(U)g_4(S) = 0. $$
Since by \eqref{eq:BesselDerivative1} the functions $f_4(U)$ and $f_4'(U)$ are linearly independent, this equation can only have a non-trivial solution for $k=0$. In this case
$$ g_4(S)=(1+S^2)^{\frac{q+2\ell-4}{2}} $$
solves the equation and we obtain the functions $f_{0,0,\ell}(\lambda,a,x)$, $\ell\geq0$. To investigate whether these functions are $K$-finite we apply $\mathfrak{k}_2\simeq\so(q)$ to $f_{0,0,\ell}$. For this we decompose $\mathfrak{k}_2$ according to Proposition~\ref{prop:SOpqKstructure} and compute the action of each part on a distribution $f(\lambda,a,x)$ of the form given in Lemma~\ref{lem:LKTSOpqStep3}.
\begin{lemma}\label{lem:LKTSOpqStep5}
For a distribution $f\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$ of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)f_3\left(S,T,I_2^q\right)\varphi(x_2) $$
with $f_3(S,T,I_2^q)$ satisfying \eqref{eq:LKTSOpqPDE1} and \eqref{eq:LKTSOpqPDE2} we have for $v\in\mathcal{J}_2$:
\begin{multline*}
id\pi_\min(Jv-\overline{v})f = (\lambda-i\sqrt{2}a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\\
\times\Bigg[\omega(x,Jv)\varphi\Big(4\partial_T^2+\sqrt{2}S\partial_S\partial_{2q}-\sqrt{2}T\partial_T\partial_{2q}+\frac{\sqrt{2}}{2}(2\ell+q-p-2)\partial_{2q}-2S^2\Big)f_3\\
+\partial_v\varphi\Big(-S\partial_S+T\partial_T+2I_2^q\partial_{2q}+\frac{p+q+2\ell-8}{2}\Big)f_3\Bigg],
\end{multline*}
\begin{multline*}
d\pi_\min(2i(B_\mu(A,Jv)+B_\mu(v,B))\mp\sqrt{2}(v+\overline{Jv}))f = (\lambda-i\sqrt{2}a)^{k\pm1}R^{-\frac{(k\pm1)+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\\
\times\Bigg[\omega(x,Jv)\varphi\Big(-2\partial_S\partial_{2q}\mp4\partial_T\pm\sqrt{2}T\partial_{2q}+2\sqrt{2}S\Big)f_3\\
+\partial_v\varphi\Big(\sqrt{2}\partial_S\mp T\Big)f_3\Bigg],
\end{multline*}
and
\begin{multline*}
d\pi_\min(-P+\overline{Q}\mp i\sqrt{2}T_0)f = (\lambda-i\sqrt{2}a)^{k\pm1}R^{-\frac{(k\pm1)+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\varphi(x_2)\\
\times\Big(\pm\sqrt{2}T\partial_T^2\pm2\sqrt{2}I_2^q\partial_T\partial_{2q}+(1+S^2)\partial_S+(-ST\pm\sqrt{2}\tfrac{p+q+2\ell-6}{2})\partial_T\\
-2SI_2^q\partial_{2q}\mp\tfrac{\sqrt{2}}{2}(1+S^2)T-\tfrac{p+q+2\ell\mp2k-8}{2}S\Big)f_3.
\end{multline*}
\end{lemma}
Applying these operators to $f_{0,0,\ell}$ suggests that the $\mathfrak{k}$-representation generated by $f_{0,0,0}$ consists of functions $f(\lambda,a,x)$ as in Lemma~\ref{lem:LKTSOpqStep3} with
\begin{equation}
f_3(S,T,I_2^q) = \sum_m p_m(S,T)(1+S^2)^{\frac{q+2\ell+2m-4}{2}}\overline{K}_{\frac{q+2\ell+2m-4}{2}}(U)\label{eq:LKTSOpqAnsatzSumPolyBessel}
\end{equation}
for some polynomials $p_m(S,T)$.
\begin{lemma}\label{lem:LKTSOpqStep6}
A function $f_3(S,T,I_2^q)$ of the form \eqref{eq:LKTSOpqAnsatzSumPolyBessel} solves the equations \eqref{eq:LKTSOpqPDE1} and \eqref{eq:LKTSOpqPDE2} in Lemma~\ref{lem:LKTSOpqStep4} if and only if the family of polynomials $p_m(S,T)$ satisfies \eqref{lem:LKTSOpqUniquePolys1} and \eqref{lem:LKTSOpqUniquePolys2} in Lemma~\ref{lem:LKTSOpqUniquePolys}.
\end{lemma}
\begin{proof}
From \eqref{eq:BesselDerivative1} and \eqref{eq:BesselDerivative2} it follows easily that
\begin{align*}
\partial_S(1+S^2)^\alpha\overline{K}_\alpha(U) &= -S(1+S^2)^{\alpha-1}\overline{K}_{\alpha-1}(U),\\
\partial_T(1+S^2)^\alpha\overline{K}_\alpha(U) &= -\frac{T}{2}(1+S^2)^{\alpha+1}\overline{K}_{\alpha+1}(U).
\end{align*}
Using these identities the proof is an easy computation.
\end{proof}
This motivates the definition of the functions $f_{j,k,\ell}$ in Theorem~\ref{thm:LKTSOpq}. We finally calculate how the Lie algebra $\mathfrak{k}_2\simeq\so(q)$ acts on $f_{j,k,\ell}$. For this note that
$$ \omega(x,Jv)\varphi = \varphi_v^++I_2^q\varphi_v^- $$
with
$$ \varphi_v^+ = \omega(x,Jv)\varphi+\frac{\sqrt{2}I_2^q\partial_v\varphi}{q+2\ell-5} \in \mathcal{H}^{\ell+1}(\mathbb{R}^{q-3}), \qquad \varphi_v^- = -\frac{\sqrt{2}\partial_v\varphi}{q+2\ell-5} \in \mathcal{H}^{\ell-1}(\mathbb{R}^{q-3}).\index{1wphivpm@$\varphi_v^\pm$} $$
\begin{proposition}\label{prop:LKTSOpqStep7}
For all $j,\ell\geq0$, $k\in\mathbb{Z}$ and $\varphi\in\mathcal{H}^\ell(\mathbb{R}^{q-3})$ the function $f_{j,k,\ell}\otimes\varphi$ is $\mathfrak{k}_1$-invariant and satisfies
\begin{align*}
d\pi_\min(T)f_{j,k,\ell}\otimes\varphi &= f_{j,k,\ell}\otimes(-\partial_{Tx}\varphi) && (T\in\mathfrak{k}_2\cap\mathfrak{g}_{(0,0)}\simeq\so(q-3)),\\
d\pi_\min(A-\overline{B})f_{j,k,\ell}\otimes\varphi &= ik\sqrt{2}f_{j,k,\ell}\otimes\varphi,
\end{align*}
and
\begin{align*}
&d\pi_\min(2i(B_\mu(A,Jv)+B_\mu(v,B))\mp\sqrt{2}(v+\overline{Jv}))f_{j,k,\ell}\otimes\varphi = 2\sqrt{2}(j\mp k)f_{j-1,k\pm1,\ell+1}\otimes\varphi_v^+\\
&\hspace{4cm}+ \Big[(q+j\mp k+2\ell-5)f_{j+1,k\pm1,\ell-1}+(j\mp k)f_{j-1,k\pm1,\ell-1}\Big]\otimes\varphi_v^-,\\
&id\pi_\min(Jv-\overline{v})f_{j,k,\ell}\otimes\varphi = (p-q-2j-2\ell)f_{j,k,\ell+1}\otimes\varphi_v^+\\
&\hspace{2.3cm}+ \tfrac{1}{2\sqrt{2}}\Big[(p-q-2\ell-2j)f_{j+2,k,\ell-1}+(p+q+2\ell-2j-10)f_{j,k,\ell-1}\Big]\otimes\varphi_v^-,\\
& d\pi_\min(-P+\overline{Q}\mp i\sqrt{2}T_0)f_{j,k,\ell}\otimes\varphi = \Big[(j\mp k)f_{j-1,k\pm1,\ell}-\tfrac{p-q-2j-2\ell}{2}f_{j+1,k\pm1,\ell}\Big]\otimes\varphi.
\end{align*}
\end{proposition}
\begin{proof}
With Lemma~\ref{lem:LKTSOpqUniquePolys} and Lemma~\ref{lem:LKTSOpqStep5} this is now an easy, though longish, computation using
\begin{equation*}
\partial_{2q}(1+S^2)^\alpha\overline{K}_\alpha(U) = -\sqrt{2}(1+S^2)^{\alpha+1}\overline{K}_{\alpha+1}(U).\qedhere
\end{equation*}
\end{proof}
This proves Theorem~\ref{thm:LKTSOpq}.
\section{The case $\mathfrak{g}=\so(p,3)$}\label{sec:LKTSOp3}
For $q=3$ we note that $\mathcal{H}^\ell(\mathbb{R}^{q-3})=\{0\}$ for $\ell>0$, so that the lowest $K$-type is spanned by $f_{j,k,0}$ ($0\leq j\leq\frac{p-3}{2}$, $-j\leq k\leq j$, $k\equiv j\mod2$). However, these functions cannot form a basis of $W\simeq\mathcal{H}^{\frac{p-3}{2}}(\mathbb{R}^3)\simeq S^{p-3}(\mathbb{C}^2)$ since $\dim W=p-2$, so the functions $f_{j,k,0}$ for fixed $k$ have to be linearly dependent.
\begin{lemma}
For $q=3$ and fixed $k\in\mathbb{Z}$ we have
$$ h_{j,k,0}(S,T) = i^{j+k}\sqrt{\frac{\pi}{2}}(1+S^2)^{-\frac{1}{2}}(\sqrt{1+S^2}-S)^k. $$
\end{lemma}
\begin{proof}
Since $q=3$ we have $I_2^q=0$ and therefore equation \eqref{eq:LKTSOpqPDE1} becomes
$$ \Big(-2\partial_T^2+(1+S^2)\Big)f_3 = 0 $$
which has the unique tempered solution $f_3(S,T)=g_4(S)e^{\sqrt{\frac{1+S^2}{2}}T}$. Plugging this into \eqref{eq:LKTSOpqPDE2} gives
$$ g_4'(S)+\left(\frac{S}{1+S^2}+\frac{k}{\sqrt{1+S^2}}\right)g_4(S)=0 $$
which has the solution
$$ g_4(S)=\const\times(1+S^2)^{-\frac{1}{2}}(\sqrt{1+S^2}-S)^k. $$
To find the constant we multiply both $h_{j,k,0}(S,T)$ and $f_3(S,T)$ by $(1+S^2)^{\frac{1}{2}}$ and let $S\to\pm i$ (for the unique analytic extensions of the functions). Using the explicit formulas for the $K$-Bessel function at half-integer parameters \eqref{eq:KBessel1/2} and \eqref{eq:KBesselHalfInt} shows that for $h_{j,k,0}(S,T)=\sum_{m=0}^jp_{j,k,m}(S,T)(1+S^2)^{m-\frac{1}{2}}\overline{K}_{m-\frac{1}{2}}(U)$ only the summand for $m=0$ survives. Comparing this with $(\sqrt{1+S^2}-S)^ke^{-\sqrt{U}}$ at $S=\pm i$. Shows the claim.
\end{proof}
\begin{remark}
It should be possible to obtain this expression for $h_{j,k,0}(S,T)$ from the explicit formulas for the $K$-Bessel function of half-integer parameters \eqref{eq:KBessel1/2} and \eqref{eq:KBesselHalfInt} once a closed formula for the polynomials $p_{j,k,m}(S,T)$ in Lemma~\ref{lem:LKTSOpqUniquePolys} is known.
\end{remark}
If we let
\begin{multline}
f_k(\lambda,a,x) = (\lambda-i\sqrt{2}a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\\
\times(1+S^2)^{-\frac{1}{2}}(\sqrt{1+S^2}+S)^{-k}\exp\left(\sqrt{\frac{1+S^2}{2}}T\right),\label{eq:LKTSOp3fkInt}\index{fk@$f_k$}
\end{multline}
then Proposition~\ref{prop:LKTSOpqStep7} can be reformulated as
\begin{equation*}
d\pi_\min(-P+\overline{Q}\mp i\sqrt{2}T_0)f_k = \left(\pm\frac{p-3}{2}-k\right)f_{k\pm1}.
\end{equation*}
We note that $(\lambda-i\sqrt{2}a)^k=\sgn(\lambda)^k(|\lambda|-i\sqrt{2}\sgn(\lambda)a)^k$ and, in view of the case $\mathfrak{g}=\sl(3,\mathbb{R})$, we put for even $p\in2\mathbb{N}$, $p\geq4$, and $k\in\mathbb{Z}+\frac{1}{2}$:
\begin{multline}
f_k(\lambda,a,x) = \sgn(\lambda)^{k-\frac{1}{2}}(|\lambda|-i\sqrt{2}\sgn(\lambda)a)^kR^{-\frac{k+1}{2}}\exp\left(-\frac{iaI_3}{\lambda R}\right)\\
\times(1+S^2)^{-\frac{1}{2}}(\sqrt{1+S^2}+S)^{-k}\exp\left(\sqrt{\frac{1+S^2}{2}}T\right).\label{eq:LKTSOp3fkHalfInt}\index{fk@$f_k$}
\end{multline}
Then it is easy to see that $f_k$ is still $\mathfrak{k}_1$-invariant. The subalgebra $\mathfrak{k}_2\simeq\so(3)\simeq\su(2)$ is spanned by the $\su(2)$-triple
$$ T_1=\tfrac{\sqrt{2}}{2}A+Q+\theta(\tfrac{\sqrt{2}}{2}A+Q), \qquad T_2=\tfrac{\sqrt{2}}{2}B-P+\theta(\tfrac{\sqrt{2}}{2}B-P), \qquad T_3=\sqrt{2}T_0-(E-F),\index{T1@$T_1$}\index{T2@$T_2$}\index{T3@$T_3$} $$
and it follows from the computations in Section~\ref{sec:LKTSOpq} that
\begin{equation*}
d\pi_\min(T_1)f_k = 2ikf_k, \qquad d\pi_\min(T_2\mp iT_3)f_k = \left(\pm(p-3)-2k\right)f_{k\pm1}.
\end{equation*}
This shows:
\begin{theorem}\label{thm:LKTSOp3}
Let $p\geq3$ be arbitrary. Then the space
$$ W={\mathrm span}\left\{f_k:k=-\frac{p-3}{2},-\frac{p-3}{2}+1,\ldots,\frac{p-3}{2}\right\}\index{W1@$W$} $$
with $f_k$ as in \eqref{eq:LKTSOp3fkInt} resp. \eqref{eq:LKTSOp3fkHalfInt} is a $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ isomorphic to the representation $\mathbb{C}\boxtimes S^{p-3}(\mathbb{C}^2)$ of $\mathfrak{k}\simeq\so(p)\oplus\su(2)$.
\end{theorem}
\chapter{$L^2$-models for minimal representations}
After having exhibited explicit $K$-finite vectors in the representation $d\pi_\min$ of $\mathfrak{g}$ on $\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$, we show in this section that these vectors generate an irreducible $(\mathfrak{g},K)$-module which integrates to an irreducible unitary representation of $G$ (or a covering) on $L^2(\mathbb{R}^\times\times\Lambda)$.
\section{Integration of the $(\mathfrak{g},K)$-module}\label{sec:Int(g,K)Module}
Let $W$ be the irreducible $\mathfrak{k}$-subrepresentation of $(d\pi_\min,\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda))$ constructed in Chapter~\ref{ch:LKT}. For $\mathfrak{g}=\sl(n,\mathbb{R})$, $W$ may be one of the $K$-types $W_{0,r}$ or $W_{1,r}$, where we now assume that $r\in i\mathbb{R}$ (in order for the Lie algebra representation $d\pi_{\min,r}$ to be infinitesimally unitary on $L^2(\mathbb{R}^\times\times\Lambda)$). For $\mathfrak{g}=\sl(3,\mathbb{R})$ we additionally allow $W_{\frac{1}{2}}$. Consider the $\mathfrak{g}$-subrepresentation generated by $W$:
$$ \overline{W}=d\pi_\min(U(\mathfrak{g}))W\subseteq\mathcal{D}'(\mathbb{R}^\times\times\Lambda).\index{W1@$\overline{W}$} $$
Then by standard arguments, $\overline{W}$ is a $(\mathfrak{g},K)$-module (see e.g. \cite[Lemma 2.23]{HKM14}).
\begin{proposition}\label{prop:WinL2}
$\overline{W}\subseteq L^2(\mathbb{R}^\times\times\Lambda)$.
\end{proposition}
\begin{proof}
We first observe that $W\subseteq L^2(\mathbb{R}^\times\times\Lambda)$. In fact, this follows from the asymptotic behavior of the $K$-Bessel function (see Section \ref{sec:BesselDiffEqAsymptotics}). For the more general statement $\overline{W}\subseteq L^2(\mathbb{R}^\times\times\Lambda)$ first note that for the Lie algebra $\mathfrak{q}$ of any standard parabolic subgroup of $G$ we have $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{q}$ which implies $U(\mathfrak{g})=U(\mathfrak{q})U(\mathfrak{k})$ by the Poincare--Birkhoff--Witt Theorem. For $\mathfrak{q}$ we may for instance choose
$$ \mathfrak{q}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{(-1,1)}\oplus\mathfrak{g}_{(0,0)}. $$
Then $\overline{W}=d\pi_\min(U(\mathfrak{q}))W$ can be computed using the identities \eqref{eq:BesselDerivative1} and \eqref{eq:BesselDerivative2}, the rest is technicality.
\end{proof}
\begin{lemma}\label{lem:WinfUnit}
The action of $\mathfrak{g}$ on $\overline{W}$ is infinitesimally unitary with respect to the inner product on $L^2(\mathbb{R}^\times\times\Lambda)$.
\end{lemma}
\begin{proof}
A heuristic proof is given by the observations in Remark \ref{rem:MotivationL2}. A rigorous proof can be obtained using the formulas for $d\pi_\min(X)$ ($X\in\mathfrak{g}$) and integrating by parts, showing directly that $d\pi_\min(X)$ is symmetric on $L^2(\mathbb{R}^\times\times\Lambda)$.
\end{proof}
To finally integrate $\overline{W}$ to a group representation, we make use of the following statement about its restriction to $\overline{P}_0$:
\begin{lemma}\label{lem:RepOfPbar}
The representation of $\overline{P}_0=M_0A\overline{N}$ on $L^2(\mathbb{R}^\times\times\Lambda)$ given by
\begin{align*}
\varpi(\overline{n}_{(z,t)})f(\lambda,x) &= e^{i\lambda t}e^{i(\omega(z'',x)+\frac{1}{2}\lambda\omega(z',z''))}f(\lambda,x-\lambda z'),\\
\varpi(m)f(\lambda,x) &= (\id_{\mathbb{R}^\times}^*\otimes\,\omega_{\met,\lambda^{-1}}(m))f(\lambda,x),\\
\varpi(\exp(sH))f(\lambda,x) &= e^{-\frac{\dim\Lambda+2}{2}s}f(e^{-2s}\lambda,e^{-s}x)\index{1pi@$\varpi$}
\end{align*}
is unitary and decomposes into the direct sum of irreducible subrepresentations $L^2(\mathbb{R}_+\times\Lambda)\oplus L^2(\mathbb{R}_-\times\Lambda)$. Moreover, if $\varpi$ extends to a unitary representation of $\overline{P}=MA\overline{N}$ on $L^2(\mathbb{R}^\times\times\Lambda)$, then this extension is irreducible.
\end{lemma}
\begin{proof}
Using the isomorphism
$$ L^2(\mathbb{R}^\times\times\Lambda) \simeq L^2(\mathbb{R}^\times,L^2(\Lambda)) $$
we observe that $\varpi|_{\overline{N}}$ is given by
$$ (\varpi(\overline{n})f)(\lambda) = \widetilde{\sigma}_\lambda[f(\lambda)] \qquad (f\in L^2(\mathbb{R}^\times,L^2(\Lambda))),$$
where $\widetilde{\sigma}_\lambda$ is the representation of $\overline{N}$ on $L^2(\Lambda)$ given by $\widetilde{\sigma}_\lambda(\overline{n}_{(z,t)})=\sigma_{\lambda^{-1}}(\overline{n}_{(\lambda z,\lambda^2t)})$. Since $\sigma_\lambda$ is irreducible for every $\lambda\in\mathbb{R}^\times$, it follows from Schur's Lemma that any intertwining operator $T:L^2(\mathbb{R}^\times,L^2(\Lambda))\to L^2(\mathbb{R}^\times,L^2(\Lambda))$ is of the form $Tf(\lambda,x)=t(\lambda)f(\lambda,x)$ for some measurable function $t$ on $\mathbb{R}^\times$. Now, $T$ also commutes with $A$ which implies $t(e^{-2s}\lambda)=t(\lambda)$ for all $s\in\mathbb{R}$, whence $t(\lambda)$ is constant on $\mathbb{R}_+$ and $\mathbb{R}_-$, respectively.\\
Now assume that $\varpi$ extends to $\overline{P}$ and let $m_0\in M$ with $\chi(m_0)=-1$ (which exists by Theorem \ref{thm:CharacterizationHermitian} since $G$ is non-Hermitian). The operator $\varpi(m_0)$ satisfies
$$ \varpi(m_0)\circ\varpi(\overline{n}_{(0,t)}) = \varpi(\overline{n}_{(0,-t)})\circ\varpi(m_0), $$
where $\varpi(\overline{n}_{(0,t)})f(\lambda,x)=e^{i\lambda t}f(\lambda,x)$. It follows that $\varpi(m_0)f(\lambda,x)=m(\lambda)U_xf(-\lambda,x)$ for some function $m(\lambda)$ and a unitary operator $U$ on $L^2(\Lambda)$. If now $T$ also commutes with $\varpi(m_0)$, it follows that $t(-\lambda)=t(\lambda)$ which implies that $t$ is constant on $\mathbb{R}^\times$ and hence $T$ is a scalar multiple of the identity.
\end{proof}
\begin{theorem}\label{thm:IntMinRep}
The $(\mathfrak{g},K)$-module $\overline{W}$ integrates to an irreducible unitary representation $\pi_\min$ of the universal cover $\widetilde{G}$ of $G$ on $L^2(\mathbb{R}^\times\times\Lambda)$. This representation is minimal in the sense that its annihilator in $U(\mathfrak{g}_\mathbb{C})$ is the Joseph ideal.
\end{theorem}
\begin{proof}
Using Proposition \ref{prop:WinL2}, Lemma \ref{lem:WinfUnit} and Lemma \ref{lem:RepOfPbar}, it follows along the same lines as in \cite[Proposition 2.27]{HKM14} that $\overline{W}$ is admissible. It therefore integrates to a representation $\pi_\min$ of $G$. This representation is unitary on a Hilbert space $\mathcal{H}\subseteq L^2(\mathbb{R}^\times\times\Lambda)$ by Proposition \ref{prop:WinL2} and Lemma \ref{lem:WinfUnit}. On the other hand, its restriction to $\overline{P}$ is given by the action in Lemma \ref{lem:RepOfPbar} which is irreducible on $L^2(\mathbb{R}^\times\times\Lambda)$. This implies $\mathcal{H}=L^2(\mathbb{R}^\times\times\Lambda)$ and $\pi_\min$ is irreducible. That the representation is minimal follows from Proposition \ref{prop:AnnihilatorJosephIdeal}.
\end{proof}
For $\mathfrak{g}=\sl(n,\mathbb{R})$ we write $\pi_{\min,\varepsilon,r}$\index{1piminepsilonr@$\pi_{\min,\varepsilon,r}$} for the representation with underlying $(\mathfrak{g},K)$-module $W_{\varepsilon,r}$, $r\in i\mathbb{R}$, and in the case $n=3$ we write $\pi_{\min,\frac{1}{2}}$\index{1pimin12@$\pi_{\min,\frac{1}{2}}$} for the representation with underlying $(\mathfrak{g},K)$-module $W_{\frac{1}{2}}$.
\section{Action of Weyl group elements}\label{sec:ActionWeylGroupElts}
The results from the previous section can also be phrased in a different way. The parabolic subgroup $\overline{P}$ acts unitarily and irreducibly on $L^2(\mathbb{R}^\times\times\Lambda)$ by the representation $\varpi$ (see Lemma~\ref{lem:RepOfPbar}). Theorem~\ref{thm:IntMinRep} shows that this representation extends to some covering group of $G$. This point of view was used in \cite[Theorem 2]{KS90} and \cite[Proposition 4.2]{Sav93} in order to construct the above $L^2$-models for the split groups $\SO(n,n)$, $E_{6(6)}$, $E_{7(7)}$, $E_{8(8)}$ and $G_{2(2)}$. There it is shown that $\varpi$ can be extended to an irreducible unitary representation $\pi_\min$ of $G$ by defining $\pi_\min$ on the representative of a certain Weyl group element $w_1$ and checking the Chevalley relations (see Section~\ref{sec:Bigrading} for the definition of $w_1$). This technique does not easily generalize to the case of non-split groups. However, after having constructed the $L^2$-model in a different way, we can obtain the action of $w_1$ as a corollary.
The answer depends on the eigenvalues of $d\pi_\min(A-\overline{B})$ on the lowest $K$-type $W$. Note that in all cases, $W$ is spanned by functions of the form
$$ f(\lambda,a,x) = (\lambda-i\sqrt{2}a)^kg(\lambda^2+2a^2,x) \qquad (\lambda>0) $$
with either $k\in\mathbb{Z}$ or $k\in\mathbb{Z}+\frac{1}{2}$. On such functions, $d\pi_\min(A-\overline{B})$ acts by $ik\sqrt{2}$. We refer to the \emph{integer case} if $k\in\mathbb{Z}$ and to the \emph{half-integer case} if $k\in\mathbb{Z}+\frac{1}{2}$. From the constructions in Chapter~\ref{ch:LKT} it follows that:
\begin{itemize}
\item For $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{6(6)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(-24)},\mathfrak{e}_{8(8)},\mathfrak{g}_{2(2)}$ the representation $\pi_\min$ belongs to the integer case.
\item For $\mathfrak{g}=\sl(n,\mathbb{R})$ the representations $\pi_{\min,0,r}$ and $\pi_{\min,1,r}$ ($r\in i\mathbb{R}$) belong to the integer case, and for $\mathfrak{g}=\sl(3,\mathbb{R})$ the representation $\pi_{\min,\frac{1}{2}}$ belongs to the half-integer case.
\item For $\mathfrak{g}=\so(p,q)$ the representation $\pi_\min$ belongs to the integer case if $p,q\geq3$, $p+q$ even, and it belongs to the half-integer case if $p\geq q=3$, $p$ even.
\end{itemize}
\begin{theorem}\label{thm:ActionW1}
The element $w_1$ acts in the $L^2$-model of the minimal representation by
\begin{equation}
\pi_\min(w_1)f(\lambda,a,x) = e^{-i\frac{n(x)}{\lambda a}}f(\sqrt{2}a,-\tfrac{\lambda}{\sqrt{2}},x)\times\begin{cases}1&\mbox{in the integer case,}\\\varepsilon(a\lambda)&\mbox{in the half-integer case,}\end{cases}\label{eq:ActionW1}
\end{equation}
where
$$ \varepsilon(x) = \begin{cases}1&\mbox{for $x>0$,}\\i&\mbox{for $x<0$.}\end{cases}\index{1epsilonx@$\varepsilon(x)$} $$
\end{theorem}
\begin{proof}
Let $A$ denote the unitary operator on $L^2(\mathbb{R}^\times\times\Lambda)$ given by \eqref{eq:ActionW1}. Then it is an easy computation to show that
$$ A\circ d\pi_\min(X)\circ A^{-1} = d\pi_\min(\Ad(w_1)X) \qquad \mbox{for all }X\in\mathfrak{g}. $$
It follows that $A\circ\pi_\min(w_1)^{-1}$ is a $\mathfrak{g}$-intertwining unitary operator on $L^2(\mathbb{R}^\times\times\Lambda)$ and therefore has to be a scalar multiple of the identity by Schur's Lemma. To find the scalar, we apply both $A$ and $\pi_\min(w_1)$ to a vector in the lowest $K$-type $W$. Both can be computed using the explicit description of $W$ in Chapter~\ref{ch:LKT}.
\end{proof}
\begin{remark}
As mentioned above, the formula for $\pi_\min(w_1)$ can be found in \cite[Theorem 2]{KS90} and \cite[Proposition 4.2]{Sav93} for the cases $G=\SO(n,n)$, $E_{6(6)}$, $E_{7(7)}$, $E_{8(8)}$ and $G_{2(2)}$. Note that these are all integer cases. In the half-integer case $G=\widetilde{\SL}(3,\mathbb{R})$ Torasso obtained the formula in \cite[Lemme 16]{Tor83}. In fact, he even obtained the action of the whole one-parameter subgroup $\exp(\mathbb{R}(A-\overline{B}))$ which should also be possible in general using the same methods as in Theorem~\ref{thm:ActionW1}.
\end{remark}
\begin{remark}
The restriction $\varpi$ of $\pi_\min$ to $\overline{P}$ together with the action $\pi_\min(w_1)$ of $w_1$ determines the representation $\pi_\min$ uniquely since $\overline{P}$ and $w_1$ generate $G$. This philosophy was advocated in \cite{KM11} where a different $L^2$-model for the minimal representation of ${\mathrm O}(p,q)$ was explicitly determined on a maximal parabolic subgroup and the representative of a non-trivial Weyl group element. Theorem~\ref{thm:ActionW1} can be seen as an analogue of their result for our $L^2$-models.
\end{remark}
\begin{remark}
It would be interesting to also find explicit formulas for the action of the Weyl group element $w_0$. One possible way to achieve this is by the help of an additional element
$$ w_2=\exp\left(\frac{\pi}{2\sqrt{2}}(B+\overline{A})\right). $$
We have the identity
$$ w_2 = w_1w_0w_1^{-1}, $$
so that $\pi_\min(w_0)$ and $\pi_\min(w_2)$ can be computed from each other using the previously obtained formula for $\pi_\min(w_1)$. Moreover, in all cases except $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ the elements $w_1$ and $w_2$ are conjugate via $M_0$ which acts in $\pi_\min$ via the metaplectic representation. It should be possible to use this in order to obtain a formula for $\pi_\min(w_2)$ and then also for $\pi_\min(w_0)$.
\end{remark}
Using the same technique as in Theorem~\ref{thm:ActionW1} we can obtain the action of $w_0^2$, $w_1^2$ and $w_2^2$ which are all contained in $M$, but may lie in different connected components of $M$.
\begin{proposition}\label{prop:ActionWeylGroupSquares}
The elements $w_0^2,w_1^2,w_2^2\in M$ act in the $L^2$-model of the minimal representation in the following way:
\begin{enumerate}[(1)]
\item In the quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ we have
\begin{align*}
\pi_\min(w_0^2)f(\lambda,a,x) &= (-1)^nf(\lambda,-a,-x),\\
\pi_\min(w_1^2)f(\lambda,a,x) &= f(-\lambda,-a,x),\\
\pi_\min(w_2^2)f(\lambda,a,x) &= (-1)^nf(-\lambda,a,-x),
\end{align*}
where $n=-s_\min-1=1,2,4$, i.e. the lowest $K$-type has dimension $2n+1$.
\item In the split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ we have
\begin{align*}
\pi_\min(w_0^2)f(\lambda,a,x) &= f(\lambda,-a,-x),\\
\pi_\min(w_1^2)f(\lambda,a,x) &= f(-\lambda,-a,x),\\
\pi_\min(w_2^2)f(\lambda,a,x) &= f(-\lambda,a,-x).
\end{align*}
\item In the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$ we have
\begin{align*}
\pi_\min(w_0^2)f(\lambda,a,x) &= -f(\lambda,-a,-x),\\
\pi_\min(w_1^2)f(\lambda,a,x) &= f(-\lambda,-a,x),\\
\pi_\min(w_2^2)f(\lambda,a,x) &= -f(-\lambda,a,-x).
\end{align*}
\item In the case $\mathfrak{g}=\sl(n,\mathbb{R})$ we have for $\varepsilon=0,1$, $r\in i\mathbb{R}$
\begin{align*}
\pi_{\min,\varepsilon,r}(w_0^2)f(\lambda,a,x) &= (-1)^\varepsilon f(\lambda,-a,-x),\\
\pi_{\min,\varepsilon,r}(w_1^2)f(\lambda,a,x) &= f(-\lambda,-a,x),\\
\pi_{\min,\varepsilon,r}(w_2^2)f(\lambda,a,x) &= (-1)^\varepsilon f(-\lambda,a,-x).
\end{align*}
\item In the case $\mathfrak{g}=\sl(3,\mathbb{R})$ we have
\begin{align*}
\pi_{\min,\frac{1}{2}}(w_0^2)f(\lambda,a) &= -i\sgn(\lambda)f(\lambda,-a),\\
\pi_{\min,\frac{1}{2}}(w_1^2)f(\lambda,a) &= if(-\lambda,-a),\\
\pi_{\min,\frac{1}{2}}(w_2^2)f(\lambda,a) &= -\sgn(\lambda)f(-\lambda,a).
\end{align*}
\item In the case $\mathfrak{g}=\so(p,q)$ with either $p\geq q\geq4$ and $p+q$ even or $p\geq q=3$ we have
\begin{align*}
\pi_\min(w_0^2)f(\lambda,a,x) &= (-i)^{p-q}\sgn(\lambda)^{p-q}f(\lambda,-a,-x),\\
\pi_\min(w_1^2)f(\lambda,a,x) &= f(-\lambda,-a,x)\times\begin{cases}1&\mbox{for $p+q$ even,}\\i&\mbox{for $p+q$ odd,}\end{cases}\\
\pi_\min(w_2^2)f(\lambda,a,x) &= (-i)^{p-q}\sgn(\lambda)^{p-q}f(-\lambda,a,-x)\times\begin{cases}1&\mbox{for $p+q$ even,}\\i&\mbox{for $p+q$ odd.}\end{cases}
\end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
As in the proof of Theorem~\ref{thm:ActionW1} we define a unitary operator $A$ on $L^2(\mathbb{R}^\times\times\Lambda)$ by the right hand side for one of the elements $m=w_0^2,w_1^2,w_2^2$ and show that it satisfies
$$ A\circ d\pi_\min(X)\circ A^{-1} = d\pi_\min(\Ad(m)X) \qquad \mbox{for all }X\in\mathfrak{g} $$
using the adjoint action of $m$ which was computed in Section~\ref{sec:W0} and Lemma~\ref{lem:W1W2}. Then, thanks to Schur's Lemma, $\pi_\min(m)=\const\times A$. To show that $\pi_\min(m)=A$ we apply $\pi_\min(m)$ and $A$ to a vector in the lowest $K$-type. Using the explicit formulas for vectors in the lowest $K$-type $W$ from Chapter~\ref{ch:LKT} one can compute $A$ on $W$, and the identification of $W$ with a finite-dimensional $\mathfrak{k}$-representation allows to compute $\pi_\min(m)=\exp(d\pi_\min(X))$ where $X=\pi(E-F)$ for $m=w_0^2$, $X=\frac{\pi}{\sqrt{2}}(A-\overline{B})$ for $m=w_1^2$ and $X=\frac{\pi}{\sqrt{2}}(B+\overline{A})$ for $m=w_2^2$. The latter only requires the representation theory of $\su(2)$, more precisely if $U_1,U_2,U_3\in\su(2)$ form an $\su(2)$-triple and $V_n$ is an irreducible representation of $\su(2)$ of dimension $n$ with basis $v_0,v_1,\ldots,v_n$ such that
$$ U_1\cdot v_k = i(n-2k)v_k, \qquad (U_2+iU_3)\cdot v_k = -2i(n-k)v_{k+1}, \quad (U_2-iU_3)\cdot v_k = -2ikv_{k-1}, $$
then
$$ \exp(\tfrac{\pi}{2}U_1)\cdot v_k = i^{n-2k}v_k, \quad \exp(\tfrac{\pi}{2}U_2)\cdot v_k = i^{-n}v_{n-k}, \quad \exp(\tfrac{\pi}{2}U_3)\cdot v_k = (-1)^{n-k}v_{n-k}. $$
In all cases except $\mathfrak{g}\simeq\so(p,q)$, $p\geq q\geq4$, $p+q$ even, the lowest $K$-type is an irreducible representation of $\su(2)$, and for $\mathfrak{g}\simeq\so(p,q)$ it is sufficient to consider the vectors $f=f_{0,0,\frac{p-q}{2}}\otimes\varphi$, $\varphi\in\mathcal{H}^{\frac{p-q}{2}}(\mathbb{R}^{q-3})$ which are invariant under $\so(p)\oplus\so(3)\subseteq\mathfrak{k}$ with $\so(3)\subseteq\so(q)$ spanned by
\begin{align*}
T_1 &= \tfrac{\sqrt{2}}{2}A+Q+\theta(\tfrac{\sqrt{2}}{2}A+Q) && \equiv \sqrt{2}(A-\overline{B}) \mod\so(p),\\
T_2 &= \tfrac{\sqrt{2}}{2}B-P+\theta(\tfrac{\sqrt{2}}{2}B-P) && \equiv \sqrt{2}(B+\overline{A}) \mod\so(p),\\
T_3 &= \sqrt{2}T_0-(E-F) && \equiv -2(E-F) \mod\so(p).\qedhere
\end{align*}
\end{proof}
The knowledge of $\pi_\min(m)$ for $m=w_0^2,w_1^2,w_2^2\in M$ allows us to obtain information about the induction parameter $\zeta$ for the corresponding degenerate principal series representation $I(\zeta,\nu)$ that contains $\pi_\min$ as a subrepresentation. We exclude the case $\mathfrak{g}=\so(p,q)$ since here $\zeta$ is infinite-dimensional (see Section \ref{sec:FTpictureMinRepSOpq} for details).
\begin{corollary}\label{cor:RelationZetaImageFT}
Assume that $\pi_\min$ is a subrepresentation of the degenerate principal series $I(\zeta,\nu)$.
\begin{enumerate}[(1)]
\item In the quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ the character $\zeta$ of $M/M_0$ satisfies
$$ \zeta(w_0^2)=1, \qquad \zeta(w_1^2)=\zeta(w_2^2)=(-1)^n, \qquad \mbox{where }n=-s_\min-1. $$
\item In the split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ the character $\zeta$ of $M/M_0$ satisfies
$$ \zeta(w_0^2)=\zeta(w_1^2)=\zeta(w_2^2)=1. $$
\item In the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$ the character $\zeta$ of $M/M_0$ satisfies
$$ \zeta(w_0^2)=1, \qquad \zeta(w_1^2)=\zeta(w_2^2)=-1. $$
\item In the case $\mathfrak{g}=\sl(n,\mathbb{R})$ with $\pi_\min=\pi_{\min,0,r}$, $r\in i\mathbb{R}$, the character $\zeta$ of $M$ satisfies
$$ \zeta(w_0^2)=\zeta(w_1^2)=\zeta(w_2^2)=1, $$
and for $\pi_\min=\pi_{\min,1,r}$, $r\in i\mathbb{R}$, it satisfies
$$ \zeta(w_0^2)=-1, \qquad \mbox{and either }\begin{cases}\zeta(w_1^2)=1\quad\mbox{and}\quad\zeta(w_2^2)=-1&\mbox{or}\\\zeta(w_1^2)=-1\quad\mbox{and}\quad\zeta(w_2^2)=1.&\end{cases} $$
\item In the case $\mathfrak{g}=\sl(3,\mathbb{R})$ with $\pi_\min=\pi_{\min,\frac{1}{2}}$ the representation $\zeta$ is the unique irreducible two-dimensional representation of the quaternion group $M\simeq\{\pm1,\pm i,\pm j,\pm k\}$.
\end{enumerate}
\end{corollary}
\begin{proof}
By Frobenius reciprocity, the lowest $K$-type $W$ as determined in Chapter \ref{ch:LKT} is contained in the degenerate principal series $\pi_{\zeta,\nu}$ if and only if
\begin{equation}
\Hom_{M\cap K}(W|_{M\cap K},\zeta|_{M\cap K}) \neq \{0\}.\label{eq:FrobeniusReciprocity}
\end{equation}
\begin{enumerate}[(1)]
\item In the quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$, the lowest $K$-type has to contain a non-zero vector $f\in W$ such that $d\pi_\min(\mathfrak{m}\cap\mathfrak{k})f=0$ and $\pi_\min(w_i^2)f=\zeta(w_i^2)f$, $i=0,1,2$. The first condition implies that $d\pi_\min(T_1)f=0$, and with \eqref{eq:SU2TripleActionQuat} it follows that
$$ f=\const\times\sum_{\substack{k=-n\\k\equiv n\mod2}}^n{n\choose\frac{k+n}{2}}f_k. $$
Acting by $\pi_\min(w_i^2)$ using Corollary \ref{cor:ActionWeylSquaresLKTquat} and comparing with $\zeta(w_i^2)$ shows the claim.
\item In the split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$, the lowest $K$-type $W$ is the trivial representation of $K$, hence the character $\zeta$ has to be trivial on $M\cap K$ by \eqref{eq:FrobeniusReciprocity}.
\item In the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$, the lowest $K$-type has to contain a non-zero vector $f\in W$ such that $d\pi_\min(\mathfrak{m}\cap\mathfrak{k})f=0$ and $\pi_\min(w_i^2)f=\zeta(w_i^2)f$, $i=0,1,2$. The first condition implies that $d\pi_\min(S_1)f=0$, and with Proposition~\ref{prop:G2Step5} it follows that $f=\const\times(f_1+f_{-1})$. Acting by $\pi_\min(w_i^2)$ using Corollary \ref{cor:ActionWeylSquaresLKTG2} and comparing with $\zeta(w_i^2)$ shows the claim.
\item In the case $\mathfrak{g}=\sl(n,\mathbb{R})$ with $\pi_\min=\pi_{\min,0,r}$ resp. $\pi_{\min,1,r}$, the lowest $K$-type is the trivial representation $\mathbb{C}$ of $K=\SO(n)$ resp. the standard representation $\mathbb{C}^n$ of $K=\SO(n)$. In the first case, it is clear that $\zeta|_{M\cap K}$ must be the trivial representation. In the second case, the lowest $K$-type must contain a non-zero vector $f$ such that $d\pi_\min(\mathfrak{m}\cap\mathfrak{k})f=0$ and $\pi_\min(w_i^2)f=\zeta(w_i^2)f$, $i=0,1,2$. The first condition implies that $f=c_0f_0+c_1f_1$. By Proposition \ref{prop:ActionWeylGroupSquares} we find
\begin{align*}
\pi_{\min,1}(w_0^2)f_0 &= -f_0, & \pi_{\min,1}(w_1^2)f_0 &= -f_0, & \pi_{\min,1}(w_2^2)f_0 &= f_0,\\
\pi_{\min,1}(w_0^2)f_1 &= -f_1, & \pi_{\min,1}(w_1^2)f_1 &= f_1, & \pi_{\min,1}(w_2^2)f_1 &= -f_1,
\end{align*}
which implies $\zeta(w_0^2)=-1$ and either $c_0=0$ or $c_1=0$. The claim follows.
\item In the case $\mathfrak{g}=\sl(3,\mathbb{R})$ the restriction of the lowest $K$-type $W\simeq\mathbb{C}^2$ to $M\subseteq K$ is irreducible and two-dimensional.\qedhere
\end{enumerate}
\end{proof}
\begin{remark}
For the adjoint group $G=G_{2(2)}$, the subgroup $M$ has two connected components (see e.g. \cite[Section 2]{Kab12}) and since $\chi(w_1^2)=\chi(w_2^2)=-1$ it follows that $w_1^2,w_2^2\in M$ are contained in the non-trivial component. Therefore, Corollary \ref{cor:RelationZetaImageFT} determines the character $\zeta$ completely in this case. We believe that a similar statement is true for the other cases. Note that even for the non-linear group $G=\widetilde{\SL}(3,\mathbb{R})$ for which $M$ has $8$ connected components, the elements $w_0^2,w_1^2,w_2^2$ generate the component group $M/M_0$ and hence any representation $\zeta$ of $M$ is uniquely determined by $\zeta(w_0^2),\zeta(w_1^2),\zeta(w_2^2)$.
\end{remark}
Finally, we are able to describe the precise principal series embedding $\pi_\min\hookrightarrow\pi_{\zeta,\nu}$. Recall that for $u\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$ (or the corresponding subrepresentations in the cases $\mathfrak{g}\simeq\sl(n,\mathbb{R})$ and $\so(p,q)$) it was shown that
$$ \widehat{u}(\lambda,x,y) = \xi_{-\lambda,0}(x)u_0(\lambda,y)+\xi_{-\lambda,1}(x)u_1(\lambda,y) $$
for some $u_0,u_1\in\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)$. Recall further the map
$$ \Phi_\delta:\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda)\to\mathcal{D}'(\mathbb{R}^\times)\otimeshat\mathcal{S}'(\Lambda), \quad \Phi_\delta u(\lambda,x) = \sgn(\lambda)^\delta|\lambda|^{-s_\min}u(\lambda,\tfrac{x}{\lambda}),\index{1Phidelta@$\Phi_\delta$} $$
then it was shown in Sections \ref{sec:FTpictureMinRep} that $u\mapsto\Phi_\delta u_\varepsilon$ is $\mathfrak{g}$-intertwining from $d\pi_{\zeta,\nu}$ to $d\pi_\min$ for any $\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$. Note that $u_\varepsilon$ could be zero. We determine for which $\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ the map $u\mapsto\Phi_\delta u_\varepsilon$ is $G$-intertwining from $\pi_{\zeta,\nu}$ to $\pi_\min$. Note that the case $\mathfrak{g}=\sl(n,\mathbb{R})$ is excluded since here $\widehat{u}(\lambda,x,y)=u_0(\lambda,y)$, so there is no $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ to determine.
\begin{corollary}\label{cor:PSEmbedding}
Let $u\in I(\zeta,\nu)^{\Omega_\mu(\mathfrak{m})}$, then
$$ \widehat{u}(\lambda,x,y) = \xi_{-\lambda,\varepsilon}(x)u_\varepsilon(\lambda,y) $$
and the map $u\mapsto\Phi_\delta u_\varepsilon$ is $G$-intertwining from $\pi_{\zeta,\nu}$ to $\pi_\min$, where
\begin{enumerate}[(1)]
\item In the quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$ we have
$$ \delta=\varepsilon=n=-s_\min-1. $$
\item In the split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$ we have
$$ \delta=\varepsilon=0. $$
\item In the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$ we have
$$ \delta=\varepsilon=1. $$
\end{enumerate}
\end{corollary}
\begin{proof}
Let $u\in I(\zeta,\nu)$ such that
$$ \widehat{u}(\lambda,x,y) = \xi_{-\lambda,0}(x)u_0(\lambda,y) + \xi_{-\lambda,1}(x)u_1(\lambda,y). $$
For $m\in M$ we have by Proposition \ref{prop:ActionFTpicture}
\begin{multline}
\widehat{\pi}_{\zeta,\nu}(m)\widehat{u}(\lambda,x,y) = \zeta(m)\Bigg(\omega_{\met,-\lambda}(m)\xi_{-\chi(m)\lambda,0}(x)\cdot\omega_{\met,\lambda}(m)u_0(\chi(m)\lambda,y)\\
+ \omega_{\met,-\lambda}(m)\xi_{-\chi(m)\lambda,1}(x)\cdot\omega_{\met,\lambda}(m)u_1(\chi(m)\lambda,y)\Bigg).\label{eq:RelationZetaImageFT1}
\end{multline}
On the other hand, if $d\rho_\min$ integrates to the group representation $\rho_\min$ it follows that
\begin{equation}
\widehat{\pi}_{\zeta,\nu}(m)\widehat{u}(\lambda,x,y) = \xi_{-\lambda,0}(x)\cdot\rho_\min(m)u_0(\lambda,y) + \xi_{-\lambda,1}(x)\cdot\rho_\min(m)u_1(\lambda,y).\label{eq:RelationZetaImageFT2}
\end{equation}
We compare \eqref{eq:RelationZetaImageFT1} and \eqref{eq:RelationZetaImageFT2} for $m=w_0^2,w_1^2,w_2^2$. Note that for $m=w_0^2,w_1^2,w_2^2$ the action $\omega_{\met,\lambda}(m)$ can be computed using \eqref{eq:DefMetaplecticRep} and \eqref{eq:DefSchroedingerModel} as well as the adjoint action $\Ad(m)$ on $V=\mathfrak{g}_{-1}$ which is known by Section~\ref{sec:W0} and Lemma~\ref{lem:W1W2}:
\begin{align*}
\omega_{\met,\lambda}(w_0^2)u(a,y) &= \pm u(-a,-y),\\
\omega_{\met,\lambda}(w_1^2)u(a,y) &= \pm u(a,-y),\\
\omega_{\met,\lambda}(w_2^2)u(a,y) &= \pm u(-a,y).
\end{align*}
Moreover, we have
$$ \chi(w_0^2)=1 \qquad \mbox{and} \qquad \chi(w_1^2)=\chi(w_2^2)=-1. $$
Finally, $\rho_\min(m)=\Phi_\delta^{-1}\circ\pi_\min(m)\circ\Phi_\delta$ with $\pi_\min(m)$ given by Proposition \ref{prop:ActionWeylGroupSquares}.
\begin{enumerate}[(1)]
\item In the quaternionic cases $\mathfrak{g}=\mathfrak{e}_{6(2)},\mathfrak{e}_{7(-5)},\mathfrak{e}_{8(-24)}$, comparing \eqref{eq:RelationZetaImageFT1} and \eqref{eq:RelationZetaImageFT2} shows that
$$ \zeta(w_0^2)=(-1)^{\varepsilon+n}, \qquad \zeta(w_1^2)=(-1)^\delta, \qquad \zeta(w_2^2)=(-1)^{\delta+\varepsilon+n}, $$
where $n=-s_\min-1$ and
\begin{align*}
\widehat{u}(\lambda,x,y) = \xi_{-\lambda,\varepsilon}(x)u_\varepsilon(y).
\end{align*}
Comparing with Corollary \ref{cor:RelationZetaImageFT} shows $\delta=\varepsilon=n$.
\item In the split cases $\mathfrak{g}=\mathfrak{e}_{6(6)},\mathfrak{e}_{7(7)},\mathfrak{e}_{8(8)}$, comparing \eqref{eq:RelationZetaImageFT1} and \eqref{eq:RelationZetaImageFT2} shows that
$$ \zeta(w_0^2)=(-1)^\varepsilon, \qquad \zeta(w_1^2)=(-1)^\delta, \qquad \zeta(w_2^2)=(-1)^{\delta+\varepsilon}. $$
Comparing with Corollary \ref{cor:RelationZetaImageFT} shows $\delta=\varepsilon=0$.
\item In the case $\mathfrak{g}=\mathfrak{g}_{2(2)}$, comparing \eqref{eq:RelationZetaImageFT1} and \eqref{eq:RelationZetaImageFT2} shows that
$$ \zeta(w_0^2)=(-1)^{\varepsilon+1}, \qquad \zeta(w_1^2)=(-1)^\delta, \qquad \zeta(w_2^2)=(-1)^{\delta+\varepsilon+1}. $$
Comparing with Corollary \ref{cor:RelationZetaImageFT} shows $\delta=\varepsilon=1$.\qedhere
\end{enumerate}
\end{proof}
|
2,877,628,089,468 | arxiv | \section{Introduction}
After they decouple from the plasma at recombination, the CMB photons move freely along the null geodesics of the curved spacetime. Obviously, the exact geodesic lines are slightly modified by cosmological perturbations as compared to the unperturbed geodesics. In general, this small change can be decomposed into a component along the line of sight causing a redshift in the frequency and a displacement perpendicular to the line of sight yielding lensing effects. The gravitational lensing of CMB photons is well studied, for a review see \cite{glcmb}, and the effect has also precise observational signatures \cite{ob1,ob2}.
Previous studies mostly focus on how the CMB temperature map on the sky is modified by the lensed angular positions of the CMB photons. The deflection angle caused by lensing becomes a pure gradient and the corresponding potential is introduced as the main statistical variable. There is also some work on the lensing shear effect that modifies the so called hot and cold spot CMB ellipticity distribution \cite{el1,el2,el3}. Since the background temperature map is uniform, these are nonlinear effects that appear at the second order. Moreover, they are also dominated by the density perturbations; the influence of the tensor modes is completely negligible. Yet, it is also possible to extract a rotational component of the shear which have contributions only from the gravitational waves \cite{gw0,gw1} (this is similar to the $B$-mode of the CMB polarization) but unfortunately the presumed signal is very small and below the noise level \cite{gw1,gw2,gw3}.
The gravitational lensing effects can also be studied using the geodesic deviation equation. In that framework, one calculates the expansion, shear and rotation parameters as the basic geometrical variables of the congruence. In a recent work \cite{ak}, we have instead determined the induced two dimensional metric on the transverse cross section of a null geodesic beam in a perturbed FRW background. We have shown that the transverse metric does not depend on the slicing and its derivative along the geodesic flow can be decomposed to yield the expansion, shear and rotation. Clearly, the induced metric offers a direct geometrical description of a photon congruence.
Consider the evolution of a CMB photon beam back in time from the moment of its capture by a telescope today to the time of decoupling. The transverse slice corresponding to the telescope collection area is mapped to another slice of the beam at decoupling. The distribution of the trajectories on the initial surface is expected to be uniform since the photons are in local thermal equilibrium. However, the distribution of the photons hitting the telescope surface today would in general be nonuniform because of the gravitational shearing. Each photon trajectory marks a point on a transverse slice of the beam and we call the distribution of these points {\it the intensity profile} of the congruence. In \cite{ak} we have shown that the CMB intensity profile is characterized by two variables that are reminiscent of the Stokes polarization parameters. In this work, we will elaborate more on these variables; specifically we will construct a pseudo scalar quantity which is only generated by the primordial gravitational waves, similar to the $B$-mode of polarization.
\section{Null geodesics on perturbed FRW}
Let us look at the null geodesics on the following perturbed FRW spacetime
\begin{equation}\label{met}
ds^2=a(\eta)^2\left[-(1+2\Psi)d\eta^2+[(1+2\Phi)\delta_{ij}+\gamma_{ij}]dx^i dx^j\right].
\end{equation}
By definition the tensor mode is traceless $\delta^{ij}\gamma_{ij}=0$ and at the moment we do not impose any gauge fixing conditions. In the present study, we will only work in the linear theory, therefore {\em all equations written below must be assumed to be valid up to the first order in perturbations.}
To determine the null geodesic trajectories, one can first solve for the {\it tangent vector field} on the spacetime obeying
\begin{equation}
p^\mu\nabla_\mu \,p^\nu=0, \hs{7} p^\mu p_\mu=0.
\end{equation}
Defining the perturbations around the unperturbed field by
\begin{eqnarray}
&&p^0=\frac{1}{a^2}+\delta p^0,\nonumber \\
&&p^i=\frac{l^i}{a^2}+\delta p^i,\hs{5}\delta_{ij}l^i l^j=l^il^i=1,\label{g}
\end{eqnarray}
one can fix $\delta p^0$ from $p^\mu p_\mu=0$ as
\begin{equation}\label{dp0}
\delta p^0=l^i\delta p^i+\frac{1}{a^2}\left(\Phi-\Psi\right)+\frac{1}{2a^2}\gamma_{ij}l^i l^j
\end{equation}
and solve for $\delta p^i$ so that
\begin{eqnarray}
&&\delta p^i(x,\eta)=-\frac{2}{a(\eta)^2}l^i\,\Phi(x,\eta)-\frac{1}{a(\eta)^2}\gamma_{ij}(x,\eta)\,l^j \nonumber\\
&&+\frac{1}{a(\eta)^2}\partial_i\int_{\eta_0}^{\eta} d\eta'\left[\Phi- \Psi+\fr12 \gamma_{jk}l^j l^k\right]\left(x_{\eta'\eta},\eta'\right)\nonumber\\
&&+\frac{1}{a(\eta)^2}\left[2l^i\,\Phi+\gamma_{ij}\,l^j\right](x_{\eta_0\eta},\eta_0)\label{dpi}
\end{eqnarray}
where $\eta_0$ is the present conformal time and we introduce $x^i_{\eta_1\eta_2}$ to be the spatial position on an unperturbed geodesic path
\begin{equation}\label{xl}
x_{\eta_1\eta_2}^i=x^i+l^i(\eta_1-\eta_2).
\end{equation}
This is the unique solution obeying the condition
\begin{equation}\label{ic0}
\delta p^i(x,\eta_0)=0
\end{equation}
hence $p^i(x,\eta_0)=l^i/a(\eta_0)^2=l^i$, where we also set $a(\eta_0)=1$. As a result, $l^i$ defines the present direction of propagation and the actual line of sight including the lensing effects. Note that the time argument of the fields in the last line in \eq{dpi} is the present time $\eta_0$. These terms arise since we demand \eq{ic0} and they vanish when the derivative operator along the unperturbed geodesic $\partial_\eta+l^i\partial_i$ is applied.
These equations are worked out for a photon having ``unit" energy and the general case can be obtained by scaling $p^\mu\to E p^\mu$. Our results below do not depend on the parameter $E$ and therefore we are not going to introduce it.
It is possible to obtain the Sachs-Wolfe effect using the above equations. The 4-velocity vector of a comoving observer in \eq{met} (obeying $u^\mu u_\mu=-1$) can be found as
\begin{equation}\label{u}
u^0=\frac{1}{a}(1-\Psi),\hs{5}u^i=0
\end{equation}
and the energy of a photon as measured by this observer is given by
\begin{equation}\label{up}
\omega=-u^\mu p_\mu.
\end{equation}
One can see that
\begin{equation}\label{w}
\omega=p=\frac{1}{a}\left[1+a^2 l^i\delta p^i+\Phi+\fr12\gamma_{ij}l^i l^j\right],
\end{equation}
where
\begin{equation}
p=\sqrt{g_{ij}p^i p^j }.
\end{equation}
As it was first observed in \cite{rd}, \eq{w} encodes the Sachs-Wolfe effect if one defines $T\propto \omega$. Indeed, by applying the derivative along the geodesic trajectory $p^\mu\partial_\mu$ one can see
\begin{equation}\label{sw}
\left(\frac{\partial}{\partial \eta}+l^j\partial_j\right)\frac{\delta T}{T}=-\Phi'-l^i\partial_i\Psi-\fr12 \gamma_{ij}'l^il^j,
\end{equation}
which exactly gives the evolution of the temperature fluctuations along the unperturbed geodesic lines. In \cite{ak} we have shown that \eq{sw} is valid for any (and not necessarily thermal) distribution function provided one reads the temperature from the average intensity by $T^4\propto I$.
One can obtain the geodesic path $x^\mu(\lambda)$ by integrating
\begin{equation}\label{tv}
\frac{dx^\mu}{d\lambda}=p^\mu(x(\lambda)),
\end{equation}
where $\lambda$ is an affine parameter. The zeroth component of the above equation can be used to relate $\lambda$ and the conformal time as
\begin{equation}
\frac{d}{d\lambda}=\left(\frac{1}{a^2}+\delta p^0\right)\frac{d}{d\eta}.
\end{equation}
Defining the perturbed geodesic
\begin{equation}\label{gx}
x^i(\eta)=x_0^i +l^i(\eta-\eta_0)+\delta x^i(\eta),
\end{equation}
\eq{tv} implies
\begin{equation}
\frac{d\delta x^i}{d\eta}=a^2\delta p^i - a^2l^i\delta p ^0.
\end{equation}
After using \eq{dp0}, one can integrate to obtain
\begin{eqnarray}
&&\delta x^i(\eta)=\left[\delta^{ij}-l^i l^j\right]\int_{\eta_0}^\eta d\eta'a^2(\eta')\,\delta p^j(x_{0\eta'\eta_0},\eta')\nonumber\\
&&\hs{3}+l^i\int_{\eta_0}^\eta d\eta'\left[\Psi-\Phi-\fr12\gamma_{jk}l^j l^k\right](x_{0\eta'\eta_0},\eta'), \label{dxi}
\end{eqnarray}
where $\delta p^j$ is found in \eq{dpi} and the spatial argument of the functions $x_{0\eta'\eta_0}$ stands for $x^i_{0\eta'\eta_0}=x^i_0+l^i(\eta'-\eta_0)$,
as in \eq{xl}. Note that \eq{dxi} obeys $\delta x^i(\eta_0)=0$ and thus \eq{gx} yields the unique null geodesic path which passes from the spatial position $x^i_0$ at time $\eta_0$ along the direction $l^i$.
\section{The geometry of the photon beam and gravitational shearing}
Eq. \eq{gx}, where $\delta x^i$ is given in \eq{dxi}, actually describes a family of geodesics parametrized by the constants $x^i_0$ and $l^i$. A photon beam observed at $\eta_0$ along direction $l^i$ corresponds to a (small) subset in that family. Let $\Delta x_0^i$ denote the coordinate difference between two nearby geodesic lines at $\eta_0$. The time evolution of this interval can be found from the solution \eq{gx}
\begin{equation}\label{tev}
\Delta x^i(\eta)=\Delta x_0^i +\Delta \delta x^i(\eta),
\end{equation}
where $\Delta \delta x^i(\eta)$ is obtained by varying \eq{dxi} with respect $x^i_0$. The corresponding physical length is given by the metric
\begin{equation}\label{tmet1}
\left|\Delta x^i(\eta)\right|=\left[g_{ij}(\eta)\Delta x^i(\eta)\Delta x^j(\eta)\right]^{1/2}.
\end{equation}
At the time of observation, the transverse cross section of the beam can be specified by the vectors $m^i$ and $n^i$, where $(l^i,m^i,n^i)$ forms an orthonormal set with respect to $\delta_{ij}$ (the impact of the metric perturbations at that instant is completely negligible). The evolution of the transverse beam cross section can be found by choosing $\Delta x_0^i=L m^i$ or $\Delta x_0^i=L n^i$ in \eq{tev}, where $L$ is the telescope size. We have checked that the two dimensional metric obtained from \eq{tmet1} (involving the displacements $\Delta x_0^i=L m^i$ and $\Delta x_0^i=L n^i$) exactly agrees with the slicing independent transverse metric obtained from the geodesic deviation equation in \cite{ak} ($l^i$ had been chosen as the geodesic cotangent at the time of decoupling in \cite{ak}).
We now compare the physical lengths of the two transverse directions $Lm^i$ and $Ln^i$ at the time of recombination $\eta_r$. Their difference equals $La(\eta_r)Q$, where we define
\begin{equation}\label{uu}
Q=\left[\fr12 \gamma_{ij}(x_{0\eta_r\eta_0},\eta_r)+\partial_{x^j_0}\delta x^i(\eta_r)\right](m^i m^j-n^i n^j).
\end{equation}
One can also determine the size difference between $\pi/4$ rotated directions $(m^i+n^i)/\sqrt{2}$ and $(m^i-n^i)/\sqrt{2}$, which can be found as $La(\eta_r)U$, where
\begin{equation}\label{vv}
U=\left[\fr12 \gamma_{ij}(x_{0\eta_r\eta_0},\eta_r)+\partial_{x^j_0}\delta x^i(\eta_r)\right](m^i n^j+n^i m^j).
\end{equation}
The two parameters $Q$ and $U$, which depend on the directions $(l^i,m^i,n^i)$, identify the shape of the initial transverse surface at the time of recombination, {\it which has a uniform photon distribution over it.} Obviously, while this initial surface evolves to become the (circular) cross section today, the photons are diluted in the direction that expands more compared to the other, see Fig \ref{fig1}.
\begin{figure}
\centerline{\includegraphics[width=9cm]{f1}}
\caption{CMB photons hitting a detector. On the left there is a uniform distribution over the area but on the right the photons are diluted along the $x$-axis as compared to the $y$-axis due to gravitational lensing.}
\label{fig1}
\end{figure}
The phase space volume element along a geodesic flow does not change by Liouville's theorem and this leads to the standard rule that gravitational lensing does not modify specific intensity, see e.g. \cite{mtw}. For the variables $Q$ and $U$, this result is avoided since these do not directly measure the intensity; instead they are related to the distribution of photons over a transverse surface (which we call the intensity profile of the beam). Obviously, the validity of the particle description is crucial for the observability of this effect. Relying on the photon picture, one can quantify the surface distribution by measuring the energy flux over narrow slits instead of the whole area. For a given wavelength, the slit width must be small enough so that the usual concept of intensity fails (note that intensity is a coarse grained concept in the photon picture). In that case, $Q$ becomes proportional to the energy flux difference between two slits extending along $m^i$ and $n^i$ directions, see Fig. \ref{fig2}. Likewise, the flux difference between $\pi/4$ rotated slits gives $U$.
\begin{figure}
\centerline{\includegraphics[width=9cm]{f2}}
\caption{The lensed photons impinging on two narrow slits instead of the whole detector area. The observed ``intensities" are not equal because the number of incident photons is different for each case.}
\label{fig2}
\end{figure}
One can simplify \eq{uu} and \eq{vv} to a very good approximation by computing the leading order contributions. From \eq{dxi}, the terms coming from $\delta x^i(\eta_r)$ can be seen to appear inside single or double time integrals, which give oscillating contributions. The tensor modes in these integrals are negligible compared to the first term in \eq{uu} and \eq{vv}. Using also $\Psi\simeq -\Phi$ and ignoring the monopole term, one can obtain
\begin{eqnarray}
&&Q\simeq\left[\fr12 \gamma_{ij}(x_{0\eta_r\eta_0},\eta_r)+K_{ij}\right](m^i m^j-n^i n^j)\nonumber \\
&&U\simeq\left[\gamma_{ij}(x_{0\eta_r\eta_0},\eta_r)+2K_{ij}\right]m^i n^j,\label{uva}
\end{eqnarray}
where
\begin{equation}\label{kij}
K_{ij}=2\frac{\partial^2}{\partial x_0^i \partial x_0^j}\int_{\eta_0}^{\eta_r}d\eta'(\eta_r-\eta')\Phi(x_{0\eta'\eta_0},\eta').
\end{equation}
The explicit $(\eta_r-\eta')$ factor in $K_{ij}$ appears after changing the order of a double time integral.
In \eq{uva} the scalar mode contributions involve an oscillating time integral but they are still expected to dominate the power spectra over gravitational waves. Therefore, it is desirable to construct a variable which only depends on the tensor modes. The doublet $(Q,U)$ rotates by $2\alpha$ when the tangent vectors $(m,n)$ are rotated by $\alpha$. Thus they constitute spin-weight-two objects on the sphere. The infinitesimal variations of $l^i$ (that respect the constraint $l^i l^i=1$) can be parametrized like
\begin{equation}
\delta l^i= (\delta a) m^i + (\delta b) n^i,
\end{equation}
where the doublet $(\delta a,\delta b)$ has spin-weight $-1$. The derivative operator $(\delta^2/\delta a^2-\delta^2/\delta b^2,2\delta^2/\delta a\delta b)$ has spin-weight 2 and by applying it on $(Q,U)$ with Kronecker delta and epsilon tensor contractions, one can obtain spin-weight-zero scalar and pseudo-scalar on the sphere
\begin{eqnarray}
&&E(l)=\left[\frac{\delta^2}{\delta a^2}-\frac{\delta^2}{\delta b^2}\right]Q+\frac{2\delta^2U}{\delta a\delta b},\nonumber\\
&&B(l)=\frac{2\delta^2Q}{\delta a\delta b}-\left[\frac{\delta^2}{\delta a^2}-\frac{\delta^2}{\delta b^2}\right]U. \label{EB1}
\end{eqnarray}
It is easy to see that $\Phi$ drops out in $B$ which indeed becomes
\begin{eqnarray}
B(l)&=&(\eta_0-\eta_r)^2\,\frac{\partial^2 \gamma_{ij}(x_{0\eta_r\eta_0},\eta_r)}{\partial x^k_0\, \partial x^l_0}\\
&&\left[m^k n^l(m^i m^j-n^i n^j)-m^in^j(m^km^l-n^kn^l)\right].\nonumber
\end{eqnarray}
Just like in the polarization case, $E$ and $B$ represent curl and divergence free field lines, this time, formed by the ``eigen-directions" of $(Q,U)$ on the sphere (the eigen-direction at a given point can be defined from one of the vectors of the basis $(m,n)$ in which $(Q,U)$ becomes proportional to $(1,0)$, i.e. one has $U=0$).
\section{The power spectra}
We work out the power spectra of these variables at superhorizon scales in a simplified model having only two epochs, inflation and radiation. The scale factor in such a model is given by
\begin{equation}\label{mct}
a(\eta)=\begin{cases}-\frac{1}{H_I\eta}&\eta\leq \eta_I,\vs{3} \\
H_0(\eta-2\eta_I)&\eta_I\leq \eta,
\end{cases}
\end{equation}
where $\eta_I=-1/\sqrt{H_IH_0}$, and $H_I$ and $H_0$ are the Hubble parameters at inflation and today, respectively. The form of \eq{mct} is fixed by demanding the continuity of the scale factor and the Hubble parameter at $\eta_I$. The present conformal time can be found from $a(\eta_0)=1$ which gives $\eta_0=1/H_0+2\eta_I$. The redshift at recombination is given by
\begin{equation}\label{zr}
z_r=\frac{\eta_0}{\eta_r}
\end{equation}
and one may take $z_r\simeq 10^3$. Note the following hierarchy $\eta_0\gg\eta_r\gg|\eta_I|$.
The mode function of a minimally coupled massless scalar field that is released in its Bunch-Davies vacuum at inflation is given by
\begin{equation}\label{bdv}
\mu_k=\begin{cases}\frac{1}{\sqrt{2k}}\left[1-\frac{i}{k\eta}\right]e^{-ik\eta} & \eta\leq\eta_I,\vs{3}\\
\mu_k^I\cos\left[k(\eta-\eta_I)\right]+\frac{\mu_k^I{}'}{k}\sin\left[k(\eta-\eta_I)\right] & \eta_I<\eta,
\end{cases}
\end{equation}
where $\mu_k^I=\mu_k(\eta_I)$, $\mu_k^I{}'=\mu_k'(\eta_I)$ and prime denotes $\eta$ derivative.
The tensor perturbation can be expanded in terms of the mode functions
\begin{equation}
\gamma_{ij}=\frac{1}{(2\pi)^{3/2}}\int d^3k\, e^{i\vec{k}.\vec{x}}\,\gamma_k(\eta)\, \epsilon_{ij}^s \tilde{a}^s_{\vec{k}}+h.c.\nonumber
\end{equation}
where $s=1,2$ and the creation-annihilation operators satisfy the usual commutation relations, e.g. $[a_k,a^\dagger_{k'}]=\delta^3(k-k')$. The polarization tensor $\epsilon^s_{ij}$ has the following properties
\begin{eqnarray}
&&k^i\epsilon^s_{ij}=0,\hs{5} e^s_{ii}=0,\hs{5}\epsilon^s_{ij}e^{s'}_{ij}=2\delta^{ss'}.\nonumber\\
&& \epsilon^s_{ij}e^{s}_{kl}=P_{ik}P_{jl}+P_{il}P_{jk}-P_{ij}P_{kl},\label{tme}
\end{eqnarray}
where $P_{ij}=\delta_{ij}-k^i k^j/k^2$. The tensor mode function $\gamma_k(\eta)$ can be determined in terms of $\mu_k(\eta)$ in \eq{bdv} as
\begin{equation}\label{cck}
\gamma_k=\frac{2}{a M_p}\mu_k,
\end{equation}
where $M_p$ is the reduced Planck mass $M_p^2=1/(8\pi G)$.
The gravitational potential $\Phi$ is determined from the curvature perturbation $\zeta$, which is conserved at super-horizon scales and can be expanded like \eq{tme}. The corresponding mode function during inflation can be taken as
\begin{equation}\label{zk}
\zeta_k=\frac{1}{a \sqrt{2\epsilon} M_p}\mu_k,
\end{equation}
where $\epsilon$ is the slow-roll parameter (for constant $\epsilon$, $\zeta_k$ is actually given by the first Hankel function but \eq{zk} is a very good approximation when $\epsilon\ll1$). The standard gauge fixing breaks down in reheating after inflation but there are alternative smooth gauges which would imply the standard results \cite{ak2}. The gravitational potential can be obtained by applying a coordinate change that sets the shift variable of the metric to zero, $N^i=0$. This yields
\begin{equation}
\Phi_k=-\frac{\dot{H}a^2}{Hk^2}\dot{\zeta}_k,
\end{equation}
where the dot denotes derivative with respect to the proper time $dt=ad\eta$ and $H=\dot{a}/a$ is the Hubble parameter after inflation.
In general, a two-point function involving the variables $Q$, $U$, $E$ and $B$ is specified by two distinct vector sets $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$. One can conveniently choose $(l,m,n)=(\hat{r},\hat{\theta},\hat{\phi})$ so that the angular integrals in the correlators become straightforward. The remaining (radial) momentum integrals contain the usual (distributional) UV infinities, which can be cured by $i\epsilon$-terms (see \cite{ak3} for the implementation of the $i\epsilon$-prescription in cosmology). In the following we take
\begin{equation}\label{th}
\theta >\frac{1}{z_r},
\end{equation}
where $\theta$ is the angle between $l_1$ and $l_2$; i.e. $\cos(\theta)=l_1^i l_2^i$. On the last scattering surface \eq{th} corresponds to superhorizon scales.
Although the oscillating time integrals diminish their power, we estimate that the scalar modes still dominate the expectation values $\left<Q_1Q_2\right>$ and $\left<U_1U_2\right>$ (one has $\left<Q_1U_2\right>=\left<Q_2U_1\right>=0$ identically) because of the slow-roll enhancement $1/\epsilon$ coming from the curvature perturbation \eq{zk}. The angular integrals in momentum space give an oscillating factor which effectively sets a (comoving) cutoff scale for the remaining radial momentum integral (the cutoff is equivalent to the UV improvement implied by the $i\epsilon$-prescription). This scale is roughly proportional to $\eta_0$ and from \eq{kij}, which encodes the contribution of the scalar perturbation, one sees that on dimensional grounds while the two spatial derivatives yield $1/\eta_0^2$ the time integrals give $\eta_0^2$. This shows that in $\left<Q_1Q_2\right>$ and $\left<U_1U_2\right>$, the order of magnitude contributions of the tensors and the scalars are similar to the amplitudes of the expectation values $\left<\gamma\cc\right>$ and $\left<\Phi\Phi\right>$, respectively. Hence the tensors are suppressed by the slow-roll parameter and only the $BB$-correlator is relevant for the gravitational waves.
As usual in the two-point function $\left<B(l_1)B(l_2)\right>$ the oscillating subhorizon modes give negligible contributions when $\theta$ obeys \eq{th}. Thus, to a very good approximation one can use the superhorizon spectrum (which can be obtained from \eq{cck} and \eq{bdv} when $k\eta_r\ll1$)
\begin{equation}\label{shs}
\left|\gamma_k\right|^2\simeq\frac{2H_I^2}{M_p^2}\,\frac{1}{k^3}.
\end{equation}
In that case, the momentum integral can be calculated exactly without any issues (the integral is convergent at IR as $k\to0$ and its UV behavior is cured by the $i\epsilon$-prescription). The result contains many terms when $i\epsilon\not=0$, but in the limit $i\epsilon\to0$ one gets a remarkably simple final formula
\begin{equation}\label{bbs}
\left<B(l_1)B(l_2)\right>\simeq -\frac{2H_I^2}{\pi^2M_p^2}\cot^2(\theta/2), \hs{5} \theta >\frac{1}{z_r}.
\end{equation}
Note that $B(l)$ is not a positive operator, hence a negative expectation value on scales \eq{th} is conceivable.
\section{Conclusions}
Eq. \eq{bbs} is the main result of this work. It gives a distinctive superhorizon signal that starts from zero at $\theta=\pi$ and increases in magnitude with decreasing $\theta$. Indeed, \eq{bbs} greatly enlarges as $\theta$ approaches the subhorizon-superhorizon border (of this simple model) at $\theta=1/z_r$. Of course, one would not expect \eq{bbs} to be correct up to that order since the subhorizon corrections to \eq{shs} become more and more important.
The amplitude in \eq{bbs} depends directly on the scale of inflation, which is encoded by the Hubble parameter $H_I$. This is an expected feature for a power spectrum involving gravitational waves. Using the typical upper limit $H_I\simeq 10^{-5} M_p$, which can be obtained from the upper observational limit on the tensor-to-scalar ratio, one finds a very small amplitude, of the order of $10^{-10}$. Of course, this is the largest estimate since $H_I$ can be much smaller. Note that as they are defined, $Q$, $U$, $E$ and $B$ are all dimensionless variables and they measure relative magnitudes, e.g. if $I_V$ and $I_H$ are the intensities corresponding to the vertical and horizontal slabs in Fig. \ref{fig2}, then $Q=(I_V-I_H)/((I_V+I_H)/2)$. Therefore the figure $10^{-10}$ estimates a dimensionless signal (related to the variations of $Q$ and $U$ on the sphere), which can be compared to the usual temperature fluctuations having relative order of magnitude $10^{-5}$.
In any case, the result is encouraging for further investigations in a realistic model including small angles. Note that \eq{bbs} does not depend on the photon frequency and it can be determined from flux measurements as discussed above. These are technical advantages in terms of observability but detecting the corresponding signal will be hard if not impossible. Nevertheless, it is valuable to have an (even in principle) alternative to the CMB polarization experiments, as observing a quantum gravitational wave effect is already expected to be quite difficult.
\begin{acknowledgments}
I am grateful for the support of IIE-SRF fellowship program and thank the colleagues at Connecticut College for their hospitality.
\end{acknowledgments}
|
2,877,628,089,469 | arxiv | \section{Introduction}
Let $\mathcal{G}$ be a group and let $(\tau,V)$ be an irreducible
representation of $\mathcal{G}$. If $\tau$ is self-dual, i.e., it
is isomorphic to its contragradient $\check{\tau}$, then there exists
a non-degenerate $\mathcal{G}$-invariant bilinear form $B:V\times V\rightarrow\mathbb{C}$
which is unique up to scalars. It is thus either symmetric or skew
symmetric. The \textit{sign} or the \textit{Frobenius\textendash Schur
indicator} $\operatorname{sgn}(\tau)$ of $\tau$ is defined to be $+1$ (resp.
$-1$) according as $B$ is symmetric (resp. skew-symmetric). When
$\mathcal{G}$ is a finite group,
\[
\operatorname{sgn}(\tau)=\frac{1}{\mid\mathcal{G}\mid}\sum_{g\in\mathcal{G}}\omega_{\tau}(g^{2}),
\]
where $\omega_{\tau}$ denotes the character of $\tau$. The expression
on the right hand side of the above equality is zero when $\tau$
is not self-dual.
Now let $G$ be a connected reductive group defined over a local or
finite field $F$ and let $\pi$ be a smooth irreducible representation
of $G(F)$. When $\pi$ is self-dual and also generic, i.e., it admits
a Whittaker model, D. Prasad introduced the idea of studying the sign
by the action of certain order two element of the center $Z(F)$ of
$G(F)$ \cite{Pra98,Pra99}. See \cite[Conjecture 8.3]{GR10} for
a possible connection of this element with the Deligne-Langlands local
root number.
Now let $G$ be unramified, $F$ be $p$-adic and let $\pi$ be an
irreducible regular depth-zero supercuspidal representation. Regular
depth-zero supercuspidal representations are the ones which arise
from certain Deligne-Lusztig cuspidal representations of finite reductive
groups. These representations were studied by DeBacker and Reeder
\cite{DeRe09}. Assuming further that $G$ has connected center and
that $\pi$ is self-dual generic, we show in Theorem \ref{thm:main}
that the sign of $\pi$ is given by the central character $\omega_{\pi}$
evaluated at a certain order two element $\epsilon$ of the center
of $G(F)$.
The main idea in the proof of Theorem \ref{thm:main} is to reduce
the problem to a question about finite reductive groups and use Prasad's
result in that setting. In Prasad's result, we first observe that
the central element $\epsilon$ has an explicit description in terms
of the root data. The assumption of the genericity of $\pi$ is used
to ensure - by a result of DeBacker and Reeder - that the finite reductive
group in the inducing data of $\pi$ has the same root system as $G$.
We finally use Kaletha's description of regular depth-zero representations
to relate $\omega_{\pi}(\epsilon)$ to the analogous element of the
finite reductive group.
When $G$ is quasi split and tamely ramified over $F$, we give in
Theorem \ref{thm:exist}, a sufficient condition for self-dual supercuspidal
representations of $G(F)$ to exist. We show that if $-1$ is in the
$F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal
representations do exist. The proof uses Kaletha's description of
regular supercuspidal representations \cite{Kal2016} and Hakim-Murnaghan's
result about dual Yu-datum \cite{HM08}.
\section{Notations}
Let $\mathcal{G}$ be a reductive group over a local or finite field
$F$. The central character of a representation $\pi$ of $\mathcal{G}(F)$
will be denoted by $\omega_{\pi}$. The contragradient of $\pi$ will
be denoted by $\check{\pi}$. If $\pi$ is irreducible self-dual,
then its Frobenius\textendash Schur indicator will be denoted by $\operatorname{sgn}(\pi)$.
When $F$ is non-archimedean local, we write $\mathcal{B}(\mathcal{G},F)$
(resp. $\mathcal{B}^{\operatorname{red}}(\mathcal{G},F)$) to denote the Bruhat-Tits
building (resp. reduced Bruhat-Tits building) of $\mathcal{G}(F)$.
We follow the standard notations (as in \cite[Sec. 2]{Kal2016} for
instance) for parahoric subgroups of $\mathcal{G}(F)$ and their Moy-Prasad
filtrations.
\section{\label{sec:finite}finite reductive group }
Let $\mathtt{G}$ be a connected reductive group defined over a finite
field $\mathbb{F}_{q}$. We assume that center $\mathtt{Z}$ of $\mathtt{G}$
is connected. Let $\mathtt{B=TU}$ be an $\mathbb{F}_{q}$-Borel subgroup
of $\mathtt{G}$, where $\mathtt{U}$ is the unipotent radical of
$\mathtt{B}$ and $\mathtt{T}$ is an $\mathbb{F}_{q}$-maximal torus
of $\mathtt{G}$ contained in $\mathtt{B}$. We denote the adjoint
torus by $\mathtt{T}_{\operatorname{ad}}$. The character lattice of $\mathtt{T}$
(resp. $\mathtt{T}_{\operatorname{ad}}$) will be denoted by $X^{*}(\mathtt{T})$
(resp. $X^{*}(\mathtt{T}_{\operatorname{ad}})$).
\begin{thm}
[Prasad]There exists an element $s_{0}$ in $\mathtt{T}(\mathbb{F}_{q})$
such that it operates by $-1$ on all the simple root spaces of $\mathtt{U}$.
Further, $t_{0}:=s_{0}^{2}$ belongs to $\mathtt{Z}(\mathbb{F}_{q})$
and $t_{0}$ acts on an irreducible, generic, self-dual representation
by $1$ iff the representation is orthogonal.
\end{thm}
From the short exact sequence
\[
\xymatrix{1\ar@{->}[r] & \mathtt{Z}\ar@{->}[r] & \mathtt{T}\ar@{->}[r] & \mathtt{T}_{\operatorname{ad}}\ar@{->}[r] & 1}
,
\]
we get the long exact sequence
\[
\xymatrix{1\ar@{->}[r] & \mathtt{Z}(\mathbb{F}_{q})\ar@{->}[r] & \mathtt{T}(\mathbb{F}_{q})\ar@{->}[r] & \mathtt{T}_{\operatorname{ad}}(\mathbb{F}_{q})\ar@{->}[r] & \operatorname{H}^{1}(\Gamma,\mathtt{Z})\ar@{->}[r] & \cdots}
.
\]
Since $\mathtt{Z}$ is connected, $\operatorname{H}^{1}(\Gamma,\mathtt{Z})$
is trivial by Lang's theorem. Therefore,
\[
\xymatrix{1\ar@{->}[r] & \mathtt{Z}(\mathbb{F}_{q})\ar@{->}[r] & \mathtt{T}(\mathbb{F}_{q})\ar@{->}[r] & \mathtt{T}_{\operatorname{ad}}(\mathbb{F}_{q})\ar@{->}[r] & 1}
\]
is exact. Let $\check{\rho}$ denote half the sum of positive co-roots.
Let $s^{\prime}$ be the element of $\mathtt{T}_{\operatorname{ad}}(\mathbb{F}_{q})=\operatorname{Hom}(X^{*}(\mathtt{T}_{\operatorname{ad}}),\mathbb{G}_{m})(\mathbb{F}_{q})$
given by
\[
\chi\in X^{*}(\mathtt{T}_{\operatorname{ad}})\mapsto(-1)^{\langle\chi,\check{\rho}\rangle}\in\mathbb{G}_{m}.
\]
Let $s$ denote any pull back of $s^{\prime}$ in $\mathtt{T}(\mathbb{F}_{q})$.
Then $s$ operates by $-1$ on all simple root spaces of $\mathtt{U}$.
The element $t:=s^{2}\in\mathtt{Z}(\mathbb{F}_{q})$ has the description
\[
\chi\in X^{*}(\mathtt{T})\mapsto(-1)^{\langle\chi,2\check{\rho}\rangle}\in\mathbb{G}_{m}.
\]
We can thus rewrite the above Theorem as
\begin{conjbis}
\label{Thm:Prasad}Let $\pi$ be an irreducible generic representation
of $\mathtt{G}(\mathbb{F}_{q})$. Then $\operatorname{sgn}(\pi)=\omega_{\pi}(t)$.
\end{conjbis}
\begin{rem}
The assumption in Theorem \ref{Thm:Prasad} that $\mathtt{Z}$ is
connected cannot be entirely dropped, as shown in the counter example
in \cite[Sec. 9]{Pra98}.
\end{rem}
\section{\label{sec:Regular}Regular depth-zero representations}
Let $G$ be an unramified connected reductive group defined over a
$p$-adic field $F$. Assume that the center $Z$ of $G$ is connected.
Let $\mathfrak{f}$ denote the residue field of $F$.
\subsection{\label{sub:Construction}Construction of regular depth-zero supercuspidal }
For the definition of \textit{regular} depth-zero supercuspidal representations
and the details of the construction in this section, see \cite[Sec. 3.2.3]{Kal2016}.
Let $S$ be an elliptic maximal torus of $G$ and let $\theta:S(F)\rightarrow\mathbb{C}^{\times}$
be a depth-zero character. Let $S(F)_{0}$ be the Iwahori subgroup
of $S(F)$. Assume that $\theta$ is \textit{regular}, i.e., the stabilizer
of $\theta\mid_{S(F)_{0}}$ in $N(S(F),G(F))/S(F)$ is trivial, where
$N(S(F),G(F))$ denotes the normalizer of $S(F)$ in $G(F)$. The
restriction of $\theta\mid_{S(F)_{0}}$ factors through a character
$\bar{\theta}$ of $S(F)_{0:0+}$. Let $x\in\mathcal{B}^{\operatorname{red}}(G,F)$
be the vertex associated to $S$. The group $G(F)_{0:0+}$ is the
$\mathfrak{f}$-points of a connected reductive $\mathfrak{f}$-group
$\mathtt{G}_{x}$ and $S(F)_{0:0+}$ is the $\mathfrak{f}$-points
of an elliptic maximal $\mathfrak{f}$-torus $\mathtt{S}^{\prime}$
of $\mathtt{G}_{x}$. Let $\kappa(S,\theta)$ denote the irreducible
cuspidal Deligne-Lusztig representation of $\mathtt{G}_{x}(\mathfrak{f})$
associated to the pair $(\mathtt{S}^{\prime},\bar{\theta})$. Denote
again by $\kappa(S,\theta)$ its inflation to $G(F)_{x,0}$. This
representation extends to a representation $\tilde{\kappa}(S,\theta)$
of $Z(F)G(F)_{x,0}=G(F)_{x}$.
\begin{lem}
\cite[Lemma 3.17, 3.22]{Kal2016} \label{lem:kal}The representation
$\pi(S,\theta):=\operatorname{c-Ind}_{G(F)_{x}}^{G(F)}\tilde{\kappa}(S,\theta)$
is irreducible (and hence supercuspidal) and every regular depth-zero
supercuspidal representation is of this form.
\end{lem}
\subsection{\label{sub:Sign}Sign}
Choose a system of positive roots $\Phi^{+}(G,S)$ for the set of
roots $\Phi(G,S)$. Let $\check{\rho}$ denote half the sum of positive
roots and let $t$ denote the element $2\check{\rho}(-1)\in$ $Z(F)$
(see \cite[Sec. 8.5]{GR10}).
\begin{thm}
\label{thm:main}Let $\pi$ be a generic regular depth-zero self-dual
supercuspidal representation of $G(F)$. Then $\operatorname{sgn}(\pi)=\omega_{\pi}(t)$.\end{thm}
\begin{proof}
By Lemma \ref{lem:kal}, the representation $\pi$ arises out of a
pair $(S,\theta)$ as in Section \ref{sub:Construction}. Let $x\in\mathcal{B}^{\operatorname{red}}(G,F)$
be the vertex associated to $S$. The normalizer in $G(F)$ of $G(F)_{x,0}$
is $G(F)_{x}$ \cite[Lemma 3.3]{Yu01}. Therefore $G(F)_{x}$ is self
normalizer. Since $\check{\pi}=\operatorname{c-Ind}_{G(F)_{x}}^{G(F)}\check{\tilde{\kappa}}$,
$\pi\cong\check{\pi}$ iff there exists $g\in G(F)$ such that $(G(F)_{x},\tilde{\kappa})$
is conjugate by an element $g\in G(F)$ to the pair $(G(F)_{x},\check{\tilde{\kappa}})$
(by \cite[Theorem 6.7 ]{HM08} for instance without any hypothesis).
But then $g\in G(F)_{x}$ since $G(F)$ is self normalizer. Therefore,
$\pi(S,\theta)$ is self-dual iff $\tilde{\kappa}(S,\theta)$ is so.
Thus, $\operatorname{sgn}(\pi)=\operatorname{sgn}(\tilde{\kappa})$. Since $\pi(S,\theta)$
is generic, the vertex $x$ associated to $S$ is hyperspecial \cite[Theorem 1.1]{DeRe10}
(also \cite[Lemma 6.1.2]{DeRe09}). Therefore the root system $\Phi(G,S)$
can be identified with $\Phi(\mathtt{G}_{x},\mathtt{S}^{\prime})$.
Let $\Phi^{+}(\mathtt{G}_{x},\mathtt{S}^{\prime})$ be the positive
roots of $\Phi(\mathtt{G}_{x},\mathtt{S}^{\prime})$ under the identification.
Let $\underline{\check{\rho}}$ be half the sum of positive roots
of $\Phi(\mathtt{G}_{x},\mathtt{S}^{\prime})$ and $\underline{t}=2\underline{\check{\rho}}(-1)\in\mathtt{Z}_{x}(\mathbb{F}_{q})$.
Since $x$ is hyperspecial, $\mathtt{Z}$ is connected implies $\mathtt{Z}_{x}$
is connected. Also, $\kappa(S,\theta)$ is generic \cite[Lemma 6.1.2]{DeRe09}.
We therefore have by Theorem \ref{Thm:Prasad} that $\operatorname{sgn}(\tilde{\kappa})=\omega_{\tilde{\kappa}}(\underline{t})$.
But $\omega_{\tilde{\kappa}}(\underline{t})=\bar{\theta}(\underline{t})=\theta(t)$.
The Theorem now follows because $\omega_{\pi}=\theta\mid_{Z(F)}$
by \cite[Fact 3.38]{Kal2016}.
\end{proof}
\section{\label{sec:Existence}Existence of self-dual representations}
Let $G$ be a quasi-split tamely ramified connected reductive group
over a $p$-adic field $F$. Let $\Omega(S,G)$ be the absolute weyl
group. In \cite[Sec. 3.2.1]{kaletha}, a vertex $x\in\mathcal{B}^{\operatorname{red}}(G,F)$
is called \textit{superspecial} if it is special in $\mathcal{B}^{\operatorname{red}}(G,E)$,
where $E$ is any finite Galois extension of $F$ splitting $G$.
\begin{thm}
\label{thm:exist}If $-1\in\Omega(S,G)(F)$, then self-dual supercuspidal
representations of $G(F)$ exist. \end{thm}
\begin{proof}
Let $(S,\theta)$ be a tame regular elliptic pair \cite[Def. 3.23]{Kal2016}
such that $S$ is \textit{relatively unramified} \cite[Sec. 3.2.1]{Kal2016}
and the point $x\in\mathcal{B}^{\operatorname{red}}(G,F)$ associated to $S$
is superspecial. Let $\pi(S,\theta)$ be the associated regular supercuspidal
representation. By \cite[Theorem 4.25]{HM08}, $\check{\pi}(S,\theta)\cong\pi(S,\theta^{-1})$.
By \cite[Lemma 3.37]{Kal2016}, $\pi(S,\theta)\cong\pi(S,\theta^{-1})$
iff $(S,\theta)$ is $G(F)$-conjugate to $(S,\theta^{-1})$. By \cite[Lemma 3.11]{Kal2016}
$\Omega(S,G)(F)\cong N(S,G)(F)/S(F)$, where $N(S,G)$ denotes the
normalizer of $S$ in $G$. Thus if $-1\in\Omega(S,G)(F)$, then it
follows that $(S,\theta)$ is $G(F)$-conjugate to $(S,\theta^{-1})$. \end{proof}
\begin{rem}
If the root system of $G$ is of type $B_{n}$, $C_{n}$, $E_{7}$,
$E_{8}$, $G_{2}$ or $D_{n}$ ($n$-even), then the longest weyl
group element of $G$ is $-1$.
\end{rem}
\begin{rem}
When $G=\operatorname{GL}_{n}$, Adler \cite{Adler97} showed that the necessary
and sufficient condition for self-dual regular supercuspidal representations
of $G(F)$ to exist is that either $n$ or the residue characteristic
of $F$ be even.
\end{rem}
\section{Acknowledgment}
The author is very thankful to Sandeep Varma, Dipendra Prasad and
Steven Spallone for many helpful conversations.
\bibliographystyle{plain}
|
2,877,628,089,470 | arxiv | \section{\@startsection{section}{1}{\z@}
\newcommand{\myvector}[1]{\boldsymbol{#1}}
\newcommand{\mymatrix}[1]{\mathrm{#1}}
\newcommand{\mytensor}[1]{\boldsymbol{\mathrm{#1}}}
\newcommand{\myset}[1]{\mathrm{#1}}
\newcommand{\gaussian}[3]{\mathcal{N}(#1 | #2, #3)}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\newcounter{num}
\newcommand{\rnum}[1]{\setcounter{num}{#1} \roman{num}}
\newcommand{\Rnum}[1]{\setcounter{num}{#1} \Roman{num}}
\pdfminorversion=7
\begin{document}
\title{Checkerboard-Artifact-Free Image-Enhancement Network
Considering Local and Global Features}
\author{%
\authorblockN{Yuma Kinoshita and Hitoshi Kiya}%
\authorblockA{Tokyo Metropolitan University, Tokyo, Japan}%
}
\maketitle
\thispagestyle{empty}
\begin{abstract}
In this paper, we propose a novel convolutional neural network (CNN)
that never causes checkerboard artifacts, for image enhancement.
In research fields of image-to-image translation problems,
it is well-known that images generated by usual CNNs
are distorted by checkerboard artifacts
which mainly caused in forward-propagation of upsampling layers.
However, checkerboard artifacts in image enhancement have never been discussed.
In this paper, we point out that applying U-Net based CNNs to image enhancement
causes checkerboard artifacts.
In contrast, the proposed network that contains fixed convolutional layers
can perfectly prevent the artifacts.
In addition, the proposed network architecture, which can handle
both local and global features,
enables us to improve the performance of image enhancement.
Experimental results show that the use of fixed convolutional layers
can prevent checkerboard artifacts and the proposed network outperforms
state-of-the-art CNN-based image-enhancement methods
in terms of various objective quality metrics: PSNR, SSIM, and NIQE.
\end{abstract}
\section{Introduction}
The low dynamic range of modern digital cameras
is a major factor that prevents cameras from capturing images as well as human vision.
This is due to the limited dynamic range that imaging sensors have,
resulting in low-contrast images.
Enhancing such images reveals hidden details.
Various kinds of research on single-image enhancement have been reported
\cite{zuiderveld1994contrast, wu2017contrast,
guo2017lime, fu2016weighted, kinoshita2018automatic_trans}.
Most image enhancement methods can be divided into two types:
histogram equalization (HE)-based methods and Retinex-based methods.
Additionally, multi-exposure-fusion (MEF)-based single-image enhancement methods
have also been proposed
~\cite{kinoshita2019scene, kinoshita2018automatic_trans, ying2017bio}.
However, these analysis-based methods cannot restore lost pixel values
due to quantizing and clipping.
The problem leads to banding artifacts in enhanced images.
Recent work has demonstrated great progress by using
convolutional neural networks (CNNs)
in preference to analytical approaches such as HE
~\cite{gharbi2017deep, shen2017msrnet, cai2018learning, chen2018learning, yang2018image,
ruixing2019underexposed, jiang2019enlighten, kinoshita2019convolutional}.
Most of these methods employ a U-Net \cite{ronneberger2015unet}-based
network architecture in order to learn the mapping from low-quality images
to high-quality ones.
However, it is well-known that CNNs having upsampling layers such as U-Net
cause images to be distorted by checkerboard artifacts
due to upsampling layers
~\cite{odena2016deconvolution, sugawara2018super,
sugawara2019checkerboard, kinoshita2020fixed}.
Despite this situation,
checkerboard artifacts have never been discussed
in the field of image enhancement so far.
In this paper, we point out that applying U-Net based CNNs to image enhancement
causes checkerboard artifacts.
To prevent the artifacts, we also propose a novel image-enhancement CNN
that never causes checkerboard artifacts.
In the proposed network,
fixed convolutional layers~\cite{kinoshita2020fixed} are applied
to upsampling and downsampling operations.
The use of fixed convolutional layers
can perfectly prevent checkerboard artifacts.
Moreover, the proposed network architecture that
can handle both local and global features
enables us to prevent distortions resulting from the lack of global image information.
We evaluate the effectiveness of the proposed image-enhancement network
in terms of the quality of enhanced images
by an experiment using a dataset from~\cite{cai2018learning},
where the peak signal-to-noise ratio (PSNR),
the structural similarity (SSIM), and discrete entropy
are utilized as quality metrics.
Experimental results show that the proposed method outperforms
state-of-the-art contrast enhancement methods
in terms of those quality metrics.
Furthermore, the proposed method does not cause checkerboard artifacts
and distortions resulting from the lack of global features.
\section{Related work}
Here, we briefly summarize image-enhancement methods
and checkerboard artifacts in CNNs.
\subsection{Image enhancement}
Many image-enhancement methods have been studied
\cite{zuiderveld1994contrast, wu2017contrast,
guo2017lime, fu2016weighted,
kinoshita2018automatic_trans, kinoshita2019scene}.
Among the methods, HE has received
the most attention because of its intuitive implementation quality and
high efficiency.
It aims to derive a mapping function such that the entropy of
a distribution of output luminance values can be maximized.
However, HE often causes over-enhancement.
To avoid this,
numerous improved methods based on HE have also been developed
\cite{zuiderveld1994contrast, wu2017contrast}.
Another way for enhancing images is to use the Retinex theory \cite{land1977retinex}.
Retinex-based methods \cite{guo2017lime, fu2016weighted} decompose images into
reflectance and illumination, and then enhance images by manipulating illumination.
Additionally, multi-exposure-fusion (MEF)-based single-image enhancement methods
were recently proposed
\cite{kinoshita2019scene, kinoshita2018automatic_trans, ying2017bio}.
One of them, a pseudo MEF scheme~\cite{kinoshita2018automatic_trans},
makes any single image applicable
to MEF methods~\cite{mertens2009exposure}
by generating pseudo multi-exposure images from a single image.
By using this scheme, images with improved quality are produced
with the use of detailed local features.
Recent work has demonstrated great progress by using data-driven approaches
in preference to traditional analytical approaches such as HE
~\cite{gharbi2017deep, shen2017msrnet, cai2018learning, chen2018learning, yang2018image,
ruixing2019underexposed, jiang2019enlighten, kinoshita2019convolutional}.
These data-driven approaches utilize high- and low-quality images
to train deep neural networks,
and the trained networks can be used to enhance color images.
Most of these methods employ a U-Net \cite{ronneberger2015unet}-based
network architecture.
For example, Cai et al. utilized U-Net for enhancing a luminance map
calculated from an input image~\cite{cai2018learning}.
However, U-Net based network architectures cause checkerboard artifacts
as described in the following section.
\subsection{Checkerboard artifacts in CNNs}
Checkerboard artifacts have been studied as a distortion
caused by using upsamplers in linear multi-rate systems
~\cite{harada1998multidimensional, tamura1998design, harada1998multidimensional_trans,
iwai2010methods}.
In research fields of image-to-image translation problems,
e.g., image super-resolution,
checkerboard artifacts are known to be
caused by forward-propagation of upsampling layers
including transposed convolutional layers
and by backward-propagation of downsampling layers including strided convolutional layers
~\cite{odena2016deconvolution}.
CNN architectures usually have upsampling layers and/or have downsampling layers,
such as VGG~\cite{simonyan2014very}, ResNet~\cite{he2016deep},
and U-Net~\cite{ronneberger2015unet},
for increasing and/or reducing the spatial sampling rate of feature maps,
respectively~\cite{goodfellow2016deep}.
For this reason,
checkerboard artifacts affect most commonly-used CNNs.
In particular, the checkerboard artifacts caused by forward-propagation
directory distort images generated by CNNs.
To overcome checkerboard artifacts caused by upsampling layers,
Sugawara et al.~\cite{sugawara2018super, sugawara2019checkerboard}
gave us two approaches to perfectly prevent checkerboard artifacts
by extending a condition for avoiding checkerboard artifacts
in linear multirate systems
~\cite{harada1998multidimensional, tamura1998design, harada1998multidimensional_trans,
iwai2010methods}.
In accordance with Sugawara's approaches,
Kinoshita et al. proposed fixed smooth convolutional layers
that can prevent checkerboard artifacts caused by both upsampling
and downsampling layers~\cite{kinoshita2020fixed}.
\subsection{Scenario}
In image enhancement,
it has already been pointed out that
CNNs that cannot handle global image information
cause images to be distorted~\cite{marnerides2018expandnet, kinoshita2019convolutional}.
In addition to this distortion,
checkerboard artifacts will appear in enhanced images because of upsampling layers.
Figure~\ref{fig:artifacts} shows an example of images enhanced
by using U-Net and iTM-Net~\cite{kinoshita2019convolutional},
where iTM-Net can handle global features but U-Net cannot.
From Fig.~\ref{fig:artifacts}(\subref{fig:unet_example}),
there are luminance inversions in many parts of the image enhanced by using U-Net.
These luminance inversions are distortions due to the lack of global features.
In addition, checkerboard artifacts also appeared in the image.
iTM-Net prevented the distortions but checkerboard artifacts still remain
[see Fig.~\ref{fig:artifacts}(\subref{fig:itm-net_example})].
\begin{figure}[!t]
\centering
\begin{subfigure}[t]{0.45\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/input_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/input_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/input_under_zoom2.png}
\caption{Input image \label{fig:input_example}}
\end{subfigure}
\begin{subfigure}[t]{0.45\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/target_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/target_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/target_under_zoom2.png}
\caption{Ground truth \label{fig:target_example}}
\end{subfigure}\\
\begin{subfigure}[t]{0.45\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/unet_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/unet_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/unet_under_zoom2.png}
\caption{U-Net \label{fig:unet_example}}
\end{subfigure}
\begin{subfigure}[t]{0.45\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/itm-net_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/itm-net_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/itm-net_under_zoom2.png}
\caption{iTM-Net \label{fig:itm-net_example}}
\end{subfigure}
\caption{Checkerboard artifacts caused by upsampling layers
and distortions due to the lack of global features.
Zoom-ins of boxed regions are shown in bottom of each image.
\label{fig:artifacts}}
\end{figure}
For these reasons,
CNN architectures for image enhancement should satisfy the following two conditions:
\begin{itemize}
\item Both local and global features of images can be handled.
\item Checkerboard artifacts caused by upsampling layers can be prevented.
\end{itemize}
However, there are no CNNs that satisfy the conditions.
Therefore, in this paper, we propose a novel image-enhancement network
that enables us not only to perfectly prevent checkerboard artifacts
but also to consider both local and global features.
\section{Proposed method}
As shown in Fig. \ref{fig:network},
the architecture of the proposed image-enhancement network
consists of three sub-networks: a local encoder, a global encoder, and a decoder.
The use of the local encoder and the global encoder makes it possible
to handle local and global features, respectively.
Furthermore, checkerboard artifacts in the network are perfectly prevented by using
fixed convolutional layers~\cite{kinoshita2020fixed} in both upsampling and downsampling.
\begin{figure*}[t]
\centering
\includegraphics[clip, width=0.95\hsize]{./figs/network.pdf}
\caption{Network architecture.
Architecture consists of local encoder, global encoder, and decoder.
For upsampling and downsampling, fixed convolutional layers are applied
to transposed convolution and strided convolution, respectively.
Each box denotes multi-channel feature map produced by each layer.
Number of channels is denoted above each box.
Feature map resolutions are denoted to the left of a box.
\label{fig:network}}
\end{figure*}
\subsection{Fixed convolutional layer}
As in~\cite{sugawara2018super, sugawara2019checkerboard, kinoshita2020fixed},
checkerboard artifacts can be perfectly prevented by inserting a fixed convolutional layer
after each upsampling layer and before each downsampling layer.
The filter kernel $\mytensor{K}^{(d)}$ of the fixed convolutional layer is obtained
by convolving a zero-order hold kernel $\mymatrix{h}_0$ multiple times, as
\begin{equation}
\mytensor{K}^{(d)}_{i, i, :, :} =
\begin{cases}
\mymatrix{h}_0 & (d = 0) \\
\mytensor{K}^{(d-1)}_{i, i, :, :} * \mymatrix{h}_0 & (d > 1)
\end{cases},
\label{eq:fixed_kernel}
\end{equation}
where a parameter $d$, referred to as the order of smoothness,
$\mytensor{K}^{(d)}_{i, i, :, :}$ is 2-D slice of 4-D tensor $\mytensor{K}^{(d)}$,
and $\mymatrix{A} * \mymatrix{B} $ means
the convolution on two matrices $\mymatrix{A}$ and $\mymatrix{B}$.
When the fixed layer is applied to an upsampling layer with an upsampling rate $U$,
the kernel size for $\mymatrix{h}_0$ is given as $U \times U$.
In contrast,
the kernel size is given as $D \times D$
when the proposed layer is applied to a downsampling layer with a downsampling rate $D$.
By using a filter kernel $\mytensor{K}^{(d)}$ and a trainable bias $\myvector{b}$,
an output feature map $\mytensor{Z}$ of the fixed layer
can be written as
\begin{equation}
\mytensor{Z}_{i, j, k} = \sum_{m, n} \mytensor{V}_{i, j+m-1, k+n-1}
\mytensor{K}^{(d)}_{i, i, m, n} + \myvector{b}_i,
\end{equation}
where $\myvector{V}$ is an input feature map with a size of
\textit{channel} $\times$ \textit{height} $\times$ \textit{width}.
\subsection{Network architecture}
Figure \ref{fig:network} shows the overall network architecture of
the proposed network.
The architecture consists of three sub-networks:
a local encoder, a global encoder, and a decoder,
which is based on iTM-Net used in
~\cite{kinoshita2019convolutional, kinoshita2019itmnet_trans}.
iTM-Net utilizes the global encoder in addition to U-Net's architecture
and combines features extracted by both encoders to prevent the distortions.
In addition to iTM-Net,
the use of the fixed convolutional layers for upsampling and downsampling
enables us to prevent checkerboard artifacts.
The input for the local encoder is an $H \times W$ pixels 24-bit color image.
For the global encoder,
the input image is resized to a fixed size ($128 \times 128$).
The proposed network have five types of layers as shown in Fig. \ref{fig:network}:
\subsubsection*{$3 \times 3$ Conv. + BN + ReLU}
which calculates a $3 \times 3$ convolution
with a stride of $1$ and a padding of $1$.
After the convolution, batch normalization \cite{ioffe2015batch}
and the rectified linear unit activation function \cite{glorot2011deep} (ReLU)
are applied.
In the local encoder and the decoder, two adjacent
$3 \times 3$ Conv. + BN + ReLU layers will have the same number $K$ of filters.
From the first two layers to the last ones,
the number of filters are
$K = 32$, $64$, $128$, $256$, $512$, $256$, $128$, $64$, and $32$,
respectively.
In the global encoder, all layers have $64$ filters.
\subsubsection*{$1 \times 1$ Conv. + ReLU}
which calculates a $1 \times 1$ convolution with a stride of $1$ and without padding.
After the convolution, ReLU is applied.
The number of filters in the layer is $3$.
\subsubsection*{$4 \times 4$ Conv. + BN + ReLU (w/o padding)}
which calculates a $4 \times 4$ convolution without padding.
The number of filters in the layer is $64$.
\subsubsection*{Downsampling}
which calculates a $2 \times 2$ convolution with a stride of $1$ and a padding of $1$,
by using a fixed kernel in Eq. (\ref{eq:fixed_kernel}).
After the convolution, it downsamples feature maps by a $3 \times 3$ strided convolution
with a stride of $2$ and a padding of $1$.
\subsubsection*{Upsampling}
which upsamples feature maps by a $4 \times 4$ transposed convolution
with a stride of $1/2$ and a padding of $1$.
After the transposed convolution,
it calculates a $2 \times 2$ convolution with a stride of $1$ and a padding of $1$,
by using a fixed kernel in Eq. (\ref{eq:fixed_kernel}).
\section{Simulation}
We evaluated the effectiveness of the proposed method
by using three objective quality metrics.
\subsection{Simulation conditions}
In our experiments, we trained the proposed network with 100 epochs
by using training data in a multi-exposure image dataset
which which constructed by Cai et al.~\cite{cai2018learning}.
For data augmentation,
we resized each original input image with a random scaling factor in the range $[0.6, 1.0]$
for every epoch.
After the resizing, we randomly cropped the resized image to
an image patch with a size of $256 \times 256$ pixels
and flipped the patch horizontally with a probability of 0.5.
Loss between an image patch generated by the proposed network
and the corresponding target image patch was calculated by the simple $\ell 1$-distance.
Here, the Adam optimizer \cite{kingma2014adam} was utilized for optimization,
where parameters in Adam were set as $\alpha=0.001, \beta_1=0.9$, and $\beta_2=0.999$.
He's method \cite{he2015delving} was used for initializing the network.
As in~\cite{cai2018learning},
we tested the proposed network by using underexposed and overexposed images
having -1 and 1 $\mathrm{[EV]}$, respectively, in test data in the dataset.
The quality of images enhanced by the proposed network
was evaluated by three objective quality metrics:
the peak signal-to-noise ratio (PSNR),
the structural similarity (SSIM)~\cite{wang2004image},
and the naturalness image quality evaluator (NIQE) \cite{mittal2013making},
where the target image corresponding to the input image utilized
as a reference for PSNR and SSIM.
\subsection{Results}
To confirm the effectiveness of the proposed network architecture,
we first compare four network architectures:
U-Net~\cite{ronneberger2015unet},
U-Net with fixed convolutional layers,
iTM-Net~\cite{kinoshita2019convolutional, kinoshita2019itmnet_trans},
and iTM-Net with fixed convolutional layers (i.e., the proposed one).
Please note that these networks were trained by using the same data and parameters
as the proposed network, and the difference among the networks
was only the network architecture.
Table \ref{tab:ablation} illustrates the average scores of the objective assessment
for 58 underexposed images and 58 overexposed ones,
in terms of PSNR, SSIM, and NIQE.
In the case of PSNR and SSIM,
a larger value means a higher similarity between an enhanced image
and a reference image.
By contrast, a smaller value for NIQE indicates that an enhanced image has
less distortions such as noise or blur.
As shown in Table \ref{tab:ablation},
the proposed method provided the highest average scores in the four networks
for both underexposed and overexposed images, under all three metrics.
\begin{table*}[!t]
\centering
\caption{Average scores of objective quality metrics for ablation study}
\begin{tabular}{l|ccc|ccc} \hline \hline
\multirow{2}{*}{Architecture} & \multicolumn{3}{c}{Underexposure}
& \multicolumn{3}{|c}{Overexposure} \\
& PSNR & SSIM & NIQE & PSNR & SSIM & NIQE \\ \hline
U-Net~\cite{ronneberger2015unet} & 16.92 & 0.8213 & 2.613
& 16.69 & 0.7961 & 2.681 \\
U-Net w/ fixed layers & 16.29 & 0.8152 & 2.475
& 16.18 & 0.7912 & 2.559 \\
iTM-Net~\cite{kinoshita2019convolutional} & 20.20 & \textbf{0.8575} & 2.584
& 20.46 & 0.8433 & 2.633 \\
iTM-Net w/ fixed layers (Proposed) & \textbf{20.52} & \textbf{0.8575} & \textbf{2.382}
& \textbf{21.00} & \textbf{0.8455} & \textbf{2.496} \\ \hline \hline
\end{tabular}
\label{tab:ablation}
\end{table*}
Figure \ref{fig:ablation} shows
images generated from an artificial gray image by the four networks
and their log-amplitude spectra,
where each spectrum was normalized in the range $[0, 1]$.
From the figure,
it is confirmed that network architectures without fixed convolutional layers
caused checkerboard artifacts in the generated images.
These artifacts can also be seen as a lattice pattern in the frequency domain.
In addition, the use of fixed convolutional layers
perfectly prevented the artifacts.
For these reasons, the architecture of the proposed network that can handle both
local and global features and can prevent checkerboard artifacts
is effective for image enhancement.
\begin{figure*}[!t]
\centering
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/input_gray.png} \\
\vspace{1pt}
\includegraphics[width=\columnwidth]{./figs/input_gray_freq.png}
\caption{Input image \label{fig:input_ablation}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/unet_gray.png} \\
\vspace{1pt}
\includegraphics[width=\columnwidth]{./figs/unet_gray_freq.png}
\caption{U-Net~\cite{ronneberger2015unet} \label{fig:unet_ablation}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/unet_wo_checker_gray.png} \\
\vspace{1pt}
\includegraphics[width=\columnwidth]{./figs/unet_wo_checker_gray_freq.png}
\caption{U-Net w/ fixed layers \label{fig:unet_wo_checker_ablation}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/itm-net_gray.png} \\
\vspace{1pt}
\includegraphics[width=\columnwidth]{./figs/itm-net_gray_freq.png}
\caption{iTM-Net~\cite{kinoshita2019convolutional} \label{fig:itm-net_ablation}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/itm-net_wo_checker_gray.png} \\
\vspace{1pt}
\includegraphics[width=\columnwidth]{./figs/itm-net_wo_checker_gray_freq.png}
\caption{iTM-Net w/ fixed layers (Proposed) \label{fig:itm-net_wo_checker_ablation}}
\end{subfigure}
\caption{Checkerboard artifacts in generated images.
Top: generated images. Bottom: log-amplitude spectra normalized in range $[0, 1]$.
\label{fig:ablation}}
\end{figure*}
The proposed network was also compared with seven conventional methods:
histogram equalization (HE),
contrast-accumulated histogram equalization (CACHE) \cite{wu2017contrast},
simultaneous reflectance and illumination estimation (SRIE) \cite{fu2016weighted},
low-light image enhancement via illumination map estimation (LIME) \cite{guo2017lime},
pseudo multi-exposure image fusion (PMEF)~\cite{kinoshita2018automatic},
deep underexposed photo enhancement (DeepUPE)~\cite{ruixing2019underexposed},
and EnlightenGAN~\cite{jiang2019enlighten},
where HE and CACHE are HE-based methods,
SRIE and LIME are Retinex-based ones,
PMEF is an MEF-based one,
and DeepUPE and EnlightenGAN are deep-learning-based ones.
Figure~\ref{fig:comparison} shows an example of images enhanced by the eight methods.
From the figure, deep-learning-based methods were demonstrated to provide
higher-quality images than conventional HE-, Retinex-, and MEF-based methods.
In particular, the proposed network and EnlightenGAN generated
images that clearly showed whole regions in the images.
\begin{figure*}[!t]
\centering
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/input_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/input_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/input_under_zoom2.png}
\caption{Input image \label{fig:input_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/target_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/target_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/target_under_zoom2.png}
\caption{Ground truth \label{fig:target_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/he_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/he_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/he_under_zoom2.png}
\caption{HE \label{fig:he_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/cache_under.png}\\
\includegraphics[width=0.47\columnwidth]{./figs/cache_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/cache_under_zoom2.png}
\caption{CACHE~\cite{wu2017contrast} \label{fig:cache_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/srie_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/srie_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/srie_under_zoom2.png}
\caption{SRIE~\cite{fu2016weighted} \label{fig:srie_under}}
\end{subfigure} \\
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/lime_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/lime_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/lime_under_zoom2.png}
\caption{LIME~\cite{guo2017lime} \label{fig:lime_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/pmef_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/pmef_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/pmef_under_zoom2.png}
\caption{PMEF~\cite{kinoshita2018automatic} \label{fig:pmef_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/DeepUPE_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/DeepUPE_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/DeepUPE_under_zoom2.png}
\caption{DeepUPE~\cite{ruixing2019underexposed} \label{fig:DeepUPE_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/EnlightenGAN_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/EnlightenGAN_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/EnlightenGAN_under_zoom2.png}
\caption{EnlightenGAN~\cite{jiang2019enlighten} \label{fig:EnlightenGAN_under}}
\end{subfigure}
\begin{subfigure}[t]{0.19\hsize}
\centering
\includegraphics[width=\columnwidth]{./figs/itm-net_wo_checker_under.png}\\
\vspace{2pt}
\includegraphics[width=0.47\columnwidth]{./figs/itm-net_wo_checker_under_zoom1.png}
\includegraphics[width=0.47\columnwidth]{./figs/itm-net_wo_checker_under_zoom2.png}
\caption{Proposed \label{fig:itm-net_wo_checker_under}}
\end{subfigure}
\caption{Example of enhanced images from underexposed image.
Zoom-ins of boxed regions are shown in bottom of each image.
\label{fig:comparison}}
\end{figure*}
Table~\ref{tab:comparison} illustrates the average scores of the objective assessment
for 58 underexposed images,
in terms of PSNR, SSIM, and NIQE.
Since compared methods aim to enhance low-light images,
we did not evaluate scores for overexposed images.
As shown in Table~\ref{tab:comparison},
the proposed network provided the highest PSNR and SSIM scores in the eight methods.
In contrast, conventional HE-, Retinex-, MEF-based methods produced
better NIQE scores than deep-learning-based methods,
because of the simplicity of these traditional analysis-based methods.
However,
the proposed network provided the lowest NIQE score in the three deep-learning-based methods.
Hence, it is confirmed that the proposed network architecture
is unlikely to cause distortions.
\begin{table*}[!t]
\centering
\caption{Average scores of objective quality metrics}
\begin{tabular}{l|cccccccc} \hline \hline
Metric & HE & CACHE~\cite{wu2017contrast} & SRIE~\cite{fu2016weighted}
& LIME~\cite{guo2017lime} & PMEF~\cite{kinoshita2018automatic}
& DeepUPE~\cite{ruixing2019underexposed}
& EnlightenGAN~\cite{jiang2019enlighten} & Proposed \\ \hline
PSNR & 17.18 & 16.30 & 17.05 & 16.47 & 16.52
& 16.59 & 15.83 & \textbf{20.52} \\
SSIM & 0.7359 & 0.7521 & 0.7983 & 0.7991 & 0.7296
& 0.7011 & 0.7677 & \textbf{0.8575} \\
NIQE & 2.768 & \textbf{2.268} & 2.524 & 2.345 & 2.309
& 2.474 & 2.494 & 2.382 \\ \hline \hline
\end{tabular}
\label{tab:comparison}
\end{table*}
Those experimental results show that
the proposed network architecture is effective for enhancing single images.
\section{Conclusion}
In this paper, we proposed a novel image-enhancement network
that never causes checkerboard artifacts.
In the proposed network,
the use of fixed convolutional layers perfectly prevents
checkerboard artifacts caused by upsampling and downsampling.
Furthermore,
the architecture of the proposed network, which
consists of a local encoder, a global encoder, and a decoder,
can effectively prevent distortions
resulting from the lack of global image information required for image enhancement.
Experimental results showed that
the use of the fixed layers enabled us to prevent checkerboard artifacts.
In addition, the proposed network was shown to outperform
state-of-the-art CNN-based image-enhancement methods,
in terms of three objective quality metrics: PSNR, SSIM, and NIQE.
In future work, we will train an image enhancement network
with the proposed architecture by using more advanced loss functions,
such as GAN-based ones,
for improving the enhancement performance.
\section*{Acknowledgment}
This work was supported by JSPS KAKENHI Grant Number JP18J20326.
\input{ref.bbl}
\end{document}
|
2,877,628,089,471 | arxiv | \section*{Introduction}
Munn algebras appeared in the theory of semigroups as semigroup algebras of \emph{completely $0$-simple} semigroups
\cite{ClifPres1,Okninski}. They were immediately used for the study of representations of such semigroups. An important input
was made by Ponizovski{\u\i} in the paper \cite{Poniz}, where he established the cases when a finite $0$-simple semigroup
is \emph{representation finite}, i.e. only has finitely many indecomposable representations, over an algebraically closed
field $\Bbbk$ whose characteristic does not divide the order of the underlying group of its Rees matrix presentation
\cite[Th.\,3.5]{ClifPres1}. He also considered the case
of semigroups that are unions of mutually annihilating $0$-simple semigroups with common zero.
The questions remained what happens if the field is not algebraically closed and when the representation type of such a
semigroup is \emph{tame}, i.e. indecomposable representations of each dimension form a finite number of
$1$-parameter families. In this article we give a complete answer to these questions (also for the fields of characteristics
that does non divide the orders of the underlying groups). Of course, in the case of an algebraically
closed field our criterion of finiteness coincides with that of Ponizovski\u\i.
Actually, we obtain criteria of finiteness and tameness for all Munn algebras with semisimple base,
even in a bit more wide context than they are considered in \cite{ClifPres1}.
To prove these results, we establish a relation of modules over Munn algebras with \emph{representations of valued graphs}
in the sense of \cite{DR} (in the algebraically closed case they are just \emph{representations of quivers} in the sense of \cite{GabindI}).
Then we apply the criteria from this paper.
It follows from \cite{tame-wild} (and can be easily checked directly) that in all other cases the Munn algebra $\mM$ (or the corresponding semigroup) is
\emph{representation wild} over the field $\Bbbk$, i.e. for every finitely generated $\Bbbk$-algebra $A$ there is an exact functor
$A\mbox{-}\mathrm{Mod}\to\mM\mbox{-}\mathrm{Mod}$ mapping non-isomorphic modules to non-isomorphic and indecomposable to indecomposable.
\section{Munn algebras}
\label{s1}
In this paper \emph{algebra} means an associative algebra over a commutative ring $\Bbbk$. We do not suppose that such an algebra is unital,
but always suppose that modules over such algebra are also $\Bbbk$-modules and the multiplication by elements of the algebra is $\Bbbk$-bilinear.
We denote by $A\mbox{-}\mathrm{Mod}$ and $\mathrm{Mod}\mbox{-} A$, respectively, the categories of left and right $A$-modules.
By $A^1$ we denote the algebra obtained from an algebra $A$ by the formal attachment of unit. Then the categories of $A$-modules
and unital $A^1$-modules are equivalent. So $A$ and $B$ are Morita equivalent if and only if so are $A^1$ and $B^1$. We consider the elements from $A^1$
as formal sums $\la+ a$, where $a\in A,\,\la\in\Bbbk$.
\begin{definition}\label{def11}
\begin{enumerate}
\item
Let $R$ be a $\Bbbk$-algebra and $\mu:N\to M$ be a homomorphism of $R$-modules. Define a multiplication on $\mathop\mathrm{Hom}\nolimits_R(M,N)$
setting $a\boldsymbol\cdot b=a\mu b$. The resulting ring is called a \emph{Munn algebra} and denoted by $\mM(R,M,N,\mu)$.\!%
%
\footnote{\,This definition is a bit more general than that from \cite{ClifPres1} or \cite{Okninski}, where only the case of free modules is considered.}
%
We say that this Munn algebra \emph{is based} on the algebra $R$.
We denote by $\mM^1(R,M,N,\mu)$ the algebra obtained from $\mM(R,M,N,\mu)$ by the formal attachment of unit.
\item A Munn algebra $\mM(R,M,N,\mu)$ is said to be \emph{regular} if the homomorphism $\mu$ is von Neumann regular, i.e. there is a
homomorphism $\theta:M\to N$ such that $\mu\theta\mu=\mu$.
For instance, this is the case if $R$ is von Neumann regular, while $M$ and $N$ are finitely generated and projective and $\mu\ne0$
(it follows from \cite[Th.\,1.7]{GdrlVNeum}).
\end{enumerate}
\end{definition}
\begin{remark}\label{rem12}
One can see that $\mM(R,M,N,\mu)$ has a unit if and only if there are decompositions $M\simeq M_1\+M_2$ and $N\simeq N_1\+N_2$ such that
$\mathop\mathrm{Hom}\nolimits_R(M_2,N)=\mathop\mathrm{Hom}\nolimits_R(M,N_2)=0$ and the map $\bar\mu=\mathrm{pr}_1\mathop{\scriptstyle\circ}\mu|_{N_1}$ is an isomorphism $N_1\stackrel\sim\to M_1$. Then the unit $u:M\to N$
coincides with $\bar{\mu}^{-1}$. Actually, in this case $\mM(R,M,N,\mu)\simeq\mM(R,M_1,N_1,\bar{\mu})\simeq\mathop\mathrm{End}\nolimits_RM_1$.
\end{remark}
\begin{proposition}\label{prop13}
Let $\mM(R,M,N,\mu)$ be a regular Munn algebra. There are isomorphisms $M\simeq L\+M'$ and $N\simeq L\+N'$ such that with respect to these decompositions
$\mu=\smtr{1_L&0\\emptyset&0}$.
\end{proposition}
\begin{proof}
Let $\theta:M\to N$ be such that $\mu\theta\mu=\mu$. Then $\mu\theta:M\to M$ and $\theta\mu:N\to N$ are idempotents. Therefore,
$M=M_1\+M_2$, where $M_1=\mathop\mathrm{Im}\nolimits\mu\theta,\,M_2=\mathop\mathrm{Ker}\nolimits\mu\theta$ and $N=N_1\+N_2$, where $N_1=\mathop\mathrm{Im}\nolimits\theta\mu,\,N_2=\mathop\mathrm{Ker}\nolimits\theta\mu$.
One easily sees that $\mathop\mathrm{Ker}\nolimits\mu=\mathop\mathrm{Ker}\nolimits\theta\mu$ and $\mathop\mathrm{Im}\nolimits\mu=\mathop\mathrm{Im}\nolimits\mu\theta$, so $\bar{\mu}=\mathrm{pr}_1\mathop{\scriptstyle\circ}\mu|_{N_1}$ is an isomorphism and
$\bar{\mu}^{-1}=\mathrm{pr}_1\mathop{\scriptstyle\circ}\theta|_{M_1}$, while $\mu|_{N_2}=0$ and $\mathrm{pr}_2\mathop{\scriptstyle\circ}\mu=0$, hence
$\mu=\smtr{\bar\mu &0\\emptyset&0}$ with respect to these decompositions. Obviously, it implies the claim.
\end{proof}
\begin{definition}\label{def14}
We write $\mM(R,L,M,N)$ instead of $\mM(R,L\+M,L\+N,\mu)$, where $\mu=\smtr{1_L&0\\emptyset&0}$, and call such a Munn algebra \emph{normal}.
Thus every regular Munn algebra is isomorphic to a normal one. As above, we denote by $\mM^1(R,L,M,N)$ the algebra obtained
from $\mM(R,L,M,N)$ by the formal attachment of unit.
\end{definition}
\begin{lemma}\label{lem15}
Let $A$ and $C$ be two rings, $P$ be a right $C$-module, $M$ be a right $A$-module and $N$ be a right $A$--left $C$-bimodule.
Define the natural map $\phi:P\*_C\mathop\mathrm{Hom}\nolimits_A(M,N)\to\mathop\mathrm{Hom}\nolimits_A(M,P\*_CN)$ mapping $p\*f$ to the homomorphism $x\mapsto p\*f(x)$.
If $P$ is projective and either $P$ or $M$ is finitely generated, $\phi$ is an isomorphism.
\end{lemma}
The proof is obvious. \qed
\begin{lemma}\label{lem16}
Let $A$ be a unital ring, $1=e_1+e_2$, where $e_1,e_2$ are orthogonal idempotents. We denote $A_i=e_iA$, $A_{ij}=e_iAe_j\simeq\mathop\mathrm{Hom}\nolimits_A(A_j,A_i)$
and identify $A$ with the ring of matrices
\begin{equation}\label{eq12}
\mtr{A_{11} & A_{12} \\ A_{21} & A_{22} }.
\end{equation}
Let $P$ be a progenerator of the category $\mathrm{Mod}\mbox{-} A_{11}$.
Then $P^\sharp=(P\*_{A_{11}}A_1)\+A_2$ is a progenerator of the category $\mathrm{Mod}\mbox{-} A$, hence $A\mbox{-}\mathrm{Mod}\simeq B\mbox{-}\mathrm{Mod}$,
where $B=\mathop\mathrm{End}\nolimits_AP^\sharp$. The ring $B$ can be identified with the ring of matrices
\begin{equation}\label{eq11}
B= \mtr{ B_{11} & B_{12} \\ B_{21} & B_{22} },
\end{equation}
where
$
B_{11}=\mathop\mathrm{End}\nolimits_{A_{11}}P,\,
B_{12}=P\*_{A_{11}}A_{12},\,
B_{21}=\mathop\mathrm{Hom}\nolimits_{A_{11}}(P,A_{21}),\,
B_{22}=A_{22}.
$
\end{lemma}
\begin{proof}
For some $m$ there is an epimorphism of $A_{11}$-modules $P^m\twoheadrightarrow A_{11}$, which induces an epimorphism $(P\*_{A_{11}}A_1)^m\twoheadrightarrow A_1$.
Hence, $A$ is a direct summand of $(P\*_{A_{11}}A_1)^m\+A_2$ and $P^\sharp$ is a progenerator of $A\mbox{-}\mathrm{Mod}$.
Using Lemma~\ref{lem15}, we obtain:
\begin{align*}
& \mathop\mathrm{Hom}\nolimits_A(P\*_{A_{11}} A_1,P\*_{A_{11}} A_1)\simeq \\
& \simeq \mathop\mathrm{Hom}\nolimits_{A_{11}}(P,\mathop\mathrm{Hom}\nolimits_A(A_1,P\*_{A_{11}}A_1) \simeq \\
& \simeq\mathop\mathrm{Hom}\nolimits_{A_{11}}(P,P\*_{A_{11}}A_{11})\simeq \mathop\mathrm{End}\nolimits_{A_{11}}P;\\
& \mathop\mathrm{Hom}\nolimits_A(A_2,P\*_{A_{11}}A_1)\simeq P\*_{A_{11}}A_{12};\\
& \mathop\mathrm{Hom}\nolimits_A(P\*_{A_{11}}A_1,A_2)\simeq \mathop\mathrm{Hom}\nolimits_{A_{11}}(P,A_{21}).
\end{align*}
It gives the presentation \eqref{eq11} for $\mathop\mathrm{End}\nolimits_AP^\sharp$.
\end{proof}
\begin{theorem}\label{th17}
Let $\mM=\mM(R,L,M,N)$ be a normal Munn algebra, $C=\mathop\mathrm{End}\nolimits_RL$ and $P$ be a progenerator of the category $\mathrm{Mod}\mbox{-} C$.
Then $\mM$ is Morita equivalent to the normal Munn algebra $\mM(R,P\*_CL,M,N)$.
\end{theorem}
\begin{proof}
Let $A=\mM^1(R,L,M,N)$. Consider the idempotents $e_1=\smtr{1&0\\emptyset&0}$ and $e_2=1-e_1$. The presentation
\eqref{eq12} of the algebra $A$ is of the form
\begin{equation}\label{eq13}
\mtr{ C & \mathop\mathrm{Hom}\nolimits_R(M,L) \\ \mathop\mathrm{Hom}\nolimits_R(L,N) & \Bbbk+\mathop\mathrm{Hom}\nolimits_R(M,N) }
\end{equation}
By Lemma~\ref{lem16}, $A$ is Morita equivalent to the algebra $B$ of the matrices of the form \eqref{eq11}, where,
due to Lemma~\ref{lem15},
\begin{align*}
B_{11}&=\mathop\mathrm{Hom}\nolimits_C(P,P)\simeq \mathop\mathrm{Hom}\nolimits_C\big(P,P\*_C\mathop\mathrm{Hom}\nolimits_R(L,L)\big)\simeq \\
&\simeq \mathop\mathrm{Hom}\nolimits_C(P,\mathop\mathrm{Hom}\nolimits_R(L,P\*_CL))\simeq \mathop\mathrm{Hom}\nolimits_R(P\*_CL,P\*_CL);\\
B_{12}&=P\*_C\mathop\mathrm{Hom}\nolimits_R(M,L)\simeq \mathop\mathrm{Hom}\nolimits_R(M,P\*_CL);\\
B_{21}&=\mathop\mathrm{Hom}\nolimits_C(P,\mathop\mathrm{Hom}\nolimits_R(L,N))\simeq\mathop\mathrm{Hom}\nolimits_R(P\*_CL,N);\\
B_{22}&=\Bbbk+ \mathop\mathrm{Hom}\nolimits_R(M,N).
\end{align*}
But it is just the matrix presentation of $\mM^1(R,P\*_CL,M,N)$.
\end{proof}
The following fact is evident.
\begin{proposition}\label{prop18}
$\prod_{k=1}^s\mM(R_k,M_k,N_k,\mu_k)\simeq \mM(R,M,N,\mu)$, where
$R=\prod_{k=1}^sR_k,\,M=\bigoplus_{k=1}^sM_k,\,N=\bigoplus_{k=1}^sN_k$ and $\mu|_{N_k}=\mu_k$.
\end{proposition}
\begin{remark}\label{rem19}
Note that $\prod_{k=1}^s\mM^1(R_k,M_k,N_k,\mu_k)\not\simeq \mM^1(R,M,N,\mu)$.
\end{remark}
Let now $R$ be a semisimple ring. Then $R=\prod_{k=1}^sR_k$, where $R_k=\mathop\mathrm{Mat}\nolimits(d_k,F_k)$ for some
integers $d_k$ and some skewfields $F_k$. So any Munn algebra based on $R$ is a product of Munn algebras based on the simple
algebras $R_k$. All of them are regular, so can be supposed normal.
\begin{proposition}\label{prop110}
Let $R=\mathop\mathrm{Mat}\nolimits(d,F)$, where $F$ is a skewfield, $U$ be the simple $R$-module, $L=U^r,\,M=U^m,\,N=U^n$. The algebra $\mM(R,L,M,N)$,
up to isomorphism, only depends on $r,m,n$ and does not depend on $d$. In particular, it is isomorphic to $\mM(F,F^r,F^m,F^l)$.
\smallskip
\emph{We denote the algebra $\mM(F,F^r,F^m,F^n)$ by $\mM(F,r,m,n)$.\!}%
%
\footnote{\,Ponizovski{\u\i} \cite{Poniz} denotes this algebra by $\mathfrak{A}(E_{m+r,n+r,r},F)$.}
%
\end{proposition}
\begin{proof}
Indeed, $\mathop\mathrm{Hom}\nolimits_R(U^k,U^l)\simeq \mathop\mathrm{Mat}\nolimits(l\times k,F)$ does not depend on $d$ and with respect to such isomorphisms $\mM(R,L,M,N)=\mathop\mathrm{Mat}\nolimits((r+n)\times(r+m),F)$
with the multiplication $a\boldsymbol\cdot b=a\mu b$, where $\mu=\smtr{I&0\\emptyset&0}$ (of size $(r+m)\times(r+n)$) and $I$ is the $r\times r$ unit matrix.
\end{proof}
\begin{theorem}\label{th111}
Let $\mM=\prod_{k=1}^s\mM(F_k,r_k,m_k,n_k)$, where $F_k$ are skewfields. Then $\mM$ is Morita equivalent to
$\prod_{k=1}^s\mM(F_k,1,m_k,n_k)$.
\end{theorem}
\begin{proof}
Let $R=\prod_{k=1}^sF_k$, $L_k=F_k^{r_k}$ and $L=\prod_{k=1}^sL_k$. Then $C_k=\mathop\mathrm{End}\nolimits_RL_k\simeq\mathop\mathrm{Mat}\nolimits(r_k\times r_k,F_k)$. Let $P_k$ be the
simple right $C_k$-module. It is a progenerator of the category $\mathrm{Mod}\mbox{-} C_k$ and $P_k\*_{C_k}L_k\simeq F_k$. Now apply Theorem~\ref{th17}.
\end{proof}
We denote the algebra $M(F,1,m,n)$ by $\mM(F,m,n)$. It is the algebra of $(n+1)\times(m+1)$ matrices over $F$ with the multiplication
$a\boldsymbol\cdot b=a\mu b$, where $\mu$ is the $(m+1)\times(n+1)$ matrix with $1$ at the $(1,1)$-place and $0$ elsewhere.
\section{Representations}
\label{s2}
In this section we consider representations of finite dimensional regular Munn algebras over a field $\Bbbk$ with a semisimple base. According to
Theorem~\ref{th111}, such an algebra is Morita equivalent to a direct product $\mM=\prod_{k=1}^s\mM_k$, where
$\mM_k=\mM(F_k,m_k,n_k)$ and $F_k$ are skewfields.
If $m_k=n_k=0$, $\mM(F_k,m_k,n_k)=F_k$ and is a direct factor of $\mM^1$. So we can and will
suppose that there are no such components in $\mM$. The algebra $\mM_k$ contains an idempotent $e_k$ which is the
$(n_k+1)\times(m_k+1)$ matrix with $1$ at the $(1,1)$-place and $0$ elsewhere. Let $e_0=1-\sum_{k=1}^se_k$. Then,
if $k\ne0$, $e_k\mM^1e_k=F_k$, $e_0\mM^1e_k=F_k^{n_k}$, $e_k\mM^1e_0=F_k^{m_k}$,
$e_0\mM^1e_0=\Bbbk+\bigoplus_{k=1}^sM_k$, where $M_k\simeq\mathop\mathrm{Mat}\nolimits(n_k\times m_k,F_k)$, and $e_k\mM^1e_l=0$ if $0\ne k\ne l\ne0$.
Choose an $F_k$-basis $\{a_{k1},a_{k2},\dots,a_{km_k}\}$ in each space $e_k\mM^1e_0$ and an $F_k$-basis $\{b_{k1},b_{k2},\dots,b_{kn_k}\}$
in each space $e_0\mM^1e_k$. Then $a_{ki}b_{lj}=0$ for all $k,l,i,j$, $b_{ki}a_{lj}=0$ if $k\ne l$ and $\{b_{ki}a_{kj}\}$ is a basis of $M_k$.
For every $\mM^1$-module $V$ set $V_k=e_kV\ (0\le k\le s)$. It is a vector space over $F_k$. The multiplication
by $a_{ki}$ gives rise to a $\Bbbk$-linear map $\al_{ki}:V_0\to V_k$ and the multiplication by $b_{kj}$ gives rise to a $\Bbbk$-linear map
$\be_{ki}:V_k\to V_0$. Since $\mathop\mathrm{Hom}\nolimits_\Bbbk(V_0,V_k)\simeq\mathop\mathrm{Hom}\nolimits_{F_k}(F_k\*_\Bbbk V_0,V_k)$ and
$\mathop\mathrm{Hom}\nolimits_\Bbbk(V_k,V_0)\simeq\mathop\mathrm{Hom}\nolimits_{F_k}(V_k,\mathop\mathrm{Hom}\nolimits_\Bbbk(F_k,V_0))$, both $\al$ and $\be$ can be considered as matrices over $F_k$
of appropriate sizes. So $V$ is defined by the set of maps (or of matrices) $\{\al_{ki},\be_{lj}\}$ such that $\al_{ki}\be_{lj}=0$ for all $k,l,i,j$.
We present it by the diagram
\[
V:\ \xymatrix{ \{ V_k\}\,{\stackrel0\circlearrowleft} \ar@<-.5ex>[rr]_{\{\be_{kj}\}} && V_0, \ar@<-.5ex>[ll]_{\{\al_{ki}\}} }
\]
A homomorphism $\phi:V\to V'$ is given by a set of $F_k$-linear maps $\phi_k:V_k\to V'_k\ (0\le k\le s)$, where $F_0=\Bbbk$,
such that $\phi_k\al_{ki}=\al'_{ki}\phi_0$ and $\phi_0\be_{kj}=\be'_{kj}\phi_k$, i.e. the following diagram is commutative:
\begin{equation}\label{eq22}
\vcenter{ \xymatrix{ \{ V_k\}\,{\stackrel0\circlearrowleft} \ar@<-.5ex>[rr]_{\{\be_{kj}\}} \ar[d]_{\{\phi_k\}} && V_0 \ar@<-.5ex>[ll]_{\{\al_{ki}\}} \ar[d]^{\phi_0} \\
\{ V'_k\}\,{\stackrel0\circlearrowleft} \ar@<-.5ex>[rr]_{\{\be'_{kj}\}} && V'_0 \ar@<-.5ex>[ll]_{\{\al'_{ki}\}} } }
\end{equation}
$\phi$ is an isomorphism if and only if so are all $\phi_k$.
Set $V_+=\sum_{l,j}\mathop\mathrm{Im}\nolimits\be_{lj}\subseteq V_0,\ V_-=V_0/V_+$. Then $\al_{ki}(V_+)=0$. Hence $\al_{ki}$ can be considered as a map $V_-\to V_k$
and we obtain a diagram
\[
\tiv:\ \vcenter{ \xymatrix@R=1ex{ && V_- \ar[dll]_{\{\al_{ki}\}} \\ \{V_k\} \ar[drr]_{\{\be_{kj}\}} \\ && V_+ } }
\]
with the condition $\sum_{k,j}\mathop\mathrm{Im}\nolimits\be_{kj}=V_+$. Such diagram can be considered as a representation of the \emph{realization
$(\dM,\Om)$ of the valued graph} $(\Ga,d)$ in the sense of \cite{DR}. Namely the vertices of the graph $\Ga$ are
$\{+,-,1,2,\dots,s\}$, $d_k=\dim_\Bbbk F_k$, $d_{k+}=(m_k,m_kd_k)$, $d_{-k}=(n_kd_k,n_k)$ and $d_{ij}=0$ otherwise.
The \emph{orientation} $\Om$ of the edge $\{k,+\}$ is $\,k\to+\,$ and that of the edge $\{-,k\}$ is $\,-\to k$. The \emph{modulation} $\dM$
of $\Ga$ is given by the algebras $F_k$ and $F_\pm=\Bbbk$, $F_k\mbox{-}F_-$-bimodules ${_kM_-}=m_kF_k$ and
$F_+\mbox{-}F_k$-bimodules ${_+M_k}=n_kF_k$. Thus a representation of this realization is indeed given by a set of
$F_k$-vector spaces $V_k$, $F_0$-vector spaces $V_{\pm}$ and a set of linear maps $\tal_k:n_kV_-\to V_k$ and $\tbe_l:m_lV_l\to V_+$.
There components are just $\al_{ki}$ and $\be_{lj}$.
\begin{theorem}\label{th21}
Let $\Rep^+(\dM,\Om)$ be the full subcategory of the category of representations of $(\dM,\Om)$ such that
$\sum_{l=1}^s\mathop\mathrm{Im}\nolimits\tbe_l=V_+$ and $\bigcap_{k}\mathop\mathrm{Ker}\nolimits\tal_k=0$.
Let also $\mM\mbox{-}\mathrm{Mod}^+$ be the full subcategory of $\mM\mbox{-}\mathrm{Mod}$ consisting of such modules $V$ that
$\sum_{l,j}\mathop\mathrm{Im}\nolimits\be_{lj}=\bigcap_{k,i}\mathop\mathrm{Ker}\nolimits\al_{ki}$. Denote by $\kI$ the ideal of the category $\mM\mbox{-}\mathrm{Mod}^+$ consisting of all morphisms
$\phi:V\to V'$ such that $\phi_k=0$ for $k\ne0$, $\phi_0(V_+)=0$ and $\mathop\mathrm{Im}\nolimits\phi_0\subseteq V'_+$.
Then $\mM\mbox{-}\mathrm{Mod}^+/\kI\simeq\Rep^+(\dM,\Om)$ and $\kI^2=0$.
\end{theorem}
\begin{proof}
We have already constructed, for any $\mM$-module $V$, the representation $\tilde{V}$. By definition, $\tilde{V}\in\Rep^+(\dM,\Om)$.
Given a homomorphism $\phi=\{\phi_k\}:V\to V'$ as in \eqref{eq22}, we obtain linear maps $\phi_+:V_+\to V'_+$ and $\phi_-:V_-\to V'_-$
such that together with the maps $\phi_k$ they give a morphism $\tilde{\phi}:\tiv\to\tiv'$. Obviously, $\tilde{\phi}=0$ if and only if $\phi\in\kI$.
Thus we obtain a functor $\Phi:\mM\mbox{-}\mathrm{Mod}^+/\kI\to\Rep^+(\dM,\Om)$. Obviously $\kI^2=0$.
Let $W=(W_k,W_+,W_-,\al_k,\be_k \mid 1\le k\le s)$ be a representation from $\Rep^+(\dM,\Om)$. Set $\tiw_0=W_+\+W_-$,
take for $\tal_{ki}:W_0\to W_k$ the maps that are $0$ on $W_+$ and coincide with the components of $\al_k$ on $W_-$, and take
for $\tbe_{lj}:W_l\to \tiw_0$ the components of $\be_l:W_l\to W_+$. It defines an $\mM$-module $\tiw\in\mM\mbox{-}\mathrm{Mod}^+$. If
$\psi:W\to W'$ is a morphism of representations, set $\tilde{\psi}(w)=\psi_+(w_+)+\psi_-(w_-)$ if $w=w_++w_-$, where $w_\pm\in W_\pm$.
It gives a homomorphism $\tilde{\psi}:\tiw\to \tiw'$. Taking its class modulo $\kI$, we obtain a functor $\Psi:\Rep^+(\dM,\Om)\to\mM\mbox{-}\mathrm{Mod}^+/\kI$.
One easily verifies that this functor is quasi-inverse to $\Phi$.
\end{proof}
\begin{remark}\label{rem22}
Since $\kI^2=0$, the isomorphism classes of objects in $\mM\mbox{-}\mathrm{Mod}^+$ are the same as in $\mM\mbox{-}\mathrm{Mod}^+/\kI$.
The only indecomposable representations not belonging to $\Rep^+(\dM,\Om)$ are two \emph{trivial representations} such that $V_+=\Bbbk$
(or $V_-=\Bbbk$) and $V_k=0$ for $k\ne+$ (respectively, for $k\ne-$).
The only indecomposable $\mM$-module not belonging to $\mM\mbox{-}\mathrm{Mod}^+$ is the $1$-dimensional vector space
with zero multiplication by the elements of $\mM$. Therefore, the \emph{representation type} of the algebra $\mM$ (finite, tame or wild)
is the same as that of the realization $(\dM,\Om)$ of the valued graph $\Ga$.
\end{remark}
It is proved in \cite{DR} that the representation type of $(\dM,\Om)$ actually only depends on the valued graph itself.
Namely, it is representation finite if and only if all its connected components are \emph{Dynkin graphs} and representation tame
if and only if all of them are Dynkin or \emph{Euclidean} (\emph{extended Dynkin}) graphs and at least one Euclidean graph occurs.
For the list of these graphs see \cite[p.\,3]{DR}. In all other cases it is representation wild.
Taking into account the construction of the valued graph $\Ga$ from the algebra $\mM$, we can establish the representation type of any
finite dimensional Munn algebra with a semisimple base. Actually it only depends on the set of triples $\{(d_k,m_k,n_k)\}$, where $d_k=\dim_\Bbbk F_k$.
We use the following notations:
\begin{align*}
& \mathfrak{T}(d_1,\dots,d_r\mid d_{r+1},\dots,d_s)=\\
&\ =\{(d_1,1,0),\dots,(d_r,1,0),(d_{r+1},0,1),\dots,(d_s,0,1)\},\\
\intertext{and, for $\mathfrak{T}=\mathfrak{T}(d_1,\dots,d_r\mid d_{r+1},\dots,d_s)$,}
& S^-(\mathfrak{T})={\sum}_{k=1}^rd_k,\\
& S^+(\mathfrak{T})={\sum}_{k=r+1}^sd_k,\\
& S(\mathfrak{T})=S^-(\mathfrak{T})+S^+(\mathfrak{T}).
\end{align*}
Certainly, maybe $r=0$ or $r=s$.
\begin{theorem}\label{th23}
Let $\mM=\prod_{k=1}^s\mM(F_k,m_k,n_k)$, $\mathfrak{T}=\{(d_k,m_k,n_k)\mid (m_k,n_k)\ne(0,0)\}$, where $d_k=\dim_\Bbbk F_k$.
\begin{enumerate}
\item\hspace*{-1ex}%
%
\footnote{\,If the field $\Bbbk$ is algebraically closed, hence all $d_k=1$, this result coincides with that of Ponizovski{\u\i} \cite[n$^\circ$\,5]{Poniz}.}
%
$\mM$ is representation finite if and only if $\mathfrak{T}=\mathfrak{T}_0\cup\mathfrak{T}_1$, where $\mathfrak{T}_0=\mathfrak{T}(d_1,\dots,d_r\mid d_{r+1},\dots,d_s)$ for
some $d_k$ and
\begin{enumerate}
\item either $\mathfrak{T}_1=\emptyset$ and $\max\{S^-(\mathfrak{T}_0),S^+(\mathfrak{T}_0)\}\le3$
\item or $\mathfrak{T}_1=\{(1,1,1)\}$, $S(\mathfrak{T}_0)\le3$ and $\max\{S^-(\mathfrak{T}_0),S^+(\mathfrak{T}_0)\}\le2$.
\end{enumerate}
\smallskip
\item $\mM$ is representation tame if and only if $\mathfrak{T}=\mathfrak{T}_0\cup\mathfrak{T}_1$, where
$\mathfrak{T}_0=\mathfrak{T}(d_1,\dots,d_r\mid d_{r+1},\dots,d_s)$ for some $r$ and $d_k$, and
\begin{enumerate}
\item either $\mathfrak{T}_0=\emptyset$ and $\mathfrak{T}$ is one of the sets $$\{(1,1,1),(1,1,1)\},\,\{(2,1,1)\},\,\{(1,2,0)\},\,\{(1,0,2)\},$$
\item or $\mathfrak{T}_1=\emptyset$ and $\max\{S^-(\mathfrak{T}_0),S^+(\mathfrak{T}_0)\}=4$,
\item or $\mathfrak{T}_1=\{(1,1,1)\}$ and $S^-(\mathfrak{T}_0)=S^+(\mathfrak{T}_0)=2$.
\end{enumerate}
\smallskip
\item In all other cases $\mM$ is representation wild.
\end{enumerate}
\end{theorem}
\begin{proof}
(1a) In this case the graph $\Ga$ is a disjoint union of $2$ graphs of the types $A_2,A_3,D_4,B_2$ or $B_3$.
(1b) In this case $\Ga$ is of one of the types $A_3,A_4,A_5,D_5,D_6,B_4$ or $B_5$.
In other cases $\Ga$ is not a disjoint union of Dynkin graphs.
From now on we only list the cases when $\mM$ is not representation finite.
(2a) In these cases $\Ga$ is, respectively, of type $\tilde{A}_3$, or $\tilde{B}_2$, or $\tilde{A}_{12}$.
(2b) In this case $\Ga$ is a disjoint union of two graphs, where either both are of types $\tilde{D}_4,\tilde{BD}_3,\tilde{B}_2,\tilde{A}_{11}$ or $\tilde{G}_2$
or one is of one of these types while the other is of a type cited in case (1a).
(2c) In this case $\Ga$ is of type $\tilde{D}_6,\tilde{BD}_5$ or $\tilde{B}_4$.
(3) In all other cases the graph $\Ga$ is not a disjoint union of Dynkin and Euclidean graphs.
\end{proof}
\section{Semigroups}
\label{s3}
We apply the obtained result to representations of finite \emph{Rees matrix semigroups}. Recall
\cite[\S3.1]{ClifPres1} that such semigroup $\kM(G,p,q,\mu)$ is given by a finite group $G$ and a matrix $\mu$ of size $p\times q$
with coefficients from the group $G$.
The elements of $\kM(G,p,q,\mu)$ are $q\times p$ matrices with coefficients from $G^0=G\sqcup\{0\}$ containing at most one non-zero
element and the multiplication is defined by the rule $a\boldsymbol\cdot b=a\mu b$. If the sandwich matrix $\mu$ is \emph{regular}, i.e.
every column and every row of $\mu$ contains a non-zero element, the semigroup $\kM(G,p,q,\mu)$ is \emph{$0$-simple}
(hence \emph{completely $0$-simple}) and every finite $0$-simple semigroup is isomorphic to a Rees matrix semigroup with a
regular sandwich matrix \cite[Th.3.5]{ClifPres1}. We always suppose that the matrix $\mu$ is non-zero; otherwise $\kM(G,p,q,\mu)$
is just a semigroup with zero multiplication.
Let $\Bbbk$ be a field, $R=\Bbbk G$ and $\kM=\kM(G,p,q,\mu)$. Obviously, $\Bbbk\kM=\mM(R,R^p,R^q,\mu)$, where $\mu$ is
considered as an element of $\mathop\mathrm{Mat}\nolimits(p\times q,R)$ and is identified with an $R$-homomorphism $R^q\to R^p$.
We suppose that $\mathop\mathrm{char}\nolimits\Bbbk\nmid \#(G)$. Then $R$ is semisimple. Namely,
let $U_1,U_2,\dots,U_s$ be all irreducible representations of $G$ over $\Bbbk$, $F_k=\mathop\mathrm{End}\nolimits_GU_k$, $d_k=\dim_\Bbbk F_k$ and
$u_k=\dim_\Bbbk U_k$. Set $c_k=\frac{u_k}{d_k}$. Then $R\simeq\prod_{k=1}^sR_k$, where $R_k=\mathop\mathrm{Mat}\nolimits(c_k\times c_k,F_k)$, and
$\mathop\mathrm{Mat}\nolimits(p\times q,R_k)= \mathop\mathrm{Mat}\nolimits(pc_k\times qc_k,F_k)$.
Denote by $\mu_k$ the projection of $\mu$ onto $\mathop\mathrm{Mat}\nolimits(pc_k\times qc_k,F_k)$ and set $r_k=\mathop\mathrm{rk}\nolimits\mu_k$.
As $\mu\ne0$, also all $\mu_k\ne0$ and the Munn algebra $\Bbbk\kM$ is regular.
Then $\Bbbk\kM\simeq\prod_{k=1}^s\mM(F_k,r_k,m_k,n_k)$, where $m_k=pc_k-r_k$ and $n_k=qc_k-r_k$.
Theorem~\ref{th111} now implies the following result.
\begin{corollary}\label{cor31}
$\Bbbk\kM$ is Morita equivalent to $\prod_{k=1}^s\mM(F_k,m_k,n_k)$.
\end{corollary}
\begin{remark}\label{rem32}
Note that $c_k\mid m_k-n_k$ and $\frac{m_k-n_k}{c_k}=p-q$ does not depend on $k$. In particular, if $m_k=n_k$, or $m_k>n_k$, or $m_k<n_k$
for some $k$, the same holds for all $k$.
\end{remark}
From Corollary~\ref{cor31} and Theorem~\ref{th23}, taking into account Remark~\ref{rem32}, we obtain a classification of
representation types of Rees matrix semigroups, in particular, of $0$-simple semigroups. In the next theorem we use the
just introduced notations.
\begin{theorem}\label{th33}
Let $\kM=\kM(G,p,q,\mu)$ be a finite Rees matrix semigroup, $\Bbbk$ be a field such that
$\mathop\mathrm{char}\nolimits\Bbbk\nmid \#(G)$. Set $\mathfrak{T}(\kM)=\{(d_k,m_k,n_k)\mid (m_k,n_k)\ne(0,0)\}$.
\begin{enumerate}
\item\hspace*{-1ex}%
%
\footnote{\,If the field $\Bbbk$ is algebraically closed, hence all $d_k=1$, this result was proved by Ponizovski{\u\i} \cite{Poniz}.}
%
$\kM$ is representation finite over the field $\Bbbk$ if and only if
\begin{enumerate}
\item either $\mathfrak{T}=\{(1,1,1)\}$
\item or $\,\#(G)\le 3$ and $\mathfrak{T}$ contains either only triples $(d_k,1,0)$ or only triples $(d_k,0,1)$.
\end{enumerate}
\item $\kM$ is representation tame over the field $\Bbbk$ if and only if
\begin{enumerate}
\item either $\mathfrak{T}(\kM)=\{(1,1,1),(1,1,1)\}$, or $\mathfrak{T}(\kM)=\{(2,1,1)\}$,
\item or $\,\#(G)=4$ and $\mathfrak{T}(\kM)$ contains either only triples $(d_k,1,0)$ or only triples $(d_k,0,1)$,
\item $G=\{1\}$ and $\mathfrak{T}(\kM)=\{(1,2,0)\}$ or $\mathfrak{T}(\kM)=\{(1,0,2)\}$.
\end{enumerate}
\smallskip
\item In all other cases $\kM$ is representation wild over the field $\Bbbk$.
\end{enumerate}
Note that in cases (1a) and (2a) $p=q$, while in cases (1b) and (2b) the group $G$ is commutative.
\end{theorem}
\begin{remark}\label{r3}
According to Proposition~\ref{prop110}, the algebra $\Bbbk\kM(G,p,q,\mu)$ only depends on the ranks $r_k$. Elementary transformations
of the matrix $\mu$ do not change these ranks. Obviously, using them one can obtain a matrix $\mu'$ such that there is a non-zero element
in every row and in every column. Therefore, $\Bbbk\kM(G,p,q,\mu)\simeq\Bbbk\kM(G,p,q,\mu')$ and $\kM(G,p,q,\mu')$ is a 0-simple
semigroup \cite[Thm.3.3]{ClifPres1}. Thus, for every Rees matrix semigroup with a non-zero sandwich matrix there is a 0-simple semigroup
with the same representation theory.
\end{remark}
If a finite semigroup $\mathcal{S}=\bigvee_{i=1}^t\kM_i$ is a union of pairwise annihilating Rees matrix semigroups $\kM_i$ with common $0$,
its semigroup algebra $\Bbbk\mathcal{S}$ is a direct product of semigroup algebras $\Bbbk\kM_i$ and all of them are Munn algebras.
So we obtain the following result.
\begin{theorem}\label{th34}
Let $\mathcal{S}=\bigvee_{i=1}^t\kM_i$, where $\kM_i=\kM(G_i,m_i,n_i,\mu_i)$ are finite Rees matrix semigroups, $\Bbbk$ be a field such that
$\mathop\mathrm{char}\nolimits\Bbbk\nmid \#(G_i)$ for all $i$. Denote
\begin{align*}
& T_>=\sum_{m_i>n_i} \#(G_i),\\
& T_<=\sum_{m_i<n_i} \#(G_i),\\
& \mathfrak{T}_0=\bigcup_{m_i\ne n_i}\mathfrak{T}(\kM_i),\\
& \mathfrak{T}_1=\bigcup_{m_i= n_i}\mathfrak{T}(\kM_i)
\end{align*}
\begin{enumerate}
\item\hspace*{-1ex}%
%
\footnote{\,If the field $\Bbbk$ is algebraically closed, this result easily follows from that of Ponizovski{\u\i} \cite[\rm{n}$^\circ$5]{Poniz} and Remark
\ref{rem32}.}
%
$\mathcal{S}$ is representation finite over the field $\Bbbk$ if and only if
\begin{enumerate}
\item either $\mathfrak{T}_1=\emptyset$, $\max\{T_>,T_<\}\le3$ and all triples from $\mathfrak{T}_0$ are either $(d_k,1,0)$ or $(d_k,0,1)$
\item or $\mathfrak{T}_1=\{(1,1,1)\}$, $T_>+T_<\le 3$, $\max\{T_>,T_<\}\le2$ and all triples from $\mathfrak{T}_0$ are either $(d_k,1,0)$ or $(d_k,0,1)$.
\end{enumerate}
\item $\mathcal{S}$ is representation tame over the field $\Bbbk$ if and only if
\begin{enumerate}
\item either $\mathfrak{T}_1=\emptyset$, $\max\{T_>,T_<\}= 4$ and all triples from $\mathfrak{T}_0$ are either $(d_k,1,0)$ or $(d_k,0,1)$,
\item or $\mathfrak{T}_1=\{(1,1,1)\}$, $T_>=T_<= 2$ and all triples from $\mathfrak{T}_0$ are either $(d_k,1,0)$ or $(d_k,0,1)$,
\item or $\mathfrak{T}_0=\emptyset$ and either $\mathfrak{T}_1=\{(1,1,1),(1,1,1)\}$ or $\mathfrak{T}_1=\{(2,1,1)\}$,
\item or $\mathfrak{T}_1=\emptyset$ and $\mathfrak{T}_0=\{(1,2,0)\}$ or $\mathfrak{T}_0=\{(1,0,2)\}$.
\end{enumerate}
In the last case there is a unique index $i$ such that $m_i\ne n_i$ and the corresponding group $G_i=\{1\}$.
\item In all other cases $\mathcal{S}$ is representation wild over the field $\Bbbk$.
\end{enumerate}
\end{theorem}
\section*{Acknowledgements}
\footnotesize
This work was supported within the framework of the program of support of priority for the state scientific researches and scientific and technical (experimental) developments of the Department of Mathematics NAS of Ukraine for 2022-2023 (Project ``Innovative methods in the theory of differential equations, computational mathematics and mathematical modeling'', No. 7/1/241). The final version of the paper was prepared during the stay of the first author in the Max-Plank-Institute for
Mathematics (Bonn) and he is grateful to the Institute for their kind support.
\bibliographystyle{acm}
|
2,877,628,089,472 | arxiv | \section{Introduction}
\label{sintro}
{\em Gravity Probe B} (\mbox{\em GP-B}) is the spaceborne relativity experiment
developed by NASA and Stanford University to test two predictions of
general relativity. The experiment used four superconducting
gyroscopes, contained in a low-altitude, polar orbiting spacecraft, to
measure the geodetic effect and the much smaller frame-dragging
effect. According to general relativity, each of these effects induces
precessions of the gyroscopes in planes perpendicular to each
other. For the geodetic effect, which depends directly on the Earth's mass,
the predicted precession is 6.6 arcsec yr$^{-1}$ and for the frame-dragging
effect, which depends directly on the angular momentum of the Earth, it is 39~\mbox{mas~yr$^{-1}$}. \mbox{\em GP-B}\ was
expected to measure each precession with a standard error $\leq$0.5
\mbox{mas~yr$^{-1}$}\ relative to the distant universe. Because of technical
limitations, the spacecraft could not measure the precessions directly
relative to the distant universe but only to an optically bright star, the guide
star, chosen to be IM Pegasi (\objectname[]{HR 8703}). We must
therefore determine IM~Peg's proper motion relative to the distant universe,
which is, for our purposes, best represented by
extragalactic radio sources.
For our part of the \mbox{\em GP-B}\ project, we determined the coordinates
and the proper motion of the guide star in the radio relative to the ``core'' of
the quasar 3C~454.3 (B2251+158). This core was tied to two other radio
sources, which are compact, extragalactic, nearby to it on the sky, and also tied to a celestial
reference frame (CRF) defined by a large suite of extragalactic sources.
These ties are the main subject of this paper. Most important for \mbox{\em GP-B},
of course, is the bound that we place on the proper motion of the core,
which serves as the principal reference for determining the proper motion of
\objectname[]{IM Peg}. The \mbox{\em GP-B}\ project needs the proper motion of the optical
source in \objectname[]{IM Peg}. The radio source in \objectname[]{IM Peg},
however, moves erratically with respect to the optical source. In order to be
able to average as well as feasible over the erratic motion, we place our VLBI
limit on the motion of the core of 3C~454.3 over as long a period as feasible.
By contrast, for astrophysical
purposes, we place a more stringent bound on the core's motion, but only
for a substantially shorter period of time.
Apart from its relevance for \mbox{\em GP-B}, our observations and astrometric
analysis are also of astrophysical
interest. The quasar 3C~454.3 is a highly active superluminal radio
source \citep[see, e.g.,][]{Pauliny-Toth+1987}. It consists of a
relatively compact region from which a bent jet emanates
\citep[e.g.,][]{Pauliny-Toth1998}. Superluminal motion refers to those
apparent transverse velocities of the components within the source
that are measured to be greater than $c$, the speed of light. On the
basis of synchrotron radiation theory, the core is generally
identified as that component which is compact and has a flat or
inverted radio spectrum. Observations at 43 and 86 GHz show that
3C~454.3 has such a component located in the eastern part of the brightness
distribution \citep{Pagels+2004}. Its characteristics are consistent
with those expected for the environment of a supermassive black hole
or the base of an associated jet, but not conclusively diagnostic of either one.
An additionally powerful probe for the location of a possible
supermassive black hole in an extragalactic source is to identify the
component in the radio structure of the source which shows the
smallest motion of all components or is stationary on the sky. Such a
component would be a strong candidate for being closely related to the
purported supermassive black hole which is likely to be both close to
the center of mass of the source and virtually stationary on the sky.
Placement of stringent limits on the proper motions of quasars and
other compact extragalactic radio sources distributed across the sky
are being made by others on a routine basis through astrometric/geodetic VLBI
observations\footnote{Apart from source positions, these observations yield
antenna coordinates and velocities, and series of Earth orientation parameters.}.
Random errors of frequently observed sources may be as low as
6~$\mu$as, although systematic errors, mainly due to unaccounted
propagation effects and source structure are believed to be in the
range of 50--1000~$\mu$as.
All geodetic VLBI measurements are based on
interferometric group delays. More accurate measurements can be made, for example,
with targeted differential VLBI observations of two or more sources
located close to each other on the sky \citep{Shapiro+1979}. In such
cases, interferometric phases or phase delays can be used, yielding
relative positions and proper motions of properly selected celestial
sources with uncertainties as low as $\sim$10~\mbox{$\mu$as}~ and
$\sim$10~\mbox{$\mu$as~yr$^{-1}$}, respectively \citep[e.g.,][]{MarcaideS1983,
Bartel+1986, RiojaP2000, BietenholzBR2001, FomalontK2003,
Brunthaler+2005}.
This paper is the third in a series of seven papers reporting on the
astrometric support for \mbox{\em GP-B}\ for the purpose of defining the
cosmological reference frame for the gyroscope precession
measurements. In the first paper of this series we gave an overview
of the astronomical support for \mbox{\em GP-B}\ \citep[Paper I]{GPB-I}. In the
second paper we focused on the characteristics of quasar 3C~454.3 and
the other two extragalactic reference sources, B2250+194 and
B2252+172, and reported on their structure and structure changes with
time and frequency \citep[Paper II]{GPB-II}. In this paper (Paper
III), we report on the degree of stationarity of the core of the
quasar 3C~454.3, which is the reference source for the guide star
\objectname[]{IM Peg} and therefore pivotal for \mbox{\em GP-B}\@. In the fourth
paper we present our VLBI astrometry analysis technique and compare it
with other such techniques \citep[Paper IV]{GPB-IV}. In the fifth
paper we present our results for the proper motion and parallax of
\objectname[]{IM Peg} relative to the core of 3C~454.3 \citep[Paper
V]{GPB-V}. In the sixth paper we report on the orbital motion of
\objectname[]{IM Peg} and interpret the radio structure of the star
\citep[Paper VI]{GPB-VI}. Finally, in the seventh paper, we focus on
the individual epochs of observation of \objectname[]{IM Peg} and
include a movie of the radio images of this star \citep[Paper
VII]{GPB-VII}.
Here we first briefly describe our observations, in \S~\ref{obs}. We
give characteristics and show representative images of 3C~454.3,
B2250+194, and B2252+172 in \S~\ref{refsources}. We describe our
astrometry program in \S~\ref{astprog}. We present astrometric
results in \S~\ref{ast1}, \ref{ast2}, and \ref{ast3}, discuss these
results in \S~\ref{discuss}, and give our conclusions in
\S~\ref{conclus}.
\section{Observations}
\label{obs}
As one of the strongest-emitting quasars at radio frequencies,
3C~454.3 has been observed in geodetic group-delay VLBI sessions since
1979. For our \mbox{\em GP-B}\ VLBI program we made use of observations from the
total of 1119 such sessions between 1980 and 2008. In addition we used
geodetic observations of B2250+194 from a total of 38 sessions between
1996 and 2008 that were made in support of \mbox{\em GP-B}\ VLBI.
The bulk of our \mbox{\em GP-B}\ VLBI efforts were devoted to phase-delay VLBI
observations of \objectname[]{IM Peg} and our three reference sources,
3C~454.3, B2250+194, and B2252+172. A detailed description of these
latter observations was given in Paper II; however, for the
convenience of the reader, we give a summary here.
We obtained 35 sets of 8.4 GHz VLBI observations in support of \mbox{\em GP-B}\
between 1997 January 16 and 2005 July 16. We used a global array of
12 to 16 radio telescopes, which most often included MPIfR's 100 m
telescope at Effelsberg, Germany; NASA/Caltech/JPL's 70m DSN
telescopes at Robledo, Spain, Goldstone, CA, and Tidbinbilla,
Australia; NRAO's ten 25 m telescopes of the VLBA, across the U.S.A;
NRAO's phased VLA, equivalent to a 130 m telescope, near Socorro, NM;
and, at early times, NRCan's 46 m Algonquin Radio Telescope near
Pembroke, ONT, Canada, and, at later times, NRAO's 110 m GBT in
WV. In each session we made interleaved observations of 3C~454.3,
\objectname[]{IM Peg}, and B2250+194 by using a sequence of typically
3C~454.3 (80~s) - IM Peg (170~s) - 2250 (80~s). For the last 12
sessions, starting 2002 November, we also observed B2252+172, but only
after every second sequence to allow greater concentration on the main
three sources. The new observing sequence was 3C~454.3 (80~s) - IM
Peg (125~s) - 2250 (80~s) - 3C~454.3 (80~s) - IM Peg (125~s) - 2250
(80~s) - 2252 (90~s)\footnote{Here and hereafter we sometimes use as abbreviations
2250 for B2250+194 and 2252 for B2252+172.}. In three sessions we also observed at 5.0 and
15.0 GHz. All observations were recorded in both right and left
circular polarizations and processed on the VLBA hardware correlator at
Socorro.
\section{The Celestial Reference Sources}
\label{refsources}
\subsection{Sky Positions and Cosmological Distances}
In Figure~\ref{f1skypos} we show the positions of
\objectname[]{3C 454.3}, \objectname[]{B2250+194}, and
\objectname[]{B2252+172} along with that of the \mbox{\em GP-B}\ guide star,
\objectname[]{IM Peg}. All four sources are located approximately
along a single north-south axis allowing us to make easier use of
interpolation to estimate and potentially reduce the contributions of
the troposphere and the ionosphere to the total error in determining
the sources' relative positions. In Table~\ref{t1gen} we give the sky
separations of the two extragalactic sources from 3C~454.3, the
sources' flux densities, redshifts (when known), and angular diameter
distances, the latter assuming an inhomogeneous
Friedmann-Lema{\^i}tre-Robertson-Walker cosmology.
For comparison, we also give the characteristics of IM Peg.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{f1skypos.eps}
\caption{A sky chart with coordinates of the three reference sources
and the guide star, IM peg. The linear scale is the same for right
ascension ($\alpha$) and declination ($\delta$) for the declination of
3C~454.3.}
\label{f1skypos}
\end{figure}
\begin{deluxetable}{l l l l l l l}
\tabletypesize{\small}
\tablecaption{Characteristics of the sources}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{Type} & \multicolumn{2}{c}{Separation} & Flux density\tablenotemark{a} & \colhead{Redshift}
& \colhead{Distance}\tablenotemark{b} \\
& & \colhead{$\Delta$\mbox{$\alpha$}(\arcdeg)} & \colhead{$\Delta$\mbox{$\delta$}(\arcdeg)} & \colhead{(Jy)}
& & \colhead{(Mpc)}
}
\startdata
3C~454.3 & quasar & \nodata & \nodata & 7 -- 10 & 0.859 & 1610 \\
B2250+194 & galaxy & $-0.2$ & 3.6 & 0.35 -- 0.45 & 0.28 & \phn880 \\
B2252+172 & unidentified & \phn 0.4 & 1.4 & 0.017 & \nodata & \nodata \\
\\
IM Peg & RS CVn & $-0.1$ & 0.7 & 0.005 -- 0.05 & 0.0 & \phn\phn\phn0.0 \\
\enddata
\tablenotetext{a}{The range gives the lowest and highest flux density we measured at 8.4 GHz with the VLA during the course of our
observations, 1997 January to 2005 July.}
\tablenotetext{b}{The angular diameter distance for a flat universe with Hubble constant,
$H_0$=70~\mbox{km s$^{-1}$ Mpc$^{-1}$}, and normalized density parameters, $\Omega_M=0.27$ and $\Omega_{\lambda}$=0.73 \citep{KantowskiKT2000}.}
\label{t1gen}
\end{deluxetable}
\subsection{Representative Images}
In Figures~\ref{f23C454}, \ref{f32250}, and \ref{f42252} we show
representative images of \objectname[]{3C 454.3},
\objectname[]{B2250+194}, and \objectname[]{B2252+172}. The source
3C~454.3 is a superluminal quasar with the highest radio flux density
of the three sources. It consists of a core region, which is primarily
extended east-west, and can be well modeled for each of our 35 epochs
by two compact components, C1 and C2, separated by $\sim$0.6 mas. When
studied at a higher resolution, e.g., at 43 and 86 GHz, the same
region has a complex structure \citep{GomezMA1999, Jorstad+2001b,
Jorstad+2005}, with C1 being essentially unresolved and having a size
at 86 GHz of $\leq$70~\mbox{$\mu$as}\ \citep{Pagels+2004}. C2 is
extended. Also, between these two components there are others that
appear to move away from C1 toward C2 with superluminal speeds
\citep{Jorstad+2001b, Jorstad+2005}; these other components are not
individually visible in our lower-resolution 8.4 GHz images.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{f23c45405may28.eps}
\caption{An image of 3C~454.3 from observations on 2005 May 28 with
components C1, C2, D1, D2, J1, and Jext indicated. The contours start
at 10 m\mbox{Jy bm$^{-1}$}\ and increase by factors of $\sqrt{2}$ towards the
peak. The peak brightness is 2.68 \mbox{Jy bm$^{-1}$}. The rms brightness of the
background noise is 0.73 m\mbox{Jy bm$^{-1}$}. The full-width at half-maximum (FWHM)
contour of the Gaussian convolving beam is given in the lower
right. North is up and east to the left.}
\label{f23C454}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{f32250+194_05jan15.eps}
\caption{An image of B2250+194 from observations on 2005 January 15.
The contours are at 0.3, 0.5, 1, 2, 5, 10, 20, 40, and 80\% of the
peak brightness of 0.43 \mbox{Jy bm$^{-1}$}. The rms brightness of the background noise in 0.08 \mbox{mJy bm$^{-1}$}. The FWHM contour of the Gaussian
convolving beam is given in the lower left. North is up and east to
the left.}
\label{f32250}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{f42252+172ref_05jan15.eps}
\caption{An image of B2252+172 from observations on 2005 January 15.
The contours are at 3, 5, 10, 20, 30, \dots\ 90\% of the peak brightness
of 0.012 \mbox{Jy bm$^{-1}$}. The rms brightness of the background noise is 0.05 \mbox{mJy bm$^{-1}$}.
The FWHM contour of the Gaussian convolving beam is
given in the lower left. North is up and east to the left.}
\label{f42252}
\end{figure}
Further to the west there are components, D1 and D2, that are visible
only at later epochs in our observations. We modeled these as compact
components. They bridge the core region to the outer part of the jet
with a more than 10 mas extent in our images. This outer part of the
jet bends toward the northwest and in fact extends as far as a few
arcseconds away from the core \citep[see also][]{CawthorneG1996,
Pauliny-Toth1998}. The brightness peak of the extended 10 mas jet in
our image is clearly visible at each of our 35 epochs, and we modeled
it as a compact component which we call J1. The extended jet can be
modeled as an elliptical Gaussian, which we call Jext.
The source \objectname[]{B2250+194} is 20 times weaker in terms of
flux density than 3C~454.3, but much more compact, consisting of a
central component with north-south extensions and an apparent length of
$\sim$5~mas. The source \objectname[]{B2252+172} is the weakest of the
three in terms of flux density but also the most compact, consisting
of one dominant component and a barely visible extension to the
west. For more detail on the structure of these sources and their
evolution or lack thereof, see Paper II.
\subsection{Selection of 3C~454.3 as a Reference Source for \mbox{\em GP-B}}
Because of these differences in characteristics and separations from
\objectname[]{IM Peg}, each of these sources has advantages and
disadvantages as a reference source for \objectname[]{IM Peg}. The
quasar 3C~454.3 has the advantage of the highest flux density and
closest proximity to the guide star. The latter point is of chief
importance, since the standard errors of astrometric VLBI measurements
scale approximately linearly with the separation of the reference
source from the target source \citep{Shapiro+1979,
PradelCL2006}. Another advantage is that 3C~454.3 was used as the
astrometric reference source for \objectname[]{IM Peg} as early as
1991 \citep{Lestrade+1999}, thus extending the time baseline of VLBI position determinations and allowing increased accuracy in our proper-motion estimates. The main disadvantage of this reference source is its complex structure.
The advantage of the other two sources is their compact
structure.
However, \objectname[]{B2250+194} is relatively far away on
the sky from \objectname[]{IM Peg}, and \objectname[]{B2252+172} has a
relatively low flux density. All things considered, we decided to use
3C~454.3 as the primary reference source for \objectname[]{IM Peg}.
Our task was then to find the component in 3C~454.3 that is most
closely associated with the dynamical center of the quasar and to test
the stationarity of this component with respect to our two other
reference sources and our CRF.
Images of 3C~454.3 at 8.4 and 15 GHz (Paper II) and at 43 and 86 GHz
\citep{Pagels+2004} show that the easternmost component, C1, remains
compact at the highest frequencies and angular resolutions yet investigated, and has a
flat or inverted spectrum in this frequency region. Other components
or condensations show structure at 86 GHz, and, in cases where it was
determinable, a steep spectral index. These characteristics indicate
that for our 8.4 GHz images, C1 is likely the component most closely
related to the putative supermassive black hole and the quasar's center of
mass.
\section{The Astrometry Program for \mbox{\em GP-B}: Goal, Strategy, and Procedure}
\label{astprog}
\begin{trivlist}
\item{\sl Goal: }Our main goal is to determine C1's position and
especially a bound on its proper motion relative to the distant
universe, to confirm its suitability as the primary reference for \mbox{\em GP-B}.
\item{\sl Strategy: }The distant universe is for our purposes most
usefully represented by compact, extragalactic radio sources. We choose
a procedure where we first determine the position and proper motion of C1
relative to our two reference sources, B2250+194 and B2252+172, and
second relative to our CRF. For the first step,
interferometric phase-delays are used exclusively. This has the advantage of
simplicity, utilization of the same type of data (phase-delays)
and analysis technique, and highest precision and possibly highest
accuracy of the results. For the second, the
results from the first are added to position and proper-motion
determinations of B2250+194 in the CRF based on interferometric group
delays from the geodetic VLBI sessions. This has the advantage of having as a
reference not only two very distant and compact sources, but $\sim$4000 sources
that define our CRF.
\item{\sl Procedure:} We determine the position and the limit on proper motion of:
\begin{enumerate}
\item C1 relative to
B2250+194 and B2252+172, and B2252+172 relative to B2250+194 in the
senses (C1 $-$ 2250), (C1 $-$ 2252), and (2252 $-$ 2250) using our
interleaved phase-delay VLBI observations. The combined result for C1
is obtained as a weighted mean of the first two
differenced solutions, with the third serving to demonstrate the
consistency of our results and zero proper motion within the errors
for the two sources relative to each other.
Due to the source's compactness, any motions or brightness
distribution changes of these two reference sources
would likely be very small and therefore have only a marginal, if any,
effect on our astrometric results for C1.
\item B2250+194 in the CRF using routine geodetic and astrometric group-delay VLBI
observations distributed by
the International VLBI Service for Geodesy and Astrometry\footnote{Available
at http://ivscc.gsfc.nasa.gov/products-data/index.html};
\item B2252+172 in the CRF
by adding the result from 2.\ to that from 1.\ in the sense (2252 $-$
2250) + 2250, and confirming the position result and proper motion limit by using recent single-epoch
geodetic group-delay VLBI observations of B2252+172; and
\item C1 in the CRF in two
ways: First by adding the results from 1.\ to those from 2.: C1 = (C1
$-$ 2250) + 2250, and second by adding the results from 1. to those of
3.: C1 = (C1 $-$ 2252) + 2252. The combined result is obtained as a
weighted mean from these two ways.
\end{enumerate}
\end{trivlist}
\section{Astrometric Results (1): Position Determinations for Each Observing Session}
\label{ast1}
\subsection{Analysis of Interleaved Phase-delay Observations of 3C 454.3, B2250+194, and B2252+172}
Our VLBI data for the reference sources, 3C~454.3, B2250+194, B2252+172,
and also from IM Peg, were analyzed
with an astrometric software package that was developed specifically
for the analysis of the \mbox{\em GP-B}\ VLBI data. It includes a
phase-connection program that automatically resolves 2$\pi$
ambiguities that exist in the set of VLBI phases for each
baseline so as to convert them to phase delays. The software also
includes in the phase-delay fitting \citep[e.g.,][]{Shapiro+1979,
Bartel+1986} a Kalman filter (see Paper IV) to model the variations of
the troposphere, the ionosphere, and the clock offsets at each VLBI
site \citep{Lebach+1999}. In addition, we used two different models
to initially correct for the effects of the ionosphere, one we call
``JPL'' which is part of NRAO's imaging package, AIPS, and based on
GPS data provided by JPL, and the other, the older parametrized
ionospheric model ``PIM,'' developed at USAF Research Labs
\citep[described by, e.g.,][]{Campbell+1999}.
To relate the phase delays to a particular reference point in
3C~454.3, namely the core component C1, all phase delays from
differential astrometry involving 3C~454.3 were corrected for the
structure of 3C 454.3, as represented by the CLEAN components produced with AIPS;
C1 served as the phase reference
point. The other two extragalactic sources were deemed sufficiently
compact for our purposes so that the reference point for each could be assumed
to be the brightness peak in its image. We elaborate on this method
of astrometric VLBI data analysis and compare it to other methods in
Paper IV.
\subsection{Positions of the Components of 3C 454.3 Relative to Those of B2250+194 and B2252+172}
To test the positional stability of 3C~454.3's component C1 relative
to our two reference sources, we determined, for each of our 35
sessions of 8.4 GHz observations, the coordinates of C1, and for
comparison also those of C2, D1, D2, J1, and Jext, all relative to the
brightness peak of B2250+194 and, for the last 12 epochs, also to that of
B2252+172. We obtained two sets of coordinates by correcting for the
effects of the ionosphere in two different ways, one set by using the
JPL model and another set by using PIM\@. Although the JPL model was
not available for our first 8 epochs but only from 1998 September 17
onward, we nevertheless in this paper use mainly phase-delay data corrected with the JPL model
since it proved to be superior in that it resulted in
smaller uncertainties of our astrometric estimates despite precluding
the use of our earliest phase-delay data. We elaborate on the comparison below. (In
Paper V we use PIM instead, because in that paper errors in modeling the
ionosphere play a less significant role than they play here, and because
PIM has the advantage that it can be used for all of
our VLBI data.)
We list our coordinate determinations for C1, C2, D1, D2, J1, and Jext relative
to the brightness peak of \objectname[]{B2250+194} in
Table~\ref{t32250} and to that of \objectname[] {B2252+172} in
Table~\ref{t42252}, all obtained with the JPL model.
\subsection{The Uncertainties of the Relative Positions of the Components of 3C 454.3}
\label{errors}
The uncertainties of the coordinates listed in Tables~\ref{t32250} and \ref{t42252} were
determined partly empirically,
namely by adding a constant in quadrature to the statistical standard
errors so as to obtain a reduced Chi-square of unity ($\mbox{$\chi^2_\nu / \nu$}=1$,
where $\nu$ is the number of degrees of freedom) in our residuals after solving for
relative position and proper motion in \mbox{$\alpha$}\ and \mbox{$\delta$}\ separately.
This constant is assumed to approximately reflect
non-statistical errors.
Accurate standard errors are difficult to estimate in any other way. They contain
contributions from noise and from systematic errors, with the latter
due mostly to deconvolution, source structure, and atmospheric and
ionospheric variations. We next discuss and approximately quantify each contribution in turn.
\subsubsection{Noise}
Noise in an image has a number of sources. The rms background brightness
in the images is dominated by contributions from statistical noise in
the radio signals and thermal noise in the receivers.
However, for our relatively high
dynamic range images (typically over 2,000 to 1; see,
e.g., Figures~\ref{f23C454} to \ref{f42252}), the various uncertainties in the bright
parts of the image are larger than their corresponding rms background brightnesses
and are dominated by contributions which are not
strictly random such as residual calibration errors and deconvolution
errors.
Given the small rms of the background brightness relative to the
peaks in the maps, we conclude that this noise causes errors in the estimate of the
separation of each of components C1 and C2 in 3C~454.3 from B2250+194
by a correspondingly small portion of the HWHM (half-width at
half-maximum) of the beam, namely by $<$5~\mbox{$\mu$as}, and of each of components D1,
D2, J1 and Jext in 3C~454.3 from B2250+194 by $<$10~\mbox{$\mu$as}. The
corresponding errors in our estimates of the separation of these components from the
weaker source, B2252+172, are dominated by the source's lower
peak-to-noise ratio but are still $<$10~\mbox{$\mu$as}.
\subsubsection{Deconvolution errors}
Deconvolution errors are caused by the visibility measurements not
filling the \mbox{$u$-$v$}~plane of the VLBI array up to its highest angular
resolution and by the resulting generation of side lobes in the image
plane, which are not completely eliminated through the deconvolution
process.
We studied
this type of error by using a noise-free model image similar to the
image of 3C~454.3 at 8.4 GHz,
Fourier-transforming the model to the \mbox{$u$-$v$}\ plane, and then using the
same \mbox{$u$-$v$}-plane sampling as in one of our typical observing sessions.
The
generated \mbox{$u$-$v$}\ model data were then used for imaging and deconvolution. We then
determined the difference between the position of each component in the
model image and the position of the corresponding component in the
deconvolved image. We found that the deconvolution
error for each coordinate of each of the six components of 3C~454.3 is
typically $30$~\mbox{$\mu$as}\ and never larger than $\sim40$~\mbox{$\mu$as}. The
deconvolution errors for sources like B2250+194 and B2252+172 with
relatively simple brightness distributions are doubtless smaller given
the same \mbox{$u$-$v$}\ coverage as for 3C~454.3. We therefore conclude that the
standard error in each coordinate of the separation of any component
in 3C~454.3 from either B2250+194 or B2252+172 is typically
$30$~\mbox{$\mu$as}.
\subsubsection{Structure errors}
Structure errors are caused by a mismatch between the Gaussian
component model and the brightness distribution of the source. If the
source were completely unresolved, a fit of a Gaussian to the image
with the parameters of the convolving beam would give the position of
the source with an essentially zero structure error. Indeed, the
sources B2250+194 and B2252+172 are rather compact, with brightness
peaks that can be clearly identified and located, and hence we expect the structure
errors to be small in comparison to those for the
components in 3C~454.3. As can be seen in Figure~\ref{f23C454},
components C1 and C2 are located close together in comparison to the
size of the beam and can barely be distinguished at several epochs
(see Paper II). In addition, there is ``confusing'' emission in the
neighborhood of these two components. These two characteristics cause
significant structure errors, difficult to estimate quantitatively,
but likely as large as a good fraction of the HWHM of the beam. The
components D1 and D2 are weaker than C1 and C2, and visible only at later
epochs, but then clearly distinguishable. Their identification is
likely as uncertain as that of C1 and C2. Component J1 is
always visible as a single peak; however, it is embedded in the
extended brightness distribution of the 10 mas long jet rendering the
identification of a component in that jet as uncertain as, for many
epochs, the identification of C1 and C2. The identification of Jext is
more uncertain than that of any of the other components since this component
represents the extended brightness distribution that J1 is embedded
in. The extension is largely toward the northwest, thus affecting both
position coordinates of Jext. A detailed analysis of these uncertainties
is difficult to carry out quantitatively, in view of the unknown
characteristics of the relevant structures. Based on our experience,
however, we expect structure errors in each
of the position coordinates to be $\sim30$~\mbox{$\mu$as}\ for components C1, C2,
D1, D2, and J1 and $\sim50$~\mbox{$\mu$as}\ for the component Jext. In comparison, the
structure errors for B2250+194 and B2252+172 are negligible.
\subsubsection{Residual propagation-medium errors}
Using our Kalman filter \citep[Paper IV]{GPB-IV} removes a large portion of the
propagation-medium errors from the estimates of separation between two
sources. Nevertheless, errors in modeling the troposphere and the
ionosphere at each site likely still represent the largest sources of
error in our estimates of relative positions. These errors scale
approximately with the separation between the sources
\citep{Shapiro+1979, PradelCL2006}. These errors also depend on the model
used for the ionosphere (JPL or PIM).
In our case the sources are oriented approximately north-south with the
separations in \mbox{$\alpha$}\ of B2250+194 and B2252+172 from C1 being about
equal, but with the separation in \mbox{$\delta$}\ of B2252+172 from C1 being only
about 40\% of that for B2250+194 from C1. Also, the position
determinations are more affected in \mbox{$\delta$}\ due to the propagation
medium's distorting effects depending strongly on the elevation angle of the source.
Therefore, any residual uncorrected ionospheric or tropospheric
effects should be most visible in comparisons of the estimated declinations
of C1 relative to B2250+194 with those relative to
B2252+172, since the angular separation of B2250+194 and B2252+172 is
predominantly north-south. To evaluate the adequacy of the models, we plot
the relative coordinates, C1 $-$ B2250+194, as a function of the
relative coordinates, C1 $-$ B2252+172, separately in \mbox{$\alpha$}\ and
\mbox{$\delta$}. For instance, a straight-line slope significantly larger than
unity would indicate uncorrected ionospheric or tropospheric
effects. We plot the relative coordinates for the JPL model and for
PIM in Figure~\ref{f7lcor}. The data corrected with the JPL model were
taken from Tables~\ref{t32250} and \ref{t42252}, whereas the data
corrected with PIM are not listed.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{lcor4panel.eps}
\caption{The coordinate differences (except for offsets) in \mbox{$\alpha$}\
(upper row) and \mbox{$\delta$}\ (lower row) of B2250+194 $-$ C1 as a function of
the coordinate differences of B2252+172 $-$ C1 for data for which the
ionospheric effects were corrected with the JPL model (left column)
and with PIM (right column). The data corrected with the JPL model
were taken from Tables~\ref{t32250} and \ref{t42252}. The data
corrected with PIM are not given in the tables. The straight lines
indicate the least-squares fits. The slope and its statistical
standard error are given for the fit in each figure.}
\label{f7lcor}
\end{figure}
Least-squares fits\footnote{For simplicity we fit the data via
least-squares only in the vertical direction. Fitting the data in both
directions would not significantly alter our results. Also, since the
errors in the vertical direction are similar to each other and the
errors in the horizontal direction are similar to each other too, we
did not weight the data in the fits and also did not plot the errors in the
figures.} to the data give slopes that are all larger than
unity, reflecting the larger separation of B2250+194 from C1 than of
B2252+172 from C1.
However, the slopes are within 1.3$\sigma$ of unity, except for the
data in \mbox{$\delta$}\ for the PIM model. Here the slope is $1.78\pm0.20$. It is
smaller than a slope of $\sim$2.5 which, if there were no ionospheric
and tropospheric correction, would likely be expected on the basis of
the $\sim$2.5:1 ratio of the separations of the two sources from
C1. However, this slope is significantly ($\sim4\sigma$) larger than unity and also larger
than the corresponding slope for the data corrected with the JPL
model. The JPL model thus seems to provide a better correction for the
ionosphere.
An approximate estimate of uncorrected errors due to the ionosphere
and troposphere for position estimates using PIM (as in Paper V for IM
Peg) and an upper limit of such errors for position estimates using
the JPL model (this paper) can be obtained from a close inspection of
the declination data corrected with PIM (lower right panel of
Figure~\ref{f7lcor}). While the $\Delta$\mbox{$\delta$}\ values of C1 $-$
2252 vary over a range of 0.19 mas, the corresponding values of
C1 $-$ 2250 vary over a range of 0.34 mas, 1.78 times larger, as
also indicated by the slope. Therefore, the difference of the
variations of 0.15 mas can likely be attributed to effects of the
ionosphere left uncorrected by PIM. Consequently, we think that for
data corrected with PIM the peak-to-peak contribution to the error of
the position difference, C1 $-$ 2250, in \mbox{$\delta$}\ is approximately
$\pm$0.08~mas. To be conservative, we take this as a standard error.
The corresponding contribution to the error of the position
difference, C1 $-$ 2252, in \mbox{$\delta$}\ approximately scales with the
separation and is therefore likely to be about 0.03 mas. The
contributions to the errors in \mbox{$\alpha$}\ that we estimated from scaling are
0.04 mas and again 0.03 mas for the two position differences,
respectively. For data corrected with the JPL model, we considered all these
estimates for PIM as still more conservative upper limits for the
standard deviations of these errors.
\subsubsection{Total errors}
Adding in quadrature the errors from noise, deconvolution, structure,
and propagation, we obtain estimated standard errors in \mbox{$\alpha$}\ of the
position differences from B2250+194 for the components C1, C2, D1, D2,
and J1 of 0.06 mas and for the component Jext of 0.07 mas. The
corresponding standard errors in \mbox{$\delta$}\ are 0.09 mas and 0.10 mas. The
corresponding standard errors of the position differences from
B2252+172, in both \mbox{$\alpha$}\ and \mbox{$\delta$}, for the components C1, C2, D1, D2,
and J1 are 0.05 mas and for the component Jext 0.07 mas.
\subsubsection{Comparison with empirically determined errors}
For the position differences of these components from B2250+194, our
estimated standard errors are somewhat smaller than the empirically
determined standard errors which were mostly between the values of
0.08 and 0.10 mas in both \mbox{$\alpha$}\ and \mbox{$\delta$}\ for these components
(see Table \ref{t32250}). The reason is the large variation of some
of the data points near epoch 2001
which increased the empirically determined errors. For the
corresponding position differences from B2252+172, all for epochs
after 2001, these estimated standard errors agree well with the
empirically determined standard errors of 0.04 to 0.06 mas in \mbox{$\alpha$}\ and
\mbox{$\delta$}, respectively, for the components C1, C2, D1, D2, and J1 and of
0.07 mas in \mbox{$\alpha$}\ and \mbox{$\delta$}\ for the component Jext (see Table
\ref{t42252}).
\begin{deluxetable}{r c r r r r r r r r r r r r}
\rotate
\tabletypesize{\tiny}
\tablecaption{Separation of 3C 454.3 components from B2250+194}
\tablewidth{0pt}
\tablehead{
\colhead{Epoch} & \colhead{Julian date} & \multicolumn{2}{c}{C1\tablenotemark{a}} & \multicolumn{2}{c}{C2\tablenotemark{a}} & \multicolumn{2}{c}{D1\tablenotemark{a}} & \multicolumn{2}{c}{D2\tablenotemark{a}} & \multicolumn{2}{c}{J1\tablenotemark{a}} & \multicolumn{2}{c}{Jext\tablenotemark{a}} \\
& \colhead{2450000.0 +} & \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}} \\
& & \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
}
\startdata
1998 Sep 17 & 1073.8 & -0.11$\pm$0.08 & 0.25$\pm$0.09 & -0.58$\pm$0.11 & 0.22$\pm$0.10 & \nodata & \nodata & \nodata & \nodata & -5.57$\pm$0.07 & 0.72$\pm$0.10 & -6.34$\pm$0.10 & 2.09$\pm$0.08 \\
1999 Mar 13 & 1251.3 & -0.21$\pm$0.08 & 0.11$\pm$0.08 & -0.68$\pm$0.12 & 0.06$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.63$\pm$0.07 & 0.61$\pm$0.09 & -6.46$\pm$0.11 & 1.99$\pm$0.08 \\
1999 May 15 & 1314.1 & -0.11$\pm$0.08 & 0.14$\pm$0.08 & -0.52$\pm$0.12 & 0.07$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.48$\pm$0.07 & 0.64$\pm$0.09 & -6.29$\pm$0.11 & 1.98$\pm$0.08 \\
1999 Sep 18 & 1440.8 & -0.27$\pm$0.08 & 0.23$\pm$0.08 & -0.68$\pm$0.12 & 0.14$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.66$\pm$0.07 & 0.74$\pm$0.10 & -6.54$\pm$0.11 & 2.14$\pm$0.08 \\
1999 Dec 09 & 1522.5 & -0.12$\pm$0.08 & 0.21$\pm$0.08 & -0.60$\pm$0.12 & 0.16$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.59$\pm$0.07 & 0.76$\pm$0.10 & -6.52$\pm$0.10 & 2.24$\pm$0.08 \\
2000 May 15 & 1680.1 & -0.10$\pm$0.08 & 0.16$\pm$0.08 & -0.71$\pm$0.12 & 0.17$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.64$\pm$0.08 & 0.87$\pm$0.08 & -6.50$\pm$0.10 & 2.05$\pm$0.08 \\
2000 Aug 07 & 1763.9 & -0.13$\pm$0.09 & 0.10$\pm$0.09 & -0.78$\pm$0.14 & 0.08$\pm$0.10 & \nodata & \nodata & \nodata & \nodata & -5.64$\pm$0.09 & 0.82$\pm$0.10 & -6.59$\pm$0.11 & 2.17$\pm$0.09 \\
2000 Nov 05 & 1854.6 & -0.26$\pm$0.08 & 0.06$\pm$0.08 & -0.84$\pm$0.13 & -0.01$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.68$\pm$0.08 & 0.77$\pm$0.09 & -6.53$\pm$0.10 & 1.99$\pm$0.08 \\
2000 Nov 06 & 1855.6 & -0.44$\pm$0.13 & -0.36$\pm$0.18 & -1.14$\pm$0.16 & -0.42$\pm$0.18 & \nodata & \nodata & \nodata & \nodata & -5.93$\pm$0.12 & 0.35$\pm$0.19 & -7.01$\pm$0.14 & 2.01$\pm$0.18 \\
2001 Mar 31 & 2000.2 & 0.06$\pm$0.09 & 0.19$\pm$0.10 & -0.37$\pm$0.12 & 0.13$\pm$0.11 & \nodata & \nodata & \nodata & \nodata & -5.42$\pm$0.08 & 0.96$\pm$0.10 & -6.48$\pm$0.11 & 2.34$\pm$0.09 \\
2001 Jun 29 & 2090.0 & -0.05$\pm$0.08 & 0.14$\pm$0.08 & -0.59$\pm$0.14 & 0.09$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.57$\pm$0.09 & 0.95$\pm$0.09 & -6.50$\pm$0.11 & 2.22$\pm$0.08 \\
2001 Oct 19 & 2202.7 & -0.12$\pm$0.11 & 0.02$\pm$0.15 & -0.70$\pm$0.14 & -0.05$\pm$0.16 & \nodata & \nodata & \nodata & \nodata & -5.65$\pm$0.11 & 0.88$\pm$0.16 & -6.68$\pm$0.13 & 2.29$\pm$0.15 \\
2001 Dec 21 & 2265.5 & -0.21$\pm$0.08 & -0.06$\pm$0.08 & -0.89$\pm$0.14 & -0.13$\pm$0.09 & \nodata & \nodata & \nodata & \nodata & -5.75$\pm$0.08 & 0.79$\pm$0.09 & -6.81$\pm$0.11 & 2.28$\pm$0.08 \\
2002 Apr 14 & 2379.2 & -0.09$\pm$0.09 & 0.12$\pm$0.11 & -0.64$\pm$0.12 & 0.01$\pm$0.12 & \nodata & \nodata & \nodata & \nodata & -5.58$\pm$0.09 & 1.06$\pm$0.13 & -6.65$\pm$0.11 & 2.41$\pm$0.11 \\
2002 Jul 14 & 2469.9 & -0.09$\pm$0.08 & 0.10$\pm$0.09 & -0.60$\pm$0.14 & 0.00$\pm$0.10 & \nodata & \nodata & \nodata & \nodata & -5.59$\pm$0.08 & 0.98$\pm$0.09 & -6.65$\pm$0.11 & 2.34$\pm$0.08 \\
2002 Nov 20 & 2599.6 & -0.12$\pm$0.09 & -0.16$\pm$0.10 & -0.69$\pm$0.13 & -0.25$\pm$0.11 & \nodata & \nodata & \nodata & \nodata & -5.71$\pm$0.09 & 0.77$\pm$0.11 & -6.79$\pm$0.11 & 2.22$\pm$0.09 \\
2003 Jan 26 & 2666.4 & -0.08$\pm$0.09 & 0.06$\pm$0.10 & -0.67$\pm$0.13 & -0.06$\pm$0.11 & -1.56$\pm$0.07 & -0.20$\pm$0.07 & \nodata & \nodata & -5.65$\pm$0.09 & 0.95$\pm$0.11 & -6.67$\pm$0.11 & 2.28$\pm$0.09 \\
2003 May 18 & 2778.1 & 0.05$\pm$0.08 & 0.05$\pm$0.08 & -0.64$\pm$0.12 & -0.01$\pm$0.09 & -1.51$\pm$0.08 & -0.24$\pm$0.06 & \nodata & \nodata & -5.64$\pm$0.08 & 1.03$\pm$0.10 & -6.59$\pm$0.10 & 2.30$\pm$0.08 \\
2003 Sep 08 & 2891.7 & 0.02$\pm$0.08 & 0.02$\pm$0.09 & -0.69$\pm$0.12 & -0.06$\pm$0.10 & -1.62$\pm$0.08 & -0.30$\pm$0.07 & \nodata & \nodata & -5.60$\pm$0.11 & 1.00$\pm$0.11 & -6.61$\pm$0.11 & 2.28$\pm$0.09 \\
2003 Dec 05 & 2979.5 & -0.01$\pm$0.08 & 0.11$\pm$0.09 & -0.73$\pm$0.11 & 0.01$\pm$0.10 & -1.70$\pm$0.09 & -0.27$\pm$0.06 & \nodata & \nodata & -5.59$\pm$0.08 & 1.03$\pm$0.10 & -6.63$\pm$0.10 & 2.37$\pm$0.08 \\
2004 Mar 06 & 3071.3 & 0.02$\pm$0.08 & 0.00$\pm$0.09 & -0.68$\pm$0.12 & -0.10$\pm$0.10 & -1.66$\pm$0.09 & -0.37$\pm$0.05 & -3.53$\pm$0.05 & -0.50$\pm$0.13 & -5.56$\pm$0.08 & 0.96$\pm$0.10 & -6.62$\pm$0.10 & 2.38$\pm$0.08 \\
2004 May 18 & 3144.1 & -0.03$\pm$0.09 & 0.09$\pm$0.09 & -0.71$\pm$0.12 & -0.06$\pm$0.10 & -1.69$\pm$0.07 & -0.29$\pm$0.07 & -3.59$\pm$0.05 & -0.41$\pm$0.12 & -5.57$\pm$0.10 & 1.05$\pm$0.12 & -6.72$\pm$0.11 & 2.53$\pm$0.09 \\
2004 Jun 26 & 3183.0 & -0.13$\pm$0.08 & -0.04$\pm$0.09 & -0.83$\pm$0.16 & -0.18$\pm$0.10 & -1.86$\pm$0.07 & -0.37$\pm$0.06 & -3.69$\pm$0.05 & -0.53$\pm$0.10 & -5.61$\pm$0.10 & 0.91$\pm$0.10 & -6.73$\pm$0.11 & 2.31$\pm$0.08 \\
2004 Dec 11 & 3351.5 & 0.01$\pm$0.08 & 0.01$\pm$0.08 & -0.67$\pm$0.12 & -0.06$\pm$0.09 & -1.66$\pm$0.10 & -0.44$\pm$0.05 & -3.70$\pm$0.04 & -0.39$\pm$0.10 & -5.62$\pm$0.09 & 0.94$\pm$0.09 & -6.77$\pm$0.10 & 2.45$\pm$0.07 \\
2005 Jan 15 & 3386.4 & 0.06$\pm$0.08 & -0.05$\pm$0.08 & -0.62$\pm$0.12 & -0.10$\pm$0.09 & -1.66$\pm$0.10 & -0.48$\pm$0.05 & -3.64$\pm$0.05 & -0.55$\pm$0.10 & -5.57$\pm$0.08 & 0.88$\pm$0.09 & -6.75$\pm$0.10 & 2.42$\pm$0.08 \\
2005 May 28 & 3519.1 & 0.00$\pm$0.08 & 0.04$\pm$0.08 & -0.69$\pm$0.12 & 0.00$\pm$0.09 & -1.82$\pm$0.09 & -0.40$\pm$0.05 & -3.71$\pm$0.05 & -0.23$\pm$0.10 & -5.63$\pm$0.08 & 1.02$\pm$0.10 & -6.75$\pm$0.10 & 2.52$\pm$0.08 \\
2005 Jul 16 & 3567.9 & -0.04$\pm$0.08 & 0.09$\pm$0.09 & -0.76$\pm$0.12 & 0.06$\pm$0.10 & -1.92$\pm$0.09 & -0.34$\pm$0.06 & -3.74$\pm$0.05 & -0.23$\pm$0.11 & -5.69$\pm$0.08 & 1.05$\pm$0.11 & -6.86$\pm$0.10 & 2.60$\pm$0.08 \\
\enddata \tablenotetext{a}{The coordinate differences of the
components of 3C 454.3 from those of B2250+194 (3C~454.3 - 2250)
for each epoch for which the JPL model for the correction of the
ionospheric effects could be used: \Ra{00}{00}{50}{3787837} +
$\Delta$\mbox{$\alpha$}\ and \dec{-3}{33}{41}{067505} + $\Delta$\mbox{$\delta$}. The
coordinate differences are based on our differential measurements of
C1 relative to B2250+194 and on the determinations of C2, D1, D2, J1,
and Jext relative to C1 (Paper II). For B2250+194 the CRF coordinates
\Ra{22}{53}{7}{3691736} and \dec{19}{42}{34}{628786} (solution
\#3, Table~\ref{t2icrf}) were used. The standard errors are the
statistical standard errors with a constant added in
quadrature so that $\mbox{$\chi^2_\nu / \nu$} = 1.$}
\label{t32250}
\end{deluxetable}
\begin{deluxetable}{r@{}c@{}c c c c c c c c c c c c}
\rotate
\tabletypesize{\tiny}
\tablecaption{Separation of 3C 454.3 components from B2252+172}
\tablewidth{0pt}
\tablehead{
\colhead{Epoch} & \colhead{Julian} & \multicolumn{2}{c}{C1\tablenotemark{a}} & \multicolumn{2}{c}{C2\tablenotemark{a}} & \multicolumn{2}{c}{D1\tablenotemark{a}} & \multicolumn{2}{c}{D2\tablenotemark{a}} & \multicolumn{2}{c}{J1\tablenotemark{a}} & \multicolumn{2}{c}{Jext\tablenotemark{a}} \\
& \colhead{date} \\
& \colhead{2450000+} & \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}}
& \colhead{$\Delta$\mbox{$\alpha$}} & \colhead{$\Delta$\mbox{$\delta$}} \\
& & \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
& \colhead{(mas)} & \colhead{(mas)}
}
\startdata
2002 Nov 20 & 2599.6 & -0.11$\pm$0.05 & -0.06$\pm$0.04 & -0.68$\pm$0.06 & -0.15$\pm$0.05 & \nodata & \nodata & \nodata & \nodata & -5.70$\pm$0.05 & 0.87$\pm$0.04 & -6.78$\pm$0.07 & 2.32$\pm$0.07 \\
2003 Jan 26 & 2666.4 & -0.05$\pm$0.05 & 0.07$\pm$0.04 & -0.64$\pm$0.07 & -0.06$\pm$0.05 & -1.53$\pm$0.05 & -0.20$\pm$0.06 & \nodata & \nodata & -5.62$\pm$0.06 & 0.95$\pm$0.05 & -6.64$\pm$0.07 & 2.28$\pm$0.07 \\
2003 May 18 & 2778.1 & 0.04$\pm$0.05 & -0.02$\pm$0.04 & -0.65$\pm$0.03 & -0.09$\pm$0.04 & -1.53$\pm$0.07 & -0.32$\pm$0.05 & \nodata & \nodata & -5.65$\pm$0.01 & 0.96$\pm$0.05 & -6.61$\pm$0.06 & 2.23$\pm$0.06 \\
2003 Sep 08 & 2891.7 & 0.02$\pm$0.05 & 0.01$\pm$0.04 & -0.69$\pm$0.04 & -0.08$\pm$0.04 & -1.62$\pm$0.06 & -0.32$\pm$0.05 & \nodata & \nodata & -5.61$\pm$0.08 & 0.99$\pm$0.05 & -6.61$\pm$0.06 & 2.27$\pm$0.07 \\
2003 Dec 05 & 2979.5 & -0.02$\pm$0.05 & 0.00$\pm$0.04 & -0.74$\pm$0.03 & -0.10$\pm$0.05 & -1.71$\pm$0.07 & -0.37$\pm$0.05 & \nodata & \nodata & -5.60$\pm$0.04 & 0.93$\pm$0.05 & -6.64$\pm$0.06 & 2.27$\pm$0.07 \\
2004 Mar 06 & 3071.3 & -0.01$\pm$0.05 & -0.02$\pm$0.04 & -0.70$\pm$0.03 & -0.12$\pm$0.05 & -1.69$\pm$0.07 & -0.39$\pm$0.05 & -3.56$\pm$0.03 & -0.52$\pm$0.11 & -5.59$\pm$0.03 & 0.93$\pm$0.05 & -6.64$\pm$0.06 & 2.36$\pm$0.07 \\
2004 May 18 & 3144.1 & -0.06$\pm$0.05 & 0.06$\pm$0.05 & -0.74$\pm$0.05 & -0.08$\pm$0.05 & -1.73$\pm$0.06 & -0.32$\pm$0.05 & -3.62$\pm$0.03 & -0.43$\pm$0.10 & -5.61$\pm$0.07 & 1.03$\pm$0.07 & -6.76$\pm$0.06 & 2.50$\pm$0.07 \\
2004 Jun 26 & 3183.0 & -0.09$\pm$0.05 & -0.04$\pm$0.04 & -0.79$\pm$0.11 & -0.18$\pm$0.05 & -1.82$\pm$0.05 & -0.38$\pm$0.05 & -3.65$\pm$0.02 & -0.53$\pm$0.08 & -5.57$\pm$0.06 & 0.90$\pm$0.04 & -6.69$\pm$0.06 & 2.30$\pm$0.06 \\
2004 Dec 11 & 3351.5 & 0.02$\pm$0.05 & -0.04$\pm$0.04 & -0.66$\pm$0.04 & -0.11$\pm$0.04 & -1.65$\pm$0.08 & -0.49$\pm$0.05 & -3.69$\pm$0.02 & -0.45$\pm$0.08 & -5.61$\pm$0.05 & 0.89$\pm$0.05 & -6.76$\pm$0.06 & 2.40$\pm$0.06 \\
2005 Jan 15 & 3386.4 & 0.04$\pm$0.05 & -0.02$\pm$0.04 & -0.64$\pm$0.04 & -0.08$\pm$0.04 & -1.69$\pm$0.09 & -0.46$\pm$0.05 & -3.66$\pm$0.03 & -0.52$\pm$0.08 & -5.59$\pm$0.04 & 0.91$\pm$0.04 & -6.77$\pm$0.06 & 2.45$\pm$0.06 \\
2005 May 28 & 3519.1 & 0.00$\pm$0.05 & 0.03$\pm$0.04 & -0.70$\pm$0.03 & -0.01$\pm$0.04 & -1.83$\pm$0.08 & -0.41$\pm$0.05 & -3.72$\pm$0.03 & -0.24$\pm$0.08 & -5.64$\pm$0.03 & 1.01$\pm$0.04 & -6.76$\pm$0.06 & 2.51$\pm$0.06 \\
2005 Jul 16 & 3567.9 & 0.00$\pm$0.05 & 0.02$\pm$0.04 & -0.73$\pm$0.03 & -0.01$\pm$0.04 & -1.89$\pm$0.07 & -0.41$\pm$0.05 & -3.71$\pm$0.02 & -0.30$\pm$0.09 & -5.66$\pm$0.03 & 0.97$\pm$0.06 & -6.82$\pm$0.06 & 2.52$\pm$0.06 \\
\enddata
\tablenotetext{a} {As in Table~\ref{t32250} but now with B2252+172 as a reference.
The coordinate differences (3C454.3 $-$ 2252) are \Ra{-00}{01}{01}{8494807} + $\Delta$\mbox{$\alpha$}\ and
\dec{-1}{24}{31}{1293578} + $\Delta$\mbox{$\delta$}. For B2252+172 the CRF coordinates
\Ra{22}{54}{59}{5974430} and \dec{17}{33}{24}{690713} (solution \#6, Table~\ref{t2icrf})
were used. The standard errors are the statistical standard errors
with a constant added in quadrature so that $\mbox{$\chi^2_\nu / \nu$} = 1$.}
\label{t42252}
\end{deluxetable}
\subsection{Analysis of Geodetic Group-delay Observations of 3C 454.3 and B2250+194}
The data from two of our sources, 3C~454.3 and B2250+194, observed in
many of the geodetic VLBI sessions, each extending over about one day,
were analyzed by one of us (L. P.) with the VTD/post-Solve software package.
We made a weighted
least-squares solution using all available geodetic VLBI observations
of 3955 sources, including \objectname[]{B2250+194}, \objectname[]{3C
454.3}, and the 212 ``defining" sources from the ICRF catalogue
\citep{FeyGJ2009}, made at 157 stations from 1979 to 2008 (dubbed
solution gpb$\_$2008a). All in all we used a total of 6.5 million
determinations of group delay from observations made simultaneously at
8.4 and 2.3 GHz. This solution forms our CRF. It is consistent with the
ICRF2 \citep{FeyGJ2009}, which, however, does not provide information that we need as described below.
In particular, we estimated the coordinates of B2250+194 and 3C~454.3
for each observing session while forcing the coordinates of each other
source to be constant. We also solved for the positions and
velocities of all stations, for polar motion and UT1 parameters, and
their rates of change, for nutation daily offsets at the middle epoch
of each session and for numerous other parameters such as those that
model clock offsets, atmosphere path delays in the zenith direction,
and tilts of the assumed atmosphere axis of symmetry.
We imposed the constraint that the net rotation of our estimates of
source positions of the 212 defining sources with respect to the
positions of these sources in the ICRF catalogue be zero. Such a choice for the
constraint provides the continuity of our solution to other VLBI
solutions, including the ICRF2 solution. For more
details, see \citet{Petrov+2009}.
The individual source positions from this solution
were estimated not with respect to a particular reference source, but
with respect to the entire ensemble of observed sources. This approach
was possible because the geodetic VLBI sessions were designed in such
a way that (1) a set of $\sim$100 core sources and $\sim$20 core
antenna sites (``stations'') were common to all sessions, and (2) the
resulting estimates of source positions, station positions, and Earth
orientation parameters would not be strongly correlated. The presence
of sources common to all sessions tended to ensure the consistency of
determination of the interferometer orientation with respect to these
core sources.
Although our list of sources has 3955 objects, the relative weight of
the $\sim$100 predominant sources ensured that they dominate the ensemble. Therefore, we interpret
the estimated positions of \objectname[]{B2250+194} and \objectname[]{3C 454.3}
as positions almost entirely with respect to the ensemble of the $\sim$100 predominant extragalactic
objects.
For the solution yielding our CRF, each source was assumed to be a point
source. However, almost all of these sources exhibit some extended
structure at the milliarcsecond scale that may also vary over time.
Since these sources were not routinely imaged and hence the group
delays not corrected for structure effects, a position determination
from geodetic VLBI cannot be identified with respect to a specific
fiducial point in the brightness distribution of a source. That
fiducial point is therefore in principle unidentified for any source
with structure. In general, however, the more compact the source, the
closer the estimated position is to the peak in the brightness
distribution of the source.
The source B2250+194 is sufficiently compact that for our purposes the
position determined is effectively that of the brightness peak. For
3C~454.3 the fiducial point for each session is less well known
because of the complexity of the source structure and its changes with
time. Moreover, the fiducial point could change due to changes of the
\mbox{$u$-$v$}\ coverage. For these reasons, we do not rely on the 3C~454.3
position from geodetic VLBI observations, but rather use our dedicated
VLBI observations and phase-referenced data to identify individual
components in the brightness distribution of 3C~454.3 and to determine
their positions relative to our two more compact extragalactic
sources, B2250+194 and B2252+172, which in turn we tie to the CRF.
\section{Astrometric Results (2): Fit for the Position at Epoch and Proper Motion}
\label{ast2}
\subsection{Fit: Components of 3C 454.3 Relative to B2250+194 and B2252+172}
We determined the position at epoch and proper motion of each of C1,
C2, D1, D2, J1, and Jext relative to the brightness peaks of B2250+194
and B2252+172 with weighted least-squares fits for \mbox{$\alpha$}\ and separately
for \mbox{$\delta$}. We list our results together with the weighted linear correlation
coefficients and the weighted post-fit rms values (wrms) in
Table~\ref{t5comp} (solutions \#1 and \#2). We used the data
corrected with the JPL ionosphere model from Tables~\ref{t32250} and
\ref{t42252}.
The data and the corresponding lines from the fits (solutions \#1 and
\#2) are plotted in Figures~\ref{f9c15052}, \ref{f10c25052},
\ref{f11d15052}, \ref{f12d25052}, \ref{f13j15052}, and
\ref{f14j25052}. Not surprisingly, the smallest wrms values, in both
\mbox{$\alpha$}\ and \mbox{$\delta$}, were obtained for the components of 3C~454.3 relative to
the close reference B2252+172 (solution \#2).
Our combined proper-motion estimates are given as solution
\#3. They were obtained as a weighted average of proper-motion
estimates relative to B2250+194 and B2252+172 for the same (short)
time range.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{c1x5052tjplslope.eps}
\caption{The coordinates of C1 in 3C~454.3 relative to those of
B2250+194 and B2252+172 (except for offsets) as a function of
time. The values of $\Delta$\mbox{$\alpha$}\ and $\Delta$\mbox{$\delta$}\ are obtained from
the entries in Tables~\ref{t32250} and \ref{t42252}. For discussion of
apparent partial correlations between the two position-difference data sets,
see text.}
\label{f9c15052}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{c2x5052tjplslope.eps}
\caption{As in Figure~\ref{f9c15052}, but now for C2 in 3C 454.3.}
\label{f10c25052}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{d1x5052tjplslope.eps}
\caption{As in Figure~\ref{f9c15052}, but now for D1 in 3C 454.3.}
\label{f11d15052}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{d2x5052tjplslope.eps}
\caption{As in Figure~\ref{f9c15052}, but now for D2 in 3C 454.3.}
\label{f12d25052}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{j1x5052tjplslope.eps}
\caption{As in Figure~\ref{f9c15052}, but now for J1 in 3C 454.3.}
\label{f13j15052}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{j2x5052tjplslope.eps}
\caption{As in Figure~\ref{f9c15052}, but now for Jext in 3C 454.3.}
\label{f14j25052}
\end{figure}
\subsection{Fit: B2252+172 Relative to B2250+194}
The position at epoch and proper motion of B2252+172 relative to
B2250+194 were determined by first differencing the position
determinations of C1 relative to B2250+194 and C1 relative to
B2252+172 from Tables~\ref{t32250} and \ref{t42252} in the sense (C1
$-$ 2250) $-$ (C1 $-$ 2252)\footnote{Differencing the phase delays for
each scan at each epoch would have given us the position of B2252+172
relative to B2250+194 directly for each epoch and likely with a
somewhat smaller uncertainty. However, our procedure proved to also
give sufficiently accurate results for our purposes.}. We then used
weighted least-squares to fit a straight line to these differences. We
list the results also in Table~\ref{t5comp} (solution \#4) and plot
the data with the fit line in Figure~\ref{f85250}. The relative
proper motion of the two sources is zero within a small portion of
1$\sigma$. The 1$\sigma$ upper limits are 11 and 24 \mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\
and \mbox{$\delta$}, respectively.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{52x50tjplslope.eps}
\caption{The coordinates of B2252+172 relative to those of B2250+194
(except for an offset in each coordinate) as determined from the
differences of the position estimates, (2252-C1)$-$(2250-C1)
given in Tables~\ref{t32250} and \ref{t42252}.}
\label{f85250}
\end{figure}
\begin{deluxetable}{c@{ }c@{ }c@{ }c r r r c c c r r@{}c@{}c@{}c}
\rotate
\tabletypesize{\tiny}
\tablecaption{Relative positions at epoch and proper motions of the components of 3C 454.3, B2250+194, and B2252+172\tablenotemark{a}}
\tablewidth{0pt}
\tablehead{
\colhead{Component} & \multicolumn{3}{c}{\mbox{$\alpha$}} & \colhead{$\mu_\alpha$} & \colhead{$\rho_{\mbox{$\alpha$}}$} & \colhead{wrms$_{\mbox{$\alpha$}}$} & \multicolumn{3}{c}{\mbox{$\delta$}} & \colhead{$\mu_{\delta}$}
& \colhead{$\rho_{\mbox{$\delta$}}$} & \colhead{wrms$_{\mbox{$\delta$}}$} & \colhead{Range of epochs\tablenotemark{b}} & \colhead{Solution$\,$\#} \\
\colhead{-reference} & \colhead{(h)} &\colhead{(m)} &\colhead{(s)} & \colhead{(\mbox{$\mu$as~yr$^{-1}$})} & & \colhead{(\mbox{$\mu$as})} & \colhead{(\arcdeg)}
&\colhead{(\arcmin)} &\colhead{(\arcsec)} & \colhead{(\mbox{$\mu$as~yr$^{-1}$})} & & \colhead{(\mbox{$\mu$as})}
}
\startdata
C1 $-$ 2250 & & & 50.3787837 (19) &$30\pm\phn8$& 0.60 & 93 &$-$3& 33& 41.067505 (28) &$-27\pm\phn8$ &$-$0.56 &\phn85 & 1998.71 -- 2005.54 & 1\tablenotemark{c} \\
C2 $-$ 2250 & & & 50.3787347 (27) &$-9\pm11$ &$-$0.16 & 119 &$-$3& 33& 41.067591 (32) &$-33\pm\phn9$ &$-$0.58 &\phn97 & 1998.71 -- 2005.54 & 1\tablenotemark{c} \\
D1 $-$ 2250 & & & 50.3786586 (25) &$-126\pm32$ &$-$0.79 & 74 &$-$3& 33& 41.067913 (23) &$-89\pm23$ &$-$0.79 &\phn52 & 2003.07 -- 2005.54 & 1\tablenotemark{c} \\
D2 $-$ 2250 & & & 50.3784787 (14) &$-115\pm38$ &$-$0.80 & 41 &$-$3& 33& 41.067960 (42) & $196\pm87$ & 0.71 &\phn88 & 2004.18 -- 2005.54 & 1\tablenotemark{c} \\
J1 $-$ 2250 & & & 50.3783932 (19) &$-4\pm\phn8$&$-$0.10 & 80 &$-$3& 33& 41.066488 (33) & $49\pm\phn9$ & 0.73 &\phn96 & 1998.71 -- 2005.54 & 1\tablenotemark{c} \\
Jext $-$ 2250 &&& 50.3783141 (24) &$-51\pm10$ &$-$0.72 & 103 &$-$3& 33& 41.065042 (27) & $70\pm\phn8$ & 0.87 &\phn83 & 1998.71 -- 2005.54 & 1\tablenotemark{c} \\
\\
C1 $-$ 2252 & &$-$01& 01.8494857 (14)&$20\pm17$ & 0.36 & 45 &$-$1& 24& 31.129358 (16) & $4\pm14$ & 0.09 &\phn37 & 2002.89 -- 2005.54 & 2\tablenotemark{d} \\
C2 $-$ 2252 & &$-$01& 01.8495343 (10)&$ -8\pm14$ &$-$0.19 & 35 &$-$1& 24& 31.129424 (18) & $25\pm15$ & 0.46 &\phn42 & 2002.89 -- 2005.54 & 2\tablenotemark{d} \\
D1 $-$ 2252 & &$-$01& 01.8496108 (20)&$-128\pm26$ &$-$0.86 & 59 &$-$1& 24& 31.129789 (20) &$-81\pm20$ &$-0.80$ &\phn46 & 2003.07 -- 2005.54 & 2\tablenotemark{d} \\
D2 $-$ 2252 & &$-$01& 01.8497412 (08)&$-93\pm21$ &$-$0.89 & 23 &$-$1& 24& 31.129750 (35) &$186\pm72$ & 0.76 &\phn72 & 2004.18 -- 2005.54 & 2\tablenotemark{d} \\
J1 $-$ 2252 & &$-$01& 01.8498757 (19)&$ 15\pm10$ & 0.43 & 26 &$-$1& 24& 31.128410 (19) &$12\pm16$ & 0.24 &\phn43 & 2002.89 -- 2005.54 & 2\tablenotemark{d} \\
Jext $-$ 2252&&$-$01& 01.8499544 (16)&$-59\pm21$ &$-$0.67 & 55 &$-$1& 24& 31.126911 (25) &$96\pm22$ & 0.81 &\phn59 & 2002.89 -- 2005.54 & 2\tablenotemark{d} \\
\\
C1 $-$ 2250/2252 &&& & $20\pm13$ & & & & & & $7\pm12$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
C2 $-$ 2250/2252 &&& &$-10\pm11$ & & & & & & $28\pm14$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
D1 $-$ 2250/2252 &&& &$127\pm20$ & & & & & &$-84\pm15$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
D2 $-$ 2250/2252 &&& &$-98\pm18$ & & & & & &$190\pm56$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
J1 $-$ 2250/2252 &&& &$12\pm\phn8$& & & & & & $15\pm14$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
Jext $-$ 2250/2252 &&& &$-59\pm16$ & & & & & &$104\pm15$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{e} \\
\\
\\
2252 $-$ 2250&00 & 01 & 52.2282694 (6) &\phn-$2\pm 9$ &$-$0.06 & 22 & $-$2& 09 & 09.938073 (20) & $5\pm19$ & 0.08 &\phn48 & 2002.89 -- 2005.54 & 4\tablenotemark{f} \\
\enddata
\tablenotetext{a}{The coordinates and proper motions of the core and
jet components of 3C~454.3 for the reference epoch -- the midpoint of the \mbox{\em GP-B}\ mission -- 2005 Feb.\ 1
(2005.08), using VLBI differential observations of 3C~454.3 relative
to B2250+194 and B2252+172. The parameters are derived from weighted
least-squares linear fits. The uncertainties are standard errors
derived from the fit on the basis of the statistical standard errors
of the individual measurements added in quadrature to constants so
that $\mbox{$\chi^2_\nu / \nu$} = 1$. We searched for any sign that a possible
correlation between consecutive data points (see, e.g.,
Fig~\ref{f9c15052}) could render our uncertainties too small by
repeating the weighted least squares fit for even and odd numbered
data points separately. However, we did not find any such sign and
therefore think that any possible correlation would only have a minor
effect on our uncertainty estimates. Also listed are the weighted linear correlation
coefficients, $\rho$, from the fits, and the weighted
rms values, wrms, for the post-fit residuals, for the two
coordinates. All data are corrected for the effects of the ionosphere
by using a model implemented in the AIPS software and based on GPS
data (JPL, see text). The earliest date at which the model could be
used is 1998 Sept. 17 (1998.71). For D1 and D2 the time range starts
with the first epoch at which a component could be identified in the
brightness distribution, that is, 2003 Jan. 26 (2003.07) and 2004 Mar.
6 (2004.18), respectively.}
\tablenotetext{b}{The range of epochs from which the data were taken
for the solution. The range 1998.71 -- 2005.54 is the time range from
1998 Sep.\ 17 to 2005 July 16 of our \mbox{\em GP-B}\ VLBI observations for which
the ionospheric model, JPL, could be used. The epoch 2002.89 refers to
2002 Nov.\ 20 when we started to include B2252+172 in our
observations. The other time ranges refer to the total ranges for
which data were collected for the respective component and source. The reference
epoch 2005.09 is 2005 Feb.\ 1, the midpoint of the time period through
which data were taken on the \mbox{\em GP-B}\ spacecraft.}
\tablenotetext{c}{The differences from B2250+194 in the coordinates of
the positions and in the corresponding components of the proper motion
for each of 3C~454.3's six core and jet components. The time range
starts for all components with the first epoch of our VLBI
observations, except for components D1 and D2, where it starts with the
first epoch at which a component could be identified in the brightness
distribution. The epochs of the time range are 1998 Sept. 17
(1998.71), 2003 Jan. 26 (2003.07), and 2005 Jul. 16 (2005.54).}
\tablenotetext{d}{As in c for solution \#1 but now relative to
B2252+172. The time range starts with the date of the first B2252+172
observations, 2002 Nov.\ 20 (2002.89).}
\tablenotetext{e}{As for solution \#1 but now starting not earlier
than at epoch 2002.89, and including the values for B2252+172. In
particular, for each component, we took the weighted average of (a)
the proper motion from a solution (3C~454.3-2250) for the short time
range (not listed), and (b) the proper motion of solution \#2
(3C~454.3-2252).
The errors from (a) and (b) were added in quadrature.}
\tablenotetext{f}{The coordinates of B2252+172 relative to those of
B2250+194 and their changes with time. They were derived from the data
(2252 $-$ C1) $-$ (2250 $-$ C1) determined for each epoch.}
\label{t5comp}
\end{deluxetable}
\subsection{Fit: 3C 454.3, B2250+194, and B2252+172 in the CRF}
\subsubsection{3C 454.3}
In Figure~\ref{f53cicrf} we plot the position determinations of
3C~454.3 from geodetic VLBI observations spanning almost 30 years.
The statistical standard errors of the coordinates vary widely in part
because of the different lengths of time this source was observed in
the various sessions, and in part because the sensitivity of the VLBI
systems used for the observations generally improved over time.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{3ccrf52005.eps}
\caption{The coordinates of 3C~454.3 as determined from
routine geodetic VLBI observations of up to $\sim$4000 extragalactic sources
scattered over the sky.
Here
and hereafter for plots of coordinates versus time, the straight lines
give the weighted least-squares fit to the data points. The slopes of
the lines and their statistical standard errors are indicated. These
errors are standard errors adjusted with a constant added in
quadrature to the statistical standard errors for each coordinate so
that $\mbox{$\chi^2_\nu / \nu$}=1$ (see text).}
\label{f53cicrf}
\end{figure}
We solved for the position and proper motion of 3C~454.3
by using weighted least-squares fits for \mbox{$\alpha$}\ and \mbox{$\delta$}\ separately,
for the whole time range of observations from 1980 to 2008.
Since these weighted least-squares fits to the data gave $\mbox{$\chi^2_\nu / \nu$}$ larger than unity,
we again added a constant in quadrature to the statistical standard
errors, separately for \mbox{$\alpha$}\ and \mbox{$\delta$}, so as to obtain $\mbox{$\chi^2_\nu / \nu$}=1$ for
each coordinate. The standard errors for the individual data points
so determined, together with the fit lines, are plotted in
Figure~\ref{f53cicrf} and the results are listed in
Table~\ref{t2icrf} (solution \#1).
In our CRF, the proper-motion component in \mbox{$\delta$}\ is zero within
0.8$\sigma$. However, in \mbox{$\alpha$}\ it is $18\pm5$~\mbox{$\mu$as~yr$^{-1}$}, non-zero at a
3.6$\sigma$ significance level.
However, since this determination does not refer consistently to any particular
component in the brightness distribution of the source, its significance
is not clear.
\subsubsection{B2250+194}
The second of our sources observed with geodetic VLBI is B2250+194.
About 12 years of such observations yielded determinations of position
and proper motion of B2250+194 in the CRF. Via this tie and our phase-delay observations,
we determined the positions of the components of 3C~454.3, of B2252+172, and of IM
Peg in the CRF. For the source B2250+194, the standard errors of the position
estimates also vary widely from session to session because of the
different spans of time over which observations of this source were
spread in the various sessions. As for 3C~454.3, we plot the
coordinates for B2250+194 from the different observing sessions and
their standard errors, computed as described above, together with the
linear fit lines (solution \#2 in Table~\ref{t2icrf})
for the whole observing period, from 1997 to 2008, in
Figure~\ref{f650icrf}.
The proper motion is zero within 0.4$\sigma$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2250crf2005.eps}
\caption{The change in coordinates of B2250+194\@. Otherwise similar
to Figure~\ref{f53cicrf}.}
\label{f650icrf}
\end{figure}
\subsubsection{B2252+172}
Using the position at epoch and proper motion of B2250+194 in the CRF,
given in Table~\ref{t2icrf} (solution \#2), we also obtain the
equivalent values for B2252+172 in the CRF\@. In particular, we take
the values from the above solution \#2 and add them to the values from
our phase-reference observations from Table~\ref{t5comp} (solution
\#4) in the sense 2250 + (2252 $-$ 2250) and list them also in
Table~\ref{t2icrf} (solution \#3). The proper motion is zero within
0.1$\sigma$.
Owing to a recent VLBA sensitivity upgrade, it became possible now
to determine group delays of sources as weak as 10~mJy. Source B2252+172
was observed in two scans each 320~s long, with a 9-station VLBA network
at epoch 2011 August 14 at 8.4 GHz during a gap in the schedule of target sources in
a VLBI survey of bright
infrared galaxies \citep{Condon+2011}.
The source appeared slightly resolved with a correlated
flux density in a range of 11--13 mJy at a range of baseline projection
lengths 3--200 megawavelengths. Its group-delay coordinates from that
experiment are $22^{\rm h} 54^{\rm m} 59\fs{597\,449} \pm 0.000\,015, \enskip
+17\arcdeg 33\arcmin 24\farcs{690\,49} \pm 0.000\,42$. These coordinates
agree with those from Table~\ref{t5comp} (solution \#3) within the formers sixfold larger uncertainties. The
corresponding proper motion for the 6.6 years between our reference epoch (see Table~\ref{t5comp})
and the observing date in 2011 is $13\pm33~\mbox{$\mu$as}$ and $-34\pm64~\mbox{$\mu$as}$, which gives $1\sigma$ upper
limits about threefold larger than those in Table~\ref{t5comp}.
\begin{deluxetable}{c c c c r c c c c c r r c c c}
\rotate
\tabletypesize{\tiny}
\tablecaption{The positions at epoch and proper motions in the celestial reference frame (CRF)\tablenotemark{a}}
\tablewidth{0pt}
\tablehead{
\colhead{Component-reference} & \multicolumn{3}{c}{\mbox{$\alpha$}} & \colhead{$\mu_\alpha$} & \colhead{$\rho_{\mbox{$\alpha$}}$} & \colhead{wrms$_{\mbox{$\alpha$}}$} & \multicolumn{3}{c}{\mbox{$\delta$}} & \colhead{$\mu_{\delta}$}
& \colhead{$\rho_{\mbox{$\delta$}}$} & \colhead{wrms$_{\mbox{$\delta$}}$} & \colhead{Range of epochs\tablenotemark{b}} & \colhead{Solution \#} \\
& \colhead{(h)} &\colhead{(m)} &\colhead{(s)} & \colhead{(\mbox{$\mu$as~yr$^{-1}$})} & & \colhead{(\mbox{$\mu$as})} & \colhead{(\arcdeg)}
&\colhead{(\arcmin)} &\colhead{(\arcsec)} & \colhead{(\mbox{$\mu$as}~yr)} & & \colhead{(\mbox{$\mu$as})} & &
}
\startdata
3C 454.3 & 22& 53 & 57.7479664 (60) & $18\pm\phn5$& 0.11 & 742 & 16 & 08 & 53.560943 (80)&$\phn4\pm\phn5$& 0.03 &\phn918 & 1980.28 -- 2008.47 & 1 \\
B2250+194 & 22& 53 &\phn7.3691736 (24)& $3\pm10$ & 0.05 & 179 & 19 & 42 & 34.628786 (62) &$-7\pm20$ &$-0.06$&\phn345 & 1996.01 -- 2008.36 & 2 \\
B2252+172 & 22& 54 & 59.5974430 (25) & $1\pm13$ & & & 17 & 33 & 24.690713 (65) &$-2\pm28$ & & & 2002.89 -- 2005.54 & 3\tablenotemark{c} \\
\\
C1 (50) &22&53& 57.7479573 (31) &$33\pm13$ &\nodata &\nodata & 16 & 08 & 53.561281 (68) &$-35\pm21$ & \nodata & \nodata & 1998.71 -- 2005.54 & 4\tablenotemark{d} \\
C2 (50) &22&53& 57.7479083 (36) &$-7\pm15$ &\nodata &\nodata & 16 & 08 & 53.561195 (70) &$-40\pm22$ & \nodata & \nodata & 1998.71 -- 2005.54 & 4\tablenotemark{d} \\
D1 (50) &22&53& 57.7478322 (35) &$-123\pm34$ &\nodata &\nodata & 16 & 08 & 53.560873 (66) &$-96\pm30$ & \nodata & \nodata & 2003.07 -- 2005.54 & 4\tablenotemark{d} \\
D2 (50) &22&53& 57.7476523 (28) &$-112\pm39$ &\nodata &\nodata & 16 & 08 & 53.560826 (75) &$188\pm89$ & \nodata & \nodata & 2004.18 -- 2005.54 & 4\tablenotemark{d} \\
J1 (50) &22&53& 57.7475668 (31) &$-1\pm13$ &\nodata &\nodata & 16 & 08 & 53.562298 (70) & $42\pm22$ & \nodata & \nodata & 1998.71 -- 2005.54 & 4\tablenotemark{d} \\
Jext (50) &22&53& 57.7474877 (34) &$-49\pm14$ &\nodata &\nodata & 16 & 08 & 53.563744 (68) & $63\pm21$ & \nodata & \nodata & 1998.71 -- 2005.54 & 4\tablenotemark{d} \\
\\
C1 (50/52) &\nodata&\nodata&\nodata&$22\pm17$ &\nodata&\nodata&\nodata&\nodata& \nodata& $4\pm26$ &\nodata&\nodata& 2002.89 -- 2005.54 & 5\tablenotemark{e} \\
C2 (50/52) &\nodata&\nodata&\nodata&$-8\pm15$ &\nodata&\nodata&\nodata&\nodata& \nodata& $25\pm27$ &\nodata&\nodata& 2002.89 -- 2005.54 & 5\tablenotemark{e} \\
D1 (50/52) &\nodata&\nodata&\nodata&$-125\pm23$ &\nodata&\nodata&\nodata&\nodata& \nodata&$-92\pm27$ &\nodata&\nodata& 2003.07 -- 2005.54 & 5\tablenotemark{e} \\
D2 (50/52) &\nodata&\nodata&\nodata&$-99\pm25$ &\nodata&\nodata&\nodata&\nodata& \nodata&$185\pm59$ &\nodata&\nodata& 2004.18 -- 2005.54 & 5\tablenotemark{e} \\
J1 (50/52) &\nodata&\nodata&\nodata& $13\pm14$ &\nodata&\nodata&\nodata&\nodata& \nodata& $10\pm24$ &\nodata&\nodata& 2002.89 -- 2005.54 & 5\tablenotemark{e} \\
Jext (50/52) &\nodata&\nodata&\nodata&$-58\pm25$ &\nodata&\nodata&\nodata&\nodata& \nodata&$100\pm26$ &\nodata&\nodata& 2002.89 -- 2005.54 & 5\tablenotemark{e} \\
\enddata
\tablenotetext{a}{The source coordinates, $\alpha$ and $\delta$, of 3C
454.3, B2250+194, and B2252+172 at the reference epoch 2005.08, and
their changes with time, $\mu_\alpha$ and $\mu_\delta$ respectively,
in the CRF. The parameters are derived from weighted least-squares
linear fits. The uncertainties are standard errors derived from the
fit on the basis of the statistical standard errors of the
individual measurements added in quadrature to constants so that
$\mbox{$\chi^2_\nu / \nu$} = 1.$. Also listed are the weighted linear correlation coefficients, $\rho$,
from the fits, as well as the weighted rms
values, wrms, for the post fit residuals, for the two coordinates.}
\tablenotetext{b}{The range of epochs as in Table~\ref{t2icrf}.}
\tablenotetext{c} {The coordinates of B2252+172 and their changes with
time derived by adding the solutions from \#2 to those of \#4 in Table~\ref{t5comp}. The
errors were added in quadrature.}
\tablenotetext{d}{The coordinates and proper motions of the components of 3C 454.3 in the CRF.
We added the values from
solution \#1 of Table~\ref{t5comp} to the values from solution \#2 of this table in the sense
(3C~454.3 comp. $-$ 2250) +2250, and added the errors in quadrature.}
\tablenotetext{e}{As for solutions \#4 but now only for the proper
motion, and at epochs not earlier than 2002.89, and with the inclusion
of the values for B2252+172. In particular, we added the values from
solution \#3 of Table~\ref{t5comp} to the values from solution \#2 of
this table in the sense (3C~454.3 comp. $-$ 2250/2252) + 2250 and
added the errors in quadrature.}
\label{t2icrf}
\end{deluxetable}
\subsection{Fit: Components of 3C 454.3 in the CRF}
We determined the position at epoch and proper motion of each of C1,
C2, D1, D2, J1, and Jext in the CRF by combining the values relative
to B2250+194 and B2252+172 with the values of the latter sources in
the CRF. In particular for the positions, we take the values from
Table~\ref{t5comp} (solution \#1) and add them to the values from
Table~\ref{t2icrf} (solution \#2) in the sense (3C~454.3 components
$-$ 2250) + 2250. For the proper motion values we include the
data involving B2252+172 since they are independent of those involving
B2250+194. We thus take all data into account, but do so only for
the restricted range of epochs for which we have B2252+172 VLBI
observations. In particular, we took the average solution \#3 from
Table~\ref{t5comp} and added to it the values of B2250+194 in the CRF
by using the solution \#2 from Table~\ref{t2icrf} in
the sense (3C~454.3 components $-$ 2250/2252) + 2250. We list the resulting
position-at-epoch and proper-motion values for each component in
Table~\ref{t2icrf} (solution \#4).
\subsubsection{Position of the Core Component, C1, of 3C 454.3}
We now discuss position determinations obtained
via our two different astrometric
techniques: the position of 3C~454.3 in the CRF obtained from geodetic
group-delay observations (solution \#1, Table~\ref{t2icrf}) and the
position of C1 in the CRF obtained through a combination of geodetic
group-delay and our phase-delay observations (solution \#4,
Table~\ref{t2icrf}).
We emphasize that these two estimates are not expected to coincide, even
in principle, because the former relates to some (ill defined) average
over source structure and the latter to a far better defined component (C1)
within that structure.
The difference between the two position determinations (pure group
delay minus combination) is $9.1 \pm6.8$~\mbox{$\mu$s}\
in \mbox{$\alpha$}\ and $-338 \pm 105$~\mbox{$\mu$as}\ in \mbox{$\delta$}. While the difference in \mbox{$\alpha$}\
is only 1.3$\sigma$, the difference in \mbox{$\delta$}\
is 3.2$\sigma$, large enough to perhaps be significant.
In this
context we compare our pure group-delay position
estimate for 3C~454.3 with other such estimates. For instance,
the ICRF2 catalog \citep{FeyGJ2009} provides position estimates of 3C 454.3
and B2250+194\footnote{This source is incorrectly referred to as B2250+190 in the
ICRF2 catalog.} which are close to ours. It is based on essentially the same set of
group delays, the same data editing, and the same software as we used but with a slightly
different reduction and estimation model.
The differences (ours $-$ ICRF2) are:
$-16\pm62$ \mbox{$\mu$s}\ and $-0.2\pm3.5$ \mbox{$\mu$s}\ in \mbox{$\alpha$}\ and $-4\pm970$ \mbox{$\mu$as}\ and $39\pm62$ \mbox{$\mu$as}\ in \mbox{$\delta$}, for 3C~454.3
and B2250+194, respectively. Here we take our estimate of an apparent proper motion in \mbox{$\alpha$}\ of $18\pm5$ \mbox{$\mu$as~yr$^{-1}$}\
into account and propagate our estimate back 11.13 years to their mean epoch of 1993.95. Our estimate
of the proper motion in \mbox{$\delta$}\ and the corresponding estimates for B2250+194 were small enough so that it
was not necessary to consider them. The uncertainties are those from the ICRF2 listings only.
Further, the latest estimate from the USNO celestial reference frame solution,
crf2009b\footnote{http://rorf.usno.navy.mil/vlbi/}, which uses essentially the same data
set as we used only extended by another year, is
different from ours in the sense (ours $-$ USNO) by $-1.7\pm6.1$~\mbox{$\mu$s}\ and $-1.4\pm1.9$~\mbox{$\mu$s}\
in \mbox{$\alpha$}\ and $-21\pm17$ \mbox{$\mu$as}\ and $52\pm31$ \mbox{$\mu$as}\ in \mbox{$\delta$}, for 3C~454.3 and B2250+194, respectively.
Here again we take our estimate of an apparent proper motion in \mbox{$\alpha$}\ for 3C~454.3 into account and
propagate our estimate back 17.79 years to
their mean epoch of 1987.29. The uncertainty is our proper motion error added in quadrature with
the USNO position error.
For the other differences, the proper motion estimates did not need to be taken into account.
The uncertainties are those
from the USNO position estimates only.
Another
estimate of the position of 3C~454.3, but not of B2250+194, was made
recently with group-delay observations at 24 GHz
\citep{Lanyi+2010}.
The corresponding differences are $-15.6\pm8$
\mbox{$\mu$s}\ and $-177\pm176$ \mbox{$\mu$as}. With our, statistically independent,
errors added in quadrature, the difference in \mbox{$\alpha$}\ reduces to
$1.6\sigma$ and in \mbox{$\delta$}\ to $0.9\sigma$. We do not consider these
differences to be significant. Nevertheless, note that the difference in
\mbox{$\delta$}\ is in the direction to reduce the difference with our determination
for C1.
Since we compare here results from different catalogs, we point out that they are almost
identical in their overall orientations. The differences in these orientations
correspond to a level of only several tens of microarcseconds, which is negligible for our purposes.
To summarize:
First there is a $-338$ \mbox{$\mu$as}\ ($3.2\sigma$) difference in \mbox{$\delta$}\ between our pure
group-delay position of 3C 454.3 and the combined
group-delay phase-delay position of C1. Second, our pure group-delay position
determinations for 3C 454.3 and, for comparison, also for B2250+194 agree within
$<1.7\sigma$ with the USNO and ICRF2 position determinations.
Third, the position of 3C~454.3 at 24 GHz in \mbox{$\delta$}, while its error is
large, cuts the above 3C~454.3/C1 discrepancy in half. We discuss
these results below in \S~\ref{discuss}.
\subsubsection{Limit on the Proper Motion of the Core Component C1 of 3C 454.3}
\label{limit}
Table~\ref{t5comp} (solution \#1) shows that from 1998 to 2005 the
proper motion of C1 relative to B2250+194 is in the southeast
direction at a significance level of 3.8$\sigma$ and 3.4$\sigma$ in \mbox{$\alpha$}\ and
\mbox{$\delta$}, respectively. In contrast, from 2002 to 2005 the motion of C1
relative to either B2250+194 (solution not listed) or B2252+172
(solution \#2) is smaller, and with an uncertainty corresponding to $<1.2~\sigma$, not
significant, although the errors are larger. In any case, it appears
that either \objectname[]{B2250+194} moved to the northwest or C1
indeed moved to the southeast, particularly during the time from 1998
to 2002.
The cause of this apparent motion is not clear. Is it possible that
the brightness peak of B2250+194 belongs to a jet that moved
northwestward during the early time period? Figure~\ref{f32250} shows
that \objectname[]{B2250+194} is elongated to the northwest and also
slightly to the south, with the brightness peak located near the
center of its curved structure. If there were a supermassive black
hole located at the southern end of the structure, then the brightness
peak would likely belong to a jet component moving away from the black
hole, in this case to the northwest. The geodetic observations which
determined the proper motion of B2250+194 in the CRF show an
insignificant northward motion of $21\pm46$ \mbox{$\mu$as~yr$^{-1}$}\ for the time from
1998 to 2005 (solution not listed). The observations over the longer
period from 1996 to 2008 are collectively more sensitive, but do not
indicate any motion to the northwest. Moreover, the geodetic
observations of 3C~454.3 itself are too insensitive for a useful
proper-motion determination for the period from 1998 to 2005.
A useful way to test this jet hypothesis would be to determine the
position of the brightness peak of \objectname[]{B2250+194} at other
frequencies \citep[see also,][]{Kovalev+2008}. The location of the
peak at the highest radio frequencies is expected to be close to the
source's core and the putative black hole there. For the similarly
compact and elongated source, M81$^*$, in the center of the nearby
galaxy M81, this method did lead to the confirmation of the
approximate location of the core of M81$^*$. This location was
earlier determined as the most stationary in the varying brightness
distribution of M81$^*$ relative to another source (the shell center
of SN 1993J) in the same galaxy (\citealt{BietenholzBR2001}; see also
\citealt{BietenholzBR2004}). However, our observations at 5 and 15 GHz
were not planned for high-precision astrometry and could not be used
for this purpose. Our 5 and 15 GHz images appear to be, respectively,
just larger and smaller versions of our 8.4 GHz images with the
brightness peak remaining in the center of the image (Paper II),
without giving any hint as to whether the core may be located in the
south of the structure.
As reported in Paper II, \objectname[]{B2250+194} undergoes slight
structure changes. When the source is modeled with an elliptical
Gaussian, the major axis of the Gaussian varies between $\sim$0.6 and
$\sim$0.8 mas along a position angle of $-11$\arcdeg\ over the time from
1997 to 2005. These structural changes could at least partly account
for the nominal proper motion of C1 relative to
\objectname[]{B2250+194}.
What is the $1\sigma$ upper limit on the proper motion of C1? From
the point of view of our IM Peg VLBI observations from 1997 to 2005
for which we use C1 as a reference, the degree of stationarity for
that period is relevant. For our longest period for the data corrected
with JPL, from 1998 to 2005, we obtain for C1 a 1$\sigma$ proper
motion limit relative to B2250+194 of 38 and 35 \mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and
\mbox{$\delta$}, respectively (solution \#1, Table~\ref{t5comp}).
The equivalent upper limit of the proper motion in the CRF over that
period is 46 and 56~\mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and \mbox{$\delta$}, respectively (solution
\#4, Table~\ref{t2icrf}). For \mbox{\em GP-B}\ (see \S~\ref{sintro}), this is
our fundamental result on
the level of stationarity of the chosen single reference point, C1,
relative to the distant universe.
For the shorter period, from 2002 to 2005, no significant proper
motion for C1 was found above the 1.5$\sigma$\ level, neither
relative to B2250 +194, nor to B2252+172, nor within the CRF. For
this period we obtain smaller limits that are interesting from an
astrophysical point of view. C1 is stationary relative to the
combination of the two reference sources within a $1\sigma$ upper
limit of 33 and 19 \mbox{$\mu$as~yr$^{-1}$}\ (solution \#3 in Table~\ref{t2icrf}) and
in the CRF within an upper limit of 39 and 30~\mbox{$\mu$as~yr$^{-1}$}, in \mbox{$\alpha$}\ and
\mbox{$\delta$}, respectively (solution \#5, Table~\ref{t2icrf}).
\subsubsection{Motion of the Jet Components of 3C 454.3}
\label{jetmotion}
What are the proper motions of the components other than C1 relative
to the two reference sources and in the CRF? Are they significantly
different from that of C1 and, moreover, can significant motion be
detected of any of them relative to the distant universe? In Paper II
we showed that C2, D1, D2, J1, and Jext are all moving on average away
from C1.
Their relative speeds are not necessarily constant over the more than 7 years
of our observations. Most dramatically, J1 moves at about twice its
average speed until 2002 and then slows down to almost
zero velocity thereafter (Paper II).
In Table~\ref{t5comp} we list the proper motions of these components
relative to each of the two reference sources and to both combined,
and in Table~\ref{t2icrf} to the CRF.
For C2 we find motion with a larger significance than that for C1 only
relative to B2252+172 (solution \#2) and to both reference sources
combined (2.0$\sigma$ in \mbox{$\delta$}, solution \#3). However, we do not
regard the significance large enough to consider the motion real.
Relative to B2250+194 and to the CRF, C2 is stationary at about the
same significance level as C1. For J1 a clear average northward motion
of 49$\pm9$ \mbox{$\mu$as~yr$^{-1}$}\ (5.4$\sigma$) relative to B2250+194 is observed
between 1998 and 2005 (solution \#1). The significance, however,
decreases to only $1.9\sigma$, due to larger uncertainties, when the
motion is measured in the CRF (solution \#4), comparable to the
significance of the motions in the CRF of each, C1 and C2. And the motion itself
decreases to (almost) zero within the errors for the period from 2002 to
2005 relative to either B2250+194 (solution not listed) or B2252+172
(solution \#2) or both (solution \#3) or to the CRF (solution \#5)
consistent with J1's motion
relative to C1. The largest and/or most significant motions, relative
to either of the two reference sources or both combined or in the CRF
were found for D1, D2, and Jext, with, for instance, D2 having a speed
in the CRF of $-99\pm25$ and $185\pm59$ \mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and \mbox{$\delta$},
respectively, for the short period from 2004 to 2005 (solution \#5).
To summarize: For the long period from 1998 to 2005, C1 and C2 are
stationary within the same small bounds relative to B2250+194 and are
joined by J1 when the stationarity is measured (less accurately) in the CRF. The other
components are stationary only within larger bounds or move
significantly. For the short period from 2002 to 2005, C1 and J1 are
stationary within the same small bounds relative to B2252+172 and to
both reference sources combined, and are joined by C2 when the
stationarity is measured in the CRF. Again, the other components are
stationary only within larger bounds or show significant proper
motion. The largest significant proper motion was found for D2.
\section{Astrometric Results (3): Analysis of Other Motions}
\label{ast3}
\subsection{Fit: Parallax, proper acceleration, and orbital motion of the core component, C1, of 3C 454.3}
Since our computation of the motion of the guide star
\objectname[]{IM Peg} includes solutions for parallax and proper
acceleration, to help us put limits on certain systematic errors we
also solve for these parameters for C1. In Table~\ref{t6res2} we list
the proper motion and the parallax, $\pi$, obtained for C1 relative to
\objectname[]{B2250+194} and separately to \objectname[]{B2252+172} by
assuming that both of the reference sources are infinitely distant
from Earth.
In addition, we solve for the acceleration components, \mbox{$\dot\mu_{\alpha}$}\ and
\mbox{$\dot\mu_{\delta}$}, for C1 and also list these results in Table~\ref{t6res2}.
\begin{deluxetable}{l c c c c c}
\tabletypesize{\small}
\tablecaption{Proper motion, parallax, and proper acceleration\tablenotemark{a}}
\tablewidth{0pt}
\tablehead{
\colhead{Source-reference}
& \colhead{\mbox{$\mu_{\alpha}$}} & \colhead{\mbox{$\mu_{\delta}$}} & \colhead{$\pi$}
& \colhead{\mbox{$\dot\mu_{\alpha}$}} & \colhead{\mbox{$\dot\mu_{\delta}$}} \\
& \colhead{(\mbox{$\mu$as~yr$^{-1}$})} & \colhead{(\mbox{$\mu$as~yr$^{-1}$})} & \colhead{(\mbox{$\mu$as})}
& \colhead{(\mbox{$\mu$as~yr$^{-2}$})} & \colhead{(\mbox{$\mu$as~yr$^{-2}$})}
}
\startdata
C1$-$2250 & 29$\pm$\phn8 &$-27\pm\phn8$ &$\phn40\pm20$ & \nodata & \nodata \\
C1$-$2252 & 21$\pm$18 &\phn\phn$4\pm14$ & $-2\pm17$ & \nodata & \nodata \\
\\
C1$-$2250 & 33$\pm$26 &\phn\phn$11\pm27$ &$\phn38\pm20$ & \phn\phn2 $\pm$ 9 & 13$\pm$ 9 \\
C1$-$2252 & \phn3$\pm$44 &\phn\phn$13\pm37$ & $-2\pm18$ & $-21\pm47$ & 11$\pm$ 40 \\
\enddata
\tablenotetext{a}{Parameter estimates for component C1
of 3C~454.3 relative to \objectname[]{B2250+194} and, separately,
to \objectname[]{B2252+172}.
Uncertainties are statistical standard errors derived from the weighted
least-squares fit, scaled to $\mbox{$\chi^2_\nu / \nu$} = 1$. The reference epoch is 2005
February 1.}
\label{t6res2}
\end{deluxetable}
Our two most accurate solutions for parallax (relative to B2252+172),
unsurprisingly, are zero (to within $0.1\sigma$), with the upper limit being
$\sim$20~\mbox{$\mu$as}, corresponding to a distance, $D$, of $>50$~kpc. Although
other data sets might well yield more accurate parallax measurements,
or bounds thereon, ours is one of the most, if not the most, accurate so far obtained,
given the assumption that the reference sources are sufficiently
distant to have negligible parallax. The
proper acceleration of C1 relative to B2250+194 is within 1.4$\sigma$
of zero, and is not significant.
We also extended the number of free parameters still further and included a
fit to orbital parameters corresponding to the period of the IM~Peg
binary system of 24.6~d, since such a fit is used in our analysis
of IM Peg (Paper V)\@. We found no indication of such orbital motion
of C1, with each of the orbital parameters being zero within 1$\sigma$.
\subsection{Non-linear motion of the core component, C1, of 3C 454.3 on the sky?}
\label{nonlinearmotion}
Is there significant motion of the core component, C1, of 3C~454.3 on
the sky that departs from the (linear) proper motion inherent in our
fit models? Inspecting Figure~\ref{f9c15052}, we see that the motion of
C1 relative to B2250+194 is somewhat correlated with C1's motion
relative to B2252+172 in both \mbox{$\alpha$}\ and \mbox{$\delta$}.
To investigate possible non-linear, or in general any unmodeled,
motion in more detail we first plot the coordinates of B2252+172
relative to those of B2250+194 in Figure~\ref{f1550x52ijpl}. We are
\begin{figure}
\centering
\includegraphics[width=\textwidth]{52x50ijploffset.eps}
\caption{The coordinate determinations, except for an offset, of
B2252+172 relative to those of B2250+194 obtained by differencing the
values in Tables~\ref{t32250} and \ref{t42252} for each epoch in the
sense (C1$-$2250)$-$(C1$-$2252). Errors are left unplotted
for clarity but can be computed from the errors given in the above
tables. The data point for the first epoch, 2002 Nov.\ 20, is
indicated by a circle around the black dot. The cross in the lower
left indicates the weighted rms of the scatter of the data points in the figure, of
23~\mbox{$\mu$as}\ in \mbox{$\alpha$}\ and 51~\mbox{$\mu$as}\ in \mbox{$\delta$}.}
\label{f1550x52ijpl}
\end{figure}
using only data corrected for ionospheric effects with the JPL model
since this model seems superior to the PIM model and since we are only
using data taken during the period of our B2252+172 observations for
which corrections with the JPL model were available, i.e. from 2002 to
2005. The plot shows quasi-random motion with wrms values of 23~\mbox{$\mu$as}\
in \mbox{$\alpha$}\ and 51~\mbox{$\mu$as}\ in \mbox{$\delta$}. In Figure~\ref{f16c1d1j2x5052} (upper
left panel), we plot the positions of C1 relative to those of B2250+194
\begin{figure}
\centering
\includegraphics[width=\textwidth]{c1d1j2ijpl6panelland.eps}
\caption{The coordinate determinations of C1, D1, and Jext, except for
an offset for each, relative to both B2250+194 and B2252+172 for each epoch
(upper row). The values were taken from Tables~\ref{t32250} and
\ref{t42252}. The lower row gives the coordinate determinations as
averages in the sense, (C1$-$2250)/2 + (C1$-$2252)/2. The data
points for the first epoch, 2002 Nov.\ 20, are indicated in the upper panels by a circle
around (i) the black dot for the motions relative to B2250+194
and (ii) the open circle for the motions
relative to B2252+172; in each of the lower panels the corresponding circle is again around a black dot. The crosses in
the lower left of each upper panel are taken from
Fig.~\ref{f1550x52ijpl} and approximately indicate the standard errors
of the data points. For the lower panels, the standard errors are
assumed to be smaller by $\sqrt[]{2}$.}
\label{f16c1d1j2x5052}
\end{figure}
and B2252+172. It is apparent that the positions of C1 are covering a
larger area than the positions of B2252+172 in the previous figure,
mainly because of a larger scatter along \mbox{$\alpha$}. The peak-to-peak
variations are 2.5 times larger in \mbox{$\alpha$}\ but only 1.2 times larger in
\mbox{$\delta$}\ than in the preceding figures.
Also, the positions of C1 relative to those of B2250+194 resemble the
positions of C1 relative to those of B2252+172. These two indications
may be evidence for C1 apparently moving on the sky along the east-west
axis above the noise level which we adopt to be the wrms values from
the B2252+172 versus B2250+194 plot and indicate by the cross in the
lower left corner of the figure. In the lower left panel we plot the
mean of each pair of position determinations (with $\sqrt[]{2}$
smaller cross bars) to display this apparent motion more clearly. The
plotted motion is confined to an area not larger than $\sim$0.2
$\times$0.2 mas$^{2}$, a small portion of the beam area but larger
than the area of ``jittery'' motion in Figure~\ref{f1550x52ijpl}. Any
east-west jittery motion of C1 is consistent with our simulations (Paper
II) of how C1 moves east-west relative to the larger
core region structure and relative to the 43-GHz core located $0.18\pm0.06$~mas
east of C1. This east-west motion simulated for C1 is confined to within
0.12 mas. Our finding of possible motion within 0.2 mas is less
precise than the result from Paper II, but in principle more accurate
since it is measured relative to physically unrelated sources nearby
on the sky.
For comparison we show the equivalent motions of the components D1 and
Jext in the middle and right columns of Figure~\ref{f16c1d1j2x5052}. As
expected from our solution for the proper motion, D1's motion has a
linear component to the west-southwest and Jext's toward the
northwest.
\section{Discussion}
\label{discuss}
\subsection{Considerations for geodetic VLBI with group delays}
By comparing the positions of C1 and 3C~454.3, both in the CRF, we
found a possibly significant difference of $-338 \pm 105$~\mbox{$\mu$as}\
(3.2$\sigma$) in \mbox{$\delta$}\ while the difference in \mbox{$\alpha$}\ is not
significant. To repeat, the position of C1 in the CRF was determined from the
position of B2250+194 in the CRF by the addition of the position of
C1 relative to that of B2250+194. Both the positions of 3C 454.3 and
B2250+194 in the CRF were determined from geodetic VLBI group delays
while the relative position (C1$-$2250) was determined from our
differential VLBI phase delays. In contrast to the latter, the
group-delay observations were not corrected for structure
effects. While such effects can be assumed to be insignificant for
B2250+194, they may well be significant for 3C~454.3. Therefore, in
contrast to that for B2250+194, the position solution for 3C~454.3
does not refer to a particular reference point in the source's
brightness distribution. The solution should therefore be influenced
by the structure of the source and its changes with time and
frequency. In particular, a shift of the position of the core with decreasing frequency
away from the foot of the jet but along the jet axis \citep[see, e.g.,][]{Sokolovsky+2011} could influence the results from group
delays and phase delays differently.
But the source is largely oriented east-west in its
brightest part, namely the core region, and oriented toward the
northwest in its low-luminosity 10-mas long jet. On first sight one might
therefore expect a larger discrepancy in \mbox{$\alpha$}\ \citep[see,
also,][]{Porcas2009} and a smaller one, if any, in \mbox{$\delta$}\ and then with
a shift to the north, not as observed to the south. If the 10-mas jet caused
the discrepancy, it would be via
some peculiar influence. The group-delay
position determined in 2007 at 24 GHz \citep{Lanyi+2010}
indeed diminishes the \mbox{$\delta$}\ discrepancy in the
position of C1 by about half although the uncertainty of the 24-GHz position
is large. In these observations, 3C~454.3 appears more compact than it does at 8.4
GHz. The most easterly component at 24 GHz appears to be point-like
and dominates the image. Low-brightness extended features appear up to
3 mas toward the west but the 10-mas jet is not visible
\citep{Charlot+2010} and should therefore have essentially no
influence on the position determination. In this context,
\citet{Marti-Vidal+2008} have also found apparently significant
discrepancies between source positions determined from geodetic VLBI
group delays without structure corrections and those determined from
differential VLBI phase-delays with structure corrections. All said,
the cause of the discrepancy is not understood.
\subsection{Astrophysical implications}
\subsubsection{Limit on the proper motion of the core of 3C 454.3 and the
proper motions of the jet components}
Since the component C1 is as close to stationary in the CRF as is any
of our other five components of 3C~454.3., and could be closely
related to the easternmost compact flat-spectrum component in 43-GHz
images (Paper II), this component is likely the closest in our images
to the expected supermassive black hole, and therefore very near the
center of mass of the quasar. The radio emission probably originates
close to the foot of the jet in the vicinity of this putative black
hole. The jet components, C2, D1, D2, J1, and Jext are all moving
away from C1 (Paper II). The motions of D1, D2, and Jext are also
significant in the CRF at our sensitivity levels. The proper-motion
values of all six components of 3C~454.3 from Table~\ref{t5comp}
(solution \#5) were converted to apparent velocities and are listed
in Table~\ref{t7vel}. Our $1\sigma$ upper limit on the proper motion of C1 in the CRF
of 39 and 30~\mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and \mbox{$\delta$}, respectively, corresponds to a
limit of 1.0 and 0.8 $c$. The speeds of the jet components in the CRF
can be faster and superluminal; for instance for D2 the speed is
$\sim5 c$.
\begin{deluxetable}{c@{\hspace{0.3in}} c@{\hspace{0.3in}} c@{\hspace{0.6in}} c}
\tabletypesize{\small}
\tablecaption{Velocity of the components of 3C 454.3 in the CRF\tablenotemark{a}}
\tablewidth{0pt}
\tablehead{
\colhead{Component} & \colhead{$v_{\alpha}/c$} &
\colhead{$v_{\delta}/c$} & \colhead{Range of epochs\tablenotemark{b}}
}
\startdata
C1 & $\phn0.6\pm0.4$ & $0.1\pm0.7$ & 2002.89 -- 2005.54 \\
C2 & $ -0.2\pm0.4$ & $0.6\pm0.7$ & 2002.89 -- 2005.54 \\
D1 & $ -3.2\pm0.6$ & $2.3\pm0.7$& 2003.07 -- 2005.54 \\
D2 & $\phn2.5\pm0.6$ & $4.7\pm1.5$& 2004.18 -- 2005.54 \\
J1 & $\phn0.3\pm0.4$ & $0.3\pm0.6$& 2002.89 -- 2005.54 \\
Jext & $ -1.5\pm0.6$ & $2.5\pm0.7$& 2002.89 -- 2005.54 \\
\enddata
\tablenotetext{a}{Velocity in $\alpha$ and $\delta$ in units of the
speed of light for angular velocities in Table~\ref{t2icrf} (solution
\#5). The velocities were computed for an angular diameter distance of
3C~454.3 of 1.6 Gpc.}
\tablenotetext{b}{Same as in Table~\ref{t5comp}.}
\label{t7vel}
\end{deluxetable}
Similar characteristics are displayed by the superluminal quasar,
3C~345. The core was found to be stationary relative to the
physically unrelated quasar, NRAO 512, within $0.4 c$ in the east-west
direction while the jet components moved away from the core in this
direction at up to $9 c$ \citep{Bartel+1986}\footnote{We assume the
same cosmological parameters as we use here for 3C~454.3.}.
Since in each of these two cases the core is compact and has an
approximately flat spectrum in the GHz frequency range, we conclude
that compactness of the component and flatness of the spectrum are
indeed, as generally assumed, indicative of the nearby presence of the
gravitational center of the quasar.
Component J1 is special partly because the limit on
its speed over the short time interval from 2002 to 2005 is subluminal and partly
because its speed was likely much larger at earlier times.
\citet{Pauliny-Toth1998} found from VLBI observations at 11 GHz that a
component, dubbed ``A,'' moved away from the core toward the northwest
from a distance of $\sim$2.8 mas in 1984 to a distance of $\sim$5.0
mas in 1992 with a speed averaging $25 c$, but varying from $15 c$ between
1984 and 1985 to
$30 c$ between 1985 and 1989 and $20 c$ between 1989 and 1992.
This component is likely to be our J1. \citet{Jorstad+2005} found
that by 2001 this component, dubbed ``D'' in their paper, has possibly
decelerated considerably to $6 c$.
The strong deceleration can also be seen in our data from 1998 onward
such that by 2002 to 2005 the component remained stationary within our
subluminal limits (see Table~\ref{t7vel}).
Larger speeds of up to $530\pm50$~\mbox{$\mu$as~yr$^{-1}$}\ were reported for jet
components that could be discerned with higher angular resolution at
43 GHz in the C1 and C2 area \citep{Jorstad+2001b}. These authors
also speculated on whether perhaps the 43-GHz core could have moved to
the northeast by 0.1 mas in \mbox{$\alpha$}\ and 0.2 mas in \mbox{$\delta$}\ between 1997.6
and 1998.2. While this time range is just before the start of our VLBI
observations, our proper-motion measurements limit any such motion for
later times.
Recently an exceptionally bright optical outburst was detected in
3C~454.3 \citep{Villata+2006, Vercellone+2010} reaching a maximum in spring
2005. It was accompanied by an increase of radio emission at 43 GHz
from the core that started in early 2005 and reached a maximum in
September 2005. If the outburst is associated with activity of the
core, perhaps with the ejection of a new jet component, then the
position of C1 may be expected to be affected.
However, inspection of our graphs of the temporal changes
in the position of C1 does not show any indication of a possible
emerging jet component near C1, a result not surprising
given the 2005 mid-July end of our data set.
To repeat, any transverse linear motion on the sky found for the core
of 3C~454.3 has a speed $\leq 1.0 c$. This limit is almost as low in magnitude
as the radial velocity of the quasar, computed from its
redshift. Transverse velocities comparable in magnitude to the
redshift velocities are not expected for quasars for a cosmological
model where the dominant motion is the redshift velocity due to the
expansion of the universe.
Any transverse velocity for the cores of quasars should be at least
two orders of magnitude smaller than $c$ based on our knowledge of
peculiar motions of galaxies and galaxy clusters.
The effect of acceleration of the solar-system barycenter toward the
galactic center is also expected to be relatively small.
However, it is plausible that a non-zero mean proper motion with
respect to the CRF of physically unrelated sources that are separated
by less than, say, a radian on the sky is significant because, in
effect, we construct the CRF using the approximation that the position
of the currently estimated solar-system barycenter is an inertial
reference. Fortunately, the estimated acceleration of that barycenter
toward the Galactic center has a relatively small effect: At its
maximum value (approximately applicable for our sources), the effect
is only about 4~\mbox{$\mu$as~yr$^{-1}$}\ \citep{Sovers+1998, Titov2010} and has recently been reported
to be observed \citep{Titov+2011}.
Moreover, any unexpectedly
large acceleration of the solar-system barycenter is less likely, given
the study by \citet{ZakamskaT2005}, who find that pulsar timing
data (from both single and binary pulsars) are inconsistent with any
unmodeled accelerations of the solar-system barycenter greater than
$\sim4\times 10^{-9}$ cm s$^{-2}$, which is only about twice the magnitude of
the galactocentric acceleration. The more recent pulsar VLBI results
of \citet{Deller+2008} for PSR J0437-4715 likely strengthen this upper limit.
In the future it may be possible to search for the proper motion of
the cores of quasars with uncertainties much smaller than $c$. Then it
could be confirmed for the first time that the Hubble flow
dominates the motion of quasar cores and that the velocity expected
from the solar-system-barycenter acceleration is indeed to a high
degree consistent with models.
\subsubsection{Non-linear motion of the core within the boundaries of the proper-motion limit}
Our observation of possibly significant jittery motion of the core
within an area of the sky as small as 0.2$\times$0.2 mas$^{2}$ would
be only the second time such motion has been recorded unambiguously
for a source with core-jet structure by using as a reference for the
motion a physically unrelated source nearby on the
sky. The first source for which such motion was detected is the core
in the core-jet structure of the nearby galaxy M81. In this case, the
center of the expanding shell of the nearby supernova 1993J was used
as a reference \citep{BietenholzBR2001}.
What caused this apparent non-linear motion? Could such motion be
indicative of orbital motion related to a binary black-hole system?
This possibility is unlikely since the motion is jittery and its
magnitude too large to be physically reasonable.
Our measurements of the jittery motion of C1 along the north-south
direction is within the noise determined from the jitter in the
positions of B2250+194 relative to B2252+172. C1's jittery motion
along the east-west direction is above the noise level. In fact, the
peak-to-peak variation is 2.5 times larger than the corresponding
variation in the positions of the two reference sources and may be
significant. If so, it is likely caused by slight
brightness-distribution changes due to activity at the foot of the jet
close to the putative supermassive black hole. Such changes may have
influenced the fit position of the Gaussian core component, C1 (Paper
II).
This result, if confirmed, has implications for high-precision astrometric
observations in general. It shows that any component, even a core
component, clearly identified in the structure of a celestial object
may still move on the sky.
\subsection{The relevance for \mbox{\em GP-B}}
The goal of \mbox{\em GP-B}\ at launch was to measure the precession of the
gyroscopes relative to distant inertial space with a standard error of
0.5 \mbox{mas~yr$^{-1}$}\ or less in each sky coordinate. To be a minor
contributor to the error budget, the proper motion of the guide star,
\objectname[]{IM Peg}, was to be determined with a standard
error no larger than 0.14~\mbox{mas~yr$^{-1}$}\ in each sky coordinate. Our reference
source, the quasar
3C~454.3 was shown to be stationary within the CRF over $\sim30$ years of
geodetic VLBI observations to within 0.023 and 0.009~\mbox{mas~yr$^{-1}$}\ in \mbox{$\alpha$}\
and \mbox{$\delta$}, respectively. More to the point, our primary reference point
for \mbox{\em GP-B}, C1 in 3C~454.3, is stationary with respect to B2250+194 to
within 0.038 and 0.035 \mbox{mas~yr$^{-1}$}\ and within the CRF to within 0.046 and
0.056~\mbox{mas~yr$^{-1}$}\ in \mbox{$\alpha$}\ and \mbox{$\delta$}, respectively, over the seven years that almost
entirely cover the period of our VLBI observations in support of the
\mbox{\em GP-B}\ mission and is therefore a negligible contributor to the error
budget of the proper motion of the guide star.
\section{Conclusions}
\label{conclus}
\noindent Here we summarize our observations and data analysis, and give our conclusions:
\begin{trivlist}
\item{1.} We made differential VLBI observations at 35 epochs of the
quasar \objectname[]{3C 454.3} and the radio galaxy
\objectname[]{B2250+194} along with 12 epochs of the extragalactic,
unidentified source, \objectname[]{B2252+172}, at 8.4 GHz between 1997
and 2005. With these sources we provided a reference frame composed
of extragalactic sources nearby on the sky to \objectname[]{IM Peg}
and, together with geodetic VLBI observations made by others we
provided a (global) CRF, the latter closely linked to the ICRF2,
for the determination of the proper motion of
the \mbox{\em GP-B}\ guide star IM Peg with respect to the distant universe.
\item{2.} We analyzed our differential VLBI observations using
phase-referenced mapping and phase-delay fitting in combination with a
Kalman filter.
\item{3.} Our $1\sigma$ upper limit of the proper motion of \objectname[]{B2252+172} relative to
\objectname[]{B2250+194} is $11$~\mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and
$24$~\mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\delta$}\ for the time from 2002 to 2005, identifying
\objectname[]{B2252+172} unequivocally as extragalactic and providing
for a highly stable reference frame of sources nearby on the sky to
\objectname[]{3C 454.3}.
\item{4.} Our $1\sigma$ upper limits of the proper motions of \objectname[]{3C 454.3},
\objectname[]{B2250+194}, and \objectname[]{B2252+172} in the CRF determined with geodetic observations,
and the latter two also partly with phase-delay observations, are $\leq30$~\mbox{$\mu$as~yr$^{-1}$}\ in each coordinate.
\item{5.} Our $1\sigma$ upper limit on the proper motion of C1, our
core component of 3C~454.3, relative to
the combination of B2250+194
and B2252+172 for the time from 2002 to 2005 is $<35$ \mbox{$\mu$as~yr$^{-1}$}\ in each of \mbox{$\alpha$}\
and \mbox{$\delta$}, indicating that C1 is
highly stable with respect to two extragalactic sources
nearby on the sky.
\item{6.} The 1$\sigma$ upper limit on the proper motion of C1 in the CRF for the
time from 1998 to 2005 is 46 and 56 \mbox{$\mu$as~yr$^{-1}$}\ in \mbox{$\alpha$}\ and \mbox{$\delta$}, respectively.. This is our
fundamental result of the stationarity of the reference point for the
guide star IM Peg in support of the \mbox{\em GP-B}\ mission. For the shorter
time from 2002 to 2005 the 1$\sigma$ upper limit on the proper motion
of C1 in the CRF is 39 and 30 \mbox{$\mu$as~yr$^{-1}$}\ for the two coordinates,
respectively, corresponding to subluminal motion of $\leq
1.0 c$ and $< 0.8 c$, for \mbox{$\alpha$}\ and \mbox{$\delta$}, respectively, for an
angular-size distance to 3C~454.3 of 1.6 Gpc, for a flat universe with
$H_0$=70~\mbox{km s$^{-1}$ Mpc$^{-1}$}, $\Omega_M=0.27$, and $\Omega_{\lambda}=0.73$.
\item{7.} The source coordinates of C1 in the CRF
differ from those of 3C~454.3 determined from geodetic group-delay data
by $131\pm98$~\mbox{$\mu$as}\ in \mbox{$\alpha$}\ and $-338\pm105$~\mbox{$\mu$as}\ in \mbox{$\delta$}, the latter coordinate being different at
the 3.2$\sigma$ level. This difference in \mbox{$\delta$}\ is not understood, except possibly as a statistical fluke.
\item{8.} C2 and J1, the latter for the period from 2002 to 2005, are
stationary in the CRF to within bounds as small as those for C1.
However, C1's
flat spectrum and compactness in contrast to the spectra and the compactness of
the other components, indicate that C1 is closest
to the putative supermassive black hole and the probable gravitational
center of the quasar.
\item{9.} The jet components, D1, D2, and Jext clearly
move in the CRF.
Their motions correspond to superluminal speeds, which for D2 is $5~c$.
\item{10.} Notwithstanding our limit on the proper motion of C1, there
is evidence for its having jittery $\sim0.2$~mas east-west motion
above the noise level, likely related to jet activity in the vicinity
of the core. This evidence is consistent with the jittery motion of C1
found in Paper II.
\item{11.} The 1$\sigma$ upper limit on the parallax of C1 relative to
\objectname[]{B2250+194} and B2252+172 is 20~\mbox{$\mu$as}, one of the most, if not the most, accurate
limit so far obtained, corresponding to an unsurprising lower limit of 50 kpc
on its distance from Earth and demonstrating the sensitivity of parallax measurements with VLBI.
\item{12.} The upper limit on the proper motion of 3C~454.3 over $\sim30$
years of geodetic VLBI observations and of C1 over $\sim8.5$ years of
our phase-delay VLBI observations
is sufficiently small to meet the goal of the \mbox{\em GP-B}\ mission and
therefore to justify use of C1 as the primary reference point for
\mbox{\em GP-B}.
\end{trivlist}
ACKNOWLEDGMENTS. This research was primarily supported by NASA,
through a contract with Stanford University to SAO, and a subcontract
from SAO to York University. The National Radio Astronomy Observatory
(NRao) is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. The DSN is
operated by JPL/Caltech, under contract with NASA\@. This research
has made use of the United States Naval Observatory (USNO) Radio
Reference Frame Image Database (RRFID)\@. We have made use of NASA's
Astrophysics Data System Abstract Service, developed and maintained at
SAO\@. Jeff Cadieux and Julie Tome helped with the data reduction
during their tenure as students at York University. We thank the
International VLBI Service for Geodesy and Astrometry
\citep[IVS;][]{SchluterB2007} for their support.
|
2,877,628,089,473 | arxiv | \section{Introduction}
Let $\mathbb{H}:=\{z=x+i y\in\mathbb{C}\,\vert\,y>0\}$ denote the upper half-plane and $\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$ a Fuchsian
subgroup of the first kind, which acts by fractional linear transformations on $\mathbb{H}$. Let $\mathcal{S}_{k}(\Gamma)$ denote the space of cusp
forms of weight $k$ for $\Gamma$ and let $\mathcal{H}_{k}(\Gamma)$ denote the space of real-analytic automorphic forms of weight $k$ for $\Gamma$,
on which the Maa\ss{} Laplacian
\begin{align*}
\Delta_{k}:=y^{2}\bigg(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\bigg)-iky\frac{\partial}{\partial x}
\end{align*}
of weight $k$ acts. Then, it is well-known that there is an isomorphism
\begin{align}
\label{1-iso}
\mathcal{S}_{k}(\Gamma)\cong\ker\bigg(\Delta_{k}+\frac{k}{2}\bigg(1-\frac{k}{2}\bigg)\mathrm{id}\bigg)
\end{align}
of $\mathbb{C}$-vector spaces, induced by the assignment $f\mapsto y^{k/2}f$, where the right-hand side consists of the Maa\ss{} forms in $\mathcal
{H}_{k}(\Gamma)$ with eigenvalue $k/2(1-k/2)$ of $\Delta_{k}$. This identification of two types of automorphic forms for $\Gamma$ has various useful
applications. For example, in the article \cite{JK2}, the isomorphism~\eqref{1-iso} was crucial in relating the sup-norm bound problem for cusp forms of
weight $k$ for $\Gamma$ to bounds for the heat kernel for $\Delta_{k}$ on the quotient space $\Gamma\backslash\mathbb{H}$.
In this paper, we attempt a generalization of the isomorphism~\eqref{1-iso} to the Siegel modular setting, which, to our surprise, we could not find in
the literature. Letting $\mathrm{Sym}_{n}(\mathbb{C})$ be the set of complex symmetric $(n\times n)$-matrices, we let $\mathbb{H}_{n}:=\{Z=X+iY\in
\mathrm{Sym}_{n}(\mathbb{C})\,\vert\,Y>0\}$ denote the Siegel upper half-space of degree $n\geq 1$ and we let $\Gamma\subset\mathrm{Sp}_{n}
(\mathbb{R})$ denote a subgroup acting by generalized fractional linear transformations on $\mathbb{H}_{n}$, which is commensurable with the Siegel
modular group $\mathrm{Sp}_{n}(\mathbb{Z})$. Then, let $\mathcal{S}_{k}^{n}(\Gamma)$ denote the space of Siegel cusp forms of weight $k$ and
degree $n$ for $\Gamma$ and let $\mathcal{H}_{k}^{n}(\Gamma)$ be the space of real-analytic automorphic forms of weight $k$ and degree $n$ for
$\Gamma$, on which the Siegel--Maa\ss{} Laplacian
\begin{align}
\label{deltak}
\Delta_{k}:=\tr\bigg(Y\bigg(\bigg(Y\dpd{}{X}\bigg)^{t}\dpd{}{X}+\bigg(Y\dpd{}{Y}\bigg)^{t}\dpd{}{Y}\bigg)-ikY\dpd{}{X}\bigg)
\end{align}
of weight $k$ acts. As the main result of this paper, we show in Corollary~\ref{kernel connection} the isomorphism
\begin{align}
\label{1-iso general}
\mathcal{S}_{k}^{n}(\Gamma)\cong\ker\bigg(\Delta_{k}+\frac{nk}{4}(n-k+1)\mathrm{id}\bigg),
\end{align}
of $\mathbb{C}$-vector spaces, induced by the assignment $f\mapsto\det(Y)^{k/2}f$, thereby generalizing the isomorphism~\eqref{1-iso} to the Siegel
modular setting. The right-hand side of~\eqref{1-iso general} now consists of the Siegel--Maa\ss{} forms in $\mathcal{H}_{k}^{n}(\Gamma)$ with
eigenvalue $nk(n-k+1)/4)$ of $\Delta_{k}$.
In case $n=1$, the isomorphism~\eqref{1-iso} is obtained as a by-product of the proof of the symmetry of the Maa\ss{} Laplacian $\Delta_{k}$ (see
\cite{Roelcke}). The most straightforward proof of the symmetry of $\Delta_{k}$ is obtained by constructing a suitable $\mathrm{SL}_{2}(\mathbb{R})
$-invariant differential form using the raising or the lowering operators on $\mathbb{H}$, and then integrating it over the quotient space $\Gamma
\backslash\mathbb{H}$ (see \cite{Bump}, p.~135). Generalizations of all these operators as well as their transformation behaviour under the action
of the symplectic group $\mathrm{Sp}_{n}(\mathbb{R})$ to the Siegel modular setting have been provided by Maa\ss{} in~\cite{Maass2}. However,
in spite of all these crucial ingredients being around for a long time, we could not find in the literature a precise proof of the symmetry of the
Siegel--Maa\ss{} Laplacian $\Delta_{k}$ of weight $k$. We provide a complete proof of the symmetry of $\Delta_{k}$ in Theorem~\ref{main theorem},
where we construct the appropriate $\mathrm{Sp}_{n}(\mathbb{R})$-invariant differential form on $\mathbb{H}_{n}$, which, while calculationally a bit
cumbersome, is conceptually a rather straightforward piecing-together of Maa\ss{}'s calculations. Our main result in Corollary~\ref{kernel connection}
is then a consequence of Theorem~\ref{main theorem}.
As indicated above, the generalization of the isomorphism~\eqref{1-iso} will perspectively allow us, among others, to relate the sup-norm bound
problem for Siegel cusp forms of weight $k$ and degree $n$ for $\Gamma$ to bounds for the heat kernel for the Siegel--Maa\ss{} Laplacian $\Delta_
{k}$ on the quotient space $\Gamma\backslash\mathbb{H}_{n}$.
This paper is organized as follows: In section~2, we provide a quick summary of the basics of the Siegel upper half-space and Siegel modular forms.
In the subsequent two sections~3 and~4, we introduce and discuss the transformation behaviour of the relevant operators in the Siegel modular setting.
This material is already present in~\cite{Maass2}, but due to sub-optimal typesetting, at places, it is hard to decipher. So we take this opportunity to
redo these calculations along Maa\ss{}'s lines and present them here for the reader's convenience. However, no claim of originality is made here on
this material. Finally in section~5, piecing together Maa\ss{}'s results, we construct the appropriate $\mathrm{Sp}_{n}(\mathbb{R})$-invariant differential
form on $\mathbb{H}_{n}$ to show the symmetry of the Siegel--Maa\ss{} Laplacian $\Delta_{k}$, and then use it to show the generalization~\eqref
{1-iso general} of the isomorphism~\eqref{1-iso}.
\section{Basic notations and definitions}
For $n\in\mathbb{N}_{>0}$ and a commutative ring $R$, let $\mathrm{M}_{n}(R)$ denote the set of $(n\times n)$-matrices with entries in $R$ and
$\mathrm{Sym}_{n}(R)$ the set of symmetric matrices in $\mathrm{M}_{n}(R)$. The Siegel upper half-space
$\mathbb{H}_{n}$ of degree $n$ is then defined by
\begin{align*}
\mathbb{H}_{n}:=\{Z=X+iY\in\mathrm{M}_{n}(\mathbb{C})\,\vert\,X,Y\in\mathrm{Sym}_{n}(\mathbb{R}):\,Y>0\}.
\end{align*}
The symplectic group $\mathrm{Sp}_{n}(\mathbb{R})$ of degree $n$ is defined by
\begin{align*}
\mathrm{Sp}_{n}(\mathbb{R}):=\{\gamma\in\mathrm{M}_{2n}(\mathbb{R})\,\vert\,\gamma^{t}J_{n}\gamma=J_{n}\},
\end{align*}
where $J_{n}\in\mathrm{M}_{2n}(\mathbb{R})$ is the skew-symmetric matrix
\begin{align*}
J_{n}:=\begin{pmatrix}0&\mathbbm{1}_{n}\\-\mathbbm{1}_{n}&0\end{pmatrix}
\end{align*}
with $\mathbbm{1}_{n}$ denoting the identity matrix of $\mathrm{M}_{n}(\mathbb{R})$. Writing an element $\gamma\in\mathrm{Sp}_{n}(\mathbb{R})$
in block form as
\begin{align*}
\gamma=\begin{pmatrix}A&B\\C&D\end{pmatrix}
\end{align*}
with $A,B,C,D\in\mathrm{M}_{n}(\mathbb{R})$, we can recast the relation $\gamma^{t}J_{n}\gamma=J_{n}$ as the set of relations
\begin{align}
\label{symprelation 1}
A^{t}C=C^{t}A,\quad B^{t}D=D^{t}B,\quad A^{t}D-C^{t}B=\mathbbm{1}_{n}.
\end{align}
Observing that with $\gamma\in\mathrm{Sp}_{n}(\mathbb{R})$, we also have $\gamma^{t}\in\mathrm{Sp}_{n}(\mathbb{R})$, the set of relations~\eqref
{symprelation 1} turns out to be equivalent to the relations
\begin{align*}
AB^{t}=BA^{t},\quad CD^{t}=DC^{t},\quad AD^{t}-BC^{t}=\mathbbm{1}_{n}.
\end{align*}
The group $\mathrm{Sp}_{n}(\mathbb{R})$ acts by the symplectic action
\begin{align}
\label{sympaction}
\mathbb{H}_{n}\ni Z\mapsto \gamma Z=(AZ+B)(CZ+D)^{-1} \qquad\big(\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_
{n}(\mathbb{R})\big)
\end{align}
on $\mathbb{H}_{n}$. Using the equality
\begin{align}
\label{actsymmetry}
(AZ+B)(CZ+D)^{-1}=(CZ+D)^{-t}(AZ+B)^{t},
\end{align}
which follows from equation~\eqref{symprelation 1}, we compute
\begin{align*}
(CZ+D)^{t}\,\mathrm{Im}(\gamma Z)(C\overline{Z}+D)=\mathrm{Im}(Z),
\end{align*}
from which we derive the important relation
\begin{align}
\label{imaginary transform}
\mathrm{Im}(\gamma Z)=(CZ+D)^{-t}\,\mathrm{Im}(Z)(C\overline{Z}+D)^{-1}
\end{align}
giving rise to the determinant relation
\begin{align*}
\det(\mathrm{Im}(\gamma Z))=\frac{\det(\mathrm{Im}(Z))}{\vert\det(CZ+D)\vert^{2}}.
\end{align*}
For the differential of the symplectic action~\eqref{sympaction}, we compute using again equation~\eqref{actsymmetry}
\begin{align*}
\mathrm{d}(\gamma Z)&=A\,\mathrm{d}Z(CZ+D)^{-1}-(AZ+B)(CZ+D)^{-1}C\,\mathrm{d}Z(CZ+D)^{-1} \\
&=(CZ+D)^{-t}((CZ+D)^{t}A\,\mathrm{d}Z-(AZ+B)^{t}C\,\mathrm{d}Z)(CZ+D)^{-1}.
\end{align*}
Using once again equation~\eqref{symprelation 1}, this gives
\begin{align}
\label{difZ calc2}
\mathrm{d}(\gamma Z)=(CZ+D)^{-t}\,\mathrm{d}Z\,(CZ+D)^{-1}.
\end{align}
The arclength $\mathrm{d}s^{2}_{n}$ and the volume form $\mathrm{d}\mu_{n}$ on $\mathbb{H}_{n}$ in terms of $Z=(z_{j,k})_{1\leq j\leq k\leq n}\in
\mathbb{H}_{n}$ are given by
\begin{align*}
\mathrm{d}s^{2}_{n}(Z)&=\tr(Y^{-1}\,\mathrm{d}Z\,Y^{-1}\,\mathrm{d}\overline{Z})\quad\qquad(Z=X+iY), \\[1mm]
\mathrm{d}\mu_{n}(Z)&=\frac{\bigwedge\limits_{1\leq j\leq k\leq n}\mathrm{d}x_{j,k}\wedge\mathrm{d}y_{j,k}}{\det(Y)^{n+1}}\qquad(z_{j,k}=x_{j,k}+
iy_{j,k}).
\end{align*}
From equations~\eqref{imaginary transform} and~\eqref{difZ calc2} it is obvious that the arclength $\mathrm{d}s^{2}_{n}$ and the volume form
$\mathrm{d}\mu_{n}$ on $\mathbb{H}_{n}$ given by the above equations are invariant under the symplectic action. Corresponding to this metric,
we have the Laplace--Beltrami operator $\Delta$ on $\mathbb{H}_{n}$, called the Siegel Laplacian, which is also invariant under the symplectic
action.
\begin{definition}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$, i.e., the intersection
$\Gamma\cap\mathrm{Sp}_{n}(\mathbb{Z})$ is a finite index subgroup of $\Gamma$ as well as of $\mathrm{Sp}_{n}(\mathbb{Z})$. We let $\gamma_
{j}\in\mathrm{Sp}_{n}(\mathbb{Z})$ ($j=1,\ldots,h$) denote a set of representatives for the left cosets of $\Gamma\cap\mathrm{Sp}_{n}(\mathbb{Z})$
in $\mathrm{Sp}_{n}(\mathbb{Z})$. Then, a \emph{Siegel modular form of weight $k$ and degree $n$ for $\Gamma$} is a function $f\colon\mathbb
{H}_{n}\longrightarrow\mathbb{C}$ satisfying the following conditions:
\begin{itemize}
\item[(i)]
$f$ is holomorphic;
\item[(ii)]
$f(\gamma Z)=\det(CZ+D)^{k}f(Z)$ for all $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$;
\item[(iii)]
given $Y_{0}\in\mathrm{Sym}_{n}(\mathbb{R})$ with $Y_{0}>0$, the quantities $\det(C_{j}Z+D_{j})^{-k}f(\gamma_{j} Z)$ are bounded in the region
$\{Z=X+iY\in\mathbb{H}_{n}\,\vert\,Y\geq Y_{0}\}$ for the set of representatives $\gamma_{j}=\big(\begin{smallmatrix}A_{j}&B_{j}\\C_{j}&D_{j}\end
{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{Z})$ ($j=1,\ldots,h$).
\end{itemize}
Moreover, a Siegel modular form $f$ as above is called a \emph{Siegel cusp form of weight $k$ and degree $n$ for $\Gamma$} if condition~(iii)
above is strengthened to the condition
\begin{itemize}
\item[(iii')]
given $Y_{0}\in\mathrm{Sym}_{n}(\mathbb{R})$ with $Y_{0}\gg 0$, the quantities $\det(C_{j}Z+D_{j})^{-k}f(\gamma_{j} Z)$ become arbitrarily small
in the region $\{Z=X+iY\in\mathbb{H}_{n}\,\vert\,Y\geq Y_{0}\}$ for the set of representatives $\gamma_{j}=\big(\begin{smallmatrix}A_{j}&B_{j}\\C_
{j}&D_{j}\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{Z})$ ($j=1,\ldots,h$).
\end{itemize}
\end{definition}
\begin{remark}
The sets of Siegel modular forms and Siegel cusp forms of weight $k$ and degree $n$ for $\Gamma$ have the structure of $\mathbb{C}$-vector
spaces, which we denote by $\mathcal{M}_{k}^{n}(\Gamma)$ and $\mathcal{S}_{k}^{n}(\Gamma)$, respectively, and which turn out to be finite
dimensional. Moreover, the space $\mathcal{S}_{k}^{n}(\Gamma)$ is equipped with the so-called Petersson inner product given by
\begin{align*}
\langle f,g\rangle:=\int_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{k}f(Z)\overline{g}(Z)\,\mathrm{d}\mu_{n}(Z)\qquad(f,g\in\mathcal{S}_{k}^{n}
(\Gamma)),
\end{align*}
making $\mathcal{S}_{k}^{n}(\Gamma)$ into a hermitian inner product space.
\end{remark}
\section{Siegel--Maa\ss{} Laplacian of weight $(\alpha,\beta)$}
In this section, we will recall from~\cite{Maass2} various differential operators acting on smooth complex valued functions defined on $\mathbb{H}_
{n}$. In particular, we will define the Siegel--Maa\ss{} Laplacian of weight $(\alpha,\beta)$, where $\alpha,\beta\in\mathbb{R}$. Letting $\alpha=k/2$
and $\beta=-k/2$ will then lead us to the Siegel--Maa\ss{} Laplacian $\Delta_{k}$ mentioned in formula \eqref{deltak} in the introduction. We point
out that the Siegel Laplacian $\Delta$ mentioned in the previous section and the Siegel--Maa\ss{} Laplacian $\Delta_{k}$ are related by the formula
\begin{align*}
\Delta_{k}=\Delta-ik\tr\bigg(Y\dpd{}{X}\bigg)
\end{align*}
with the symmetric $(n\times n)$-matrix $\partial/\partial X$ of partial derivatives being defined below.
Given $Z=X+iY\in\mathbb{H}_{n}$, we start by introducing the following symmetric $(n\times n)$-matrices of partial derivatives:
\begin{align*}
\mathrm{(i)}&\quad\bigg(\dpd{}{X}\bigg)_{j,k}:=\frac{1+\delta_{j,k}}{2}\dpd{}{x_{j,k}}, \\
\mathrm{(ii)}&\quad\bigg(\dpd{}{Y}\bigg)_{j,k}:=\frac{1+\delta_{j,k}}{2}\dpd{}{y_{j,k}}, \\
\mathrm{(iii)}&\quad\dpd{}{Z}:=\frac{1}{2}\bigg(\dpd{}{X}-i\dpd{}{Y}\bigg), \\
\mathrm{(iv)}&\quad\dpd{}{\overline{Z}}:=\frac{1}{2}\bigg(\dpd{}{X}+i\dpd{}{Y}\bigg),
\end{align*}
where $\delta_{j,k}$ is the Kornecker delta symbol.
\begin{definition}
\label{maass operators}
Following Maa\ss{}~\cite{Maass2}, we define, using the above notations, for arbitrary real numbers $\alpha,\beta\in\mathbb{R}$, the following $(n
\times n)$-matrices of differential operators acting on smooth complex valued functions on~$\mathbb{H}_{n}$:
\begin{align*}
\mathrm{(i)}&\quad K_{\alpha}:=(Z-\overline{Z})\dpd{}{Z}+\alpha\mathbbm{1}_{n}, \\
\mathrm{(ii)}&\quad\Lambda_{\beta}:=(Z-\overline{Z})\dpd{}{\overline{Z}}-\beta\mathbbm{1}_{n}, \\[2mm]
\mathrm{(iii)}&\quad\Omega_{\alpha,\beta}:=\Lambda_{\beta-(n+1)/2}K_{\alpha}+\alpha(\beta-(n+1)/2)\mathbbm{1}_{n}, \\[3mm]
\mathrm{(iv)}&\quad\widetilde{\Omega}_{\alpha,\beta}:=K_{\alpha-(n+1)/2}\Lambda_{\beta}+\beta(\alpha-(n+1)/2)\mathbbm{1}_{n}.
\end{align*}
\end{definition}
Next, we want to expand $\Omega_{\alpha,\beta}$ and $\widetilde{\Omega}_{\alpha,\beta}$ in terms of $Z,\overline{ Z},\partial/\partial Z$, and
$\partial/\partial{\overline{Z}}$. For that we need the following lemma.
\begin{lemma}
\label{operator derivative identity general}
Let $C,D\colon\mathbb{H}_{n}\longrightarrow\mathrm{M}_{n}(\mathbb{C})$ be smooth matrix valued functions depending on $Z$ and $\overline{Z}$.
Then, the following matrix operator identities hold: \\[1mm]
\emph{(i)} Assuming that $\partial C/\partial Z=0$ and $\partial D/\partial Z=0$, we have
\begin{align*}
\dpd{}{Z}(CZ+D)^{t}=\bigg((CZ+D)\dpd{}{Z}\bigg)^{t}+\frac{1}{2}(n+1)C^{t}.
\end{align*}
\emph{(ii)} Assuming that $\partial C/\partial\overline{Z}=0$ and $\partial D/\partial\overline{Z}=0$, we have
\begin{align*}
\dpd{}{\overline{Z}}(C\overline{Z}+D)^{t}=\bigg((C\overline{Z}+D)\dpd{}{\overline{Z}}\bigg)^{t}+\frac{1}{2}(n+1)C^{t}.
\end{align*}
\end{lemma}
\begin{proof}
Since part~(ii) follows from part~(i) by conjugation, we prove only (i). Let $\Phi$ be a matrix depending on $Z$ and $\overline{Z}$ such that the
product $(CZ+D)^{t}\Phi$ makes sense. Then, writing the $(j,k)$-th entry of the matrix $\partial/\partial Z(CZ+D)^{t}\Phi$ as the sum
\begin{align*}
\bigg(\dpd{}{Z}(CZ+D)^{t}\Phi\bigg)_{j,k}=\sum\limits_{l,m=1}^{n}\bigg(\dpd{}{Z}\bigg)_{j,l}\big((CZ+D)^{t}_{l,m}\Phi_{m,k}\big)
\end{align*}
and noting that $\partial Z/\partial z_{j,l}=(1-\delta_{j,l})E_{j,l}+E_{l,j}$, where $E_{j,k}\in\mathrm{M}_{n}(\mathbb{C})$ is the matrix with its $(j,k)$-th
entry being $1$ and the remaining entries being $0$, elementary calculations lead us to the operator identity
\begin{align*}
\dpd{}{Z}(CZ+D)^{t}=\bigg((CZ+D)\dpd{}{Z}\bigg)^{t}+\frac{1}{2}(n+1)C^{t},
\end{align*}
which is what we needed to prove.
\end{proof}
\begin{corollary}
\label{operator derivative identity}
For $Z\in\mathbb{H}_{n}$, the following operator identities hold:
\begin{align*}
\mathrm{(i)}\quad\dpd{}{Z}(Z-\overline{Z})=\bigg((Z-\overline{Z})\dpd{}{Z}\bigg)^{t}+\frac{1}{2}(n+1)\mathbbm{1}_{n}, \\
\mathrm{(ii)}\quad\dpd{}{\overline{Z}}(Z-\overline{Z})=\bigg((Z-\overline{Z})\dpd{}{\overline{Z}}\bigg)^{t}-\frac{1}{2}(n+1)\mathbbm{1}_{n}.
\end{align*}
\end{corollary}
\begin{proof}
As $\partial\overline{Z}/\partial Z=0$, we can choose $C=\mathbbm{1}_{n}$ and $D=-\overline{ Z}$ in Lemma~\ref{operator derivative identity
general}~(i), from which the first claimed formula follows. The second formula follows analogously.
\end{proof}
Using the above corollary, one can expand $\Omega_{\alpha,\beta}$ and $\widetilde{\Omega}_{\alpha,\beta}$ as
\begin{align*}
\Omega_{\alpha,\beta}&=(Z-\overline{ Z})\bigg((Z-\overline{ Z})\dpd{}{\overline{Z}}\bigg)^{t}\dpd{}{Z}+\alpha(Z-\overline{Z})\dpd{}{\overline{Z}}-
\beta(Z-\overline{Z})\dpd{}{Z}, \\
\widetilde{\Omega}_{\alpha,\beta}&=(Z-\overline{Z})\bigg((Z-\overline{Z})\dpd{}{Z}\bigg)^{t}\dpd{}{\overline{Z}}+\alpha(Z-\overline{Z})\dpd{}
{\overline{Z}}-\beta(Z-\overline{Z})\dpd{}{Z}.
\end{align*}
Then, $\Omega_{\alpha,\beta}$ and $\widetilde{\Omega}_{\alpha,\beta}$ are related by the identity
\begin{align*}
\widetilde{\Omega}_{\alpha,\beta}=(Z-\overline{ Z})\big((Z-\overline{Z})^{-1}\Omega_{\alpha,\beta}\big)^{t}.
\end{align*}
\begin{definition}
The operator $\Delta_{\alpha,\beta}:=-\tr(\Omega_{\alpha,\beta})=-\tr(\widetilde{\Omega}_{\alpha,\beta})$ is called the \emph{Siegel--Maa\ss{}
Laplacian of weight $(\alpha,\beta)$}.
\end{definition}
\section{Transformation behaviour of Maa\ss{} operators}
Recall that the symplectic action of $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$ on the
point $Z\in\mathbb{H}_{n}$ is given by
\begin{align*}
\gamma Z=(AZ+B)(CZ+D)^{-1}=(CZ+D)^{-t}(AZ+B)^{t};
\end{align*}
to avoid cumbersome notation, we will use in this paper sometimes the shorthand $Z^{\gamma}:=\gamma Z$. In this section, we will then study
the transformation behaviour of the Maa\ss{} operators introduced in Definition~\ref{maass operators} by expressing the operators $K^{\gamma}_
{\alpha}$, $\Lambda^{\gamma}_{\beta}$, $\Omega^{\gamma}_{\alpha,\beta}$ obtained by replacing $Z,\overline{Z}$ by $Z^{\gamma},\overline
{Z}^{\gamma}$ in ${K}_{\alpha}$, ${\Lambda}_{\beta}$, ${\Omega}_{\alpha,\beta}$, respectively, as they operate on smooth complex valued
functions defined on $\mathbb{H}_{n}$.
We begin by investigating how the matrix differential operators $\partial/\partial Z$ and $\partial/\partial\overline{Z}$ transform under the symplectic
action of $\gamma$ on $Z$. From equation~\eqref{difZ calc2}, we know that the differential $\mathrm{d}Z$ transforms like
\begin{align*}
\mathrm{d}Z^{\gamma}=(CZ+D)^{-t}\,\mathrm{d}Z\,(CZ+D)^{-1}.
\end{align*}
Therefore, as the differential of a smooth function $\varphi\colon\mathbb{H}_{n}\longrightarrow\mathbb{C}$ depending only on $Z$ is given by
$\mathrm{d}\varphi=\tr(\partial\varphi/\partial Z\,\mathrm{d}Z)$, we have
\begin{align*}
\dpd{\varphi}{Z}=(CZ+D)^{-1}\dpd{\varphi}{{Z^{\gamma}}}(C Z+D)^{-t},
\end{align*}
i.e., the operator $\partial/\partial Z$ transforms as
\begin{align}
\label{partial transform}
\dpd{}{{Z^{\gamma}}}=(C Z+D)\bigg((CZ+D)\dpd{}{Z}\bigg)^{t}.
\end{align}
By conjugation, the operator $\partial/\partial\overline{Z}$ transforms as
\begin{align}
\label{partial transform 2}
\dpd{}{{\overline{Z}^{\gamma}}}=(C\overline{Z}+D)\bigg((C\overline{Z}+D)\dpd{}{\overline{Z}}\bigg)^{t}.
\end{align}
Next we need to know how to differentiate $\det(Z-\overline{Z})$ and $\det(CZ+D)$ with respect to $Z$, which we carry out in the next two lemmas.
\begin{lemma}
\label{determinant differentiate}
The matrix identity
\begin{align*}
\dpd{\det(Z-\overline{Z})}{Z}=\det(Z-\overline{Z})(Z-\overline{Z})^{-1}
\end{align*}
holds.
\end{lemma}
\begin{proof}
Since $Y=\frac{1}{2i}(Z-\overline{Z})\in\mathrm{Sym}_{n}(\mathbb{R})$, it can be diagonalized with orthogonal matrices. Thus, let $Y=U^{t}\Lambda
U$, where $\Lambda\in\mathrm{M}_{n}(\mathbb{R})$ is a diagonal matrix with diagonal entries equal to the eigenvalues $\lambda_{1},\ldots,\lambda_
{n}$ of $Y$ and $U\in\mathrm{O}_{n}(\mathbb{R})$. Therefore, differentiating $\det(Y)$ with respect to the entries of $Y=(y_{j,k})_{1\leq j,k\leq n}$,
we have
\begin{align*}
\dpd{\det(Y)}{y_{j,k}}=\dpd{\prod\limits_{l=1}^{n}\lambda_{l}}{y_{j,k}}=\det(Y)\tr\bigg(\Lambda^{-1}\dpd{\Lambda}{y_{j,k}}\bigg).
\end{align*}
Now writing $\tr(\Lambda^{-1}\,\partial\Lambda/\partial y_{j,k})$ as $\tr(Y^{-1}\,\partial Y/\partial y_{j,k})$ and using the fact that
\begin{align*}
\dpd{Y}{y_{j,k}}=(1-\delta_{j,k})E_{j,k}+E_{k,j},
\end{align*}
we obtain
\begin{align*}
\dpd{\det(Y)}{y_{j,k}}=\det (Y)(2-\delta_{j,k})(Y^{-1})_{j,k}.
\end{align*}
Now since $((1+\delta_{j,k})/2)(2-\delta_{j,k})=1$, we have $\partial\det(Y)/\partial Y=\det(Y)Y^{-1}$. This is equivalent to the identity
\begin{align*}
\dpd{\det(Z-\overline{Z})}{Z}=\det(Z-\overline{Z})(Z-\overline{Z})^{-1},
\end{align*}
which is what we needed to prove.
\end{proof}
\begin{lemma}
\label{gen determinant differentiate}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$. Then, the matrix identity
\begin{align*}
\dpd{\det(CZ+D)}{Z}=\det(CZ+D)(CZ+D)^{-1}C=\det(CZ+D)C^{t}(C Z+D)^{-t}
\end{align*}
holds.
\end{lemma}
\begin{proof}
For $1\leq j,k\leq n$, let $u_{j,k}\colon\mathbb{H}_{n}\longrightarrow\mathbb{C}$ be smooth scalar valued functions; then, $U:=(u_{j,k})_{1\leq j,
k\leq n}$ becomes a smooth matrix valued function on $\mathbb{H}_{n}$. Moreover, let $\varphi\colon\mathrm{M}_{n}(\mathbb{C})\longrightarrow
\mathbb{C}$ be a smooth scalar valued function. Then, differentiating the function $\varphi\circ U\colon\mathbb{H}_{n}\longrightarrow\mathbb{C}$
with respect to the entries $z_{j,k}$ of $Z$, we have
\begin{align*}
\dpd{\varphi(U(Z))}{z_{j,k}}=\sum\limits_{l,m=1}^{n}\dpd{\varphi}{u_{l,m}}\dpd{u_{l,m}}{z_{j,k}}=\sum\limits_{l,m=1}^{n}\bigg(\dpd{\varphi}{U}\bigg)^
{t}_{m,l}\bigg(\dpd{U}{z_{j,k}}\bigg)_{l,m}=\sum\limits_{m=1}^{n}\bigg(\bigg(\dpd{\varphi}{U}\bigg)^{t}\dpd{U}{z_{j,k}}\bigg)_{m,m}.
\end{align*}
Thus, the chain rule of differentiation in this case takes the form
\begin{align*}
\dpd{\varphi(U(Z))}{ z_{j,k}}=\tr\bigg(\bigg(\dpd{\varphi}{U}\bigg)^t\dpd{U}{ z_{j,k}}\bigg).
\end{align*}
Note that since we do not assume $U$ to be symmetric beforehand, in this case we have $(\partial/\partial U)_{j,k}=\partial/\partial u_{j,k}$ instead
of $(\partial/\partial U)_{j,k}=((1+\delta_{j,k})/2)\partial/\partial u_{j,k}$.
Now putting $\varphi(Z)=\det(Z)$ and $U(Z)=CZ+D$, by the above formula, we have
\begin{align*}
\dpd{\det(CZ+D)}{z_{j,k}}=\tr\bigg(\bigg(\dpd{\det(CZ+D)}{(CZ+D)}\bigg)^{t}\dpd{(CZ+D)}{z_{j,k}}\bigg).
\end{align*}
We already derived a formula for differentiating the determinant of a symmetric matrix by its entries. The structure of symmetry actually complicates
the calculation as its entries are no longer independent. For a square matrix $U$, not assumed to be symmetric beforehand, the calculation can be
simplified by considering the cofactor expansion of a determinant. Let $\widetilde{U}=(\widetilde{u}_{j,l})_{1\leq j,l\leq n}$ be the cofactor matrix of
$U$. Then, we have
\begin{align*}
\dpd{\det(U)}{u_{j,k}}=\dpd{}{u_{j,k}}\sum\limits_{l=1}^{n}u_{j,l}\widetilde{u}_{j,l}=\widetilde{u}_{j,k}.
\end{align*}
Here we exploit the property that since we delete the $j$-th row (and the $l$-th column) to build the cofactor $\widetilde{u}_{j,l}$, it must be independent
of $u_{j,k}$. This does not hold for a symmetric matrix. Now, since $U^{-1}=1/\det (U)\,\widetilde{U}^{t}$, we have
\begin{align*}
\dpd{\det(U)}{U}=\widetilde{U}=\det(U)\,U^{-t}.
\end{align*}
Thus, going back to our initial calculation, we arrive at
\begin{align*}
\dpd{\det(CZ+D)}{z_{j,k}}&=\det(CZ+D)\tr\bigg((CZ+D)^{-1}\dpd{(CZ+D)}{z_{j,k}}\bigg) \\
&=\det(CZ+D)\sum\limits_{l,m=1}^{n}(CZ+D)^{-1}_{l,m}\dpd{(CZ+D)_{m,l}}{z_{j,k}}.
\end{align*}
Now, entrywise partial differentiation with respect to entries $z_{j,k}$ of $Z$ followed by an elementary calculation with taking care of the ensuing
Kronecker delta symbols leads us to the matrix identity
\begin{align*}
\dpd{\det(CZ+D)}{Z}=\det(CZ+D)(CZ+D)^{-1}C=\det(CZ+D)C^{t}(CZ+D)^{-t},
\end{align*}
which is what we needed to prove.
\end{proof}
Lemmas~\ref{determinant differentiate} and~\ref{gen determinant differentiate} prepare the groundwork for calculating the transformation behaviour
of the Maa\ss{} operators, which we undertake one by one in the subsequent three propositions.
\begin{proposition}
\label{Ktransformation}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$ and $\varphi\colon\mathbb{H}_{n}\longrightarrow
\mathbb{C}$ be a smooth function. Then, the operator $K^{\gamma}_{\alpha}$ obtained by replacing $Z\in\mathbb{H}_{n}$ in ${K}_{\alpha}$ by $Z^
{\gamma}=\gamma Z$ is related to the operator ${K}_{\alpha}$ by the identity
\begin{align*}
&K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t}.
\end{align*}
\end{proposition}
\begin{proof}
From the definition of $K^{\gamma}_{\alpha}$, we have
\begin{align*}
&K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\
&\qquad=\bigg((Z^{\gamma}-\overline{Z}^{\gamma})\frac{\partial}{\partial Z^{\gamma}}+\alpha\mathbbm{1}_{n}\bigg)\det(CZ+D)^{\alpha}\det(C\overline{Z}+
D)^{\beta}\varphi(Z).
\end{align*}
Then, expanding $\partial/\partial Z^{\gamma}$ by means of equation~\eqref{partial transform} gives
\begin{align}
\label{longKstar}
\notag
&K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)=\alpha\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi
(Z)\mathbbm{1}_{n} \\[1mm]
&\qquad+(C\overline{Z}+D)^{-t}(Z-\overline{Z})\dpd{}{Z}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)(CZ+D)^{t}.
\end{align}
Now, focusing on the second line of the above equality and using Lemma~\ref{gen determinant differentiate}, we get
\begin{align*}
&\dpd{}{Z}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\bigg(\alpha\varphi(Z)C^{t}(CZ+D)^{-t}+\dpd{\varphi}{Z}\bigg).
\end{align*}
Multiplying the above equation from the left by $(Z-\overline{Z})$ gives
\begin{align*}
&(Z-\overline{Z})\dpd{}{Z}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\bigg(\alpha\varphi(Z)(ZC^{t}-\overline{Z}C^{t})(CZ+D)^{-t}+(Z-\overline{Z})\dpd{\varphi}{Z}\bigg).
\end{align*}
Now writing $(ZC^{t}-\overline{Z}C^{t})=(CZ+D)^{t}-(C\overline{Z}+D)^{t}$ and using the definition of $K_{\alpha}$ on the right-hand side of the above equation,
we have
\begin{align*}
&(Z-\overline{Z})\dpd{}{Z}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(C Z+D)^{\alpha}\det(C\overline{ Z}+D)^{\beta} \big(K_{\alpha}\varphi( Z)-\alpha\varphi( Z)(C\overline{ Z}+D)^t(C Z+D)^{-t}\big).
\end{align*}
Therefore, multiplying on the left by $(C\overline{Z}+D)^{-t}$ and on the right by $(CZ+D)^{t}$, we obtain
\begin{align*}
&(C\overline{Z}+D)^{-t}(Z-\overline{Z})\dpd{}{Z}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)(CZ+D)^{t} \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\big((C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t}-\alpha\varphi(Z)\mathbbm{1}_{n}\big).
\end{align*}
Combining the last equality with equation~\eqref{longKstar}, leads to the identity
\begin{align*}
&K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(C\overline{Z}+D)^{-t}{K}_{\alpha}\varphi(Z)(CZ+D)^{t},
\end{align*}
which is what we had set out to prove.
\end{proof}
\begin{proposition}
\label{Ltransformation}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$ and $\varphi\colon\mathbb{H}_{n}\longrightarrow
\mathbb{C}$ be a smooth function. Then, the operator $\Lambda^{\gamma}_{\beta}$ obtained by replacing $Z\in\mathbb{H}_{n}$ in ${\Lambda}_{\beta}$ by
$Z^{\gamma}=\gamma Z$ is related to the operator ${\Lambda}_{\beta}$ by the identity
\begin{align*}
&\Lambda^{\gamma}_{\beta}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t}{\Lambda}_{\beta}\varphi(Z)(C\overline{Z}+D)^{t}.
\end{align*}
\end{proposition}
\begin{proof}
Since $\overline{K}_{\beta}=-\Lambda_{\beta}$, the required identity follows from Proposition~\ref{Ktransformation} by complex conjugation.
\end{proof}
\begin{proposition}
\label{Omegatransform}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$ and $\varphi\colon\mathbb{H}_{n}\longrightarrow
\mathbb{C}$ be a smooth function. Then, the operator $\Omega^{\gamma}_{\alpha,\beta}$ obtained by replacing $Z\in\mathbb{H}_{n}$ in ${\Omega}_{\alpha,
\beta}$ by $Z^{\gamma}=\gamma Z$ is related to the operator ${\Omega}_{\alpha,\beta}$ by the identity
\begin{align}
\label{Omegatransformeq}
\notag
&\Omega^{\gamma}_{\alpha,\beta}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t}{\Omega}_{\alpha,\beta}\varphi(Z)(CZ+D)^{t}.
\end{align}
\end{proposition}
\begin{proof}
To prove the proposition, we first need to calculate
\begin{align*}
\dpd{}{{\overline{Z}^{\gamma}}}\big(K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)\big),
\end{align*}
which, upon expanding $\partial/\partial\overline{Z}^{\gamma}$ by means of equation~\eqref{partial transform 2} and using Proposition~\ref{Ktransformation},
becomes
\begin{align*}
(C\overline{Z}+D)\bigg((C\overline{Z}+D)\dpd{}{\overline{Z}}\bigg)^{t}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(C\overline{Z}+D)^{-t}{K}_{\alpha}
\varphi(Z)(CZ+D)^{t}\big).
\end{align*}
Using Lemma~\ref{gen determinant differentiate}, the above expression becomes
\begin{align*}
&\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\bigg(\beta(C\overline{Z}+D)C^{t}(C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t} \\
&\qquad+(C\overline{Z}+D)\bigg((C\overline{Z}+D)\dpd{}{\overline{Z}}\bigg)^{t}\big((C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t}\big)\bigg).
\end{align*}
Now, using Lemma~\ref{operator derivative identity general}~(ii), we have
\begin{align*}
& \bigg((C\overline{Z}+D)\dpd{}{\overline{ Z}}\bigg)^{t}\big((C\overline{Z}+D)^{-t}{K}_{\alpha}\varphi(Z)(CZ+D)^{t}\big) \\
&\qquad=\dpd{}{\overline{ Z}}\big({K}_{\alpha}\varphi(Z)\big)(CZ+D)^{t}-\frac{1}{2}(n+1)C^{t}(C\overline{Z}+D)^{-t}{K}_{\alpha}\varphi(Z)(CZ+D)^{t},
\end{align*}
and thus arrive from the above calculation at
\begin{align*}
&\dpd{}{{\overline{Z}^{\gamma}}}\big(K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)\big) \\
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\bigg((\beta-(n+1)/2)(C\overline{Z}+D)C^{t}(C\overline{Z}+D)^{-t}\times \\
&\qquad\quad\times K_{\alpha}\varphi(Z)(CZ+D)^{t}+(C\overline{Z}+D)\dpd{}{\overline{Z}}\big(K_{\alpha}\varphi(Z)\big)(CZ+D)^{t}\bigg).
\end{align*}
Then, multiplying both sides from the left by $(Z^{\gamma}-\overline{Z}^{\gamma})$ and using
\begin{align*}
(Z^{\gamma}-\overline{Z}^{\gamma})=(C{Z}+D)^{-t}(Z-\overline{Z})(C\overline{Z}+D)^{-1}
\end{align*}
on the right-hand side, we get
\begin{align*}
&(Z^{\gamma}-\overline{Z}^{\gamma})\dpd{}{{\overline{Z}^{\gamma}}}\big(K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi
(Z)\big)\big) \\
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\bigg((\beta-(n+1)/2)(CZ+D)^{-t}(Z-\overline{Z})C^{t}(C\overline{Z}+D)^{-t}\times \\
&\qquad\quad\times K_{\alpha}\varphi(Z)(CZ+D)^{t}+(CZ+D)^{-t}(Z-\overline{Z})\dpd{}{\overline{Z}}\big({K}_{\alpha}\varphi(Z)\big)(CZ+D)^{t}\bigg).
\end{align*}
Next, writing the expression $(Z-\overline{Z})C^{t}$ on the right-hand side of the above equation as $(CZ+D)^{t}-(C\overline{Z}+D)^{t}$, we can rewrite the
above equation as
\begin{align*}
&(Z^{\gamma}-\overline{Z}^{\gamma})\dpd{}{{\overline{Z}^{\gamma}}}\big(K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi
(Z)\big)\big)=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\times \\
&\qquad\times\bigg((\beta-(n+1)/2)\big((C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t}-(CZ+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)^{t}\big) \\
&\qquad\quad+(CZ+D)^{-t}(Z-\overline{Z})\dpd{}{\overline{Z}}\big(K_{\alpha}\varphi(Z)\big)(CZ+D)^{t}\bigg).
\end{align*}
Now shifting the first term on the right-hand side to the left and using the transformation behaviour of $K_{\alpha}$ derived in Proposition~\ref{Ktransformation},
we arrive at
\begin{align*}
&\bigg((Z^{\gamma}-\overline{Z}^{\gamma})\dpd{}{{\overline{Z}^{\gamma}}}-(\beta-(n+1)/2)\mathbbm{1}_{n}\bigg)\big(K^{\gamma}_{\alpha}\big(\det(CZ+D)^
{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)\big) \\
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t} \bigg((Z-\overline{Z})\dpd{}{\overline{Z}}-(\beta-(n+1)/2)\mathbbm{1}_{n}\bigg)\times
\\[1mm]
&\qquad\quad\times K_{\alpha}\varphi(Z)(CZ+D)^{t},
\end{align*}
which, by Definition~\ref{maass operators} and the transformation behaviour of $\Lambda_{\beta}$ given in Proposition~\ref{Ltransformation}, gives
\begin{align*}
&\Lambda^{\gamma}_{\beta-(n+1)/2}K^{\gamma}_{\alpha}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t}\Lambda_{\beta-(n+1)/2}K_{\alpha}\varphi(Z)(CZ+D)^{t},
\end{align*}
which, by definition of $\Omega^{\gamma}_{\alpha,\beta}$, yields the desired identity.
\end{proof}
\begin{remark}
\label{functional equation}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$ and $\varphi\colon\mathbb{H}_{n}\longrightarrow
\mathbb{C}$ be a smooth function. Taking traces on both sides of equation~\eqref{Omegatransformeq} leads to the following transformation behaviour of
the Siegel--Maa\ss{} Laplacian $\Delta_{\alpha,\beta}$
\begin{align*}
\Delta^{\gamma}_{\alpha,\beta}\big(\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)\big)=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\Delta_
{\alpha,\beta}\varphi(Z).
\end{align*}
Now, if the smooth function $\varphi$ satisfies the functional equation
\begin{align*}
\varphi(Z^{\gamma})=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z),
\end{align*}
the transformation behaviour of $\Delta_{\alpha,\beta}$ leads to the identity
\begin{align*}
\Delta^{\gamma}_{\alpha,\beta}\varphi(Z^{\gamma})=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\Delta_{\alpha,\beta}\varphi(Z).
\end{align*}
\end{remark}
\begin{definition}
\label{siegel-maass definition}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$, i.e., the intersection $\Gamma\cap
\mathrm{Sp}_{n}(\mathbb{Z})$ is a finite index subgroup of $\Gamma$ as well as of $\mathrm{Sp}_{n}(\mathbb{Z})$. We let $\gamma_{j}\in\mathrm{Sp}_{n}
(\mathbb{Z})$ ($j=1,\ldots,h$) denote a set of representatives for the left cosets of $\Gamma\cap\mathrm{Sp}_{n}(\mathbb{Z})$ in $\mathrm{Sp}_{n}(\mathbb
{Z})$. We then let $\mathcal{V}_{\alpha,\beta}^{n}(\Gamma)$ denote the space of all functions $\varphi\colon\mathbb{H}_{n}\longrightarrow\mathbb{C}$
satisfying the following conditions:
\begin{itemize}
\item[(i)]
$\varphi$ is real-analytic;
\item[(ii)]
$\varphi(\gamma Z)=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)$ for all $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in
\Gamma$;
\item[(iii)]
given $Y_{0}\in\mathrm{Sym}_{n}(\mathbb{R})$ with $Y_{0}>0$, there exist $M\in\mathbb{R}_{>0}$ and $N\in\mathbb{N}$ such that the inequalities
\begin{align*}
\vert\det(C_{j}Z+D_{j})^{-\alpha}\det(C_{j}\overline{Z}+D_{j})^{-\beta}\varphi(\gamma_{j} Z)\vert\leq M\tr(Y)^{N}
\end{align*}
hold in the region $\{Z=X+iY\in\mathbb{H}_{n}\,\vert\,Y\geq Y_{0}\}$ for the set of representatives $\gamma_{j}=\big(\begin{smallmatrix}A_{j}&B_{j}\\C_{j}&
D_{j}\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{Z})$ ($j=1,\ldots,h$).
\end{itemize}
\end{definition}
\begin{remark}
\label{boundedness}
For $\varphi\in\mathcal{V}_{\alpha,\beta}^{n}(\Gamma)$, we set
\begin{align*}
\Vert{\varphi}\Vert^{2}:=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\vert\varphi(Z)\vert^{2}\,\mathrm{d}\mu_{n}(Z),
\end{align*}
whenever it is defined. In this way we obtain the Hilbert space
\begin{align*}
\mathcal{H}_{\alpha,\beta}^{n}(\Gamma):=\big\{\varphi\in\mathcal{V}_{\alpha,\beta}^{n}(\Gamma)\,\big\vert\,\Vert\varphi\Vert<\infty\big\}
\end{align*}
equipped with the inner product
\begin{align*}
\langle\varphi,\psi\rangle=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\varphi(Z)\overline{\psi}(Z)\,\mathrm{d}\mu_{n}(Z)\qquad(\varphi,
\psi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)).
\end{align*}
We note that in order to enable $\Vert\varphi\Vert<\infty$, the exponent $N\in\mathbb{N}$ in part (iii) of Definition~\ref{siegel-maass definition} has to be~$0$.
Moreover, we note that Remark~\ref{functional equation} shows that the Siegel--Maa\ss{} Laplacian $\Delta_{\alpha,\beta}$ acts on the Hilbert space $\mathcal
{H}_{\alpha,\beta}^{n}(\Gamma)$.
\end{remark}
\begin{definition}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$. The elements of the Hilbert space
$\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ are called \emph{automorphic forms of weight $(\alpha,\beta)$ and degree $n$ for $\Gamma$}. Moreover, if
$\varphi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ is an eigenform of $\Delta_{\alpha,\beta}$, it is called a \emph{Siegel--Maa\ss{} form of weight $(\alpha,
\beta)$ and degree $n$ for $\Gamma$}.
\end{definition}
\begin{corollary}
\label{maass form transform}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$ and $\varphi\in\mathcal{H}_{\alpha,
\beta}^{n}(\Gamma)$. Then, we have for all $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$
\begin{align*}
\mathrm{(i)}&\quad K^{\gamma}_{\alpha}\varphi(Z^{\gamma})=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(C\overline{Z}+D)^{-t}K_{\alpha}\varphi(Z)(CZ+D)
^{t}, \\[1mm]
\mathrm{(ii)}&\quad\Lambda^{\gamma}_{\beta}\varphi(Z^{\gamma})=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t}{\Lambda}_{\beta}\varphi(Z)(C
\overline{Z}+D)^{t}, \\[1mm]
\mathrm{(iii)}&\quad\Omega^{\gamma}_{\alpha,\beta}\varphi(Z^{\gamma})=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}(CZ+D)^{-t}{\Omega}_{\alpha,\beta}
\varphi( Z)(CZ+D)^{t}.
\end{align*}
\end{corollary}
\begin{proof}
The proof is an immediate consequence of Propositions \ref{Ktransformation}--\ref{Omegatransform} and the definition of the Hilbert space $\mathcal{H}_{\alpha,
\beta}^{n}(\Gamma)$.
\end{proof}
\section{Symmetry of the Siegel--Maa\ss{} Laplacian of weight $(\alpha,\beta)$}
Let $\mathrm{d}Z:=(\mathrm{d}z_{j,k})_{1\leq j,k\leq n}$ denote the $(n\times n)$-matrix consisting of differential forms of degree $1$ and let $[\mathrm{d}Z]:=
\bigwedge_{1\leq j\leq k\leq n}\mathrm{d}z_{j,k}$ denote the differential form of degree $n(n+1)/2$ at $Z\in\mathbb{H}_{n}$. We introduce an $(n\times n)$-matrix $
\{\mathrm{d}Z\}$ consisting of differential forms of degree $(n(n+1)/2-1)$, namely
\begin{align*}
\{\mathrm{d}Z\}_{j,k}:=\frac{1+\delta_{j,k}}{2}\,\varpi_{j,k},
\end{align*}
where $\varpi_{j,k}$ is defined by
\begin{align*}
\varpi_{j,k}:=\varepsilon_{j,k}\bigwedge\limits_{\substack{1\leq l\leq m \leq n\\(l,m)\neq(j,k)}}\mathrm{d}z_{l,m}\qquad(1\leq j\leq k\leq n)
\end{align*}
in case $j\leq k$ and $\varpi_{j,k}=\varpi_{k,j}$ in case $j>k$ with the sign $\varepsilon_{j,k}=\pm1$ determined by $\mathrm{d}z_{j,k}\wedge\varpi_{j,k}=[\mathrm
{d}Z]$. It is easy to see that
\begin{align*}
\mathrm{d}Z\wedge\{\mathrm{d}Z\}=\frac{1}{2}(n+1)[\mathrm{d}Z]\mathbbm{1}_{n}.
\end{align*}
Let now $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{R})$. Since we have $\mathrm{d}Z^{\gamma}=(CZ+D)^{-t}
\,\mathrm{d}Z\,(CZ+D)^{-1}$ and $[\mathrm{d}Z^{\gamma}]=\linebreak\det(CZ+D)^{-(n+1)}[\mathrm{d}Z]$, we derive from the relation
\begin{align*}
\mathrm{d}Z^{\gamma}\wedge\{\mathrm{d}Z^{\gamma}\}=\frac{1}{2}(n+1)[\mathrm{d}Z^{\gamma}]\mathbbm{1}_{n}
\end{align*}
that the matrix $\{\mathrm{d}Z\}$ has the transformation behaviour
\begin{align*}
\{\mathrm{d}Z^{\gamma}\}=\det(C Z+D)^{-(n+1)}(CZ+D)\{\mathrm{d}Z\}(CZ+D)^{t}.
\end{align*}
Next we shall use these differential forms to show that the Siegel--Maa\ss{} Laplacian $\Delta_{\alpha,\beta}$ acts as a symmetric operator on the Hilbert space
$\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$.
\begin{theorem}
\label{main theorem}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$ and let $\varphi,\psi\in\mathcal{H}_{\alpha,
\beta}^{n}(\Gamma)$ be compactly supported. Then, we have the formula
\begin{align*}
\langle-\Delta_{\alpha,\beta}\varphi,\psi\rangle=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\Lambda}_
{\beta}\overline{\psi}(Z)\big)\,\mathrm{d}{\mu}_{n}(Z)+n\beta(\alpha-(n+1)/2)\langle\varphi,\psi\rangle.
\end{align*}
In particular, this formula establishes the relation
\begin{align*}
\langle\Delta_{\alpha,\beta}\varphi,\psi\rangle=\langle\varphi,\Delta_{\alpha,\beta}\psi\rangle,
\end{align*}
which shows that the Siegel--Maa\ss{} Laplacian $\Delta_{\alpha,\beta}$ acts as a symmetric operator on the Hilbert space $\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$.
\end{theorem}
\begin{proof}
We start by considering the differential form
\begin{align*}
\omega(Z):=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\overline{\psi}(Z)\tr\big(\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\{\mathrm{d}Z\}\big)\wedge[\mathrm{d}\overline
{Z}].
\end{align*}
Let $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$. Then, the transformation formulas
\begin{align*}
\mathrm{(a})&\quad\det(Z^{\gamma}-\overline{Z}^{\gamma})^{\alpha+\beta-(n+1)} \\
&\qquad=\det(CZ+D)^{-(\alpha+\beta-(n+1))}\det(C\overline{Z}+D)^{-(\alpha+\beta-(n+1))}\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}, \\[1mm]
\mathrm{(b})&\quad\overline{\psi}(Z^{\gamma})=\det(CZ+D)^{\beta}\det(C\overline{Z}+D)^{\alpha}\overline{\psi}(Z), \\[1mm]
\mathrm{(c})&\quad\tr\big(\Lambda^{\gamma}_{\beta}\varphi(Z^{\gamma})(Z^{\gamma}-\overline{Z}^{\gamma})\{\mathrm{d}Z^{\gamma}\}\big) \\[1mm]
&\qquad=\det(CZ+D)^{\alpha-(n+1)}\det(C\overline{Z}+D)^{\beta}\tr\big(\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\{\mathrm{d}Z\}\big), \\[1mm]
\mathrm{(d})&\quad[\mathrm{d}\overline{Z}^{\gamma}]=\det(C\overline{Z}+D)^{-(n+1)}[\mathrm{d}\overline{Z}]
\end{align*}
show that $\omega(Z^{\gamma})=\omega(Z)$ for all $\gamma\in\Gamma$, i.e., $\omega(Z)$ is a $\Gamma$-invariant differential form on $\mathbb{H}_{n}$, and
hence can be considered as a differential form on the quotient space $\Gamma\backslash\mathbb{H}_{n}$. Since the automorphic forms $\varphi,\psi$ are
real-analytic, the differential form $\omega$ is a smooth differential form. Therefore, by Stokes' theorem, we have
\begin{align*}
\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\mathrm{d}\omega(Z)=\int\limits_{\partial\Gamma\backslash\mathbb{H}_{n}}\omega(Z).
\end{align*}
As $\varphi,\psi$ are compactly supported, the integral on the right-hand side of the above equation vanishes, which gives
\begin{align}
\label{int zero}
\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\mathrm{d}\omega(Z)=0.
\end{align}
As we shall see, by explicitly computing $\mathrm{d}\omega(Z)$, the vanishing of the above integral will lead to the formula claimed in the theorem.
For the computation of $\mathrm{d}\omega(Z)$, we set $\rho:=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\overline{\psi}(Z)$, $P:=\Lambda_{\beta}\varphi(Z)$, and
$Q:=(Z-\overline{Z})$. Then, we obtain
\begin{align*}
\omega(Z)=\rho\tr(P\,Q\,\{\mathrm{d}Z\})\wedge[\mathrm{d}\overline{Z}]=\sum\limits_{j,k,l=1}^{n}\rho\,p_{j,k}\,q_{k,l}\,\{\mathrm{d}Z\}_{l,j}\wedge[\mathrm{d}\overline
{Z}].
\end{align*}
Taking exterior derivatives on both sides leads to
\begin{align}
\notag
\mathrm{d}\omega(Z)&=\sum\limits_{j,k,l=1}^{n}\dpd{}{z_{l,j}}(\rho\,p_{j,k}\,q_{k,l})\,\mathrm{d}z_{l,j}\wedge\frac{1+\delta_{l,j}}{2}\varpi_{l,j}\wedge[\mathrm{d}\overline
{Z}] \\
\notag
&=\sum\limits_{j,k,l=1}^{n}\frac{1+\delta_{l,j}}{2}\dpd{}{z_{l,j}}(\rho\,p_{j,k}\,q_{k,l})\,[\mathrm{d}Z]\wedge[\mathrm{d}\overline{Z}] \\
\label{dabba}
&=\sum\limits_{j,k,l=1}^{n}\bigg(\dpd{}{Z}\bigg)_{l,j}(\rho\,p_{j,k}\,q_{k,l})\,[\mathrm{d}Z]\wedge[\mathrm{d}\overline{Z}].
\end{align}
Now a term by term differentiation in the last expression on the right-hand side of the above equation allows us to write it as the sum of the three traces
\begin{align}
\label{rhopq0}
\sum\limits_{j,k,l=1}^{n}\bigg(\dpd{}{Z}\bigg)_{l,j}(\rho\,p_{j,k}\,q_{k,l})=\tr\bigg(\dpd{\rho}{Z}P\,Q\bigg)+\rho\tr\bigg(\dpd{}{Z}P\,Q\bigg)+\rho\tr\bigg(P^t\dpd{}{Z}Q\bigg),
\end{align}
which we calculate one by one next.
\noindent
(i) We begin by considering
\begin{align*}
\dpd{\rho}{Z}=\dpd{}{Z}\big(\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\overline{\psi}(Z)\big),
\end{align*}
which, by Lemma~\ref{determinant differentiate}, calculates to
\begin{align*}
\dpd{\rho}{Z}=(\alpha+\beta-(n+1))\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}(Z-\overline{Z})^{-1}\,\overline{\psi}(Z)+\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\dpd
{\overline{\psi}(Z)}{Z}.
\end{align*}
Now multiplying both sides of the above equation on the right by $P\,Q=\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})$ and taking the trace gives
\begin{align*}
\tr\bigg(\dpd{\rho}{Z}P\,Q\bigg)&=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\bigg((\alpha+\beta-(n+1))\tr\big((Z-\overline{Z})^{-1}\overline{\psi}(Z)\Lambda_{\beta}
\varphi(Z)(Z-\overline{Z})\big) \\
&\hspace*{35mm}+\tr\bigg(\dpd{\overline{\psi}(Z)}{Z}\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\bigg)\bigg),
\end{align*}
which, upon rearranging the terms inside the traces on the right-hand side by cyclically permuting them, becomes
\begin{align}
\notag
&\tr\bigg(\dpd{\rho}{Z}P\,Q\bigg)=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\bigg((\alpha+\beta-(n+1))\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\psi}(Z)\big) \\
\label{rhopq1}
&\hspace*{57mm}+\tr\bigg(\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\dpd{\overline{\psi}(Z)}{Z}\bigg)\bigg).
\end{align}
\noindent
(ii) Next, we consider the second trace
\begin{align*}
\rho\tr\bigg(\dpd{}{Z}P\,Q\bigg)=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\overline{\psi}(Z)\tr\bigg(\dpd{}{Z}\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\bigg)
\end{align*}
in equation~\eqref{rhopq0}, which, again through rearrangement of the terms inside the trace by a cyclical permutation, takes the form
\begin{align}
\label{rhopq2}
\rho\tr\bigg(\dpd{}{Z}P\,Q\bigg)=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\tr\bigg((Z-\overline{Z})\dpd{}{Z}\Lambda_{\beta}\varphi(Z)\overline{\psi}(Z)\bigg).
\end{align}
\noindent
(iii) Finally, we consider the third trace
\begin{align*}
\rho\tr\bigg(P^{t}\dpd{}{Z}Q\bigg)=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\overline{\psi}(Z)\tr\bigg(\big(\Lambda_{\beta}\varphi(Z)\big)^{t}\bigg(\dpd{}{Z}(Z-\overline
{Z})\bigg)\mathbbm{1}_{n}\bigg)
\end{align*}
in equation~\eqref{rhopq0}. By the first operator identity in Corollary~\ref{operator derivative identity}, we have the matrix identity
\begin{align*}
\bigg(\dpd{}{Z}(Z-\overline{Z})\bigg)\mathbbm{1}_{n}=\bigg((Z-\overline{Z})\dpd{}{Z}\bigg)^{t}\mathbbm{1}_{n}+\frac{1}{2}(n+1)\mathbbm{1}_{n}=\frac{1}{2}(n+1)
\mathbbm{1}_{n},
\end{align*}
which gives, upon rearrangement of the scalar quantities, the identity
\begin{align}
\label{rhopq3}
\rho\tr\bigg(P^{t}\dpd{}{Z}Q\bigg)=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\frac{1}{2}(n+1)\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\psi}(Z)\big).
\end{align}
\noindent
Now, adding up equations \eqref{rhopq1}--\eqref{rhopq3}, it follows from equation~\eqref{rhopq0} that
\begin{align*}
&\sum\limits_{j,k,l=1}^{n}\bigg(\dpd{}{Z}\bigg)_{l,j}(\rho\,p_{j,k}\,q_{k,l})=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\bigg((\alpha+\beta-(n+1)/2)\tr\big(\Lambda_{\beta}
\varphi(Z)\overline{\psi}(Z)\big) \\
&\hspace*{35mm}+\tr\bigg(\Lambda_{\beta}\varphi(Z)(Z-\overline{Z})\dpd{\overline{\psi}(Z)}{Z}\bigg)+\tr\bigg((Z-\overline{Z})\dpd{}{Z}\Lambda_{\beta}\varphi(Z)
\overline{\psi}(Z)\bigg)\bigg).
\end{align*}
Rearranging terms on the right-hand side of the last expression, leads to
\begin{align*}
\sum\limits_{j,k,l=1}^{n}\bigg(\dpd{}{Z}\bigg)_{l,j}(\rho\,p_{j,k}\,q_{k,l})&=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\bigg(\tr\bigg(\Lambda_{\beta}\varphi(Z)\bigg
((Z-\overline{Z})\dpd{}{Z}+\beta\mathbbm{1}_{n}\bigg)\overline{\psi}(Z)\bigg) \\
&\quad+\tr\bigg((Z-\overline{Z})\dpd{}{Z}+(\alpha-(n+1)/2)\mathbbm{1}_{n}\bigg)\Lambda_{\beta}\varphi(Z)\overline{\psi}(Z)\bigg).
\end{align*}
Identifying the operator $(Z-\overline{Z})\partial/\partial Z+\beta\mathbbm{1}_{n}$ on the right-hand side of the above equation as $-\overline{\Lambda}_{\beta}$
and the operator $(Z-\overline{Z})\partial/\partial Z+(\alpha-(n+1)/2)\mathbbm{1}_{n}$ as $K_{\alpha-(n+1)/2}$, we can rewrite the right-hand side of the above
equation as
\begin{align*}
\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\Big(-\tr\big(\Lambda_{\beta}\varphi(Z)\,\overline{\Lambda}_{\beta}\overline{\psi}(Z)\big)+\tr\big(K_{\alpha-(n+1)/2}\,\Lambda_
{\beta}\varphi(Z)\overline{\psi}(Z)\big)\Big),
\end{align*}
which, by definition of $\widetilde{\Omega}_{\alpha,\beta}$, is equal to
\begin{align*}
\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\Big(\tr\big(\widetilde{\Omega}_{\alpha,\beta}-\beta(\alpha-(n+1)/2)\mathbbm{1}_{n}\big)\varphi(Z)\overline{\psi}(Z)-\tr\big
(\Lambda_{\beta}\varphi(Z)\overline{\Lambda}_{\beta}\overline{\psi}(Z)\big)\Big).
\end{align*}
In total, we get
\begin{align*}
\sum\limits_{j,k,l=1}^{n}\bigg(\dpd{}{Z}\bigg)_{l,j}(\rho\,p_{j,k}\,q_{k,l})&=\det(Z-\overline{Z})^{\alpha+\beta-(n+1)}\Big(-\Delta_{\alpha,\beta}\,\varphi(Z)\overline{\psi}
(Z)-\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\Lambda}_{\beta}\overline{\psi}(Z)\big) \\
&\quad-n\beta(\alpha-(n+1)/2)\,\varphi(Z)\overline{\psi}(Z)\Big).
\end{align*}
Thus, substituting $\sum\limits_{j,k,l=1}^{n}(\partial/\partial Z)_{l,j}(\rho\,p_{j,k}\,q_{k,l})$ back into equation \eqref{dabba}, we arrive at
\begin{align*}
\mathrm{d}\omega(Z)&=\det(Z-\overline{Z})^{\alpha+\beta}\Big(-\Delta_{\alpha,\beta}\,\varphi(Z)\overline{\psi}(Z)-\tr\big(\Lambda_{\beta}\varphi(Z)\overline
{\Lambda}_{\beta}\overline{\psi}(Z)\big) \\[1mm]
&\quad-n\beta(\alpha-(n+1)/2)\,\varphi(Z)\overline{\psi}(Z)\Big)\frac{[\mathrm{d}Z]\wedge[\mathrm{d}\overline{Z}]}{\det(Z-\overline{Z})^{n+1}}.
\end{align*}
Now, noting that the volume form
\begin{align*}
\det(Z-\overline{Z})^{\alpha+\beta}\frac{[\mathrm{d}Z]\wedge[\mathrm{d}\overline{Z}]}{\det(Z-\overline{Z})^{n+1}}
\end{align*}
is just a constant multiple of $\det(Y)^{\alpha+\beta}\mathrm{d}\mu_{n}(Z)$, it follows readily from the vanishing result~\eqref{int zero} that
\begin{align*}
\langle-\Delta_{\alpha,\beta}\varphi,\psi\rangle=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\Lambda}_
{\beta}\overline{\psi}(Z)\big)\,\mathrm{d}{\mu}_{n}(Z)+n\beta(\alpha-(n+1)/2)\langle\varphi,\psi\rangle,
\end{align*}
which is the claimed formula.
Using the latter formula, we compute
\begin{align*}
\langle\varphi,-\Delta_{\alpha,\beta}\psi\rangle&=\overline{\langle-\Delta_{\alpha,\beta}\psi,\varphi\rangle} \\
&=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\,\overline{\tr\big(\Lambda_{\beta}\psi(Z)\overline{\Lambda}_{\beta}\overline{\varphi}(Z)\big)}\,
\mathrm{d}\mu_{n}( Z)+n\beta(\alpha-(n+1)/2)\overline{\langle\psi,\varphi\rangle} \\
&=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\tr\big(\Lambda_{\beta}\varphi(Z)\overline{\Lambda}_{\beta}\overline{\psi}(Z)\big)\,\mathrm{d}
\mu_{n}(Z)+n\beta(\alpha-(n+1)/2)\langle\varphi,\psi\rangle \\[1mm]
&=\langle-\Delta_{\alpha,\beta}\varphi,\psi\rangle,
\end{align*}
which proves the claimed symmetry of the Siegel--Maa\ss{} Laplacian $\Delta_{\alpha,\beta}$.
\end{proof}
\begin{corollary}
\label{characterization}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$ and let $\varphi\in\mathcal{H}_{\alpha,
\beta}^{n}(\Gamma)$ be a Siegel--Maa\ss{} form of weight $(\alpha,\beta)$ and degree $n$ for $\Gamma$. Then, if $\varphi$ is an eigenform of $\Delta_{\alpha,
\beta}$ with eigenvalue $\lambda$, we have $\lambda\in\mathbb{R}$ and $\lambda\geq n\beta(\alpha-(n+1)/2)$.
Furthermore, $\varphi$ has eigenvalue $\lambda=\beta(\alpha-(n+1)/2)$ if and only if $\varphi(Z)=\det(Y)^{-\beta}f(Z)$, where $f\colon\mathbb{H}_{n}\longrightarrow
\mathbb{C}$ is a holomorphic function satisfying
\begin{align*}
f(\gamma Z)=\det(CZ+D)^{\alpha-\beta} f(Z)
\end{align*}
for all $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$. Moreover, if $\beta<0$, then $f$ is a Siegel cusp form of weight $\alpha-
\beta$ and degree $n$ for~$\Gamma$.
\end{corollary}
\begin{proof}
Since $\varphi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ is an eigenform of $\Delta_{\alpha,\beta}$ with eigenvalue $\lambda$, i.e., we have $(\Delta_{\alpha,\beta}+
\lambda\,\mathrm{id})\varphi=0$, we compute using Theorem~\ref{main theorem}
\begin{align*}
\lambda\langle\varphi,\varphi\rangle&=\langle-\Delta_{\alpha,\beta}\varphi,\varphi\rangle \\
&=\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\tr\big(\vert\Lambda_{\beta}\varphi(Z)\vert^{2}\big)\,\mathrm{d}\mu_{n}(Z)+n\beta(\alpha-(n+1)/2)
\langle\varphi,\varphi\rangle.
\end{align*}
This immediately implies that $\lambda\in\mathbb{R}$. Furthermore, since $\tr(\vert\Lambda_{\beta}\varphi(Z)\vert^{2})\geq 0$, we conclude that
\begin{align*}
\lambda\geq n\beta(\alpha-(n+1)/2).
\end{align*}
To prove the second part of the corollary, we observe that the above equation shows that the equality $\lambda=n\beta(\alpha-(n+1)/2)$ is equivalent to
\begin{align*}
\int\limits_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{\alpha+\beta}\tr\big(\vert\Lambda_{\beta}\varphi(Z)\vert^{2}\big)\,\mathrm{d}\mu_{n}(Z)=0.
\end{align*}
Since $\tr(\vert\Lambda_{\beta}\varphi(Z)\vert^{2})\geq 0$, the above integral vanishes if and only if $\tr(\vert\Lambda_{\beta}\varphi(Z)\vert^{2})=0$. Now, as
the matrix
\begin{align*}
\Lambda_{\beta}\varphi(Z)=(Z-\overline{Z})\dpd{\varphi}{\overline{Z}}-\beta\varphi(Z)\mathbbm{1}_{n}
\end{align*}
is similar to the complex symmetric matrix
\begin{align*}
S(Z):=2i\,Y^{1/2}\dpd{\varphi}{\overline{Z}}Y^{1/2}-\beta\varphi(Z)\mathbbm{1}_{n},
\end{align*}
as we have the relation $\Lambda_{\beta}\varphi(Z)=Y^{1/2}S(Z)Y^{-1/2}$, the matrix $\vert\Lambda_{\beta}\varphi(Z)\vert^{2}$ becomes similar to the positive
semidefinite hermitian matrix $S(Z)\overline{S}(Z)$, which is diagonalizable with non-negative real eigenvalues. Therefore, the condition $\tr(S(Z)\overline{S}(Z))=
\tr(\vert\Lambda_{\beta}\varphi(Z)\vert^{2})=0$ is equivalent to the vanishing of all the eigenvalues of $S(Z)\overline{S}(Z)$, which is equivalent to the vanishing
of $S(Z)$ and hence of $\Lambda_{\beta}\varphi(Z)$. All in all, this proves that the equality $\lambda=n\beta(\alpha-(n+1)/2)$ is equivalent to the vanishing condition
$\Lambda_{\beta}\varphi=0$.
Continuing, we now set $f(Z):=\det(Y)^{\beta}\varphi(Z)$, and compute
\begin{align*}
\dpd{f}{\overline{Z}}=\beta\det(Y)^{\beta-1}\dpd{\det(Y)}{\overline{Z}}\varphi(Z)+\det(Y)^{\beta}\dpd{\varphi}{\overline{Z}}.
\end{align*}
Since we have
\begin{align*}
\dpd{\det(Y)}{\overline{Z}}=\frac{1}{2}\bigg(\dpd{}{X}+i\dpd{}{Y}\bigg)\det(Y)=\frac{i}{2}\dpd{\det(Y)}{Y}=\frac{i}{2}\det(Y)Y^{-1},
\end{align*}
the above equality becomes
\begin{align*}
\dpd{f}{\overline{Z}}&=\frac{i\beta}{2}\det(Y)^{\beta}Y^{-1}\varphi(Z)+\det(Y)^{\beta}\dpd{\varphi}{\overline{Z}} \\
&=-\frac{i}{2}\det(Y)^{\beta}Y^{-1}\bigg(-\beta\varphi(Z)\mathbbm{1}_{n}+2iY\dpd{\varphi}{\overline{Z}}\bigg) \\
&=-\frac{i}{2}\det(Y)^{\beta}Y^{-1}\bigg((Z-\overline{Z})\dpd{\varphi}{\overline{Z}}-\beta\varphi(Z)\mathbbm{1}_{n}\bigg) \\
&=-\frac{i}{2}\det(Y)^{\beta}Y^{-1}\Lambda_{\beta}\varphi(Z).
\end{align*}
In total, this shows that $\partial f/\partial\overline{Z}=0$, i.e., the function $f$ is holomorphic, if and only if $\Lambda_{\beta}\varphi(Z)=0$, which, by the previous
argument, is equivalent to $\varphi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ being a Siegel--Maa\ss{} form with eigenvalue $\lambda=\beta(\alpha-(n+1)/2)$.
Furthermore, as the function $\varphi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ has the transformation behaviour
\begin{align*}
\varphi(\gamma Z)=\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z)
\end{align*}
for all $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$, the function $f(Z)=\det(Y)^{\beta}\varphi(Z)=\det(\im(Z))^{\beta}\varphi(Z)$
has the transformation behaviour
\begin{align*}
f(\gamma Z)&=\det(\im(\gamma Z))^{\beta}\varphi(\gamma Z) \\
&=\bigg(\frac{\det(\im(Z))}{\vert\det(CZ+D)\vert^{2}}\bigg)^{\beta}\det(CZ+D)^{\alpha}\det(C\overline{Z}+D)^{\beta}\varphi(Z) \\[1mm]
&=\det(CZ+D)^{\alpha-\beta}\det(\im(Z))^{\beta}\varphi(Z) \\[2mm]
&=\det(CZ+D)^{\alpha-\beta}f(Z),
\end{align*}
as claimed.
Finally, letting $\gamma_{j}=\big(\begin{smallmatrix}A_{j}&B_{j}\\C_{j}&D_{j}\end{smallmatrix}\big)\in\mathrm{Sp}_{n}(\mathbb{Z})$ ($j=1,\ldots,h$) be a set of
representatives for the left cosets of $\Gamma\cap\mathrm{Sp}_{n}(\mathbb{Z})$ in $\mathrm{Sp}_{n}(\mathbb{Z})$, Remark~\ref{boundedness} shows that
given $Y_{0}\in\mathrm{Sym}_{n}(\mathbb{R})$ with $Y_{0}>0$, the quantities
\begin{align*}
\vert\det(C_{j}Z+D_{j})^{-\alpha}\det(C_{j}\overline{Z}+D_{j})^{-\beta}\varphi(\gamma_{j} Z)\vert
\end{align*}
have to be bounded in the region $\{Z=X+iY\in\mathbb{H}_{n}\,\vert\,Y\geq Y_{0}\}$. Therefore, if $\beta<0$, this implies that given $Y_{0}\in\mathrm{Sym}_{n}
(\mathbb{R})$ with $Y_{0}\gg 0$, the quantities
\begin{align*}
\vert\det(C_{j}Z+D_{j})^{-(\alpha-\beta)}f(\gamma_{j} Z)\vert=\vert\det(C_{j}Z+D_{j})^{-\alpha}\det(C_{j}\overline{Z}+D_{j})^{-\beta}\det(\im(\gamma_{j} Z))^{\beta}
\varphi(\gamma_{j} Z)\vert
\end{align*}
will become arbitrarily small in the region $\{Z=X+iY\in\mathbb{H}_{n}\,\vert\,Y\geq Y_{0}\}$. In other words, $f$ is indeed a Siegel cusp form of weight $\alpha-
\beta$ and degree $n$ for $\Gamma$.
With all this, the proof of the corollary is complete.
\end{proof}
\begin{remark}
For $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$ and $\alpha=k/2$, $\beta=-k/2$ with
$k\in\mathbb{N}_{>0}$, we denote the Hilbert space $\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)$ simply by $\mathcal{H}_{k}^{n}(\Gamma)$. Similarly, we write
for the operator $\Omega_{\alpha,\beta}$ simply $\Omega_{k}$, which becomes
\begin{align*}
\Omega_{k}&=(Z-\overline{Z})\bigg((Z-\overline{Z})\dpd{}{\overline{Z}}\bigg)^{t}\dpd{}{Z}+\frac{k}{2}(Z-\overline{Z})\dpd{}{\overline{Z}}+\frac{k}{2}(Z-\overline{Z})
\dpd{}{Z} \\
&=-Y\bigg(\bigg(Y\dpd{}{X}\bigg)^{t}\dpd{}{X}+\bigg(Y\dpd{}{Y}\bigg)^{t}\dpd{}{Y}\bigg)+ikY\dpd{}{X}.
\end{align*}
Finally, we write for the operator $\Delta_{\alpha,\beta}$ simply $\Delta_{k}$ and call it the Siegel--Maa\ss{} Laplacian of weight $k$; it is given as
\begin{align*}
\Delta_{k}=\tr\bigg(Y\bigg(\bigg(Y\dpd{}{X}\bigg)^{t}\dpd{}{X}+\bigg(Y\dpd{}{Y}\bigg)^{t}\dpd{}{Y}\bigg)-ikY\dpd{}{X}\bigg).
\end{align*}
We note that the transformation behaviour of a Siegel--Maa\ss{} form $\varphi$ of weight $k$ and degree $n$ for $\Gamma$ takes the form
\begin{align*}
\varphi(\gamma Z)=\bigg(\frac{\det(CZ+D)}{\det(C\overline{Z}+D)}\bigg)^{k/2}\varphi(Z),
\end{align*}
where $\gamma=\big(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\big)\in\Gamma$.
\end{remark}
In the last corollary, we summarize the main results about Siegel--Maa\ss{} forms of weight $k$ and degree~$n$ for $\Gamma$.
\begin{corollary}
\label{kernel connection}
Let $\Gamma\subset\mathrm{Sp}_{n}(\mathbb{R})$ be a subgroup commensurable with $\mathrm{Sp}_{n}(\mathbb{Z})$ and let $\varphi\in\mathcal{H}_{k}^{n}
(\Gamma)$ be a Siegel--Maa\ss{} form of weight $k$ and degree $n$ for $\Gamma$. Then, if $\varphi$ is an eigenform of $\Delta_{k}$ with eigenvalue $\lambda$,
we have $\lambda\in\mathbb{R}$ and
\begin{align*}
\lambda\geq \frac{nk}{4}(n-k+1),
\end{align*}
with equality attained if and only if the function $\varphi$ is of the form $\varphi(Z)=\det(Y)^{k/2}f(Z)$ for some Siegel cusp form $f\in\mathcal{S}_{k}^{n}(\Gamma)$
of weight $k$ and degree $n$ for $\Gamma$. In other words, there is an isomorphism
\begin{align*}
\mathcal{S}_{k}^{n}(\Gamma)\cong\ker\bigg(\Delta_{k}+\frac{nk}{4}(n-k+1)\mathrm{id}\bigg)
\end{align*}
of $\mathbb{C}$-vector spaces, induced by the assignment $f\mapsto\det(Y)^{k/2}f$.
\end{corollary}
\begin{proof}
The proof is an immediate consequence of Corollray~\ref{characterization} by setting $\alpha=k/2$ and $\beta=-k/2$.
\end{proof}
\newpage
\bibliographystyle{amsplain}
|
2,877,628,089,474 | arxiv | \section{Algorithm for \textsf{MAX-2-LIN($k$)}\label{sec:recursion}}
\newcommand{\textsc{RecursiveConstruct}}{\textsc{RecursiveConstruct}}
Theorem~\ref{thm:sparsification} tells us that, given an instance $\mathcal{I}^*$, we can find a sparse instance $\mathcal{I}$ so that the quadratic forms of the corresponding Laplacians $\mathcal{L}_{\mathcal{I}^*}$ and $\mathcal{L}_\mathcal{I}$ are related by (\ref{eq:gua}). Therefore throughout this section we assume that the input instance $\mathcal{I}$ for \textsf{MAX-2-LIN($k$)}\ with $n$ variables has $m = \widetilde{O}\left((1 / \delta^2)\cdot nk\right)$ equations for some parameter $\delta>0$.
Recall that Theorem~\ref{thm:cheeger_k} shows that, for any \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{I}$, given an eigenvector for the smallest eigenvalue $\lambda_1(\mathcal{L}_\mathcal{I})$, we can obtain a partial assignment $\phi$ satisfying
\begin{align}
\label{eq:cheegerprecise}
p^\phi \leq
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} \sqrt{2 \lambda_1}.
\end{align}
Now we show that, by a repeated application of Theorem~\ref{thm:cheeger_k} on the subset of the equations of $\mathcal{I}$ for which both variables are unassigned, we can obtain a full assignment of $\mathcal{I}$. Our algorithm closely follows the one by Trevisan~\cite{Trevisan09} and is described in Algorithm~\ref{algo1}.
\begin{algorithm}
\caption{ $\textsc{RecursiveConstruct}(\mathcal{I},\delta)$\label{algo1}}
\label{influx}
\begin{algorithmic}[1]
\State Compute vector $z\in \mathbb{C}^n$ satisfying
\begin{align}\label{eq:cond}
\frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z} \leq (1 + 2 \delta) \lambda_1(\mathcal{L}_\mathcal{I});
\end{align}
\State Apply the algorithm from Theorem~\ref{thm:cheeger_k} to
compute $\phi: V \to [k]\cup\setof{\bot}$ such that
\begin{align}
p^\phi \leq
(1 + \delta)
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} \sqrt{2 \lambda_1};
\end{align}
\If{$2 p^\phi \geq \kh{1 - 1/k} \mathsf{Vol}(\phi)$}
\State return random full assignment $\phi': V\to [k]$; \\
\Comment{the case where the current assignment is worse than a random assignment}
\ElsIf{$\phi$ \rm{is a full assignment (i.e.} $\phi(V) \subseteq [k]$\rm{)}}
\State return $\phi$; \\
\Comment{The recursion terminates if every variable's assignment is determined}
\Else
\State $\mathcal{I}' \gets$ set of equations from $\mathcal{I}$ in which both variables' assignments are not determined;
\If{$\mathcal{I}'=\emptyset$}
\State set $\phi(u)$ to be an arbitrary assignment if $\phi(u) =\bot$ for any $u$;
\State return $\phi$;
\Else
\State $\phi_1 \gets \textsc{RecursiveConstruct}(\mathcal{I}',\delta)$;
\State return $\phi \cup \phi_1$;
\EndIf
\EndIf
\end{algorithmic}
\end{algorithm}
To achieve the guarantees of (\ref{eq:cheegerprecise}), however, we would need to compute the eigenvector corresponding to $\lambda_1(\mathcal{L}_\mathcal{I})$ \emph{exactly}. To obtain a nearly-linear time algorithm, instead, we relax this requirement and compute a vector $z$ that well-approximates this eigenvector. In particular, the following lemma shows that, for any $\delta$, we can compute a vector $z \in \mathbb{C}^n$
satisfying (\ref{eq:cond}) in nearly-linear time.
\begin{lemma}
\label{lem:powermethod}
For any given error parameter $\delta$, there is an $\tilde{O}\left( \left(1/\delta^{3}\right) \cdot kn\right)$ time algorithm that returns $z\in\mathbb{C}^n$ satisfying (\ref{eq:cond}).
\end{lemma}
\begin{proof} Following the discussion in
\cite[Section 8.2]{vishnoilx}, we compute a vector $z\in\mathbb{C}^n$ satisfying (\ref{eq:cond}) in $O\left((1/\delta)\cdot \log({n/\delta})\right)$ iterations by the power method, where each iteration consists in solving a linear system of the form $\mathcal{L}_\mathcal{I} x = b$ for some vector $b$. This can be done up to $\delta$ precision in $O\left(\left(m + n \log^2n\right)\log(1/\delta)\right)$-time using a nearly-linear time solver for connection Laplacians~\cite{kyng16,KyngS16}. The total running time follows from our assumption on $m$.
\end{proof}
To analyse Algorithm~\ref{algo1}, we introduce some notation. Let $t$ be the number of recursive executions of Algorithm~\ref{algo1}. For any $1\leq j \leq t+1$, let $\mathcal{I}_{j}$ be the instance of \textsf{MAX-2-LIN($k$)}\ in the $j$-th execution. We indicate with $\rho_j m$ the number of equations in $\mathcal{I}_j$, where $0\leq \rho_j\leq 1$. Notice that $\mathcal{I}_{1} = \mathcal{I}$ and $\mathcal{I}_{t+1} = \emptyset$.
We assume that the maximum number of equations in $\mathcal{I}_{j}$ that can be satisfied by an assignment is $(1-\varepsilon_j) \rho_j m$, with $\varepsilon=\varepsilon_1$. Also notice that it holds for any $1\leq j\leq t$ that $\varepsilon_j \rho_j m \leq \varepsilon m$,
which implies
\begin{equation}\label{eq:eq11}
\varepsilon_j \leq \varepsilon / \rho_j.
\end{equation}
The next theorem presents the performance of our algorithm, whose informal version is Theorem~\ref{thm:main1}
\begin{theorem}\label{thm:algoanalysis}
Given an instance $\mathcal{I}$ of \textsf{MAX-2-LIN($k$)}\
whose optimum is $1 - \varepsilon$
and a parameter $\delta > 0$,
the algorithm $\textsc{RecursiveConstruct}(\mathcal{I},\delta)$
returns in $\widetilde{O}\left( \left(1/\delta^{3}\right)\cdot k n^2\right)$ time
an assignment $\phi$ satisfying at least
$1 - 8 \nu \sqrt{\varepsilon}$ fraction of the equations, where
\[
\nu \stackrel{\mathrm{def}}{=} (1 + \delta)
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} = O(k).
\]
\end{theorem}
\begin{proof}
Suppose we are now at the $j$-th iteration.
By Theorem~\ref{thm:cheeger_k}, we know that
the total weight of unsatisfied equations in $\mathcal{I}_j \setminus \mathcal{I}_{j+1}$
is at most
\begin{align*}
\lefteqn{2\cdot (\rho_j - \rho_{j+1}) m \left( 2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k)}\right)\cdot \sqrt{2\cdot (1+2\delta) \cdot\lambda_1\left(\mathcal{L}_{\mathcal{I}_j} \right) }}\\
& \leq 2\cdot(\rho_j - \rho_{j+1}) m (1+\delta) \left( 2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k)}\right) \sqrt{2\cdot\lambda_1\left(\mathcal{L}_{\mathcal{I}_j} \right) } \\
& = 2 \cdot(\rho_j - \rho_{j+1}) m \nu \sqrt{2\cdot\lambda_1\left(L_{\mathcal{I}_j} \right) } \\
& \leq 4 \cdot(\rho_j - \rho_{j+1}) m \nu \sqrt{\varepsilon_j }\\
& \leq 4 \cdot(\rho_j - \rho_{j+1}) m \nu \sqrt{\varepsilon/\rho_j }\\
& \leq 4\cdot m \nu \sqrt{\varepsilon} \int_{\rho_{j+1}}^{\rho_j}
\sqrt{\frac{1}{r}} \ \mathrm{d} r,
\end{align*}
Therefore, the total weight of unsatisfied equations in $\mathcal{I}$
can be upper bounded by
\[
4 m \nu \sqrt{\varepsilon}\ \sum_{j=1}^{t}
\int_{\rho_{j+1}}^{\rho_j}
\sqrt{\frac{1}{r}} \mathrm{d} r
\leq
4 m \nu \sqrt{\varepsilon}
\int_{0}^{1}
\sqrt{\frac{1}{r}}\ \mathrm{d} r
= 8m \nu \sqrt{\varepsilon},
\]
which implies that the total weight of satisfied equations
is at least $(1 - 8\nu \sqrt{\varepsilon})m$. The runtime follows by Lemma~\ref{lem:powermethod} and the fact that we perform at most a linear number of recursive iterations.
\end{proof}
The following corollary
which states how much our algorithm beats a random assignment
follows from Theorem~\ref{thm:main1}.
\begin{cor}\label{cor:k3}
Given a \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{I}$
whose optimum is $\xi$
and a constant $\delta > 0$,
Algorithm~\ref{algo1}
returns in $\tilde{O}\left( \delta^{-3} n^2\right)$ time
an assignment $\phi$ satisfying at least
$\kh{1/k + \tau}\xi$ fraction of the equations, where
$
\tau = \Omega\kh{\frac{1}{k^3}}$.
\end{cor}
\begin{proof}
We define the parameter \[
\varepsilon' = \frac{(1 - \frac{1}{k})^2}{64\nu^2},\]
which implies that $1 - 8\nu \sqrt{\varepsilon'} = 1/k$.
Since Algorithm~\ref{algo1} always chooses the best between
the assignment found by recursively applications of Theorem~\ref{thm:cheeger_k} and a random assignment, the algorithm's approximation ratio is at least
\begin{align*}
& \frac{\max\setof{1 - 8\nu \sqrt{\varepsilon}, 1/k}}{1 - \varepsilon}
\geq \frac{1/k}{1 - \varepsilon'}
\geq \frac{1/k}{1 - \frac{1}{256\nu^2}}
\geq \frac{1/k}{1 - \frac{1}{512(1 + \delta)^2 k^2}}
\geq \frac{1}{k} + \frac{1}{(1 + \delta)^2 k^3}
= \frac{1}{k} + \Omega\kh{\frac{1}{k^3}},
\end{align*}
where the first inequality follows by the fact that
$\frac{1 - 8\nu\sqrt{\varepsilon}}{1-\varepsilon}$ is a monotone decreasing function in $\varepsilon$,
the third inequality follows from the definition of $\nu$,
and the last inequality follows from that $\delta$ is a constant. \end{proof}
\section{A Cheeger inequality for $\lambda_1(\mathcal{L}_\mathcal{I})$ and \textsf{MAX-2-LIN($k$)} \label{sec:cheeger}}
The discrete Cheeger inequality~\cite{Alon86}
shows that, for any undirected graph $G$, the conductance $h_G$ of $G=(V,E)$ can be approximated by the second smallest eigenvalue of $G$'s normalised Laplacian matrix $\mathcal{L}_G$, i.e.,
\begin{equation}\label{eq:cheeger}
\frac{\lambda_2(\mathcal{L}_G)}{2} \leq h_G\leq \sqrt{2\cdot\lambda_2(\mathcal{L}_G)}.
\end{equation}
Moreover, the proof of the second inequality above is constructive, and indicates that a subset $S\subset V$ with conductance at most $\sqrt{2\cdot \lambda_2(\mathcal{L}_G)}$
can be found by using the second bottom eigenvector of $\mathcal{L}_G$ to embed vertices on the real line.
As one of the most fundamental results in spectral graph theory, the Cheeger inequality has found applications in the study of a wide range of optimisation problems, e.g., graph partitioning~\cite{journals/jacm/LeeGT14}, max-cut~\cite{Trevisan09}, and many practical problems like image segmentation~\cite{ShiM00} and web search~\cite{Kleinberg99}.
In this section, we develop connections between $\lambda_1(\mathcal{L}_\mathcal{I})$
and \textsf{MAX-2-LIN($k$)}\ by proving a Cheeger-type inequality. Let
\[
\phi: \setof{x_1,\ldots,x_n} \to [k]\cup\setof{\bot}
\]
be an arbitrary \emph{partial assignment} of an instance $\mathcal{I}$, where $\phi(x_i)=\bot$ means that the assignment of $x_i$ has not been decided. These variables' assignments will be determined through some recursive construction, which will be elaborated in Section~\ref{sec:recursion}. We remark that this framework of recursively computing a partial assignment was first
introduced by Trevisan~\cite{Trevisan09}, and our theorem can be viewed as a generalisation of the one in \cite{Trevisan09},
which corresponds to the $k = 2$ case of ours.
To relate quadratic forms of $\mathcal{L}_\mathcal{I}$ with the objective function of the \textsf{MAX-2-LIN($k$)}\ problem, we introduce a \emph{penalty} function as follows:
\begin{definition}\label{def:pnt} Given a partial assignment $\phi: \setof{x_1,\ldots,x_n} \to [k]\cup\setof{\bot}$ and a directed edge $(u,v)$, the penalty of $(u,v)$ with respect to $\phi$ is defined by
\begin{align}
p^\phi_{uv}(\mathcal{I}) \stackrel{\mathrm{def}}{=}
\begin{cases}
0 & \phi(u)\neq \bot,\phi(v)\neq \bot, \phi(u) - \phi(v) \equiv c_{uv} \hspace{-5pt}\mod k \\
1 & \phi(u)\neq \bot,\phi(v)\neq \bot, \phi(u) - \phi(v) \not\equiv c_{uv} \hspace{-5pt}\mod k \\
0 & \phi(u) = \phi(v) = \bot \\
1 - 1/k \qquad & \text{exactly one of $\phi(u),\phi(v)$ is $\bot$.}
\end{cases}
\end{align}
For simplicity, we write $p^\phi_{uv}$ when the underlying instance $\mathcal{I}$ is clear from the context.
\end{definition}
The values of $ p^\phi_{uv} $ from Definition~\ref{def:pnt} are chosen according to the following facts: (1) If both $u$ and $v$'s values are assigned, then their penalty is $1$ if the equation $\phi(u) - \phi(v) \not\equiv c_{uv} \hspace{-5pt}\mod k$ associated with $(u,v)$ is unsatisfied, and $0$ otherwise; (2) If both $u$ and $v$'s values are $\bot$, then their penalty is temporally set to $0$. Their penalty will be computed when $u$ and $v$'s assignment are determined during a later recursive stage;
(3) If exactly one of $u,v$ is assigned, $p^\phi_{uv}$ is set to $1 - 1/k$,
since a random assignment
to the other variable makes the edge $(u,v)$ satisfied with probability $1/k$.
Without loss of generality, we only consider $\phi$ for which $\phi(u)\ne \bot$ for at least one vertex $u$, and define the penalty of assignment $\phi$ by
\begin{align}
p^\phi \stackrel{\mathrm{def}}{=} \frac{2 \sum_{u\leadsto v} b_{uv} p^\phi_{uv}}{\mathsf{Vol}(\phi)},
\end{align}
where $\mathsf{Vol}(\phi) \stackrel{\mathrm{def}}{=} \sum_{\phi(u)\neq \bot} d_u$.
Notice that the $p^\phi_{uv}$'s value is multiplied by $b_{uv}$ in accordance with the objective of \textsf{MAX-2-LIN($k$)}\ which is to maximise the total weight of satisfied assignments. Also, we multiply $p^\phi_{uv}$ by $2$ in the numerator
since edges with at least one assigned endpoint are counted at most twice in $\mathsf{Vol}(\phi)$.
Notice that, as long as $G$ is weakly connected,
$p^\phi=0$ if and only if all edges are satisfied by $\phi$ and, in general, the smaller the value of $p^{\phi}$, the more edges are satisfied by $\phi$. With this in mind, we
define the \textit{imperfectness} $p(\mathcal{I})$ of $\mathcal{I}$ to quantify
how close $\mathcal{I}$ is
to an instance where all equations can be satisfied by a single assignment.
\begin{definition}\label{def:perfect} Given any \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{I} = (G, k)$, the imperfectness of $\mathcal{I}$ is defined by
\begin{align}
p(\mathcal{I}) \stackrel{\mathrm{def}}{=} \min_{\phi\in \kh{[k]\cup\setof{\bot}}^V\setminus \setof{\bot}^V}\, p^\phi.
\end{align}
\end{definition}
The main result of this section is a Cheeger-type inequality that relates
$p(\mathcal{I})$ and $\lambda_1(\mathcal{L}_\mathcal{I})$, which is summarised in Theorem~\ref{thm:cheeger_k}.
Note that, since $\sin(x)\geq (2/\pi)\cdot x$ for $x\in [0, \pi/2]$,
the factor before $\sqrt{2\lambda_1}$ in the theorem statement
is at most $(2 +k/4)$ for $k\geq 2$.
\begin{theorem}\label{thm:cheeger_k}
Let $\lambda_1$ be the smallest eigenvalue of $\mathcal{L}_\mathcal{I}$.
It holds that
\begin{align}\label{eq:cheegerd2}
\frac{\lambda_1}{2} \leq p(\mathcal{I}) \leq \kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} \sqrt{2 \lambda_1}.
\end{align}
Moreover, given the eigenvector associated with $\lambda_1$,
there is an $O(m +n\log n)$-time algorithm that returns
a partial assignment $\phi$
such that
\begin{align}\label{eq:cheegereq4}
\frac{\lambda_1}{2} \leq p^\phi \leq
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} \sqrt{2 \lambda_1}.
\end{align}
\end{theorem}
Our analysis is based on the following fact about the relations about the angle between two vectors and their Euclidean distance. For some $a,b\in \mathbb{C}$, we write $\theta(a,b)\in[-\pi,\pi)$ to denote the angle from $b$ to $a$, i.e., $\theta(a,b)$ is the unique real number in $[-\pi,\pi)$ such that
\[
\frac{a}{\norm{a}} = \frac{b}{\norm{b}} \exp\left(i \theta(a,b)\right).
\]
\begin{fact}\label{fact:arc2}
Let $a,b$ be complex numbers
such that $\theta = \theta(a,b)$. The following statements hold:
\begin{enumerate}
\item If $\theta\in \left[-\frac{2\pi}{k}, \frac{2\pi}{k} \right)$, then it holds that
\begin{align}\label{eq:arc2}
\sizeof{\theta} \cdot \min\setof{\norm{a},\norm{b}} \leq \frac{\pi}{k\cdot\sin (\pi/k)}\cdot \norm{a - b}.
\end{align}
\item If $\theta \in \left[-\pi, -\frac{2\pi}{k}\right) \cup \left[\frac{2\pi}{k}, \pi\right)$, then it holds that
\begin{align}\label{eq:arc3}
\min\setof{\norm{a},\norm{b}} \leq
\frac{1}{2\cdot \sin(\pi/k) }\cdot \norm{a - b}.
\end{align}
\end{enumerate}
\end{fact}
\begin{proof}
We assume $\theta\in \left[-\frac{2\pi}{k}, \frac{2\pi}{k} \right)$ and prove the first statement.
Let
\[
a' = \frac{a}{ \| a\|}\cdot\min \{ \|a\|, \|b\|\},
\]
\[
b' = \frac{b}{ \| b\|}\cdot\min \{ \|a\|, \|b\|\}.
\]
Then we have that
\begin{align}
\norm{a - b} & \geq \|a'-b' \| \notag \\
& \geq 2\cdot \sin\left(\frac{\sizeof{\theta}}{2}\right) \cdot \min\setof{\norm{a},\norm{b}} \notag \\
& \geq 2\cdot \frac{\sizeof{\theta}}{2}\cdot \frac{\sin(\pi/k) }{\pi/k} \cdot \min\setof{\norm{a},\norm{b}} \notag \\
& = \sizeof{\theta}\cdot \frac{k \sin(\pi/k) }{\pi} \cdot \min\setof{\norm{a},\norm{b}},\nonumber
\end{align}
where the last inequality follows by the fact that
for any $\alpha \in \left[0, \frac{\pi}{2}\right]$
and $x \in [0, \alpha]$ it holds that
$\sin x \geq x\cdot \frac{\sin \alpha}{\alpha}$.
Multiplying $\frac{\pi}{k \sin\frac{\pi}{k}}$ on the both sides of the inequality above gives us (\ref{eq:arc2}).
Now we prove the second statement. We have
\begin{align}
\norm{a - b} & \geq \norm{a'-b'} \notag \\
& \geq 2\cdot \sin\left(\frac{\sizeof{\theta}}{2}\right) \cdot \min\setof{\norm{a},\norm{b}} \notag \\
& \geq 2\cdot \sin \left( \frac{\pi}{k} \right)\cdot \min\setof{\norm{a},\norm{b}}\nonumber
\end{align}
where the last inequality follows from the fact that
$\theta \in \left[-\pi, -\frac{2\pi}{k}\right) \cup \left[\frac{2\pi}{k}, \pi\right)$.
Dividing both sides of the inequality above by $2\cdot \sin(\pi/k) $
gives us~(\ref{eq:arc3}).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:cheeger_k}]
We first prove $\frac{\lambda_1}{2}\leq p(\mathcal{I})$.
For a partial assignment $\phi: V\to [k]\cup\setof{\bot}$,
we construct a vector $x_{\phi} \in \mathbb{C}^n$ by
\begin{align}
\kh{x_{\phi}}_u =
\begin{cases}
\omega_k^{j} \quad & \phi(u) = j\in[k],\\
0 & \phi(u) = \bot.
\end{cases}
\end{align}
Then, we have
\begin{align}
p(\mathcal{I}) = & \min_{\phi\in\kh{[k]\cup\setof{\bot}}^V \setminus \setof{\bot}^V } \frac{2 \sum_{u\leadsto v} b_{uv}p^\phi_{uv}}{\mathsf{Vol}(\phi)} \notag \\
\geq & \min_{\phi\in\kh{[k]\cup\setof{\bot}}^V\setminus \setof{\bot}^V} \frac{\sum_{u\leadsto v}
b_{uv}\norm{\kh{x_\phi}_u - \omega_k^{c_{uv}} \kh{x_\phi}_v}^2}{2 \cdot \mathsf{Vol}(\phi)} \notag \\
= & \min_{\phi\in\kh{[k]\cup\setof{\bot}}^V \setminus \setof{\bot}^V} \frac{x_{\phi}^* L_\mathcal{I} x_{\phi}}
{2 \cdot x_{\phi}^* D_\mathcal{I} x_{\phi}} \notag \\
\geq & \frac{1}{2}\cdot \min_{x\in \mathbb{C}^n, x\neq 0^n} \frac{x^* L_\mathcal{I} x}{x^* D_\mathcal{I} x} = \frac{\lambda_1(\mathcal{L}_\mathcal{I})}{2},
\end{align}
where the second line follows from the fact that
\begin{align}
\norm{\kh{x_{\phi}}_u - \omega_k^{c_{uv}} \kh{x_{\phi}}_v}^2
\leq 4 \cdot p^\phi_{uv}
\end{align}
always holds for all $(u,v)\in E$,
and the third line follows from
Lemma~\ref{lem:qL} and that
\begin{align}
x_{\phi}^* D_\mathcal{I} x_{\phi} =
\sum_{u\in V} d_u \norm{\kh{x_{\phi}}_u}^2 =
\sum_{u: \phi(u)\neq \bot} d_u = \mathsf{Vol}(\phi).
\end{align}
This proves that $\lambda_1/2 \leq p(\mathcal{I})$.
Secondly, we assume that $z\in\mathbb{C}^n$ is the vector such that
\[
\frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z} = \lambda_1,
\]
and prove the existence of an assignment $\phi$ based on $z$ satisfying
\[
p^{\phi}\leq \left( 2- \frac{2}{k} + \frac{1}{2\sin(\pi/k)} \right) \sqrt{2\lambda_1},
\]
which will imply (\ref{eq:cheegerd2}) and (\ref{eq:cheegereq4}). We scale each coordinate of $z$ and without loss of generality assume that $\max_{u\in V} \| z_u\| ^2=1$.
For real numbers $t \geq 0$ and $\eta \in [0, \frac{2\pi}{k})$,
we define $k$ disjoint sets of vertices indexed by $j\in[k]$ as follows:
\begin{align}
S^{(j)}_{t,\eta} =
\setof{ u\ \left|\ \norm{z_u} \geq t\ \mathrm{and}\
\theta(z_u, \mathrm{e}^{i \eta }) \in \left[j \cdot \frac{2\pi}{k}, (j+1)\cdot \frac{2\pi}{k}\right) \right.}.
\end{align}
We then define an assignment $\phi_{t,\eta}$ where
\begin{align}
\phi_{t,\eta}(u) =
\begin{cases}
j \quad & \exists j\in [k]: u\in S^{(j)}_{t,\eta}, \\
\bot \quad & \mathrm{otherwise.}
\end{cases}
\end{align}
By definition, the $k$ vertex sets correspond to the vectors in the $k$ regions of the unit ball
after each vector is rotated by $\eta$ radians counterclockwise.
The role of $t$ is to only consider the coordinates $z_u$ with $\| z_u\|\geq t$. This is illustrated in Figure~\ref{fig1}.
\definecolor{ashgrey}{rgb}{0.7, 0.75, 0.71}
\definecolor{burlywood}{rgb}{0.87, 0.72, 0.53}
\definecolor{cadetblue}{rgb}{0.37, 0.62, 0.63}
\definecolor{carolinablue}{rgb}{0.6, 0.73, 0.89}
\definecolor{cinereous}{rgb}{0.6, 0.51, 0.48}
\definecolor{coolblack}{rgb}{0.0, 0.18, 0.39}
\definecolor{darkcerulean}{rgb}{0.03, 0.27, 0.49}
\definecolor{dollarbill}{rgb}{0.52, 0.73, 0.4}
\definecolor{graynew}{rgb}{0.75, 0.75, 0.75}
\definecolor{grullo}{rgb}{0.66, 0.6, 0.53}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=2]
\coordinate (A) at (1.149,0.954);
\coordinate (B) at (-1.410,0.513);
\coordinate (C) at (0.260, -1.478);
\draw[draw=white] (B) to[arc through ccw=(A)] (C) -- (arc through center) -- cycle;
\draw[draw=white] (A) to[arc through ccw=(C)] (B) -- (arc through center) -- cycle;
\draw[draw=burlywood] (C) to[arc through ccw=(B)] (A) -- (arc through center) -- cycle;
\draw[ultra thick,draw=white] (0,0) -- (A);
\draw[ultra thick,draw=white] (0,0) -- (C);
\draw[thick,dashed] (0,0) -- (A);
\draw[thick,dashed] (0,0) -- (B);
\draw[thick,dashed] (0,0) -- (C);
\draw[very thick, ->, draw=dollarbill,rotate=40] (1.5,0) arc(0:120:1.5cm and 1.5cm);
\draw[ultra thick, ->, draw=darkcerulean, rotate=160 ] (1.5,0) arc(0:120:1.5cm and 1.5cm);
\draw[ultra thick, ->, draw=orange, rotate=280 ] (1.5,0) arc(0:120:1.5cm and 1.5cm);
\draw (0.6, 1.6) node[right] {{Set} $S^{(1)}_{t,\eta}$};
\draw (-2.2, -0.85) node[right] {{Set} $S^{(2)}_{t,\eta}$};
\draw (1, -1.3) node[right] {{Set} $S^{(3)}_{t,\eta}$};
\draw[very thick, ->, draw=red, rotate=0] (0.7,0) arc(0:40:0.7cm and 0.7cm);
\draw [fill=white] (0.82,0.1) rectangle (2.6,0.55);
\draw (0.9, 0.45) node[right] {Random rotation};
\draw (0.9, 0.23) node[right] {by $\eta\in[0, 2\pi/k)$};
\draw[very thick, draw=black, ->] (-1.8,0) -- (1.8,0) coordinate (x axis);
\draw[very thick,draw=black, ->] (0,-1.8) -- (0,1.82) coordinate (y axis);
\draw[fill=cinereous!10] (0,0) circle (0.5cm);
\draw[fill=grullo] (0.4,0) circle (0.02cm);
\draw[fill=grullo] (0.8,0.8) circle (0.02cm);
\draw[fill=grullo] (0.76,0.2) circle (0.02cm);
\draw[fill=grullo] (-0.36,-0.2) circle (0.02cm);
\draw[fill=grullo] (-0.36,-0.5) circle (0.02cm);
\draw[fill=grullo] (-0.76,-1) circle (0.02cm);
\draw[fill=grullo] (-0.66,-0.7) circle (0.02cm);
\draw[fill=grullo] (-0.63534,-0.62) circle (0.02cm);
\draw[fill=grullo] (-0.72,-0.734) circle (0.02cm);
\draw[fill=grullo] (-0.7342,-0.6) circle (0.02cm);
\draw[fill=grullo] (0.8,-0.45) circle (0.02cm);
\draw[fill=grullo] (0.345,-0.3) circle (0.02cm);
\draw[fill=grullo] (0.81,-0.7) circle (0.02cm);
\draw[fill=grullo] (0.79,-0.64) circle (0.02cm);
\draw[fill=grullo] (0.72,-1.2) circle (0.02cm);
\draw[fill=grullo] (0.9,-1.1) circle (0.02cm);
\draw[fill=grullo] (0.8,-0.81) circle (0.02cm);
\draw[fill=grullo] (0.9,-0.7) circle (0.02cm);
\draw[fill=grullo] (-0.6,0.85) circle (0.02cm);
\draw[fill=grullo] (-0.8,0.75) circle (0.02cm);
\draw[fill=grullo] (-0.76,0.72) circle (0.02cm);
\draw[fill=grullo] (-0.7,0.85) circle (0.02cm);
\draw[fill=grullo] (-0.5,1) circle (0.02cm);
\draw[fill=grullo] (-0.8,0.9) circle (0.02cm);
\draw[fill=grullo] (-1.1,0.6) circle (0.02cm);
\draw[fill=grullo] (-1.2,0.75) circle (0.02cm);
\draw[fill=grullo] (-0.3,0.25) circle (0.02cm);
\draw[fill=grullo] (0.12,0.1) circle (0.02cm);
\draw[ultra thick,draw=black, ] (-0.07,0.5) -- (0.07,0.5) coordinate (y axis);
\draw (-0.3, 0.6) node[right] {\textbf{\small $t$}};
\end{tikzpicture}
\end{center}
\caption{Illustration of the proof for Theorem~\ref{thm:cheeger_k} for the case of $k=3$. The gray circle is obtained by sweeping $t\in[0,1]$, and the red arrow represents a random angle $\eta\in[0, 2\pi/k)$. A partial assignment is determined by the values of $\eta$ and $t$.\label{fig1}}
\end{figure}
Our goal is to construct probability distributions for $t$ and $\eta$ such that
\begin{align}\label{eq:cheegermainobj4}
\frac{\expec{t,\eta}{2\sum_{u\leadsto v} b_{uv} p^\phi_{uv} }}{\expec{t,\eta}{\mathsf{Vol}(\phi_{t,\eta})}}
\leq
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }} \cdot \sqrt{2\cdot \frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z}}.
\end{align}
This implies by linearity of expectation that
\begin{align}
\expec{t,\eta}{2\sum_{u\leadsto v} b_{uv} p^\phi_{uv} -
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }}\cdot \mathsf{Vol}(\phi_{t,\eta})\cdot
\sqrt{2\cdot \frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z}}}
\leq
0,
\end{align}
and existence of an assignment $\phi$ satisfying (\ref{eq:cheegereq4}).
Now let us assume that $t\in[0,1]$ is chosen such that $t^2$ follows from a uniform distribution over $[0,1]$, and $\eta$ is chosen uniformly at random from $[0, 2 \pi/k)$. We analyse the numerator and denominator in the left-hand side of (\ref{eq:cheegermainobj4}).
For the denominator, it holds that
\begin{align}\label{eq:x2}
\expec{t,\eta}{\mathsf{Vol}(\phi_{t,\eta})} & =
\sum_{u\in V} d_u\cdot \prob{}{ \phi(u) \neq \bot} = \sum_{u\in V} d_u\cdot \prob{}{ \norm{z_u}\geq t} \notag \\
& = \sum_{u\in V} d_u \norm{z_u}^2 = z^* D_\mathcal{I} z.
\end{align}
For the numerator,
it holds by linearity of expectation that
\begin{align}
\expec{t,\eta}{2\sum_{u\leadsto v} b_{uv} p^\phi_{uv} }
= 2 \sum_{u\leadsto v} b_{uv}\, \expec{t,\eta}{p^\phi_{uv}}.
\end{align}
Then we look at $\expec{t,\eta}{p^\phi_{uv}}$ for every edge $(u,v)\in E$.
The analysis is based on the value of $\theta=\theta(z_u, \omega_k^{c_{uv}} z_v)$,
the angle from $z_v$ rotated by $2c_{uv}\pi/k$ radians clockwise
to $z_u$.
\begin{itemize}
\item Case~1:
$\theta = \theta(z_u, \omega_k^{c_{uv}}z_v) \in \left[-\frac{2\pi}{k}, \frac{2\pi}{k}\right)$. It holds that
\begin{align}
\expec{t,\eta}{p^\phi_{uv}}
= & \kh{1 - \frac{1}{k}} \cdot \prob{}{ \| z_u\| < t\leq \|z_v\| \mbox{\ or\ } \| z_v\| < t\leq \|z_u\|} \notag \\
& \qquad \qquad +1\cdot \prob{}{\norm{z_u}\geq t, \norm{z_v}\geq t, \phi(u) -\phi(v) \not\equiv c_{uv}\hspace{-5pt}\mod k} \notag \\
= & \kh{1 - \frac{1}{k}} \sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
\frac{\sizeof{\theta}}{ 2\pi/k} \cdot \min\setof{\norm{z_u}^2, \norm{z_v}^2} \notag \\
\leq & \kh{1 - \frac{1}{k}}\sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
\frac{{\pi}/\kh{k \sin(\pi/k)}}{2\pi/ k}\cdot \norm{z_u - \omega_k^{c_{uv}} z_v}
\cdot \min\setof{\norm{z_u}, \norm{z_v}} \notag \\
\leq & \kh{1 - \frac{1}{k}} \sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
\frac{1}{4\cdot\sin (\pi/k) } \cdot \norm{z_u - \omega_k^{c_{uv}} z_v} \cdot \kh{\norm{z_u} + \norm{z_v}}
\nonumber \\
\leq & \kh{1 - \frac{1}{k} + \frac{1}{4\cdot \sin (\pi/k) }}
\norm{z_u - \omega_k^{c_{uv}} z_v} \kh{\norm{z_u} + \norm{z_v} } \nonumber,
\end{align}
where the second equality follows from that
\begin{align}
\lefteqn{\prob{}{\norm{z_u}\geq t, \norm{z_v}\geq t, \phi(u) -\phi(v) \not\equiv c_{uv}\hspace{-5pt}\mod k}} \notag \\
= & \prob{}{\norm{z_u}\geq t, \norm{z_v}\geq t} \cdot
\prob{}{\phi(u) -\phi(v) \not\equiv c_{uv}\hspace{-5pt}\mod k\ |\ \norm{z_u}\geq t, \norm{z_v}\geq t} \notag \\
= & \min\setof{\norm{z_u}^2, \norm{z_v}^2}\cdot\frac{\sizeof{\theta}}{2\pi / k}, \notag
\end{align}
the third inequality follows by Fact~\ref{fact:arc2} and
that $\sizeof{\theta}$ equals exactly the angle between $z_u$ and $\omega_k^{c_{uv}} z_v$.
\item Case~2: $\theta = \theta(z_u, \omega_k^{c_{uv}}z_v) \in \left[-\pi, -\frac{2\pi}{k}\right)\cup \left[\frac{2\pi}{k},\pi\right)$.
It holds that
\begin{align}
\expec{t,\eta}{p^\phi_{uv}}
= & \kh{1 - \frac{1}{k}} \cdot \prob{}{ \| z_u\| < t\leq \|z_v\| \mbox{\ or\ } \| z_v\| < t\leq \|z_u\|} \notag\\
& \qquad\qquad +1\cdot \prob{}{\norm{z_u}\geq t, \norm{z_v}\geq t, \phi(u) -\phi(v) \not\equiv c_{uv}\hspace{-5pt}\mod k} \notag \\
= & \kh{1 - \frac{1}{k}}\cdot \sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
1\cdot \min\setof{\norm{z_u}^2, \norm{z_v}^2} \notag \\
\leq & \kh{1 - \frac{1}{k}}\cdot \sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
\frac{1}{2\sin(\pi/k) }\cdot \norm{z_u - \omega_k^{c_{uv}} z_v}
\cdot \min\setof{\norm{z_u}, \norm{z_v}} \notag \\
\leq & \kh{1 - \frac{1}{k}}\cdot \sizeof{\norm{z_u}^2 - \norm{z_v}^2} +
\frac{1}{4\sin (\pi/k) } \cdot \norm{z_u - \omega_k^{c_{uv}} z_v} \cdot \kh{\norm{z_u} + \norm{z_v}}
\nonumber \\
\leq & \kh{1 - \frac{1}{k} + \frac{1}{4\cdot\sin(\pi/k) }}
\norm{z_u - \omega_k^{c_{uv}} z_v} \kh{\norm{z_u} + \norm{z_v} } \nonumber,
\end{align}
where
the second equality follows from the fact that
edge $(u,v)$ can not be satisfied when $\theta$ is in this range,
the first inequality follows by Fact~\ref{fact:arc2}
and that the angle between $z_u$ and $\omega_k^{c_{uv}}z_v$ is
at least $\frac{2\pi}{k}$,
and the last line follows by the triangle inequality.
\end{itemize}
Combining these two cases gives us that
\begin{align}
\expec{t,\eta} {2\sum_{u\leadsto v} b_{uv}\, {p^\phi_{uv}} }
&\leq \kh{2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k) }}
\sum_{u\leadsto v} b_{uv} \norm{z_u - \omega_k^{c_{uv}} z_v}\kh{\norm{z_u} + \norm{z_v} } \notag \\
&\leq \kh{2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k) }}
\sqrt{\sum_{u\leadsto v} b_{uv} \norm{z_u - \omega_k^{c_{uv}} z_v}^2}
\sqrt{\sum_{u\leadsto v} b_{uv} \kh{\norm{z_u} + \norm{z_v} }^2}\notag \\
&\leq \kh{2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k) }}
\sqrt{\sum_{u\leadsto v} b_{uv} \norm{z_u - \omega_k^{c_{uv}} z_v}^2}
\sqrt{2\sum_{u} d_u \norm{z_u}^2 } \notag \\
& = \kh{2 - \frac{2}{k} + \frac{1}{2\sin(\pi/k) }}
\cdot \sqrt{z^* L_\mathcal{I} z} \cdot \sqrt{2 z^* D_\mathcal{I} z},
\end{align}
where
the second inequality follows
by the Cauchy-Schwarz inequality.
Combining this with (\ref{eq:x2}) finishes the proof of the inequality (\ref{eq:cheegerd2}).
Finally, let us look at the time needed to
find the desired partial assignment. Notice that, by the law of total expectation, we can write \begin{align}
& \expec{t,\eta}{2\sum_{u\leadsto v} b_{uv} p^\phi_{uv} -
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }}\cdot \mathsf{Vol}(\phi_{t,\eta})\cdot
\sqrt{2\cdot \frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z}}} \notag\\
= & \expec{t_0}{\expec{t,\eta}{\left. 2\sum_{u\leadsto v} b_{uv} p^\phi_{uv} -
\kh{2 - \frac{2}{k} + \frac{1}{2\sin (\pi/k) }}\cdot \mathsf{Vol}(\phi_{t,\eta})\cdot
\sqrt{2\cdot \frac{z^* L_\mathcal{I} z}{z^* D_\mathcal{I} z}}\ \right|\ t = t_0}}.
\end{align}
As a preparation step, we build
two ordered sequences of coordinates of $\{ z_u\}$:
the first ordered sequence is based on $z_u$'s norm, and the other ordered sequence is based on $z_u$'s angle. This step takes $O(n\log n)$ time.
Now we construct two sequences of sweep sets: the first is based on $t$, and the second is based on $\eta$. For constructing the sweep sets based on $t$, the algorithm increases $t$ from $0$ to $1$,
and updates the conditional expectation of the edges incident with $u$ whenever $t$ exceeds $\norm{z_u}$. Notice that each edge $(u,v)$ will be updated at most twice, i.e., in the step when $t$ reaches $\norm{z_u}$ and when it reaches $\norm{z_v}$.
Hence, the total runtime for constructing the sweep sets on $t$ is $O(m)$. The runtime analysis for constructing the sweep sets on $\eta$ is similar: the algorithm increases $\eta$ from $0$ to $2\pi/k$, and updates the penalties $p^\phi_{uv}$ of the edges $(u,v)$ only if the assignment of $u$ of $v$ changes. Since every edge will be updated at most twice, the total runtime for constructing the sweep sets on $\eta$ is $O(m)$ as well. The algorithm terminates if the
assignment $\phi$ satisfying~(\ref{eq:cheegereq4}) is found. The total runtime of the algorithm is $O(m+n\log n)$.
\end{proof}
\begin{remark}
We remark that the factors $\lambda_1/2$ and $\sqrt{\lambda_1}$ in
Theorem~\ref{thm:cheeger_k} are both tight
within constant factors.
The tightness can be derived directly from Section 5 of~\cite{Trevisan09},
since when $k=2$, our inequality is the same as the one in~\cite{Trevisan09} up to constant factors.
We also remark that the factor of $k$ in Theorem~\ref{thm:cheeger_k} is necessary, which is shown by the following instance: the linear system has $nk$ variables where every variable belongs to one of $k$ sets $S_0,\dots,S_{k-1}$ with $|S_i| = n$ for any $0 \le i \le k-1$. Now, for any $i$, we add $n$ equations of the form $x_u - x_v = 1 \mod k$ with $x_u \in S_i$, $x_v \in S_j$, and $j = (i+1) \mod k$, and $n$ equations of the form $x_u - x_v = 1 \mod k$ with $x_u \in S_i$, $x_v \in S_j$, and $j = (i+2) \mod k$. This instance is constructed such that the underlying graph is regular, and every assignment could only satisfy at most half of the equations, implying that the imperfectness is $p(\mathcal{I}) = \Omega(1)$. However, mapping each variable in $S_i$ to the root of unity $\omega_k^i$, it's easy to see that $\lambda_1(\mathcal{L}_{\mathcal{I}}) = O(1/k^2)$. Hence Theorem~\ref{thm:cheeger_k} is tight with respect to $k$.
\end{remark}
\begin{remark}
We notice that this factor of $k$ originates from the relation between the quadratic forms of the Hermitian Laplacian and the penalty function $p$ of Definition~\ref{def:pnt}. Indeed, we could re-define our penalty function such that, for an equation of the form $x_u - x_v = c \mod k$ and assignment $\phi(u) -\phi(v) = d \neq c \mod k$, the value of the penalty to this equation with respect to $\phi$ is proportional to $\min\setof{|c-d|,k-|c-d|}$, i.e., the distance between $c$ and $d$. Based on this new penalty function, we could obtain the same Cheeger inequality without any dependency on $k$. However, with this new penalty function we would end up solving a different version of the original \textsf{MAX-2-LIN($k$)}\ problem.
\end{remark}
Finally, we compare the proof techniques of Theorem~\ref{thm:cheeger_k} with other Cheeger-type inequalities in the literature: first of all, most of the Cheeger-type inequalities~(e.g.,~\cite{Alon86,Trevisan09,journals/jacm/LeeGT14,KwokLLGT13}) consider the case where every eigenvector is in $\mathbb{R}^n$ and are only applicable for undirected graphs, while for our problem the graph $G$ associated with $\mathcal{I}$ is directed and eigenvectors of $\mathcal{L}_{\mathcal{I}}$ are in $\mathbb{C}^n$. Therefore, constructing sweep sets in $\mathbb{C}$ is needed, which is more involved than proving similar Cheeger-type inequalities~(e.g.,~\cite{Alon86, Trevisan09}). Secondly,
by dividing the complex unit ball into $k$ regions, we
are able to show that a partial assignment corresponding to $k$ disjoint subsets can be found using a single eigenvector. This is quite different from the
techniques used for finding $k$ vertex-disjoint subsets of low conductance in an undirected graph, where $k$ eigenvectors are usually needed~(e.g.~\cite{journals/jacm/LeeGT14,KwokLLGT13, PSZ17}).
It is also worth mentioning that a Cheeger-like inequality was shown in \cite{BandeiraSS13} for a synchronisation problem which has some connections to \textsf{MAX-2-LIN($k$)}.~Their analysis, however, cannot be adapted in our setting.
We also remark that, while sweeping through values of $t$ is needed to obtain \emph{any} guarantee on the penalty of the partial assignment computed, we could in principle just choose a random angle $\eta$: in this way, however, the partial assignment returned would satisfy (\ref{eq:cheegereq4}) only in expectation.
\section{NP-hardness of Maximum 3-way Directed Cut}
\section{Algorithm for \textsf{MAX-2-LIN($k$)}\ on expanders \label{sec:expander}}
In this section we further develop techniques for analysing Hermitian Laplacian matrices by presenting a
subquadratic-time approximation algorithm for the \textsf{MAX-2-LIN($k$)}\ problem on expander graphs.
Our proof technique is inspired by Kolla's algorithm~\cite{kolla11}.
However, in contrast to the algorithm in \cite{kolla11},
we use the Hermitian Laplacian to represent a \textsf{MAX-2-LIN($k$)}\ instance and show that,
when the underlying graph
has good expansion,
a good approximate solution is encoded in the eigenvector associated with $\lambda_1(\mathcal{L}_\mathcal{I})$.
We assume that $G$ is a $d$-regular graph, and hence $\mathcal{I} = (G,k)$ is a \textsf{MAX-2-LIN($k$)}\ instance
with $n$ variables and $nd/2$ equations whose optimum
is $1 - \varepsilon$. One can view $\mathcal{I}$ as an instance generated by modifying $\varepsilon$ fraction of the constraints~(i.e., edges) from a completely satisfiable instance $\widehat{\ists} = (\widehat{G},k)$. Hence,
a satisfiable assignment $\psi: V\to [k]$ for $\widehat{\ists}$
will satisfy at least a $(1 - \varepsilon)$-fraction of equations in $\mathcal{I}$.
Now we discuss the techniques used to prove Theorem~\ref{thm:res2}.
Let $y_{\psi} \in \mathbb{C}^n$ be the normalised ``indicator vector'' of $\psi$, i.e.,
$\kh{y_{\psi}}_u = \frac{1}{\sqrt{n}} \omega_k^{\psi(u)}$.
Then it holds that
\begin{align}
\kh{y_\psi}^* \mathcal{L}_{\widehat{\ists}} y_{\psi} =
\frac{1}{d}
\sum_{u\leadsto v} b_{uv} \norm{(y_{\psi})_u - \omega_k^{c_{uv}} (y_{\psi})_v}^2
= 0.\notag
\end{align}
This implies that $y_{\psi}$ is an eigenvector associated with
$\lambda_1\left(\mathcal{L}_{\widehat{\ists}} \right)=0$.
We denote by $\mathcal{U}$ the underlying undirected graph of $G$,
and denote by $\mathcal{L}_{\mathcal{U}}$ the normalised Laplacian of
$\mathcal{U}$.
Note that since $\mathcal{U}$ is undirected, $\mathcal{L}_{\mathcal{U}}$
only contains real-valued entries.
We first show that the eigenvalues of
$\mathcal{L}_{\widehat{\ists}}$, the normalised Laplacian of the completely satisfiable instance, and of $\mathcal{L}_{\mathcal{U}}$, the normalised Laplacian of the underlining undirected graph $\mathcal{U}$, coincide.
Since $\mathcal{L}_{\mathcal{U}}$ is the Laplacian matrix of an expander graph, this implies that there is a gap between
$\lambda_1\left( \mathcal{L}_{\widehat{\ists}}\right)$ and $\lambda_2\left( \mathcal{L}_{\widehat{\ists}}\right)$.
\begin{lemma}\label{lem:lambdas}
It holds for all $1\leq i\leq n$ that
$\lambda_i\left(\mathcal{L}_{\widehat{\ists}}\right) = \lambda_i\left(\mathcal{L}_{\mathcal{U}}\right)$.
\end{lemma}
\begin{proof}
For any unit-norm eigenvector $f_i \in \mathbb{R}^n$ corresponding to the eigenvalue $\lambda_i(\mathcal{L}_{\mathcal{U}})$,
we construct another unit vector $g_i\in\mathbb{C}^n$ such that
\[
(g_i)_u = (y_{\psi})_u (f_i)_u = \frac{1}{\sqrt{n}}\ \omega_k^{\psi(u)} \cdot (f_i)_u.
\]
Then, it follows that
\begin{align*}
\kh{\mathcal{L}_{\widehat{\ists}} g_i}_u &=
\frac{1}{d} \kh{ \sum\nolimits_{u\leadsto v} b_{uv} \kh{(g_i)_u - \omega_k^{c_{uv}} (g_i)_v}
+ \sum\nolimits_{v\leadsto u} b_{vu} \kh{(g_i)_u - \conj{\omega_k}\,^{c_{uv}}(g_i)_v}} \notag \\
&=
\frac{(y_{\psi})_u}{d}
\kh{ \sum\nolimits_{u\leadsto v} b_{uv} \kh{(f_i)_u - (f_i)_v}
+ \sum\nolimits_{v\leadsto u} b_{vu} \kh{(f_i)_u - (f_i)_v}} \notag \\
&= (y_{\psi})_u\cdot \kh{\mathcal{L}_{\mathcal{U}} f_i}_u \notag \\
&= \lambda_i(\mathcal{L}_{\mathcal{U}}) (y_{\psi})_u (f_i)_u \notag \\
&= \lambda_i(\mathcal{L}_{\mathcal{U}}) (g_i)_u,
\end{align*}
which implies that $\mathcal{L}_{\widehat{\ists}} g_i = \lambda_i(\mathcal{L}_{\mathcal{U}}) g_i$.
Since by construction
$g_i$ is orthogonal to $g_j$ for any $i\neq j$,
the lemma follows.
\end{proof}
Next we bound the perturbation of the bottom eigenspace of $\mathcal{L}_{\widehat{\ists}}$ when the latter is turned into $\mathcal{L}_{\mathcal{I}}$.
In particular,
Lemma~\ref{lem:6.2} below proves that this perturbation does not affect too much to the vectors that have norm spreads out uniformly over all their coordinates.
\begin{lemma}\label{lem:6.2}
Let $f\in \mathbb{C}^n$ be
a vector such that $\norm{f_u} = \frac{1}{\sqrt{n}}$
for all $u\in V$.
It holds that
\begin{align}\label{eq:6.1}
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}f} \leq 2\sqrt{\varepsilon}.
\end{align}
\end{lemma}
\begin{proof}
Let $R\in\mathbb{R}^{n\times n}$ be a matrix defined by
\begin{align}
R_{uv} =
\begin{cases}
b_{uv}/d & \mbox{if}\ \kh{\mathcal{L}_\mathcal{I}}_{uv} \neq \kh{\mathcal{L}_{\widehat{\ists}}}_{uv}, \\
0 & \mbox{otherwise}.
\end{cases}\notag
\end{align}
Then, it holds that
\begin{align}\label{eq:sqr}
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}f} &=
\sqrt{\sum_{u\in V}
\norm{\sum_{v\in V} \kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}_{uv} f_v}^2} \notag \\
&\leq
\sqrt{\sum_{u\in V}
\kh{\sum_{v\in V} \norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}_{uv} f_v}}^2} \notag \\
&\leq
\sqrt{\sum_{u\in V}
\kh{\sum_{v\in V} 2R_{uv}\norm{f_v}}^2} \notag \\
& =\frac{2}{\sqrt{n}} \sqrt{\sum_{u\in V}
\kh{\sum_{v\in V} R_{uv}}^2}.
\end{align}
Since $\mathcal{I}$ can be viewed as modifying an $\varepsilon$-fraction of the edges from $\widehat{\ists}$, the sum of the entires
of $R$ is at most $\varepsilon n d / d = \varepsilon n$,
and the sum of each row of $R$ is at most $1$.
Since (\ref{eq:sqr})
is maximised when there are $\varepsilon n$ rows of $R$ whose sum is $1$,
we obtain (\ref{eq:6.1}).
\end{proof}
Based on Lemma~\ref{lem:6.2}, we prove that the change from $\mathcal{L}_{\widehat{\ists}}$ to $\mathcal{L}_\mathcal{I}$ doesn't have too much influence on the eigenvector associated with $\lambda_1(\mathcal{L}_\mathcal{I})$.
For simplicity, let $\lambda_2 = \lambda_2(\mathcal{L}_{\widehat{\ists}}) = \lambda_2(\mathcal{L}_{\mathcal{U}})$.
\begin{lemma}\label{lem:dsk}
Let $f_1\in \mathbb{C}^n$
be a unit eigenvector associated with $\lambda_1(\mathcal{L}_\mathcal{I})$.
Then we have
\begin{align} \notag
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}f_1} \leq 20\sqrt{\frac{\varepsilon}{\lambda_2}}.
\end{align}
\end{lemma}
\begin{proof}
We first show that $f_1$ is close to a unit vector
whose coordinates are all of the same norm $\frac{1}{\sqrt{n}}$.
Let $g\in\mathbb{C}^n$ and $h\in\mathbb{R}^n$ be defined by
\begin{align} \notag
g_u = \frac{(f_1)_u}{\sqrt{n}\norm{(f_1)_u}}\quad\text{and}\quad
h_u = \norm{(f_1)_u}.
\end{align}
Then we have
\begin{align}\label{eq:6.6}
h^\intercal \mathcal{L}_{\mathcal{U}} h =
\frac{1}{d} \sum_{u\leadsto v} \kh{\norm{(f_1)_u} - \norm{(f_1)_v}}^2 \leq
\frac{1}{d} \sum_{u\leadsto v} \norm{(f_1)_u - \omega_k^{c_{uv}} (f_1)_v}^2 =
f_1^* \mathcal{L}_\mathcal{I} f_1 \leq 2\varepsilon,
\end{align}
where the last inequality follows from the easy direction of our Cheeger inequality~(Theorem~\ref{thm:cheeger_k}). We introduce parameters $a,b$ such that
\begin{align}
h = a \vec{1}
+ b \vec{1}_\bot, \notag
\end{align}
where $\vec{1}$ is the normalised all-ones (i.e., with all $\frac{1}{\sqrt{n}}$ entries) vector and $\vec{1}_\bot$ is a unit vector orthogonal to $\vec{1}$. Since $\vec{1}$ is the eigenvector associated with $\lambda_1(\mathcal{L}_{\mathcal{U}}) = 0$,
it holds that
\begin{align}
h^{\intercal} \mathcal{L}_{\mathcal{U}} h = b^2\kh{\kh{\vec{1}_\bot}^\intercal \mathcal{L}_{\mathcal{U}} \vec{1}_\bot} \geq b^2\lambda_2 , \notag
\end{align}
which coupled with (\ref{eq:6.6}) gives us that $b \leq \sqrt{\frac{2\varepsilon}{\lambda_2}}$.
Hence, we can upper bound the distance between $h$ and $\vec{1}$ by \begin{align}
\norm{h - \vec{1}} =
\sqrt{(1 - a)^2 + b^2} \leq
\sqrt{1 - a^2 + b^2}
= \sqrt{2}b \leq
2\sqrt{\frac{\varepsilon}{\lambda_2}} \notag
\end{align}
where the first inequality holds since
$h$ is a unit vector and thus $a \in [0,1]$.
This gives us that
\begin{align*}
\norm{f_1 - g} = \sqrt{\sum_{u\in V} \norm{(f_1)_u - \frac{(f_1)_u}{\sqrt{n}\norm{(f_1)_u}}}^2}
= \sqrt{\sum_{u\in V} \kh{\norm{(f_1)_u} - \frac{1}{\sqrt{n}}}^2}
= \norm{h - \vec{1}}\leq 2\sqrt{\frac{\varepsilon}{\lambda_2}}.
\end{align*}
We can use this to derive the upper bound in this lemma by
\begin{align}
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}}f_1} &\leq
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}} g} +
\norm{\kh{\mathcal{L}_\mathcal{I} - \mathcal{L}_{\widehat{\ists}}} \kh{f_1 - g}} \notag \\
&\leq 2\sqrt{\varepsilon} + \norm{\mathcal{L}_\mathcal{I} \kh{f_1 - g}} + \norm{\mathcal{L}_{\widehat{\ists}} \kh{f_1 - g}} \notag\\
&\leq 2\sqrt{\varepsilon} + 4\norm{f_1 - g} \notag\\
&\leq 20 \sqrt{\frac{\varepsilon}{\lambda_2}}, \notag
\end{align}
where the second inequality follows from Lemma~\ref{lem:6.2},
and the third inequality follows from the fact that the eigenvalues
of $\mathcal{L}_\mathcal{I}$ and $\mathcal{L}_{\widehat{\ists}}$ are at most $2$.
\end{proof}
We then prove the following lemma which shows that
the eigenvector $f_1$ corresponding to $\lambda_1(\mathcal{L}_\mathcal{I})$ is close to $y_{\psi}$,
the indicator vector of the optimal assignment $\psi$.
\begin{lemma}\label{lem:span}
Let $f_1\in \mathbb{C}^n$
be a unit eigenvector associated with $\lambda_1(\mathcal{L}_\mathcal{I})$. Then, there exist $\alpha,\beta \in \mathbb{C}$ and a unit vector $y_\bot\in\mathbb{C}^n$
orthogonal to $y_{\psi}$ (i.e. $\kh{y_\bot}^* y_{\psi} = 0$) such that
$ f_1 = \alpha y_{\psi} + \beta y_\bot $
and
$
\norm{\beta} \leq 30 \sqrt{\varepsilon/ \lambda_2^3}$.
\end{lemma}
\begin{proof}
The proof essentially corresponds to the Davis-Kahan theorem~\cite{DavisKahan} for 1-dimensional eigenspaces. Let $y_{\psi} = v_1,\ldots,v_n\in\mathbb{C}^n$ be the orthonormal eigenvectors
associated with eigenvalues $0 = \lambda_1\left(\mathcal{L}_{\widehat{\ists}}\right) \le \cdots \le \lambda_n\left(\mathcal{L}_{\widehat{\ists}}\right)$ of $\mathcal{L}_{\widehat{\ists}}$,
which means $\mathcal{L}_{\widehat{\ists}}$ can be diagonalised by
$\mathcal{L}_{\widehat{\ists}} = \sum_{i=2}^n \lambda_i\left(\mathcal{L}_{\widehat{\ists}}\right) v_1 v_1^*$.
Then it holds that
\begin{align}\label{eq:6.14}
\norm{\mathcal{L}_{\widehat{\ists}} f_1}^2 =
f_1^{\intercal} \mathcal{L}_{\widehat{\ists}}^2 f_1
= \sum_{i=2}^n \lambda_i^2\left(\mathcal{L}_{\widehat{\ists}}\right) \norm{f_1^* v_i}^2
\geq \lambda_2^2\left(\mathcal{L}_{\widehat{\ists}}\right) \kh{1 - \norm{f_1^* y_{\psi}}^2}
= \lambda_2^2 \norm{\beta}^2.
\end{align}
By Lemma~\ref{lem:dsk},
the square root of this quantity can be upper bounded by
\begin{align}\label{eq:6.15}
\norm{\mathcal{L}_{\widehat{\ists}} f_1}
\leq
\norm{\mathcal{L}_{\mathcal{I}} f_1} +
\norm{\kh{\mathcal{L}_{\mathcal{I}} - \mathcal{L}_{\widehat{\ists}}} f_1}
\leq \lambda_1\kh{\mathcal{L}_{\mathcal{I}}} + 20 \sqrt{\frac{\varepsilon}{\lambda_2}}
\leq 30\sqrt{\frac{\varepsilon}{\lambda_2}},
\end{align}
where the last inequality follows by noting
$\lambda_1(\mathcal{L}_{\mathcal{I}}) \leq 2\varepsilon$ by the easy direction
of our Cheeger inequality and $\lambda_2\leq 2$.
Combining~(\ref{eq:6.14}) and~(\ref{eq:6.15}) proves the statement.
\end{proof}
Based on Lemma~\ref{lem:span},
$f_1$ is close to the indicator vector of an optimal assignment rotated by some angle.
In particular, we have that
\begin{align} \label{eq:abc}
\norm{f_1 - \frac{\alpha}{\norm{\alpha}} y_{\psi}} =
\sqrt{(1 - \norm{\alpha})^2 + \norm{\beta}^2} \leq
\sqrt{1 - \norm{\alpha}^2 + \norm{\beta}^2}
= \sqrt{2}\norm{\beta} \leq
30\sqrt{\frac{2\varepsilon}{\lambda_2^3}},
\end{align}
where $\frac{\alpha}{\norm{\alpha}} y_{\psi}$ is the vector that encodes the information of
an assignment that satisfies
all the equations in $\widehat{\ists}$
and at least $1 - \varepsilon$ fraction
of equations in $\mathcal{I}$.
Therefore, our goal is to recover $\frac{\alpha}{\norm{\alpha}} y_{\psi}$
from $f_1$.
\begin{proof}[Proof of Theorem~\ref{thm:res2}]
Let $\psi$ be the optimal assignment of $\mathcal{I}$
satisfying $1 - \varepsilon$ fraction of equations,
which is also a completely satisfying assignment of
$\widehat{\ists}$.
Let $f_1$ be a unit eigenvector associated with $\lambda_1(\mathcal{L}_{\mathcal{I}})$.
By Lemma~\ref{lem:span},
there exists $\alpha,\beta\in \mathbb{C}$ such that
$f_1 = \alpha y_{\psi} + \beta y_{\bot}$
where $\norm{\beta} \leq 30 \sqrt{\varepsilon /\lambda_2^3}$.
Our goal is to find a
vector $z_\phi \in \mathbb{C}^n$,
which equals the indicator vector of $\phi$ ratoted by some angle
and satisfies
\begin{align}\label{eq:sat}
\norm{f_1 - z_\phi } \leq
\norm{f_1 - \frac{\alpha}{\| \alpha\|} y_{\psi}} \leq
30\sqrt{\frac{2\varepsilon}{\lambda_2^3}},
\end{align}
where the last inequality follows by (\ref{eq:abc}).
The assignment $\phi$ corresponding to
such a $z_\phi$ will give us
that
the fraction of unsatisfied equations by
$\phi$ is
\begin{align}
p^{\phi}(\mathcal{I}) &\leq 10 k^2 z_{\phi}^* \mathcal{L}_{\mathcal{I}} z_{\phi} \notag \\
&= 10k^2 (z_{\phi} - f_1 + f_1)^* \mathcal{L}_{\mathcal{I}} (z_{\phi} - f_1 + f_1) \notag\\
&\leq k^2 \kh{ (z_\phi - f_1)^* \mathcal{L}_\mathcal{I} (z_\phi - f_1)
+ f_1^* \mathcal{L}_\mathcal{I} f_1
+ 2 \norm{ (z_\phi - f_1)^* \mathcal{L}_\mathcal{I} f_1} }
\notag\\
&\leq 10k^2 \kh{ 2 \norm{z_\phi - f_1}^2 + \lambda_1(\mathcal{L}_\mathcal{I}) +
2 \norm{z_\phi - f_1} \sqrt{\lambda_1{(\mathcal{L}_\mathcal{I})}} } \notag \\
& \leq 10k^2 \left( 2\cdot 900\cdot \frac{2\varepsilon}{\lambda_2^3} + 2\varepsilon + 2\cdot 30\cdot \sqrt{\frac{2\varepsilon}{\lambda_2^3} }\cdot
\sqrt{2\varepsilon} \right) \notag\\
&\leq 100000 k^2 \cdot \frac{\varepsilon}{\lambda_2^3}, \notag
\end{align}
where the factor $10 k^2$ above follows from the fact that
$\norm{1 -\omega_k^j}^2$ is at least $1/(10k^2)$
for $j = 1,\ldots,k-1$.
To find such vector $z_{\phi}$ satisfying (\ref{eq:sat}), we define $\phi_\eta: V \rightarrow [k]$ by
\begin{align}\notag
\phi_\eta(u) = \mathrm{arg\,min}_{j\in[k]} \norm{(f_1)_u - \mathrm{e}^{\eta i}\omega_k^j}.
\end{align}
Notice that, since $\frac{\alpha}{\norm{\alpha}}$ is equal to $\mathrm{e}^{\eta i}$ for some $\eta\in [0,2\pi)$, by defining
$
(z_{\phi_\eta})_u = \mathrm{e}^{\eta i} \omega_k^{\phi_\eta(u)}
$ the solution to the following optimisation problem
\begin{align}\notag
\min_{\eta\in [0,2\pi)} \norm{z_{\phi_\eta} - f_1}
\end{align}
gives us a vector that satisfies (\ref{eq:sat}).
To solve this optimisation problem,
we notice that it suffices to consider $\eta$ in the range $[0, 2\pi / k)$.
Therefore, we simply enumerate all $\eta$'s over the following discrete set:
\begin{align}\notag
\setof{\frac{t\sqrt{\varepsilon}}{\sqrt{n}}\ \left|\ t = 0,1,\ldots,\ceil{\frac{2\pi \sqrt{n}}{k \sqrt{\varepsilon}}}\right.}.
\end{align}
By enumerating this set, we can find an assignment $\phi$ and an $\eta$ such that
\begin{align}\notag
\norm{f_1 - z_{\phi_\eta} } \leq \norm{ f_1 - \frac{\alpha }{\| \alpha \|}y_{\psi} } + O(\sqrt{\varepsilon}),
\end{align}
which is enough to get our desired approximation.
Since the size of this set is $O\left(\frac{\sqrt{n}}{k\sqrt{\varepsilon}}\right)$,
the total running time is $O\left(\frac{n^{1.5}}{k\sqrt{\varepsilon}}\right)$
plus the running time needed to compute the eigenvector $f_1$.
\end{proof}
\section{Introduction}
In the \textsf{MAX-2-LIN($k$)}\ problem, we are given a system of $m$ linear equations of the form
\begin{equation}\label{eq:general}
u_i - v_i \equiv c_i\hspace{-5pt}\mod k
\end{equation}
where $u_i, v_i \in \setof{x_1,\ldots,x_n}$
and each equation has weight $b_i$.
The objective is to find an assignment to the variables $x_i$ that maximises the total weight of satisfied equations. As an important case of Unique Games~\cite{feigelovasz92,Kho02a}, the \textsf{MAX-2-LIN($k$)}\ problem has been extensively studied in theoretical computer science.
This problem is known
to be \textsf{NP}-hard to approximate within a ratio of $11/12+\delta$ for any constant $\delta> 0$~\cite{feige04,Hastad01}, and it is conjectured to be hard to distinguish between \textsf{MAX-2-LIN($k$)}\ instances for which a $(1-\varepsilon)$-fraction of equations can be satisfied versus
instances for which only an $\varepsilon$-fraction can be satisfied~\cite{KhotKMO07}.
On the algorithmic side, there has been a number of LP and SDP-based algorithms proposed for the \textsf{MAX-2-LIN($k$)}\ problem~(e.g., \cite{Kho02a,trevisan05,charikar06,gupta06}), and
the case of $k=2$, which corresponds to the classical \textsf{MAX-CUT} problem for undirected graphs~\cite{GW95, Karp72},
has been widely studied over the past fifty years.
In this paper we investigate efficient spectral algorithms for \textsf{MAX-2-LIN($k$)}. For any \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{I}$ with $n$ variables, we express $\mathcal{I}$ by a Hermitian Laplacian matrix $L_\mathcal{I}\in\mathbb{C}^{n\times n}$, and analyse the spectral properties of $L_{\mathcal{I}}$. In comparison to the well-known Laplacian matrix for undirected graphs~\cite{chung1}, complex-valued entries in $L_{\mathcal{I}}$ are able to express directed edges in the graph associated with $\mathcal{I}$, and at the same time ensure that all the eigenvalues of $L_{\mathcal{I}}$ are real-valued. We demonstrate the power of our Hermitian Laplacian matrices by relating the maximum number of satisfied equations of $\mathcal{I}$ to the spectral properties of $L_{\mathcal{I}}$. In particular, we develop a Cheeger inequality that relates partial assignments of $\mathcal{I}$ to $\lambda_1(L_{\mathcal{I}})$, the smallest eigenvalue of $L_{\mathcal{I}}$.
Based on a recursive application of the algorithm behind our Cheeger inequality, as well as a spectral sparsification procedure for \textsf{MAX-2-LIN($k$)}\ instances, we present an approximation algorithm for \textsf{MAX-2-LIN($k$)}\ that runs in $\widetilde{O}(k\cdot n^2)$ time. Our algorithm is easy to implement, and is
significantly faster than most SDP-based algorithms for this problem in the literature, while achieving similar guarantees for constant values of $k$.
The formal statement of our result is as follows:
\begin{theorem}
\label{thm:main1}
There is an $\widetilde{O}(k\cdot n^2)$-time algorithm such that, for any given \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{I}$ with optimum $1 - \varepsilon$, the algorithm returns
an assignment $\phi$ satisfying at least
a $(1 - O(k) \sqrt{\varepsilon})$-fraction of the equations\footnote{
An instance $\mathcal{I}$ has optimum $1-\varepsilon$, if the maximum fraction of the total weights of satisfied equations is $1-\varepsilon$.}.
\end{theorem}
Our result can be viewed as a generalisation of the \textsf{MAX-CUT} algorithm by Trevisan~\cite{Trevisan09}, who derived a Cheeger inequality that relates the value of the maximum cut to the smallest eigenvalue of an undirected graph's adjacency matrix. The proof of Trevisan's Cheeger inequality, however, is based on constructing sweep sets in $\mathbb{R}$, while in our setting constructing
sweep sets in $\mathbb{C}$ is needed, as the underlying graph defined by $L_{\mathcal{I}}$ is directed and eigenvectors of $L_{\mathcal{I}}$ are in $\mathbb{C}^n$.
The other difference between our result and the one in \cite{Trevisan09} is that
the goal of the \textsf{MAX-CUT} problem is to find a \emph{bipartition} of the vertex set, while for the \textsf{MAX-2-LIN($k$)}\ problem we need to
use an eigenvector to find $k$ vertex-disjoint subsets, which corresponds to subsets of variables assigned to the same value.
Our approach also shares some similarities with the one by Goemans and Williamson~\cite{GW04}, who presented a $0.793733$-approximation algorithm for \textsf{MAX-2-LIN(3)} based on Complex Semidefinite Programming. The objective function of their SDP relaxation is, in fact, exactly the quadratic form of our Hermitian Laplacian matrix $L_{\mathcal{I}}$, although this matrix was not explicitly defined in their paper. In addition, their rounding scheme divides the complex unit ball into $k$ regions according to the angle with a random vector, which is part of our rounding scheme as well. Therefore, if one views Trevisan's work~\cite{Trevisan09} as a spectral analogue to the celebrated SDP-based algorithm for \textsf{MAX-CUT} by Goemans and Williamson~\cite{GW95}, our result can be seen as a spectral analogue to the Goemans and Williamson's algorithm for \textsf{MAX-2-LIN($k$)}.
We further prove that, when the undirected graph associated with a \textsf{MAX-2-LIN($k$)}\ instance is an expander,
the approximation ratio from Theorem~\ref{thm:main1}
can be improved. Our result is formally stated as follows:
\begin{theorem}\label{thm:res2}
Let $\mathcal{I}$ be an instance of \textsf{MAX-2-LIN($k$)}\ on a $d$-regular graph with $n$ vertices\
and suppose its optimum is $1 - \varepsilon$.
There is an $\widetilde{O}\left(nd + \frac{n^{1.5}}{k\sqrt{\varepsilon}}\right)$-time algorithm that
returns an
assignment $\phi: V\to [k]$
satisfying at least a
\begin{align}\label{eq:thm2eq}
1 - O(k^2)\cdot \frac{\varepsilon}{\lambda_2^3(\mathcal{L}_{\mathcal{U}})}
\end{align}
fraction of equations in $\mathcal{I}$,
where $\lambda_2(\mathcal{L}_{\mathcal{U}})$
is the second smallest eigenvalue of the normalised Laplacian matrix of the underlying
undirected graph $\mathcal{U}$.
\end{theorem}
Our technique is similar to the one by Kolla~\cite{kolla11},
which was used to show that solving the \textsf{MAX-2-LIN($k$)}\ problem on expander graphs is easier.
In \cite{kolla11}, a \textsf{MAX-2-LIN($k$)}\ instance is represented by the label-extended graph, and the algorithm is
based on an exhaustive search
in a subspace spanned by eigenvectors associated with eigenvalues close to $0$.
When the underlying graph of the \textsf{MAX-2-LIN($k$)}\ instance has good expansion,
this subspace is of dimension $k$.
Therefore, the exhaustive search runs in
time $O\left(2^k + \mathrm{poly}(n\cdot k)\right)$, which is polynomial-time when $k = O(\log n)$.
Comparing with the work in \cite{kolla11}, we show that,
when the underlying graph has good expansion, the eigenvector associated with the smallest eigenvalue $\lambda_1(\mathcal{L}_\mathcal{I})$ of the Hermitian Laplacians suffices to give a good approximation.
We notice that Arora et al.~\cite{stoc/AroraKKSTV08} already showed that, for expander graphs, it is possible to
satisfy a $1-O(\varepsilon \log(1/\varepsilon))$ fraction of equations in polynomial time without any dependency on $k$. Their algorithm is based on
an SDP relaxation.
\paragraph{Other related work.}
There are many research results for the \textsf{MAX-2-LIN($k$)}\ problem~(e.g., \cite{Kho02a,trevisan05,charikar06,gupta06}), and we briefly discuss the ones most closely related to our work.
For the \textsf{MAX-2-LIN($k$)}\ problem and Unique Games, spectral techniques are usually employed to analyse the Laplacian matrix of the so-called Label-Extended graphs. Apart from the above-mentioned result~\cite{kolla11}, Arora, Barak and Steurer~\cite{abs}
obtained an $\mathrm{exp}\left((kn)^{O(\varepsilon)}\right)\mathrm{poly}(n)$-time algorithm for Unique Games, whose algorithm makes use of Label-Extended graphs as well.
We also notice that the adjacency matrix corresponding to our Hermitian Laplacian was considered by Singer~\cite{singer11} in relation to an angular synchronisation problem. The connection between the eigenvectors of such matrix and the \textsf{MAX-2-LIN($k$)}\ problem was also mentioned, but without offering formal approximation guarantees.
\section{Concluding remarks}
Our work leaves several open questions for further research: while the factor of $k$ in our Cheeger inequality (Theorem~\ref{thm:cheeger_k}) is needed, it would be interesting to see if it's possible to construct a different Laplacian for which a similar Cheeger inequality holds with a smaller dependency on $k$.
For example, instead of embedding vertices in $\mathbb{C}$ and mapping assignments to roots of unity, one could consider embedding vertices in higher dimensions using the bottom $k$ eigenvectors of the Laplacian of the label extended graph, and see if a relation between the imperfectness ratio of Definition~\ref{def:perfect} and the $k$-th smallest eigenvalue of this Laplacian still holds
Finally, we observe that several cut problems in directed graphs can be formulated as special cases of \textsf{MAX-2-LIN($k$)}\ (see, e.g., \cite{AnderssonEH01,GW04}). Because of this, we believe the Hermitian Laplacians studied in our paper will have further applications in the development of fast algorithms for combinatorial problems on directed graphs, and might have further connections to Unique Games.
\section{Hermitian Matrices for \textsf{MAX-2-LIN($k$)} \label{sec:preliminiares}}
We can write an instance of \textsf{MAX-2-LIN($k$)}\ by $\mathcal{I} = (G, k)$, where $G = (V,E,b,c)$ denotes
a directed graph with an edge weight function $b: E\to \mathbb{R}^+$
and an edge color function $c: E\to [k]$, where $[k]\stackrel{\mathrm{def}}{=} \setof{0,1,\ldots,k-1}$.
More precisely, every equation $u_i - v_i \equiv c_i \hspace{-5pt}\mod k$ with weight $b_i$
corresponds to a directed edge $(u_i, v_i)$ with weight $b(u_i,v_i) = b_{u_iv_i} = b_i$ and color $c(u_i,v_i) = c_{u_iv_i} = c_i$.
In the rest of this paper, we will assume that $G$ is weakly connected, and write $u\leadsto v$ if there is a directed edge from $u$ to $v$. The conjugate transpose of any vector $x\in\mathbb{C}^n$ is denoted by $x^*$
We define the Hermitian adjacency matrix $A_\mathcal{I}\in\mathbb{C}^{n\times n}$ for instance $\mathcal{I}$ by
\begin{align}
(A_\mathcal{I})_{uv} \stackrel{\mathrm{def}}{=}
\begin{cases}
b_{uv}\omega_k^{c_{uv}} & u\leadsto v, \\
b_{vu}\conj{\omega_k}\,^{c_{vu}} & v\leadsto u, \\
0 & \mathrm{otherwise},
\end{cases}
\end{align}
where $\omega_k = \exp\left(\frac{2\pi i}{k}\right)$ is the complex
$k$-th root of unity,
and $\conj{\omega_k} = \exp\left(-\frac{2\pi i}{k}\right)$ is its conjugate.
We define the degree-diagonal matrix $D_\mathcal{I}$ by
$(D_\mathcal{I})_{uu} = d_u$ where $d_u$ is the weighted degree given by
\begin{align}
d_u \stackrel{\mathrm{def}}{=} \sum_{u\leadsto v} b_{uv}
+ \sum_{v\leadsto u} b_{vu}.
\end{align}
The Hermitian Laplacian matrix is then defined by $L_{\mathcal{I}} = D_{\mathcal{I}} - A_{\mathcal{I}}$,
and the corresponding normalised Laplacian matrix by
$\mathcal{L}_{\mathcal{I}} = D_{\mathcal{I}}^{-1/2} L_{\mathcal{I}} D_{\mathcal{I}}^{-1/2} = I - D_{\mathcal{I}}^{-1/2} A_{\mathcal{I}} D_{\mathcal{I}}^{-1/2}$. The eigenvalues of
any matrix $A$ are expressed by $\lambda_1(A)\leq \ldots\leq \lambda_n(A)$.
The quadratic forms of $L_{\mathcal{I}}$ can be related to the corresponding instance of \textsf{MAX-2-LIN($k$)}\ by the following lemma.
\begin{lemma}\label{lem:qL}
For any vector $x \in \mathbb{C}^n$, we have
\begin{align}
x^* L_\mathcal{I} x = \sum_{u\leadsto v} b_{uv} \norm{x_u - \omega_k^{c_{uv}} x_v}^2
\end{align}
and
\begin{align}
x^* L_\mathcal{I} x = 2\sum_{u\in V} d_u \norm{x_u}^2 - \sum_{u\leadsto v} b_{uv} \norm{x_u + \omega_k^{c_{uv}} x_v}^2.
\end{align}
\end{lemma}
\begin{proof}
For any vector $x \in \mathbb{C}^n$,
we can write
\begin{align}
x^* A_\mathcal{I} x &=
\sum_{u\leadsto v}
b_{uv} \kh{ \conj{x_u} \omega_k^{c_{uv}} x_v
+ \conj{x_v}\,\conj{\omega_k}^{c_{uv}} x_u } \notag \\
&= - \sum_{u\leadsto v}b_{uv} \kh{
\kh{ \conj{x_u} - \conj{x_v}\, \conj{\omega_k}\,^{c_{uv}}}
\kh{ x_u - \omega_k^{c_{uv}} x_v }
- \norm{x_u}^2 - \norm{x_v}^2 } \notag \\
&= \sum_{u\in V} d_u \norm{x_u}^2-
\sum_{u\leadsto v} b_{uv} \norm{x_u - \omega_k^{c_{uv}} x_v}^2.
\end{align}
We can also write
\begin{align}
x^* A_\mathcal{I} x &=
\sum_{u\leadsto v}
b_{uv} \kh{ \conj{x_u} \omega_k^{c_{uv}} x_v
+ \conj{x_v}\,\conj{\omega_k}^{c_{uv}} x_u } \notag \\
&=\sum_{u\leadsto v}b_{uv}
\kh{ \kh{ \conj{x_u} + \conj{x_v}\, \conj{\omega_k}\,^{c_{uv}}} \kh{ x_u + \omega_k^{c_{uv}} x_v }
- \norm{x_u}^2 - \norm{x_v}^2 } \notag \\
&= - \sum_{u\in V} d_u \norm{x_u}^2+
\sum_{u\leadsto v} b_{uv} \norm{x_u + \omega_k^{c_{uv}} x_v}^2.
\end{align}
Combining these with $x^* D_\mathcal{I} x = \sum\nolimits_{u\in V} d_u\norm{x_u}^2$ finishes
the proof.
\end{proof}
The lemma below presents a qualitative relationship between the eigenvector associated with $\lambda_1(\mathcal{L}_\mathcal{I})$ and an assignment of $\mathcal{I}$.
\begin{lemma}\label{lem:spectrum}
All eigenvalues of $\mathcal{L}_{\mathcal{I}}$ are in the range $[0,2]$.
Moreover, $\lambda_1(\mathcal{L}_\mathcal{I}) = 0$ if and only if
there exists an assignment satisfying
all equations in $\mathcal{I}$.
\end{lemma}
\begin{proof}
To bound the eigenvalues of $\mathcal{L}_{\mathcal{I}}$, we look at the
following Rayleigh quotient
\[
\frac{x^* L_\mathcal{I} x}{x^* D_\mathcal{I} x},
\]
where $x \neq 0$.
By Lemma~\ref{lem:qL}, the numerator satisfies
\[
x^* L_\mathcal{I} x = \sum_{u\leadsto v} b_{uv} \norm{x_u - \omega_k^{c_{uv}} x_v}^2 \geq 0
\]
and also
\[
x^* L_\mathcal{I} x = 2\sum_{u\in V} d_u \norm{x_u}^2 - \sum_{u\leadsto v} b_{uv} \norm{x_u + \omega_k^{c_{uv}} x_v}^2
\leq 2\sum_{u\in V} d_u \norm{x_u}^2 = 2 x^* D_\mathcal{I} x.
\]
Therefore, the eigenvalues of $\mathcal{L}_\mathcal{I}$ lie in the range $[0,2]$.
Moreover, $\lambda_1(\mathcal{L}_\mathcal{I}) = 0$ if and only if
there exists an $x \in \mathbb{C}^n$
such that $x^* L_\mathcal{I} x = 0$, i.e.,
\[
\norm{x_u - \omega_k^{c_{uv}} x_v}^2 = 0
\]
holds for all $u\leadsto v$.
The existence of such an $x$ is equivalent to the existence
of an assignment satisfying all equations in $\mathcal{I}$.
\end{proof}
\section{Sparsification for \textsf{MAX-2-LIN($k$)} \label{sec:sparsification}}
We have seen in Section~\ref{sec:cheeger} that, given any vector in $\mathbb{C}^n$ whose quadratic form in $\mathcal{L}_{\mathcal{I}}$ is close to $\lambda_1(\mathcal{L}_\mathcal{I})$, we can compute a partial assignment of $\mathcal{I}$ with bounded approximation guarantee. In Section~\ref{sec:recursion} we will show that a total assignment can be found by recursively applying this procedure on variables for which an assignment has not yet been fixed. In particular, we will show that every iteration takes a time nearly-linear in the number of equations of our instance, which can be quadratic in the number of variables. To speed-up each iteration and obtain a time per iteration that is nearly-linear in the number of variables, we need to sparsify our input instance $\mathcal{I}$.
In this section we show that the construction of spectral sparsifiers by
effective resistance sampling
introduced by Spielman and Srivastava~\cite{SpielmanS11} can be generalised to sparsify \textsf{MAX-2-LIN($k$)}\ instances.
In particular,
given an instance $\mathcal{I}$ of \textsf{MAX-2-LIN($k$)}\ with $n$ variables and $m$ equations,
we can find in nearly-linear time
a sparsified instance $\mathcal{J}$ with $O(nk\log(nk))$ equations
such that for any partial assignment $\phi: V\to [k]\cup\setof{\bot}$,
the number of unsatisfied equations in $\mathcal{J}$ is preserved within a constant factor.
This means that we can apply our algorithm for \textsf{MAX-2-LIN($k$)}\ to a sparsified instance $\mathcal{J}$,
and any dependency on $m$ in our runtime can be replaced by $nk\log(nk)$.
We remark that we could simply apply uniform sampling to obtain a sparsified instance. However, this would in the end result in an additive error in the fraction of unsatisfied equations, much like in the case of the original Trevisan's result for \textsf{MAX-CUT}~\cite{Trevisan09}. With our construction, instead, we only lose a small multiplicative error. For completeness of discussion, we first recall the definition of a spectral sparsifier.
\begin{definition}
Let $G=(V,E,w)$ be an arbitrary undirected graph with $n$ vertices and $m$ edges. We call a sparse subgraph $H$ of $G$, with proper reweighting of the edges, a $(1+\delta)$-spectral sparsifier of $G$ if
\[
(1-\delta)x^{\intercal}L_Gx\leq x^{\intercal} L_{H} x\leq (1+\delta) x^{\intercal} L_Gx
\]
holds for any $x\in\mathbb{R}^n$,
where $L_G$ and $L_{H}$ are the respective Laplacian matrices of $G$ and $H$.
\end{definition}
To construct a sparsified instance $\mathcal{J}$, we introduce label-extended graphs and their Laplacian matrices to characterise the original \textsf{MAX-2-LIN($k$)}\ instance.
Let $P \in \mathbb{R}^{k\times k}$ be the permutation matrix where
$P_{ij}=1$ if $i \equiv j + 1 \hspace{-5pt}\mod k$, and $P_{ij}=0$ otherwise.
We define the adjacency matrix $\widetilde{A}_{\mathcal{I}} \in \kh{\mathbb{R}^{k\times k}}^{n\times n}$
for the label-extended graph of instance $\mathcal{I}$,
where each entry of $\widetilde{A}_{\mathcal{I}}$ is a matrix in $\mathbb{R}^{k\times k}$ given by
\begin{align}
(\widetilde{A}_\mathcal{I})_{uv} \stackrel{\mathrm{def}}{=}
\begin{cases}
b_{uv}P^{c_{uv}} & u\leadsto v, \\
b_{vu}\kh{P^{\intercal}}^{c_{vu}} & v\leadsto u, \\
0 & \mathrm{otherwise}.
\end{cases}
\end{align}
We then define the degree-diagonal matrix $\widetilde{D}_\mathcal{I}\in\kh{\mathbb{R}^{k\times k}}^{n\times n}$
by $(\widetilde{D}_\mathcal{I})_{uu} = d_u\cdot I_{k\times k}$,
where $I_{k\times k}$ is the $k\times k$ identity matrix,
and define the Laplacian matrix by
\begin{align}
& \widetilde{L}_{\mathcal{I}} = \widetilde{D}_{\mathcal{I}} - \widetilde{A}_{\mathcal{I}}.
\end{align}
Notice that the Hermitian Laplacian $L_{\mathcal{I}}$ is a \emph{compression} of $\widetilde{L}_{\mathcal{I}}$, i.e., there exists an orthogonal projection $U$ such that $U^*\widetilde{L}_{\mathcal{I}}U = L_{\mathcal{I}}$.
We further write
$\widetilde{L}_{\mathcal{I}} = \widetilde{D}_{\mathcal{I}} - \widetilde{A}_{\mathcal{I}}$ as a sum of matrices, each one
corresponding to a single equation.
More precisely, for an equation $u_i - v_i \equiv c_{i} \hspace{-5pt}\mod k$ with weight $b_{u_iv_i}$,
we define a matrix $\widetilde{B}_{uv}\in\kh{\mathbb{R}^{k\times k}}^{n\times 1}$ by
\begin{align}
\kh{\widetilde{B}_{uv}}_{w} =
\begin{cases}
I_{k\times k}& w = u, \\
-\kh{P^{\intercal}}^{c_{uv}} & w = v, \\
0 & \mathrm{otherwise.}
\end{cases}
\end{align}
Then it is easy to verify that
\begin{equation}\label{eq:llll}
\widetilde{L}_\mathcal{I} = \sum_{u\leadsto v} b_{uv} \widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal},
\end{equation}
and it holds for any $x \in \kh{\mathbb{R}^{k}}^n$ that
\begin{align}\label{eq:qua}
x^{\intercal} \widetilde{L}_{\mathcal{I}} x = \sum_{u\leadsto v} b_{uv} \norm{x_u - P^{c_{uv}} x_v}^2.
\end{align}
For any assignment $\phi : V \to [k]$,
we construct an indicator vector $\widetilde{x}_\mathcal{I}\in\kh{\mathbb{R}^k}^n$ by
$
\kh{\widetilde{x}_\mathcal{I}}_u = e_{\phi(u) + 1}
$,
where $e_j \in \mathbb{R}^k$ is the $j$-th standard basis vector.
Then it is easy to see that the total weight of unsatisfied equations for $\phi$ is $(1/2)\cdot \widetilde{x}_\mathcal{I}^{\intercal} \widetilde{L}_\mathcal{I} \widetilde{x}_\mathcal{I}$\footnote{
We remark that, if we use the Hermitian Laplacian matrices $L_\mathcal{I}$ directly instead, this relation only holds up to an $O(k)$ factor. That is why we sparsify the matrix $\widetilde{L}_{\mathcal{I}}$ instead. }.
Next we will present an algorithm that produces a sparse \textsf{MAX-2-LIN($k$)}\ instance $\mathcal{J}$ from $\mathcal{I}$ such that the total weight of unsatisfied equations is preserved\footnote{Notice that we can decide whether there is an assignment satisfying all the equations in $\mathcal{I}$ by fixing the assignment of an arbitrary vertex and determining assignments for other vertices accordingly, and therefore we only need to consider the case when $\mathcal{I}$ is unsatisfiable.}. Our algorithm can be described as follows:
first we sample every edge $(u,v)$ in $\mathcal{I}$ with a certain probability $ p_{uv}$, and set the weight of every sampled edge $(u,v)$ as its original weight multiplied by $1/ p_{u,v}$. Then, we output an instance $\mathcal{J}$ which consists of all the sampled edges. Notice that this sampling scheme ensures that $\mbox{{\bf E}}[\widetilde{L}_\mathcal{J}] = \widetilde{L}_\mathcal{I}$, but we need to choose $p_{uv}$ properly to ensure that (1) $\widetilde{L}_\mathcal{J}$ is sparse, and (2) $\widetilde{L}_\mathcal{J}$
approximates $\widetilde{L}_\mathcal{I}$ with high probability.
We remark that, while our algorithm and analysis closely follow the one by Spielman and Srivastava~\cite{SpielmanS11}, the requirement of our output is slightly stronger: in addition to the sparsity constraint for $\mathcal{J}$, we need to ensure that the output $\mathcal{J}$ is a valid \textsf{MAX-2-LIN($k$)}\ instance.
To analyse the algorithm, for every edge $(u,v)$ let $X_{uv}$ be a random matrix defined by
\begin{align}
X_{uv} =
\begin{cases}
\frac{b_{uv}}{p_{uv}}\cdot \widetilde{L}_\mathcal{I}^{-1/2}\widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1/2}&
\text{with probability $p_{uv}$,} \\
0 & \text{with probability $1 - p_{uv}$.}
\end{cases}
\end{align}
We set the probabilities to
\begin{align}
p_{uv} = \min\setof{1, 10 \left(1/\delta^2\right) \log(nk) \ell_{uv} }
\end{align}
where $\ell_{uv}$ is defined by
\begin{align}
\ell_{uv} = b_{uv} \trace{\widetilde{L}_\mathcal{I}^{-1/2} \widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1/2}}.
\end{align}
Notice that by the definition of $\ell_{uv}$ we have that
\begin{equation}\label{eq:boundluv}
\sum_{u\leadsto v} \ell_{uv} = \sum_{u\leadsto v}b_{uv}\cdot \trace{\widetilde{L}_\mathcal{I}^{-1/2} \widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1/2}} = \sum_{u\leadsto v}b_{uv}\cdot \trace{\widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal}} =nk.
\end{equation}
We also assume without loss of generality that
$p_{uv}< 1$ holds for all edges $(u,v)$. Otherwise, we split every edge $(u,v)$ with
\begin{align}
\ell_{uv} \geq \frac{1}{10\cdot (1/\delta^2)\cdot \log(nk)}
\end{align}
into $K = \ceil{10\cdot (1/\delta^2)\cdot \log(nk) }$ parallel edges,
each of which has weight $b_{uv} / K$.
By (\ref{eq:boundluv}) there are at most $O\left( (1/\delta^2)\cdot nk\log(nk) \right)$ such edges.
The following
matrix Chernoff bound will be used in our analysis.
\begin{lemma}[\cite{Tro12}]\label{lem:chernoff}
Let $X_1,\ldots,X_m$ be independent random $n$-dimensional symmetric positive semidefinite matrices
such that
\begin{itemize}
\item $\expec{}{X} = I$ where $X = \sum_{i=1}^m X_i$ and
\item $\norm{X_i} \leq R$ holds for all $i = 1,\ldots,m$.
\end{itemize}
Then, for any $0< \delta < 1$,
\begin{align}
& \prob{}{\lambda_{\mathrm{min}}\kh{\sum\nolimits_{i=1}^m X_i} \leq (1 - \delta)}
\leq n \cdot \exp\kh{- \frac{\delta^2}{2R}}, \notag \\
& \prob{}{\lambda_{\mathrm{max}}\kh{\sum\nolimits_{i=1}^m X_i} \geq (1 + \delta)}
\leq n \cdot \exp\kh{- \frac{\delta^2}{3R}}.
\end{align}
\end{lemma}
\begin{theorem}\label{thm:sparsification}
There is an algorithm that,
given an unsatisfiable instance $\mathcal{I}$ of \textsf{MAX-2-LIN($k$)}\ with
$n$ variables and $m$ equations
and parameter $0 < \delta < 1$,
returns in $\widetilde{O}(mk)$ time
an instance $\mathcal{J}$ with
the same set of variables and $O\left((1/\delta^2)\cdot nk \log(nk)\right)$ equations.
Furthermore,
with high probability
it holds for any vector $x\in\kh{\mathbb{R}^{k}}^n$ that
\begin{align}\label{eq:gua}
(1 - \delta) x^{\intercal} \widetilde{L}_\mathcal{I} x \leq
x^{\intercal} \widetilde{L}_\mathcal{J} x \leq
(1 + \delta) x^{\intercal} \widetilde{L}_\mathcal{I} x.
\end{align}
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:sparsification}]
We first prove the approximation guarantee~(\ref{eq:gua}). Since it holds that
\begin{align}
\expec{}{\sum_{u\leadsto v} X_{uv}}
= \sum_{u\leadsto v}\expec{}{X_{uv}}
= \sum_{u\leadsto v} b_{uv}\widetilde{L}_\mathcal{I}^{-1/2}\widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1/2}
= I,
\end{align}and
\begin{align}
\norm{X_{uv}} \leq & \frac{\delta^2}{10\log(nk)}
\cdot
\frac{\norm{\widetilde{L}_\mathcal{I}^{-1/2} \widetilde{B}_{uv} \widetilde{B}_{uv}^T \widetilde{L}_\mathcal{I}^{-1/2}}}
{\trace{\widetilde{L}_\mathcal{I}^{-1/2} \widetilde{B}_{uv} \widetilde{B}_{uv}^T \widetilde{L}_\mathcal{I}^{-1/2}}}
\leq
\frac{\delta^2}{10\log(nk)}.
\end{align}
By applying Lemma~\ref{lem:chernoff}, the approximation guarantee holds.
The number of edges in $\mathcal{J}$ follows from (\ref{eq:boundluv}) and Markov's inequality.
It remains to analyse the runtime needed to compute $\ell_{uv}$ for all edges.
To this end, we will need the Johnson-Lindenstrauss lemma~\cite{JL84,Ach03}
and nearly-linear time Laplacian solvers~\cite{ST14,CKM+14}.
Specifically, we write $\ell_{uv}$ as
\begin{align}
\ell_{uv} = & b_{uv}\cdot \trace{\widetilde{L}_\mathcal{I}^{-1/2} \widetilde{B}_{uv} \widetilde{B}_{uv}^{\intercal}\widetilde{L}_\mathcal{I}^{-1/2}} \notag \\
= & b_{uv}\cdot \trace{\widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}} \notag \\
= & b_{uv}\cdot \trace{\widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1} \kh{\sum_{x\leadsto y} b_{xy} \widetilde{B}_{xy}\widetilde{B}_{xy}^{\intercal}} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}}
\notag \\
= & b_{uv} \cdot \trace{\widetilde{B}_{uv}^{\intercal} \widetilde{L}_\mathcal{I}^{-1} \widetilde{F}^{\intercal} \widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}} \notag \\
= & b_{uv} \cdot \norm{\widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}}_\mathrm{F}^2,
\end{align}
where the second equality follows from the cyclicality of trace,
and in the last two lines we write $\widetilde{F}$ to denote the matrix whose rows
are $\sqrt{b_{xy}}\ \widetilde{B}_{xy}^{\intercal}$.
Now we generate a matrix $Q$ of size $q\times mk$ with random $\pm 1/\sqrt{q}$ entries,
where $q = 100 \log(nk)$.
By the Johnson-Lindenstrauss lemma,
with high probability it holds for all edges $(u,v)$ that
\begin{align}
\frac{1}{2} \norm{\widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}}_\mathrm{F}^2 \leq
\norm{Q \widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}}_\mathrm{F}^2 \leq
\frac{3}{2} \norm{\widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}}_\mathrm{F}^2.
\end{align}
Therefore, our runtime follows by computing
every row of $Q \widetilde{F} \widetilde{L}_\mathcal{I}^{-1} \widetilde{B}_{uv}$
by a nearly-linear time Laplacian solver.
\end{proof}
|
2,877,628,089,475 | arxiv | \section{Introduction}
One of the major focuses in the study of nuclear matter has recently been
to understand the equation of state (EOS) of asymmetric nuclear matter and the
density dependence of the nuclear symmetry energy.
The nuclear symmetry energy is a fundamental quantity
which determines several important properties of very small
entities such as the atomic nuclei as well as very large objects such as neutron stars
\cite{prakash}.
In fact, the behavior of nuclear symmetry energy is most uncertain among all
properties of dense nuclear matter. Furthermore, the symmetry energy is important
for modeling nuclear matter by probing the isospin part of nuclear
interactions. Recent \cite{tili} experimental studies of
isospin-sensitive observables in intermediate-energy
nuclear reactions involving radioactive beams have been quite useful in
providing some constraints on the density dependence of nuclear
symmetry energy at subsaturation densities. The effects of symmetry energy and
its slope on neutron star properties is an important area of study.
Another area of relevance in
the study of asymmetric nuclear matter is the instabilities associated with possible
liquid-gas phase transitions at subsaturation densities. Such
liquid-gas phase transition plays an important role in the
description of the crust of compact star matter at densities between
$0.03$ fm$^{-3}$ and saturation density ($~0.15$ fm$^{-3}$).
Here, we would like to address these two relevant aspects in the study of
asymmetric nuclear matter in a phenomenological model that we have used in
our earlier work for symmetric nuclear matter.
There has been a proliferation of phenomenological models
to describe infinite nuclear matter and also properties of
finite nuclei. These are in fact
essential steps in the development of this area of study for which realistic
first-principles theoretical descriptions as well as adequate experimental
or observational data are not available. Variations in different phenomenological
approaches stretching from the nonrelativistic to the relativistic are
tried incorporating some further aspects of theoretical requirements in the
model. All these models are usually set in terms of parameters that are fit
to reproduce the properties of either finite nuclei \cite{nl3} or bulk nuclear matter.
As a result, most of the models behave more or less similarly as far as the
equation of state
is concerned around the saturation density and at zero temperature. However, when these
models are used to describe nuclear matter at subsaturation densities to explain
the liquid-gas phase transition or at high densities to explain neutron star matter,
they yield very different results. Therefore, it has been seen as essential
to incorporate some constraints related to symmetry energy and its derivatives
by using up-to-date theoretical and experimental information. Most of the
relativistic-mean-field (RMF) models are attempts in these directions. However, in these
models nucleons are treated as structureless point objects. Therefore, as a next step
in the requirement of incorporating the quark structure of the nucleon with
meson couplings at the basic level; quark-meson-coupling (QMC) models have been proposed
\cite{guichon} and properties of nuclear matter have been studied in great detail
in a series of works \cite{ST,recent,temp,phase}. In these models nucleons are
described as a system of nonoverlapping MIT bags which interact through effective
scalar and vector meson exchanges at the quark level. However, it has been argued
that the hadronic structure described by the MIT-bag model suffers from some
theoretical inadequacy due to the sharp bag boundary in breaking chiral
symmetry, which is a good symmetry of strong interactions within the partially
conserved axial current (PCAC) limit.
Therefore, more sophisticated versions such as the Cloudy Bag Model (CBM) have
been proposed for the study of hadronic structure. So, to further include this
aspect of the physics requirement, it would be more appropriate to develop a
quark-meson coupling model where nucleon structure is described by models
like the CBM instead of by MIT bags. As an alternative approach
\cite{barik,frederico89,batista}
to the CBM, the relativistic independent quark model with a phenomenologically
averaged confining potential in equally mixed scalar-vector harmonic form in
the Dirac frame work has been used extensively with remarkable consistency
in the baryonic as well as the mesonic sector \cite{bdd}. This model has provided a
very suitable alternative to the otherwise successful cloudy bag model in
describing hadronic structure with its static properties and various
decay properties.
We therefore proposed in our earlier work \cite{rnm} a modified quark-meson
coupling model (MQMC), which is based on a suitable
confining relativistic independent quark potential rather than a bag
to address the nucleon structure in vacuum as an alternative
approach to QMC for the study of symmetric nuclear matter.
This attempt was not so much as to plead superiority of MQMC over QMC at
this level. It has only incorporated an essential aspect of the physics
requirement missing in MIT-bag model to stand as an alternative to the more
appropriate CBM. Further investigations are
necessary to check its consistency and its predictability for any
new physical features. In the MQMC model, we have
studied the bulk nuclear properties such as the compressibility, the
structure of EOS, and also discussed
some implications of chiral symmetry in nuclear matter along with the nucleon
and nuclear $\Sigma$ term and the sensitivity of nuclear
matter binding energy with variations in the light quark mass. The results
obtained in such a picture for symmetric nuclear matter were quite encouraging.
In the present attempt, we study the bulk properties of asymmetric nuclear
matter
and also the low-density instabilities of the system in such a model. To treat
the asymmetric nuclear matter, we incorporate in our model the contribution of
the isovector vector meson $\rho$ in addition to those of the isoscalar scalar
meson $(\sigma)$ and isoscalar vector meson$(\omega)$ considered earlier
for symmetric nuclear matter \cite{rnm}. Such
studies are also useful to discuss the systems such as neutron stars with
$N \neq Z$.
A correlation between the symmetry energy ${\cal E}_{sym}$ and its slope $L$
has been verified recently by Ducoin {\it{et al.}} \cite{ducoin} for a set of
effective relativistic and nonrelativistic nuclear models. Such a study was
based on numerical results for ${\cal E}_{sym}$ and $L$ obtained from
different parametrizations.
Theoretically ${\cal E}_{sym}$ and $L$ are constrained \cite{stone,chen}.
In a recent paper, Santos {\it{et al}.} \cite{correl} have established an
analytic relationship between these quantities.
In this context QMC-based models have not been studied. We have made an
attempt to set up a relationship between these two quantities analytically.
The paper is organized as follows: In Sec. II, a brief outline
of the model describing the nucleon structure in vacuum is discussed.
The nucleon mass is then
realized by appropriately taking into account the center-of-mass correction,
pionic correction, and gluonic correction. The EOS is then developed.
In Sec. III, we discuss the nuclear symmetry energy, its slope and
incompressibility, and observe its density dependence. The thermodynamic
instabilities of the system are analyzed in Sec. IV.
We establish the analytic relationship between ${\cal E}_{sym}$ and $L$
and discuss the results in Sec. V.
\section{Modified quark meson coupling model}
Recently, the modified quark-meson coupling model was adopted
for symmetric nuclear matter where the $NN$ interaction was realized in a
mean-field approach through the exchange of effective $(\sigma,\omega)$
mesonic fields coupling to the quarks inside the nucleon \cite{rnm}. We now
extend this model to asymmetric nuclear matter and include the contribution of
the isovector vector meson, $\rho$, in addition to $\sigma$ and $\omega$ mesons.
In view of this, we briefly present the outlines of our approach
\cite{rnm} in the present context.
We first consider nucleons as a composite of constituent quarks confined in
a phenomenological flavor-independent confining potential, $U(r)$ in an equally
mixed scalar and vector harmonic form inside the nucleon \cite{rnm}, where
\[
U(r)=\frac{1}{2}(1+\gamma^0)V(r),
\]
with
\begin{equation}
V(r)=(ar^2+V_0),~~~~~ ~~~ a>0.
\label{eq:1}
\end{equation}
Here $(a,~ V_0)$ are the potential parameters. The confining interaction here
provides the zeroth-order quark dynamics of the hadron. In the medium, the
quark field $\psi_q({\mathbf r})$ satisfies
the Dirac equation
\begin{equation}
[\gamma^0~(\epsilon_q-V_\omega-
\frac{1}{2} \tau_{3q}V_\rho)-{\vec \gamma}.{\vec p}
-(m_q-V_\sigma)-U(r)]\psi_q(\vec r)=0
\end{equation}
where $V_\sigma=g_\sigma^q\sigma_0$, $V_\omega=g_\omega^q\omega_0$ and
$V_\rho=g_\rho^q b_{03}$; where $\sigma_0$, $\omega_0$, and $b_{03}$ are the
classical meson fields, and
$g_\sigma^q$, $g_\omega^q$, and $g_\rho^q$ are the quark couplings to
the $\sigma$, $\omega$, and $\rho$ mesons, respectively. $m_q$ is the quark
mass and $\tau_{3q}$ is the third component of the Pauli matrices. In the
present paper, we consider nonstrange $q=u$ and $d$ quarks only.
We can now define
\begin{equation}
\epsilon^{\prime}_q= (\epsilon_q^*-V_0/2)~~~
\mbox{and}~~~ m^{\prime}_q=(m_q^*+V_0/2),
\label{eprim}
\end{equation}
where the effective quark energy,
$\epsilon_q^*=\epsilon_q-V_\omega-\frac{1}{2}\tau_{3q} V_\rho$ and
effective quark mass, $m_q^*=m_q-V_\sigma$. We now introduce $\lambda_q$
and $r_{0q}$ as
\begin{equation}
(\epsilon^{\prime}_q+m^{\prime}_q)=\lambda_q~~
~~\mbox{and}~~~~r_{0q}=(a\lambda_q)^{-\frac{1}{4}}.
\label{eq:8}
\end{equation}
The ground-state quark energy can be obtained from the eigenvalue condition
\begin{equation}
(\epsilon^{\prime}_q-m^{\prime}_q)\sqrt \frac{\lambda_q}{a}=3.
\label{eq:11}
\end{equation}
The solution of equation \eqref{eq:11} for the quark energy
$\epsilon^*_q$ immediately leads to
the mass of the nucleon in the medium in zeroth order as
\begin{equation}
E_N^{*0}=\sum_q~\epsilon^*_q
\label{eq:12}
\end{equation}
We next consider the spurious center-of-mass
correction $\epsilon_{c.m.}$, the pionic correction $\delta M_{N}^\pi$
for restoration of chiral symmetry, and the
short-distance one-gluon exchange contribution $(\Delta E_N)_g$
to the zeroth-order nucleon mass in the medium.
The center-of-mass correction $\epsilon_{c.m.}$ and the pionic corrections
$\delta M_{N}^\pi$ in the present model are found, respectively, as \cite{rnm}
\begin{equation}
\epsilon_{c.m.}=\frac{(77\epsilon_u^{\prime}+31m_u^{\prime})}
{3(3\epsilon_u^{\prime}+m_u^{\prime})^2r_{0u}^2} \,
\end{equation}
and
\begin{equation}
\delta M_{N}^\pi=- \frac{171}{25}I_{\pi}f_{NN\pi}^2.
\label{pion-corr}
\end{equation}
Here,
\begin{equation}
I_{\pi}=\frac{1}{\pi{m_{\pi}}^2}\int_{0}^{\infty}dk
\frac{k^4u^2(k)}{w_k^2},
\end{equation}
with the axial vector nucleon form factor given as
\begin{equation}
u(k)=\Big[1-\frac{3}{2} \frac{k^2}{{\lambda}_q(5\epsilon_q^{\prime}+
7m_q^{\prime})}\Big]e^{-k^2r_0^2/4} \ .
\end{equation}
The pseudovector nucleon pion coupling constant $f_{NN{\pi}}$ can be obtained
from the familiar Goldberg Triemann relation by using the axial-vector coupling-constant value $g_A$ in the model,
as discussed in Ref. \cite{rnm}.
The color-electric and color-magnetic contributions to the gluonic correction
which arises due to one-gluon exchange at short distances are given as:
\begin{equation}
(\Delta E_N)_g^E={\alpha_c}(b_{uu}I_{uu}^E+b_{us}I_{us}^E+b_{ss}I_{ss}^E) \ ,
\label{enge}
\end{equation}
and due to color-magnetic contributions, as
\begin{equation}
(\Delta E_N)_g^M={\alpha_c}(a_{uu}I_{uu}^M+a_{us}I_{us}^M+a_{ss}I_{ss}^M) \ ,
\label{engm}
\end{equation}
where $a_{ij}$ and $b_{ij}$ are the numerical coefficients depending on each
baryon.
The color-electric contributions to the correction of baryon masses due to one gluon
exchange are calculated in a field-theoretic manner \cite{rnm}. It can be found that
the numerical coefficient for color-electric contributions such as
$b_{uu}, b_{us}$, and $b_{ss}$ comes out to be zero.
From calculations we have $a_{uu}=-3$ and
$a_{us}=a_{ss}=b_{uu}=b_{us}=b_{ss}=0$ for the nucleons.
The quantities $ I_{ij}^{E} $ and $ I_{ij}^{M} $ are given in the following equation
\begin{eqnarray}
I_{ij}^{E}=\frac{16}{3{\sqrt \pi}}\frac{1}{R_{ij}}\Bigl[1-
\frac{\alpha_i+\alpha_j}{R_{ij}^2}+\frac{3\alpha_i\alpha_j}{R_{ij}^4}
\Bigl],
\nonumber\\
I_{ij}^{M}=\frac{256}{9{\sqrt \pi}}\frac{1}{R_{ij}^3}\frac{1}{(3\epsilon_i^{'}
+m_{i}^{'})}\frac{1}{(3\epsilon_j^{'}+m_{j}^{'})} \ ,
\end{eqnarray}
where
\begin{eqnarray}
R_{ij}^{2}&=&3\Bigl[\frac{1}{({\epsilon_i^{'}}^2-{m_i^{'}}^2)}+
\frac{1}{({\epsilon_j^{'}}^2-{m_j^{'}}^2)}\Bigl]
\nonumber\\
\alpha_i&=&\frac{1}{ (\epsilon_i^{'}+m_i^{'})(3\epsilon_i^{'}+m_{i}^{'})} \ .
\end{eqnarray}
In the calculation we have taken $\alpha_c= 0.58$ as the strong-coupling
constant in QCD at the nucleon scale \cite{barik}. The color-electric
contribution is zero here, and the gluonic corrections to the mass of the
nucleon are due to color-magnetic contributions only.
Finally, treating all these corrections independently, the
mass of the nucleon in the medium becomes
\begin{equation}
M_N^*=E_N^{*0}-\epsilon_{c.m.}+\delta M_N^\pi+(\Delta E_N)^E_g+
(\Delta E_N)^M_g.
\label{mass}
\end{equation}
The total energy density and pressure at a
particular baryon density for the nuclear matter becomes
\begin{eqnarray}
\label{engd}
{\cal E} &=&\frac{1}{2}m_\sigma^2 \sigma_0^2+\frac{1}{2}m_\omega^2 \omega^2_0
+\frac{1}{2}m_\rho^2 b^2_{03} \nonumber\\ &+&
\frac{\gamma}{(2\pi)^3}\sum_{N=p,n}\int ^{k_{f,N}} d^3 k \sqrt{k^2+{M_N^*}^2},\\
P&=&-~\frac{1}{2}m_\sigma^2 \sigma_0^2+\frac{1}{2}m_\omega^2 \omega^2_0+
\frac{1}{2}m_\rho^2 b_{03}^2\nonumber\\
&+&\frac{\gamma}{3(2\pi)^3}\sum_{N=p,n}\int ^{k_{f,N}} \frac{k^2~ d^3 k}
{\sqrt{k^2+{M_N^*}^2}},
\end{eqnarray}
where $\gamma=2$ is the spin degeneracy factor for nuclear matter.
The nucleon density becomes
\begin{equation}
\rho_N= \frac{\gamma}{(2\pi)^3}\int_0^{k_{f,N}} d^3 k
=\frac{\gamma k_{f,N}^3}{6\pi^2}~~~\mbox{where} ~~N=p,n.
\end{equation}
Therefore, the total baryon density becomes $\rho_B=\rho_p+\rho_n$
and the (third component of) isospin density $\rho_3=\rho_p-\rho_n $.
The proton fraction, $y_p$ is defined as
\begin{equation}
y_p=\frac{\rho_p}{\rho_B}
\end{equation}
where $\rho_p$ and $\rho_n$ are the proton and neutron densities.
The vector mean-fields $\omega_0$ and $b_{03}$ are determined through
\begin{equation}
\omega_0=\frac{g_\omega}{m_\omega^2} \rho_B~~~~~~~~~~
b_{03}=\frac{g_\rho}{2m_\rho^2} \rho_3,
\label{omg}
\end{equation}
where $g_\omega=3 g_\omega^q$ and $g_\rho= g_\rho^q$.
Finally, the scalar mean-field $\sigma_0$ is fixed by
\begin{equation}
\frac{\partial {\cal E }}{\partial \sigma_0}=0.
\label{sig}
\end{equation}
The iso-scalar scalar and iso-scalar vector couplings $g_\sigma^q$ and $g_\omega$
are fit to the saturation density and binding energy for nuclear
matter. The isovector vector coupling $g_\rho$ is set by fixing
the symmetry energy.
For a given baryon density, $\omega_0$, $b_{03}$, and $\sigma_0$ are
calculated from Eqs. \eqref{omg} and \eqref{sig}, respectively.
\section{The symmetry energy}
We may define the neutron-excess parameter $t=
\frac{\rho_n-\rho_p}{\rho_n+\rho_p}=(1-2y_p)$ so that the nuclear
symmetry energy ${\cal E}_{sym}$ can be obtained as the difference between
the total energy per baryon ${\cal E}/\rho_B=E(\rho_B,t)$ of pure neutron
matter and that of isopspin-symmetric matter at baryon density
$\rho_B$. Here we consider the nuclear matter consisting of protons and neutrons
only with $y_p$ as the proton fraction. An expansion of the total energy per
baryon, $E(\rho_B,t)$, with respect to the neutron-excess parameter, becomes
\cite{steiner}
\begin{equation}
E(\rho_B,t)=E(\rho_B,0)+tE_1(\rho_B)+\frac{t^2}{2!}E_2(\rho_B)+
\frac{t^3}{3!}E_3(\rho_B)+\cdots,
\label{esymt}
\end{equation}
where $E_1, E_2, E_3, \cdots$, etc. are the first-, second- and third- order
derivatives with respect to $t$ in a Taylor's expansion. However, neglecting Coulomb contributions near the isospin symmetry of QCD, demands the total energy
of pure neutron matter to be same as that of pure proton matter,
for which the odd powers in $t$ are to be forbidden in the above
expansion. Again
for densities near or below the saturation density ($\rho_B=\rho_0$), truncation
of this expansion to quadratic terms in $t$ is considered to be a good
approximation. In view of that, the coefficient of the quadratic term in $t$
can be identified as the symmetry energy
\begin{equation}
{\cal E}_{sym}(\rho_B)=\frac{1}{2}\left[\dfrac{\partial^2E(\rho_B)}
{\partial {t^2}}\right]_{t=0}=\frac{k_{f,N}^2}{6E_{f,N}^*}
+\frac{g_\rho^2} {8m_\rho^{2}}\rho_B,
\label{engs}
\end{equation}
where $E_{f,N}^*=(k_{f,N}^2+M_N^{*2})^{1/2}$.
This may be considered to be a good approximation even for small proton
fraction $y_p$, which can be valid for finite nuclei. But for
nuclear matter at densities in excess of the saturation density $\rho_0$,
effects of higher order than quadratic in the expansion may be
important. Therefore, in order to study the density dependence of
${\cal E}_{sym}(\rho_B)$, one may
expand this as a function of $\rho_B$ around saturation density $\rho_0$ in
terms of a parameter $x=\frac{(\rho_B-\rho_0)}{3\rho_0}$ to obtain
\begin{equation}
{\cal E}_{sym}(\rho_B)=J+xL^0+\frac{x^2}{2!}K_{sym}^0+
\frac{x^3}{3!}Q_{sym}^0+\cdots,
\label{engs1}
\end{equation}
so as to consider the symmetry-energy parameters as follows:
\begin{eqnarray}
J&=&{\cal E}_{sym}(\rho_0)\nonumber\\
L^0&=&3\rho_0\dfrac{\partial {\cal E}_{sym}(\rho_B)}{\partial \rho_B}
\Bigg|_{\rho_B=\rho_0}~{\mbox(Slope ~ of ~ {\cal E}_{sym})}\nonumber\\
K_{sym}^0&=&9\rho_0^2\dfrac{\partial^2 {\cal E}_{sym}(\rho_B)}{\partial
\rho_B^2} \Bigg|_{\rho_B=\rho_0}{\mbox(Curvature ~ of ~ {\cal E}_{sym})}\nonumber\\
Q_{sym}^0&=&27\rho_0^3\dfrac{\partial^3 {\cal E}_{sym}(\rho_B)}{\partial
\rho_B^3} \Bigg|_{\rho_B=\rho_0}{\mbox(Skewness ~ of ~ {\cal E}_{sym})}\nonumber\\
\label{cesym}
\end{eqnarray}
Apart from the quantities in Eq. \eqref{cesym}, the following quantities
calculated from pressure $P$ and energy density ${\cal E}$ for the
consideration of constraints and correlations studies are
\begin{eqnarray}
K_0&=&9\left[\frac{dP}{d\rho_B}\right]_{\rho_B=\rho_0,y_p=1/2}
~{\mbox(Compressibility)}\nonumber \\
Q_0&=&27\rho_0^3\dfrac{\partial^3 {\cal E}/\rho_B}{\partial
\rho_B^3} \Bigg|_{\rho_B=\rho_0,y_p=1/2}{\mbox(Skewness ~ coefficient)}\nonumber\\
\label{ceng}
\end{eqnarray}
and the volume part of the iso-spin incompressibility
\begin{equation}
K_{{\tau},v}=K_{sym}^0-6L^0-\frac{Q_0}{K_0}L^0.
\end{equation}
We have assumed $K_{{\tau},v}=K_{\tau}$ since the
volume term is dominant \cite{dutraetal}.
These parameters characterize the density dependence of nuclear symmetry
energy around normal nuclear matter density and thus provide important
information on the behavior of nuclear symmetry energy at both high and
low densities. Also, the curvature parameter $K_{sym}^0$
distinguishes the different parametrizations.
A more significant measurement would be the evaluation of
the shift of the incompressibility with asymmetry, which is given by
\begin{equation}
K_{asy}=K_{sym}^0-6L^0
\end{equation}
because this value can be correlated to experimental observations of the
giant monopole resonance (GMR) of neutron-rich nuclei. Recent observations
of the GMR \cite{tili} on even-$A$ Sn isotopes give a quite stringent value of
$K_{asy}=-550 \pm 100$ MeV. In the present model we determine this value
for three quark masses of $300$, $40$, and $5$ MeV and observe that
they are consistent with the GMR measurements.
The compressibility $K_0$ at saturation density can be determined analytically, from Eq. \eqref{ceng}:
\begin{eqnarray}
K_0&=&9\left(\frac{g_\omega}{m_\omega}\right)^2\rho_0+\frac{3k_{f,N}^2}{E_{f,N}^*}
+\frac{3k_{f,N}M_N^*}{E_{f,N}^*}\cdot\frac{dM_N^*}{dk_{f,N}}.
\end{eqnarray}
The study of the correlation between symmetry energy and its slope can be
performed analytically. For this purpose we use the Eqs. \eqref{engd} and
\eqref{engs} to find ${\cal E}_{sym}$.
In this model, we get the closed-form expression
\begin{equation}
L^0=3J+f(M_N^*,\rho_0,B_0,K_0),
\end{equation}
where
\begin{eqnarray}
f(M_N^*,\rho_0,&&B_0,K_0)=\frac{1}{2}\left(\frac{3\pi^2}{2}\right)^{2/3}
\frac{1}{E_{f,N}^*}\times\nonumber\\
&&\left[\left(\frac{g_\omega}
{m_\omega}\right)^2\frac{\rho_0^{5/3}}{E_{f,N}^*}-\frac{\rho_0^{2/3}K_0}
{9E_{f,N}^*}-\frac{\rho_0^{2/3}}{3}\right].
\end{eqnarray}
The correlation function $f(M_N^*,\rho_0,B_0,K_0)$ exhibits the dependence on
the different bulk parameters $M_N^*, \rho_0, B_0$, and $K_0$.
\section{Stability Conditions}
Nuclear forces have an attractive long-range part and a repulsive hard core
similar to a Van der Waals fluid. It is expected to present a
liquid and a gas phase characterized by the respective densities. Nucleons
can be either protons or neutrons. Such a two-component system undergoes
liquid-gas phase transition. The asymmetric nuclear matter (ANM) shows
two types of instabilities \cite{chomaz}: a mechanical instability conserving the proton
concentration and a chemical instability occurring at constant density.
We consider asymmetric nuclear matter characterized by proton and neutron
densities $\rho_N=\rho_p,~\rho_n$ and transform these into a set of two mutually
commuting charges $\rho_i=\rho_B,~\rho_3$ \cite{unique}.
In infinite matter the extensivity of free-energy implies that it can be
reduced to a free energy density: ${\cal F} (T,\rho_i)={\cal E}-TS$ which
at $T=0$ reduces to energy density ${\cal E}$ only.
Since, we deal with a two-component nuclear medium, spinodal instabilities are
intimately related to phase equilibria and phase transitions. Although it
consists of unstable states, the spinodal region of the phase diagram can
be addressed by standard thermodynamics.
The condition for stability implies that the free energy density is a
convex function of the densities $\rho_i$. A local necessary condition is
the positivity of the curvature matrix:
\begin{equation}
{\cal F}_{ij}=\left(\frac{\partial^2{\cal F}}{\partial \rho_i\partial\rho_j}
\right)_T\equiv\left(\frac{\partial \mu_i}{\partial\rho_{j}}\right)_T.
\label{stability}
\end{equation}
Here we used $\left.\frac{\partial{\cal F}}{\partial \rho_i}
\right|_{T,\rho_{j\ne i}}=\mu_i$,
where the effective chemical potentials in the present context are given by
\begin{eqnarray}
\mu_p&=&\sqrt{k_{f,p}^2+{M^*_p}^2}+V_\omega+\frac{1}{2} V_\rho,\nonumber\\
\mu_n&=&\sqrt{k_{f,n}^2+{M^*_n}^2}+V_\omega-\frac{1}{2} V_\rho.
\end{eqnarray}
Since we consider a two-fluid system, $\left[{\cal F}_{ij}\right]$ is a
$2\times 2$
symmetric matrix with two real eigenvalues $ \lambda_{\pm}$ \cite{ms}. The two
eigenvalues are given by,
\begin{equation}
\lambda_{\pm}=\frac{1}{2}\left(\mbox{Tr}({\cal F})\pm\sqrt{\mbox[{Tr}({\cal
F})]^2-4\mbox{Det}({\cal F})}\right),
\end{equation}
and the eigenvectors $\boldsymbol{\delta\rho_\pm}$ by
\begin{equation}
\frac{\delta\rho^\pm_i}{\delta\rho^\pm_j}=\frac{\lambda_\pm-{\cal F}_{jj}}
{{\cal F}_{ji}}, \quad i,j=p,n.
\end{equation}
The largest eigenvalue is always positive whereas the
other can take on negative value. We are interested in the
latter, because it defines the spinodal surface,
which is determined by the values of $T, \rho,$, and $y_p$.
The smallest eigenvalue of ${\cal F}_{ij}$ becomes negative.
The associated eigenvector defines the instability direction of the system,
in isospin space.
\section{Results and Discussion}
We set the model parameters $(a,V_0)$ by fitting the nucleon mass $M_N=939$ MeV
and charge radius of the proton $\langle r_N\rangle=0.87$ fm in free space.
Taking standard values for the meson masses; namely, $m_\sigma=550$ MeV,
$m_\omega=783$ MeV and $m_\rho=763$ MeV and fitting the quark-meson coupling
constants self-consistently, we obtain the correct saturation
properties of nuclear matter binding energy, $E_{B.E.}\equiv B_0={\cal E}/\rho_B-M_N=-15.7$
MeV, pressure, $P=0$, and
symmetry energy $J=32.0$ MeV at $\rho_B=\rho_0=0.15$ fm$^{-3}$.
The values of $g_\sigma^q$, $g_\omega$, and $g_\rho$
obtained this way and the values of the model parameters at quark
masses $5$, $40$, and $300$ MeV are given in Table \ref{table2}.
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{3pt}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$m_q$ (MeV)& $g^q_\sigma$& $g_\omega$& $g_\rho$& $a($fm$^{-3})$& $V_0($MeV$)$\\
\hline
5 &6.44071 &2.39398 &9.04862 &0.978629 &111.265238 \\
\hline
40 &5.46761 &3.96975 &8.99036 &0.892380 &100.187229 \\
\hline
300 &4.07565 &9.09078 &8.51458 &0.534296 &-62.257187 \\
\hline
\end{tabular}
\caption{\label{table2}Parameters for nuclear matter. They are determined
from the binding energy per nucleon, $E_{B.E}=B_0 \equiv{\cal E} /\rho_B - M_N
= -15.7$~MeV and pressure, $P=0$ at saturation density
$\rho_B=\rho_0=0.15$~fm$^{-3}$.}
\end{table}
\begin{table*}[t]
\centering
\caption{Nuclear matter properties of the models used in the present work.
The quantities presented are at saturation density.}
\begin{tabular}{lclccccccccc}
\hline
Model & $B_0$ & $~\rho_0$ & $M^*/M$ & $K_0$ & $J$ & $L^0$
& $K_{\rm{sym}}^0$ & $K_{\rm{asy}}$ & $Q_0$ &$~K_\tau$ \\
& (MeV)&(fm$^{-3}$) & & (MeV) & (MeV)& (MeV)& (MeV) & (MeV) & (MeV) &(MeV)\\
\hline
MQMC (5 MeV) &-15.7 &0.151 & 0.93 &159 & 32.0 & 84.7 & -27.7 & -535.9& 103.2 &-590.8 \\
MQMC (40 MeV) &-15.7 &0.151 & 0.91 &208 & 32.0 & 84.9 & -28.4 & -537.6& 94.2 &-575.9\\
MQMC (300 MeV) &-15.7 &0.151 & 0.76 &349 & 32.0 & 89.1 & -14.5 & -549.0& -15.6 &-545.1\\
DD~\cite{typel} &-16.0 &0.149 & 0.56 &239 & 31.6 & 55.9 & -95.3 & -431.1& 576.8 &-462.57\\
QMC~\cite{alex,alextemp} &-15.7 &0.150 & 0.77 &291 & 33.7 & 93.5 & -10.0 & -570.8& 29.4 &-580.24 \\
FSUGold~\cite{todd} &-16.2 &0.148 & 0.61 &229 & 32.6 & 60.4 & -51.4 & -414.0& 425.7 &-276.07\\
BKA24~\cite{bka24} &-15.9 &0.147 & 0.60 &227 & 34.2 & 84.8 & -14.9 & -523.7& 112.4 &-421.55\\
BSR12~\cite{bsr12} &-16.1 &0.147 & 0.61 &232 & 34.0 & 77.9 & -44.2 & -511.6& 324.2 &-414.30\\
\hline
\end{tabular}
\label{tab:properties}
\end{table*}
\begin{figure*}
\includegraphics[width=8.cm,angle=0]{fig1a.eps}
\includegraphics[width=8.cm,angle=0]{fig1b.eps}
\caption{(Color online) Nuclear matter binding energy as a function of density
for (a) different $y_p$ values for quark mass $m_q=40$ MeV and (b) quark mass
$m_q=300$ MeV.}
\label{fig1}
\end{figure*}
In Figs. \ref{fig1}(a) and \ref{fig1}(b), we plot the binding energy per
nucleon for nuclear matter as a function of density corresponding to $m_q=40$
MeV and $m_q=300$ MeV, respectively, for different $y_p$ values. In Fig. \ref{fig1a}, we
compare the variation of the binding energy per nucleon for quark mass $40$ and
$300$ MeV with that of QMC and observe that,
for $300$ MeV, MQMC compares well with that of QMC.
It is observed from Figs. \ref{fig1} and \ref{fig1a} that, at low quark
mass, the equation of state is softer. In Table \ref{tab:properties},
we compare the
nuclear matter properties at saturation for quark masses $5$, $40$, and
$300$ MeV, respectively, in the present model to QMC \cite{alex,alextemp},
and some of the approved models as suggested in Ref. \cite{dutraetal}.
The value of the compressibility
$K_0$ is determined to be $159$, $208$, and $349$ MeV respectively, for
quark masses $5$, $40$, and $300$ MeV. A recent calculation \cite{stonemos} has
predicted $K_0$ to be in the range $250~<~K_0~<~315$ from the experimental
GMR energies in even-even $^{112-124}$Sn and $^{106,100-116}$Cd. Furthermore,
the value of the effective mass calculated in the present model at quark mass
$300$ MeV is $0.76$ which compares well with the empirical value of the
effective mass, which is $0.74$ \cite{mahaux}.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig1c.eps}
\caption{(Color online) Nuclear matter binding energy as a function of
density for quark mass $m_q=40$ MeV and quark mass $m_q=300$ MeV for $y_p=0.3$.
A comparison is made between the MQMC for quark mass $40$ MeV and $300$ MeV
with that of QMC.}
\label{fig1a}
\end{figure}
We compare the symmetry energy, its slope, and incompressibility from our model
with the QMC \cite{alex,alextemp} results respectively in Figs. \ref{fig2}(a),
\ref{fig2}(b), and \ref{fig2}(c).
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig2a.eps}
\includegraphics[width=8.cm,angle=0]{fig2b.eps}
\includegraphics[width=8.cm,angle=0]{fig2c.eps}
\caption{(Color online) (a) Symmetry energy, (b) its slope parameter
$L$, and (c) $K_{sym}$ (c) in the MQMC and QMC models as functions of
baryon density $\rho_B$}.
\label{fig2}
\end{figure}
We observe that the symmetry
energy shows an extremely linear behavior.
This is further justified from the plot for the slope parameter $L$.
This is based on equation \eqref{engs} for ${\cal E}_{sym}$. However,
if we consider terms higher than the quadratic one in defining the relation
in Eq. \eqref{esymt}, it would be more appropriate to use the expression as in
Eq. \eqref{engs1} to show the density dependence of ${\cal E}_{sym}$
for higher nuclear densities $(\rho_B > \rho_0)$. This has been shown in Fig.
\ref{fig3a} in comparison with several other models, as noted there,
including QMC.
\begin{figure}
\vspace{-0.5in}
\includegraphics[width=8.cm,angle=0]{fig3a.eps}
\vspace{-0.5in}
\caption{(Color online) Density dependence of ${\cal E}_{sym}(\rho_B)$
according to Eq. \eqref{engs1}.}
\label{fig3a}
\end{figure}
\subsection{Correlation between the symmetry energy and its slope}
\begin{figure}
\includegraphics[width=8.5cm,angle=0]{fig9.eps}
\caption{\label{fig9}(Color online) Correlation function
$f(M_N^*,\rho_0,B_0,K_0)$ at various quark masses.}
\end{figure}
We study the correlation function $f(M_N^*,\rho_0,B_0,K_0)$ with the
variation of quark masses in Fig. \ref{fig9}.
We observe that the function $f(M_N^*,\rho_0,B_0,K_0)$ increases with quark
masses.
\begin{figure}
\includegraphics[width=8.5cm,angle=0]{fig10.eps}
\caption{\label{fig10}(Color online) The slope of symmetry energy, $L$ at
various quark mass.}
\end{figure}
The established value of binding energy $B_0$ and the saturation density $\rho_{0}$ in the nuclear
mean-field models cannot be applied to incompressibility $K_0$ and effective mass
$M_N^*$ since the latter are found as output in this model where the
coupling constants are fixed in a self consistent manner by taking into
consideration the binding energy and saturation density.
Therefore, we have taken the variation of $f$ with different quark masses
at the same $B_0$ and $\rho_{0}$. We observe that, because there is only one isovector parameter $g_{\rho}^2$
in the expression for ${\cal E}_{sym}$ and $L$, the variation is linear. Such linearity in the behavior
was also observed in nonrelativistic models \cite{correl}. It indicates one of the
limitations of the model parameters. We expect a nonlinear behavior
between ${\cal E}_{sym}$ and $L$ for the models with more than one isovector
parameter. In the Fig. \ref{fig10}, we have shown the slope of symmetry
energy, $L$ at various quark masses. It is interesting to note
that there is a linear relationship of the slope of the symmetry energy $L$
with quark mass. This is a direct
consequence of the dependence of the symmetry energy on $g_\rho^{2}$.
\subsection{Instability}
We next study the mechanical instability and its
dependence on the isospin asymmetry of the system by plotting the pressure as
a function of density and the asymmetry parameter $y_p$. In Fig. \ref{fig3},
we show that the mechanical instability occurs in the region where the slope
of the pressure with respect to density is negative. We observe that the
mechanical-instability region shrinks when the isospin-asymmetry increases.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig3.eps}
\caption{(Color online) Pressure as a function of density at various isospin asymmetries.}
\label{fig3}
\end{figure}
The system is stable under separation of two phases if the free energy of a
single phase is lower than the free energy in all two-phase configurations.
In the spinodal area we can get the signature of the mechanical
instability by finding the velocity of sound determined in the model as
\begin{equation}
\beta^2=\dfrac{dP}{d\cal E }=
\dfrac{dP}{d\rho_B}/{\dfrac{d\cal E}{d\rho_B}},
\end{equation}
where $\beta^2=v_s^2/c^2$, $v_s$ is the velocity of sound and $c$ is the
speed of light.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig4.eps}
\caption{\label{fig4}(Color online) Sound velocity as a function of density.}
\end{figure}
In Fig. \ref{fig4} we show the sound velocity as a function of density
by changing the
asymmetry parameter. There is a reduction in the instability when we move away
from the symmetric nuclear matter. Moreover, the velocity becomes imaginary
when we enter into the spinodal area \cite{unique}.
The positivity of the local curvature matrix is equivalent to the condition
that both the trace $Tr\left[{\cal F}_{ij}\right]=\lambda_++\lambda_-$ and the
determinant $det\left[{\cal F}_{ij}\right]=\lambda_+\lambda_-$are positive.
In the present model the above condition is violated and the system is in
the unstable region of a phase transition. Further it is to be pointed out that
for a two component, n-p thermodynamical system, the stability parameter is
given by the condition;
\begin{equation}
S_P=\left(\frac{\partial P}{\partial \rho_B}\right)_{T,y_p}\cdot
\left(\frac{\partial \mu_p}{\partial y_p}\right)_{T,P} > 0,
\label{stab}
\end{equation}
but in charge symmetric matter the isoscalar (total density) $\delta\rho_n+
\delta\rho_p$ and isovector (concentration) $\delta\rho_n-\delta\rho_p$
oscillations are not coupled and there are two separate conditions for
instability \cite{asym}. These conditions are for mechanical instability
\begin{equation}
\left(\frac{\partial P}{\partial \rho_B}\right)_{T,y_p}\leq 0,
\label{pom1}
\end{equation}
and for chemical instability
\begin{equation}
\left(\frac{\partial \mu_p}{\partial y_p}\right)_{T,P}\leq 0.
\label{pom2}
\end{equation}
In the ANM the isoscalar and isovector modes are coupled and the
two separate inequalities do not select the nature of instability. Moreover, we
observe in Fig. \ref{fig5}, a large difference in the behavior of the
stability parameter $S_P$ in Eq. \eqref{stab} inside the instability
region. For higher asymmetry, the range of the stability parameter is smaller
than at lower asymmetries.
To understand this effect we follow the
Landau-dispersion-relation approach for small-amplitude oscillations in
Fermi liquids \cite{landau}. For a two component $(n,p)$ matter, the interaction is
characterized by the Landau parameters $F^{q,q'}_0$ which is defined by the
relationship
\begin{equation}
N_q(T)\frac{\partial \mu_q}{\partial \rho_{q'}} \equiv \delta_{q,q'}+F_0^{q,q'},
\end{equation}
where $q=(n,p)$ and $N_q(T)$ represents the single-particle level density at the Fermi
energy. At zero temperature it has the simple form
\begin{equation}
N_q = \frac{k_{Fq}E^{*}_{Fq}}{\pi^2}.
\end{equation}
In the symmetric case ($F^{nn}_0=F^{pp}_0,F^{np}_0=F^{pn}_0 $), the Eqs. \eqref{pom1} and \eqref{pom2}
correspond to the two Pomeranchuk instability conditions
\begin{eqnarray}
F^{s}_0 &=& F^{nn}_0+F^{np}_0 < -1~~~~~~{\mbox mechanical} \nonumber\\
F^{a}_0 &=& F^{nn}_0-F^{np}_0 > -1~~~~~~{\mbox chemical}
\end{eqnarray}
The dispersion relations $F^{s}_0$ give the properties
of density (isoscalar) modes and $F^{a}_0$ gives the concentration
(isovector) modes.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig5.eps}
\caption{\label{fig5}(Color online) The stability parameter
$S_p$ [Eq. \eqref{stab}]
as a function of $\rho_B$ for $y_p=0.25$ and $y_p=0.1$ in the instability sector.}
\end{figure}
In the unstable region of dilute asymmetric nuclear matter
we have isoscalar-like unstable modes, hence $1+F^{s}_0<0$, while the
combination $1+F^{a}_0>0$. In the Fig. \ref{fig6} we plot the values
obtained from the calculation of these two quantities in the unstable
region at zero temperature for $y_p=0.25$. An important observation we
make from the comparison of Figs. \ref{fig5} and \ref{fig6} is
the shift in the maximum instability density region.
In Fig. \ref{fig5} the largest instability (the most negative value) is at
$\rho_B=0.065 fm^{-3}$. However, the most negative Pomeranchuk condition
$1+F_0^s$, which corresponds to the fastest unstable mode, is present
in more dilute matter at $\rho_B=0.03 fm^{-3}$.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig6.eps}
\caption{\label{fig6}(Color online) Behavior of generalized Landau parameters
(a) $1+F^{a}_0$ (b) $1+F^{s}_0$ with respect to baryon density $\rho_B$
in the instability sector for $y_p=0.25$.}
\end{figure}
In the following we study the direction of instability of the system. In Fig.
\ref{fig7}, we show the ratio of the proton versus neutron density fluctuations
corresponding to the unstable mode
which defines the direction of instability of the system. We plot the results
for different proton fractions and observe that the instabilities tend to
restore the isospin symmetry for the dense (liquid) phase leading to the
fractionation of the ANM.
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig7a.eps}
\includegraphics[width=8.cm,angle=0]{fig7b.eps}
\includegraphics[width=8.cm,angle=0]{fig7c.eps}
\caption{\label{fig7}(Color online) Ratio of proton-neutron density fluctuation
corresponding to the unstable mode showing the direction of instability for
(a) $y_p=0.05$, (b) $y_p=0.1$, and (c) $y_p=0.3$.}
\end{figure}
\begin{figure}
\includegraphics[width=8.cm,angle=0]{fig8.eps}
\caption{\label{fig8}(Color online) Proton-neutron density fluctuation ratio
versus proton fraction $y_p$ for a fixed nuclear density.}
\end{figure}
Figure \ref{fig8} shows the proton-neutron density fluctuation ratio as a function
of the isospin asymmetry for a fixed nuclear density, $\rho=0.06~fm^{-3}$ and
compares it to the QMC and Brueckner-Hartree-Fock (BHF) calculations.
The relativistic models give
larger fluctuation ratios than the corresponding value of $\rho_p/\rho_n$.
We also observe that the fluctuation ratio in the present model is larger
compared to the nonrelativistic BHF model. A pure mechanical disturbance
would occur \cite{chomaz}
if the instability preserves the ratio between protons and neutrons, i.e.,
$\frac{\delta\rho^-_p}{\delta\rho^-_n}=\frac{\rho_p}{\rho_n}$. Conversely if
$\delta\rho^-_p=-\delta\rho^-_n$ then we should observe pure chemical
disturbance. In the present case we observe that the disturbance along
the unstable eigen direction conserves neither $\rho$ nor $y_p$ but has mixed
character with both chemical and mechanical contents.
\subsection{Constrain on neutron star radii}
The symmetry energy plays an
important role in describing the mass-radius relationship in neutron stars.
Neutron stars are compact objects maintained by the equilibrium of
gravity and the degenerecy pressure of the fermions together with a
strong nuclear repulsion force due to the high density
reached in their interior. The slope of the symmetry energy, $L$,
constrains the neutron star radii. It is confirmed that the
radii for the neutron stars with canonical mass $1.4 M_{\odot}$
are not affected by the
symmetry energy at saturation density \cite{menezes}. However,
in some cases the radii increase with $L$, while in others,
there is a decrease. In fact the radii are correlated
with a variation of the slope $L$. The radii increase
up to a maximum value, then drop again. This behavior
can be associated with a maximum theoretical value of $L$,
and provide a possible constraint to nuclear matter. In the present model the
value of $L^0$ comes out to be $89$ MeV which is very close to the experimental
observation \cite{tili}.
The most direct connection between the astrophysical observations and the
nuclear symmetry energy concerns neutron star radii ($R$) which are
highly correlated with neutron star pressures near $\rho_0$. It is to be noted
that Lattimer and Prakash \cite{constrain} found the radii of neutron stars
for masses near the canonical mass $1.4 M_{\odot}$, obey a power-law relation:
\begin{equation}
R(M)=C(\rho,M)(P(\rho)/MeV fm^{-3})^{1/4}
\end{equation}
where $R(M)$ is the radius of a star, $P(\rho)$ is the pressure of neutron
star matter at density $\rho$, and $C(\rho,M)$ is a constant for a given
density and mass. Considering the structure of a neutron star as pure neutron
matter, the value of this constant at quark mass $300$ MeV in
our model comes out to be
\begin{equation}
C(2\rho_0,1.4M_{\odot})=5.90 km
\end{equation}
which is very near to $5.68 \pm 0.14$ km predicted by Lattimer {\it et al.}
\cite{constrain}. It is to be noted that the mass of the neutron star for $pn$
matter with $\beta$ equilibrium
comes out to be 2.7 $M_\odot$ with quark mass $300$ MeV and 1.64 $M_\odot$ with quark mass
$40$ MeV. The details of such calculations incorporating the hyperons in the composition
of neutron stars is in progress.
\section{Conclusion}
In the present work we have studied the EOS for asymmetric nuclear matter by
using a modified quark-meson coupling model (MQMC). Self-consistent
calculations were made by using a relativistic quark model with chiral
symmetry along with the spurious
center-of-mass correction, pionic correction for restoration of chiral symmetry,
and short-distance correction for one-gluon exchange to realize different bulk
nuclear properties. The instability in the two-component nuclear system is
then analyzed. In asymmetric matter the isoscalar and isovector modes are
coupled and the two separate inequalities for density oscillations and
concentration oscillations no longer maintain a physical meaning for
the selection of the nature of the instabilities.
The symmetry energy, its slope $(L)$, and curvature parameter $(K_{sym})$
were found in reasonable agreement with experimental values.
Without considering self
interactions in the scalar field, we found
an analytic expression for the symmetry energy ${\cal E}_{sym}$
as a function of its slope $L$. Our result establishes a linear correlation
between $L$ and ${\cal E}_{sym}$. We also study the variation of correlation
function $f(M_N^*,\rho_0,B_0,K_0)$ with the variation of quark masses.
The symmetry energy is
correlated with neutron star radii. In this model we observe that, at
twice the saturation density $(~0.3 fm^{-3})$, the constant $C(2\rho_0, M)$
is found 5.90 km in the canonical-mass region of $1.4 M_{\odot}$.
\section*{ACKNOWLEDGMENTS}
The authors would like to acknowledge the financial assistance from
BRNS, India for the Project No. 2013/37P/66/BRNS.
|
2,877,628,089,476 | arxiv |
\section{Introduction}
The use of mathematical models to study the spreading of infectious diseases within a population has a long history within the field of epidemiology~\cite{brauer2008mathematical}. One key use of such models is to determine the long term dynamics of the disease of interest as a function of various model parameters, such as the rate of infection and rate of recovery. Deterministic compartmental models have proved especially popular due to their balance between analytic tractability, low cost for numerical simulations, and reasonable accuracy in capturing epidemics. The Susceptible--Infected--Susceptible (SIS) paradigm is a fundamental compartmental paradigm, where each individual in a population is assumed to be healthy and susceptible to infection, or infected and capable of transmitting to susceptible others. Infected individuals can recover, but have no immunity (temporary or permanent) and can be immediately reinfected.
While SIS models for the spread of a single disease have received significant attention~\cite{lajmanovich1976SISnetwork,shuai2013epidemic_lyapunov}, there has been in the past two decades an increasing interest in models that capture the spread of multiple diseases~\cite{wang2019coevolution}. Cooperative diseases are those in which infection with one disease increases the likelihood of infection with another disease, while competitive diseases are those in which an individual can only be infected with one disease at any one time. The networked bivirus SIS model is one of the more widely studied competitive multivirus models. This model assumes that two competitive diseases, termed virus~$1$ and virus~$2$, spread through a set of nodes, with potentially two different network topologies capturing the different spreading patterns of the two viruses. Virtually all studies have assumed that a form of connectivity holds for the two network topologies. The same bivirus model, i.e., the same set of ordinary differential equations, has been studied in different contexts, with nodes representing gendered groups (two female and one male group) in \cite{carlos2}, individuals in \cite{sahneh2014competitive}, and large populations of constant size in \cite{ye2021_bivirus,santos2015bi}.
There has been a significant amount of literature devoted to the networked bivirus SIS model~\cite{carlos2,sahneh2014competitive,ye2021_bivirus,santos2015bi,prakash2012winner,liu2019bivirus,pare2021multi,janson2020networked,yang2017bi}-- see \cite{ye2022competitive} for a brief survey. In summary, a complete understanding is available for the case of $n=1$~\cite{prakash2012winner} and $n=2$~\cite{ye2021_bivirus}, and for a special $n=3$ case in \cite{carlos2}, but there are still important gaps in understanding the bivirus dynamics for arbitrary $n \geq 3$ node networks. In \cite{ye2021_bivirus}, it was established that for generic model parameters, the number of equilibria are finite, and convergence to an equilibrium occurred for almost all initial conditions. There are nongeneric parameters which instead yield a connected continuum of equilibria. Assuming however genericity of the parameters, there are three \textit{types} of equilibria: i) the ``healthy equilibrium'' (in which both viruses are extinct in every node) and this is always unique, ii) equilibria where virus~$1$ is present and virus~$2$ is extinct (or vice versa), these being termed ``boundary equilibria'', and iii) ``coexistence equilibria'' in which both viruses are present. It turns out that there are at most two boundary equilibria (one for each virus being extinct)~\cite{ye2022competitive}.
Much of the literature has focused on elimination of both viruses and ``survival-of-the-fittest'' outcomes, in which one virus goes extinct while the other persists. Necessary and sufficient conditions for global stability of the healthy equilibrium~\cite{liu2019bivirus,pare2021multi} and local exponential stability of boundary equilibria~\cite{ye2021_bivirus,sahneh2014competitive} are well documented, as are some sufficient conditions for global stability of boundary equilibria~\cite{santos2015bi,ye2021_bivirus}. On the other hand, coexistence equilibria are understudied except for highly nongeneric parameter values~\cite{liu2019bivirus,pare2021multi,ye2021_bivirus}. In fact, while some sufficient conditions for there to be \textit{no} coexistence equilibria are known~\cite{ye2021_bivirus,santos2015bi,liu2019bivirus}, there are only a few sufficient conditions for the existence of coexistence equilibria~\cite{janson2020networked}. More importantly, there are no general results that allow one to determine, given generic model parameter values, the number of coexistence equilibria and their local stability properties. This is in part because, with arbitrary $n$ nodes, the dynamics are captured by $2n$ coupled nonlinear differential equations (with the network topologies adding further complexities to the analysis). As a demonstration of the potential complexity of the equilibria patterns, we will in the sequel present two simple $n=4$ examples which have four attractive equilibria (two boundary equilibria and two coexistence equilibria) and two attractive equilibria (one boundary and one coexistence), respectively.
The limitations discussed motivate us to develop new ``counting'' results for studying coexistence equilibria and their stability properties. We do this using a tool, the Poincar\'e-Hopf Theorem, see e.g. \cite{guillemin2010differential,milnor1997topology}, which enables us to derive one of our main results: a counting result involving the number of equilibria of different indices. Namely, we obtain a constraint on the number and types of coexistence equilibria, (e.g. stable attractor, source, saddle point of certain index). We remark that the Poincar\'e-Hopf Theorem allows an elegant derivation of the principal results applying to the single virus networked SIS model, see \cite{ye2021applications,anderson2020extrema}. However, as will later be clear, the region of interest relevant to a networked bivirus system cannot be so straightforwardly treated as in the single virus case, due to the fact that this region is not a manifold in the sense to which the Poincar\'e-Hopf Theorem applies. In other words, as far as equilibria of the underlying equations are concerned, bivirus is not a simple extension of single virus, and we employ two key ideas to overcome this challenge. The first is that the region of interest can be distorted if necessary at its boundary, to incorporate any stable equilibria located on the boundary, and the second is that a homeomorphism can be established between the modified region of interest and an even-dimensional sphere excluding a single point of that sphere (a fact which then makes it much more straightforward to use the Poincar\'e-Hopf Theorem). As part of our application of the Poincar\'e-Hopf Theorem, we also prove that for generic model parameters, each equilibrium is isolated (and hence there are a finite number), nondegenerate, and hyperbolic. We then make a reasoned argument as to why the networked bivirus SIS model is a so-called Morse-Smale system \cite{smale1967differentiable} (but offer no rigorous proof), before exploiting Morse-Smale inequalities to further strengthen our ability to count the number of coexistence equilibria and determine their stability properties.
The paper is structured as follows. In \Cref{sec:background}, after notational, linear algebra and graph theoretical preliminaries we review the single virus and bivirus equations, and present a motivating example with multiple attractive coexistence equilibria.
We also introduce the Poincar\'e-Hopf Theorem and indicate the difficulties in immediately applying it. \Cref{sec:main} establishes the first main result of the paper using the Poincar\'e-Hopf Theorem. The use of Morse-Smale inequalities to develop further counting results occurs in \Cref{sec:morse_smale}. A brief illustration of the results is presented in \Cref{sec:examples}, by considering several bivirus systems with two nodes. Finally, conclusions are drawn in \Cref{sec:conclusions}.
\section{Background Material, Epidemic Dynamics and Poincar\'e-Hopf Theory}\label{sec:background}
\subsection{Notation, Linear Algebra and Graph Theory Background}\label{ssec:background}
For real vectors $x,y\in\mathbb R^n$ with entries $x_i$ and $y_i$, $x\geq y$ denotes $x_i\geq y_i$, $x>y$ denotes $x\geq y$ but $x\neq y$, and $x\gg y$ denotes $x_i>y_i$ for all $i=1,2,\dots,n$. For matrices $A,B$ of the same dimensions, $A\geq B, A>B, A\gg B$ denote the same thing as the corresponding inequalities for $\text{vec}\, A$, $\text{vec}\, B$. A matrix $A \in \mathbb R^{n\times n}$ is said to be nonnegative if $\text{vec}\, A \geq \vect 0_{n^2}$. The $n$-vectors of all 1's and all 0's are denoted by $\vect 1_n$ and $\vect 0_n$ respectively.
For a square matrix $A$, $\rho(A)$ denotes the spectral radius and $\sigma(A)$ the spectral abscissa, or greatest real part of any eigenvalue of $A$. The matrix is termed reducible (irreducible) if there exists (does not exist) a permutation matrix $P$ such that $P^{\top}AP$ is block triangular. For a nonnegative and nonzero $A$, $\sigma(A)=\rho(A)$ and corresponds to a real eigenvalue for which there exist associated left and right eigenvectors which can be taken to be nonnegative; if in addition, $A$ is irreducible, $\rho(A)$ is a unique eigenvalue and these eigenvectors can be taken to be positive and unique up to a scaling. A matrix $-D+A$ with $A$ nonnegative and irreducible and $D$ diagonal has $\sigma(-D+A)$ as a simple eigenvalue, and the associated left and right eigenvectors can be taken to be positive. These facts come from the Perron-Frobenius Theorem and its corollaries, see e.g \cite{horn1994topics_matrix}. Additionally, when $D$ is also positive definite, $\sigma(-D+A)>0\iff\rho(D^{-1}A)>1, \sigma(-D+A)=0 \iff \rho(D^{-1}A)=1$ and $\sigma(-D+A)<0\iff \rho(D^{-1}A)<1$~\cite[Proposition~1]{liu2019bivirus}. Irreducible nonnegative $A$ have the following property: if $Ax=y$ for $x>{\bf{0}}_n$, $y>{\bf{0}}_n$, then $y$ cannot have a zero entry in every position where $x$ has a zero entry (but it may have zero entries in some of those positions).
A weighted directed graph $\mathcal G$ is a triple $\mathcal G=(\mathcal V,\mathcal E, W)$ with $\mathcal V=\{1,2,\dots,n\}$ denoting the vertex (or node) set, $\mathcal E\subset\mathcal V\times \mathcal V$ denoting the edge set, and $A$ a nonnegative $n\times n$ matrix with $a_{ij}>0$ if and only if $(j,i)\in \mathcal E$, implying the existence of a directed edge from vertex $j$ to vertex $i$. A path is a sequence of edges of the form $(i_0,i_1),(i_1,i_2),\dots,(i_{p-1},i_{p})$ with the vertices distinct, except possibly for the first and last. A (directed) graph is strongly connected if and only if any one vertex can be joined to any other vertex by a path starting from the first and ending at the second. Further, $\mathcal G$ is strongly connected if and only if $A$ is irreducible~\cite{berman1979nonnegative_matrices}.
For a set $X$, typically in this paper a subset of $\mathbb R^n$, the closure, interior and boundary are denoted by $\bar X, X^{\circ}$ and $\partial X$ respectively.
\subsection{Dynamics of single virus networks}\label{ssec:single_virus}
The spreading dynamics of a single virus have been studied using deterministic susceptible-infectious-susceptible (SIS) network models for a long time, see e.g. \cite{lajmanovich1976SISnetwork,fall2007SIS_model,vanMeighem2009_virus,mei2017epidemics_review,shuai2013epidemic_lyapunov}. We summarize the results in a manner relevant to the treatment of bivirus problems below. Assume there are $n$ populations, corresponding to vertices or nodes of a directed graph, each of a large and constant size. Individuals in each population can exist in and move between one of two mutually exclusive \textit{health compartments}: individuals may be healthy and susceptible (S) to becoming infected by the virus, or infected (I) with the virus and able to transmit it. The edges of the graph capture virus transmission possibilities; there is an edge from node $j$ to node $i$ precisely when virus transmission can occur from the infected individuals in the $j$-th population to the susceptible individuals in the $i$-th population. The rates of infection are captured by nonnegative $\beta_{ij}$, so that $\beta_{ij} > 0$ if and only if $(j,i)$ is an edge in the graph (and thus the $\beta_{ij}$ can be regarded as weights on the edges of the underlying graph). Individuals infected with the virus can recover (with no immunity and immediate susceptibility to infection again): the recovery (healing) rate of each population $i$ is captured by the positive parameter $\delta_i$. Let $x_i(t)$, the $i$-th entry of a vector $x(t)\in\mathbb R^n$, denote the fraction of individuals of population $i$ infected with the virus, and let $D={\rm{diag}}(\delta_1,\delta_2,\dots,\delta_n), B=(\beta_{ij})$ and $X(t)={\rm{diag}}(x_1(t),x_2(t),\dots,x_n(t))$. The dynamical equations describing the SIS network system then become
\begin{equation}\label{eq:singlevirusunlabelled}
\dot x(t)=[-D+(I-X(t))B]x(t).
\end{equation}
The key properties of \eqref{eq:singlevirusunlabelled} established by the literature~\cite{lajmanovich1976SISnetwork,fall2007SIS_model,mei2017epidemics_review} include a focus on asymptotic behavior and can be summed up as follows:
\begin{theorem}\label{thm:singlevirus}
With notation as given above, consider the single virus equation~\eqref{eq:singlevirusunlabelled}. Suppose that ${\bf{0}}_n\leq x(0)\leq {\bf{1}}_n$, and the graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, B)$ is strongly connected. Then ${\bf{0}}_n\leq x(t)\leq {\bf{1}}_n,\,\forall\,t\geq 0$. Moreover:
\begin{enumerate}
\item If $\rho(D^{-1}B)\leq 1$, all trajectories converge asymptotically to the healthy equilibrium $\vect 0_n$ as $t\to\infty$. Convergence is exponentially fast iff \mbox{$\rho(D^{-1}B)< 1$.}
\item If $\rho(D^{-1}B)> 1$, then there is precisely one other equilibrium, $\bar x$, besides the healthy equilibrium $\vect 0_n$. All trajectories converge exponentially fast to $\bar x$ as $t\to\infty$ except if $x(0) = \vect 0_n$. The equilibrium $\bar x$ satisfies $\vect 0_n \ll \bar x \ll \vect 1_n$ and is called the endemic equilibrium.
\end{enumerate}
\end{theorem}
Limit cycles and chaos are precluded. The bounds on $x(t)$ reflect its physical interpretation as a vector of proper fractions, and, with one exception, all trajectories go to the same equilibrium irrespective of initial conditions, viz. the endemic equilibrium if it exists, or the healthy equilibrium. The exception occurs if an endemic equilibrium exists, but the initial condition lies at the healthy equilibrium.
The literature often defines $\mathcal{R} = \rho(D^{-1}B)$ as the reproduction number of the SIS network. In epidemiology, the reproduction number is the expected number of secondary infections generated in a population of entirely susceptible individuals, by a single infectious individual. Conveniently, $\mathcal{R}$ for \eqref{eq:singlevirusunlabelled} identifies whether the virus will be eliminated or persist as $t\to\infty$, corresponding closely with the epidemiological definition~\cite{anderson1991_virusbook,hethcote2000mathematics}. We remark that \eqref{eq:singlevirusunlabelled} describes a deterministic system, while real-world virus propagation is a stochastic process. In fact, \eqref{eq:singlevirusunlabelled} can be viewed as the mean-field approximation of a stochastic SIS process. However, we do not provide further comment on this issue, which is beyond the scope of our focus, and instead refer the reader to established literature, e.g.~\cite{nowzari2016epidemics,li2012susceptible,van2015accuracy}.
\subsection{Dynamics of bivirus networks}
Moving from the single virus SIS model, we again assume there are $n$ populations but now two competing viruses may be present, called virus~1 and virus~2 for convenience, and there are three mutually exclusive health compartments, see Fig.~\ref{fig:transitions_bivirus}. Individuals may be susceptible, or infected with virus~1, or infected with virus~2, but cannot be infected with both viruses at the same time due to their competing nature. Individuals that recover from infection by either virus becomes immediately susceptible to infection from either virus again.
Associated with virus~1 and virus~2 are two graphs, $\mathcal{G}^1 = (\mathcal{V},\mathcal{E}^1, B^1)$ and $\mathcal{G}^2 = (\mathcal{V},\mathcal{E}^2, B^2)$, respectively, which share the same node set but need not have the same edge set or edge weights for both viruses, see Fig.~\ref{fig:two_layer}. The nonnegative infection rates $\beta^1_{ij},\beta^2_{ij}$ capture the associated transmission rates, and are weights for the edges in the two graphs, so that $B^1=(\beta^1_{ij})$ and $B^2 = (\beta^2_{ij})$. Each population $i$ has associated with it positive healing rates $\delta ^1_i$ and $\delta^2_i$ for virus~1 and virus~2, respectively. Let $x^1_i(t),x^2_i(t)$ denote the fraction of individuals in population $i$ infected with virus~1 and virus~2 respectively. Because the viruses are competitive, the total fraction of individuals in population $i$ affected with either virus is $x_i^1(t)+x_i^2(t)$ and the fraction of susceptible individuals is $1-x_i^1(t)-x_i^2(t)$. Let $x^1(t),x^2(t)$ denote the associated vectors of fractions of infected individuals through the $n$ populations, and set $D^1={\rm{diag}}(\delta^1_1,\delta ^1_2,\dots,\delta^1_n)$ and $D^2$ similarly; set $X^1(t)={\rm{diag}}(x^1_1(t),x^1_2(t),\dots,x^1_n(t))$ and $X^2(t)$ similarly. Then the dynamical equations for the bivirus network system become
\begin{subequations}\label{eq:bivirus}
\begin{equation}\label{eq:bivirusa}
\dot x^1(t)=[-D^1+\big(I_n-X^1(t)-X^2(t)\big)B^1]x^1(t)
\end{equation}
\begin{equation}\label{eq:bivirusb}
\dot x^2(t)=[-D^2+\big(I_n-X^1(t)-X^2(t)\big)B^2]x^2(t)
\end{equation}
\end{subequations}
\begin{figure}
\centering
\subfloat[]{\def0.9\linewidth{0.5\linewidth}
\input{transitions_bivirus.pdf_tex}\label{fig:transitions_bivirus}}
\hfill
\subfloat[]{\def0.9\linewidth{0.5\linewidth}
\input{two_layer_bivirus.pdf_tex}\label{fig:two_layer}}
\caption{Schematic of the compartment transitions and infection network. (a) Each individual exists in one of three health compartments: Susceptible ($S$), Infected with virus~$1$ ($I^1$, orange), or Infected with virus $2$, ($I^2$, purple). Arrows represent possible transition paths between health states. (b) The network through which the viruses can spread between populations (nodes) is captured by two graphs, $\mathcal{G}^1$ and $\mathcal{G}^2$. Note that the edge sets of the two graphs do not need to match, although the node sets are the same.
}
\label{fig:epidemic_schematic}
\end{figure}
On physical grounds, the fractions of infected individuals should never move outside the interval $[0,1]$. In fact, that this property is captured by the model is one of the early results in \cite{liu2019bivirus}:
\begin{lemma}
With the above notation and sign constraints on the entries of $D^i$ and $B^i$, suppose that the initial conditions for \eqref{eq:bivirus} satisfy ${\bf{0}}_n\leq x^i(0)\leq {\bf{1}}_n$ for $i=1,2$ and $x^1(0)+x^2(0)\leq {\bf{1}}_n$. Then for all $t>0$, there holds ${\bf{0}}_n\leq x^i(t)\leq {\bf{1}}_n$ for $i=1,2$ and $x^1(t)+x^2(t)\leq {\bf{1}}_n$.
\end{lemma}
In the sequel, the term `region of interest' will be used to denote the set $$\Xi_{2n}=\{(x^1,x^2) : {\bf{0}}_n\leq x^i\leq {\bf{1}}_n, i=1,2 \text{ and } x^1+x^2\leq {\bf{1}}_n \}.$$
The prime interest in this paper is in the limiting behavior of the equations, and particularly the nature of the equilibria in the region of interest. The situation is certainly more complicated than in the single virus case. To make progress, it is appropriate to impose further restrictions on $D^i,B^i$, through the following standing assumption.
\begin{assumption}\label{ass:constraints}
The matrices $D^i$ are positive definite and the matrices $B^i$ are irreducible.
\end{assumption}
This ensures that each population could recover from either virus if infection transmissions were totally prevented, which is a natural assumption. The irreducibility of $B^i$ implies that for each virus, there is a path of transmission from any population to any other population. Such assumptions are standard in the bivirus network literature, see e.g.~\cite{liu2019bivirus,ye2021_bivirus,santos2015bi,sahneh2014competitive}. We remark that $B^1$ and $B^2$ are assumed separately irreducible, i.e. $\mathcal{G}^1$ and $\mathcal{G}^2$ are both strongly connected but may not share the same edge set or edge weights.
We now introduce further standing assumptions on the matrices $(D^i)^{-1}B^i$ in order for the bivirus dynamics to be meaningful and interesting.
\begin{assumption}\label{ass:unstablehealthy}
For $i=1,2$, there holds $\rho((D^i)^{-1}B^i)>1$
\end{assumption}
We briefly describe the reason for these assumptions, but refer the reader to \cite[Section~2.2]{ye2021_bivirus} or \cite[Theorems~1 and 2]{liu2019bivirus} for a detailed treatment and formal proofs. Briefly, if the above assumption fails, we are effectively back in a single virus situation, which has been well explored in the literature, and thus does not deserve further treatment here. For example, suppose the inequality fails for $i = 1$, i.e., $\rho((D^1)^{-1}B^1)\leq 1$. In this case, virus~1 becomes asymptotically extinct\footnote{Exponentially fast in fact if the inequality is strict.}, {\textit{irrespective of the presence of virus 2}}, i.e. $x^1(t)\rightarrow 0$ as $t\rightarrow\infty$. Indeed, presence of virus~2 simply results in a faster extinction for virus~1.
Once $x^1(t)$ has converged to $\vect 0_n$, unsurprisingly given the form of \eqref{eq:bivirusb}, the system behaves like the single virus system with only virus~2 present.
\begin{equation}\label{eq:univirus}
\dot x^2(t)=[-D^2+\big(I-X^2(t)\big)B^2]x(t)
\end{equation}
From \Cref{thm:singlevirus}, we either have $\lim_{t\to\infty }x^2(t) = \vect 0_n$ if $\rho((D^2)^{-1}B^2)\leq 1$ or $\lim_{t\to\infty }x^2(t) = \bar x^2$ if $\rho((D^2)^{-1}B^2)>1$, where $\bar x^2$ is the unique endemic equilibrium for \eqref{eq:univirus}.
In summary then, \Cref{ass:unstablehealthy} provides us the opportunity to study genuine bivirus dynamics, where the persistence and extinction of a virus is tied to the overall networked system, and not to the reproduction number defining it at the single virus level.
With \Cref{ass:unstablehealthy} in place, the equilibria of the bivirus system can all be characterized as one of three types, namely `healthy', `boundary' or `coexistence', as follows.
\begin{enumerate}
\item The healthy equilibrium is $(x^1 = \vect 0_n, x^2 = \vect 0_n)$, and it is unstable.
\item There are precisely two equilibria where one virus is extinct and the other is present: $(\bar x^1, \vect 0_n)$ and $(\vect 0_n, \bar x^2)$, where $\bar x^1$ and $\bar x^2$ correspond to the unique endemic equilibria of the single virus system for virus~1 and virus~2, respectively. Because $(\bar x^1, \vect 0_n)$ and $(\vect 0_n, \bar x^2)$ are on the boundary of the region of interest $\Xi_{2n}$, we shall refer to them loosely as `boundary equilibria', thus being consistent with the literature~\cite{carlos2,ye2021_bivirus}.
\item Any other equilibrium $(\tilde x^1, \tilde x^2)$ (if it exists) is termed a coexistence equilibrium, as it must satisfy $\vect 0_n \ll \tilde x^i \ll \vect 1_n$ and $\tilde x^1 + \tilde x^2 \ll \vect 1_n$ (see \cite[Lemma~3.1]{ye2021_bivirus} for the details on the inequality conditions on coexistence equilibria).
\end{enumerate}
Our recent work in \cite{ye2021_bivirus} used monotone systems theory~\cite{smith1988monotone_survey} to establish that there is no chaotic behavior for \eqref{eq:bivirus}. From all initial conditions except possibly a set of measure zero, trajectories will converge to an equilibrium point, and limit cycles, if they exist, must be nonattracting. (To the best of our knowledge, no bivirus system has been demonstrated to have a limit cycle). Necessary and sufficient conditions for the boundary equilibria to be locally exponentially stable and unstable were presented in~\cite[Theorem~3.10]{ye2021_bivirus}. Sufficient conditions on the $D^i$ and $B^i$ have been identified yielding all the possible \textit{stability configurations} of the boundary equilibria, that is, conditions such that the resulting bivirus system has neither, one or two of the boundary equilibria being stable~\cite{carlos2,ye2021_bivirus,santos2015bi,santos2015bivirus_conference,janson2020networked}.
With convergence assured and the local stability properties of boundary equilibria fully characterised, key open questions, including our efforts in this work, center around describing the nature and stability properties of the coexistence equilibria and their number. This is a highly nontrivial challenge, partly owing to the network dynamics: the stability configuration of the boundary equilibria does not immediately establish the existence and stability properties (if any) of coexistence equilibria, without further assumptions and conditions on $D^i$ and $B^i$. For example~\cite{ye2021_bivirus}, $D^1 = D^2 = I_n$ and $B^1 > B^2$ is a \textit{sufficient} (but not necessary) condition for i) $(\bar x^1, \vect 0_n)$ to be locally stable and $(\vect 0_n, \bar x^2)$ to be unstable, and ii) no coexistence equilibria\footnote{It turns out this also guarantees that $(\bar x^1, \vect 0_n)$ is globally stable in $\Xi_{2n}\setminus \{(\vect 0_n, \vect 0_n),(\vect 0_n. \bar x^2)\}$.}. It is not clear whether the particular stability configuration of one stable and one unstable boundary equilibria implies the non-existence of coexistence equilibria, or whether non-existence is a consequence of the further constraint $B^1 > B^2$. The aim of this paper is to shed light on how the stability configurations of boundary equilibria help to determine the coexistence equilibria and their stability properties.
For future reference, we note that the Jacobian associated with \eqref{eq:bivirus} evaluated at $(\bar x^1,{\bf{0}}_n)$ is given by
\begin{equation}\label{eq:boundaryjacobian}
J(\bar x^1,{\bf{0}}_n)=\begin{bmatrix}
-D^1+(I_n-\bar X^1)B^1-{\rm{diag}}(B^1\bar x^1)&-{\rm{diag}}(B^1\bar x^1)\\
0&-D^2+(I_n-\bar X^1)B^2
\end{bmatrix}
\end{equation}
The 1-1 block of the Jacobian is stable, being the same as the Jacobian applying to the steady state (equilibrium) solution of the single virus problem \eqref{eq:singlevirusunlabelled} associated with the parameters $D^1,B^1$, while the 2-2 block may or may not be stable. The same is true \textit{mutatis mutandis} for $J({\bf{0}_n},\bar x^2)$. One can even construct a special example where both boundary equilibria have 0 as the eigenvalue with largest real part~\cite{liu2019bivirus,ye2021_bivirus}.
\subsection{A motivating example}\label{ssec:n4_example}
We now present an $n=4$ example bivirus system to motivate the need for tools to derive additional insight into the equilibria.
We set $D^1 = D^2 = I$, and
\begin{equation}
B^1=\begin{bmatrix} 1.6&1& 0.001 & 0.001 \\ 1&1.6 & 0.001& 0.001 \\ 0.001 & 0.001 & 2.1&0.156\\0.001& 0.001 & 3.0659&1.1
\end{bmatrix},
\quad
B^2=\begin{bmatrix} 2.1&0.156& 0.001 & 0.001 \\ 3.0659&1.1 & 0.001& 0.001 \\ 0.001 & 0.001 & 1&1.6\\0.001& 0.001 & 1.6&1
\end{bmatrix}.
\end{equation}
With $n = 4$, directly solving the equations in \eqref{eq:bivirus} to identify all equilibria in $\Xi_{2n}$ is nontrivial, even with the aid of numerical solvers. Based on simulations of multiple initial conditions, there are four locally stable equilibria, of which two are boundary equilibria and two are co-existence equilibria. With $(\tilde x^1, \tilde x^2)$ being an equilibrium, four equilibria are given up to four decimal points by
\begin{equation*}
\left(\begin{bmatrix}
0.6157\\
0.6157\\
0.5652\\
0.7160
\end{bmatrix},\vect 0_4\right)
\left(\vect 0_4, \begin{bmatrix}
0.5652\\
0.7160\\
0.6157\\
0.6157
\end{bmatrix}\right)
\left(\begin{bmatrix}
0.0111\\
0.0065\\
0.5540 \\
0.7076
\end{bmatrix},
\begin{bmatrix}
0.5540 \\
0.7076 \\
0.0111\\
0.0065
\end{bmatrix}\right)
\left(\begin{bmatrix}
0.6063\\
0.6005\\
0.0077\\
0.0164\\
\end{bmatrix},
\begin{bmatrix}
0.0077\\
0.0164\\
0.6063\\
0.6005
\end{bmatrix}\right)
\end{equation*}
Sample trajectories are given in \Cref{fig:example}.
\begin{figure*}[!htp]
\begin{minipage}{0.475\linewidth}
\centering
\subfloat[Boundary equilibrium with virus~1 endemic\label{fig:n4_boundary1}]{\includegraphics[width=\columnwidth]{n4_boundary1.pdf}}
\end{minipage}
\hfill
\begin{minipage}{0.475\linewidth}
\centering\subfloat[Boundary equilibrium with virus~2 endemic]{\includegraphics[width=\columnwidth]{n4_boundary2.pdf}\label{fig:n4_boundary2}}
\end{minipage}
\vfill
\begin{minipage}{0.475\linewidth}
\centering
\subfloat[First attractive coexistence equilibrium]{\includegraphics[width=\columnwidth]{n4_coexistence1.pdf}\label{fig:n4_coexistence1}}
\end{minipage}
\hfill
\begin{minipage}{0.475\linewidth}
\centering\subfloat[Second attractive coexistence equilibrium]{\includegraphics[width=\columnwidth]{n4_coexistence2.pdf}\label{fig:n4_coexistence2}}
\end{minipage}
\caption{Sample trajectories of the $n=4$ example, with different initial conditions. As evident, convergence occurs to four different attractive (locally exponentially stable) equilibria, dependent on the initial conditions. Two are boundary equilibria, where one virus is endemic and the other extinct. Two are coexistence equilibria, where both viruses infect a nonzero fraction of every population. }\label{fig:example}
\end{figure*}
The number of attractor equilibria may be obtained via extensive numerical simulations exploring a variety of initial conditions (though one can never obtain guarantees on this number using this approach). However, it is not viable to use numerical simulations to identify the number of unstable coexistence equilibria as the stable manifold of each such equilibrium has measure zero (or is nonexistent for a source)~\footnote{If the unstable equilibrium was a source, it would be possible to run the dynamics \eqref{eq:bivirus} backward in time to locate the equilibrium, but there is no guarantee that it is a source.}. These numerical simulations highlight the fact that the equilibria patterns can be complex, and strongly motivate the need for theoretical tools that allow us to count the number of coexistence equilibria and even possibly determine the number of stable eigenvalues of the Jacobian matrix at the equilibrium.
\subsection{Poincar\'e-Hopf Theorem}\label{ssec:ph_theory}
The version of the Poincar\'e-Hopf Theorem that we will use, which is drawn from \cite[see p. 35 and Lemma 4, p.37]{milnor1997topology}, is as follows (we explain the terminology in detail immediately below):
\begin{theorem}\label{thm:PH}
Consider a smooth vector field on a compact $m$-dimensional manifold $\mathcal M$, defined by the map $f:\mathcal M\rightarrow T\mathcal M$. If $\mathcal M$ is a manifold-with-boundary, then f must point outward at every point on the boundary, denoted by $\partial \mathcal M$. Suppose that every zero $x_k \in\mathcal M$ of $f$ is nondegenerate.\footnote{The most common statement of the Poincar\'e-Hopf Theorem hypothesizes that all zeros are isolated and makes no assumption of nondegeneracy of the vector field zeros or, equivalently, nonsingularity of the Jacobian at the zero; nor does it involve the signs of the determinants of $df_{x_k}$. Our statement both imposes a tighter condition on the zeros, since nondegeneracy of a zero implies it is isolated, \cite[see p.139] {guillemin2010differential} and obtains a more precise result. Note further that the facts that $\mathcal M$ is compact and every zero is nondegenerate together ensure the number of zeros is finite, so there is no need for a separate explicit requirement for this property in the theorem hypothesis.} Then
\begin{equation}\label{eq:PHmainequation}
\sum_k{\rm{ind}}_{x_k}(f)=\sum_k {\rm{sign \; det}}(df_{x_k})=\chi(\mathcal M)
\end{equation}
\end{theorem}
This theorem statement of course uses language of topology, see e.g. \cite{lee2013introduction,guillemin2010differential}. We provide minimal decoding remarks here. One can think of $f$, a vector field, as the right side of a differential equation
\begin{equation}\label{eq:general_system}
\dot x=f(x).
\end{equation}
A point $x_k$ is said to be a zero of $f$ if $f(x_k) = \vect 0$, and thus a zero $x_k$ of $f$ is equivalent to $x_k$ being an equilibrium of \eqref{eq:general_system}. The symbol $T\mathcal M$ denotes the tangent space of the manifold $\mathcal M$. The term `manifold-with-boundary' does have a technical meaning. More precisely, a point $x\in \mathcal M\setminus \partial\mathcal M$ must have the property that there exists a neighborhood of $x$ in $\mathcal M$ that is diffeomorphic to a neighborhood of a point in $\mathbb R^m$, where $m$ is the dimension of $\mathcal M$. Meanwhile, a point $x\in \partial\mathcal M$ (for a manifold-with-boundary) must have a neighborhood in $\mathcal M$ which is diffeomorphic to a half space in $\mathbb R^m$~\cite{guillemin2010differential, lee2013introduction}. As a consequence, a region such as a square (including its edges and corners) in $\mathbb R^2$ would {\textit{not}} qualify, due to the corners not having the requisite property for their neighborhood. Nor could we work with the interior of a square, since it would fail the compactness requirement of the statement of \Cref{thm:PH}. The notion of pointing outward can be rigorously defined using the notion of a `tangent-cone'~\cite{blanchini1999set_invariance,ye2021applications}, but for our purposes the intuitive interpretation is adequate. The notation $df_{x_k}$ denotes the Jacobian of $f$ evaluated at $x_k$ and a nondegenerate zero is defined as one at which the Jacobian is nonsingular, see \cite[p.~37]{milnor1997topology}, with such a zero being necessarily isolated. (Separately, note that nondegeneracy is a property independent of the choice of coordinate system, see e.g. \cite[see p.~42]{matsumoto2002introduction}, where the independence of the sign of the Jacobian determinant is also demonstrated). The index of a vector field is the sum of the indices of its zeros, and the index of a zero of a vector field is a standard concept, see e.g. \cite[Section 3.5]{guillemin2010differential}. Even in as simple a manifold as $\mathbb R^2$, any positive or negative integer value for the index at a zero is possible; however, with the nondegeneracy assumption, the only possibilities for an arbitrary manifold are $\pm 1$, see~\cite[p.~37, Lemma 4]{milnor1997topology} and \cite[p.~139]{guillemin2010differential}. The Euler characteristic of $\mathcal{M}$, denoted by $\chi(\mathcal{M})$, is determined by the shape of the manifold, and values are known for common manifolds, e.g. ball, sphere, torus, $\mathbb R^n$, etc.
For future reference we note that if $n_k$ is the number of eigenvalues of the Jacobian $df_{x_k}$ with negative real part, then
\begin{equation}\label{eq:ph_sum}
{\rm{sign\; det}}(df_{x_k})=(-1)^{n_k}
\end{equation}
Our aim is to use the above version of the Poincar\'e-Hopf Theorem in a bivirus setting. However, seeking to apply the Theorem by identifying the region of interest $\Xi_{2n}$ with $\mathcal M$ immediately creates some problems, including the following.
\begin{enumerate}
\item
The theorem requires zeros of the vector field to be nondegenerate (which implies they are isolated, \cite[[see p. 139]{guillemin2010differential}. This implies that with $f$ denoting the vector field, if the manifold in question is bounded, the number of zeros of $f$ or equilibrium points of \eqref{eq:general_system} must be finite. It is clearly preferable for a property such as finiteness of the number of zeros, or nondegeneracy of each zero, to be demonstrated, rather than just assumed. Awkwardly however, for the bivirus dynamics in \eqref{eq:bivirus}, special networks have been identified with an infinite number of equilibria that form a segment of a line in $\Xi_{2n}$~\cite{liu2019bivirus,ye2021_bivirus}.
\item
The region of interest $\Xi_{2n}$ is not a manifold without boundary, nor a manifold-with-boundary. It is a compact region, and it does have a boundary in a set theory sense, but it fails to be a manifold-with-boundary, essentially because it has corners. It can in fact be studied as a manifold-with-corners~\cite{lee2013introduction}, but a version of the Poincar\'e-Hopf Theorem for a manifold-with-corners does not appear to be available in the literature.
\item
The theorem requires the vector field defined by $f$ to be outward pointing on the boundary $\partial \mathcal M$. If one believed that the theorem should apply to a manifold-with-corners such as $\Xi_{2n}$, this requirement would preclude the existence of any zeros of the vector field on the boundary, since the direction is simply not defined at a zero (and in an arbitrary neighborhood many directions occur, a fact which probably precludes a limiting argument). There are however three equilibria on the boundary of $\Xi_{2n}$, viz. the healthy equilibrium, and the two boundary equilibria.
\end{enumerate}
Our paper will address all three issues. The first issue is comparatively easy to deal with, as it turns out. Our strategy for dealing with the second issue depends on two major steps; in the first, the region of interest $\Xi_{2n}$ will be perturbed with an enlargement so that any stable equilibrium on its boundary becomes an interior point of the perturbed region. Second, we will exhibit a diffeomorphic map from the interior of the modified region of interest to an even-dimensional sphere, less a single point; because this is a region to which the Poincar\'e-Hopf Theorem is obviously almost applicable, with some further massaging a result can be obtained for the bivirus system. To deal with the third issue, we will actually apply the Poincar\'e-Hopf THeorem to a modified manifold, viz. the $2n$-dimensional sphere, which has no boundary. There are then no trajectory directions that have to be checked.
\section{Main Result}\label{sec:main}
In this section, we return to the direct study of \eqref{eq:bivirus}. As remarked previously, use of Poincar\'e-Hopf theory presupposes that the number of equilibria, at least in $\Xi_{2n}$, of the differential equations (equivalently the zeros of the vector field $f$) is finite, and indeed that all zeros have a nondegeneracy property. The first novel contribution of the paper establishes that such nondegeneracy is normally to be expected, using differential topology ideas for the proof. It is a fundamental precursor for the two main `counting' results of the paper.
\begin{theorem}\label{thm:finiteness}
With notation as previously defined, consider the equation set \eqref{eq:bivirus} in the region $\Xi_{2n}$ and assume that \Cref{ass:constraints} and \Cref{ass:unstablehealthy} both hold. Then with any fixed $B^1,B^2$, and the exclusion of a set of values for the entries of $D^1,D^2$ of measure zero, the number of equilibrium points (equivalently zeros of the associated vector field) is finite and each zero is nondegenerate. Similarly, with any fixed $D^1,D^2$, and the exclusion of a set of values for the entries of $B^1,B^2$ of measure zero, the same property for the equilibrium points is assured.
\end{theorem}
This second claim of the theorem can be established by appealing to ideas of algebraic geometry~\cite[Theorem~3.6]{ye2021_bivirus} and indeed a different proof can be found mixing manifold ideas and algebraic geometry ideas in~\cite{robbin1981algebraic}. We provide in \Cref{app:pf_thm_finiteness} a third proof avoiding algebraic geometry entirely, instead appealing to the Parametric Transversality Theorem in manifold theory, see e.g.~\cite[p.~145]{lee2013introduction} and \cite[p.~68]{guillemin2010differential}, and the proof also covers the first part of the theorem as well. Approaches relying on algebraic geometry require the system equations in \eqref{eq:bivirus} to be polynomial in the state variables, while the Parametric Transversality Theorem approach does not. The latter thus offers an advantage for extending analysis to bivirus network dynamics with non-polynomial terms, for instance if we were to consider introduction of feedback control~\cite{ye2021applications}, or certain small smooth variations to the right side of the differential equation, or even the larger variations provided in a modification to the quadratic terms suggested in \cite{yang2017bi}. The requirement that, when the $D^i$ are fixed, the $B^i$ are excluded from a set of measure zero is indeed needed: as we mentioned in \Cref{ssec:ph_theory}, sets of specially structured $B^i$ exist for which there is a continuum of equilibria.
Going along with the immediately preceding theorem is a strengthening of the nondegeneracy condition that we will need. Generically, zeros of the vector field are not just nondegenerate, but also hyperbolic. That is, the Jacobian matrix at a zero is free of eigenvalues with zero real part.
\begin{theorem}\label{app:pf_thm_hyperbolicity}
Adopt the same hypothesis as Theorem~\ref{thm:finiteness}. With any fixed matrices $B^1,B^2$, and the exclusion of a set of values for the entries of $D^1,D^2$ of measure zero, the number of equilibrium points is finite and the associated vector field zero is hyperbolic. Similarly, with any fixed $D^1,D^2$, and the exclusion of a set of values for the entries of $B^1,B^2$ of measure zero, the same property for the equilibrium points is assured.
\end{theorem}
The proof of this theorem can be found in Appendix~\ref{sec:app b}. The first main result of the paper can now be stated as follows:
\begin{theorem}\label{thm:main}
With notation as previously defined, consider the equation set \eqref{eq:bivirus} and suppose that Assumptions \ref{ass:constraints} and \ref{ass:unstablehealthy} both hold. Suppose that the equilibria of \eqref{eq:bivirus} in the region of interest $\Xi_{2n}=\{{\bf{0}}_n\leq x^i\leq {\bf{1}}_n, i=1,2 \}\cap\{ x^1+x^2\leq {\bf{1}}_n\}$ are all hyperbolic and thus finite in number. Excluding the healthy equilibrium $({\bf{0}}_n,{\bf{0}}_n)$ and any unstable boundary equilibrium, let $n_k$ denote the number of open left half plane eigenvalues of the Jacobian associated with the $k$-th equilibrium in $\Xi_{2n}$. Then there holds
\begin{equation}\label{eq:count}
\sum_k(-1)^{n_k}=1
\end{equation}
\end{theorem}
We remark that the appendix of a paper by Glass \cite{glass1975combinatorial} also suggests in an equilibrium classification problem a modification of the Poincar\'e-Hopf formula applied to that problem obtained through similar exclusion of unstable vector field zeros on the boundary of a region of interest. The argument of that paper is more in outline form than provided in full detail, whereas we provide a rigorous treatment.
The proof of this theorem will be developed through some preliminary results, dealing with the three issues described earlier below \eqref{eq:ph_sum}. First, we focus on the behavior of the vector field on the boundary of $\Xi_{2n}$, and demonstrate that an appropriate perturbation of the region of interest will ensure the vector field `points inward' to $\Xi_{2n}$ as required to apply the Poincar\'e--Hopf Theorem. It is convenient to look at three different types of boundary points of $\Xi_{2n}$:
\begin{enumerate}
\item
Boundary points where $x^1_i+x^2_i=1$ for one or more $i$;
\item
Boundary points where $x^2_i=0$ for at least one but not all $i$ (with the same conclusions applying in respect of $x^1_i=0$ for some but not all $i$);
\item
Boundary points where $x^2={\bf{0}}_n$ (with the same conclusions applying to $x^1={\bf{0}}_n$).
\end{enumerate}
The above three categories include all boundary points apart from the healthy equilibrium $({\bf{0}}_n,{\bf{0}}_n)$.
\subsection{Trajectories from the first two types of boundary point}\label{ssec:traj_01}
The outcomes for the first two cases are summarised in the following two lemmas.
\begin{lemma}
Consider a boundary point of $\Xi_{2n}$ where $x^1_i+x^2_i=1$ for some $i$. Then trajectories of \eqref{eq:bivirus} are inward-pointing.
\end{lemma}
\begin{proof}
Suppose $x^1_i+x^2_i=1$. The differential equation for $x^1_i$ then immediately yields $\dot x^1_i=-\delta^1_i x^1_i$ and similarly for $x^2_i$, and the claim is immediate.
\end{proof}
\begin{lemma}
Suppose that $x^1(0)+x^2(0)\ll {\bf{1}}_n$ and that (with inessential reordering if necessary), there holds $x^2_1(0)=0, x^2_2(0)=0,\dots, x^2_p(0)=0,x^2_{p+1}(0)>0,\dots,x^2_{n}(0)>0$ for some $0<p<n$. Then for some $\bar i\in\{1,2,\dots,p \}$, there holds $\dot{\bar x}^2_i>0$, and at an arbitrarily small time $t_p>0$, trajectories obey $\dot x^2(t_p)\gg {\bf 0}_n$.
\end{lemma}
\begin{proof}
Since the matrix $B^2$ is irreducible, and $I-X^1(0)-X^2(0)$ is nonsingular, the matrix $[I-X^1(0)-X^2(0)]B^2$ is also irreducible. This means that the vector $[I-X^1(0)-X^2(0)]B^2x^2(0)$ cannot have a zero in every position where $x^2(0)$ has a zero, i.e. for one or more $i\in\{1,2,\dots,p\}$, say $i=\bar i$, there holds $[\big (I-X^1(0)-X^2(0)\big )B^2x^2(0)]_{\bar i}>0$, whence also from \eqref{eq:bivirus}, there holds $\dot x^2_{\bar i}>0$. Note that for all other $i\in\{1,2,\dots,p\}$, there necessarily holds $\dot x^2_{ i}(0)=[[I-X^1(0)-X^2(0)]B^2x^2(0)]_i \geq 0$.
As a consequence of the first part of the lemma, we see that for a time $t_1$ that is arbitrarily small and positive, fewer than $p$ entries of $x^2(t_1)$ will be zero. And then for a time $t_2$ for which $t_2-t_1$ is arbitrarily small and positive, fewer than $p-1$ entries will be zero. Continuing the argument, there exists an arbitrarily small but positive time $t_p$ for which $\dot x^2(t_p)\gg {\bf 0}_n$, which is equivalent to saying that trajectories are inward pointing. \hfill$\qed$
\end{proof}
\subsection{Trajectories from the third type of boundary point}\label{ssec:traj_02}
We now consider boundary points where $x^2={\bf{0}}_n$ (with any conclusions drawn also applying to boundary points where $x^1 = {\bf 0}_n$). As explained below \Cref{ass:unstablehealthy}, for any initial condition for which $x^2(0)={\bf{0}}_n$, there will hold $x^2(t)={\bf{0}}_n$, for all $t$. Thus trajectories are neither inward or outward pointing with respect to $\Xi_n$, but remain on the boundary. Further, on the boundary where $x^2={\bf{0}}_n$, there is precisely one equilibrium point, viz. the boundary equilibrium $(\bar x^1, {\bf 0}_n)$ with $\bar x^1\gg {\bf 0}_n$. Below we consider first the case where this is a stable equilibrium (all eigenvalues of the Jacobian have negative real part), and subsequently the case where it is unstable (one or more eigenvalues of the Jacobian have positive real part).
\subsubsection{Perturbation of the region of interest around a stable boundary equilibrium} \label{sssec:stable_boundary}
With the boundary equilibrium $(\bar x^1, {\bf 0}_n)$ locally exponentially stable (as an equilibrium of \eqref{eq:bivirus}), we shall explain how to make a perturbation of the boundary of $\Xi_{2n}$ in the vicinity of $(\bar x^1, {\bf 0}_n)$, defined by a hemisphere extending outwards from the boundary, and joined smoothly to the boundary by a $C^{\infty}$ bump function \cite[see p.127]{tu2011introduction}.
\begin{figure}
\centering
\def0.9\linewidth{0.9\linewidth}
\input{bivirus_perturbation.pdf_tex}
\caption{Illustration of perturbation of $\Xi_{2n}$ for $n = 1$, with the triangles indicating boundary equilibria. In this example $(\bar x^1, 0)$ is locally exponentially stable, and $(0, \bar x^2)$ is unstable. Blue arrows indicate the trajectory directions, i.e., the flow of the vector field. At $(\bar x^1, 0)$, a perturbation is introduced, adding a hemisphere (semicircle for scalar $x^1, x^2$) of radius $\epsilon$ to cover points with $|x^2|<\epsilon$ but smoothly joined to $x^2=0$. By picking $\epsilon$ suitably small, all points in the perturbed region are in the region of attraction of $(\bar x^1, 0)$ and hence on the boundary of the hemisphere, the trajectory points `inward'. This is detailed in \cref{sssec:stable_boundary}. At $(0, \bar x^2)$, we make so such perturbation. Trajectories beginning with $x^1(0) = 0$ will have $x^1(t) = 0$ for all $t$, i.e. the trajectories move along the $x^2$ axis towards $(0, \bar x^2)$. However, in the interior, all trajectories point away from the unstable equilibrium, as detailed in \cref{sssec:unstable_boundary}.} \label{fig:bivirus_perturbation}
\end{figure}
To introduce the idea, suppose for the moment there is a single dimension, i.e. $x^2$ is a scalar, and we are working with a bivirus system with just one population. Now the boundary of $\Xi_{2}$ which has $x^2=0$ defines a line, along which $x^1$ varies. At some point on this line lies the equilibrium $(\bar x^1, 0)$, which is locally exponentially stable by hypothesis. Perturb the boundary of $\Xi_2$ along the line $x^2=0$ using a bump function in the vicinity of $(\bar x^1,0)$ so that for an arbitrary fixed but suitably small positive $\epsilon$, the perturbation occurs within the interval $(\bar x^1-\epsilon,\bar x^1+\epsilon)$, the perturbation is in the direction $x^2<0$, and the perturbation ensures that $|x^2|<\epsilon$ is within the new perturbed boundary.\footnote{In more detail, suppose $f(t)$ is the function $\exp(-1/t), t>0$ and $0$ for $t\leq 0$. Define $g(t)=f(t)/[f(t)+f(1-t)]$; this is a function which transitions smoothly and monotonically from value 0 at $t=0$ to value 1 at $t=1$. Define $h(t)=g\big((t +\epsilon)(2\epsilon)^{-1}\big)$. This function transitions smoothly and monotonically from 0 at $t=-\epsilon$ to 1 at $t=\epsilon$. The function $\phi_{\epsilon}(t)=4\epsilon h(t)(1-h(t))$ is smooth and transitions with monotone increase from 0 at $t=-\epsilon$ to $\epsilon$ at $t=0$ and transitions with monotone decrease from $\epsilon$ at $t=0$ to zero at $t=\epsilon$. It is zero outside $(-\epsilon, \epsilon)$.
}
The boundary $x^2=0$ is replaced $x^2=-\phi_{\epsilon}(x_1-\bar x_1)$.
The detailed definition of $\phi_{\epsilon}$ is contained in the footnote, and the boundary perturbation is like a semicircle extending into the left half plane, but smoothly connected to the axis $x^2=0$. An illustration of this is presented in \Cref{fig:bivirus_perturbation}.
Here is how to generalize this idea to the case when $x^1,x^2$ are both $n$-dimensional, using the same function $\phi_{\epsilon}$.
\begin{lemma}\label{lem:stableboundary}
Suppose the region $\Xi_{2n}$ is as defined earlier and with $\epsilon$ an arbitrary positive constant, denote by $\phi_{\epsilon}$ a smooth bump function that is zero outside of $(-\epsilon,\epsilon)$, positive in $(-\epsilon,\epsilon)$, and taking the value $1$ at $0$. Consider that part of the boundary of $\Xi_{2n}$ defined by $x^2={\bf{0}}_n$, with $(\bar x^1, {\bf{0}}_n)$ being the boundary equilibrium of the bivirus system. Now with $\epsilon$ a suitably small arbitrary but fixed positive constant, expand the region by defining a new boundary via
\begin{equation}\label{eq:perturb}
x^2_i=-\frac{1}{\sqrt n}\phi_{\epsilon}(\|x^1-\bar x^1\|)\,,\;\forall\, i\in\mathcal{V}
\end{equation}
Then points on the new boundary either have all entries of $x^2$ negative, or all entries zero, and those with $x^2\ll {\bf{0}}_n$ are all no greater than a distance $\epsilon$ from $(\bar x^1,{\bf{0}}_n)$ and obey $\|x^2\| \leq \epsilon$.
\end{lemma}
Now choose $\epsilon$ so that all points in the perturbed region at a distance $\epsilon$ from the equilibrium $(\bar x^1,{\bf{0}}_n)$ are in its region of attraction. Then all points on the modified boundary which have $x^2\ll {\bf{0}}_n$ lie within the region of attraction of $(\bar x^1, {\bf{0}}_n)$. All other points on the boundary (i.e. those which are also on the boundary of the unmodified region) have an inward pointing trajectory (in the case where one or more entries of $x^2$ are positive), or a trajectory pointing along the boundary towards $(\bar x^1,{\bf{0}}_n)$ (in the case where $x^2={\bf{0}}_n$), by the arguments given previously.
Henceforth, we shall use the notation $\tilde{\Xi}_{2n}$ to denote the perturbation of $\Xi_{2n}$ to encompass any locally exponentially stable boundary equilibrium, achieved using \eqref{eq:perturb} in the case of $(\bar x^1, {\bf{0}}_n)$ and/or a similar perturbation in the case of $({\bf 0}_n, \bar x^2)$.
\subsubsection{Behavior in the vicinity of an unstable boundary equilibrium}\label{sssec:unstable_boundary}
We now examine trajectories in the vicinity of a boundary equilibrium that is unstable. Our exposition will consider $(\bar x^1, {\bf{0}}_n)$, but the same conclusions can be drawn if $({\bf 0}_n, \bar x^2)$ is unstable. No perturbation of the region of interest is made. Unless all eigenvalues of the associated Jacobian matrix have positive real parts, there are always some trajectories which can approach the unstable equilibrium (with such trajectories defining the `stable manifold' of this equilibrium). Those trajectories starting on the boundary of $\Xi_{2n}$ with $x^2={\bf{0}}_n$ (including those starting in a neighborhood of the equilibrium) evolve according to the single virus equation, and thus approach the equilibrium. See \cref{ssec:single_virus} above for details. We can further ask whether there is any trajectory starting in a neighborhood of $(\bar x^1, {\bf{0}}_n)$ and in the interior of $\Xi_{2n}$ that might also converge to $(\bar x^1, {\bf{0}}_n)$ (even if some trajectories will not, on account of the instability property). The key conclusion is as follows. (An illustrative example is presented in \Cref{fig:bivirus_perturbation}, but assuming $(0, \bar x^2)$ is the unstable boundary equilibrium.)
\begin{lemma}
Suppose that the boundary equilibrium $(\bar x^1,{\bf{0}}_n)$ of \eqref{eq:bivirus} is unstable, in the sense that one or more eigenvalues of the associated Jacobian have positive real part. Then there exists no trajectory beginning in the interior of $\tilde \Xi_{2n}$ which approaches this unstable equilibrium, with $\tilde \Xi_{2n}$ is defined above. Namely, $\tilde \Xi_{2n}$ is equal to $\Xi_{2n}$ with perturbation to encompass the boundary equilibrium $({\bf 0}_n, \bar x^2)$ if it is locally exponentially stable.
\end{lemma}
\begin{proof}
Suppose, to obtain a contradiction, there is a trajectory, call it $\mathcal T$, starting inside the region $\tilde \Xi_{2n}$ which has the equilibrium $(\bar x^1,{\bf{0}}_n)$ as its limit. If $({\bf 0}_n, \bar x^2)$ is locally exponentially stable, then any trajectory beginning in $\tilde \Xi_{2n}\setminus\Xi_{2n}$ (i.e. in the perturbed region) is in the region of attraction of $({\bf 0}_n, \bar x^2)$, and thus cannot converge to $(\bar x^1,{\bf{0}}_n)$. Therefore without loss of generality we can assume that the initial condition for the trajectory $\mathcal T$ satisfies $x^2(0)\gg {\bf{0}}_n$. Now for sufficiently large values of time, $t\geq T_0$ say, the trajectory will be arbitrarily close to the limit and thus its evolution from time $T_0$ onwards can be modelled (to first order) using the linearized equation
\begin{equation}\label{eq:linfull}
\dot x=J(\bar x^1,{\bf{0}}_n)x
\end{equation}
which in the light of \eqref{eq:boundaryjacobian} means that the projection $x^2(t)$ satisfies
\begin{equation}
\dot x^2=[-D^2+(I-\bar X^1)B^2]x^2
\end{equation}
with $x^2(t)\gg 0$ for all finite $t$ and converging to $\vect 0_n$ as $t\to\infty$.
Note that hyperbolicity is required to justify the validity of the linearization approximation and in particular, the drawing of stability conclusions using the linearized equation; hyperbolicity
is guaranteed via Theorem~\ref{app:pf_thm_hyperbolicity}.
Suppose that $x^2(T)$ with $T>T_0$ is some point on the projection of the trajectory $\mathcal T$. Suppose initially that the eigenvalues of $[-D^2+(I-\bar X^1)B^2]$ are distinct, in which case there are $n$ linearly independent eigenvectors, call them $v_1,v_2,\dots,v_n$. Further suppose that $v_1\gg {\bf 0}_n$ corresponds to the eigenvalue $\sigma[-D^2+(I-\bar X^1)B^2] $, which is simple since $-D^2+(I-\bar X^1)B^2$ is an irreducible Metzler matrix. Because the equilibrium is unstable, one or more eigenvalues of this matrix must have positive real part (else the entire Jacobian of \eqref{eq:boundaryjacobian} would have strict left half plane eigenvalues, a property which is guaranteed for its block 11 entry). Hence the eigenvalue $\sigma[-D^2+(I-\bar X^1)B^2] > 0$, necessarily real and simple by the Metzler property, Write $x^2(T)=\sum_i \zeta_iv_i$ for some scalars $\zeta_i$. Because of the convergence of $\mathcal T$ to $(\bar x^1, {\bf 0}_n)$, there must hold $\zeta_1=0$, and in fact $\zeta_i=0$ for any $i$ for which the associated eigenvalue of $[-D^2+(I-\bar X^1)B^2]$ has nonnegative real part. Hence $x^2(T)=\sum_{i\in\mathcal I}\zeta_iv_i$, where $\mathcal I$ is the set of indices for which the $i$-th eigenvalue of $[-D^2+(I-\bar X^1)B^2]$ has negative real part. Now suppose $u_1^{\top}$ is the left eigenvector corresponding to eigenvalue $\sigma[-D^2+(I-\bar X^1)B^2]$ with $u_1^{\top}v_1=1$ and $u_1\gg {\bf 0}_n$. Then because $u_1^{\top}v_i=0$ for $i\neq 1$, it follows that $u_1^{\top}x^2(T)=0$. This is a contradiction to the fact that $x^2(T)\gg {\bf 0}_n$ and the fact that $u_1\gg {\bf 0}_n$. In other words, no trajectory starting in the interior of $\tilde \Xi_{2n}$ can reach the equilibrium $(\bar x^1, {\bf 0}_n)$. Equivalently, in the vicinity of the equilibrium, all trajectories lying in $\Xi_{2n}$ are pointing away from the equilibrium.
The case when the eigenvalues of $[-D^2+(I-\bar X^1)B^2]$ are not distinct can be handled by a notationally messy argument involving Jordan blocks; note that the uniqueness of the eigenvalue $\sigma[-D^2+(I-\bar X^1)B^2]$ will be critical.
\end{proof}
Despite the replacement of $\Xi_{2n}$ by $\tilde\Xi_{2n}$ to ensure an inward-pointing property for trajectories at the boundary, we cannot apply the Poincar\'e-Hopf Theorem to $\tilde \Xi_{2n}$ because (a) there are trajectories confined to the boundaries and (b) there are zero(s) of the vector field lying on the boundary (corresponding to the always present unstable healthy equilibrium and any boundary equilibria that are unstable). Nor, in an attempt to avoid these problems, can we apply the Theorem to the interior of $\tilde\Xi_{2n}$ since this is not a compact manifold. We are however in a position to introduce some transformations of the manifold $\tilde\Xi_{2n}$ to resolve these issues.
\subsection{Diffeomorphism involving a sphere and use of Poincar\'e-Hopf Theorem}
We suppose that, if there is any stable boundary equilibrium (with at most two being able to occur), the region of interest $\Xi_{2n}$ has been perturbed so as to make such equilibria lie in the interior of the perturbed region $\tilde\Xi_{2n}$, as described in \cref{sssec:stable_boundary}. We will introduce diffeomorphic transformations starting with the interior $\tilde\Xi_{2n}^{\circ}$ of $\tilde\Xi_{2n}$, and subsequently deal with what happens to the boundaries. In so doing, we will be drawing on an argument used by \cite{glass1975combinatorial} for a related but simpler problem (in which $\tilde\Xi_{2n}$ is replaced by the positive orthant and no trajectories confined to boundaries can exist).
A summation of what we are about to prove concerning the interiors of the various regions is in the following Lemma.
\begin{lemma}\label{lem:diffeo}
With notation as previously, there exists a diffeomorphism $F_0$ between the interior $\tilde\Xi_{2n}^{\circ}$ of $\tilde\Xi_{2n}$ and a punctured sphere $S^{2n}\setminus N$, where $N$ denotes the north pole. This diffeomorphism maps the bivirus vector field defined by \eqref{eq:bivirus} on the manifold $\tilde\Xi_{2n}^{\circ}$ onto a unique smooth vector field on $S^{2n}\setminus N$ with zeros which are the images under $F_0$ of the vector field zeros in $\tilde\Xi_{2n}^{\circ}$, and with the indices of corresponding zeros the same.
\end{lemma}
\begin{proof}
Observe first that the interior of the new region $\tilde\Xi_{2n}$ is obviously diffeomorphic to the interior of a solid ball $\mathcal B_R$ of arbitrary radius $R$ and of dimension $2n$. One could construct such a diffeomorphism, call it $F_1$, by picking a point in the interior of $\Xi_{2n}$ and mapping it to the origin of the ball, and by mapping each line joining that point to a boundary point of $\tilde\Xi_{2n}$ to a line in the same direction joining the origin to the boundary of the ball. We remark that the boundary of $\tilde \Xi_{2n}$ corresponds to the boundary of the ball.
Next, observe that the interior of the ball is diffeomorphic to $\mathbb R^{2n}$ under the mapping $F_2: \mathcal B_R^{\circ}\to \mathbb R^{2n}, p\mapsto (\tan\frac{\pi\|p\|}{2R})p$. Note that points on the boundary of the ball are mapped to infinity in $\mathbb R^{2n}$.
Under a further diffeomorphism $F_3$, $\mathbb R^{2n}$ corresponds to a sphere $S^{2n}$ excluding one point, say the north pole of the sphere. Such a diffeomorphism is standard, e.g. \cite[see p.6]{guillemin2010differential} or \cite[see p. 29]{ballmann2018introduction}. Points at infinity in $\mathbb R^{2n}$ effectively correspond to the north pole of $S^{2n}$.
The three diffeomorphisms combine to give a single diffeomorphism $F_0=F_3\circ F_2\circ F_1$ from the interior of $\tilde\Xi_{2n}$ to the punctured sphere $S^{2n}\setminus N$ where $N$ denotes the north pole.
The vector field defining the bivirus equations in \eqref{eq:bivirus}, is, on the manifold $\tilde\Xi_{2n}$, also transformed by $F_0$. Because the mapping is a diffeomorphism, a unique smooth vector field on $S^{2n}\setminus N$ is guaranteed to exist, \cite[see Proposition 8.19]{lee2013introduction}{\footnote{Two potential difficulties can arise when a vector field on one manifold $\mathcal M$ is mapped to another manifold $\mathcal N$: when the mapping is not surjective, the vector field is not defined at some points of $\mathcal N$, and if the mapping is not injective, one point in $\mathcal N$ may have a nonunique vector field, \cite[see p.181]{lee2013introduction}. The diffeomorphism property of $F_0$ rules out such problems.}}. Further, the index of the vector field at an isolated zero is preserved (together with the isolation of the zero) by a diffeomorphism, \cite[see p.33, Lemma 1]{milnor1997topology}. At a nondegenerate zero, as pointed out by \cite[see p. 136]{guillemin2010differential}, for a diffeomorphism $y=F(x)$, the vector field at $y$ corresponding to $df_x$ at $x$ is given by the similarity transformation $dg_y=(dF_x)^{-1}df_x(dF_x)$, from which it is evident that ${\rm{sign\; det}}(df_{x})={\rm{sign\; det}}(dg_y)$. Not only is hyperbolicity preserved, but even the actual eigenvalues of the Jacobian at the equilibrium
\end{proof}
The entire boundary of $\tilde\Xi_{2n}$ can be identified with the north pole of the sphere $S^{2n}$ under the mapping $F$. Note that this pole amalgamates so to speak the healthy equilibrium, any unstable boundary equilibrium and trajectories confined to the boundary, and the initial point of trajectories which start on the boundary but immediately leave it (corresponding to i) single-virus behavior of the bivirus system where $x^i(0) = \vect 0_n$ for some $i$, or ii) $x^1(0)+x^2(0) = \vect 1_n$), as such points are a subset of the boundary $\partial \tilde\Xi_{2n}$. On the surface of the sphere (including the north pole) there will be trajectories for which the north pole is a source, and no trajectories will approach the north pole. This is in the light of our analysis in \cref{ssec:traj_01,ssec:traj_02}, and the properties we established concerning trajectories at and adjacent to the boundary of $\tilde\Xi_{2n}$. This means one can add the north pole to the punctured sphere, and the associated vector field is well defined at the north pole, with a zero there (in fact, this zero is a source).
Crucially, the sphere itself (without the puncture) is a compact manifold to which the Poincar\'e-Hopf Theorem in principle can be applied.
\subsection{Completion of proof of Theorem~\ref{thm:main}}
We apply the Poincar\'e-Hopf Theorem to the sphere. The argument is as follows. Let $n_k$ denote the number of open left half plane eigenvalues of the Jacobian associated with the $k$-th equilibrium or vector field zero in $\tilde\Xi_{2n}^{\circ}$. As noted above in \Cref{lem:diffeo}, this is the same as the number of eigenvalues associated with the Jacobian of the corresponding vector field zero (using the result of \cite{milnor1997topology} and \cite{guillemin2010differential}) on the sphere, any such zero being away from the north pole. The index for the north pole, corresponding to the boundary of $\tilde\Xi_{2n},$ which is simply a source from the point of view of trajectories on the sphere, is $1$, since there are no left half plane eigenvalues of the Jacobian. Now it is standard that the Euler characteristic of $S^{2n}$ is 2, see e.g. \cite[pg.~134]{hirsch2012differential}. Hence using the Poincar\'e-Hopf Theorem, i.e. \Cref{thm:PH}, for the sphere, we have $\sum_k(-1)^{n_k}+1=2$,
or
\begin{equation}
\sum_k (-1)^{n_k}=1
\end{equation}
This equation, although obtained by studying the sphere, is also the equation which relates the vector field zeros for the bivirus problem, with the understanding that the healthy equilibrium is not counted, a boundary equilibrium which is stable (eigenvalues of the Jacobian in the open left half plane) is counted, and a boundary equilibrium which is not stable (one or more eigenvalues of the Jacobian in the right half plane) is not counted.
\subsection{Consequences of \texorpdfstring{\Cref{thm:main}}{}}
An immediate and important consequence of the main result, \Cref{thm:main}, is the following. It treats all three of the possible configurations of boundary equilibria that can occur.
\begin{corollary}\label{cor:ph_count}
Adopt the hypotheses of Theorem \ref{thm:main}. Then
\begin{enumerate}
\item
if there is one locally exponentially stable and one unstable boundary equilibrium of the bivirus system, there need not be any coexistence equilibrium (though there may be);
\item if both boundary equilibria of the bivirus system are unstable, then there exists an odd number $k\geq 1$ of coexistence equilibria. There are $(k+1)/2\geq 1$ equilibria with the associated Jacobian having an even number of open left-half plane eigenvalues, and $(k-1)/2$ equilibria (none of which can be stable) with the associated Jacobian having an odd number of open left-half plane eigenvalues;
\item
if both boundary equilibria of the bivirus system are locally exponentially stable, then there exists an odd number $k\geq 1$ of coexistence equilibria. There are $(k+1)/2\geq 1$ equilibria (none of which can be stable) with the associated Jacobian having an odd number of open left-half plane eigenvalues, and $(k-1)/2$ equilibria with the associated Jacobian having an even number of open left-half plane eigenvalues.
\end{enumerate}
\end{corollary}
\begin{proof}
The fact that the first configuration of boundary equilibria is possible with no coexistence equilibrium has been known in the literature for some time, e.g.~\cite[Corollary~3.11]{ye2021_bivirus} and \cite{santos2015bi}. An example below will establish the possibility that such a configuration of boundary equilibria can also allow for the presence of coexistence equilibria.
For the second claim, suppose there are two unstable boundary equilibria. Since we do not count any unstable boundary equilibria in computing \eqref{eq:count}, there must be at least one equilibrium in $\tilde \Xi_{2n}^\circ$ contributing to the sum on the left hand side of \eqref{eq:count} in order that the sum be positive, and the associated Jacobian must have an even number of eigenvalues in the open left-half plane. Thus $k \geq 1$. Suppose that $k_e,(k_0) $ denote the number of equilibria with an even (odd) count of open left-half plane eigenvalues, with $k_e+k_0=k$. Since an equilibrium with a Jacobian having an even (respectively, odd) number of eigenvalues in the open left-half plane contribute $+1$ (respectively, $-1$) to the right hand side of \eqref{eq:count}, there holds $k_e-k_0=1$. The remaining conclusions of item 2) of the corollary are easily established.
For the third claim, suppose there are two locally exponentially stable boundary equilibria. The argument is the same as that for the second claim, save that \eqref{eq:count} yields $k_e-k_0=-1$, in light of the two stable boundary equilibria.
\end{proof}
If the bivirus system has two unstable boundary equilibria (Item~2 of \Cref{cor:ph_count}), then one can further exploit known properties of monotone systems to conclude that among the $k$ coexistence equilibria, at least one of them is locally exponentially stable~\cite[Theorem~2.8]{smith1988monotone_survey}.
We will develop further counting conditions in the next section, based on Morse inequalities for Morse-Smale systems, which provides an alternative method to show that there is necessarily a stable coexistence equilibrium if both boundary equilibria are unstable.
A weaker version of the second claim of the lemma can be found in \cite{janson2020networked}, where it is established that there must be at least one coexistence equilibrium (but no stability properties are provided for the equilibrium).
\subsection{Application of results to example in \texorpdfstring{\Cref{ssec:n4_example}}{}}
Recall that the $n=4$ example in \Cref{ssec:n4_example} had two stable boundary equilibria and two stable coexistence equilibria, but we could not determine via numerical simulations the number of unstable coexistence equilibria. With the results obtained using the Poincar\'e-Hopf Theorem (i.e. \Cref{thm:main} and \Cref{cor:ph_count}), we can obtain a lower bound on the number of unstable coexistence equilibria. Evidently, the two stable coexistence equilibria have an even number of open left-half plane eigenvalues (being 8, the system dimension). Thus, the total number of coexistence equilibria with an even number of open left-half plane eigenvalues must be $(k-1)/2 \geq 2$, which implies $k \geq 5$. In other words, there are at least $5$ coexistence equilibria, of which at least $3$ must be unstable with an odd number of eigenvalues in the open left-half plane. Not only does this underscore the complexity of the equilibria patterns for networked bivirus SIS systems, but highlights the additional insights provided through Poincar\'e-Hopf Theory. While monotone systems theory (\cite[Proposition~2.9]{smith1988monotone_survey}) can allow one to conclude the presence of $3$ unstable coexistence equilibria, the eigenvalue properties cannot be so obtained.
\section{Further counting results, involving inequalities}\label{sec:morse_smale}
Well after the original work establishing the Poincar\'e-Hopf formula of \cref{thm:PH}~\cite{hopf27}, further counting results were obtained involving the equilibria of an equation $\dot x=f(x)$ defined on a compact manifold $\mathcal M$, which were additional to (though also incorporating) that formula. We briefly summarize below those aspects of the results needed for use on a sphere, and then demonstrate its application to the bivirus system.
Two major sequential developments provided the results. The first built on the work of Morse \cite{morse1934calculus} on critical points of a smooth scalar function, call it $g(x)$, defined on a $n$-dimensional manifold. For a summary, see~\cite[pp. 28-31]{milnor2016morse} and \cite[pp. 290-291]{mukherjee2015differential}, while \cite{matsumoto2002introduction} contains a more leisurely treatment. On an $n$-dimensional manifold, the nondegenerate critical points of a scalar function (nondegenerate critical points being those where the gradient is zero and the Hessian is nonsingular) may be minima, maxima, or saddle points, corresponding to the number of negative eigenvalues of the Hessian being $0$, $n$ or any integer in between, respectively. The number of such eigenvalues is termed the Morse index. Morse obtained a set of $n+1$ inequalities (including one equality) relating the numbers of saddle points with different Morse indices, assuming that the number of critical points is finite and all are nondegenerate; the set of inequalities also involves the values of certain topological indices termed Betti numbers, see e.g. \cite{matsumoto2002introduction}, in addition to the Euler characteristic of the manifold.
The second and further major advance on this work can be attributed to Smale, who studied the equilibria of systems $\dot x=f(x)$ defined on a manifold $\mathcal M$; the work essentially gave identical results to Morse Theory in the special case when $f(x)={\mbox{grad}}_{\mathcal M}g(x)$ is a gradient of some smooth scalar function $g(x)$, given that further conditions are imposed on $f(x)$, as described further below. One such restriction is that all equilibria are hyperbolic. For introductory remarks on such systems, see \cite{zomorodian2005topology}, while the key reference for our use of such ideas is~\cite{smale1967differentiable}. For an $n$-dimensional manifold, let $c_{\lambda}$ denote the number of equilibrium points for which the associated Jacobian has precisely $\lambda$ eigenvalues with negative real part. Then a set of inequalities involving the $c_{\lambda}$ can written down, which include an equality that is equivalent to the equality arising in the Poincar\'e-Hopf Theorem. The Euler characteristic and the Betti numbers of the manifold appear in the inequalities. More details are now offered relevant to their application to the bivirus problem.
\subsection{Definition and Properties of Morse-Smale systems}
The definition of a Morse-Smale system requires an understanding of the concepts of the stable manifold and unstable manifold of an equilibrium point of a dynamical system \cite{smale1960morse,sastry2013nonlinear, wiggins2003introduction}. Roughly speaking, the stable manifold of an equilibrium point is the set of points from which forward-time trajectories will converge to the point, and the unstable manifold is the set of points from which backward-time trajectories will converge to the point. While such sets do not always constitute manifolds, they do so when equilibrium points are hyperbolic \cite[pp. 289-290]{sastry2013nonlinear}. The dimension of a stable (unstable) manifold is then the number of left (right) half plane eigenvalues of the Jacobian $df_x$ evaluated at the equilibrium. Note that a hyperbolic equilibrium point which is not stable will have an associated stable manifold unless it is actually a source (i.e. the associated Jacobian matrix has all eigenvalues with positive real parts). The following defines the properties characterizing a Morse-Smale dynamical system.
\begin{definition}\label{ass:MS}
A smooth dynamical system $\dot x=f(x)$ existing on some $n$-dimensional manifold $\mathcal M$ is a \textit{Morse-Smale system} when the following conditions hold:
\begin{enumerate}
\item
Trajectories have no finite escape times in the forward or backward directions, i.e. $\|x(t)\| \to \infty$ when $t\to T$ for some finite $T$ is not possible, and for any initial condition, solutions exist in both directions on $(-\infty,\infty)$.
\item
Equilibrium points are hyperbolic (i.e. at an equilibrium point $x_k$, the matrix $df_{x_k}$ has no eigenvalues with zero real part).
\item
If the stable manifold $\mathcal W_s(x_j)$ of the equilibrium point $x_j$ intersects the unstable manifold $\mathcal W_u(x_k)$ of a second equilibrium point $x_k$, the intersection is transverse, that is, if $x\in\mathcal W_s(x_j)\cap\mathcal W_u(x_k)$, then the span of the corresponding tangent space is $\mathbb R^n$, i.e. $T_x(\mathcal W_s(x_j))+T_x(\mathcal W_u(x_k))=\mathbb R^n$. This condition may alternatively be stated as ${\rm{dim}} \mathcal W_s(x_j)+{\rm{dim}}W_u(x_k)-n={\rm{dim}}(T_x(\mathcal W_s(x_j))\cap T_x(\mathcal W_u(x_k)))$.
\item
If there are periodic orbits, they are hyperbolic.\footnote{For an explanation of hyperbolicity of periodic orbits (which is a generalization of the idea of hyperbolicity of an equilibrium point), see e.g. \cite{robinson1998dynamical}. This paper however will make virtually no use of this notion.} In particular then, nonattractive limit cycles are not permitted.
\end{enumerate}
\end{definition}
The key counting result for general Morse-Smale systems, though simplified by exclusion of the possibility of periodic orbits (since such an exclusion will be justified in applying the result to bivirus sytems), is presented below, first in general form and then specialised to the case of motion on a sphere $S^{2n}$:
\begin{theorem}\label{thm:morsesmale}
Suppose that $\dot x=f(x)$ is a Morse-Smale system without limit cycles defined on an $n$-dimensional compact manifold $\mathcal M$. Let $c_{\lambda}$ denote the number of equilibrium points\footnote{The fact that $c_{\lambda}$ must be defined using a particular choice of coordinate basis at an equilibrium point but assumes a value that is independent of the choice is not explicitly demonstrated in \cite{smale1967differentiable} but is implicitly assumed.} for which the associated Jacobian has precisely $\lambda$ eigenvalues with negative real part, or equivalently the associated stable manifold has dimension $\lambda$.
Let $r_{\lambda}$ denote the rank of the $\lambda$-th homology group of $\mathcal M$ (the $\lambda$-th Betti number). Then the following inequalities hold:
\begin{align}\label{eq:morsesmaleineq}
c_0 &\geq r_0 \\\notag
c_1-c_0 &\geq r_1-r_0 \\\notag
c_2-c_1+c_0 &\geq r_2-r_1+r_0 \\\notag
&\vdots\\\notag
c_{n-1}-c_{n-2}+\cdots+(-1)^{n-1}c_0&
\geq r_{n-1}-r_{n-2}+\cdots+(-1)^{n-1}r_0\\\notag
c_n-c_{n-1}+\cdots+(-1)^nc_0&=r_n-r_{n-1}+\cdots+(-1)^nr_0=(-1)^n\chi(\mathcal M)
\end{align}
\end{theorem}
The last equation above can be rewritten as
\[
(-1)^kc_k=\chi(M)
\]
This is the same as the equality \eqref{eq:PHmainequation} resulting from the Poincar\'e-Hopf Theorem. To see this, recall that any equilibrium whose Jacobian has an odd number of eigenvalues with negative real part has a negative index i.e. the sign of the determinant of the Jacobian is negative. Hence the left hand side of the last equation adds together with correct sign the indices of all the equilibrium points, and is simply $\sum_k{\mbox{ind}}_{x_k}(f)$, which from \eqref{eq:PHmainequation} is $\chi(M)$.
For a sphere $S^n$, the only nonzero homology groups are $H_0=H_n=\mathbb Z$, and so $r_0=r_n=1$, but otherwise $r_{\lambda}=0$. Further $\chi(S^n)=1+(-1)^n$. These properties are set out in \cite[see p.141]{matsumoto2002introduction}. For our purposes, we record what happens for the even dimension sphere $S^{2n}$:
\begin{corollary}
Adopt the same hypotheses as for Theorem \ref{thm:morsesmale}, save regarding the dimension of $\mathcal M$, and suppose that $\mathcal M$ is $S^{2n}$. Then there holds
\begin{align}\label{eq:MSsphere}
c_0&\geq 1\\\notag
c_1-c_0&\geq -1\\\notag
c_2-c_1+c_0&\geq 1\\\notag&\vdots\\\notag
c_{2n-1}-c_{2n-2}+\dots-c_0&\geq -1\\\notag
c_{2n}-c_{2n-1}+\dots-c_1+c_0&=2
\end{align}
\end{corollary}
\subsection{Application to the bivirus problem}
We now indicate how these inequalities affect the bivirus equation results. We actually apply them to the system obtained by transforming the bivirus equations in the region $\tilde\Xi_{2n}$ to the sphere $S^{2n}$. Such a development must rest on an assumption the bivirus system is Morse-Smale. However, as far as the authors are aware, there are no formal results which establish that the property holds for a bivirus system, and this paper does not provide an explicit proof. Rather, as we now argue, the known properties of the bivirus system makes it \textit{reasonable to assume} that it is Morse-Smale, and we do so for the purposes of advancing our counting approach.
Recall that boundary points of the bivirus equations correspond to the north pole of the sphere, which is a source. Condition 1 of the Morse-Smale system definition is trivially fulfilled. Condition 2 is effectively covered by genericity of $D^i$ and $B^i$, see Theorem \ref{app:pf_thm_hyperbolicity}. Formal demonstration of Condition 3 is however \textit{not} possible. However, Condition 3 also appears as if it is generically satisfied. We note that it is well understood that smooth dynamical systems defined on a manifold are generic in a particular sense, see~\cite{smale1967differentiable}. Genericity here actually refers to the notion of possibility perturbing the vector field in a small region by arbitrarily small bumps, rather than changing the numerical values of the parameters appearing in the differential equations.
Interpreted for the bivirus system as opposed to the system defined on the sphere, the genericity results mean that the precise models containing for example quadratic terms in the state within the vector field might not be Morse-Smale, though an arbitrarily small perturbation will be Morse-Smale. We also note that the reference~\cite{robbin1981algebraic} proves that polynomial vector fields of a given degree are generically Morse-Smale. In the case of bivirus systems, despite the fact that the matrices $D^i,B^i$ are generic, the associated system is however not generic within the set of \textit{all} quadratic vector fields: observe that the component of the vector field associated with, for example, $\dot x^1_i$ includes $x^1_i\beta^1_{ij}x^1_j$ and $x^2_i\beta^1_{ij}x^1_j$, i.e two of the quadratic terms have the same coefficient. Apart from this, allowing $B^i$ with zero entries (albeit with an irreducibility assumption) is also a form of specialization moving the system outside the scope of those covered in \cite{robbin1981algebraic}.
As for the fourth requirement characterising a Morse-Smale system, recall that because the bivirus system is a monotone system, the only limit cycles possible are those which are nonattractive, see \cite[see p.95] {smith1988monotone_survey}. However, the mere occurrence of any type of limit cycle in a bivirus model appears to be a nongeneric property: limit cycles, nonattractive or otherwise, have never been observed in the bivirus literature. Hence we will assume that for a generic bivirus system (and its mapping onto the sphere $S^{2n}$), there are no limit cycles of any type.
\subsection{Additional insights for the bivirus problem}
Previously, we showed how a counting result involving equilibria for the bivirus problem could be obtained by relating the problem to trajectories on a sphere and appealing to the Poincar\'e-Hopf formula. The key adjustment was to use a single source equilibrium on the sphere to account for the healthy equilibrium in the bivirus problem, together with any unstable boundary equilibrium, of which there can be zero, one or two. Stable equilibria on the sphere include those corresponding to stable boundary equilibria for the bivirus problem, of which there can be two, one or zero (corresponding to zero, one or two unstable boundary equilibria).
We sum up the second main counting result of the paper, flowing from Morse-Smale theory and which we have just proved, as follows.
\begin{theorem}\label{thm:bivirusMS}
With notation as previously defined, consider the equation set \eqref{eq:bivirus} and suppose that Assumptions \ref{ass:constraints} and \ref{ass:unstablehealthy} hold. Suppose further that the four conditions in \Cref{ass:MS} all hold and hence \eqref{eq:bivirus} is a Morse-Smale system, thereby guaranteeing that equilibria of the equations in the region of interest $\Xi_{2n}$ are all nondegenerate and thus finite in number, and hyperbolic.
Let $c_{\lambda}$ denote the number of equilibria in the region of interest whose associated Jacobian has $\lambda$ open left-half plane eigenvalues.
The healthy equilibrium $({\bf{0}}_n,{\bf{0}}_n)$ and any unstable boundary equilibrium together contribute an allowance of 1 to $c_0$, and any coexisting source makes a further contribution of 1. Then the equation set \eqref{eq:MSsphere} holds.
\end{theorem}
Using these counting conditions, we can obtain further insights into the nature of the equilibria.
\begin{corollary}\label{cor:MScount}
Adopt the hypotheses of Theorem \ref{thm:bivirusMS}. Then
\begin{enumerate}
\item
There always exists a stable equilibrium.
\item
If there are two unstable boundary equilibria there is a stable coexistence equilibrium.
\item
If every inequality in the set \eqref{eq:MSsphere} is an equality, then $c_0=1,c_1=c_2=\dots=c_{2n-1}=0,c_{2n}=1$ and conversely, and the only corresponding equilibrium configurations are one stable and one unstable boundary equilibrium with no interior equilibrium, or two unstable boundary equilibrium and one interior stable equilibrium; both configurations are possible.
\end{enumerate}
\end{corollary}
\begin{proof}
To establish the first claim, observe that addition of the last two equations in \eqref{eq:MSsphere} yields $c_{2n}\geq 1$, implying there is at least one stable equilibrium. The second claim is a consequence of the first, and the allowed patterns of stability for the boundary equilibria.
For the third claim, the values of $c_i$ are trivial to establish if the inequalities are in fact all equalities. The consequential configuration restrictions are immediate.
\end{proof}
Examples illustrating the above claims are provided in the sequel.
We can easily make some specific remarks applying to the case $n=2$.
It is shown in \cite{ye2021_bivirus} that in addition to the two boundary equilibria, there can be zero, one or two coexistence equilibria.
Unstable boundary equilibria make no difference to the value of $c_0$ while stable boundary equilibria add to the value of $c_4$. Coexistence equilibria may add to the value of any $c_i$.
We now set out the full range of possibilities for different types of equilibria.
\begin{enumerate}
\item
Suppose there are no coexistence equilibria. Necessarily, $c_0=1$, corresponding to the healthy equilibrium, and any unstable boundary equilibria. However, with no coexistence equilibria, the first claim of Corollary \ref{cor:MScount} then implies there must be at least one stable (and could be two) boundary equilibrium, corresponding to $c_4$ being $1$ or $2$. It is easily verified that only the value $c_4=1$ is consistent with \eqref{eq:MSsphere}. Thus one boundary equilibrium is stable and the other is unstable. (This conclusion is also available in \cite[see Corollary 3.16]{ye2021_bivirus}.) There holds $\{c_0,c_1,c_2,c_3,c_4\}=\{1,0,0,0,1\}$.
\item
Suppose there is one coexistence equilibrium. First, this is consistent with there being two unstable boundary equilibria and a stable coexistence equilibrium, i.e. $\{c_0,c_1,c_2,c_3,c_4\}=\{1,0,0,0,1\}$. There are no other possibilities with two unstable boundary equilibria. Second, it is also consistent with there being two stable boundary equilibria, and one coexistence equilibrium which is unstable with precisely one unstable eigenvalue for its Jacobian, implying $\{c_0,c_1,c_2,c_3,c_4\}=\{1,0,0,1,2\}$. There are no other possibilities with two stable boundary equilibria. There are no possibilities at all with one stable and one unstable boundary equilibrium. Conversely, if there are two stable boundary equilibria, or two unstable boundary equilibria, there is precisely one coexistence equilibrium. Thus the stability properties of the single coexistence equilibrium are governed by the stability properties of the boundary equilibria, which must be both stable or both unstable.
\item
Now suppose there are two coexistence equilibria. By the preceding point, there is necessarily one stable and one unstable boundary equilibrium. One can check the various possibilities to conclude that the following possibilities exhaust those which are consistent with \eqref{eq:MSsphere}
\begin{align*}
\{c_0,c_1,c_2,c_3,c_4\}\in \{\{1,0,0,1,2\},\{2,1,0,0,1\},\{1,1,1,0,1\},\{1,0,1,1,1\} \}
\end{align*}
The first possibility corresponds to one stable coexistence equilibrium, and the second coexistence equilibrium having a single positive eigenvalue of its Jacobian. The second possibility corresponds to two unstable coexistence equilibria, with one being a source, and the other having three unstable (positive real part) Jacobian eigenvalues. The third possibility corresponds to both coexistence equilibrium being unstable, with three and two unstable eigenvalues of the Jacobian. The last possibility captures a situation with two unstable coexistence equilibria, with one and two unstable eigenvalues of the Jacobian.
\end{enumerate}
\section{Numerical examples}\label{sec:examples}
We now use two numerical examples to illustrate some complex equilibria patterns and the Morse inequalities developed in \Cref{thm:bivirusMS}.
\subsection{Example 1} We return to the $n=4$ example presented in \Cref{ssec:n4_example}, recalling that we concluded below \Cref{cor:ph_count} there exist at least 5 coexistence equilibria (two being stable and identified in \Cref{ssec:n4_example}, and at least three being unstable).
We constructed this example by taking two separate $n=2$ systems (in which case it is possible to analytically determine all equilibria~\cite{ye2021_bivirus}) and weakly coupling them together via the $1\times 10^{-3}$ terms in $B^1$ and $B^2$ --- more precisely, via a homotopy. One can use a gradient descent algorithm to locate an \textit{unstable} coexistence equilibrium, provided one starts the algorithm sufficiently close to it --- our knowledge that one expects isolated and hyperbolic equilibria, viz. \Cref{thm:finiteness,app:pf_thm_hyperbolicity}, establishes that the joined $n=4$ system will have equilibria which are perturbations of the separate $n=2$ systems, providing the critical starting location information. The technical details are beyond the scope of this paper, and thus omitted.
Using such a gradient descent algorithm, we located 5 unstable coexistence equilibria $(\tilde x^1, \tilde x^2)$ which are given by
\begin{align*}
&\left(\begin{bmatrix}
0.3478\\
0.2662\\
0.2298\\
0.3899
\end{bmatrix},\begin{bmatrix}
0.2298\\
0.3899\\
0.3478\\
0.2662
\end{bmatrix}\right),\;
\left(\begin{bmatrix}
0.3574\\
0.2761\\
0.0039\\
0.0084
\end{bmatrix}, \begin{bmatrix}
0.2211\\
0.3785\\
0.6109\\
0.6079
\end{bmatrix}\right),\;
\left(\begin{bmatrix}
0.0055\\
0.0032\\
0.2388 \\
0.4016
\end{bmatrix},
\begin{bmatrix}
0.5596 \\
0.7119 \\
0.3379\\
0.2563
\end{bmatrix}\right),\nonumber \\
&\left(\begin{bmatrix}
0.3379\\
0.2563\\
0.5596\\
0.7119\\
\end{bmatrix},
\begin{bmatrix}
0.2388\\
0.4016\\
0.0055\\
0.0032
\end{bmatrix}\right),\;
\left(\begin{bmatrix}
0.6109\\
0.6079\\
0.2211\\
0.3785\\
\end{bmatrix},
\begin{bmatrix}
0.0039\\
0.0084\\
0.3574\\
0.2761
\end{bmatrix}\right).
\end{align*}
The Jacobian of the first listed equilibrium has two unstable eigenvalues, while the Jacobian of all other equilibria have just one unstable eigenvalue.
In terms of the Morse inequalities for the $8$ dimensional system, we thus have $c_0 = 1$ (the healthy equilibrium), $c_8 = 4$ (the two stable boundary equilibria and two stable coexistence equilibria from \Cref{ssec:n4_example}), $c_7 = 4$, $c_6 = 1$, and $c_i = 0$ for $i = 1,2,\hdots ,5$. It is easily verified that all inequalities (and the final equality) in \eqref{eq:MSsphere} hold.
\subsection{Example 2} We next present another $n=4$ example differing from that presented in \Cref{ssec:n4_example}, with $D^1 = D^2 = I$ and
\begin{equation}
B^1=\begin{bmatrix} 1.6&1& 0.001 & 0.001 \\ 1&1.6 & 0.001& 0.001 \\ 0.001 & 0.001 & 1.7&1\\0.001& 0.001 & 1.2&0.5
\end{bmatrix},
\quad
B^2=\begin{bmatrix} 2.1&0.156& 0.001 & 0.001 \\ 3.0659&1.1 & 0.001& 0.001 \\ 0.001 & 0.001 & 1.6&1\\0.001& 0.001 & 1.2&0
\end{bmatrix}.
\end{equation}
The boundary equilibria $(\bar x^1, \vect 0_4)$ and $(\vect 0_4, \bar x^2)$ are locally exponentially stable and unstable, respectively. There are two coexistence equilibria given by
\begin{align*}
&\left(\begin{bmatrix}
0.0095\\
0.0056\\
0.5965\\
0.4875
\end{bmatrix},\begin{bmatrix}
0.5555\\
0.7089\\
0.0062\\
0.0044
\end{bmatrix}\right),\;
\left(\begin{bmatrix}
0.3391\\
0.2576\\
0.5998\\
0.4901
\end{bmatrix}, \begin{bmatrix}
0.2376\\
0.4001\\
0.0031\\
0.0022
\end{bmatrix}\right).
\end{align*}
The former is locally exponentially stable, while the latter is unstable and its Jacobian has a single unstable eigenvalue. Sample trajectories for convergence to the stable boundary equilibrium and stable coexistence equilibrium are given in \Cref{fig:example2}.
\begin{figure*}[!htp]
\begin{minipage}{0.475\linewidth}
\centering
\subfloat[Boundary equilibrium with virus~1 endemic\label{fig:n4_example2_boundary}]{\includegraphics[width=\columnwidth]{n4_example2_boundary.pdf}}
\end{minipage}
\hfill
\begin{minipage}{0.475\linewidth}
\centering\subfloat[Stable coexistence equilibrium]{\includegraphics[width=\columnwidth]{n4_example2_coexistence.pdf}\label{fig:n4_example2_coexistence}}
\end{minipage}
\caption{Sample trajectories of \textit{Example~2} system, with different initial conditions. As is evident, convergence occurs to two different attractive (locally exponentially stable) equilibria, dependent on the initial conditions. }\label{fig:example2}
\end{figure*}
In terms of the Morse inequalities for the $8$ dimensional system, we thus have $c_0 = 1$ (the healthy equilibrium and the unstable boundary equilibrium $(\vect 0_4, \bar x^2)$), $c_8 = 2$ (the stable boundary equilibrium $(\bar x^1, \vect 0_4)$ and the stable coexistence equilibrium), $c_7 = 1$ (the unstable coexistence equilibrium), and $c_i = 0$ for $i = 1,2,\hdots ,6$. Again, all inequalities (and the final equality) in \eqref{eq:MSsphere} hold.
We conclude by remarking that a recent preprint provided a numerical example of a bivirus system (modified to have additional nonlinearities in the dynamics) with multiple attractive coexistence equilibria~\cite{doshi2022convergence_bivirus}. To the best of our knowledge, our work and \cite{doshi2022convergence_bivirus} are the first to demonstrate multiple coexistence equilibria for networked bivirus models. However, \cite{doshi2022convergence_bivirus} only provides a single numerical example and is limited to simulations of convergence to stable coexistence equilibria. Here, we provide significant theoretical advances that establish counting results on how many coexistence equilibria there may be, and their stability properties (including unstable equilibria and the number of unstable eigenvalues of their Jacobian).
\section{Conclusions}\label{sec:conclusions}
In this paper, we applied the Poincar\'e-Hopf Theorem to the SIS networked bivirus model, which required significant adaptation due to various complexities of the bivirus dynamics. We then applied Morse inequalities, under the assumption that the bivirus system was a Morse-Smale system. Through these methods, we obtained a set of counting results which, given different stability configurations of boundary equilibria, yield lower bounds on the number of coexistence equilibria, and importantly, information about the number of stable eigenvalues of their Jacobian matrices. We provided numerical examples to illustrate the results, and provide evidence of the highly complex coexistence equilibria patterns possible. In future work, we aim to extend our approach to multivirus models with three or more competing viruses, and identify explicit relations between the parameter matrices $D^i, B^i$ and the number of coexistence equilibria.
|
2,877,628,089,477 | arxiv | \section{Introduction}
The {\it fuzzball} program (see \cite{Mathur0} for reviews) is an approach to understand the microscopic origin of the entropy of black holes, and also to provide a solution of the {\it Black Hole Information Loss} problem. Fuzzballs are the microstates inside of the black hole, and the number of possible microstates is argued to lead to the famous Beckenstein formula for the entropy of a black hole \cite{BHentropy}. The fuzzball states have support everywhere inside the black hole horizon. In this approach, the individual black hole microstates do not have a horizon. The horizon arises at the level of the effective theory after coarse graining. Hence, information can escape from the black hole, and there is no black hole information loss problem. Note that fuzzballs carry positive energy.
More recently, the concept of VECROs (Virtual Extended Compression-Resistant Objects) has been introduced \cite{Mathur1} to understand what happens when a shell of matter collapses to form a black hole. The idea is that when a shell of regular matter collapses and falls into its Schwarzschild radius, the vacuum responds by generating a set of virtual extended objects stretching out beyond the location of the mass shell. VECRO formation is closely related to fuzzball formation, in a way which we will come back to later on. VECROs are also argued to cap off the geometry at the center of the mass concentration, thus eliminating the singularity of the metric.
In this short note we speculate that VECROs will also form in a contracting cosmology and will resist the contraction to a singularity. Rather, they will mediate a nonsingular cosmological bounce.
Key to reaching our conclusion is the realization that, in order for VECROs to eliminate the black hole singularity, the effective energy density with which they couple to gravity needs to be negative. We then argue that a gas of VECROs in a contracting cosmology will have the equation of state $p = \rho$ of a stiff fluid \footnote{$\rho$ and $p$ denote energy density and pressure, respectively.}. Combining these two arguments, the Friedmann equations then immediately yield a nonsingular bounce.
Our work is motivated in part by proposals \cite{BF, Ven, Jerome} that the early universe could be described by a gas of black holes \cite{BF} or ``string holes'' \cite{Ven, Jerome} which has a stiff equation of state.
\section{Key Arguments}
We will make use of two key arguments. The first is that, in order to resolve the black hole singularity in the context of the effective field theory of Einstein gravity, the VECRO energy density must be negative. To show this, we start with the metric of a vacuum black hole without charge and without rotation
\be
ds^2 \, = \, - (1 - \frac{2GM}{r}) dt^2 + (1 - \frac{2GM}{r})^{-1} dr^2
\ee
(we drop the angular variables), where $t$ is the time of an observer far from the center of the black hole, and $r$ is the radial distance from the hole, chosen such that the area of the sphere surrounding the hole at radius $r$ is $4 \pi r^2$. In the presence of matter, the metric of a static, spherically symmetric space-time can be written in the form
\be
ds^2 \, = \, - A(r) dt^2 + B(r) dr^2 \, ,
\ee
with coefficient functions $A(r)$ and $B(r)$ which are determined by the distribution of matter. If we use the parametrization
\be
A(r) \, \equiv \, e^{2 h(r)} f(r)
\ee
and
\be
B(r) \, \equiv \, f(r)^{-1} \, ,
\ee
then the Einstein equations lead to the following equations (see e.g. \cite{Blau}) which determine the coefficient functions in terms of the components of the energy-momentum tensor $T_{\mu \nu}$:
\be \label{E1}
h^{\prime}(r) \, = \, 4 \pi G r f(r)^{-1} \bigl( {T^r}_r - {T^t}_t \bigr)
\ee
and
\be \label{E2}
\bigl[ r ( f(r) - 1 ) \bigr]^{\prime} \, = \, 8 \pi G r^2 ( - {T^t}_t ) \, ,
\ee
where a prime denotes a derivative with respect to $r$. Redefining
\be
r ( f(r) - 1 ) \, \equiv \, - 2 m(r)
\ee
such that
\be
f(r) \, = \, 1 - \frac{2 m(r)}{r} \, ,
\ee
Eq. (\ref{E2}) becomes
\be
m(r)^{\prime} \, = \, 4 \pi G r^2 (- {T^t}_t ) \, = \, 4 \pi G r^2 \rho(r)
\ee
and hence
\be
m(r) \, = \, 4 \pi G \int_0^r dr' r'^2 \rho(r') \, .
\ee
Let us now consider a shell of mass $m_0$ collapsing to form a black hole, and assume that the mass shell is not held up by some pressure. . Then in order to avoid the singularity at $r = 0$, it is necessary that a compensating energy component with negative energy density $\rho_v(r) < 0$ builds up just outside of the collapsing shell. This is the ``VECRO'' component.
\be
m(r) \, = \, m_0 + 4 \pi G \int_0^r dr^{\prime} (r^{\prime})^2 \rho_v(r^{\prime}) \, .
\ee
If $r_m$ is the radius of the collapsing shell, then the VECRO component has to be localized just outside of $r_m$, and its contribution to the mass must cancel $m_0$ at the radius $r_m$. This implies that, as $r_m$ tends to zero, the VECRO energy density diverges. For $r > r_s$ (where $r_s$ is the Schwarzschild radius associated with the mass $m_0$) we require $m(r) = m_0$. This implies that there is a positive energy density component $\rho_f(r)$ which at values $r \rightarrow r_s$ cancels the negative energy contribution of the VECROs. This component is the fuzzball component \cite{Mathur0}. Note that $\rho_f(r)$ has support between $r_m$ and $r_s$, and does not diverge as $r_m$ tends to zero \footnote{This discussion illustrates the relationship between VECRO and fuzzball formation.}.
The second key argument in our analysis is the claim that the equation of state of a gas of VECROs in a contracting universe is $w = 1$, i.e.
\be \label{EoS}
p_v \, = \, \rho_v \, .
\ee
This is the same equation of state as that of the black hole gas discussed in \cite{BF} and the gas of string holes analyzed in \cite{Ven}. To demonstrate this,we can follow the discussion in \cite{Jerome}. The crucial input in the analysis of \cite{Jerome} is that the entropy $S$ of a gas of VECROs is given in terms of the mass of a single VECRO in the same way as for a black hole, namely holographically
\be
S \, = \, N M_v^2 G \, ,
\ee
where $M_v$ is the mass of a single VECRO and $N$ is the number of these objects in some fixed comoving volume $V$. The energy $E$ of a gas of VECROs is
\be
E \, = \, N M_v \, .
\ee
Using the thermodynamic identities
\be
P \, = \, T \bigl( \partial S / \partial V \bigr)_E
\ee
(where $P$ is the total pressure and $T$ is the temperature) and
\be
T^{-1} \, = \bigl( \partial S / \partial E \bigr)_V
\ee
we then immediately obtain (\ref{EoS}).
\section{Vecros and a Nonsingular Bounce}
In analogy to how we have argued in the previous section that the negativity of the VECRO energy density can cap off the singularity at the center of a collapsing mass shell, we now argue that the negative energy density of VECROs, combined with their stiff equation of state, will naturally lead to a nonsingular cosmology.
We consider a homogeneous and isotropic contracting universe filled with cold matter (equation of state $p = 0$), radiation (equation of state $p = 1/3 \rho$) and vecros (equation of state $p = \rho$). To be specific, we will consider a spatially flat universe. From the respective continuity equations, it immediately follows that the energy density in cold matter scales as $\rho_m(t) \sim a(t)^{-3}$, where $a(t)$ is the cosmological scale factor, that in radiation as $\rho_r(t) \sim a^{-4}$, and that in VECROs as $\rho_v \sim a^{-6}$. As we approach $a(t) = 0$, radiation will dominate over cold matter, and thus in the following we will neglect the cold matter component.
The Friedmann equation reads
\be \label{FRW1}
H^2 \, = \, \frac{8 \pi G}{3} \rho \, ,
\ee
where $H \equiv {\dot{a}}/a$ is the Hubble expansion rate, and $G$ is Newton's gravitational constant, and
\be \label{FRW2}
\frac{\ddot{a}}{a} \, = \, - \frac{4 \pi G}{3} \bigl( \rho + 3 p \bigr) \, .
\ee
We assume that early in the contracting phase the energy density in VECROs
\be
\rho_v(t) \, = \, - A_v \bigl( \frac{a(t)}{a(t_i)} \bigr)^{-6}\, ,
\ee
where $t_i$ can be taken to be some initial time, and $A_v$ is a positive amplitude, is smaller than the energy density in radiation
\be
\rho_r(t) \, = \, A_r \bigl( \frac{a(t)}{a(t_i)} \bigr)^{-4}\, ,
\ee
where $A_r$ is the initial radiation density amplitude. As the Universe contracts and $a(t)$ decreases, the absolute value of the energy density in VECROs will catch up to the energy density in radiation, and they will become equal at some time $t_b$ and we will have (from (\ref{FRW1}))
\be
H(t_b) \, = \, 0 \, .
\ee
Since the equation of state $p / \rho$ for VECROs is larger than for radiation, then at the time $t_b$ the abolute value of the VECRO pressure will be larger than the radiation pressure, and by (\ref{FRW2}) it then follows that
\be
{\dot{H}}(t_b) \, > \, 0 \, .
\ee
Thus, we see that the addition of the VECRO component will lead to a smooth cosmological bounce.
\section{Discussion}
In this note we have argued that a gas of VECROs in a contracting univere can yield a nonsingular cosmological bounce. VECROs can be viewed as a quantum response of the vacuum which works to counteract a developing singularity, in the same way that VECROs form in the case of spherical infall to prevent the formation of a Schwarzschild singularity.
One key ingredient in our argument is the assumption that the energy density of VECROs is negative when considering its coupling to classical gravity. We have argued for this based on the role which VECROs play in removing the singularity at the center of a black hole. We also propose an analogy with the back-reaction of infrared modes of cosmological perturbations \footnote{See \cite{Abramo} for original work, \cite{BRreview} for a review, and \cite{entropy} for studies demonstrating the need for the presence of entropy fluctuations in order to obtain a nonvanishing effect.}. A matter fluctuation will create a gravitational potential well, and on super-Hubble scales the negativity of the gravitational energy overwhelms the positivity of the matter energy, thus leading to a net negative energy density which leads to a reduction of the locally observable expansion rate. In analogy, space inside a contracting matter shell will respond gravitationally to produce a negative effective energy density which tends to reduce the locally measured curvature inside the black hole.
A uniform contracting cosmology will create a packed gas of VECROS. The holographic entropy scaling of VECROs then leads to the conclusion that the equation of state is that of a stiff fluid $p = \rho$. This is the second key ingredient of our argument.
These two arguments then immediately imply that a homogeneous collapsing cosmology with undergo a nonsingular bounce because of the formation of a gas of VECROs which resist the contraction.
Bouncing cosmologies form a class of interesting resolutions of the Big Bang singularity. Bouncing cosmologies (see e.g. \cite{BounceRev} for a review) can also provide an alternative to cosmological inflation as a solution to the horizon problem of Standard Big Bang Cosmology \footnote{The {\it emergent scenario} is another alternative to cosmological inflation which may naturally arise in string theory, as discussed in the context of {\it String Gas Cosmology} \cite{BV} or, more recently, matrix model cosmology \cite{us} (see also \cite{Vafa}). In this case, it is thermal fluctuations in the emergent phase which lead to the curvature fluctuations and gravitational waves which are observed today \cite{Nayeri}.}. In the same way that in certain accelerating expanding cosmologies quantum vacuum perturbations can develop into an approximately scale-invariant spectrum of curvature fluctuations \cite{ChibMukh} and gravitational waves \cite{Starob}, in certain classes of decelerating contracting cosmologies quantum vacuum perturbations can also develop into approximately scale-invariant perturbations after the bounce. One example is the {\it matter bounce} scenario \cite{Fabio} (defined as a model in which the equation of state is that of cold matter during contraction when scales which are currently observed exit the Hubble horizon). This model, however, is unstable towards the development of anisotropies \cite{Peter}. Another possibility is to have a phase of Ekpyrotic contraction (given by an equation of state $w \gg 1$) \cite{Ekp} in which case initial vacuum perturbations evolve into a scale-invariant spectrum of metric fluctuations which can also lead to scale-invariant curvature perturbations after the bounce \cite{Durrer} (see e.g. \cite{Ziwei} for a concrete realization). A key challenge for bouncing cosmologies is how to obtain the cosmological bounce. Here, we are proposing a mechanism which yields such a bounce. In the way we have presented the argument, it works as long as there is no other component of matter with an equation of state $w > 1$. Thus, the VECRO mechanism will automatically resolve the singularity in the matter bounce scenario. In the case of the Ekpyrotic scenario, we need to require that the phase of Ekpyrotic contraction ends at sufficiently high energy densitites. This appears to be a rather mild requirement since before the string energy density is reached, the negative exponential potential of the scalar field which yields Ekpyrotic contraction is expected to flatten out and hence the equation of state of the the scalar field will revert to $w < 1$.
Note that if the energy scale at the bounce point is lower than the Planck scale, then the amplitude of the cosmological fluctuations will remain in the linear regime (if, as is commonly assumed in bouncing cosmoloiges, the fluctuations originate as quantum vacuum perturbations on sub-Hubble scales in the far past).
The VECRO mechanism which we are proposing in this note addresses a key challenge for bouncing cosmologies. This is all the more important since recent arguments indicate that cosmologies with a phase of sufficient length of accelerated expansion to realize the inflationary scenario are hard to realize in string theory (they are in tension with the ``swampland'' criteria \cite{swamp}) and also face conceptual problems from unitarity considerations (the ``Trans-Planckian Censorship Conjecture'' (TCC) \cite{TCC1, TCC2} (see also \cite{RHBrevs} for reviews)), while bouncing and emergent scenarios are consistent with the TCC as long as the energy density at the bounce or in the emergent phase is smaller than the Planck scale.
A key open issue is a clear derivation of VECROs in cosmology from superstring theory. At a superficial level, our proposed gas of VECROs is in tension with energy conditions for an effective four space-time dimensional theory derived from string theory \cite{Bernardo}, where it is shown that the Null Energy Condition (NEC) must be obeyed. While our total energy density obeys the NEC, the gas of VECROs does not since it has negative energy density. The assumptions made in \cite{Bernardo} on the continuity properties of the scale factor are obeyed in our model, our VECROs are quantum objects, and hence the analysis of \cite{Bernardo} does not directly apply. Nevertheless, this issue requires further study.
\section*{Acknowledgments}
We are grateful to Heliudson Bernardo, Keshav Dasgupta, Jerome Quintin and (in particular) Samir Mathur for discussions. The research at McGill is supported in part by funds from NSERC and from the Canada Research Chair program.
|
2,877,628,089,478 | arxiv | \section{Introduction}
Let $K$ be a number field and $A/K$ an abelian variety. The completed $L$-series, $L^\star(A/K,s)$, of $A/K$ conjecturally has an analytic continuation to the whole of the complex plane and satisfies a functional equation
\[L^\star(A/K,s)=w(A/K)L^\star(A/K,2-s)\]
where $w(A/K)\in\{\pm 1\}$ is the global root number of $A/K$. The Birch and Swinnerton-Dyer conjecture asserts that the Mordell-Weil rank of $A/K$ agrees with the order of vanishing at $s=1$ of $L^\star(A/K,s)$;
\[\text{ord}_{s=1} L^\star(A/K,s)=\text{rk}(A/K).\]
If $w(A/K)=1$ (resp. -1), then $L^\star(A/K,s)$ is an even (resp. odd) function around $s=1$ and as such its order of vanishing there is even (resp. odd). Thus a consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture;
\[w(A/K)=(-1)^{\text{rk}(A/K)}.\]
Essentially all progress towards the parity conjecture has proceeded via the $p$-parity conjecture. For a fixed prime $p$, we denote by $\text{rk}_p(A/K)$ the $p$-infinity Selmer rank of $A/K$. Under the conjectural finiteness of the Shafarevich-Tate group (or indeed, under the weaker assumption that its $p$-primary part is finite), $\text{rk}_p(A/K)$ agrees with $\text{rk}(A/K)$. The $p$-parity conjecture is the assertion that \[w(A/K)=(-1)^{\text{rk}_p(A/K)}.\] Note that, without knowing finiteness of the Shafarevich-Tate group, these conjectures for different $p$ are inequivalent, and so there is interest in knowing the $p$-parity conjecture for multiple values of $p$.
Dokchitser and Dokchitser have shown that for all elliptic curves over $\mathbb{Q}$, the $p$-parity conjecture is true for all primes $p$ \cite[Theorem 1.4] {MR2680426}. More recently, Nekov{\'a}{\v{r}} has extended this result to replace $\mathbb{Q}$ by any totally real number field, excluding some elliptic curves with potential complex multiplication \cite[Theorem A]{MR3101073}. For a general number field $K$, \v{C}esnavi\v{c}ius \cite[Theorem 1.4]{Kes14B} has shown that the $p$-parity conjecture holds for elliptic curves possessing a $p$-isogeny whilst work of Kramer-Tunnell \cite{MR664648} and Dokchitser-Dokchitser \cite{MR2831512} proves that the $2$-parity conjecture holds for all elliptic curves $E/K$, not over $K$ itself, but over any quadratic extension of $K$.
For higher dimensional abelian varieties, much less is known. The most general result at present is due to Coates, Fukaya, Kato and Sujatha, who prove in \cite[Theorem 2.1]{MR2551757} that for odd $p$, the $p$-parity conjecture holds for a $g$-dimensional abelian variety with an isogeny of degree $p^g$, providing some additional technical conditions are satisfied.
In the present paper, following on from the work of Kramer-Tunnell and Dokchitser-Dokchitser, we consider the $2$-parity conjecture for Jacobians of hyperelliptic curves over quadratic extensions of the field of definition. Specifically, we prove the following result, which provides essentially the first examples of the $2$-parity conjecture in dimension greater than 1 (see \Cref{convention} for our conventions regarding hyperelliptic curves).
\begin{theorem} \label{cases of the parity conjecture}
Let $K$ be a number field and $L/K$ a quadratic extension. Let $C/K$ be the hyperelliptic curve $y^2=af(x)$ with $f\in \mathcal O_K[x]$ a monic seperable polynomial of degree $2g+1$ or $2g+2$, and let $J$ be the Jacobian of $C$. Suppose that
\begin{itemize}
\item[{(i)}] for each prime $\mathfrak{p}\vartriangleleft \mathcal{O}_K$ not dividing $2$ that ramifies in $L/K$, either $J$ has good reduction at $\mathfrak{p}$ or the reduction $\bar{f}(x)$ mod $\mathfrak{p}$ is cube free,
\item[{(ii)}] for each prime $\mathfrak{p}\vartriangleleft \mathcal{O}_K$ dividing $2$ which does not split in $L/K$, $J$ has good reduction at $\mathfrak{p}$, and moreover if such a prime $\mathfrak{p}$ ramifies in $L/K$ then $J$ has good ordinary reduction at $\mathfrak{p}$ and $f(x)$ splits over an odd degree Galois extension of $K_\mathfrak{p}$.
\end{itemize}
Then the $2$-parity conjecture holds for $J/L$.
\end{theorem}
In fact, as will be detailed in \Cref{compatibility results}, we need only assume that $J$ satisfies the above conditions over an odd degree Galois extension $F/K$ (relative to the extension $FL/F$). Moreover, if the genus of $C$ is $2$, one can weaken the assumption that $J$ has good reduction at each prime dividing 2 and unramified in $L/K$, to assume only that $J$ has semistable reduction at such primes (see \Cref{genus 2 unramified}).
\Cref{cases of the parity conjecture} gives a large supply of hyperelliptic curves satisfying the $2$-parity conjecture over every quadratic extension of their field of definition. In fact (again see \Cref{compatibility results}) if a hyperelliptic curve $C$ has a quadratic twist satisfying the conditions of \Cref{cases of the parity conjecture} for every quadratic extension of $K$, then $C$ also satisfies the $2$-parity conjecture over every quadratic extension of $K$. As an indication of the generality in which these conditions hold, we have
\begin{theorem} \label{positive proportion}
For any $g\geq2$, a positive proportion of genus $g$ hyperelliptic curves over $\mathbb{Q}$ satisfy the $2$-parity conjecture over every quadratic extension.
\end{theorem}
(Of course, for this one needs to make rigorous what a positive proportion of hyperelliptic curves means; there are many ways to do this in such a way that the theorem holds. The setup of \cite[Section 9]{MR1740984} is particularly suited to the conditions of \Cref{cases of the parity conjecture}, though most natural orderings will lead to \Cref{positive proportion} being true. For explicit conditions on the polynomial defining $C$ that ensure the conditions of \Cref{cases of the parity conjecture} at the prime $2$ are satisfied, see \Cref{explicit ordinary cor}.)
Since the root number $w(J/L)$ decomposes as a product of local terms,
\[w(J/L)=\prod_{v\in M_K} w(J/K_v)w(J^L/K_v),\]
(here $J^L$ denotes the quadratic twist of $J$ by $L$ and $M_K$ is the set of all places of $K$) the strategy to prove \Cref{cases of the parity conjecture} is to similarly decompose the parity of the $2$-infinity Selmer rank of $J$ over $L$ into local terms and compare these place by place. To this end, we give a decomposition of the parity of $\text{rk}_{2}(J/L)$ into local terms, generalising a theorem of Kramer \cite[Theorem 1]{MR597871}. Before stating the theorem we need to introduce some notation. We define, for each place $v$ of $K$, the \textit{local norm map} $N_{L_{w}/K_v}:J(L_{w})\rightarrow A(K_{v})$ by
\[
P\mapsto N_{L_{w}/K_v}(P):=\sum_{\sigma\in\text{Gal}(L_{w}/K_{v})}\sigma(P)
\]
where $w$ is any place of $L$ extending $v$ (by definition, this is the identity map on $A(K_{v})$ in the case that $L_{w}/K_{v}$ is trivial). Moreover, we define $i_d(C_{v})$ to be -1 if $C$ is deficient over $K_v$, that is, if it has no $K_v$-rational divisor of degree $g-1$, and 1 otherwise. The reason for the appearance of $i_d(C_v)$ is a result of Poonen and Stoll \cite[Theorem 8]{MR1740984} which characterises the failure of the Shafarevich-Tate group of $J/K$ to have square order (if finite) in terms of the $i_d(C_v)$. Denoting by $C^L$ the quadratic twist of $C$ by $L$, we define $i_d(C_{v}^{L})$ similarly. We then have
\begin{theorem} \label{selmer decomposition}
Let $C/K$ be a hyperelliptic curve, $J/K$ its Jacobian and $L/K$ a quadratic extension of number fields.
Then
\[
(-1)^{\textup{rk}_{2}(J/L)}=\prod_{v\in M_K}i_d(C_{v})i_d(C_{v}^{L})(-1)^{\dim_{\mathbb{F}_{2}}J(K_{v})/N_{L_w/K_v}J(L_{w})}.
\]
\end{theorem}
\Cref{selmer decomposition} (or rather \Cref{selmer parity formula} from which we deduce it) is likely known to experts though we have not found it in the literature in this generality. Klagsbrun, Mazur and Rubin give an alternate proof of the elliptic curves case, originally due to Kramer, in \cite[Theorem 3.9]{MR3043582} and \v{C}esnavi\v{c}ius generalises their setup to higher dimension in \cite[Theorem 5.9]{KES14}. This would likely give an alternate approach to proving \Cref{selmer decomposition}. Since we shall deduce it from results in \Cref{local norm section} which we have need for elsewhere, we have not taken this approach.
Ideally, one might hope that the local terms contributing to $w(J/L)$ and $\text{rk}_{2}(J/L)$ simply agree place by place. However, this is not the case, and thus the strategy hinges on computing the discrepancy between the local terms and showing that it vanishes globally. To this end, we conjecture the following relationship between the local terms, generalising that for elliptic curves due to Kramer and Tunnell \cite{MR664648}.
\begin{conjecture} \label{Kramer Tunnell}
Let $K$ be a local field of characteristic zero, $L/K$ a quadratic extension, $C/K$ a hyperelliptic curve, and $J$ its Jacobian. Then
\[
w(J/K)w(J^L/K)=((-1)^g\Delta_C,L/K)i_d(C)i_d(C^L)(-1)^{\dim_{\mathbb{F}_2}J(K)/\N J(L)}.
\]
\end{conjecture}
Here $\left(\cdot,\cdot \right)$ is the Hilbert (or more properly Artin) symbol with respect to $L/K$ and $\Delta_C$ is the discriminant of $f$ for any equation $y^2=f(x)$ defining $C$ (the discriminant of two such polynomials differ by a square, so the Hilbert symbol is independent of this choice).
Since the Hilbert symbol appearing in the conjecture vanishes globally by the product formula, it is immediate from the discussion above that verifying this conjecture implies the 2-parity conjecture. We will prove \Cref{Kramer Tunnell} under the assumptions on the reduction of $C$ given by \Cref{cases of the parity conjecture}, hence proving the theorem. Moreover, these cases (and in fact substantially fewer) will be sufficient to deduce the conjecture from the 2-parity conjecture. More precisely, we prove
\begin{theorem} \label{global to local}
Let $K$ be a number field, $C/K$ a hyperelliptic curve, $J/K$ its Jacobian and $v_0$ a place of $K$. If the $2$-parity conjecture holds for $J/F$ over every quadratic extension $F/K$, then \Cref{Kramer Tunnell} holds for $J/K_{v_0}$ and every quadratic extension $L/K_{v_0}$.
\end{theorem}
Interestingly, \Cref{Kramer Tunnell} also holds in genus 0. Indeed, for a local field $K$ of characteristic 0, given a curve $C:y^2=f(x)$ over $K$, where $f$ has degree 1 or 2 and is separable, the Jacobian is trivial, hence the root numbers and cokernel of the local norm map are too, and $C$ (or its twist) are deficient if and only if they have no $K$-rational point. It is then easy to check that $i_d(C)i_d(C^L)=(\Delta_C,L/K)$ for any quadratic extension $L/K$.
The layout of the paper is as follows. We begin by proving some basic properties of the local norm map which will be of use later. We then combine these results with a global duality theorem due to Tate and Milne to prove \Cref{selmer decomposition}. Having done this, we examine \Cref{Kramer Tunnell} in two cases, namely when the Jacobian has good reduction, and when the local field in question is archimedean. This provides the first cases of \Cref{Kramer Tunnell} and enables us to prove \Cref{global to local}. In the last part of the paper, we prove \Cref{Kramer Tunnell} in the remaining cases needed to establish \Cref{cases of the parity conjecture}. In odd residue characteristic we prove \Cref{Kramer Tunnell} completely when the extension $L/K$ is unramified. We do this by analysing the minimal regular model of $C$ over $\mathcal{O}_K$. The key fact we use is that the formation of the minimal regular model commutes with unramified base change; this facilitates a comparison between invariants of $C$ and those of its unramified quadratic twist. We finally turn to ramified quadratic extensions with odd residue characteristic and sketch a general method for deducing \Cref{Kramer Tunnell} from knowledge of the minimal regular model of $C$ and its various quadratic twists, before specialising to the case where the reduction $\bar{f}(x)$ (as in \Cref{cases of the parity conjecture}) is cube free.
\begin{convention}[Hyperelliptic Curves]\label{convention}
\textup{Throughout the paper, a \textit{hyperelliptic curve} $C$ over a field $K$ will mean a smooth, geometrically connected curve of genus $g \geq 2$, defined over $K$, and admitting a finite separable morphism $C\rightarrow \mathbb{P}^1_K$ of degree $2$ (the assumption that $g\geq 2$ is made since this is the only case of interest and allows us to avoid dealing separately with some special cases in an \text{ad. hoc.} manner). When $K$ has characteristic zero, one can always find a separable polynomial $f(x)\in K[x]$ of degree $2g+1$ or $2g+2$ such that $C$ is the union of the two affine open subschemes
\[U_1=\text{Spec}\frac{K[x,y]}{y^2-f(x)}\]
and
\[U_2=\text{Spec}\frac{K[u,v]}{v^2-g(u)}\]
where $g(u)=u^{2g+2}f(1/u)$ and the schemes glue via the relations $x=1/u$ and $y=x^{g+1}v$. Conversely, for any such $f$, the two schemes defined above glue to give a hyperelliptic curve of genus $g$, the degree 2 morphism to $\mathbb{P}^1_K$ being given by the $x$-coordinate. By an abuse of notation, we will often say that such a hyperelliptic curve is given by the equation $y^2=f(x)$. We refer to the points in $U_2(\bar{K}) \setminus U_1(\bar{K})$ as the $\textit{points at infinity}$. There are 2 such points if $\text{deg}(f)$ is even, and 1 otherwise.}
\end{convention}
\subsection*{Acknowledgements}
I would like to thank Tim Dokchitser for suggesting the problem, for constant encouragement and many helpful discussions. I also thank K\k{e}stutis \v{C}esnavi\v{c}ius and Qing Liu for correspondence, particularly regarding \Cref{valuation of discriminant}, and Vladimir Dokchitser for helpful conversations.
\subsection*{Notation}
Local fields will have characteristic $0$ throughout. For a number field $K$, $M_K$ will denote the set of all places of $K$. For each place $v\in M_K$, $K_v$ will denote the corresponding completion. For $K$ a local field or number field, $L/K$ will almost always denote a quadratic extension.
Notation for a local field $K$:
$\begin{array}{ll}
\mathcal{O}_\K & \text{ring of integers of}~\K \\
k & \text{residue field of}~\K \\
\bar{\K} & \text{algebraic closure of}~\K \\
\bar{k} & \text{algebraic closure of}~k \\
\K^\text{nr} & \text{maximal unramified extension of}~\K \\
(a,\L/\K) & \text{Artin symbol of}~ a\in \K^\times~\text{in}~\L/\K\text{. We will always take}~\L/\K~\text{quadratic}\\
& \text{in which case we regard this symbol as}~1~\text{or}~-1~\text{in the obvious way.}\\
& \text{We conflate this with the Hilbert symbol}~(a,b)_{\K}~\text{where}~\L=\K(\sqrt{b}).
\end{array}$
Notation for a hyperelliptic curve $C$ over a field $\K$:
$\begin{array}{ll}
\Delta_C & \text{the discriminant of (any) Weierstrass equation for}~C. ~\text{See \cite[Section 2]{MR1363944}.} \\
& \text{We will always consider}~\Delta_C~\text{only up to squares in}~\K~\text{so does not} \\
& \text{depend on the choice of Weierstrass equation. If}~C~\text{is given by an equation}\\
& y^2=f(x)~\text{we can equivalently take}~\Delta_C~\text{to be the discriminant of}~f.\\
i_d(C) & \text{Defined to be} -1 ~\text{if}~C~\text{is deficient over a local field}~\K~\text{and}~1~\text{otherwise.}\\
& \text{See \cite[Section 8]{MR1740984}. }\\
C^L & \text{the quadratic twist of}~C~\text{by the extension}~L/K\\
J/K & \text{the Jacobian of} ~C/K\\
J^L/K & \text{the quadratic twist of}~J~\text{by the extension}~L/K\\
c(J/\K) & \text{the local Tamagawa number (for}~K~\text{a local field)}\\
\Sha(J/K) & \text{the Shafarevich-Tate group of}~J/K~\text{(for}~K~\text{a number field)}\\
\Sha_0(J/K) & \text{the quotient of}~\Sha(J/K)~ \text{by its maximal divisible subgroup}\\
w(J/K) & \text{the global root number of}~J/K ~\text{for}~K~\text{a number field, or the local root}\\
& \text{number of}~J/\K ~\text{for}~K~\text{a local field.}
\end{array}$
\section{Basic properties of the local norm map} \label{local norm section}
In this section we prove some basic properties of the cokernel of the local norm map. We work with arbitrary principally polarised abelian varieties as everything in this section works in this generality. We extend the definition of the local norm map to this case in the obvious way. We begin by reviewing a general method for constructing isogenies between twists of abelian varieties, due to Milne \cite[Section 2]{MR0330174}, and apply it to construct an isogeny which will be of particular interest in forthcoming sections. A similar summary is given in \cite[Section 4.2]{MR2680426}. Most of the results in this section are standard but we give proofs for convenience. We will have particular need of \Cref{norm as isogeny,norm map as Tamagawa numbers}.
\subsection{Isogenies between twists of products of abelian varieties} \label{Twists}
Let $K$ be a field of characteristic zero (in all applications it will either be a number field or the completion of a number field), $L/K$ a finite Galois extension (soon to be quadratic) with Galois group $G$, and $A/K$ a principally polarised abelian variety.
Let $n\geq1$ be an integer and view $\text{Mat}_{n}(\mathbb{Z})$ and $\text{GL}_{n}(\mathbb{Z})$ inside $\text{End}_{K}(A^{n})$
and $\text{Aut}_{K}(A^{n})$ respectively in the obvious way. If $M$ is a free $\mathbb{Z}$-module of rank $n$, equipped with a linear action of $G$, then specifying a basis for $M$ we obtain a homomorphism
$\rho:G\rightarrow GL_{n}(\mathbb{Z})$
which we view as a one cocycle $\rho_{\sigma}$ from $G$ to $\text{Aut}_{L}(A^{n})$. Thus there is an $L/K$-twist $B$ of $A$, equipped with an $L$-isomorphism $\psi:A\rightarrow B$, such that $\psi^{-1}\psi^{\sigma}=\rho_{\sigma}$ for all $\sigma\in G$.
We denote this twist by $(A\otimes M,\psi)$, or just $A\otimes M$.
If $M_{1}$ and $M_{2}$ are two $G$-modules of rank $n$, and $f:M_{1}\rightarrow M_{2}$ is an injective $\mathbb{Z}[G]$-module homomorphism, then fixing bases for $M_{1}$ and $M_{2}$, we view $f$ as a matrix $X_{f}\in\text{Mat}_{n}(\mathbb{Z})$ and hence as an element of $\text{End}_{K}(A^{n})$. Then
$\phi_{f}=\psi_{M_{2}}X_{f}\psi_{M_{1}}^{-1}$ is an isogeny from $A\otimes M_{1}$ to $A\otimes M_{2}$ defined over $K$ and this association is functorial; given $f:M_{1}\rightarrow M_{2}$ and $g:M_{2}\rightarrow M_{3}$ as above, we have $\phi_{g}\circ\phi_{f}=\phi_{gf}$.
Now let $A^{t}/K$ denote the dual abelian variety of $A$ and $\mu:A\rightarrow A^{t}$ a principal polarisation, defined over $K$. If $\phi:A^{n}\rightarrow A^{n}$ is an isogeny represented by the matrix $\phi=(\phi_{ij})$ for $\phi_{ij}\in\text{End}(A)$, then the Rosati involution $\phi^{\dagger}=\lambda^{-1}\phi^{t}\lambda$ (where $\phi^{t}$ is the dual isogeny) of $\phi$ with respect to the product polarisation $\lambda=\mu^n$ is given by $\phi^{\dagger}=(\phi_{ji}^{\dagger})$ where $\phi_{rs}^{\dagger}$ denotes the Rosati involution of $\phi_{rs}$ with respect to $\mu$ \cite[Lemma 3]{MR1995864}. For an $L/K$-twist $(A\otimes M,\psi)$ of $A^{n}$, cocycle $\rho_{\sigma}$, the principal polarisation $\lambda_{M}=(\psi^{t})^{-1}\lambda\psi^{-1}$ on $A\otimes M$ is defined over $K$ if and only if $\rho_{\sigma}^{\dagger}\rho_{\sigma}=\text{id}_{A^{n}}$ for all $\sigma\in G$ \cite[Proposition 2.2]{MR1827021}.
If we have $(A\otimes M_{1},\psi_{M_{1}})$ and $(A\otimes M_{2},\psi_{M_{2}})$ with $\lambda_{M_{1}}$ and $\lambda_{M_{2}}$ defined over $K$, and $f:M_{1}\rightarrow M_{2}$ corresponds to the isogeny $\phi_{f}:A\otimes M_{1}\rightarrow A\otimes M_{2}$, then we have an isogeny $\phi_{f}^{\dagger}:A\otimes M_{2}\rightarrow A\otimes M_{1}$ defined by $\phi_{f}^{\dagger}=\lambda_{M_{1}}^{-1}\phi_{f}^{t}\lambda_{M_{2}}$, which we refer to as the \textit{Rosati involution} of $\phi_{f}$ with respect to $\lambda_{M_{1}}$ and $\lambda_{M_{2}}$. One has $\phi_{f}^{\dagger}= \psi_{M_{1}}X_{f}^{\dagger}\psi_{M_{2}}^{-1}$ and thus $\phi_{f}^{\dagger}$ is the isogeny (defined over $K$) corresponding to the matrix transpose of $X_{f}$.
\subsection{An isogeny between twists of $A^2$}
Retaining the setup of \Cref{Twists}, suppose now that $L/K$ is quadratic and let $\sigma$ denote the generator of $G=\text{Gal}(L/K)$. Then $A\otimes\mathbb{Z}[G]=\text{Res}_{L/K}(A)$ is the Weil restriction of $A/L$ to $K$, and letting $M$ be the $G$-module $\mathbb{Z}\oplus\mathbb{Z}$ where $\sigma$ acts trivially on the first factor and as multiplication by $-1$ on the second factor, we have $A\otimes M=A\times A^{L}$.
Now let $\rho$ and $\tau$ be the cocycles corresponding to $M$ and $\mathbb{Z}[G]$ respectively (with respect to the obvious bases). We have an injective homomorphism of $G$-modules $f:\mathbb{Z}[G]\rightarrow M$
defined by
\[
f(a+b\sigma)=\left(a+b,a-b\right),
\]
corresponding to the matrix
\[
X_{f}=\left(\begin{array}{cc}
1 & 1\\
1 & -1
\end{array}\right).
\]
We thus obtain isogenies
\[
\phi_{f}:\text{Res}_{L/K}(A)\longrightarrow A\times A^{L}
\]
and
\[
\phi_{f}^{\dagger}:A\times A^{L}\longrightarrow\text{Res}_{L/K}(A)
\]
where the latter is the Rosati involution of $\phi_{f}$ with respect to the polarisations above. Since the product of $X_{f}$ with its transpose in either direction is twice the identity matrix, we see that the composition of $\phi_{f}$ and $\phi_{f}^{\dagger}$ in either direction is just multiplication by two.
\subsection{The local norm map}
Now take $K$ to be a local field of characteristic zero and $L/K$ quadratic. Let $G$ be the Galois group of $L/K$ and $\sigma$ its generator.
\begin{lemma} \label{cokernel of mult by 2}
We have
\[\left|A(K)/2A(K)\right|=\left|A^{L}(K)/2A^{L}(K)\right|.\]
\end{lemma}
\begin{proof}
From the definition of $A^{L}$, we have an $L$-isomorphism $\psi:A\rightarrow A^{L}$
such that $\psi^{-1}\psi^{\sigma}=[-1]_{A}$. Thus the restriction of $\psi$ to $A(L)[2]$ gives an isomorphism $A(L)[2]\stackrel{\sim}{\longrightarrow}A^{L}(L)[2]$ which is Galois equivariant. Consequently, we have $A(K)[2]\cong A^L(K)[2]$.
Now suppose that $K$ is a finite extension of $\mathbb{Q}_p$ for some prime $p$, denote by $g$ the dimension of $A$ and let $r=\text{ord}_p(2)$. Then by \cite[Proposition 3.9]{MR1370197} we have $\left|A(K)/2A(K)\right|=p^{gr[K:\mathbb{Q}_{p}]}\left|A(K)[2]\right|$. Since $|A(K)[2]|=|A^L(K)[2]|$ this gives the result.
Finally, suppose $K=\mathbb{R}$. Since $A$ is an abelian variety over the reals, we have an isomorphism of real lie groups
\begin{equation*}
A(\mathbb{R})\cong\left(\mathbb{R}/\mathbb{Z}\right)^{g}\times\left(\mathbb{Z}/2\mathbb{Z}\right)^{k}
\end{equation*}
where $g$ is the dimension of $A$ and $0\leq k\leq g$ (see, for example, \cite[ Proposition 1.9~and~Remark 1.12]{Silhol}). It follows immediately that $\left| A(\mathbb{R})/2A(\mathbb{R})\right|=\left|A(\mathbb{R})[2]\right|/2^g$ and we conclude as before. (If $K=\mathbb{C}$ then both quantities are trivial.)
\end{proof}
Via the (inverse of the) isomorphism $\psi:A\rightarrow A^{L}$ of \Cref{Twists}, we identify $A^{L}(L)$ with $A(L)$ and $A^{L}(K)$ as a subgroup of $A(L)$. Explicitly, the identification is
\[
A^{L}(K)=\left\{ P\in A(L)\ :\ \sigma(P)=-P\right\} =\text{ker}\left(\N:A(L)\rightarrow A(K)\right).
\]
The map $\N:A^{L}(L)\rightarrow A^{L}(K)$ then becomes the map sending $P\in A(L)$ to $P-\sigma(P)$. To avoid confusion, we denote this map by $\N^{L}$.
\begin{lemma} \label{basic properties of norm map}
The group $A(K)/\N A(L)$ is a finite dimensional $\mathbb{F}_{2}$-vector space. Moreover, we have
\[
\left|A(K)/\N A(L)\right|=\left|A^{L}(K)/\N^L A(L)\right|.
\]
\end{lemma}
\begin{proof}
We have $2A(K)\subseteq \N A(L)$ and so $A(K)/\N A(L)$ is a quotient of $A(K)/2A(K)$, whence $A(K)/\N A(L)$ is annihilated by two and is finite (by the proof of \Cref{cokernel of mult by 2}). Moreover, the map
\[\theta: (\N A(L))/2A(K)\longrightarrow (\N^{L}A(L))/2A^{L}(K)\]
sending $\N(P)$ to $\N^{L}(P)$ is easily checked to be a (well defined) group isomorphism. The result is now clear.
\end{proof}
\begin{remark}
The group $A(K)/\N\left(A(L)\right)$ is isomorphic to the Tate cohomology group $\hat{\textup{H}}^\textup{0}\left(G,A(L)\right)$, whilst the group $A^{L}(K)/\N^L\left(A(L)\right)$ is isomorphic to $\textup{H}^{1}\left(G,A(L)\right)$. Thus the second statement of \Cref{basic properties of norm map} is the statement that the $G$-module $A(L)$ has trivial Herbrand quotient. One could also (in the case $K$ is non-archimedean) consider the formal group to obtain this.
\end{remark}
We now relate the cokernel of the local norm map to the cokernel (on $K$-points) of the isogeny $\phi_{f}$ of \Cref{Twists}. We continue to identify $A^{L}(K)$ as a subgroup of $A(L)$ and identify $(A\times A^{L})(K)$ with a subgroup of $A(L)\times A(L)$ similarly. Moreover, we have an isomorphism $\psi':A\times A\rightarrow\text{Res}_{L/K}(A)$ such that $(\psi')^{-1}(\psi')^{\sigma}$ is the endomorphism of $A\times A$ sending $(P,Q)$ to $(Q,P)$. We may thus identify $\text{Res}_{L/K}(A)(K)$ with the subgroup
\[\left\{ (P,Q)\in A(L)\times A(L)\ :\ Q=\sigma(P)\right\}\]
and projection onto the first factor gives an isomorphism onto $A(L)$. With the identifications above, we view $\phi_{f}$ as a homomorphism from $A(L)$ into $A(K)\times A^{L}(K)$ and see that this map is just given by $P\mapsto\left(\N(P),\N^{L}(P)\right)$.
\begin{lemma} \label{norm as isogeny}
Viewing $\phi_f$, $\textup{[2]}$ as maps on $K$-points, we have \textup{
\[
\frac{\left|\text{coker}(\phi_{f})\right|}{\left|\ker(\phi_{f})\right|}=\left|A(K)/\N A(L)\right|\frac{\left|\text{coker}\left([2]_{A}\right)\right|}{\left|\ker\left([2]_{A}\right)\right|}.
\]}
\end{lemma}
\begin{proof}
Clearly $\text{im}(\phi_{f})\subseteq \N A(L)\times \N^{L} A(L)$. Moreover, one checks that the map
\begin{equation*}
\theta:\N A(L)\times \N^{L} A(L)\longrightarrow (\N A(L))/2A(K)
\end{equation*}
sending $(\N(P),\N^{L}(Q))$ to $\N(P-Q)$ induces an isomorphism
\[(\N A(L)\times \N^L A(L))/\text{im}(\phi_f) \stackrel{\sim}{\longrightarrow}(\N A(L))/2A(K),\]
from which it follows that
\begin{equation*}
\left|\text{coker}(\phi_{f})\right|=\left|\frac{A^{L}(K)}{\N^{L} A(L) }\right|\left|\text{coker}\left([2]_{A}\right)\right|.
\end{equation*}
Applying \Cref{basic properties of norm map} and noting that $\text{ker}(\phi_{f}) \cong A(K)[2] $ gives the result.
\end{proof}
The final lemma of this section expresses the cokernel of the local norm map in terms of Tamagawa numbers. The special case of this for elliptic curves is due to Kramer and Tunnell \cite[Corollary 7.6]{MR664648}. Here and in what follows, we denote by $c(A/K)$ the Tamagawa number of $A/K$. That is, the order of the $k$-rational points in the group of components of the Neron model of $A$ over $\mathcal{O}_K$.
\begin{lemma} \label{norm map as Tamagawa numbers}
Assume the residue characteristic of $K$ is odd. Then
\begin{equation*}
\dim_{\mathbb{F}_2}A(K)/\N A(L)=\textup{ord}_2 \frac{c(A/K)c(A^L/K)}{c(A/L)}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $X=\text{Res}_{L/K}A$ and $Y=A\times A^{L}$ and as above let $\psi$ be the isogeny $X\rightarrow Y$ of \Cref{Twists}. Since $K$ has odd residue characteristic, it follows from \Cref{norm as isogeny} and a formula of Schaefer \cite[Lemma 3.8]{MR1370197} that
\[\dim_{\mathbb{F}_2}A(K)/\N A(L)=\textup{ord}_2 \frac{c(Y/K)}{c(X/K)}.\]
The behaviour of Tamagawa numbers under Weil restriction is studied by Lorenzini in \cite{MR2961846} whose Proposition 3.19 gives $c(X/K)=c(A/L)$. Since we also have $c(Y/K)=c(A/K)c(A^L/K)$ (see part (c) of the proof of the aforementioned proposition) we obtain the result.
\end{proof}
\section{2-Selmer Groups in Quadratic Extensions} \label{2-Selmer Groups in Quadratic Extensions}
In this section we use the description of the cokernel of the local norm map given in \Cref{norm as isogeny} to prove \Cref{selmer decomposition}. The main ingredient is a global duality result of Tate-Milne used in the proof of isogeny invariance of the Birch and Swinnerton-Dyer conjecture \cite[Section 1.7]{MR2261462}.
Suppose now that $K$ is a number field and $L/K$ is a quadratic extension. Following the notation
of \cite[Section 4.2]{MR2680426} we define, for an isogeny $\psi:X\rightarrow Y$
of abelian varieties over $K$,
\[
Q(\psi)=\left|\text{coker}\left(\psi:X(K)/X(K)_{\text{tors}}\rightarrow Y(K)/Y(K)_{\text{tors}}\right)\right|\cdot\left|\ker\left(\psi:\Sha_{\text{div}}(X/K)\rightarrow\Sha_{\text{div}}(Y/K)\right)\right|.
\]
Here $\Sha_{\text{div}}$ is the maximal divisible subgroup of $\Sha$.
We then have $Q(\psi'\psi)=Q(\psi')Q(\psi)$ for isogenies $\psi:X\rightarrow Y$
and $\psi':Y\rightarrow Z$, and if $\psi:X\rightarrow X$ is multiplication by
a prime number $p$ then $Q(\psi)=p^{\text{rk}_{p}(A/K)}$ \cite[Lemma 4.2]{MR2680426}. Now denote by $\psi^t:Y^t\rightarrow X^t$ the dual isogeny to $\psi$. As part of the proof of \cite[Theorem 4.3]{MR2680426} T. and V. Dokchitser show that for a sufficiently large set of places $S$, one has
\begin{equation} \label{duality result}
\text{ord}_2 \prod_{v\in S} \frac{|\text{coker}(\psi_v)|}{|\ker{\psi_v}|} =\text{ord}_{2}\frac{Q(\psi)}{Q(\psi^{t})}\frac{|Y(K)_{\text{tors}}|}{|X(K)_{\text{tors}}|}\frac{|Y^t(K)_{\text{tors}}|}{|X^t(K)_{\text{tors}}|}\frac{\left|\Sha_{0}(X/K)[2^{\infty}]\right|}{\left|\Sha_{0}(Y/K)[2^{\infty}]\right|}.
\end{equation}
Here $\psi_v$ denotes the map on $K_v$-points induced by $\psi$.
(The proof makes crucial use of the global duality result mentioned above. We remark that here `sufficiently large' includes at least those primes at which $X$ and $Y$ have bad reduction, all primes dividing the degree of $\psi$ and all archimedean primes.)
\begin{theorem}\label{selmer parity formula}
Let $K$ be a number field, $A/K$ a principally polarised abelian variety and $L/K$ a quadratic
extension. Then
\begin{eqnarray*}
\text{rk}_{2}(A/L) & \equiv & \sum_{v\in M_{K}}\dim_{\mathbb{F}_{2}}(A(K_{v})/N_{L_w/K_v}A(L_{w}))\\
& & +\dim_{\mathbb{F}_{2}}\Sha_{0}(A/K)[2]+\dim_{\mathbb{F}_{2}}\Sha_{0}(A^{L}/K)[2]\ (\textup{mod 2}).
\end{eqnarray*}
\end{theorem}
\begin{proof}
Let $X=\text{Res}_{L/K}A$ and $Y=A\times A^{L}$. As above let $\psi$
be the isogneny $X\rightarrow Y$ of \Cref{Twists}. Then for each place $v\in M_K$ we have
\[
\frac{\left|\text{coker}(\psi_{v})\right|}{\left|\ker(\psi_{v})\right|}=\left|\frac{A(K_{v})}{N_{L_w/K_v} A(L_{w}) }\right|\frac{\left|\text{coker}\left([2]_{v,A}\right)\right|}{\left|\ker\left([2]_{v,A}\right)\right|}
\]
where $w$ is any choice of place extending $v$ to $L$. Indeed, in the case that $L_{w}/K_{v}$ is quadratic, the map $\phi_{v}$ coincides with that induced from the isogeny $\text{Res}_{L_{w}/K_{v}}A\rightarrow A\times A^{L_{w}}$ and this is \Cref{norm as isogeny}. On the other hand, if $L_{w}/K_{v}$ is trivial, $\phi_{v}$ is the map
\begin{eqnarray*}
& \phi_{v}:A(K_{v})\times A(K_{v})\longrightarrow A(K_{v})\times A(K_{v})\\
& (P,Q)\longmapsto\left(P+Q,P-Q\right).
\end{eqnarray*}
One checks that the kernel is $A(K_{v})[2]$ and that the map
$\theta:\text{coker}(\phi_{v})\rightarrow A(K_{v})/2A(K_{v})$ sending $(P,Q)$ to $P+Q$ is an isomorphism. Since $N$ is surjective by definition when $L_{w}/K_{v}$ is trivial, this gives the required result.
Using the induced principal polarisations on $X$ and $Y$ to identify $X$ and $Y$ with their duals, \cref{duality result} gives
\[\text{ord}_2\prod_{v\in S}\left|\frac{A(K_{v})}{N_{L_w/K_v} A(L_{w})}\right| =\text{ord}_{2}\frac{Q(\phi)}{Q(\phi^{t})}\frac{|Y(K)_{\text{tors}}|^{2}}{|X(K)_{\text{tors}}|^{2}}\frac{\left|\Sha_{0}(X/K)[2^{\infty}]\right|}{\left|\Sha_{0}(Y/K)[2^{\infty}]\right|}
\]
(noting that a futher application of \cref{duality result} gives $\text{ord}_2\prod_{v\in S}\frac{|\text{coker}([2]_{v,A})|}{|\ker([2]_{v,A})|}=0$ (in fact, the product itself is equal to 1 which follows immediately from the cited result of Tate and Milne)).
Working modulo 2 we have
\[
\text{ord}_{2}\prod_{v}\left|\frac{A(K_{v})}{N_{L_w/K_v} A(L_{w})}\right|\equiv\text{ord}_{2}Q(\phi)Q(\phi^{t})\left|\Sha_{0}(X/K)[2^{\infty}]\right|\left|\Sha_{0}(Y/K)[2^{\infty}]\right|\ \text{(mod 2) }.
\]
Now $Q(\phi)Q(\phi^{t})=Q(\phi^{t}\phi)=2^{\text{rk}_{2}(X/K)}=2^{\text{rk}_{2}(A/L)}$.
Moreover, we have $\Sha_{0}(X/K)[2^{\infty}]\cong\Sha_{0}(A/L)[2^{\infty}]$
whilst $\Sha_{0}(Y/K)[2^{\infty}]\cong\Sha_{0}(A/K)[2^{\infty}]\times\Sha_{0}(A^{L}/K)[2^{\infty}]$
. Finally,
\begin{equation*}
\dim_{\mathbb{F}_{2}}\Sha_{0}(A/K)[2^{\infty}]\equiv\dim_{\mathbb{F}_{2}}\Sha_{0}(A/K)[2]\ \text{(mod 2)}
\end{equation*}
and similarly for all the other Shafarevich-Tate groups \cite[Theorem 8]{MR1740984}. By \Cref{sha in extensions} we have $\dim_{\mathbb{F}_{2}}\Sha_{0}(A/L)[2]\equiv 0 ~\text{(mod 2)}$. This completes the proof.
\end{proof}
\begin{proof}[Proof of \Cref{selmer decomposition}]
Let $L/K$ be a quadratic extension of number fields, $C/K$ a hyperelliptic curve and $J/K$ its (canonically principally polarised) Jacobian. Let $C^L$ denote the quadratic twist of $C$ by $L/K$, again a hyperelliptic curve over $K$. Then the Jacobian of $C^L$ coincides with the quadratic twist $J^L$ of $J$ by $L/K$. Thus by \cite[Theorem 11]{MR1740984} we have
\[(-1)^{\dim_{\mathbb{F}_{2}}\Sha_{0}(J/K)[2]}=\prod_{v\in M_K}i_d(C_v)\]
and similarly
\[(-1)^{\dim_{\mathbb{F}_{2}}\Sha_{0}(J^L/K)[2]}=\prod_{v\in M_K}i_d(C^L_v).\]
The result now follows immediately by applying \Cref{selmer parity formula} with $A=J$.
\end{proof}
\section{Compatibility Results} \label{compatibility results}
In this section we prove several compatibility results regarding the terms of \Cref{Kramer Tunnell}. This provides some evidence in favour of the conjecture, and will also be used to make some reductions as part of the proof of \Cref{cases of the parity conjecture}.
\subsection{Odd Degree Galois Extensions}
Consider a quadratic extension $L/K$ of local fields, and let $F$ be an odd degree Galois extension of $K$. First, we show that \Cref{Kramer Tunnell} for $L/K$ is equivalent to the corresponding statement for $LF/F$.
\begin{lemma}\label{odd degree extension}
Every individual term in \Cref{Kramer Tunnell} is unchanged under odd degree Galois extension of the base field. In particular, if $L/K$ is a quadratic extension of local fields and $F/K$ is an odd degree Galois extension, then \Cref{Kramer Tunnell} holds for $L/K$ if and only if it holds for $LF/F$.
\end{lemma}
\begin{proof}
That the Hilbert symbol and the terms involving the deficiency of $C$ and its twist are individually invariant under odd degree extensions (not necessarily even Galois) is clear. The statement for each of the root numbers is also standard. See, for example, \cite[Lemma A.1~and~Proposition A.2]{MR2534092} or \cite[Proposition 3.4]{MR664648}. For the cokernel of the local norm map, the statement for elliptic curves is \cite[Proposition 3.5]{MR664648}. The argument for general abelian varieties is identical.
\end{proof}
\subsection{Quadratic Twist I}
\begin{lemma} \label{twist 1}
Let $L/K$ be a quadratic extension of local fields and $C/K$ a hyperelliptic curve with Jacobian $J$. Then \Cref{Kramer Tunnell} holds for $J/K$ (and the extension $L/K$) if and only if it holds for $J^L/K$ (and the same extension).
\end{lemma}
\begin{proof}
Since the root numbers and terms involving deficiency appear symmetrically between $J$ and $J^L$ in \Cref{Kramer Tunnell}, it suffices to show that
\[(\Delta_C,L/K)=(\Delta_C^L,L/K)\]
and
\[\dim_{\mathbb{F}_2}J(K)/\N J(L)\equiv \dim_{\mathbb{F}_2}J^L(K)/\N J(L) ~ (\text{mod 2}).\]
The second statement is \Cref{basic properties of norm map} (and in fact the dimensions are equal as opposed to just congruent modulo 2). Finally, one readily sees that $\Delta_C$ and $\Delta_{C^L}$ lie in the same class in $K^\times/K^{\times 2}$.
\end{proof}
\subsection{Quadratic Twist II}
The second compatibility result involving quadratic twists is more subtle. That such a compatibility result should exist for elliptic curves is discussed in the original paper \cite{MR664648} of Kramer and Tunnell (remark following Proposition 3.3) and is proved (again for elliptic curves) by Klagsbrun, Mazur and Rubin \cite[Lemma 5.6]{MR3043582}. To motivate the statement, fix a non-archimedean local field $K$ (the archimedean case is not relevant here as the compatibility relation involves multiple quadratic extensions) and (distinct) quadratic extensions $L_1/K$ and $L_2/K$. Let $L_3/K$ be the third quadratic subextension of $L_1L_2/K$. \Cref{Kramer Tunnell} for $J/K$ and the extensions $L_1/K$ and $L_2/K$ is the assertion that
\begin{equation} \label{L1 expression}
w(J/K)w(J^{L_1}/K)=((-1)^g\Delta_C,L_1/K)i_d(C)i_d(C^{L_1})(-1)^{\dim_{\mathbb{F}_2}J(K)/N_{L_1/K} J(L_1)}
\end{equation}
and
\begin{equation} \label{L2 expression}
w(J/K)w(J^{L_2}/K)=((-1)^g\Delta_C,L_2/K)i_d(C)i_d(C^{L_2})(-1)^{\dim_{\mathbb{F}_2}J(K)/N_{L_2/K} J(L_2)}.
\end{equation}
Multiplying together \Cref{L1 expression,L2 expression}, noting that $J^{L_2}=(J^{L_1})^{L_3}$ and that (as above) the class of $\Delta_C$ in $K^{\times}/K^{\times 2}$ does not change under quadratic twist, we obtain
\begin{eqnarray*}
w(J^{L_{1}}/K)w((J^{L_{1}})^{L_{3}}/K) & = & \left((-1)^{g}\Delta_{C^{L_{1}}},L_{3}/K\right)i_d(C^{L_{1}})i_d((C^{L_{1}})^{L_{3}})\\
& & \times(-1)^{\dim_{\mathbb{F}_{2}}J(K)/N_{L_1/K} J(L_{1})+\dim_{\mathbb{F}_{2}}J(K)/N_{L_2/K} J(L_{2})}.
\end{eqnarray*}
We thus see that \Cref{Kramer Tunnell} predicts the congruence
\[\dim_{\mathbb{F}_2}J(K)/N_{L_1/K}J(L_1)+\dim_{\mathbb{F}_2}J(K)/N_{L_2/K}J(L_2) \equiv \dim_{\mathbb{F}_2}J^{L_1}(K)/N_{L_3/K}J^{L_1}(L_3)~~\text{(mod 2)}.\]
Conversely, if the above congruence holds and we know \Cref{Kramer Tunnell} for $J/K$ and both extensions $L_1/K$ and $L_2/K$, then we obtain \Cref{Kramer Tunnell} for $J^{L_1}/K$ and the extension $L_3/K$. We now prove this congruence.
\begin{lemma}
Let $K$ be a non-archimedean local field and $A/K$ a principally polarised abelian variety. Let $L_1/K$ and $L_2/K$ be distinct quadratic extensions and $L_3/K$ be the third quadratic subextension of $L_1L_2/K$. Then
\[\dim_{\mathbb{F}_2}A(K)/N_{L_1/K}A(L_1)+\dim_{\mathbb{F}_2}A(K)/N_{L_2/K}A(L_2) \equiv \dim_{\mathbb{F}_2}A^{L_1}(K)/N_{L_3/K}A^{L_1}(L_3)~~\textup{(mod 2)}.\]
\end{lemma}
\begin{proof}
The case where $A/K$ is an elliptic curve is \cite[Lemma 5.6]{MR3043582} and the argument is the same. Let $L_0=K$ and for each $i=0,1,2,3$, identify $A^{L_i}[2]$ with $A[2]$ as $G_K$-modules (see the proof of \Cref{cokernel of mult by 2}). We may then view each $X_i:=A^{L_i}(K)/2A^{L_i}(K)$ as a subgroup of the finite dimensional $\mathbb{F}_2$-vector space $V:=\text{H}^1(K,A[2])$ by identifying them with their image under the connecting homomorphism from the Kummer sequence. By \cite[Proposition 5.2]{MR2373150} we have
\[A(K)/N_{L_i/K}A(L_i)=X_0/(X_0\cap X_i)\]
for $i=1,2,3$, and
\[A^{L_1}(K)/N_{L_3/K}A^{L_1}(L_3)=X_1/(X_1\cap X_2).\]
In \cite[Section 4]{MR2833483} Poonen and Rains construct a quadratic form on $V$ whose induced bilinear form agrees with that induced by the Weil pairing on $A[2]$. The key observation is that each $X_i$ is a maximal isotropic subspace of $V$ with respect to this quadratic form \cite[Proposition 4.11]{MR2833483}. The result now follows from \cite[Corollary 2.5]{MR3043582} which is a general result concerning the dimension of the intersection of maximal isotropic subspaces. The one difference from the case of elliptic curves is that now the quadratic form (in general) takes values in $\mathbb{Z}/4\mathbb{Z}$ rather than just in $\mathbb{F}_2$ as was previously the case. However, one readily verifies that this assumption is not used in the proof of \cite[Corollary 2.5]{MR3043582}.
\end{proof}
From the discussion preceding the above lemma, we have the following.
\begin{cor} \label{twist 2}
Let $K$ be a non-archimedean local field, $C/K$ a hyperelliptic curve, and $J/K$ its Jacobian. Further,
let $L_1/K$ and $L_2/K$ be distinct quadratic extensions and $L_3/K$ be the third quadratic subextension of $L_1L_2/K$. Then \Cref{Kramer Tunnell} for $J/K$ and the extensions $L_1/K$ and $L_2/K$, implies \Cref{Kramer Tunnell} for $J^{L_1}/K$ and the extension $L_3/K$.
\end{cor}
\begin{remark}
For a local field $K$ and hyperelliptic curve $C/K$, by \Cref{twist 1} and \Cref{twist 2}, if we seek to prove \Cref{Kramer Tunnell} for $C/K$ and all quadratic extensions of $K$, then we may first make an arbitrary quadratic twist and prove the corresponding result for the new curve.
\end{remark}
\section{Two Torsion in the Jacobian of a hyperelliptic curve} \label{2-tors sect}
Let $C:y^2=f(x)$ be a hyperelliptic curve of genus $g$ over a field $K$ of characteristic zero and let $J/K$ be its Jacobian. Let $\W=\{\alpha_1,...,\alpha_{2g+2}\}$ be the $G_K=\text{Gal}(\bar{K}/K)$-set of roots of $f$ in $\bar{K}$ (if $\text{deg}(f)=2g+1$, replace $\alpha_{2g+2}$ with the $K$-rational point at infinity on $C$). Then as $G_K$-modules we have
\[J[2]\cong \ker\left(\mathbb{F}_2[\W]\stackrel{\Sigma}{\longrightarrow}\mathbb{F}_2\right)/\mathbb{F}_2[\Delta]\]
where $\Sigma:\mathbb{F}_2[\W]\rightarrow \mathbb{F}_2$ is the sum map and $\Delta=\sum_{w\in\W} w$ (see \cite[Section 6]{MR1465369}).
In particular, one sees that as $g\geq2$, $K(J[2])/K$ is the splitting field of $f$.
We now compute the $\mathbb{F}_2$-dimension of the rational $2$-torsion $J(K)[2]$. The case where $K(J[2])/K$ is cyclic is treated already in \cite[Theorem 1.4]{MR1865865} (but note the erratum \cite{MR2169307}) whilst the case where $f$ has an odd degree factor over $K$ is \cite[Lemma 12.9]{MR1465369}. We will require a slightly more general statement however.
\begin{lemma} \label{two torsion}
Let $n$ be the number of irreducible factors of $f$ over $K$. Then if $f$ has an odd degre factor over $K$, \[\dim_{\mathbb{F}_2}J(K)[2]=n-2.\] On the other hand, if each irreducible factor of $f$ over $K$ has even degree, let $F/K$ be the splitting field of $f$ and let $m$ be the number of quadratic subextensions of $F/K$ over which $f$ factors into $2$ distinct, conjugate polynomials. Then
\[\dim_{\mathbb{F}_{2}}J(K)[2]= \begin{cases}
n-1 & ~~g~\text{even}\\
n+m-1 & ~~g~\text{odd}.
\end{cases}\]
(If $\text{deg}(f)$ is odd, the rational point at infinity on $C$ is to be interpreted as an odd degree factor of $f$ over $K$.)
\end{lemma}
\begin{proof}
Denote by $G$ the Galois group of $F/K$ and let $M$ be the $G$-module $M=\ker\left(\mathbb{F}_2[\W]\stackrel{\Sigma}{\longrightarrow}\mathbb{F}_2\right)$. Then we have a short exact sequence
\begin{equation}\label{exact sequence}
0\longrightarrow \mathbb{F}_2[\Delta] \longrightarrow M^G \longrightarrow J[2]^G\longrightarrow \ker\left(\text{H}^1(G,\mathbb{F}_2[\Delta])\rightarrow \text{H}^1(G,M)\right).
\end{equation}
Now
\[\dim_{\F_2}M^G=\dim_{\F_2}\ker\left(\F_2[\W]^G\stackrel{\Sigma}{\longrightarrow}\F_2\right)=\begin{cases} n-1 & ~~f ~ \text{has an odd degree factor over} ~K\\ n & ~~\text{else}\end{cases}\]
and so we must show that $\dim_{\F_2} \ker\left(\text{H}^1(G,\mathbb{F}_2[\Delta])\rightarrow \text{H}^1(G,M)\right)$ is equal to $0$ or $m$ according to whether $g$ is even or odd respectively.
Now $\text{H}^1(G,\F_2[\Delta])=\text{Hom}(G,\F_2[\Delta])$ and the non-trivial homomorphisms from $G$ into $\F_2[\Delta]$ correspond to the quadratic subextensions of $F/K$. Now let $\phi$ be such a homomorphism, corresponding to a quadratic extension $E/K$. Then $\phi$ maps to $0$ in $\text{H}^1(G,M)$ if and only if there is $x\in M$ with $\sigma(x)+x=\phi(\sigma)\Delta$ for each $\sigma\in G$. Now the $x \in \mathbb{F}_2[\W]$ satisfying this equation correspond to factors $f_1$ of $f$ over $E$ for which $f=f_1\sigma(f_1)$ and $f_1\neq \sigma(f_1)$. Finally, note that any such $x$ is in the sum-zero part of $\F_2[\W]$ if and only if $g$ is odd (since $|\W|=2g+2$).
\end{proof}
Now let $\Delta_{f}$ be the discriminant of $f$. It is a square in $K$ if and only if the Galois group of $f(x)$ is a subgroup of the alternating group $A_{n}$ where $n=\deg f$. As a corollary of \Cref{two torsion}, we observe that if $K(J[2])/K$ is cyclic then whether or not the discriminant of $f$ is a square in $K$ may be detected from the rational $2$-torsion in $J$ as follows.
\begin{cor}\label{two torsion cor}
Suppose $K\left(J[2]\right)/K$ is cyclic. Then $\Delta_{f}$ is a square in $K$ if and only if one of the following holds
\begin{enumerate}
\item[\textit{(i)}] $(-1)^{\dim{J(K)[2]}}=1$ and either $g$ is odd, $\text{deg}(f)$ is odd, or deg$(f)$ is even and $f$ has an odd degree factor over $K$.
\item[\textit{(ii)}] $(-1)^{\dim{J(K)[2]}}=-1$, $g$ is even, and all irreducible factors of $f$ over $K$ have even degree.
\end{enumerate}
\end{cor}
\begin{proof}
Let $\sigma$ be a generator of $\text{Gal}\left(K(J[2])/K\right)$. Then $\Delta_{f}$ is a square in $K$ if and only if $\epsilon(\sigma)=1$, where $\epsilon(\sigma)$ is the sign of $\sigma$ as a permutation on the roots of $f$. Suppose $\sigma$ has cycle type $(d_{1},...,d_{n})$, so that the $d_{i}$ are also the degrees of the irreducible factors of $f$ over $K$. Then we have $\epsilon(\sigma)=(-1)^{\sum_{i=1}^{n}(d_{i}-1)}=(-1)^{\text{deg}f-n}$. Noting that $J[2]^{\sigma}=J(K)[2]$, and that $K(J[2])/K$ contains at most one quadratic extension, which yields a factorisation into $2$ distinct conjugate polynomials if and only if each $d_i$ is even, the result follows from \Cref{two torsion}.
\end{proof}
\section{Deficiency} \label{deficiency section}
Recall \cite[Section 8]{MR1740984} that $C$ is said to be \textit{deficient} over a local field $K$ if $\text{Pic}^{g-1}(C)=\emptyset$.
In this section we give two criteria for determining whether or not the hyperelliptic curve $C$ is deficient over $K$. The first covers the case where the two-torsion of its Jacobian is defined over a cyclic extension of $K$. This works for archimedean fields and non-archimedean fields alike. The second criterion concerns non-archimedean fields and characterises deficiency in terms of the components of the special fibre of the minimal regular model of $C$. This last criterion is not new and is observed by Poonen and Stoll in the remark following \cite[Lemma 16]{MR1740984}.
We first remark that as $C$ has $K$-rational divisors of degree 2 (arising as the pull back of points on $\mathbb{P}^1_K$) if $g$ is odd then $C$ is never deficient.
We have short exact sequences of $G_K=\text{Gal}(\bar{K}/K)$-modules
\begin{equation*}
0 \longrightarrow \bar{K}(C)^{\times}/\bar{K}^{\times}\stackrel{\text{div}}{\longrightarrow}\text{Div}(C_{\bar{K}})\longrightarrow \text{Pic}(C_{\bar{K}}) \longrightarrow 0
\end{equation*}
and
\begin{equation*}
0 \longrightarrow \bar{K}^{\times}\longrightarrow \bar{K}(C)^{\times}\longrightarrow \bar{K}(C)^{\times}/\bar{K}^{\times} \longrightarrow 0.
\end{equation*}
Combining the associated long exact sequences for Galois cohomology we obtain an exact sequence
\begin{equation*}
0\longrightarrow \text{Pic}(C) \longrightarrow \text{Pic}(C_{\bar{K}})^{\text{G}_K} \longrightarrow \text{Br}(K).
\end{equation*}
Denote by $\phi ~\text{:}~\text{Pic}(C_{\bar{K}})^{\text{G}_K}\rightarrow \mathbb{Q}/\mathbb{Z} $ the composition of the map above and the local invariant map $\text{inv}:\text{Br}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$.
In (the proof of) \cite[Theorem 11]{MR1740984}, Poonen and Stoll show that if $\mathcal{L}\in \text{Pic}(C_{\bar{K}})^{\text{G}_K}$ is a rational divisor class of degree $n$ on $C$ then $\text{Pic}^n(C)$ is empty (resp. non-empty) according to $n\phi (\mathcal{L})=\frac{1}{2}$ (resp. $0$) in $\mathbb{Q}/\mathbb{Z}$.
We will apply this to the case where the polynomial defining $C$ has an \textit{odd factorisation}. That is, when it factors as a product of two odd degree polynomials which are either $K$-rational, or conjugate over a quadratic extension of $K$ (the notation comes from \cite[Section 1]{BG14}).
\begin{proposition}\label{deficiency lemma}
Let $K$ be a local field of characteristic zero and $C$ be the hyperelliptic curve $y^2=df(x)$ of even genus $g$, where $f \in K[x]$ is monic. Suppose further that $f$ has an odd factorisation and let $L$ be the etale algebra $L=K[x]/(f)$. Then $C$ is deficient over $K$ if and only if $d\in N_{L/K}(L^{\times})K^{\times2}$.
\end{proposition}
\begin{proof}
First, we deal with some trivial cases. Since $C$ has $K$-rational divisors of degree 2, having a $K$-rational divisor of degree $g-1$ is equivalent to having a $K$-rational divisor of any odd degree, which in turn is equivalent to having a rational point over some odd degree extension of $K$. If $f$ has an odd degree factor over $K$ then $C$ has a rational Weierstrass point over an odd degree extension and is not deficient. Moreover, $N_{L/K}(L^{\times})K^{\times2}=K^{\times}$ in this case. Thus we assume $f$ factors into two conjugate, odd degree polynomials over a quadratic extension $F/K$. Then over $F$, we may write $f=f_1f_2$ where each $f_i$ has degree $g+1$ and $f_1$ and $f_2$ are conjugate over $F$. Denote the roots of $f_1$ by $\alpha_1,\alpha_3,...,\alpha_{2g+1}$ and the roots of $f_2$ by $\alpha_2,\alpha_4,...,\alpha_{2g+2}$. For each $i$, let $P_i=(\alpha_i,0)\in C(\bar{K})$ and let $D$ be the degree $g+1$ divisor
\begin{equation*}
D=\sum_{i~ \text{odd}} (P_i).
\end{equation*}
If $\chi:\text{G}_K\rightarrow {\{\pm1\}}$ is the quadratic character associated to the extension $F/K$ then for all $\tau \in \text{G}_K$ we have $\tau(D)=D$ or $\tau(D)=\sum_{i~ \text{even}} (P_i)$ according to $\chi(\tau)=1$ or $-1$. Since
\begin{equation*}
\text{div}\left(\frac{y}{\prod_{i~\text{odd}}(x-\alpha_i)}\right)=\sum_{i~\text{even}}(P_i)-\sum_{i~\text{odd}}(P_i)
\end{equation*}
it follows that $D$ represents a $K$-rational divisor class. Since $\text{Pic}^{g-1}(C)=\emptyset$ if and only if $\text{Pic}^{g+1}(C)=\emptyset$, it follows that $C$ is deficient if and only if $(g+1)\phi(D)=\frac{1}{2}$ in $\mathbb{Q}/\mathbb{Z}$ (here we write $D$ for the divisor class of $D$ also). The discussion above shows that the cocycle $f_{\tau}:\text{G}_K\rightarrow \bar{K}(C)^{\times}/\bar{K}^{\times}$ that sends $\tau$ to 1 if $\chi(\tau)=1$ and (the class of) $\frac{y}{\prod_{i~\text{odd}}(x-\alpha_i)}$ if $\chi(\tau)=-1$, represents the image of $D$ under the connecting homomorphism
\[\delta:\text{Pic}(C_{\bar{K}})^{\text{G}_K}\rightarrow \text{H}^1(K,\bar{K}(C)^{\times}/\bar{K}^{\times}).\] Viewing $f_{\tau}$ instead as a cochain with values in $\bar{K}(C)^{\times}$ in the obvious way, the image of $f_\tau$ in $\text{Br}(K)$ is represented by the 2-cocycle $a_{\tau,\rho}=f_\tau{} ^{\tau}f_\rho f_{\tau\rho}^{-1}$. A straightforward computation gives $a_{\tau,\rho}=1$ unless $\chi(\tau)=-1=\chi(\rho)$ in which case it is equal to
\begin{equation*}
\frac{y}{\prod_{i~\text{odd}}(x-\alpha_i)}\cdot \frac{y}{\prod_{i~\text{even}}(x-\alpha_i)}=d.
\end{equation*}
Under $\text{inv}_K:\text{Br}(K)\rightarrow\ \mathbb{Q}/\mathbb{Z}$, $a_{\tau,\rho}$ is mapped to 0 if $d$ is a norm from $F^{\times}$ and $1/2$ otherwise, so $(g+1)\phi(D)$ is 0 if and only if $d\in N_{F/K}(F^{\times})$. It remains to show that $N_{L/K}(L^{\times})K^{\times2}=N_{F/K}(F^{\times})$. Indeed, factorise $f$ into irreducibles $g_1,...,g_r$ over $K$ and let $L_i=K[x]/(g_i)$. The assumption on the $K$-factorisation of $f$ implies that $F$ is contained in each $L_i$. Thus $N_{L/K}(L^{\times})K^{\times2}\subseteq N_{F/K}(F^{\times})$. Conversely, since $\text{deg}(f)\equiv 2~\text{(mod 4)}$, there is some $i$ for which $g_i \equiv 2~\text{(mod 4)}$ also. Then $[L_i:F]=2m+1$ is odd and for any $x\in F^{\times}$ we have $N_{L_i/K}(x)=N_{F/K}(x)N_{F/K}(x)^{2m}$. Thus $N_{F/K}(x)\in N_{L/K}(L^{\times})K^{\times2}$ and we are done.
\end{proof}
\begin{cor}\label{cyclic deficiency lemma}
Let $K$ be a local field of characteristic zero and $C$ be the hyperelliptic curve associated to the equation $y^2=df(x)$ where $f \in K[x]$ is monic and separable of degree $2g+1$ or $2g+2$ where $g$ is even. Suppose moreover that $K(J[2])/K$ is cyclic. Then $C$ is deficient over $K$ if and only if all irreducible factors of $f$ over $K$ have even degree, and $(d,F/K)=-1$ where $F/K$ is the unique quadratic subextension of $K(J[2])/K$.
\end{cor}
\begin{proof}
As before we may assume each irreducible factor of $f$ over $K$ has even degree, in which case $f$ has degree $2g+2$. Then the assumption that $K(J[2])/K$ is cyclic ensures that there is indeed a unique quadratic subextension of $K(J[2])/K$ and that $f$ factors into two conjugate odd degree polynomials over $F$. The claimed result now follows from \Cref{deficiency lemma}.
\end{proof}
We conclude the section by characterising deficiency in terms of the minimal regular model of $C$. We will make extensive use of this criterion later. Since at times we will work with curves that are not necessarily hyperelliptic, we state the result in this generality here.
\begin{lemma}\label{min reg def lemma}
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $k$. Let $X/K$ be a smooth, proper, geometrically integral curve of genus $g$, $\mathcal{X}/\mathcal{O}_K$ its minimal regular model and $\Gamma_1,...,\Gamma_n$ the irreducible components of the special fibre $\mathcal{X}_{k}$ of $\mathcal{X}$. For each component $\Gamma_i$, let $d_i$ be the multiplicity of $\Gamma_i$ in $\mathcal{X}_{k}$ and $r_i=[\bar{k}\cap k(\Gamma_i):k]$. Then $X$ is deficient over $K$ if and only if $\textup{gcd}_{1\leq i \leq n}\{r_i d_i\}$ does not divide $g-1$.
\end{lemma}
\begin{proof}
This is remarked by Poonen and Stoll in \cite{MR1740984}, immediately after the proof of Lemma 16.
\end{proof}
\begin{remark} \label{min reg def remark}
We can rephrase \Cref{min reg def lemma} as follows. Let $\mathcal{X}_{\bar{k}}$ be the special fibre of the minimal regular model of $X$ over $\mathcal{O}_K$, base-changed to $\bar{k}$ (this coincides with the special fibre of the minimal regular model of $X$ over $K^{\text{nr}}$). If a component $\bar{\Gamma}$ of $\mathcal{X}_{\bar{k}}$ with multiplicity $\bar{d}$ lies over a component $\Gamma$ of $\mathcal{X}_k$ of multiplicity $d$, then $\bar{d}=d$. Moreover, under the natural action of $\textup{Gal}(\bar{k}/k)$ on the components of $\mathcal{X}_{\bar{k}}$, we have \[[\bar{k}\cap k(\Gamma):k]= |\textup{orb}_{\textup{Gal}(\bar{k}/k)}(\bar{\Gamma})|.\]
Consequently, letting $\bar{\Gamma}_1,...,\bar{\Gamma}_m$ be the irreducible components of $\mathcal{X}_{\bar{k}}$, multiplicities $\bar{d}_i$, we see that $X$ is deficient over $K$ if and only if
\[\textup{gcd}_{1\leq i \leq m}\{\bar{d}_i \cdot |\textup{orb}_{\textup{Gal}(\bar{k}/k)}(\bar{\Gamma}_i)|\}\]
does not divide $g-1$.
\end{remark}
\begin{remark}
When $X$ is hyperelliptic, the quantity
\[\textup{gcd}\{\bar{d}_i \cdot |\textup{orb}_{\textup{Gal}(\bar{k}/k)}(\bar{\Gamma}_i)|\}\]
is either $1$ or $2$ (this is because any hyperelliptic curve has rational divisors of degree $2$ as remarked previously; this is also discussed by Poonen and Stoll in \cite{MR1740984} when they explain \Cref{min reg def lemma}). Thus in this case, $X$ is deficient over $K$ if and only if $g$ is even and
\[\textup{gcd}\{\bar{d}_i \cdot |\textup{orb}_{\textup{Gal}(\bar{k}/k)}(\bar{\Gamma}_i)|\}=2.\]
\end{remark}
\section{First Cases of \Cref{Kramer Tunnell}} \label{first cases}
In this section we prove \Cref{Kramer Tunnell} in two cases, namely for archimedean places and for places of good reduction and odd residue characteristic. It will turn out that these are the only cases needed to prove \Cref{global to local} (in fact, even the archimedean places are not necessary for this).
\subsection{Archimedean Places}
Here we consider archimedean local fields. Clearly the only case of interest is the extension $\mathbb{C}/\mathbb{R}$. In this case, we can answer \Cref{Kramer Tunnell} completely.
\begin{proposition} \label{kramer-tunnell for reals places}
\Cref{Kramer Tunnell} holds for the extension $\mathbb{C}/\mathbb{R}$ and every hyperelliptic curve $C/\mathbb{R}$.
\end{proposition}
\begin{proof}
Let $J/\mathbb{R}$ be the Jacobian of $C$. As remarked in (the proof of) \Cref{cokernel of mult by 2}, we have an isomorphism of real lie groups
\begin{equation}\label{isomorphism of lie groups}
J(\mathbb{R})\cong\left(\mathbb{R}/\mathbb{Z}\right)^{g}\times\left(\mathbb{Z}/2\mathbb{Z}\right)^{k}
\end{equation}
where $g$ is the genus of $C$ and $0\leq k\leq g$. Now $N_{\mathbb{C}/\mathbb{R}}$ is a continuous map from the connected group $J(\mathbb{C})$ to $J(\mathbb{R})$ and it follows that the image of $N_{\mathbb{C}/\mathbb{R}}$ is contained in the connected component of the identity in $J(\mathbb{R})$, denoted $J^{0}(\mathbb{R})$. Under the isomorphism \cref{isomorphism of lie groups}, $J^{0}(\mathbb{R})$ is the factor corresponding to $\left(\mathbb{R}/\mathbb{Z}\right)^{g}$. On the other hand, we have $2J(\mathbb{R})\subseteq N_{\mathbb{C}/\mathbb{R}} A(\mathbb{C})$ and we see again from \cref{isomorphism of lie groups} that multiplication by $2$ is surjective on $J^{0}(\mathbb{R})$. Thus $N_{\mathbb{C}/\mathbb{R}} J(\mathbb{C})=J^{0}(\mathbb{R})$. In particular, $|J(\mathbb{R})/N_{\mathbb{C}/\mathbb{R}} J(\mathbb{C}) |=2^{-g}|J(\mathbb{R})[2]|$. Denote by $J_{-1}$ the quadratic twist of $J$ by $\mathbb{C}/\mathbb{R}$. We have $w(J/\mathbb{R})w(J_{-1}/\mathbb{R})=1$ (see, for example, \cite[Lemma 2.1]{MR2309184}), and to verify \Cref{Kramer Tunnell} we must show that
\begin{equation*}
(-1)^{\dim_{\mathbb{F}_2}J(\mathbb{R})[2]}=(\Delta_C,-1)i_d(C)i_d(C_{-1}).
\end{equation*}
(Here the $(-1)^g$ in the expression for $|J(\mathbb{R})/N_{\mathbb{C}/\mathbb{R}} J(\mathbb{C}) |$ has canceled with the Hilbert symbol $((-1)^g,-1)$ appearing in \Cref{Kramer Tunnell}.)
Now certainly $K(J[2])/K$ is cyclic and, moreover, $(\Delta_C,-1)=1$ if and only if $\Delta_C$ is a square in $\mathbb{R}$. Consequently, \Cref{two torsion cor} gives $(-1)^{\dim_{\mathbb{F}_2}J(\mathbb{R})[2]}=(\Delta_C,-1)$ except when $g$ is even and all irreducible factors of $f$ over $\mathbb{R}$ have even degree, in which case the two expressions differ by a sign. Since by \Cref{cyclic deficiency lemma} this is exactly the case where $i_d(C)i_d(C_{-1})=-1$, we have verified \Cref{Kramer Tunnell}.
\end{proof}
\subsection{Good Reduction in Odd Residue Characteristic} \label{odd good}
Suppose now that $K$ is a finite extension of $\mathbb{Q}_p$ with $p$ odd and that $J$ has good reduction over $K$. Let $L/K$ be a quadratic extension. The following lemma describes the cokernel of the norm map from $J(L)$ to $J(K)$.
\begin{lemma}\label{good norm lemma}
If $L/K$ is unramified then $J(K)/\N J(L)$ is trivial whilst if $L/K$ is ramified we have $\N J(L)=2J(K)$. In particular,
\begin{equation*}
\left|J(K)/\N J(L)\right|=\begin{cases}
1 & L/K\ \text{unramified}\\
\left|J(K)[2]\right| & L/K\ \text{ramified}.
\end{cases}
\end{equation*}
\end{lemma}
\begin{proof}
The case $L/K$ is unramified is a result of Mazur \cite[Corollary 4.4]{MR0444670} along with the fact that the Herbrand quotient $q\left(J(L)\right)$ is trivial. The case $L/K$ ramified is essentially Corollary 4.6 in \textit{loc. cit.~}. The argument is as follows. Let $\mathcal{J}$ be the Neron model of $J$ over $K$. Since $J$ has good reduction, the Neron model of $J$ over $L$ is given by $\mathcal{J}\times_{\mathcal{O}_{K}}\mathcal{O}_{L}$. If $J_{1}(K)$ and $J_{1}(L)$ denote the kernels of reduction modulo $\pi_{K}$ and $\pi_{L}$ respectively, then we have an exact sequence of $G=\text{Gal}(L/K)$-modules
\begin{equation} \label{reduction exact sequence}
0\rightarrow J_{1}(L)\rightarrow J(L)\rightarrow\tilde{J}(k)\rightarrow0
\end{equation}
where $\tilde{J}:=\mathcal{J}\times_{\mathcal{O}_{K}}k$ and $ $has trivial G-action, and $J_{1}(L)^{G}=J_{1}(K)$. Since $\text{char}(k)\neq2$, multiplication by 2 is an isomorphism on $J_{1}(K)$. It follows that $J_{1}(K)/\N J_{1}(L)$ is trivial and that reduction gives an isomorphism
\[\hat{\text{H}}^0\left(G,J(L)\right)\stackrel{\sim}{\longrightarrow}\hat{\text{H}}^0(G,\tilde{J}(k)).\] On the other hand, considering \cref{reduction exact sequence} with $L$ replaced by $K$ (as a sequence of trivial $G$-modules) and noting that the actions on $\tilde{J}(k)$ in each sequence coincide, we similarly obtain an isomorphism \[J(K)/2J(K)\stackrel{\sim}{\longrightarrow} \hat{H}^0(G,\tilde{J}(k))\] again induced by reduction. The result now follows easily.
\end{proof}
\begin{cor} \label{kramer-tunnell for good reduction in odd residue char}
Let $K$ be a finite extension of $\mathbb{Q}_p$ for $p$ odd and $L/K$ a quadratic extension. Let $C/K$ be a hyperelliptic curve and suppose that the Jacobian $J$ of $C$ has good reduction over $K$. Then \Cref{Kramer Tunnell} holds for $C$ and the extension $L/K$.
\end{cor}
\begin{proof}
The assumptions on the reduction and residue characteristic imply that
\begin{equation*}
w(J/K)w(J^L/K)=((-1)^g,L/K)
\end{equation*}
(see, for example, \cite[Proposition 3.23]{MR2534092}).
Thus we are reduced to showing that
\begin{equation*}
(-1)^{\dim_{\mathbb{F}_2}J(K)/\N J(L)}=(\Delta_C,L/K)i_d(C)i_d(C^L).
\end{equation*}
If $L/K$ is unramified then \Cref{good norm lemma} gives $(-1)^{\dim_{\mathbb{F}_2}J(K)/\N J(L)}=1$. Moreover, the assumptions on the reduction mean $K(J[2])/K$ is unramified and so adjoining a square root of $\Delta_C$ to $K$ yields an unramified extension. In particular, $(\Delta_C,L/K)=1$. Finally, \Cref{cyclic deficiency lemma} gives $i_d(C)i_d(C^L)=1$ also.
Now suppose $L/K$ is ramified. \Cref{good norm lemma} gives $(-1)^{\dim_{\mathbb{F}_2}J(K)/\N J(L)}=(-1)^{\dim_{\mathbb{F}_2}J(K)[2]}$. Moreover, as $v_K(\Delta_C)$ is even, $\Delta_C$ is a unit modulo squares in $K$ and hence $(\Delta_C,L/K)=1$ if and only if $\Delta_C$ is a square in $K$. \Cref{two torsion cor} and \Cref{cyclic deficiency lemma} then give the desired result, noting that if $u$ is a non-square unit then $(u,L/K)=-1$.
\end{proof}
\section{Deducing \cref{Kramer Tunnell} from global conjectures}
We have now proved enough cases of \Cref{Kramer Tunnell} to prove \Cref{global to local}.
\begin{proof}[Proof of \Cref{global to local}]
Write $F=K_{v_0}(\sqrt{\alpha})$ with $\alpha\in K$. Let $S$ be a finite set of places of $K$ containing all places where $J$ has bad reduction, all places dividing 2 and all archimedean places. Set $T=S-\{v_0\}$.
Now let $L/K$ be a quadratic extension such that each place $v\in T$ splits in $L/K$ and such that there is exactly one place $w_0|v_0$ and which satisfies $L_{w_0}=F$. Explicitly, we may take $L=K(\sqrt{\beta})$ where $\beta \in K$ is chosen, by weak approximation, to be sufficiently close to $\alpha$ $v_0$-adically, and sufficiently close to $1$ $v$-adically for all $v\in T$. Then the products
\[
\prod_{v\in M_K} w(J/K_v)w(J^L_w/K_v) ~~\text{and}~~ \prod_{v\in M_K} ((-1)^g\Delta_C,L_w/K_v)i_d(C_v)i_d(C^{L_w}_v)(-1)^{\dim_{\mathbb{F}_2}J(K_v)/N_{L_w/K_v}J(L_w)}
\]
multiply to $w(J/L)$ and $\text{rk}_2(J/L)$ respectively and hence agree under the assumption that the $2$-parity conjecture holds over $L$. On the other hand, by \Cref{kramer-tunnell for good reduction in odd residue char} the contributions to each product from a single place $v$ agree save possible at $v=v_0$ (if a place $v$ splits in $L/K$ then each of the contributions from $v$ are trivial). Thus the contributions from $v=v_0$ must agree too.
\end{proof}
\section{Residue characteristic 2} \label{residue characteristic 2}
In this section we consider \Cref{Kramer Tunnell} when $K$ is a non-archimedean local field with residue characteristic 2. Here the norm map and root numbers become considerably more complicated and we assume for the rest of this section that $J/K$ has good reduction over $K$, and will impose additional assumptions if $L/K$ is ramified.
\subsection{Unramified extensions}
Suppose that $L/K$ is unramified. The result of Mazur \cite[Corollary 4.4]{MR0444670} used in the proof of \Cref{good norm lemma} did not in fact assume that the residue characteristic of $K$ was odd. In particular, we still have $|J(K)/\N J(L)|=1$ in this case. Moreover since $L/K$ is unramified, the quadratic twist $J^L$ of $J$ also has good reduction. Consequently, both $w(J/K)$ and $w(J^L/K)$ are equal to 1, and we have $(-1,L/K)=1$ also. To verify \Cref{Kramer Tunnell} we must therefore prove that $(\Delta_C,L/K)i_d(C)i_d(C^L)=1$. In fact, both $(\Delta_C,L/K)$ and $i_d(C)i_d(C^L)$ are equal to 1 individually. That the first of these quantities is, is the result of the next proposition, which may be of independent interest.
\begin{proposition} \label{valuation of discriminant}
Suppose $J$ has good reduction over $K$. Then the discriminant $\Delta_C$ of any Weierstrass equation for $C$ has even valuation. In particular, for $L/K$ the unique unramified quadratic extension of $K$, $(\Delta_C,L/K)=(-1)^{v_K(\Delta_C)}=1$.
\end{proposition}
\begin{proof}
Let $\mathcal{J}/\mathcal{O}_{K}$ be the Neron model of $J$. The assumption that $J$ has good reduction over
$K$ implies that $\mathcal{J}[2]$ is a finite flat group scheme over $\mathcal{O}_{K}$ \cite[Proposition 20.7]{MR861974}. Letting $e$ denote the absolute ramification index of $K$, it is a theorem of Fontaine that $G_{K}^{u}$ acts trivially on $\mathcal{J}[2](\bar{K})=J[2]$ provided $u>2e-1$ \cite[Th{\'e}or{\`e}me A]{MR807070}. Note that we are using Serre's upper numbering for the higher ramification groups. Let $L=K\left(\sqrt{\Delta_{C}}\right)$ and $G=\text{Gal}(L/K)$. By Herbrand's theorem (see, for example, \cite[IV, Lemma 3.5]{MR554237} ), $G^{u}$ is trivial for $u\geq2e$. In particular, the conductor $\mathfrak{f}(L/K)$ satisfies $\mathfrak{f}(L/K)\leq2e$. On the other hand, supposing $v_{K}(\Delta_{C})$ is odd, we have $L=K\left(\sqrt{\pi_{K}}\right)$ for some uniformiser $\pi_{K}$ of $K$. Letting $\sigma$ be the
non-trivial element of $G$, we obtain \[v_{L}\left(\sigma\left(\sqrt{\pi_{K}}\right)-\sqrt{\pi_{K}}\right)=v_{L}(2)+1=2e+1,\] whence $\mathfrak{f}(L/K)=2e+1$, a contradiction.
\end{proof}
\begin{remark}
This proposition is trivially true also when the residue characteristic of $K$ is odd. Indeed, then $J[2]$ is unramified and hence $\Delta_C$ is a square in $K^{\text{nr}}$, thus has even valuation.
\end{remark}
\begin{lemma}
Suppose $J$ has good reduction over $K$ and let $L/K$ be the unique quadratic unramified extension. Then $C$ is deficient over $K$ if and only if its quadratic twist $C^L$ is. That is, we have $i_d(C)i_d(C^L)=1$.
\end{lemma}
\begin{proof}
Let $\mathcal{C}_{\bar{k}}$ and $\mathcal{C}^L_{\bar{k}}$ denote the base-change to $\bar{k}$ of the special fibre of the minimal regular model over $K$ of $C$ and $C^L$ respectively. Since the formation of minimal regular models commutes with unramified base-change, we may identify $\mathcal{C}_{\bar{k}}$ and $\mathcal{C}^L_{\bar{k}}$, the only difference being that the natural action of $\text{Gal}(\bar{k}/k)$ differs by twisting by the hyperelliptic involution (which extends uniquely to an automorphism of $\mathcal{C}$). Thus by \Cref{min reg def remark} it suffices to show that the hyperelliptic involution acts trivially on the components of $\mathcal{C}_{\bar{k}}$. Since $J$ has good reduction, the dual graph of $\mathcal{C}_{\bar{k}}$ is a tree \cite[Proposition 10.1.51]{MR1917232}, and as there are no exceptional curves in $\mathcal{C}_{\bar{k}}$, each leaf corresponds to a positive genus component. Since the quotient of $\mathcal{C}_{\bar{k}}$ by the hyperelliptic involution has arithmetic genus zero, the hyperelliptic involution necessarily fixes every leaf, and consequently acts trivially on the dual graph.
\end{proof}
We have thus shown
\begin{cor} \label{char 2 unramified}
Suppose that $K$ is a finite extension of $\mathbb{Q}_2$, $L/K$ is an unramified quadratic extension and $J$ has good reduction over $K$. Then \Cref{Kramer Tunnell} holds for $J$ and $L/K$.
\end{cor}
\subsection{Ramified extensions}
Suppose now that $L/K$ is ramified and that $J/K$ has good ordinary reduction. Let $J_1(K)$ denote the kernel of reduction on $J(K)$, and likewise for $J_1(L)$. We begin by considering the norm map from $J_1(L)$ to $J_1(K)$.
\begin{lemma} \label{Lubin lemma}
We have
\[\left|J_1(K)/\N J_1(L)\right|=|J_1(K)[2]|.\]
\end{lemma}
\begin{proof}
Let $G=\text{Gal}(L/K)\cong \mathbb{Z}/2\mathbb{Z}$. Let $g$ be the genus of $C$ so that by \cite[Theorem 1]{MR0491735}, there is a $g\times g$ matrix $u$ over $\mathbb{Z}_2$ (the $\textit{twist matrix}$ associated to the formal group of $J$) such that
\[J_1(K)/\N J_1(L)\cong G^g/(I-u)G^g\]
(here $I$ is the $g\times g$ identity matrix).
On the other hand (see Lemma in \textit{loc.cit.}), denoting by $T$ the completion of $K^\text{nr}$, we have \[J_1(K)\cong \left\{\alpha \in \left(\mathcal{O}_T^\times\right)^g~:~\alpha^\phi=\alpha^u\right\},\]
where $\phi$ denotes the $K$-Frobenius automorphism of $T$. In particular, we obtain
\[J_1(K)[2]\cong \left\{\alpha \in \left\{\pm1\right\}^g~:~\alpha^{I-u}=1\right\}.\]
Identifying the groups $G$ and $\{\pm1\}$ in the obvious way, we see that $J_1(K)[2]$ becomes identified with the kernel of multiplication by $I-u$ on $G$. We now conclude by noting that the cokernel and kernel of an endomorphism of a finite group always have the same order.
\end{proof}
\begin{lemma} \label{ordinary norm size}
Suppose that all the $2$-torsion of $J$ is defined over an odd degree Galois extension of $K$. Then we have
\[\textup{dim}_{\mathbb{F}_2} J(K)/\N J(L) \equiv 0 ~~\textup{(mod 2)}.\]
\end{lemma}
\begin{proof}
By \Cref{odd degree extension}, we can actually assume that all the 2-torsion is rational and we will show that then
\[\textup{dim}_{\mathbb{F}_2} J(K)/\N J(L) =2g.\]
Consider the commutative diagram with exact rows
\[
\xymatrix{0\ar[r] & J_{1}(L)\ar[r]\ar[d]^{\N} & J(L)\ar[r]\ar[d]^{\N} & \tilde{J}(k)\ar[r]\ar[d]^{2} & 0\\
0\ar[r] & J_{1}(K)\ar[r] & J(K)\ar[r] & \tilde{J}(k)\ar[r] & 0.
}
\]
\[
\]
The assumption that all the 2-torsion is defined over $K$ means that reduction is a surjection from $J(K)[2]$ to $\tilde{J}(k)[2]$. In particular, in the exact sequence arising from applying the snake lemma to the diagram above, the connecting homomorphism is trivial. Thus we deduce the short exact sequence
\[0\longrightarrow J_1(K)/\N J_1(L)\longrightarrow J(K)/\N J(L)\longrightarrow \tilde{J}(k)/2\tilde{J}(k)\longrightarrow 0.\]
As $J$ is ordinary (and again using that all the 2-torsion is rational), we have
\[\left| \tilde{J}(k)/2\tilde{J}(k)\right|=|\tilde{J}(k)[2]|=2^g.\]
On the other hand, by \Cref{Lubin lemma} we have
\[\left|J(K)/\N J(L)\right|=|J_1(K)[2]|=2^g\]
also (where again in the last equality we are using that all the 2-torsion is rational). The result is now clear.
\end{proof}
\begin{cor} \label{good ordinary case}
Suppose that $K$ is a finite extension of $\mathbb{Q}_2$, $L/K$ is a ramified quadratic extension, $J$ has good ordinary reduction over $K$ and all of $J(\bar{K})[2]$ is defined over an odd degree Galois extension of $K$. Then \Cref{Kramer Tunnell} holds for $J$ and $L/K$.
\end{cor}
\begin{proof}
Again by \Cref{odd degree extension} we assume that all the $2$-torsion of $J$ is in fact rational. In particular, $f$ splits over $K$ and hence $(\Delta_C,L/K)=1$. Similarly, both $C$ and $C^L$ have a rational Weierstrass point and so $i_d(C)i_d(C^L)=1$. By \Cref{ordinary norm size} we have $(-1)^{\text{ord}_2J(K)/\N J(L)}=1$ also and by \cite[Proposition 3.23]{MR2534092} we have $w(J/K)w(J^L/K)=\left((-1)^g,L/K\right)$.
\end{proof}
For the purpose of giving examples we now describe how to construct hyperelliptic curves over $\mathbb{Q}$ whose Jacobians satisfy the conditions of \Cref{good ordinary case} over $\mathbb{Q}_2$.
\begin{lemma} \label{constructing ordinary curves in char 2}
Let $f(x)\in\bar{\mathbb{F}}_2[x]$ be a monic separable polynomial of degree $g+1$ and $h(x)\in \bar{\mathbb{F}}_2[x]$ a polynomial of degree $\leq g$, coprime to $f$. Then the Jacobian $J$ of the hyperelliptic curve
\[C:y^2-f(x)y=h(x)f(x)~~/\bar{\mathbb{F}}_2\]
has $2$-torsion group scheme $J[2]\cong (\mathbb{Z}/2\mathbb{Z}\oplus \mu_2)^g$. In particular, $J$ is ordinary.
\end{lemma}
\begin{proof}
Let $\alpha_1,...,\alpha_{g+1}\in \bar{\mathbb{F}}_2$ be the distinct roots of $f$ and $c_1,...,c_{g+1} \in \bar{\mathbb{F}}_2^\times$ be arbitrary. Then by \cite[Theorem 1.3]{MR3095219}, the Jacobian of the hyperelliptic curve
\[y^2-y = \sum_{i=1}^{g+1}\frac{c_i}{x-\alpha_i}\\
= \frac{1}{f(x)}\sum_{i=1}^{g+1}c_i\prod_{j\neq i}(x-\alpha_i)\]
over $\bar{\mathbb{F}}_2$ has 2-torsion group scheme of the required form. Now the polynomials $f_i(x)=\prod_{j\neq i}(x-\alpha_i)$ are $g+1$ in number and of degree $g$. Since the $\alpha_i$ are distinct, it follows that they form a basis for the $\bar{\mathbb{F}}_2$-vector space of polynomials of degree $\leq g$ over $\bar{\mathbb{F}}_2$. In particular, we may choose $c_1,...,c_{g+1}$ such that
\[h(x)=\sum_{i=1}^{g+1}c_if_i(x)\]
and the assumption that $f(x)$ and $h(x)$ are coprime ensures that none of the $c_i$ are zero. Thus the hyperelliptic curve
\[y^2-y=\frac{h(x)}{f(x)} ~~/\bar{\mathbb{F}}_2\]
has $2$-torsion group scheme $ (\mathbb{Z}/2\mathbb{Z}\oplus \mu_2)^g$. A simple change of variables shows that this is the curve $C$ in the statement.
\end{proof}
\begin{cor} \label{explicit ordinary cor}
Suppose $f(x)\in \mathbb{Z}[x]$ has odd leading coefficient and degree $g+1$, and suppose that the reduction of $f$ $\textup{(mod}~ \textup{2)}$ is separable with each irreducible factor having odd degree. Further, let $h(x)\in \mathbb{Z}[x]$ have degree $\leq g$ be such that the reduction of $h$ $\textup{(mod}~ \textup{2)}$ is coprime to that of $f$. Then the Jacobian of the hyperelliptic curve
\[C: y^2=f(x)(f(x)+4h(x))\]
has good ordinary reduction over $\mathbb{Q}_2$, and moreover has all its $2$-torsion defined over an odd degree Galois extension of $\mathbb{Q}_2$.
\end{cor}
\begin{proof}
On easily sees that a change of variables over $\mathbb{Q}_2$ brings $C$ into the form $y^2-f(x)y=h(x)f(x)$ and so the Jacobian has good ordinary reduction over $\mathbb{Q}_2$ by \Cref{constructing ordinary curves in char 2}. Moreover, both $f(x)$ and $f(x)+4h(x)$ reduce to separable polynomials over $\mathbb{F}_2$ whose irreducible factors have odd degree. It follows from Hensel's lemma that $f(x)(f(x)+4h(x))$ splits over an odd degree unramified (and hence cyclic) extension of $\mathbb{Q}_2$ and hence all the $2$-torsion of $J$ is also defined over such an extension.
\end{proof}
\section{Unramified Extensions} \label{main unramified section}
In this section we prove \Cref{Kramer Tunnell} in the case that the quadratic extension $L/K$ is unramified and the residue characteristic of $K$ is odd. The main reason that we can say more when $L/K$ is unramified is that the formation of Neron models and minimal regular models commutes with unramified base-change. This makes the relevant Tamagawa numbers easier to describe and relate to other quantities.
We begin by studying \Cref{Kramer Tunnell} without insisting that the residue characteristic of $K$ is odd, though we will eventually do this. Whilst we only prove a very small number of additional cases of \Cref{Kramer Tunnell} in residue characteristic 2 (see \Cref{genus 2 unramified}), we make a substantial reduction in all residue characteristics (the precise statement of which is \Cref{unramifed reduction}). In particular, we reduce \Cref{Kramer Tunnell} to a statement which only depends on the curve $C$ considered over the maximal unramified extension of $K$. In odd residue characteristic we then give a proof of this.
Fix now a non-archimedean local field $K$ and let $L/K$ be its unique quadratic unramified extension.
\begin{lemma} \label{unram root numbers and norm}
We have
\[w(J/K)w(J^L/K)=(-1)^{\mathfrak{f}(J/K)}\]
and
\[\dim_{\mathbb{F}_2}J(K)/\N J(L)= \dim_{\mathbb{F}_2} \textup{H}^1(k_L/k,\Phi(k_L)),\] where $\mathfrak{f}(J/K)$ denotes the conductor of $J$ and $\Phi$ is the component group of (the special fibre of the Neron model of) $J$.
\end{lemma}
\begin{proof}
For the statement about root numbers see \cite[Proposition 2.4(c)]{MR664648} which proves the result for elliptic curves, and \cite[Corollary A.6]{KES14} which proves a more general statement for arbitrary abelian varieties. The statement about the norm map follows from \Cref{basic properties of norm map} and \cite[Proposition 4.3]{MR0444670}.
\end{proof}
\Cref{unram root numbers and norm} describes two of the terms appearing in \Cref{Kramer Tunnell} and we also note that as $L/K$ is unramified, we have
\[\left((-1)^g\Delta_C,L/K\right)=(-1)^{v_K(\Delta_C)}.\]
To ease notation in subsequent formulas, we define
\[\epsilon(C,K)=\frac{1-i_d(C_K)}{2}\]
so that $\epsilon(C,K)$ is equal to $1$ if $C$ is deficient over $K$, and 0 else. (We have added explicit dependence on $K$ when defining $\epsilon(C,K)$ as we shall shortly wish to vary the base field).
The discussion above shows that \Cref{Kramer Tunnell} for $L/K$ is the assertion that
\begin{equation} \label{unramified kramer tunnell}
\mathfrak{f}(J/K)\equiv v_K(\Delta_C)+\dim_{\mathbb{F}_2} \text{H}^1(k_L/k,\Phi(k_L))+\epsilon(C,K)+\epsilon(C^L,K)~~~\text{ (mod 2)}.
\end{equation}
Since the conductor and valuation are unchanged under unramified extensions, this predicts that the quantity
\begin{equation} \label{unramified norm quantity}
\dim_{\mathbb{F}_2} \text{H}^1(k_L/k,\Phi(k_L))+\epsilon(C,K)+\epsilon(C^L,K)
\end{equation}
is also unchanged modulo 2 upon replacing $K$ by a finite unramified extension $F$, and replacing $L$ by the unique quadratic unramified extension $F'/F$. If $F$ is chosen to be sufficiently large, then $\text{Gal}(\bar{k}/k_F)$ will act trivially on $\Phi(\bar{k})$ whence \[\text{H}^1(k_{F'}/k_F,\Phi(k_{F'}))=\Phi(\bar{k})[2].\]
Moreover, as soon as $F$ contains a quadratic extension of $K$ and $\text{Gal}(\bar{k}/k_F)$ acts trivially on the components of $\mathcal{C}_{\bar{k}}$ (where as usual $\mathcal{C}$ denotes the minimal regular model of $C$ over $\mathcal{O}_K$), it follows from \Cref{deficiency in extensions} and \Cref{min reg def remark} that $\epsilon(C,F)=0$ and $\epsilon(C^{F'},F)$ is equal to 1 if and only if $C$ has even genus and every orbit of the hyperelliptic involution on the odd multiplicity components of $\mathcal{C}_{\bar{k}}$ has even length. To emphasize that this last statement is independent of the field, we set $\epsilon(C)$ to be 1 if this happens, and 0 otherwise. (In fact, since $\mathcal{C}_{\bar{k}}$ coincides with the special fibre of the minimal regular model of $\mathcal{C}$ over $\mathcal{O}_{K^{nr}}$, $\epsilon(C)$ depends only on $C$ through its base change to $K^{nr}$.)
Thus instead of just predicting that \Cref{unramified norm quantity} is unchanged modulo 2 in unramified extensions, we can write down a (so far conjectural) expression for it which clearly has this property. Since we shall subsequently prove this prediction, we state it now as a lemma.
\begin{lemma} \label{unchangedness of norm}
Let $K$ be a local field of characteristic zero and $L/K$ an unramified quadratic extension. Then
\[\dim_{\mathbb{F}_2} \textup{H}^1(L/K,\Phi(k_L))+\epsilon(C,K)+\epsilon(C^L,K) \equiv \dim_{\mathbb{F}_2}\Phi(\bar{k})[2]+ \epsilon(C)~~\textup{ (mod 2)}.\]
\end{lemma}
The proof of \Cref{unchangedness of norm} that we will give is somewhat lengthy and we postpone it to \Cref{proof of compatibility}.
\begin{remark} It is not simply true that \[\dim_{\mathbb{F}_2} \textup{H}^1(L/K,\Phi(k_L)) \equiv \dim_{\mathbb{F}_2}\Phi(\bar{k})[2] ~~\textup{ (mod 2)}\]
and \[\epsilon(C,K)+\epsilon(C^L,K) \equiv \epsilon(C)~~\textup{ (mod 2)}\] individually. Indeed, the genus \textup{2} curve
\[C:y^2=(x^2+3)((x-i)^2-3^2)((x+i)^2-3^2)\]
over $\mathbb{Q}_3$ (here $i$ is a square root of -1 in $\bar{\mathbb{Q}}_3)$ has $\epsilon(C,\mathbb{Q}_3)=\epsilon(C)=0$, yet $\epsilon(C^L,\mathbb{Q}_3)=1$ (where $L=\mathbb{Q}_3(i)$ is the unique quadratic unramified extension of $\mathbb{Q}_3$). This follows easily from the description in \Cref{double roots subsection} of the minimal regular model, along with action of Frobenius, of hyperelliptic curves (in odd residue characteristic) of the form $y^2=f(x)$ where $f(x)$ is monic and has cube free reduction.
\end{remark}
\begin{cor} \label{unramifed reduction}
Let $K$ be a non-archimedean local field, $L/K$ its unique quadratic unramified extension, $C/K$ a hyperelliptic curve and $J/K$ its Jacobian. Then \Cref{Kramer Tunnell} holds for $C$ and the extension $L/K$ if and only if
\begin{equation} \label{unramified claim}
\mathfrak{f}(J/K) \equiv v_K(\Delta_C) +\dim_{\mathbb{F}_2}\Phi(\bar{k})[2]+ \epsilon(C)~~\textup{ (mod 2)}
\end{equation}
where here $\epsilon(C)$ is equal to 1 if $C$ has even genus and every orbit of the hyperelliptic involution on the odd multiplicity components of $\mathcal{C}_{\bar{k}}$ ($\mathcal{C}$ is the minimal regular model of $C$ over $\mathcal{O}_K$) has even length, and 0 else.
\end{cor}
\begin{remark}
In follows from \Cref{first cases,residue characteristic 2} that \Cref{unramified claim} holds for hyperelliptic curves whose Jacobian has good reduction, irrespective of the residue characteristic.
\end{remark}
\subsection{Establishing \Cref{unramified claim} in odd residue characteristic}
Assume now that the residue characteristic of $K$ is odd. Under this assumption, we now establish the congruence \Cref{unramified claim}.
\begin{lemma} \label{KES lemma}
We have
\[\mathfrak{f}(J/K)=\mathfrak{f}(J[2])+\dim_{\mathbb{F}_2}\Phi(\bar{k})[2]\]
where here $\mathfrak{f}(J[2])$ denotes the Artin conductor of $J[2]$.
\end{lemma}
\begin{proof}
This is observed by \v{C}esnavi\v{c}ius in \cite[Lemma 4.2]{KES14}. Note that this requires the assumption that the residue characteristic of $K$ is odd.
\end{proof}
It thus remains to show that
\[\mathfrak{f}(J[2]) \equiv v_K(\Delta_C)+\epsilon(C) ~~ \text{ (mod 2)}.\]
Denote by $K^{\text{nr}}$ the maximal unramified extension of $K$ and let $v$ be the normalised valuation on $K^{\text{nr}}$. As usual, let $C$ be given by the equation $y^2=f(x)$ where $f\in K[x]$ is a separable polynomial of (without loss of generality) even degree $2g+2$ for $g \geq 2$. Let $E/K^{\text{nr}}$ be the field extension $E=K^{\text{nr}}(J[2])$, and set $G=\text{Gal}(E/K)$. As in \Cref{2-tors sect}, $E$ coincides with the splitting field of $f$ over $K^{\text{nr}}$. Let $G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright ...$ be the ramification filtration of $G$, and $g_i=|G_i|$. Thus $G_1$ is the wild inertia group of $E/K^{\text{nr}}$ and is a $p$-group, where $p=\text{char}(k)$ (so in particular has odd order) and $G/G_1$ is cyclic. Let $\W$ denote the $G$-set of roots of $f$ in $E$. Then by definition we have
\[\mathfrak{f}(J[2])=\sum_{i=0}^{\infty} \frac{g_i}{g_0} \textup{codim}_{\mathbb{F}_2}J[2]^{G_i}.\]
\begin{proposition} \label{2-tors conductor}
Define $\epsilon$(f) to be $1$ if the genus $g$ of $C$ is even and each irreducible factor of $f$ over $K^{\text{nr}}$ has even degree, and $0$ else. Then
\[\mathfrak{f}(J[2])\equiv v(\Delta_f)+\epsilon(f) ~~\textup{ (mod 2)}.\]
\end{proposition}
\begin{proof}
This is an immediate consequence of the following 2 lemmas.
\end{proof}
\begin{lemma} \label{comparison of conductors}
Let $\epsilon(f)$ be as above, and let $V=\mathbb{C}[\mathcal{W}]$ be the complex permutation representation for $G$ associated to $\mathcal{W}$. Then we have
\[\mathfrak{f}(J[2])=\mathfrak{f}(V)+\epsilon(f).\]
\end{lemma}
\begin{proof}
This will follow from the definition of $\mathfrak{f}(J[2])$ and $\mathfrak{f}(V)$, along with a comparison between $\text{codim}_\mathbb{C}V^{G_i}$ and $\text{codim}_{\mathbb{F}_2}J[2]^{G_i}$ for each $i$ (afforded by \Cref{two torsion}).
First let $i\geq1$ so that $G_i$ has odd order. Then necessarily $f$ has an odd degree factor over $E^{G_i}$ and it follows from \Cref{two torsion} that $\dim_{\mathbb{F}_2}J[2]^{G_i}=\dim_{\mathbb{C}}V^{G_i}-2$. Since also $\dim_{\mathbb{F}_2}J[2]=\dim_{\mathbb{C}}V-2$, we see that
\[\sum_{i=1}^{\infty}\frac{g_i}{g_0}\text{codim}_{\F_2}J[2]^{G_i}=\sum_{i=1}^{\infty}\frac{g_i}{g_0}\text{codim}_{\mathbb{C}}V^{G_i}\]
and all that remains is to show that
\[\text{codim}_{\F_2}J[2]^G\equiv\text{codim}_{\mathbb{C}}V^G+\epsilon(f)~~~\textup{(mod ~2)}.\]
If $g$ is even, \Cref{two torsion} gives $\dim_{\F_2}J[2]^G=\dim_{\mathbb{C}}V^G-2+\epsilon(f)$ and we are done. Thus suppose that $g$ is odd. If $f$ has an odd degree factor over $K^{\text{nr}}$ then again we conclude immediately from \Cref{two torsion}.
Finally, suppose each irreducible factor of $f$ over $K^{\text{nr}}$ has even degree. By one last application of \Cref{two torsion} it suffices to show that there is a unique quadratic subextension of $E/K^\text{nr}$ over which $f$ factors into $2$ distinct, conjugate polynomials. To see this, first note that there is a unique quadratic subextension of $E/K^\text{nr}$. Indeed, any such extension must necessarily be contained in $E^{G_1}$, yet $E^{G_1}/K^{\text{nr}}$ is cyclic and has even order by the assumption on the degrees of the irreducible factors of $f$ over $K$. To see that $f$ admits the required factorisation over this extension, let $S=\{h_1,...,h_l\}$ be the set of irreducible factors of $f$ over $E^{G_1}$, each of which necessarily has odd degree. The cyclic group $G/G_1$ acts on $S$ and as each factor of $f$ over $K^{\text{nr}}$ has even degree, each orbit of $G/G_1$ on $S$ has even order. Denote these disjoint orbits by $S_1,...,S_k$, and write $S_i=\{h_{i,1},...,h_{i,d_i}\}$. Fix a generator $\sigma$ of $G/G_1$ and assume without loss of generality that $\sigma(h_{i,j})=h_{i,j+1~(\text{mod}~d_i)}$.Then the polynomial
\[h=\prod_{i=1}^{k}\prod_{j~\text{odd}}h_{i,j}\]
is fixed by $\sigma^2$, has $\sigma(h)\neq h$, and $f=h\sigma(h)$.
\end{proof}
\begin{lemma}
Let $K$ be a finite extension of $\mathbb{Q}_p$, $f(x) \in K[x]$ a separable polynomial, $E/K$ its splitting field and $G=\textup{Gal}(E/K)$. Let $R$ be the set of roots of $f(x)$ in $E$, $V=\mathbb{C}[R]$ the corresponding complex permutation module,
\[\mathfrak{f}(V)=\sum_{i=0}^{\infty} \frac{g_i}{g_0} \textup{codim}V^{G_i}\]
the Artin conductor of $V$ as a G-representation,
and $\Delta$ the discriminant of $f(x)$. Then
\[\mathfrak{f}(V) \equiv v_K(\Delta) ~~~\textup{(mod 2)}.\]
\end{lemma}
\begin{proof}
Let the disjoint orbits of $G$ on $R$ be denoted $S_1,...,S_k$, corresponding to the factorisation of $f$ as $f_1...f_k$ into irreducibles over $K$. Then $V$ is a direct sum of the permutation modules $V_i=\mathbb{C}[S_i]$, and $\mathfrak{f}(V)$ is the sum of the $\mathfrak{f}(V_i)$. Let $H_i$ be the stabiliser in $G$ of a (arbitrarily chosen) root $x_i \in S_i$. Then $V_i \cong \mathbb{C}[G/H_i]$ and so by the conductor-discriminant formula \cite[VI.2 corollary to Proposition 4]{MR554237}, $\mathfrak{f}(V_i)=v_K(\Delta_{E^{H_i}/K})$, where $\Delta_{E^{H_i}/K}$ denotes the discriminant of $E^{H_i}/K$. Now as a subfield of $E$ we have $E^{H_i}=K(x_i)$ and consequently $v_K(\Delta_{E^{H_i}/K}) \equiv v_K(\Delta(f_i)) ~\text{(mod 2)}$. Finally, since for polynomials $h_1,h_2$ we have $\Delta(h_1h_2)=\Delta(h_1)\Delta(h_2) \text{Res}(h_1,h_2)^2$, the discriminant of $f$ is, up to squares in K, the product of the discriminants of the $f_i$ (here $\text{Res}(h_1,h_2)$ denotes the resultant of $h_1$ and $h_2$).
\end{proof}
Having established \Cref{2-tors conductor} we now seek to reinterpret the `correction' term $\epsilon(f)$.
\begin{lemma} \label{geom def}
For any sufficiently large finite unramified extension $F/K$, and $F'/F$ the unique unramified quadratic extension, we have $\epsilon(f)=1$ if and only if the quadratic twist $C^{F'}/F$ of $C$ by $F'$ is deficient over $F$. In particular, $\epsilon(f)=\epsilon(C)$ as defined previously.
\end{lemma}
\begin{proof}
The last paragraph of the proof of \Cref{comparison of conductors} applies equally well to the even genus case and shows that $f$ has an odd factorisation over $K^{\text{nr}}$ (since $g$ is even, the polynomial $h$ constructed there is forced to have odd degree). Thus also for every sufficiently large unramified extension $F/K$, $f$ has an odd factorisation over a totally ramified quadratic extension of $F$. By enlarging $F/K$ if necessary, we may also assume that the leading coefficient of $f$ is a norm from this quadratic extension. The result now follows from \Cref{deficiency lemma}.
\end{proof}
\begin{cor} \label{unramified kramer tunnell}
Let $\K$ be a finite extension of $\Q_p$ for $p$ odd, let $\L/\K$ be the unique quadratic unramified extension and $C/\K$ be a hyperelliptic curve. Then \Cref{Kramer Tunnell} holds for $C$ and $\L/\K$.
\end{cor}
\begin{proof}
\Cref{geom def} shows that $\epsilon(f)=\epsilon(C)$ and the result now follows from \Cref{2-tors conductor}, \Cref{KES lemma} and \Cref{unramifed reduction}.
\end{proof}
\begin{remark} \label{genus 2 unramified}
If the genus of $C$ is $2$ then one can hope to establish \Cref{unramified claim}, and hence additional cases of \Cref{Kramer Tunnell}, by using Liu's generalisation to genus \textup{2} of Ogg's formula \cite[Theoreme 1]{MR1302311}. Indeed, by combining Theoreme 1, Theoreme 2 and Proposition 1 of \textit{loc. cit.}, one obtains, independently of the residue characteristic of $K$,
\[f(J/K) \equiv v_K(\Delta_C)+n-1+\frac{d-1}{2}~~\textup{ (mod 2)}\]
where $n$ is the number of irreducible components of $\mathcal{C}_{\bar{k}}$ ($\mathcal{C}$ is the minimal regular model of $C$ over $\mathcal{O}_K$) and $d$ is a more complicated expression involving the minimal regular model and is defined in the statement of Liu's Theoreme 1. In Section 5.2 of \textit{loc. cit.}, Liu computes the term $\frac{d-1}{2}$ in a large number of cases (but not all if the residue characteristic is \textup{2}) depending on the structure of $\mathcal{C}_{\bar{k}}$ (that is, on the `type' of the special fibre as classified in \cite{MR0369362} and \cite{MR0201437}). This includes all cases where $C$, or equivalently $J$, has semistable reduction and it is then easy to establish \Cref{unramified claim} for all semistable curves of genus $2$ from the description, given by Liu in \cite[Section 8]{MR1285783}, of the component group of a genus $2$ curve in terms of its type. Thus \Cref{Kramer Tunnell} holds for unramified quadratic extensions in residue characteristic $2$, and semistable hyperelliptic curves of genus $2$.
\end{remark}
\subsection{Proof of \Cref{unchangedness of norm}} \label{proof of compatibility}
We now turn to proving \Cref{unchangedness of norm}. The main ingredient is the description of $\Phi(\bar{k})$, along with action of $\text{Gal}(\bar{k}/k)$, in terms of the minimal regular model of $C$ (see, for example, \cite[Section 1]{MR1717533}). We begin by summarising this description for general curves over $K$ since we will deduce \Cref{unchangedness of norm} from a result (\Cref{homomorphism theorem}) which does not only apply to hyperelliptic curves, and may be of independent interest.
Let $X$ be a smooth, proper, geometrically integral curve over $K$, let $\mathcal{X}/\mathcal{O}_K$ be its minimal regular model, and let $\mathcal{X}_{\bar{k}}$ denote the special fibre of $\mathcal{X}$, base-changed to $\bar{k}$. Let $\Gamma_i$, $i\in I$, be the irreducible components of $\mathcal{X}_{\bar{k}}$ and let $d_i$ be their multiplicities. Let $\mathbb{Z}[I]$ denote the free $\mathbb{Z}$-module on the $\Gamma_i$ and define $\alpha:\mathbb{Z}[\text{I}]\rightarrow\mathbb{Z}[\text{I}]$ by
\begin{equation*}
\alpha(D)=\sum_{i}(D\cdot\Gamma_i)\Gamma_i
\end{equation*}
(here $D\cdot\Gamma_i$ is the intersection number of $D$ and $\Gamma_i$) and $\beta:\mathbb{Z}[\text{I}]\rightarrow\mathbb{Z}$ by setting $\beta(\Gamma_i)=d_i$ and extending linearly.
The natural action of $\text{Gal}(\bar{k}/k)$ on $\mathcal{X}_{\bar{k}}$ makes $\mathbb{Z}[I]$ into a $\text{Gal}(\bar{k}/k)$-module, and $\alpha$ respects this action. Endowing $\mathbb{Z}$ with trivial $\text{Gal}(\bar{k}/k)$ action, the same is true of $\beta$. Thus both $\text{im}(\alpha)$ and $\ker(\beta)$ become $\text{Gal}(\bar{k}/k)$-modules and $\ker(\beta)\subseteq \text{im}(\alpha)$.
Let $J/K$ be the Jacobian of $X$, and $\Phi$ its component group. We then have \cite[Theorem 1.1]{MR1717533} an exact sequence of $\text{Gal}(\bar{k}/k)$-modules
\begin{equation}
0\longrightarrow \text{im}(\alpha)\longrightarrow \ker(\beta) \longrightarrow \Phi(\bar{k}) \longrightarrow 0.
\end{equation}
(Note that the geometric multiplicities $e_i$ of the components defined in \cite{MR1717533} are all trivial as the residue field of $K$ is perfect.)
Now $\text{Gal}(\bar{k}/k)$ acts on $\mathbb{Z}[I]$ through a finite cyclic quotient, corresponding to a field extension $k'/k$ (for example, we can take $k'$ to be the compositum over $i\in I$ of the fields $k(\Gamma_i)\cap \bar{k}$). Letting $\sigma$ denote a generator of $\text{Gal}(k'/k)$, we have
\[c(J/K)=\left| \Phi(k) \right|=\left| \left(\ker(\beta)/ \text{im}(\alpha)\right)^{\sigma}\right|\]
(where here $c(J/K)$ is the Tamagawa number of $J$).
Now $\sigma$ acts on $\mathbb{Z}[I]$ as a permutation of $I$, commuting with $\alpha$ and $\beta$. Moreover, as in \Cref{min reg def remark}, the curve $X$ is deficient over $K$ if and only if
\[\text{gcd}_{i\in I}\left\{d_i \cdot |\text{ord}_{\sigma}(\Gamma_i)|\right\}\]
does not divide $g-1$, where $g$ is the genus of $X$.
We will obtain \Cref{unchangedness of norm} as a consequence of the following.
\begin{theorem} \label{homomorphism theorem}
Let $\mathfrak{G}$ be the group of all permutations of $I$ commuting with the maps $\alpha$ and $\beta$ and for each $\rho\in \mathfrak{G}$, let $q(\rho)$ be $1$ if $\textup{gcd}_{i\in I}\{d_i \cdot |\textup{orb}_{\rho}(\Gamma_i)|\}$ divides $g-1$, and $2$ otherwise. Then the map
\[D:\mathfrak{G}\rightarrow \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}\]
defined by
\[D(\rho)=\frac{\left|\ker(\beta)/\textup{im}(\alpha)\right|}{\left|\left(\ker(\beta)/\textup{im}(\alpha)\right)^\rho\right|}\cdot q(\rho)\]
is a homomorphism.
\end{theorem}
Before proving \Cref{homomorphism theorem}, we explain how to deduce \Cref{unchangedness of norm} from this.
\begin{proof}[Proof of \Cref{unchangedness of norm}]
Maintaining the notation above, suppose $X=C$ is a hyperelliptic curve over $K$. The hyperelliptic involution $\iota$ on $C$ extends to an automorphism of the minimal regular model of $C$ and may therefore be viewed as an element of $\mathfrak{G}$. Moreover, as the induced automorphism $\iota_{*}$ of the Jacobian of $C$ is multiplication by $-1$, the action on $\Phi$ induced by $\iota \in \mathfrak{G}$ is multiplication by $-1$ also (see the proof of \cite[Theorem 1.1]{MR1717533}). Thus
\[\Phi(\bar{k})[2]=\left(\ker(\beta)/\text{im}(\alpha)\right)^\iota.\]
Letting $L$ denote the unique quadratic unramified extension of $K$, we choose, without loss of generality, the extension $k'/k$ detailed above in such a way that it contains $k_L/k$ as a subextension. Then again denoting by $\sigma$ a generator of $\text{Gal}(k'/k)$, we have
\[\left| \textup{H}^1\left(\text{Gal}(k_L/k),\Phi(k_L)\right) \right|=\left| \frac{\ker\left(1+\sigma | \Phi(\bar{k})^{\sigma^2}\right)}{\text{im}\left(1-\sigma|\Phi(\bar{k})^{\sigma^2}\right)}\right|=\frac{\left|\Phi(\bar{k})^{-\sigma}\right|\cdot \left|\Phi(\bar{k})^{\sigma}\right|}{\left|\Phi(\bar{k})^{\sigma^2}\right|}.\]
(Incidentally, this shows that \Cref{norm map as Tamagawa numbers} continues to hold in residue characteristic 2, as long as the quadratic extension is unramified.) Moreover, we have $\epsilon(C)=\text{ord}_2\left(q(\iota)\right)$, $\epsilon(C,K)=\text{ord}_2\left( q(\sigma)\right)$ and $\epsilon(C^L,K)=\text{ord}_2\left( q(\iota \circ \sigma) \right)$. Indeed, for this last equality, since the formation of minimal regular models commutes with unramified base-change, we may identify the base-change to $\bar{k}$ of the special fibre of the minimal regular model of $C^L$ over $K$ with that of $C$, except now the action of $\text{Gal}(\bar{k}/k)$ has been twisted by the hyperelliptic involution. Finally, \Cref{deficiency in extensions} gives $\epsilon(C,L)=\text{ord}_2\left( q(\sigma^2) \right)=0$. On the other hand, it follows from \Cref{homomorphism theorem} that
\[D(\sigma)D(\iota \circ \sigma)D(\sigma^2)=D(\iota)\]
as elements of $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$. Taking 2-adic valuations of this equation gives the congruence of \Cref{unchangedness of norm}.
\end{proof}
\begin{proof}[Proof of \Cref{homomorphism theorem}]
Fix $\rho\in \mathfrak{G}$, define $d=\text{gcd}_{i\in I} \{d_i\}$ and $d'(\rho)=\text{gcd}_{i\in I} \{|\textup{orb}_{\rho}(\Gamma_i)|d_i\}.$ By \cite[Theorem 1.17]{MR1717533} (or, more precisely, its proof) we have
\[ \left|\left(\ker(\beta)/\textup{im}(\alpha)\right)^\rho\right|q(\rho)=\left|\frac{\ker(\beta)^\rho}{\text{im}(\alpha)^\rho}\right|\cdot \frac{d'(\rho)}{d}.\]
Now \[\ker(\beta)^\rho/\text{im}(\alpha)^\rho\cong \ker\left(\beta:\mathbb{Z}[I]^\rho/(\alpha(\mathbb{Z}[I])^\rho\rightarrow d'(\rho)\mathbb{Z}\right)\]
and applying the snake lemma to the commutative diagram with exact rows
\[
\xymatrix{0\ar[r] & \frac{\mathbb{Z}[I]^{\rho}}{\alpha(\mathbb{Z}[I])^{\rho}}\ar[d]^{\beta_{1}}\ar[r] & \frac{\mathbb{Z}[I]}{\alpha(\mathbb{Z}[I])}\ar[d]^{\beta_{2}} \ar[r] & \frac{\mathbb{Z}[I]}{\alpha(\mathbb{Z}[I])+\mathbb{Z}[I]^{\rho}}\ar[r]\ar[d]^{\beta_{3}} & 0\\
0\ar[r] & d'(\rho)\mathbb{Z}\ar[r] & d\mathbb{Z}\ar[r] & d\mathbb{Z}/d'(\rho)\mathbb{Z}\ar[r] & 0
}
\]
(where each vertical arrow is induced by $\beta$) yields, upon noting that all vertical arrows are surjective,
\[D(\rho)=\frac{dq(\rho)^2}{d'(\rho)}|\ker(\beta_3)|=q(\rho)^2\frac{d^2}{d'(\rho)^2}\left| \frac{\mathbb{Z}[I]}{\alpha(\mathbb{Z}[I])+\mathbb{Z}[I]^{\rho}} \right|.\]
Thus as a function $\mathfrak{G} \rightarrow \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$ we have
\[D(\rho)=\left| \frac{\mathbb{Z}[I]}{\alpha(\mathbb{Z}[I])+\mathbb{Z}[I]^{\rho}} \right|.\]
Now $\rho-1$ yields an isomorphism
\[\frac{\mathbb{Z}[I]}{\alpha(\mathbb{Z}[I])+\mathbb{Z}[I]^{\rho}} \stackrel{\sim}{\longrightarrow}\frac{(\rho-1)\mathbb{Z}[I]}{(\rho-1)\alpha \left(\mathbb{Z}[I]\right)}=\frac{(\rho-1)\mathbb{Z}[I]}{\alpha\left((\rho-1)\mathbb{Z}[I]\right)}\]
where for the last equality we use that $\rho$ commutes with $\alpha$.
Now $(\rho-1)\mathbb{Z}[I]$ is a free $\mathbb{Z}$-module of finite rank and $\alpha$ is a linear endomorphism of this group. By properties of Smith normal form, the order of the group
\[\frac{(\rho-1)\mathbb{Z}[I]}{\alpha\left((\rho-1)\mathbb{Z}[I]\right)}\]
is equal to the absolute value of the determinant of $\alpha$ as a linear map on the $\mathbb{Q}$-vector space $(\rho-1)\mathbb{Q}[I]$. That is, we have shown that
\[D(\rho)=\left|\text{det}\left(\alpha | (\rho-1)\mathbb{Q}[I] \right)\right|.\]
The passage from $\mathbb{Z}$-modules with $\mathfrak{G}$-action to $\mathbb{Q}$-vector spaces with $\mathfrak{G}$-action now allows us to make use of representation theory in characteristic zero. Noting that the matrix representing $\alpha$ on $\mathbb{Q}[I]$ (with respect to the natural permutation basis) is symmetric (it's just the intersection matrix associated to $\mathcal{X}_{\bar{k}}$) we see that the minimal polynomial of $\alpha$ as an endomorphism of $\mathbb{Q}[I]$ splits over $\mathbb{R}$. Moreover, the kernel of $\alpha$ is $\mathbb{Q}[\sum_{i\in I}d_i \Gamma_i]$ which is fixed by $\mathfrak{G}$. The result is now a consequence of the following two lemmas.
\end{proof}
\begin{lemma} \label{rep lemma 1}
Let $G$ be a finite cyclic group, generator $\sigma$, and let $V$ be a $\mathbb{Q}[G]$-representation. Let $\alpha\in \textup{End}_{\mathbb{Q}[G]}V$ be a $G$-endomorphism of $V$ whose minimal polynomial splits over $\mathbb{R}$ and such that $\ker(\alpha) \subseteq V^G$. Then
\[\textup{det}\left(\alpha | (\sigma-1)V \right)=\textup{det}\left(\alpha | V_{-1,\sigma}\right)\]
where $V_{-1,\sigma}$ is the $(-1)$-eigenspace for $\sigma$ on $V$. (Here if $V_{-1,\sigma}=0$ we define $\textup{det}\left(\alpha | V_{-1,\sigma}\right)=1$.)
\end{lemma}
\begin{proof}
Let $V=\bigoplus_{i=1}^{n}V_i^{d_i}$ be an isotypic decomposition of $V$, so that each $V_i$ is an irreducible $\mathbb{Q}[G]$-representation and $V_i\ncong V_j$ for $i\neq j$. Suppose, without loss of generality, that $V_1$ is the trivial representation. Then $\alpha$ preserves this decomposition and $(\sigma-1)V=\bigoplus_{i=2}^{n}V_i^{d_i}$. By assumption, we see that the restriction of $\alpha$ to each $V_i^{d_i}$ is non-singular. Thus we are reduced to showing that if $V=W^d$ for an irreducible $\mathbb{Q}[G]$-representation, $\chi$ is the character of a complex irreducible constituent of $W$, and $\chi$ is non-real (so $\chi(\sigma)\notin \{\pm 1\}$), then $\textup{det}\left(\alpha \right) \in \mathbb{Q}^{\times 2}$. Now $A=\text{End}_{\Q[G]}V\cong \text{M}_d\left(\text{End}_{\Q[G]}W\right)$ is a finite dimensional simple algebra over $\Q$. Set $D=\text{End}_{\Q[G]}W$ so that $D$ is a division algebra. Let $K/\Q$ be the centre of $D$. Note that if $\chi$ is the character of a complex irreducible component of $W$ then (up to isomorphism over $\Q$) we have $K\cong \Q(\chi)$ where $\Q(\chi)$ is the character field of $\chi$ (see, for example, \cite{MR0122892} for proofs of representation theoretic facts used). Note that $K/\Q$ is abelian.
Now via the diagonal embedding of $K$ into $\text{End}_{\Q[G]}V$, $V$ becomes a $K[G]$-module. Since $K$ is the centre of $\text{End}_{\Q[G]}V$, each $\Q[G]$-endomorphism of $V$ is in fact $K$-linear so the natural inclusion $\text{End}_{K[G]}V \subseteq \text{End}_{\Q[G]}V$ is an equality. In particular, we may view $\alpha$ as a $K[G]$-endomorphism of $V$. Let $\text{det}_K$ denote the determinant of a $K$-endomorphism of $V$ and $\text{det}_{\Q}$ denote the determinant of the same endomorphism now viewed as a $\mathbb{Q}$-endomorphism. Then we have
\[\text{det}_{\Q} \left( \alpha \right)=N_{K/\Q} \left(\text{det}_{K}\left(\alpha \right)\right)\]
(see, for example, \cite[Theorem A.1]{MR0222054} ).
As $K$ in not totally real, there is an index $2$ totally real subfield $K^{+}$ of $K$. I claim that for each $\sigma \in G$, $\text{det}\left(\alpha \right)$ is in $K^{+}$. Indeed, since the minimal polynomial of $\alpha$ as a $\mathbb{Q}$-endomorphism of $V$ splits over $\mathbb{R}$, each root of the minimal polynomial of $\alpha$ as a $K$-endomorphism of $V$ is totally real. It follows that $\text{det}\left(\alpha \right)$ is a product of totally real numbers and hence in $K^{+}$. Thus
\[\text{det}_{\Q}( \alpha)=N_{K/\Q} \left(\text{det}_{K}(\alpha)\right)=\left(N_{K^+/\Q}(\text{det}_{K}(\alpha))\right)^2\in \Q^{\times 2}\]
as desired.
\end{proof}
\begin{lemma}
Let $G$ be a finite group and $V$ a $\mathbb{Q}[G]$-representation. Let $\alpha\in \textup{End}_{\mathbb{Q}[G]}V$ be a $G$-endomorphism of $V$ whose minimal polynomial splits over $\mathbb{R}$ and such that $\ker(\alpha) \subseteq V^G$. For each $\sigma \in G$, let $V_{-1,\sigma}$ denote the $(-1)$-eigenspace for $\sigma$ on $V$. Then the function
\[\phi:G\rightarrow \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}\]
defined by
\[\phi(\sigma)=\textup{det}\left(\alpha | V_{-1,\sigma}\right)\]
is a homomorphism. (Here if $V_{-1,\sigma}=0$ we define $\phi(\sigma)=1$.)
\end{lemma}
\begin{proof}
We begin similarly to the proof of \Cref{rep lemma 1}. By considering an isotypic decomposition of $V$ we may assume that $V=W^d$ where $W$ is an irreducible $\mathbb{Q}[G]$-representation and that $\alpha$ is non-singular. Let $A=\text{End}_{\Q[G]}V\cong \text{M}_d\left(\text{End}_{\Q[G]}W\right)$, a finite dimensional simple algebra over $\Q$, let $D$ be the division algebra $D=\text{End}_{\Q[G]}W$, and $K/\Q$ be the centre of $D$. Again, if $\chi$ is the character of a complex irreducible component of $W$ then (up to isomorphism over $\Q$) we have $K\cong \Q(\chi)$, and $K/\Q$ is abelian.
As before we view $V$ as a $K[G]$-module and note that the natural inclusion $\text{End}_{K[G]}V \subseteq \text{End}_{\Q[G]}V$ is an equality so that we may view $\alpha$ as a $K[G]$-endomorphism of $V$. For all $\sigma \in G$ we have
\[\text{det}_{\Q} \left( \alpha | V_{-1,\sigma} \right)=N_{K/\Q} \left(\text{det}_{K}\left(\alpha | V_{-1,\sigma}\right)\right).\] That is, if $\phi_{\mathbb{Q}}$ is the map $\phi$ defined previously and $\phi_K$ is the function $G\rightarrow K^{\times}/K^{\times2}$ defined by
\[\sigma \mapsto \text{det}_K\left(\alpha | V_{-1,\sigma}\right),\]
we have $\phi_\mathbb{Q}=N_{K/\mathbb{Q}}\circ\phi_K$.
First suppose that $K$ in not totally real. Then there is an index $2$ totally real subfield $K^{+}$ of $K$ and as in the proof of \Cref{rep lemma 1}, for each $\sigma \in G$, $\text{det}\left(\alpha | V_{-1,\sigma}\right)$ is in $K^{+}$. Thus for each $\sigma\in G$,
\[\phi_{\mathbb{Q}}(\sigma)=N_{K/\mathbb{Q}}\phi_K(\sigma)=\left(N_{K^{+}/\mathbb{Q}}\phi_K(\sigma)\right)^2 \in \mathbb{Q}^{\times 2}.\]
Thus $\phi_{\mathbb{Q}}$ is trivial in this case.
We may now assume that $K$ is totally real, or equivalently that $\chi$ is real valued. Let $m$ be the Schur index of $\chi$ (over $\mathbb{Q}$ or equivalently $K$). Suppose first that $\chi$ is realisable over $\mathbb{R}$. Then, via a chosen embedding $K\hookrightarrow \mathbb{R}$, we have $V\otimes_K \mathbb{R} \cong U^{md}$ for some irreducible real representation $U$. Fix $\sigma \in G$. Then $V_{-1,\sigma} \otimes_{K} \mathbb{R} = (V \otimes_{K} \mathbb{R})_{-1,\sigma}$ and viewing $\alpha$ as an element of $\text{End}_{\mathbb{R}[G]}\left(U^{md}\right)\cong \text{M}_{md}(\mathbb{R})$ we wish to compute the determinant of $\alpha$ on $(U_{-1,\sigma})^{md}$. Viewing $\alpha$ as a $md \times md$ matrix $M$ over the reals via the identification discussed above, we see that the required determinant is given by $\text{det}(M)^{\text{dim}U_{-1,\sigma}}$. In fact, one sees that $\text{det}(M)$ is equal to $\text{Nrd}(\alpha)\in K^{\times}$ where here $\text{Nrd}$ denotes the reduced norm on the central simple algebra $A=\text{End}_{\Q[G]}V$ over $K$. Thus to show that $\phi_K$ is a homomorphism (and hence $\phi_\Q$ as $N_{K/\mathbb{Q}}$ is), we want to show that the congruence
\[\text{dim}U_{-1,\sigma}+\text{dim}U_{-1,\tau}\equiv\text{dim}U_{-1,\sigma \tau}~~\text{(mod 2)}\]
holds for all $\sigma$ and $\tau$ in $G$. However, $U$ is a real vector space and each $\sigma \in G$ acts on $U$ as a finite order matrix which is hence diagonalisable over $\mathbb{C}$. Base-changing to $\mathbb{C}$, diagonalising $\sigma$ and noting that the eigenvalues of $\sigma$ are roots of unity appearing in conjugate pairs, one sees that for each $\sigma \in G$ we have
\[\text{det}(\sigma)=(-1)^{\text{dim}U_{-1,\sigma}},\]
which proves the desired congruence.
Finally, suppose that $\chi$ is not realisable over $\mathbb{R}$. Then we have $V\otimes_K \mathbb{\mathbb{C}} \cong U^{md}$ where $U$ is an irreducible representation over $\mathbb{C}$ and, by assumption, $U$ and hence $U^{md}$ possesses a non-degenerate $G$-invariant alternating form, which we denote by $\left\langle \cdot ,\cdot \right\rangle$. The argument for the previous case again gives $\text{det}_K\left(\alpha | V_{-1,\sigma}\right)= \text{Nrd}(\alpha)^{\text{dim}U_{-1,\sigma}}$. I claim that now $\text{dim}U_{-1,\sigma}$ is even for each $\sigma \in G$, from which it follows that $\phi_K$, and hence $\phi_\Q$, is trivial. Indeed, fix $\sigma\in G$. We may suppose that $\sigma$ has even order $2k$ else $U_{-1,\sigma}=0$. One easily verifies that the linear map
\[\pi=\frac{1}{2k}\sum_{i=0}^{2k-1}(-1)^i \sigma^i\]
gives a projection from $U$ onto $U_{-1,\sigma}$. Thus we may write $U=U_{-1,\sigma} \oplus \ker(\pi)$. Now let $u \in U_{-1,\sigma}$ and $w \in \ker(\pi)$. Then we have
\begin{eqnarray*}
\left\langle u,w\right\rangle & = & \frac{1}{2k}\sum_{i=0}^{2k-1}\left\langle \sigma^{i}u,\sigma^{i}w\right\rangle =\left\langle u,\frac{1}{2k}\sum_{i=0}^{2k-1}(-1)^{i}\sigma^{i}w\right\rangle =\left\langle u,\pi w\right\rangle =0.
\end{eqnarray*}
Thus $U_{-1,\sigma}$ and $\text{ker}(\pi)$ are orthogonal with respect to $\left\langle \cdot ,\cdot \right\rangle$ and so the pairing must be non-degenerate on each of them. In particular, $U_{-1,\sigma}$ admits a non-degenerate alternating pairing and hence has even dimension.
This completes the proof of the lemma.
\end{proof}
The proof of \Cref{homomorphism theorem} facilitates the computation of Tamagawa numbers of hyperelliptic curves, at least up to squares, and to end the section we record this in the following proposition.
\begin{proposition} \label{tam computations}
Let $\K$ be a finite extension of $\Q_p$, $X/K$ a smooth, proper, geometrically integral curve of genus $g$, $\mathcal{X}/\mathcal{O}_\K$ its minimal regular model and $\mathcal{X}_{\bar{k}}$ the special fibre of $\mathcal{X}$, base-changed to $\bar{k}$. Let $\Phi$ be the component group (of the Jacobian of) $X$ and $I=\{\Gamma_1,...,\Gamma_n\}$ be the (set of) irreducible components of $\mathcal{X}_{\bar{k}}$. For each $\Gamma_i$ let $d(\Gamma_i)$ be its multiplicity in $\mathcal{X}_{\bar{k}}$. Write $d=\textup{gcd}_{i\in I}\{d(\Gamma_i)\}$, $d'=\textup{gcd}_{i\in I}\{d(\Gamma_i) r(\Gamma_i)\}$ and define $q$ to be $2$ if $d'$ does not divide $g-1$, and $1$ otherwise. Finally, let $S_1,...,S_k$ be the even sized orbits of $\textup{Gal}(\bar{k}/k)$ on $\{\Gamma_1,...,\Gamma_n\}$, let $r_i=|S_i|$ and for each $1 \leq i \leq k$, write
\[\epsilon_i=\sum_{i=0}^{r_i-1}(-1)^i \sigma^i (\Gamma_{i,1})\]
where $\sigma \in \textup{Gal}(\bar{k}/k)$ denotes the Frobenius element and $\Gamma_{i,1}$ is a representative of the orbit $S_i$.
Then
\[q\frac{|\Phi(\bar{k})|}{|\Phi(k)|} \equiv \left| \textup{det}\left(\frac{1}{r_j} \left\langle \epsilon_{i} ,
\epsilon_{j}\right\rangle \right)_{1\leq i,j \leq k}\right|~~(\textup{mod }~\mathbb{Q}^{\times 2})\]
where $\left\langle \cdot, \cdot \right \rangle$ denotes the intersection pairing on $\mathcal{X}_{\bar{k}}$.
\end{proposition}
\begin{proof}
The (proof of) \Cref{homomorphism theorem}, along with \Cref{rep lemma 1}, gives
\[q\frac{|\Phi(\bar{k})|}{|\Phi(k)|} \equiv \left| \textup{det}\left(\alpha | \Q[I]_{-1}\right)\right| ~~(\text{mod }~\mathbb{Q}^{\times 2})\]
where $\mathbb{Q}[I]_{-1}$ denotes the $(-1)$-eigenspace of $\text{Gal}(\bar{k}/k)$ on the permutation module $\Q[I]$ and $\alpha$ is the linear map given by $\Gamma_i \mapsto \sum_{j}(\Gamma _i \cdot\Gamma_j)\Gamma_j $. One easily sees that $\{\epsilon_1,...,\epsilon_k\}$ forms a basis for $\Q[I]_{-1}$ (so the dimension of this space is the number of even sized orbits on $\{\Gamma_1,...,\Gamma_n\}$). Moreover, using $\text{Gal}(\bar{k}/k)$-invariance of the intersection pairing, one computes
\[\alpha(\epsilon_i)=\sum_{j=1}^{k}(\epsilon_i \cdot \Gamma_{j,1})\epsilon_j=\sum_{j=1}^{k}\frac{1}{r_j}(\epsilon_i \cdot \epsilon_j)\epsilon_j\]
and the result follows.
\end{proof}
\section{Ramified Extensions; Generalities} \label{min reg model ramified}
Let $L/K$ be a totally ramified extension of local fields where $K$ has odd residue characteristic. To complete the proof of \Cref{cases of the parity conjecture} it remains to prove \Cref{Kramer Tunnell} (with respect to $L/K$) when $C$ is given by an equation of the form $y^2=f(x)$ where $f$ is integral, has unit leading coefficient and the reduction of $f$ is cube free (to go from an arbitrary leading coefficient to a unit one we apply \Cref{odd degree extension}). By \Cref{norm map as Tamagawa numbers} we have
\[\dim_{\mathbb{F}_2} J(K)/\N J(L) = \text{ord}_2 \frac{c(J/K)c(J^L/K)}{c(J/L)}.\]
Moreover, the assumptions on $f$ mean that $C$ is semistable over $K$ (see, for example, \cite[Example 10.3.29]{MR1917232}). We begin by describing a method for computing the ratio $\frac{c(J/L)}{c(J/K)}$, at least up to squares, for general semistable curves and then apply it to our particular case. Secondly, we compute $c(J^L/K)$ (again up to squares) by analysing the minimal regular model of the quadratic twist $C^L$ of $C$ by $L$. Since $C^L$ will no longer be semistable, we use results from \Cref{main unramified section} instead.
As we shall see, the terms of \Cref{Kramer Tunnell} involving deficiency and root numbers will naturally appear as part of the computations detailed above.
\subsection{The Minimal Regular Model of a Semistable Curve} \label{min reg model}
Suppose $C/K$ has semistable reduction over $K$ and denote by $\mathcal{C}/\mathcal{O}_K$ its minimal regular model over $\mathcal{O}_K$, and by
$\mathcal{C}_{\bar{k}}$ the special fibre of $\mathcal{C}$, base changed to $\bar{k}$. If $\mathcal{C}'$ denotes the minimal regular model of $C$ over $K^\text{nr}$, then $\mathcal{C}_{\bar{k}}$ is the special fibre of $\mathcal{C}'$. The following two paragraphs and \Cref{lattice correspondence} essentially summarise \cite[Section 3.4]{MP2013}, to which we refer for more details. The only difference is that we wish to consider in addition the action of $\text{Gal}(\bar{k}/k)$ on the objects involved.
The set $S$ of singular points, and $I$ of irreducible components, of $\mathcal{C}_{\bar{k}}$ both carry natural actions of $\text{Gal}(\bar{k}/k)$ and we define the \textit{dual graph} $\mathcal{G}$ of $\mathcal{C}_{\bar{k}}$ to be the graph whose vertices are the irreducible components of $\mathcal{C}_{\bar{k}}$ and such that $\Gamma_{1}$, $\Gamma_2$ $\in I$ are joined by one edge for each singular point of $\mathcal{C}_{\bar{k}}$ lying on both $\Gamma_1$ and $\Gamma_2$ (thus $G$ may have both loops and multiple edges).
Associated to $\mathcal{G}$ we have an exact sequence (coming from simplicial homology)
\[
0\rightarrow H_1\left(\mathcal{G},\mathbb{Z}\right) \longrightarrow \bigoplus_{x\in S} \mathbb{Z}[x] \stackrel{\partial}{\longrightarrow} \bigoplus_{\Gamma \in I} \mathbb{Z}[\Gamma] \stackrel{\epsilon}{\longrightarrow} \mathbb{Z} \rightarrow 0.
\]
The map $\epsilon$ sends $\sum_{\Gamma \in I} n_\Gamma \Gamma$ to $\sum_{\Gamma \in I} n_\Gamma$. To define the map $\partial : \bigoplus _{x\in S} \mathbb{Z}[x] \rightarrow \bigoplus_{\Gamma \in I} \mathbb{Z}[\Gamma]$, we fix once and for all an orientation on the edges of $\mathcal{G}$. Then if $x \in S$ gives rise to an edge $e$ of $\mathcal{G}$ from $\Gamma_1$ to $\Gamma_2$, we set $\partial(x)=\Gamma_2-\Gamma_1$. Then $H_1(\mathcal{G},\mathbb{Z})$, as defined by the exact sequence, is a free $\mathbb{Z}$-module of rank $\beta(\mathcal{G}):=|S|-|I|+1$, which coincides with the toric rank of $\mathcal{C}$. We view $\bigoplus_{\Gamma \in I} \mathbb{Z}[\Gamma]$ as a $\text{Gal}(\bar{k}/k)$-module by extending $\mathbb{Z}$-linearly the action of $\text{Gal}(\bar{k}/k)$ on $I$. Moreover, we view $\bigoplus_{x\in S} \mathbb{Z}[x]$ as a $\text{Gal}(\bar{k}/k)$-module in the same way, except we include $\pm$ signs to take account of the orientation of the edges. Thus the $\text{Gal}(\bar{k}/k)$-action on $I$ and $S$ determines the action on $\bigoplus_{x\in S} \mathbb{Z}[x]$ save in the case where there are loops in the graph. To define the action here, note that each loop corresponds to a node lying on a single component. To each such singular point $x$ we associate the two tangents $t_x^{\pm}$ where we make a choice as to which tangent we associate the sign `+' to. Then if $\sigma \in \text{Gal}(\bar{k}/k)$ maps $x$ to $x'$, $x'$ is also a node and $\sigma$ maps the tangents $t_x^{\pm}$ at $x$ onto the tangents $t_{x'}^{\pm}$ at $x'$. If $t_x^{+}$ is mapped by $\sigma$ to $t_x'^{+}$ then we associate a `+' sign in the action on $\bigoplus_{x\in S} \mathbb{Z}[x]$, and associate a `-' sign otherwise. Whilst this depends on a choice of which tangent to identify which signs to, the resulting $\mathbb{Z}[\text{Gal}(\bar{k}/k)]$-module structure on $\bigoplus_{x\in S} \mathbb{Z}[x]$ is well-defined up to isomorphism, and the same is true for the choice of orientation on the edges of $\mathcal{G}$. The exact sequence above then becomes $\text{Gal}(\bar{k}/k)$-equivariant and $H_1(\mathcal{G},\mathbb{Z})$ inherits a natural action of $\text{Gal}(\bar{k}/k)$.
Now define a pairing on $\bigoplus _{x\in S} \mathbb{Z}[x]$ by setting
\[ \left\langle s_i , s_j \right \rangle = \begin{cases} 1 & ~~ i=j \\ 0 & ~~\text{else} \end{cases} \]
and extending bilinearly. The restriction of this pairing to $H_1(\mathcal{G},\mathbb{Z})$ induces a $\mathbb{Z}$-valued symmetric non-degenerate $\text{Gal}(\bar{k}/k)$-equivariant pairing on $H_1(\mathcal{G},\mathbb{Z})$. Denote by $\Lambda$ the dual lattice of $H_1(\mathcal{G},\mathbb{Z})$ inside $H_1(\mathcal{G},\mathbb{Z})\otimes \mathbb{Q}$. We will henceforth denote $H_1(\mathcal{G},\mathbb{Z})$ by $\Lambda^{\vee}$. Note that $\Lambda^{\vee} \subseteq \Lambda $.
\begin{theorem} \label{lattice correspondence}
Let $J/K$ be the Jacobian of $C$ and $X(T)$ the character group of the toric part of the Raynaud parametrisation of $J$, so that $X(T)$ is a free $\mathbb{Z}$-module of rank equal to the toric rank of $J$. $X(T)$ carries a natural action of $\textup{Gal}(K^\textup{nr}/K)\cong \textup{Gal}(\bar{k}/k)$ and the monodromy pairing gives a symmetric, bilinear, non-degenerate pairing on $X(T)$. Then
\begin{itemize}
\item [{\textit{(i)}}] $X(T)$ is isomorphic to $H_1(\mathcal{G},\mathbb{Z})$ as $\mathbb{Z}[\textup{Gal}(\bar{k}/k)]$-lattices equipped with a pairing.
\item [{\textit{(ii)}}] If $F/K$ it totally ramified of degree $e$, then $\Phi_F(\bar{k})$ is isomorphic to $\frac{\Lambda}{e\Lambda^{\vee}}$ as $\text{Gal}(\bar{k}/k)$-modules.
\item [{\textit{(iii)}}]
Suppose $J$ attains split semistable reduction over the (without loss of generality) unramified extension $E/K$ (so that $\textup{Gal}(K^{nr}/K)$ acts on $H_1(\mathcal{G},\mathbb{Z})$ through $\textup{Gal}(E/K)$), let $F/K$ be a Galois extension containing $E$ and $\tau$ a complex representation of $\textup{Gal}(F/K)$. Then we have
\[w(J/K,\tau)=w(\tau)^{2g}(-1)^{\left\langle \tau,H_1(G,\mathbb{Z}) \otimes \mathbb{C}\right\rangle }\]
where $w(J/K,\tau)$ denotes the root number of $J/K$ twisted by $\tau$, $w(\tau)$ denotes the root number of the representation $\tau$ of $\text{Gal}(F/K)$ and $\left\langle~,~\right\rangle$ denotes the usual representation-theoretic inner product.
\end{itemize}
\end{theorem}
\begin{proof}
Part (i) follows from \cite[Theorem 3.8]{MP2013} and the references therein (see, in particular, \cite[Section 9.2]{MR1045822}). For parts (ii) and (iii), see \cite[Section 3.v]{MR2534092}.
\end{proof}
\subsection{Computing the ratio $\frac{c(J/L)}{c(J/K)}$ }
Let $\Lambda^{\vee}$ and $\Lambda$ be as in the previous section. By \Cref{lattice correspondence} we have
\[ \frac{c(J/L)}{c(J/K)} =\frac{ \left|\left(\frac{\Lambda}{2\Lambda^{\vee}}\right)^{F}\right|}{\left|\left(\frac{\Lambda}{\Lambda^{\vee}}\right)^{F}\right|} ,\]
where here $F$ denotes the Frobenius automorphism viewed as a finite order endomorphism of the lattice $\Lambda$. Let $D=F-1$ and $N=1+F+F^{2}+...+F^{n-1}$, where $n$ is the order of $F$. Moreover, set $V=\Lambda\otimes\mathbb{Q}$, let $G$ be the finite cyclic subgroup of $\text{Aut}(\Lambda)$ generated by $F$, and define the group
\[\mathcal{B}=\mathcal{B}_{\Lambda,\Lambda^{\vee}}:= \text{im}\left( \text{H}^1 \left(G,\Lambda^{\vee}\right) \longrightarrow \text{H}^1 \left(G,\Lambda\right) \right).\]
The following Theorem is due to Betts and Dokchitser \cite{BD2012}.
\begin{theorem} \label{lattice behaviour}
Let $e\geq1$. Then we have
\[
\left|\left(\frac{\Lambda}{e\Lambda^{\vee}}\right)^{F}\right|=\left|\left(\frac{\Lambda}{\Lambda^{\vee}}\right)^{F}\right|\cdot\left|\mathcal{B}[e]\right|\cdot e^{r}
\]
where $r:=\textup{rk}\Lambda^{F}.$\end{theorem}
\begin{proof}
The group $\mathcal{B}$ is introduced by Betts and Dokchitser in \cite{BD2012} (in part with the purpose of studying the ratio of Tamagawa numbers that we are also interested in). Whilst
their definition of $\mathcal{B}$ differs slightly from ours, the equivalence of the two definitions in our context follows from \cite[Lemma 2.3.6]{BD2012} upon noting that, since $V$ is uniquely divisible, $\text{H}^1(G,\Lambda^{\vee})$ is trivial whence $\ker(N|V)=\text{im}(D|V)$. The statement of the theorem now follows immediately from \cite[Theorem 1.1.1]{BD2012}.
\end{proof}
\begin{cor} \label{betts group cor}
We have
\[(-1)^{\textup{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)((-1)^g,L/K)=(-1)^{\dim_{\mathbb{F}_2}\mathcal{B}[2]}.\]
\end{cor}
\begin{proof}
\Cref{lattice correspondence} gives $w(J/K)=(-1)^r$ where $r$ is $\textup{rk}\Lambda^{F}$ and $w(J^L/K)=((-1)^g,L/K)$. Now apply \Cref{lattice behaviour} with $e=2$.
\end{proof}
The following lemma will be useful in the computation of the group $\mathcal{B}$.
\begin{lemma} \label{poly lattice lemma}
Fix $r\geq1$ and let $p(x)\in \mathbb{Z}[x]$ be a monic polynomial dividing $x^r-1$. Let $G$ be a cyclic group of order $r$, generator $\sigma$, and suppose that $G$ acts on the free $\mathbb{Z}$-module $\Lambda = \frac{\mathbb{Z}[x]}{(p(x))}$ with $\sigma$ acting as multiplication by $x$. Then $\text{H}^1\left(G,\Lambda\right)$ is cyclic of order $p(1)$, generated by $1\in \mathbb{Z}[x]$.
\end{lemma}
\begin{proof}
We have
\[\text{H}^1\left(G,\Lambda\right)=\frac{\ker(1+x+...+x^{r-1})}{\text{im}(x-1)},\]
and it's clear that
\[\ker\left(1+x+...+x^{r-1}\right)=\frac{\frac{p(x)}{\text{gcd}(p(x),1+x+...+x^{r-1})}\mathbb{Z}[x]}{p(x)\mathbb{Z}[x]}.\]
Since $p(x) \mid x^r-1$, we have
\[\text{gcd}(p(x),1+x+...+x^{r-1})= \begin{cases} p(x) & p(1)\neq0 \\ p(x)/(x-1) & p(1)=0\end{cases}.\]
Thus
\[\text{H}^1\left(G,\Lambda\right)= \begin{cases} \frac{\mathbb{Z}[x]}{(x-1,p(x))} \cong \mathbb{Z}/p(1)\mathbb{Z} & p(1)\neq0 \\ 0 & p(1)=0\end{cases}\]
as desired.
\end{proof}
\section{Ramified Extensions; The Case of Cube Free Reduction} \label{cube free}
Let $K$ be a finite extension of $\mathbb{Q}_p$ for $p$ odd and let $\L/\K$ be a ramified quadratic extension. Let $C/\K$ be a hyperelliptic curve. We now apply the results of the previous section to compute the ratio $\frac{c(J/L)}{c(J/K)}$ (up to squares) in the case that $C$ is given by an integral equation $y^2=f(x)$ where $f$ has unit leading coefficient and cube free reduction. We'll begin by fixing some notation.We assume henceforth the degree is even to ease notation. The odd degree case follows from an easy adaptation of the arguments of this section (in fact, it is substantially easier as, amongst other things, deficiency never enters in this case).
Let $C$ be given by an equation
\[C:y^2=f(x).\]
Let $u\in \mathcal{O}_{\K}^\times$ be the leading coefficient of $f$. The assumption on the reduction $\bar{f}(x)$ of $f(x)$ means that, over $\bar{k}$, we may write
\[\bar{f}(x)=\bar{u}\prod_{i=1}^{l}(x-\bar{u}_i)^2\prod_{j=l+1}^{2(g-2l+1)}(x-\bar{w}_j)\]
where the $\bar{u}_i$ and $\bar{w}_j$ are all pairwise distinct. Since we may lift coprime factorisations over $\bar{k}$ to $\K^{nr}$ by Hensel's lemma, after completing the square of each (lifted) quadratic factor, we may factor $f(x)$ over $\K^{nr}$ as
\begin{equation} \label{explicit equation}
f(x)=u\prod_{i=1}^l \left((x-u_i)^2-v_i\pi_{\K}^{n_i}\right)\prod_{j=l+1}^{2(g-l+1)}(x-w_j)
\end{equation}
where each $n_i\geq 1$, each $u_i$ reduces to $\bar{u}_i$, each $w_i$ reduces to $\bar{w}_i$ and each $v_i\in \mathcal{O}_{\K^{\text{nr}}}^\times$.
For each double root $\bar{u}_i$ of $\bar{f}$, we associate the two `tangents'
\[t_{i}^{\pm}=\pm \sqrt{g_i(\bar{u}_i)}\in \bar{k}\]
where \[g_i(x)=\bar{u}\prod_{j\neq i}(x-\bar{u}_j)^2 \prod_{j=l+1}^{2(g-l+1)}(x-\bar{w}_j)\in \bar{k}[x].\]
By convention, we choose the square roots so that, for all $\sigma \in \text{Gal}(\bar{k}/k)$, if $\sigma (\bar{u}_i)=\bar{u}_j$ and $i\neq j$, then $\sigma (t_i^+)=t_j^+$ (this is possible as $\text{Gal}(\bar{k}/k)$ is procyclic).
The two schemes
\[U_1=\text{Spec}\frac{\mathcal{O}_{\K}[x,y]}{(y^2-f(x))}\]
and
\[U_2=\text{Spec}\frac{\mathcal{O}_{\K}[u,v]}{(v^2-h(u))}\]
where $h(u)=u^{2g+2}f(1/u)$ glue via the relations $x=1/u$ and $y=x^{g+1}v$ to define a proper model $\mathcal{C}_0$ of $C$ over $\mathcal{O}_\K$. Moreover, the assumption on the reduction of $f(x)$ means that this model is semistable and, in fact, is the unique (up to isomorphism) stable model of $C$ over $\mathcal{O}_\K$ (see \cite[Example 10.3.29]{MR1917232}). Now consider the base change, $\mathcal{C}_{0,\K^{nr}}$, of $\mathcal{C}_0$ to $\K^{nr}$. The special fibre is smooth away from the singular points $(x,y)=(\bar{u}_i,0)$. If every root of $\bar{f}$ is a double root (i.e. if $l=g+1$) then the special fibre consists of $2$ irreducible components, intersecting in the $g+1$ points $(\bar{u}_i,0)$. On the other hand, if $l<g+1$ so that $\bar{f}$ has at least one single root, then the special fibre is irreducible, and each of the $l$-many points of the form $(\bar{u}_i,0)$ is a node. The scheme $\mathcal{C}_{0,\K^{nr}}$ is regular, save possibly at the singular points $(\bar{u}_i,0)$ on the special fibre. In fact, these points are easily seen to be regular if and only if $n_i=1$. To obtain a regular model of $C$ (over $\mathcal{O}_{\K^{nr}}$), one needs to blow up ($n_i-1$)-times at each of the singular points, which serves to replace these points with a chain of $n_i-1$ copies of $\mathbb{P}^1_{\bar{k}}$. This gives a regular model $\mathcal{C}_{\K^{nr}}$ of $C$ over $\mathcal{O}_{\K^{nr}}$ which is easily seen to be the minimal regular model. Moreover, each of the copies of $\mathbb{P}^1_{\bar{k}}$ introduced have multiplicity $1$ in the special fibre, self-intersection -2, and all their intersections with other irreducible components are transversal.
Since the form of the minimal regular model differs depending on whether or not each root of $\bar{f}$ is a double root, we now split into cases.
\subsection{The case where $\bar{f}(x)$ has at least one single root} \label{single root}
Keep the notation of the previous discussion, and suppose further that the reduction $\bar{f}\in k[x]$ of $f$ has at least one single root in $\bar{k}$. If $\mathcal{C}$ denotes the minimal regular model of $C$ over $\mathcal{O}_\K$, and $\mathcal{C}_{\bar{k}}$ denotes its special fibre, base-changed to $\bar{k}$, then, as remarked previously, $\mathcal{C}_{\bar{k}}$ coincides with the special fibre of the minimal regular model of $C$ over $\K^{nr}$. Thus $\mathcal{C}_{\bar{k}}$ is as shown in Figure 1 below. The corresponding dual graph is shown in Figure 1 also. For what follows, we also label the edges of the dual graph.
\begin{figure} [!htb] \label{figure 1}
\caption{Special fibre and dual graph}
\includegraphics[angle=0,scale=0.4]{figure_1_latexed_2}
\end{figure}
The elements $h_1:=\sum_{i=1}^{n_{1}}\eta_{1,i},...,h_{l}:=\sum_{i=1}^{n_{l}}\eta_{l,i}$ form a $\mathbb{Z}$-basis for $\text{H}_1(\mathcal{G},\mathbb{Z})$. Moreover, the pairing on $\text{H}_1(\mathcal{G},\mathbb{Z})$ is given by
\[ \left\langle h_i , h_j \right \rangle = \begin{cases} n_i & i=j\\
0 & \ \text{else}.
\end{cases}\]
Then (in the notation of \Cref{min reg model ramified}) we have $\Lambda = \mathbb{Z}\left[\frac{1}{n_1}h_1,...,\frac{1}{n_l}h_l\right]$ and $\Lambda^{\vee} = \mathbb{Z}\left[ h_1,...,h_l \right]$. Let $S_1,...,S_k$ be the disjoint orbits of $G$ on the $h_i$, $n_i'$ be the common value of the $n_j$ on $S_i$, and let $d_i=|S_i|$. Now $\text{Gal}(\bar{k}/k)$ acts on $H_1(\mathcal{G},\mathbb{Z})$, and on the irreducible components of $\mathcal{C}_{\bar{k}}$, through a finite cyclic quotient, $G$ say. Setting $\Lambda_i=\mathbb{Z}\left[\frac{1}{n_i}S_i\right]$ and $\Lambda_i^{\vee}=\mathbb{Z}\left[S_i \right]=n_i'\Lambda_i$ (with signed $G$-action), we have
\[\mathcal{B}=\mathcal{B}_{\Lambda,\Lambda^{\vee}}=\bigoplus_{i=1}^{k}\mathcal{B}_{\Lambda_i,\Lambda_i^{\vee}}.\]
Moreover, as $\mathbb{Z}[G]$-modules, either $\Lambda_i \cong \frac{\mathbb{Z}[x]}{(x^{d_i}-1)}$ or $\Lambda_i \cong \frac{\mathbb{Z}[x]}{(x^{d_i}+1)}$, with a chosen generator of $G$ acting as multiplication by $x$. These $\mathbb{Z}[G]$-modules are non-isomorphic and in the first case we call the orbit $S_i$ $\textit{split}$ and in the second we call $S_i$ $\textit{non-split}$. Noting that $\mathcal{B}_{\Lambda_i,\Lambda_i^{\vee}}=\mathcal{B}_{\Lambda_i,n_i'\Lambda_i}$ is just given by $n_i'\text{H}^1\left(G,\Lambda_i\right)$, a simple application of \Cref{poly lattice lemma} and \Cref{betts group cor} yields the following result.
\begin{lemma} \label{one single root lemma1}
Let $r$ be the number of non-split orbits $S_i$ for which $n_i'$ is odd. Then we have $\mathcal{B}\cong \left(\mathbb{Z}/2\mathbb{Z}\right)^{r}$.
In particular, since by \Cref{min reg def remark} $C$ can never be deficient over $K$ in this case, we have
\[(-1)^{\textup{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)i_d(C)((-1)^g,L/K)=(-1)^r.\]
\end{lemma}
Finally, we wish to describe the number of non-split orbits in terms of the explicit form of $f(x)$ given in \Cref{explicit equation}. Now each $h_i$ corresponds to the double root $\bar{u}_i$ of $\bar{f}(x)$ and it's clear that the $\textit{unsigned}$ action of $G$ on $S$ is identical to the action of $\text{Gal}(\bar{k}/k)$ on the set $\mathcal{U}=\{\bar{u}_1,...,\bar{u}_l\}$. Thus the total number of orbits of $G$ on $S$ is equal to the number of orbits of $\text{Gal}(\bar{k}/k)$ on $\mathcal{U}$. Moreover, from the construction of the minimal regular model, one can determine if a given orbit is split or not by looking at the corresponding tangents. Specifically, one sees that an orbit $S_i$ containing $h_j$ is split if and only if (either of the tangents) $t_j^{\pm}$ is in the field $k(\bar{u}_j)$ ($\textit{a priori}$ it is only in a quadratic extension of this field). One easily verifies that this does not depend on the choice of $h_j$ in the orbit $S_i$. That is, we have the following restatement of \Cref{one single root lemma1}.
\begin{cor} \label{ramified single root ratio}
Suppose $f(x)$ has at least one single root in $\bar{k}$. For each orbit $O_i$ of $\textup{Gal}(\bar{k}/k)$ on the set $\mathcal{U}=\{\bar{u}_1,...,\bar{u}_l\}$, pick $\bar{u}_i\in O_i$. Let $r$ be the number of such orbits for which
\[\bar{u}\prod_{j\neq i}(\bar{u}_i-\bar{u}_j)^2 \prod_{j=l+1}^{2(g-l+1)}(\bar{u}_i-\bar{w}_j)\]
is a non-square in $k(\bar{u}_i)$, and $n_i$ is odd. Then \[(-1)^{\textup{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)i_d(C)((-1)^g,L/K)=(-1)^r.\]
\end{cor}
\subsection{The case where $\bar{f}(x)$ has $g+1$ distinct double roots} \label{double roots subsection}
Suppose now that $\bar{f}(x)$ is a product of $g+1$ distinct double roots (over $\bar{k}$). Then now $\mathcal{C}_{\bar{k}}$ is as shown in Figure 2 below. As before, the corresponding dual graph, along with a choice of orientation on the edges, is also depicted.
\begin{figure} [!htb] \label{figure 2}
\caption{Special fibre and dual graph}
\includegraphics[angle=0,scale=0.6]{figure_2_latexed_2}
\end{figure}
Again, $\text{Gal}(\bar{k}/k)$ acts on $H_1(\mathcal{G},\mathbb{Z})$, and on the irreducible components of $\mathcal{C}_{\bar{k}}$, through a finite cyclic quotient, say $G$. We choose $G$ such that, without loss of generality, $G$ has order divisible by $2$.
Let $h_1=\sum_{i=1}^{n_1}\eta_{1,i},...,h_l=\sum_{i=1}^{n_l}\eta_{l,i}$. The (signed) action of $G$ on the $\eta_{i,j}$ gives $\mathbb{Z}[h_1,...,h_l]$ the structure of a $G$-module. Note that either $\Gamma_1$ and $\Gamma_2$ are both fixed by $G$, in which case $G$ acts on the $h_i$ by permutation, or else (any) generator $\sigma$ for $G$ maps $\Gamma_1$ to $\Gamma_2$ and then $G$ acts on the $h_i$ as the unsigned permutation, twisted by the unique order 2 character of $G$ (thus $G$ acts on the $h_i$ by signed permutation, but unlike the previous case, the sign is controlled `globally').
Now define a (non-degenerate, $G$-invariant) pairing on $\mathbb{Z}[h_1,...h_l]$ by setting
\[ \left\langle h_i , h_j \right \rangle = \begin{cases} n_i & i=j\\
0 & \ \text{else}.
\end{cases}\]
Then we have
\[ \text{H}_1 \left(\mathcal{G},\mathbb{Z}\right)= \ker \left(\mathbb{Z}[h_1,...,h_l] \stackrel{\Sigma}{\longrightarrow}\mathbb{Z} \right)\]
with the pairing and $G$-action being that induced from those defined above on $\mathbb{Z}[h_1,...,h_l]$ (here $\Sigma$ is the sum-map sending each $h_i$ to $1$ and extending linearly).
Define $\Lambda$ to be the dual lattice of $\text{H}_1\left(\mathcal{G},\mathbb{Z}\right)$ inside $\text{H}_1\left(\mathcal{G},\mathbb{Z}\right)\otimes \mathbb{Q}$, so that $\Lambda^{\vee}= \text{H}_1\left(\mathcal{G},\mathbb{Z}\right)$.
Set $S=\left\{h_1,...,h_l\right\}$, endowed with unsigned $G$-action, so that $G$ acts by permutation on $S$. Now $S$ splits into $G$-orbits $S=S_1 \sqcup ... \sqcup S_k$ of sizes $d_1,...,d_k$. We write $S_i=\left\{h_{i,1},...,h_{i,d_i}\right\}$. If $h_i$ and $h_j$ are in the same orbit, $S_m$ say, then we write $n_m'$ for the common value of the $n_m$ on this orbit.
Write $\mathbb{Z}[S]$ for the $G$-module $\mathbb{Z}[h_1,...,h_l]$ (with its signed action). As in \Cref{single root}, the dual lattice of $\mathbb{Z}[S]\otimes \mathbb{Q}$ is given by $\mathbb{Z}[S]^{\vee}=\left[\frac{1}{n_1}h_1,...,\frac{1}{n_l}h_l\right]$. We identify this with $\mathbb{Z}[S]$ as a $G$-module in the obvious way so that the inclusion $\mathbb{Z}[S] \rightarrow \mathbb{Z}[S]^{\vee}$ corresponds to the map
\begin{eqnarray*}
\mathbb{Z}[S] & \longrightarrow & \mathbb{Z}[S]\\
h_{i} & \mapsto & n_{i}h_{i}.
\end{eqnarray*}
We saw in \Cref{single root} that the group $\mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]}$ is easy to compute. Thus to compute $\mathcal{B}_{\Lambda,\Lambda^{\vee}}$ we wish to relate this to the former group. We see in the following lemma that the difference between these two groups is controlled by whether or not $C$ is deficient over $K$.
\begin{lemma} \label{deficient Betts group}
With the notation of the previous discussion, we have
\[(-1)^{\textup{dim}_{\mathbb{F}_2}\mathcal{B}_{\Lambda,\Lambda^{\vee}}}=i_d(C)(-1)^{\textup{dim}_{\mathbb{F}_2}\mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]}}.\]
\end{lemma}
(Implicit in this is the statement that $\mathcal{B}_{\Lambda,\Lambda^{\vee}}$ is in fact a $\mathbb{F}_2$-vector space; we already know that $\mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]}$ is.)
\begin{proof}
We have a commutative diagram with exact rows
\[
\xymatrix{0\ar[r] & \Lambda^{\vee}\ar[r]\ar[d] & \mathbb{Z}[S]\ar[r]^{\Sigma}\ar[d]^{h_{i}\mapsto n_{i}h_{i}} & \mathbb{Z}\ar[r] & 0\\
0 & \Lambda\ar[l] & \mathbb{Z}[S]\ar[l] & \mathbb{Z}\ar[l] & 0.\ar[l]
}\]
Here the bottom row is the dual of the top, the map $\Lambda^{\vee} \rightarrow \Lambda$ is the natural inclusion and the map $\mathbb{Z}\rightarrow \mathbb{Z}[S]$ on the bottom row is easily seen to take $1$ to $\sum_{i=1}^{l}h_i$.
This induces the following commutative diagram for group cohomology, again having exact rows
\[
\xymatrix{\text{H}^{1}(G,\Lambda^{\vee})\ar[r]^{f}\ar[d]^{\phi} & \text{H}^{1}(G,\mathbb{Z}[S])\ar[r]\ar[d]^{g} & \text{H}^{1}(G,\mathbb{Z})\\
\text{H}^{1}(G,\Lambda) & \text{H}^{1}(G,\mathbb{Z}[S])\ar[l]_{h} & \text{H}^{1}(G,\mathbb{Z}).\ar[l]
}
\]
(Here we define the action of $G$ on $\mathbb{Z}$ so as to make the sequence $G$-equivariant.)
By definition, we have \[\mathcal{B}_{\Lambda,\Lambda^{\vee}}=\text{im}(\phi)=\text{im}(h\circ g \circ f)\]
whilst
\[\mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]}=\text{im}(g).\]
If both $\Gamma_1$ and $\Gamma_2$ are fixed by $G$ then $G$ acts on $\mathbb{Z}[S]$ by permutation and $\text{H}^{1}(G,\mathbb{Z}[S])=0$ (this follows from Shapiro's lemma). In particular, both $\mathcal{B}_{\Lambda,\Lambda^{\vee}}$ and $\mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]}$ are trivial. Since moreover $C$ is not deficient in this case, we are done.
Suppose now that some (equivalently any) generator $\sigma$ of $G$ maps $\Gamma_1$ onto $\Gamma_2$. Then one sees that for each $i$, as $G$-modules, we have
\[\mathbb{Z}[S_i]\cong \begin{cases} \frac{\mathbb{Z}[x]}{(x^{d_i}-1)} & ~~ d_i ~\text{even}\\ \frac{\mathbb{Z}[x]}{(x^{d_i}+1)} &~~ d_i ~ \text{odd} \end{cases}\]
where $\sigma$ acts on the right as multiplication by $x$ in both cases. Explicitly, this isomorphism is given by sending $h_{i,1}$ to $1$. Clearly we have a $G$-module decomposition
\[\mathbb{Z}[S]\cong \bigoplus_{i=1}^{k} \mathbb{Z}[S_i].\]
Moreover, the action of $G$ on each copy of $\mathbb{Z}$ is such that $\sigma$ acts as multiplication by $-1$.
\Cref{poly lattice lemma} now gives
\[\text{H}^1(G,\mathbb{Z}[S])=\bigoplus_{|S_i| ~ \text{odd}} \mathbb{F}_2[f_i]\]
where $f_i=\sum_{j=1}^{d_i}h_{i,j}$ and the sum ranges over all odd-sized orbits of $G$ on $S$. Moreover (with action as above), each of the $\text{H}^1(G,\mathbb{Z})$ are isomorphic to $\mathbb{F}_2$, generated by $1\in \mathbb{Z}$.
By exactness of the diagram, we obtain
\[f\left(\text{H}^{1}(G,\Lambda^{\vee})\right)=\ker\left(\bigoplus_{|S_i| ~ \text{odd}} \mathbb{F}_2[f_i] \stackrel{\Sigma}{\longrightarrow} \mathbb{F}_2\right).\]
The map $g:\bigoplus_{|S_i| ~ \text{odd}} \mathbb{F}_2[f_i] \rightarrow \bigoplus_{|S_i| ~ \text{odd}} \mathbb{F}_2[f_i]$ sends $f_i$ to $n_i' f_i$. From this, one sees that $\text{im}(g\circ f)=\text{im}(g)$ unless $n_i'$ is odd whenever $|S_i|$ is odd, in which case $\text{im}(g\circ f)$ has index $2$ in $\text{im}(g)$.
On the other hand, we have
\[\text{ker}(h)=\text{im}\left(\text{H}^1(G,\mathbb{Z})\rightarrow \text{H}^1(G,\mathbb{Z}[S]) \right)= \mathbb{F}_2,\]
the latter group being generated by $\sum_{|S_i|~\text{odd}}f_i$.
Now $\sum_{|S_i|~\text{odd}}f_i$ is in the image of $g$ if and only if each $n_i'$ is odd whenever $|S_i|$ is odd, in which case it is in the image of $g\circ f$ if and only if the number of odd-sized orbits on $S$ is even. That is, if and only if $g=l-1$ is odd.
Putting everything together, we have
\[\dim_{\mathbb{F}_2}\mathcal{B}_{\Lambda,\Lambda^{\vee}}= \begin{cases} \dim_{\mathbb{F}_2} \mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]} -2 & ~~ \sigma(\Gamma_1)=\Gamma_2,~\text{each}~n_i'~\text{odd whenever}~ |S_i| ~ \text{odd,}~g~\text{odd} \\ \dim_{\mathbb{F}_2} \mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]} -1 & ~~ \sigma(\Gamma_1)=\Gamma_2,~\text{each}~n_i'~\text{odd whenever}~ |S_i| ~ \text{odd,}~g~\text{even} \\ \dim_{\mathbb{F}_2} \mathcal{B}_{\mathbb{Z}[S]^{\vee},\mathbb{Z}[S]} & ~~ \text{else.} \end{cases}\]
Since, by \Cref{min reg def remark}, $C$ is deficient over $K$ if and only if $g$ is even, $\sigma(\Gamma_1)=\Gamma_2$ and $n_i'$ is odd whenever $|S_i|$ is, we are done.
\end{proof}
\begin{remark}
From the compatibility of \Cref{Kramer Tunnell} under quadratic twist given in \Cref{twist 2}, we could have additionally assumed that the leading coefficient of $f(x)$ was monic, in which case we would only have needed the $\sigma(\Gamma_1)=\Gamma_1$ case of \Cref{deficient Betts group}. However, as it is interesting to see where deficiency enters the computations in the general case, we have left this in.
\end{remark}
In the previous section, we called an orbit $S_i$ of $G$ on the set $\left\{h_1,...,h_l\right\}$ split if the corresponding \textup{signed} permutation module $\mathbb{Z}[S_i]$ was isomorphic to $\frac{\mathbb{Z}[x]}{(x^{d_i}-1)}$ and non-split otherwise. Now with the same convention applied here, every orbit is split if $\sigma(\Gamma_1)=\Gamma_1$, whilst the non-split orbits when $\sigma(\Gamma_1)=\Gamma_2$ are exactly those of odd size. An immediate corollary is then the following.
\begin{cor} \label{all double roots Betts group}
Define $r$ to be the number of non-split orbits $S_i$ for which $n_i'$ is odd. Then \[(-1)^{\text{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)i_d(C)((-1)^g,L/K)=(-1)^r.\]
\end{cor}
Again, we wish to reinterpret the integer $r$ in terms of the explicit form of $f(x)$ given in \Cref{explicit equation}. First note that $\sigma(\Gamma_1)=\Gamma_1$ if and only if the (reduction of the) leading coefficient $\bar{u}$ is a square in $k$. Thus if $\bar{u}$ is a square in $k$, every orbit is split, whilst if $u$ is a non-square in $k$, the even length orbits on $S$ are split, whilst the odd length orbits are not. Again, the orbits on $S$ correspond to the orbits of $\text{Gal}(\bar{k}/k)$ on the set $\mathcal{U}=\{\bar{u}_1,...,\bar{u}_l\}$ of double roots of $\bar{f}$. Since every root of $\bar{f}$ is a double root, one sees that each tangent $t_i^{\pm}$ is a square in $k(\bar{u}_i)$ if and only if $\bar{u}$ is a square in $k(\bar{u}_i)$, which happens if and only if the $\text{Gal}(\bar{k}/k)$-orbit of $\bar{u}_i$ is even in length. Thus we see that the description of $r$ in terms of \Cref{explicit equation} exactly the same as in the previous case. Specifically, we have the following Corollary.
\begin{cor} \label{ramified double root ratio}
Suppose $f(x)$ has $g+1$ distinct double roots in $\bar{k}$. For each orbit $O_i$ of $\textup{Gal}(\bar{k}/k)$ on the set $\mathcal{U}=\{\bar{u}_1,...,\bar{u}_l\}$ pick $\bar{u}_i\in O_i$. Let $r$ be the number of such orbits for which
\[\bar{u}\prod_{j\neq i}(\bar{u}_i-\bar{u}_j)^2 \prod_{j=l+1}^{2(g-l+1)}(\bar{u}_i-\bar{w}_j)\]
is a non-square in $k(\bar{u}_i)$ and $n_i$ is odd. Then \[(-1)^{\textup{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)i_d(C)((-1)^g,L/K)=(-1)^r.\]
\end{cor}
\subsection{Computing $c(J^L/K)$}
Fix a uniformiser $\pi_\K$ in $K$ such that $L=K(\sqrt{\pi_K})$. Then $C^L$ is given by the equation
\begin{equation} \label{twist equation}
C^\L:y^2=u\pi_\K\prod_{i=1}^l \left((x-u_i)^2-v_i\pi_{\K}^{n_i}\right)\prod_{j=l+1}^{2(g-l+1)}(x-w_j).
\end{equation}
Since the equation defining $C^\L$ above, and the corresponding one for the chart at infinity, are integral, they define a proper model of $C^L$ over $\mathcal{O}_\K$ in the obvious way. Let $\mathcal{C^\L}_{0,\K^{nr}}$ be the base change of this model to $\mathcal{O}_{\K^{nr}}$. Its special fibre consists of one irreducible component having multiplicity $2$ in the special fibre. We now perform the necessary blow-ups to obtain the minimal regular model. The procedure is (an easy generalisation of) the proof of steps 6 and 7 of Tate's algorithm as described in \cite[IV.9]{MR1312368}. One sees that the special fibre of the minimal regular model $\mathcal{C}^L$ of $C^L$, after base changing to $\bar{k}$, has the form given in Figure 3, where we set $k=g+1-l$ (the number of single roots of the reduction of $f(x)$). Here the numbers in the picture indicate the multiplicity of the components in the special fibre. If no number is given the multiplicity is 1.
\begin{figure} [!htb]
\caption{Special fibre of $\mathcal{C^\L}_{\K^{nr}}$ \bigskip}
\centering
\includegraphics[angle=0,scale=0.4]{figure_4_latexed_final}
\end{figure}
We wish to compute the Tamagawa number $c(J^L/K)=\Phi(k)$. In fact, we need only compute $(-1)^{\text{ord}_2 c(J^L/K)}$ and this will prove easier to compute thanks to the results of \Cref{main unramified section}. Now $\text{Gal}(\bar{k}/k)$ acts the the components of $\mathcal{C}^L_{\bar{k}}$ through a finite cyclic quotient $G$. Fix a generator $\sigma$ of $G$. Then $G$ acts on the set $S=\{\Gamma_{i,1}~:~1\leq i \leq l\}$ by permutation (this follows by symmetry, and the assumption that $g\geq2$). Moreover, on each orbit on $S$, the corresponding value of $n_i$ is constant. Suppose $O=\{\Gamma_{i_1,1},...,\Gamma_{i_r,1}\}$ is an orbit, and that $\sigma$ acts as the cycle $\left( \Gamma_{i_1,1}~...~\Gamma_{i_r,1}\right)$ on this. Then we have two possibilities. Either (after relabeling if necessary), $\{R_{i_1,1},...,R_{i_r,1}\}$ and $\{R_{i_1,2},...,R_{i_r,2}\}$ are two $G$-orbits of length $r$ with $\sigma$ acting as the obvious $r$-cycle on each, or $\{R_{i_1,1},...,R_{i_r,1},R_{i_1,2},...,R_{i_r,2}\}$ is an orbit of length $2r$ and $\sigma$ cycles the $2r$ elements of this orbit in the order they are written. In the first case, we call the orbit $O$ $\textit{small}$, and in the second we call it $\textit{large}$.
\begin{lemma} \label{tam twist lemma}
Keeping the notation of the previous paragraph, we have
\[(-1)^{\textup{ord}_2c\left(J^L/K\right)}i_d(C^L)=(-1)^{\# \text{orbits on}~\mathcal{W}+\#\text{large orbits on} ~S}.\]
\end{lemma}
\begin{proof}
We will compute the quantity $(-1)^{\text{ord}_2c\left(J^L/K\right)}i_d(C^L)$ in two stages. We first compute the order of the $\bar{k}$-points in the component group (i.e. we compute $|\Phi(\bar{k})|$) using \cite[Proposition 9.6.6]{MR1045822}, which applies in much greater generality. Secondly, we compute the quantity
\[ (-1)^{\text{ord}_2 \frac{|\Phi(\bar{k})|}{|\Phi(k)|}} i_d(C^L)\]
by an application of \Cref{tam computations}.
Let $Y$ be the graph associated to $\mathcal{C}^L_{\bar{k}}$ by taking the vertices to be the irreducible components, and joining 2 distinct vertices by a single edge if the corresponding components have non-trivial intersection number. It is clear that $Y$ is a tree. Moreover, the intersection number of any 2 distinct components is either 0 or 1, and the greatest common divisor of the multiplicities of the components is 1. Let $I$ denote the set of all irreducible components of $\mathcal{C}^L_{\bar{k}}$, and for each component $\Gamma \in I$ let $d(\Gamma)$ denote the multiplicity of $\Gamma$ in $\mathcal{C}_{\bar{k}}$, and $s(\Gamma)$ be the number of components (distinct from $\Gamma$) that meet $\Gamma$. Then \cite[Proposition 9.6.6]{MR1045822} gives
\[ |\Phi(\bar{k})|=\prod_{i \in I} d(\Gamma)^{s(\Gamma)-2}=2^{2k+2l-2}=2^{2g}.\]
In particular, we see that $\text{ord}_2|\Phi(\bar{k})|$ is even and we now wish to show that
\[ (-1)^{\text{ord}_2 \frac{|\Phi(\bar{k})|}{|\Phi(k)|}} i_d(C^L)=(-1)^{\# \text{orbits on}~\mathcal{W}+\#\text{large orbits on} ~S}.\]
Let $O_1,...,O_k$ be the even sized orbits of $\text{Gal}(\bar{k}/k)$ on $I$, let $r_i=|O_i|$ and for each $1 \leq i \leq k$, write
\[\epsilon_i=\sum_{i=0}^{r_i-1}(-1)^i \sigma^i (\Gamma^{(i)})\]
where $\sigma$ generates $G$ (we can take $\sigma$ to be the Frobenius element in $G$) and $\Gamma^{(i)}$ is a representative of the orbit $O_i$.
Then \Cref{tam computations} gives
\[(-1)^{\text{ord}_2 \frac{|\Phi(\bar{k})|}{|\Phi(k)|}} i_d(C^L)= (-1)^{\text{ord}_2 | \textup{det}(\frac{1}{r_j}\epsilon_{i} \cdot \epsilon_{j})_{ i,j}|}.\]
Now the matrix $A=(\frac{1}{r_j}\epsilon_{i} \cdot \epsilon_{j})_{ i,j}$ is block diagonal, with a block corresponding to each even sized orbit on $\W$, and a block corresponding to each orbit of $S$, save for the odd sized small orbits on $S$ which do not contribute. From this, one sees immediately that each even sized orbit on $\W$ contributes a factor of $-2$ to the determinant of $A$, as does each odd sized large orbit on $S$. The contribution from each small even orbit on $S$ is $\lambda_n$ where $n$ is the common value of the $n_i$ on the orbit, and $\lambda_k$ denotes the determinant of the $k+2$ by $k+2$ matrix
\[
\left(\begin{array}{cccccccc}
-2 & 1\\
1 & -2 \\
& 1\\
& & \ddots & 1\\
& & & -2 & 1\\
& & & 1 & -2 & 1 & 1\\
& & & & 1 & -2 & 0\\
& & & & 1 & 0 & -2
\end{array}\right).
\]
Noting that $\lambda_k$ satisfies the recurrence relation $\lambda_k=-2\lambda_{k-1}-\lambda_{k-2}$, we obtain $\lambda_k=(-1)^k \cdot 4$. For the large even orbits, the contribution is the determinant of the $n+1$ by $n+1$ matrix (with $n$ as above)
\[
\left(\begin{array}{cccccc}
-2 & 1\\
1 & -2\\
& 1\\
& & \ddots & 1\\
& & & -2 & 1\\
& & & 1 & -2 & 1\\
& & & & 2 & -2
\end{array}\right).
\]
This may be treated similarly to the previous case, and we obtain a contribution of $(-1)^n \cdot 2$.
In total, we have
\[(-1)^{\text{ord}_2 \frac{|\Phi(\bar{k})|}{|\Phi(k)|}} i_d(C^L)=(-1)^{{\# \text{even sized orbits on}~\mathcal{W}+\#\text{large orbits on} ~S}}.\]
Finally, we conclude by noting that, as $|\W|$ is even, the number of even sized orbits on $\W$ is congruent modulo $2$ to the total number of orbits on $\W$.
\end{proof}
We now seek to describe the number of orbits on $\W$ and the number of large orbits of $S$ in terms of the explicit equation for $C^L$ given in \Cref{twist equation}. From the construction of the minimal regular model, one sees that the set $\{A_1,...,A_{2k}\}$ corresponds exactly (as a set with $\text{Gal}(\bar{k}/k)$-action) to the set $\{\bar{w}_{l+1},...,\bar{w}_{2(g-l+1)}\}$ of single roots of $\bar{f}(x)$. Thus the number of orbits on these two sets coincide. Moreover, one sees that the orbits on $S$ correspond similarly to the orbits on $\{\bar{u}_1,...,\bar{u}_l\}$. Finally, one sees that an orbit, corresponding to the orbit of $\bar{u}_i$ say, is large if and only if the product $-\bar{v_i} (t_i^{\pm})^2$ is a non-square in $k(\bar{u}_i)$ if $n_i$ is odd, and if and only if $\bar{v_i}$ alone is a non-square in $k(\bar{u}_i)$ if $n_i$ is even (compare with the formula for $\tilde{\mathcal{V}}_n$ in step 7 of Tate's algorithm in \cite[page 374]{MR1312368}).
We may now put everything together to prove \Cref{Kramer Tunnell} in this case.
\begin{cor} \label{ramified cube free}
Let $K$ be a finite extension of $\mathbb{Q}_p$ for $p$ odd and let $C/K$ be a hyperelliptic curve given by the equation $y^2=f(x)$. Let $L/K$ be a ramified quadratic extension and suppose that $f(x)$ has unit leading coefficient and that the reduction of $f(x)$ is cube free. Then \Cref{Kramer Tunnell} holds for $C$ and the extension $L/K$.
\end{cor}
\begin{proof}
We are assuming that $C$, over $K^{\text{nr}}$, is given by an equation of the form
\begin{equation} \label{the equation}
C:y^2=f(x):=u\prod_{i=1}^l \left((x-u_i)^2-v_i\pi_{\K}^{n_i}\right)\prod_{j=l+1}^{2(g-l+1)}(x-w_j)
\end{equation}
where the reduction $\bar{u}_i$ and $\bar{w}_j$ are all pairwise distinct and, without loss of generality, the uniformiser $\pi_K$ is chosen such that $L=K(\sqrt{\pi_K})$.
Let $\mathcal{U}$ be the $\text{Gal}(\bar{k}/k)$-set
\[\mathcal{U}=\left\{\bar{u}_1,...,\bar{u}_l\right\},\]
let $O_1,...,O_k$ be the disjoint orbits on $\mathcal{U}$, and for each orbit $O_i$, fix $\bar{u}_i\in O_i$, along with associated $v_i$, and set
\[t'_i:=\bar{u}\prod_{j\neq i}(\bar{u}_i-\bar{u}_j)^2 \prod_{j=l+1}^{2(g-l+1)}(\bar{u}_i-\bar{w}_j)\]
(so that $t'_i$ is the square of the tangents $t_i^{\pm}$ defined previously). For each orbit $O_i$, let $n'_i$ be the common value of the $n_j$ in \Cref{the equation} associated to the $u_j \in O_i$. Moreover, let $\mathcal{W}$ be the $\text{Gal}(\bar{k}/k)$-set
\[\mathcal{W}=\left\{\bar{w}_{l+1},...,\bar{w}_{2(g-l+1)}\right\}.\]
Finally, for a finite field $\mathbb{F}$ of characteristic not 2, let $\rho_{\mathbb{F}}:\mathbb{F}^{\times}\rightarrow \{\pm1\}$ be the homomorphism whose kernel consists of the squares in $\mathbb{F}^{\times}$.
By \Cref{ramified single root ratio,ramified double root ratio}, we have
\[(-1)^{\textup{ord}_2 \frac{c(J/K)}{c(J/L)}}w(J/K)w(J^L/K)i_d(C)((-1)^g,L/K)=\prod_{\stackrel{i=1}{n_i'~\text{odd}}}^{k}\rho_{k(\bar{u}_i)}(t_i').\]
Moreover, by \Cref{tam twist lemma} (in conjunction with the discussion immediately preceding this corollary), we have
\[(-1)^{\textup{ord}_2 c(J^L/K)}i_d(C^L)=(-1)^{\# \text{orbits on}~\mathcal{W}}\times\prod_{\stackrel{i=1}{n_i'~\text{odd}}}^{k}\rho_{k(\bar{u}_i)}\left((-1)^{n_i'}\bar{v}_i t_i'\right) \prod_{\stackrel{i=1}{n_i'~\text{even}}}^{k}\rho_{k(\bar{u}_i)}\left((-1)^{n_i'}\bar{v}_i \right).\]
As each $\rho_{k(\bar{u}_i)}$ is a homomorphism, to complete the proof we must show that
\[(\Delta_C,L/K)=(-1)^{\# \text{orbits on}~\mathcal{W}}\times\prod_{i=1}^k \rho_{k(\bar{u}_i)}\left((-1)^{n_i'}\bar{v}_i \right).\]
To see this, observe that, as $-\pi_K$ is a norm from $L$, $(\Delta_C,L/K)=1$ if and only if $\Delta_C$ is a square in $F=K(\sqrt{-\pi_K})$. Moreover, it's clear from \Cref{the equation} that all roots of $f(x)$ lie in $F^{\text{nr}}=FK^{\text{nr}}$. Thus $\text{Gal}(F^{\text{nr}}/F)$ acts on the roots of $f(x)$ and letting $\sigma \in \text{Gal}(F^{\text{nr}}/F)$ denote the Frobenius element, we deduce that $(\Delta_C,L/K)$ is equal to the sign of $\sigma$ as a permutation on the roots of $f(x)$. Now the roots of $f(x)$ in $\bar{K}$ are
\[\left\{u_i \pm \sqrt{(-1)^{n_i}v_i}\cdot \sqrt{(-\pi_K)^{n_i}}~:~1\leq i \leq l \right\}\cup \left \{w_j ~:~ l+1\leq j \leq 2(g-l+1) \right \}.\]
Since $\sqrt{(-\pi_K)^{n_i}}\in F$ for each $i$, the action of $\sigma$ on these roots is the same as the action of the Frobenius element in $\text{Gal}(\bar{k}/k)$ on the set
\[\left\{ \bar{u}_i \pm \sqrt{(-1)^{n_i}\bar{v}_i}\right\} \cup \mathcal{W}\]
and the result now follows easily.
\end{proof}
\Cref{cases of the parity conjecture} now follows from \Cref{selmer decomposition}, \Cref{kramer-tunnell for reals places} and \Cref{kramer-tunnell for good reduction in odd residue char,char 2 unramified,good ordinary case,unramified kramer tunnell,ramified cube free}.
|
2,877,628,089,479 | arxiv | \section{Introduction}
Experiments at high-energy particle colliders have been integral to unraveling
the structure of our universe and have confirmed the validity of the Standard Model
of particle physics at an unprecedented accuracy. Going beyond the current level
of precision and possibly revealing new fundamental particles and forces
will require ever more detailed experimental analyses and theoretical calculations.
Monte-Carlo simulations by means of event generators play a vital role in this context,
as they link experiment and theory through the detailed description of fully exclusive
final states~\cite{Webber:1986mc,Buckley:2011ms}. They are required to describe the
dynamics of a large number of hadrons originating from QCD Bremsstrahlung,
which is modeled in the simulation through so-called parton showers. Modern parton showers
are typically based on a unified description of QCD radiative effects in a dipole picture,
which encompasses both the leading-order spin-averaged collinear radiation pattern,
and the leading-order color-averaged soft radiation pattern. The predictions generated
by such algorithms accurately describe many experimental measurements.
A notable exception to the success of the parton-shower method arises from its limited
phase-space coverage. This problem is alleviated by the matching and merging techniques
that allow to correct parton showers to any known fixed-order result at limited
final-state multiplicity, and that have been in the focus of interest of the theoretical
particle physics community in the past decade~\cite{Alwall:2007fs,Nason:2012pr}.
Currently the most pressing problem in the context of parton-shower simulations is the
lack of options to assess the intrinsic uncertainty of the method itself. The precision
of fixed-order perturbative QCD calculations is conventionally quantified by varying the renormalization and factorization
scales, and the dependence on these scales is reduced at higher orders in the perturbative
expansion if the perturbative series converges. No such technique is currently available
for parton showers, essentially because parton showers at higher precision do not yet
exist or their practical implementation is incomplete. First steps towards the construction
of next-to-leading order (NLO) parton showers have been made in~\cite{
Kato:1986sg,Kato:1988ii,Kato:1990as,Kato:1991fs,
Giele:2011cb,Hartgring:2013jma,Li:2016yez,Nagy:2017ggp,Hoche:2017iem,Hoche:2017hno},
but no method has yet been presented
that is capable of simulating fully exclusive final states at hadron colliders.
At the same time, it should be expected that a difference exists between parton showers
and analytical resummation. While it should be reduced at higher perturbative precision,
it cannot be completely eliminated due to differences in the treatment of momentum
and probability conservation~\cite{Hoeche:2017jsi}.
In this publication we address one of the most important aspects of next-to-leading
order parton showers, namely the simulation of the higher-order corrections to soft gluon
radiation, and we show how to implement these corrections in a fully differential form
in practice. In integrated form, they lead to the well-known two-loop cusp anomalous
dimension~\cite{Kodaira:1981nh,Davies:1984hs,Davies:1984sp,Catani:1988vd}, which is
included in improved leading-order parton showers by means of redefining the strong
coupling. This is known as the CMW method~\cite{Catani:1990rr}. At the differential level,
the corrections to soft gluon radiation induce spin correlations and sub-leading color corrections
that are not included in leading-order parton showers. As part of the extension to the
next-to-leading order, we adapt the algorithm in~\cite{Hoche:2015sya} to include these effects.
Moreover, the construction of a local modified subtraction procedure as anticipated
in~\cite{Hoche:2017iem} mandates the computation of the two-loop cusp anomalous
dimension as an endpoint contribution corresponding to the iterated soft times collinear limit.
The resulting algorithm will be a key ingredient in the construction of a fully differential
technique for matching parton showers to next-to-next-to leading order calculations.
This paper is organized as follows: Section~\ref{sec:analytic} present an analytic
calculation of the local K-factor due to NLO corrections to soft-gluon radiation.
Based on this calculation, Sec.~\ref{sec:four_dimensions} introduces the modified subtraction
method and presents our approach to implementing the required changes in the dipole-like
parton shower. Section~\ref{sec:results} presents a numerical validation of the new
Monte-Carlo techniques and an assessment of the effect of the fully differential
simulation compared to the CMW method. A summary is given in Sec.~\ref{sec:conclusions}.
\section{Analytic computation of double-soft corrections}
\label{sec:analytic}
We employ the formalism for the construction of parton showers at next-to-leading
order accuracy originally proposed in~\cite{Hoche:2017iem}. This technique is based
on a modified subtraction method combined with a new algorithm for mapping
$n$-particle on-shell momentum configurations to $n+2$-particle on-shell
momentum configurations and the replacement of explicit symmetry factors
by appropriate light-cone momentum fractions that can be identified as
``tags'' for evolving partons. The extension of this method to soft evolution
at next-to-leading order requires the removal of overlap between the explicitly
included higher-order corrections in the CMW scheme~\cite{Catani:1990rr} and the
potentially included triple-collinear splitting functions~\cite{Hoche:2017iem}.
In this section we will first derive analytic results for the double-soft corrections
at next-to-leading order. We define the kinematics in Sec.~\ref{sec:kinematics},
present the individual corrections in Sec.~\ref{sec:nlo_contributions} and collect
the results in Sec.~\ref{sec:nlo_corrections}. Based on this calculation,
Sec.~\ref{sec:four_dimensions} introduces a modified subtraction technique
and addresses the overlap removal.
\begin{figure}[t]
\subfigure[]{\includegraphics[scale=0.33]{Fig1a.pdf}\label{fig:lo_virt}}
\subfigure[]{\includegraphics[scale=0.33]{Fig1b.pdf}\label{fig:lo_real}}
\caption{Leading order contributions to dipole-shower evolution in the soft limit.
The double solid lines represent hard (identified) partons i.e.\ Wilson lines.
\label{fig:lo}}
\end{figure}\noindent
The leading-order contributions to the soft function, which described the interaction
between two hard jets of collinear particles through soft gluon
exchange~\cite{Sterman:1986aj,Collins:1988ig,Collins:1989gx}
are shown in Fig.~\ref{fig:lo}. The double solid lines represent the hard legs,
and the dashed line indicates the cut. The virtual correction is given by a scaleless
integral and vanishes in dimensional regularization~\cite{Monni:2011gb}. The diagram
in Fig.~\ref{fig:lo_real} and its mirror conjugate generate the eikonal factor
\begin{equation}\label{eq:lo_eik_fc}
{\bf S}_{ij}^{(0)}(q)=-{\bf T}_i{\bf T}_j\,\mc{S}_{ij}^{(0)}(q)\;,
\qquad\text{where}\qquad
\mc{S}_{ij}^{(0)}(q)=g_s^2\mu^{2\varepsilon}\,\frac{p_ip_j}{2\,(p_iq)(q\,p_j)}
=g_s^2\mu^{2\varepsilon}\,\frac{s_{ij}}{s_{iq}s_{jq}}\;.
\end{equation}
Here and in the following we will label the Wilson lines by $i$ and $j$,
while the soft momenta will be denoted by $1$ and possibly $2$. We also refer
to the combined soft momentum as $q$, where $q=p_1$ and $q=p_1+p_2$ in one-
and two-emission configurations, respectively. We restrict our analysis to the
improved leading-color approximation typically used in parton-shower simulations.
In processes with $n$ possibly color-connected partons, the eikonal term,
Eq.~\eqref{eq:lo_eik_fc}, is first partial-fractioned~\cite{Catani:1996vz},
and subsequently the color-insertion operator ${\bf T}_i{\bf T}_j$ is
approximated by assuming independence of the kinematics. This leads
to the replacement
\begin{equation}\label{eq:leading_color}
\sum_{\substack{i=1\\j=i+1}}^{n}{\bf S}_{ij}^{(0)}(q)=
-\sum_{\substack{i,j=1\\j\neq i}}^n{\bf T}_i{\bf T}_j\,\mc{D}_{i,j}^{(0)}(q)\quad\to\quad
\sum_{\substack{i,j=1\\j\neq i}}^n\frac{C_i}{n}\,\mc{D}_{i,j}^{(0)}(q)\;,
\qquad\text{where}\qquad
\mc{D}_{i,j}^{(0)}(q)=g_s^2\mu^{2\varepsilon}\,\frac{1}{s_{iq}}\frac{s_{ij}}{s_{iq}+s_{jq}}\;.
\end{equation}
As the partial fraction $\mc{D}_{i,j}(q)$ can be matched to the collinear limit
unambiguously, the corresponding color Casimir operator, $C_i$, should indeed be
associated with the emission in the soft-collinear limit. This approximation
proves to be very accurate in practice. We therefore postpone the exact treatment
of the color insertion operators to future work and perform our analysis based on
$\mc{S}_{ij}^{(0)}(q)$. We also point out that including the full next-to-leading
order corrections to Eq.~\eqref{eq:leading_color} requires that the first sub-leading
color correction be implemented in the parton shower if the two-loop cusp
anomalous dimension is to be recovered in the fully differential calculation.
These terms are related to color factors of the form $C_F-C_A/2$, where the
first contribution is absorbed into the exponentiated leading-order soft result, and the
second term becomes part of the genuine two-loop result~\cite{Cornwall:1975ty,Frenkel:1976bj}.
This will be discussed in detail in Sec.~\ref{sec:four_dimensions},
and related numerical comparisons will be made in Sec.~\ref{sec:results}.
The virtual corrections to the single emission have been computed in~\cite{Bern:1999ry,Catani:2000pi}.
They are given by
\begin{equation}\label{eq:virt_soft}
\mc{S}_{ij}^{\rm(virt)}(q)=-C_A\frac{g_s^4}{8\pi^2}
\frac{(4\pi\mu^4)^\varepsilon}{\varepsilon^2}\frac{\Gamma^4(1-\varepsilon)\Gamma^3(1+\varepsilon)}{
\Gamma^2(1-2\varepsilon)\Gamma(1+2\varepsilon)}\left(\frac{s_{ij}}{s_{iq}s_{jq}}\right)^{1+\varepsilon}\;.
\end{equation}
\begin{figure}[t]
\subfigure[]{\includegraphics[scale=0.33]{Fig2a.pdf}\label{fig:nlo_box1}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2b.pdf}\label{fig:nlo_box2}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2c.pdf}\label{fig:nlo_tgc1}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2d.pdf}\label{fig:nlo_tgc2}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2e.pdf}\label{fig:nlo_vpsg}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2f.pdf}\label{fig:nlo_vpsq}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2g.pdf}\label{fig:nlo_vpcg}}
\subfigure[]{\includegraphics[scale=0.33]{Fig2h.pdf}\label{fig:nlo_vpcq}}
\caption{Next-to-leading order real-emission contributions to dipole-shower evolution
in the soft limit. The double solid lines represent hard (identified) partons i.e.\
Wilson lines.
\label{fig:nlo}}
\end{figure}\noindent
The diagrams contributing to the gluonic real-emission corrections are schematically
displayed in Fig.~\ref{fig:nlo_box1}-\subref{fig:nlo_vpsg}, while the quark contribution
is shown in Fig.~\ref{fig:nlo_vpsq}. The vacuum polarization diagrams with gluons have
corresponding ghost diagrams, and all terms also occur in the mirror symmetric configuration.
Their sum is given by the soft insertion operators computed in~\cite{Catani:1999ss
\begin{equation}\label{eq:real_soft}
\begin{split}
\mc{S}_{ij}^{(q\bar{q})}(1,2)=&\;T_R\;\frac{s_{i1}s_{j2}+s_{i2}s_{j1}-s_{12}s_{ij}}{
s_{12}^2(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\\
\mc{S}_{ij}^{(gg)}(1,2)=&\;C_A\;\frac{(1-\varepsilon)[s_{i1}s_{j2}+s_{i2}s_{j1}]-2s_{12}s_{ij}}{
s_{12}^2(s_{i1}+s_{i2})(s_{j1}+s_{j2})}+\mc{S}_{ij}^{\rm(s.o.)}(1,2)\,\frac{C_A}{2}
\left(1+\frac{s_{i1}s_{j1}+s_{i2}s_{j2}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\right)\;.
\end{split}
\end{equation}
In the limit of strongly ordered soft emissions, $\mc{S}_{ij}^{(gg)}(1,2)$
reduces to $C_A\,\mc{S}_{ij}^{\rm(s.o.)}(1,2)$, where
\begin{equation}\label{eq:soft_so}
\mc{S}_{ij}^{\rm(s.o.)}(1,2)=
\frac{s_{ij}}{s_{i1}s_{12}s_{j2}}+\frac{s_{ij}}{s_{j1}s_{12}s_{i2}}
-\frac{s_{ij}^2}{s_{i1}s_{j1}s_{i2}s_{j2}}\;.
\end{equation}
The full real-emission corrections are obtained by adding the cut vacuum polarization diagrams
displayed in Fig.~\ref{fig:nlo_vpcg} and~\subref{fig:nlo_vpcq}, as well as the corresponding terms
with the gluons attached to the other Wilson line. They are given by~\cite{Monni:2011gb}
\begin{equation}\label{eq:real_collinear}
\begin{split}
\mc{C}_{ij}^{(q\bar{q})}(1,2)=&\;-\frac{T_R}{s_{12}^2}
\left(\frac{s_{i1}s_{i2}}{(s_{i1}+s_{i2})^2}+\frac{s_{j1}s_{j2}}{(s_{j1}+s_{j2})^2}\right)\\
\mc{C}_{ij}^{(gg)}(1,2)=&\;-(1-\varepsilon)\,\frac{C_A}{s_{12}^2}
\left(\frac{s_{i1}s_{i2}}{(s_{i1}+s_{i2})^2}+\frac{s_{j1}s_{j2}}{(s_{j1}+s_{j2})^2}\right)\\
\end{split}
\end{equation}
To simplify the integration, we define the soft remainder as well as
two collinear coefficients
\begin{equation}\label{eq:soft_helper_funcs}
\begin{split}
\mc{S}_{ij}^{\rm(rem)}(1,2)=&\;\mc{S}_{ij}^{\rm(s.o.)}(1,2)\,
\frac{s_{i1}s_{j2}+s_{i2}s_{j1}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\\
\mc{S}_{ij,B}^{\rm(coll)}(1,2)=&\;\frac{s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}
\frac{1}{s_{12}}\\
\mc{S}_{ij,A}^{\rm(coll)}(1,2)=&\;\mc{S}_{ij,B}^{\rm(coll)}(1,2)\,
4\,z_1z_2\cos^2\phi_{12}^{\,ij}\,
\qquad\text{where}\qquad
4\,z_1z_2\cos^2\phi_{12,ij}=\frac{(s_{i1}s_{j2}-s_{i2}s_{j1})^2}{
s_{12}s_{ij}(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\;.
\end{split}
\end{equation}
The precise meaning of $z$ and $\phi$ will be discussed in Sec.~\ref{sec:subtraction}.
In terms of the above functions we can write
\begin{equation}\label{eq:real_soft_2}
\begin{split}
\mc{S}_{ij}^{(q\bar{q})}(1,2)+\mc{C}_{ij}^{(q\bar{q})}(1,2)
=&\;T_R\left(\mc{S}_{ij,A}^{\rm(coll)}(1,2)-\mc{S}_{ij,B}^{\rm(coll)}(1,2)\right)\\
\mc{S}_{ij}^{(gg)}(1,2)+\mc{C}_{ij}^{(gg)}(1,2)=&\;
C_A\left((1-\varepsilon)\,\mc{S}_{ij,A}^{\rm(coll)}(1,2)-2\,\mc{S}_{ij,B}^{\rm(coll)}(1,2)
+\mc{S}_{ij}^{\rm(s.o.)}(1,2)-\frac{1}{2}\,\mc{S}_{ij}^{\rm(rem)}(1,2)\right)\;.
\end{split}
\end{equation}
\subsection{Kinematics}
\label{sec:kinematics}
We perform the calculation in a scheme that is applicable to both initial-
and final-state evolution. We parametrize the final-state momenta using two
light-like momenta $l$ and $n$ as
\begin{equation}\label{eq:def_sudakov_decomposition}
p^\mu=\alpha_p\,l^\mu+\beta_p\,n^\mu+p^\mu_T
\qquad\text{where}\qquad
\alpha_p=\frac{pn}{ln}\;,\qquad
\beta_p=\frac{pl}{ln}\;.
\end{equation}
The component along $l$ is denoted as $p^+$ and the component along $n$ as $p^-$.
The reference momenta for the Sudakov decomposition are defined in terms of
rescaled hard momenta,
\begin{equation}\label{eq:def_lc_momenta}
l^\mu=\frac{p_i^\mu}{\sqrt{1-\alpha_q-\beta_q-q^2/Q^2}}
\qquad\text{and}\qquad
n^\mu=\frac{p_j^\mu}{\sqrt{1-\alpha_q-\beta_q-q^2/Q^2}}\;,
\end{equation}
where $q=p_1$ in configurations with one, and $q=p_1+p_2$ in configurations with
two soft gluons, and where $Q^2=(p_i+p_j+q)^2$. This implies in particular that
$2ln=Q^2$, irrespective of the number of gluons in the final state, and that
$0<\alpha,\beta<1$ for any of the final-state momenta. We parametrize the integrations
over the soft momenta $p_1$ and $p_2$ as follows~\cite{Monni:2011gb}
\begin{equation}
d^Dp=\frac{1}{2}\,dp^+dp^-d^{D-2}p_T
=\frac{Q^2}{2}\,d\alpha_p d\beta_p\,d^{D-2}p_T\;.
\end{equation}
The transverse momentum integrals can be written as
\begin{equation}\label{eq:pt_int_1p}
\int d^{D-2}p_{T1}\,\delta^+(p_1^2)=
\Omega(2-2\varepsilon)\,Q^{-2\varepsilon}\big(\alpha_1\beta_1\big)^{-\varepsilon}\;,
\end{equation}
and
\begin{equation}\label{eq:pt_int_2p}
\int d^{D-2}p_{T1} d^{D-2}p_{T2}\,\delta^+(p_1^2)\delta^+(p_2^2)=
\Omega(2-2\varepsilon)\,Q^{-4\varepsilon}\big(\alpha_1\beta_1\,\alpha_2\beta_2\big)^{-\varepsilon}\,
\Omega(1-2\varepsilon)\int_0^\pi(\sin^2\phi)^{-\varepsilon}d\phi\;,
\end{equation}
where $\Omega(n)=2\pi^{n/2}/\Gamma(n/2)$ and where we have used the relation
$p^2=Q^2\alpha_p\beta_p-\mr{p}_T^2$ to perform the integrals over the magnitudes
of the transverse momenta. The remaining angular integral has to be carried out
differently for different powers of the invariant
$s_{12} = Q^2(\alpha_1\beta_2+\alpha_2\beta_1-2\sqrt{\alpha_1\beta_1\,\alpha_2\beta_2}\cos\phi_{12})$
that appears in the expressions of the soft current.
To parametrize the measurement as well as the mapping from four- to three-particle
topologies, we introduce the observables
\begin{equation}
\begin{split}
\mc{F}_{ij}(1)=&\;\delta(\alpha-2np_1/Q^2)\,\delta(\beta-2lp_1/Q^2)\\
\mc{F}_{ij}(1,2)=&\;\delta(\alpha-2np_{12}/Q^2)\,\delta(\beta-2lp_{12}/Q^2)\;.
\end{split}
\end{equation}
\subsection{Contributions at leading and next-to-leading order}
\label{sec:nlo_contributions}
The leading order momentum space soft function is given by the integral
of Eq.~\eqref{eq:lo_eik_fc}
\begin{equation}
\begin{split}
S_{ij}^{\rm(1)}(q)=&\;\int \frac{d^Dp_1}{(2\pi)^{D-1}}\,\delta^+(p_1^2)\,
\mc{S}_{ij}^{(0)}(1)\,\mc{F}_{ij}(1)\\
=&\;\frac{\Omega(2-2\varepsilon)}{(2\pi)^{3-2\varepsilon}}\,\frac{Q^2}{2}\,
(Q^2\alpha\beta)^{-\varepsilon}\,g_s^2\mu^{2\varepsilon}\,\frac{1}{Q^2\alpha\beta}
=\frac{\alpha_s^0(4\pi)^\varepsilon}{2\pi\,\Gamma(1-\varepsilon)}\,
\Big(\frac{\mu}{\kappa}\Big)^{2\varepsilon}\,\frac{Q^2}{\kappa^2}\;.
\end{split}
\end{equation}
To simplify the notation we have defined $\kappa^2=Q^2\alpha\beta$.
Next we replace the bare coupling, $\alpha_s^0$, by the renormalized one
in the $\overline{\rm MS}$ scheme,
\begin{equation}\label{eq:alphas_msbar}
\alpha_s^0=\alpha_s(\mu)\,\frac{e^{\varepsilon\gamma_E}}{(4\pi)^\varepsilon}
\left(1-\frac{1}{\varepsilon}\frac{\alpha_s(\mu)}{2\pi}\beta_0+\mc{O}(\alpha_s^2)\right)\;,
\qquad\text{where}\qquad
\beta_0=\frac{11}{6}\,C_A-\frac{2}{3}\,T_R n_f\;.
\end{equation}
Thus the leading-order soft function in the dipole shower scheme reads
\begin{equation}\label{eq:lo_soft}
S_{ij}^{\rm(0)}(q)=\frac{\bar{\alpha}_s}{2\pi}\,\frac{Q^2}{\kappa^2}\;,
\qquad\text{where}\qquad
\bar{\alpha}_s=\alpha_s(\mu)\,\frac{e^{\varepsilon\gamma_E}}{\Gamma(1-\varepsilon)}\,
\Big(\frac{\mu}{\kappa}\Big)^{2\varepsilon}\;.
\end{equation}
Similarly, the contribution from the virtual corrections, Eq.~\eqref{eq:virt_soft},
to the next-to-leading order soft dipole evolution is given by
\begin{equation}\label{eq:nlo_virt}
\begin{split}
S_{ij}^{\rm(virt)}(q)=&\;\int \frac{d^Dp_1}{(2\pi)^{D-1}}\,\delta^+(p_1^2)\,
\mc{S}_{ij}^{\rm(virt)}(1)\,\mc{F}_{ij}(1)\\
=&\;-\frac{\Omega(2-2\varepsilon)}{(2\pi)^{3-2\varepsilon}}\,\frac{Q^2}{2}\,
(Q^2\alpha\beta)^{-\varepsilon}\,C_A\frac{g_s^4}{8\pi^2}
\frac{(4\pi\mu^4)^\varepsilon}{\varepsilon^2}\frac{\Gamma^4(1-\varepsilon)\Gamma^3(1+\varepsilon)}{
\Gamma^2(1-2\varepsilon)\Gamma(1+2\varepsilon)}\Big(\frac{1}{Q^2\alpha\beta}\Big)^{1+\varepsilon}\\
=&\;-C_A\,\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{1}{\varepsilon^2}\,
\frac{\Gamma^5(1-\varepsilon)\Gamma^3(1+\varepsilon)}{\Gamma^2(1-2\varepsilon)\Gamma(1+2\varepsilon)}\,
\frac{Q^2}{\kappa^2}\;.
\end{split}
\end{equation}
The calculation of the real-emission contributions is straightforward but tedious.
We discuss the details in App.~\ref{sec:real_corrections}. The contribution
from the strong ordering approximation, Eq.~\eqref{eq:soft_so}, reads
\begin{equation}\label{eq:soft_so_int}
\begin{split}
S_{ij}^{\rm(so)}(q)=&\;
\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\left(\frac{1}{\varepsilon^2}-\frac{2}{3}\,\pi^2-14\,\varepsilon\,\zeta_3+\mc{O}(\varepsilon^2)\right)\;,\\
\end{split}
\end{equation}
The contributions from the soft remainder and the collinear terms,
Eq.~\eqref{eq:soft_helper_funcs}, are given by
\begin{equation}\label{eq:soft_rem_coll}
\begin{split}
S_{ij}^{\rm(rem)}(q)
=&\;\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\left(-\frac{2}{\varepsilon}-4-\frac{\pi^2}{3}
+\varepsilon\left(\frac{2}{3}\,\pi^2-8-10\,\zeta_3\right)+\mc{O}(\varepsilon^2)\right)\;,\\
S_{ij,gg}^{\rm(coll)}(q)
=&\;\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\left(\frac{5}{6\varepsilon}+\frac{31}{18}
+\varepsilon\left(\frac{94}{27}-\frac{5}{18}\pi^2\right)+\mc{O}(\varepsilon^2)\right)\;,\\
S_{ij,q\bar{q}}^{\rm(coll)}(q)
=&\;\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\left(\frac{2}{3\varepsilon}+\frac{10}{9}
+\varepsilon\left(\frac{56}{27}-\frac{2}{9}\pi^2\right)+\mc{O}(\varepsilon^2)\right)\;.
\end{split}
\end{equation}
In Sec.~\ref{sec:four_dimensions} we will devise a modified subtraction method
that allows to compute the coefficients of the above functions in four dimensions.
The results obtained here are used as a cross-check on the new technique.
\subsection{Complete next-to-leading order corrections}
\label{sec:nlo_corrections}
The complete Born-local next-to-leading order corrections to the soft function
in the dipole approach are given by the sum of Eqs.~\eqref{eq:nlo_virt},
\eqref{eq:soft_so_int} and~\eqref{eq:soft_rem_coll}.
The coupling renormalization, Eq.~\eqref{eq:alphas_msbar}, contributes
an additional counterterm
\begin{equation}\label{eq:alphas_ren}
\begin{split}
S_{ij}^{\rm(ren)}(q)=&\;-\frac{\alpha_s^2(\mu)}{(2\pi)^2}\,
\frac{e^{\varepsilon\gamma_E}}{\Gamma(1-\varepsilon)}\Big(\frac{\mu}{\kappa}\Big)^{2\varepsilon}\frac{Q^2}{\kappa^2}
\frac{\beta_0}{\varepsilon}\;.
\end{split}
\end{equation}
We finally obtain the fully differential two-loop momentum space soft function
\begin{equation}\label{eq:nlo}
\begin{split}
S_{ij}^{\rm(2)}(q)=&\;S_{ij}^{\rm(virt)}(q)+S_{ij}^{\rm(ren)}(q)+
C_A\left(S_{ij}^{\rm(s.o.)}(q)-\frac{S_{ij}^{\rm(rem)}(q)}{2}
+S_{ij,gg}^{\rm(coll)}(q)\right)-T_R\,n_f\;S_{ij,qq}^{\rm(coll)}(q)\\
=&\;\frac{\alpha_s^2(\mu)}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}\,
\frac{e^{2\varepsilon\gamma_E}}{\Gamma(1-\varepsilon)^2}\Big(\frac{\mu}{\kappa}\Big)^{4\varepsilon}
\left[\,\beta_0\left(\frac{1}{\varepsilon}\left(1-\frac{\Gamma(1-\varepsilon)}{e^{\varepsilon\gamma_E}}
\Big(\frac{\kappa}{\mu}\Big)^{2\varepsilon}\right)-\varepsilon\,\frac{\pi^2}{6}\right)
+\Gamma_{\rm cusp}^{(2)}+2\,\varepsilon\,\Gamma_{\rm soft}^{(2)}+\mc{O}(\varepsilon^2)\,\right]\\
\end{split}
\end{equation}
Note that Eq.~\eqref{eq:nlo} only depends on $\alpha$ and $\beta$ through $\kappa^2$,
which is a consequence of rescaling invariance in the soft limit~\cite{Gardi:2009qi,Becher:2009cu}.
The constant $\Gamma_{\rm cusp}^{(2)}$ is the well known two-loop cusp anomalous
dimension~\cite{Kodaira:1981nh,Davies:1984hs,Davies:1984sp,Catani:1988vd}
\begin{equation}\label{eq:two_loop_cusp}
\Gamma_{\rm cusp}^{(2)}=\left(\frac{67}{18}-\frac{\pi^2}{6}\right)C_A-\frac{10}{9}\,T_R\,n_f\;,
\end{equation}
and the constant $\Gamma_{\rm soft}^{(2)}$ is the two-loop soft anomalous dimension
computed in~\cite{Belitsky:1998tc,Li:2011zp},
\begin{equation}\label{eq:two_loop_soft}
\Gamma_{\rm soft}^{(2)}=\left(\frac{101}{27}-\frac{11}{72}\,\pi^2-\frac{7}{2}\,\zeta_3\right)C_A
-\left(\frac{28}{27}-\frac{\pi^2}{18}\right)\,T_R\,n_f\;.
\end{equation}
Using Eq.~\eqref{eq:plus_expansion} to expand Eqs.~\eqref{eq:lo_soft} and~\eqref{eq:nlo}
about the poles in the light-cone momenta $q^+$ and $q^-$, defined according to
Eq.~\eqref{eq:def_sudakov_decomposition}, we obtain
\begin{equation}\label{eq:lo_expanded}
\begin{split}
S_{ij}^{\rm(1)}(q)=&\;\frac{\alpha_s(\mu)}{2\pi}\,Q^2
\Bigg[\,L_{0,0}\left(\frac{1}{\varepsilon^2}-\frac{\pi^2}{12}\right)
+\frac{L_{0,1}}{\varepsilon}+(L_{0,2}+L_{1,1})+\mc{O}(\varepsilon)\,\Bigg]\;,
\end{split}
\end{equation}
and
\begin{equation}\label{eq:nlo_expanded}
\begin{split}
S_{ij}^{\rm(2)}(q)=&\;\frac{\alpha_s^2(\mu)}{(2\pi)^2}\,Q^2
\Bigg[\,\frac{L_{0,0}}{2}\left(-\frac{3\beta_0}{2\varepsilon^3}
+\frac{\Gamma_{\rm cusp}^{(2)}}{2\varepsilon^2}+\frac{\Gamma_{\rm soft}^{(2)}}{\varepsilon}
-\frac{\pi^2}{12}\,\Gamma_{\rm cusp}^{(2)}+\frac{\zeta_3}{3}\beta_0\right)\\
&+\frac{L_{0,1}}{2}\left(-\frac{\beta_0}{\varepsilon^2}+\frac{\Gamma_{\rm cusp}^{(2)}}{\varepsilon}
+2\Gamma_{\rm soft}^{(2)}-\frac{\pi^2}{6}\beta_0\right)
+\Big(L_{0,2}+L_{1,1}\Big)\Gamma_{\rm cusp}^{(2)}
+\Big(L_{0,3}+L_{1,2}\Big)\beta_0+\mc{O}(\varepsilon)\,\Bigg]\;.
\end{split}
\end{equation}
In this context we have defined the functions
\begin{equation}\label{eq:def_log_expansion}
\begin{split}
L_{0,0}=&\;\delta(q_+)\delta(q_-)\;,\\
L_{0,n}=&\;\frac{(-1)^n}{\mu}\left[\frac{\ln^{n-1}(q_+/\mu)}{q_+/\mu}\right]_+\delta(q_-)
+\frac{(-1)^n}{\mu}\left[\frac{\ln^{n-1}(q_-/\mu)}{q_-/\mu}\right]_+\delta(q_+)\;,\\
L_{n,m}=&\;\frac{(-1)^{n+m}}{1+\delta_{nm}}\bigg(\,
\frac{1}{\mu^2}\left[\frac{\ln^{n-1}(q_+/\mu)}{q_+/\mu}\right]_+
\left[\frac{\ln^{m-1}(q_-/\mu)}{q_-/\mu}\right]_+
+\frac{1}{\mu^2}\left[\frac{\ln^{n-1}(q_-/\mu)}{q_-/\mu}\right]_+
\left[\frac{\ln^{m-1}(q_+/\mu)}{q_+/\mu}\right]_+\bigg)\;.\\
\end{split}
\end{equation}
Note that only the two terms proportional to $L_{1,n}$ in Eq.~\eqref{eq:nlo_expanded}
contribute to the differential radiation pattern as $\kappa>0$. They correspond to a
next-to-leading order K-factor modifying the soft eikonal, such that the soft-gluon
emission probability becomes
\begin{equation}\label{eq:nlo_kappa}
S_{ij}^{\rm(2)}(q)\big|_{\kappa>0}=\frac{\alpha_s^2(\mu)}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\left[\,\beta_0\ln\frac{\mu^2}{\kappa^2}+\Gamma_{\rm cusp}^{(2)}+\mc{O}(\varepsilon)\,\right]\;.
\end{equation}
In the CMW scale scheme~\cite{Catani:1990rr} the $\Gamma_{\rm cusp}^{(2)}$ contribution
is absorbed into the definition of the strong coupling as
\begin{equation}\label{eq:cmw}
\alpha_s(\mu)\to\alpha_s(\mu)\left(1+\frac{\alpha_s(\mu)}{2\pi}\,\Gamma^{(2)}\right)\;.
\end{equation}
Upon setting $\mu_R=\kappa$ we can further eliminate the explicit $\beta_0$ term
in Eq.~\eqref{eq:nlo_kappa}~\cite{Amati:1980ch}. In this scheme, which is commonly
used in parton showers and dipole showers~\cite{Buckley:2011ms}, the Monte-Carlo simulation
correctly accounts for the effects of next-to-leading order soft QCD corrections
at the inclusive level, i.e.\ integrated over all real-emission configurations.
This approximation is valid in principle only for finite $\kappa$, whereas in the
double-soft limit additional corrections arise from the $L_{0,0}$ and $L_{0,n}$
terms in Eq.~\eqref{eq:nlo_expanded}. However, we will detail in the following
that the net effect of implementing two-loop soft corrections fully differentially
in the parton shower phase space indeed reduces to generating Eq.~\eqref{eq:nlo_kappa}
at the inclusive level, thereby confirming the findings of~\cite{Catani:1990rr}.
The connection to analytic soft-gluon resummation is established in App.~\ref{sec:resummation}.
\section{Implementation of the calculation in four dimensions}
\label{sec:four_dimensions}
A general scheme to implement higher-order corrections in parton showers in the form
of a modified local subtraction method was suggested in~\cite{Hoche:2017iem}.
Here we proceed to work out the details of the method in the double soft limit.
Regarding the divergence structure of the full double real corrections, this is
one of the most demanding regions due to the overlap between various singular
configurations, and it can be viewed as a part of the complete solution which
will include the simulation of higher-order corrections also in all
triple-collinear limits.
\subsection{Modified subtraction method}
\label{sec:ps_correspondence}
Our technique is based on the modified subtraction method discussed in~\cite{Frixione:2002ik}.
We identify the parton-shower splitting kernels with generalized factorization terms
in the $\overline{\rm MS}$ scheme. These terms can be computed by expanding the differential
cross section for a particular final state of interest in terms of plus distributions
corresponding to light-cone singularities along the directions of the fast partons.
Schematically, for a process with no infrared divergences at the leading order,
we can use the next-to-leading order factorization formula~\cite{Catani:1996vz} for
real-emission corrections
\begin{equation}\label{eq:cs_real}
d\sigma_{n+1}=d\Phi^{(n)}\sum_{i<j,k}\mc{D}_{ij,k}\big(\Phi^{(n)}\big)\;,
\end{equation}
where
\begin{equation}\label{eq:cs_dipole}
\mc{D}_{ij,k}\big(\Phi^{(n)}\big)=
d\Phi^{(+1)}_{ij,k}\,\big|M_{n}^{ij,k}\big(\Phi^{(n)},\Phi^{(+1)}_{ij,k}\big)\big|^2\;
\frac{\alpha_s}{2\pi}\,\frac{1}{s_{ij}}\,\hat{V}_{ij,k}\big(\Phi^{(+1)}_{ij,k}\big)\;.
\end{equation}
In this context, $|M_{n}^{ij,k}|^2$ are the color-correlated Born matrix elements for
the $n$-particle final state, and $d\Phi^{(n)}$ is the corresponding differential
phase-space element. The $\hat{V}_{ij,k}$ are the dipole insertion operators defined
in~\cite{Catani:1996vz}. They reduce to $-{\bf T}_{ij}{\bf T}_k\,\mc{S}_{ik}^{(0)}(j)$
in the soft limit, cf.\ Eq.~\eqref{eq:leading_color}. The corresponding one-emission
differential phase-space element is given by $d\Phi^{(+1)}_{ij,k}$.
The Monte-Carlo integration of NLO real-emission corrections in four dimensions can now
be performed by subtracting Eq.~\eqref{eq:cs_real} from the real-emission corrections
and computing only the remainder, while the subtraction terms $\mc{D}_{ij,k}$ are
usually integrated over $d\Phi^{(+1)}_{ij,k}$ analytically to extract the poles in $\varepsilon$.
We will instead perform these integrals in a Monte-Carlo fashion. We first parametrize
the emission phase space in the collinear limit $s_{ij}\to 0$ in terms of the virtuality
$t=s_{ij}$ and the light-cone momentum fraction $z=s_{ik}/(s_{ik}+s_{jk})$ for final state
radiation and $z=1-s_{jk}/s_{ik}$ for initial-state radiation
\begin{equation}\label{eq:coll_phasespace_ddim}
\begin{split}
d\Phi^{\rm(+1),F/I}_{ij,k}=\Omega(1-2\varepsilon)\,dt\,dz\;t^{-\varepsilon}(1-z)^{-\varepsilon}z^{\mp\varepsilon}(\sin^2\phi_i)^{-\varepsilon}\;.
\end{split}
\end{equation}
Note the sign of the exponent of the $z^{\mp\varepsilon}$ term, which is negative for emissions
from final-state particles and positive for initial-state radiation. The integrand
in Eq.~\eqref{eq:cs_dipole} can now be expanded in in powers of the dimensional regularization
parameter, $\varepsilon$, using the relation
\begin{equation}\label{eq:plus_expansion}
\frac{1}{v^{1+\varepsilon}}=-\frac{1}{\varepsilon}\,\delta(v)+\sum_{i=0}^\infty\frac{\varepsilon^n}{n!}\left(\frac{\ln^n v}{v}\right)_+\;,
\end{equation}
which is applied to both the $t$ and the $z$ integral. The $1/\varepsilon$ poles generated
in this manner will cancel against the virtual corrections and renormalization terms.
This produces a non-locality of the finite remainder which is corrected by the resummation,
as the first-order expansion of the parton-shower generates the complementary distribution
of the real-emission corrections in phase space~\cite{Frixione:2002ik}.
In order to compute the finite remainder, we simply need to compute the $\mc{O}(\varepsilon^0)$
terms of Eq.~\eqref{eq:plus_expansion} applied to Eq.~\eqref{eq:cs_dipole}. This can be done
fully differentially in the remaining phase-space variables, however we need to take into
account that the underlying $n$-particle phase space and matrix element have an $\varepsilon$
dependence that contributes finite terms when combined with the poles from real and
virtual corrections. This technique was used in~\cite{Hoche:2017iem} to obtain the matching
coefficients for the flavor-changing splitting functions. In the following we will describe
how it is implemented in the context of the two-loop soft corrections.
\subsection{Separation of iterated double-collinear endpoints}
\label{sec:ep_separation}
\begin{figure}[t]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.5]{Fig3a.pdf}
\label{fig:ep_separation_lo}}\hskip 5mm
\subfigure[]{
\includegraphics[scale=0.5]{Fig3b.pdf}
\label{fig:ep_separation_nlo}}
\end{center}
\caption{Illustration of the kinematical configurations corresponding to the
endpoint contributions $L_{0,n}$ in Eq.~\eqref{eq:lo_nlo_expanded} in one-particle
(left) and two-particle (right) emissions. Note in particular that all partons
in the two-particle configuration are forced to be collinear to the same Wilson line,
cf.\ the explanation in Sec.~\ref{sec:ep_separation}.
\label{fig:ep_separation}}
\end{figure}
First, we must account for the fact that there is no equivalent of $L_{n,m}$
in the parton shower. The factorized plus distributions are instead replaced by a
double-plus distribution and two related endpoint terms. We define the double plus
distribution by its action on a test function
\begin{equation}\label{eq:double_plus}
\big[f(x,y)\big]_{++}\,g(x,y)=f(x,y)\Big(g(x,y)-g(0,0)\Big)\;.
\end{equation}
Using this relation, we can write Eqs.~\eqref{eq:lo_expanded} and~\eqref{eq:nlo_expanded} as
\begin{equation}\label{eq:lo_nlo_expanded}
\begin{split}
S_{ij}^{\rm(1)}(q)=&\;\frac{\alpha_s(\mu)}{2\pi}\,Q^2
\Bigg[\,\frac{L_{0,0}}{\varepsilon^2}+\frac{L_{0,1}}{\varepsilon}-\frac{\pi^2}{12}\,L_{0,0}
+\Big(L_{0,2}-L_{0,1}\Big)+\tilde{L}_{1,1}
+\mc{O}(\varepsilon)\,\Bigg]\;,\\
S_{ij}^{\rm(2)}(q)=&\;\frac{\alpha_s^2(\mu)}{(2\pi)^2}\,Q^2
\Bigg[\,\frac{L_{0,0}}{2}\left(-\frac{3\beta_0}{2\varepsilon^3}
+\frac{\Gamma_{\rm cusp}^{(2)}}{2\varepsilon^2}+\frac{\Gamma_{\rm soft}^{(2)}}{\varepsilon}\right)
+\frac{L_{0,1}}{2}\left(-\frac{\beta_0}{\varepsilon^2}+\frac{\Gamma_{\rm cusp}^{(2)}}{\varepsilon}\right)
+\frac{L_{0,0}}{2}\left(-\frac{\pi^2}{12}\,\Gamma_{\rm cusp}^{(2)}+\frac{\zeta_3}{3}\beta_0\right)\\
&+L_{0,1}\Gamma_{\rm soft}^{(2)}+\left(L_{0,3}-L_{0,2}-\frac{\pi^2}{12}\,L_{0,1}\right)\beta_0
+\Big(L_{0,2}-L_{0,1}\Big)\,\Gamma_{\rm cusp}^{(2)}
+\tilde{L}_{1,2}\,\beta_0+\tilde{L}_{1,1}\Gamma_{\rm cusp}^{(2)}+\mc{O}(\varepsilon)\,\Bigg]\;,
\end{split}
\end{equation}
where
\begin{equation}\label{eq:def_log_expansion_2}
\begin{split}
\tilde{L}_{n,m}=&\;\frac{(-1)^{n+m}}{1+\delta_{nm}}\,
\frac{1}{\mu^2}\left[\frac{\mu^2}{q_+q_-}
\left(\ln^{n-1}\frac{q_+}{\mu}\ln^{m-1}\frac{q_-}{\mu}
+\ln^{m-1}\frac{q_+}{\mu}\ln^{n-1}\frac{q_-}{\mu}\right)\right]_{++}\;.
\end{split}
\end{equation}
Note that because $L_{0,n}$ is located at either $q_-=0$ or $q_+=0$,
the corresponding terms are not included in a standard parton shower.
In order to add these contributions we will need to implement endpoint
terms in the (iterated) double collinear limit. The relevant kinematical
configurations are depicted in Fig.~\ref{fig:ep_separation}. They can
be explained as follows: Suppose that a single soft momentum $q$ is emitted
off the two light-like partons $i$ and $j$ as in Fig~\ref{fig:ep_separation_lo}.
As $q_-=0$ or $q_+=0$, Eqs.~\eqref{eq:def_sudakov_decomposition}
and~\eqref{eq:def_lc_momenta} imply $q\,||\,p_i$ or $q\,||\,p_j$,
hence all terms $L_{0,n}$ are related to single soft gluon radiation
in the collinear limit. In the case of double-soft radiation of momenta
$p_1$ and $p_2$, depicted in Fig.~\ref{fig:ep_separation_nlo},
the situation is similar, but slightly more involved. Because the
radiated partons are both in the final state, $s_{i1}$ and $s_{i2}$
must have the same sign. The limit $q_-=0$ then implies that
$s_{i1}+s_{i2}=0$, which can only be fulfilled if $s_{i1}=s_{i2}=0$.
Therefore, $p_i||\,p_1$ and $p_i||\,p_2$, which leads to $p_1||\,p_2$,
such that $s_{12}=0$ and $s_{i12}=0$. The conclusion is that all
$L_{0,n}$ terms correspond to the regions where soft emissions
are collinear to one of the Wilson lines. If there are two emissions,
they must be collinear to the same Wilson line.
The change in color flow generated by the soft radiation then reduces
the phase space available to subsequent gluon radiation by a factor
proportional to the light-cone momentum fraction of the gluon that is
color-adjacent to the anti-collinear Wilson line. Let us assume that
the corresponding dipole is spanned by $p_j$ and $p_1$, then the phase
space for subsequent gluon radiation is $\alpha_1Q^2$. As we are
interested in the soft gluon limit, $\alpha_1\to 0$, the remaining
phase space is typically close to zero. Any further QCD radiation
from the $j1$-dipole will be suppressed by $\alpha_1$, and radiation
from the remaining dipoles cannot occur because $s_{i1}=s_{i2}=s_{12}=0$.
It follows that the effect of the collinear configurations
corresponding to $L_{0,n}$ is to generate a radiator of reduced
invariant mass, oriented along the light-cone directions of the
original Wilson lines. The phenomenologically relevant branching probability
for such configurations cannot be determined in the double soft limit alone,
but requires in addition the computation of endpoint contributions
in the triple-collinear limit. We therefore postpone the discussion
of these terms to a forthcoming publication.
\subsection{Differential subtraction terms}
\label{sec:subtraction}
We will now derive the modified subtraction terms needed for implementing
the two-loop soft corrections in the parton-shower. Due to the non-abelian
exponentiation theorem~\cite{Cornwall:1975ty,Frenkel:1976bj} it is sufficient
to consider the gluon splitting functions and include the complete soft eikonal
instead of the partial-fractioned term, Eq.~\eqref{eq:leading_color}.
However, we must include sub-leading color configurations, corresponding
to double soft-gluon radiation off the hard Wilson lines in order to account
for coherence effects. The subtraction terms related to the functions
$\tilde{L}_{n,m}$ in Eq.~\eqref{eq:lo_nlo_expanded} can be defined as
\begin{equation}\label{eq:real_ps}
\begin{split}
\mc{S}_{ij}^{(q\bar{q})\rm ct}(1,2)=&\;T_R\left(\mc{S}_{ij}^{\rm(coll),2}(1,2)
-2\,\mc{S}_{ij}^{\rm(coll),1}(1,2)\right)\;,\\
\mc{S}_{ij}^{(gg)\rm ct}(1,2)=&\;C_A\left(\mc{S}_{ij}^{\rm(s.o.)}(1,2)
-\frac{1}{2}\,\mc{S}_{ij}^{\rm(rem)}(1,2)-2\,\mc{S}_{ij,B}^{\rm(coll)}(1,2)
+(1-\varepsilon)\,\mc{S}_{ij}^{\rm(coll),1}(1,2)\right)\;.
\end{split}
\end{equation}
where $\mc{S}_{ij}^{\rm(s.o.)}$, $\mc{S}_{ij}^{\rm(rem)}$ and
$\mc{S}_{ij}^{\rm(coll)}$ are given by Eqs.~\eqref{eq:soft_so}
and~\eqref{eq:soft_helper_funcs}, respectively.
In the collinear limit, Eqs.~\eqref{eq:real_ps} reduce to
\begin{equation}\label{eq:real_ps_coll}
\mc{S}_{ij}^{\rm\, ct}(1,2)=g_{\mu\rho}g_{\nu\sigma}\,
J_{ij}^{\mu}(1,2)J_{ij}^{*\nu}(1,2)\,
\frac{P^{\rho\sigma}(z_1)}{s_{12}}\;,
\end{equation}
where $z_1$ is the light-cone momentum fraction of $p_1$ in the direction of
$p_1+p_2-s_{12}/(s_{n1}+s_{n2})n$, with $n$ an auxiliary light-like vector
satisfying $(p_1+p_2)n\neq 0$. The spin-dependent DGLAP splitting functions,
$P^{\mu\nu}(z)$, are given by
\begin{equation}\label{eq:dglap_corr}
\begin{split}
P_{gq}^{\mu\nu}(z)=&\;T_R\left(-g^{\mu\nu}
+4\,z(1-z)\,\frac{k_\perp^\mu k_\perp^\nu}{k_\perp^2}\right)\;,\\
P_{gg}^{\mu\nu}(z)=&\;2C_A\left(-g^{\mu\nu}\left(\frac{z}{1-z}+\frac{1-z}{z}\right)
-2\,(1-\varepsilon)z(1-z)\,\frac{k_\perp^\mu k_\perp^\nu}{k_\perp^2}\right)\;.
\end{split}
\end{equation}
The soft gluon current $J_{ij}^\mu$ is given by the standard expression in the eikonal limit.
\begin{equation}\label{eq:eikonal_current_1}
J_{ij}^{\mu}(q)=\frac{p_i^\mu}{2p_iq}-\frac{p_j^\mu}{2p_jq}\;.
\end{equation}
Note the minus sign in this expression, which arises from color conservation along
the hard Wilson line, i.e.\ ${\bf T}_i=-{\bf T}_j$. In processes with a non-trivial
color structure this condition holds only at leading color.
We rewrite Eq.~\eqref{eq:eikonal_current_1} such that the transversality of the current
becomes manifest:
\begin{equation}\label{eq:eikonal_current_2}
J_{ij}^{\mu}(q)=\sqrt{\frac{p_ip_j}{2\,(p_iq)(p_jq)}}\;j_{ij,\perp}^\mu(q)\;,
\qquad\text{where}\qquad
j_{ij,\perp}^{\mu}(q)=\frac{(p_jq)\,p_i^\mu-(p_iq)\,p_j^\mu}{
\sqrt{2\,(p_ip_j)(p_iq)(p_jq)}}\;.
\end{equation}
The transverse momentum in Eq.~\eqref{eq:dglap_corr} can be parametrized as
$k_\perp^\mu=j_{12,\perp}^{\mu}(n)$.
We can now prove that $\phi_{12}^{ij}$ defined in Eq.~\eqref{eq:soft_helper_funcs}
is indeed an azimuthal angle, as $\cos\phi_{12}^{ij}=k_\perp\,j_{ij,\perp}(p_1+p_2)$,
and we can replace $z_1z_2\to s_{n1}s_{n2}/(s_{n1}+s_{n2})^2$. In order to obtain
the correct differential radiation pattern in the leading-order simulation,
we implement $2\cos^2\phi_{12}^{ij}$ as a correction factor applied to the
purely collinear parts of the $g\to q\bar{q}$ and $g\to gg$ splitting functions,
see Sec.~\ref{sec:mc} for details.
The pure soft terms of Eq.~\eqref{eq:real_ps} can be rewritten as
\begin{equation}\label{eq:psct_soft_sum}
\mc{S}_{ij}^{\rm(sct)}(1,2)=
\frac{1}{2}\left(\mc{S}_{ij,A}^{\rm(sct)}(1,2)+
\mc{S}_{ij}^{\rm(s.o.)}(1,2)\right)\;.
\end{equation}
where
\begin{equation}\label{eq:psct_soft}
\mc{S}_{ij,A}^{\rm(sct)}(1,2)=
\mc{S}_{ij}^{\rm(s.o.)}(1,2)-\mc{S}_{ij}^{\rm(rem)}(1,2)=
\frac{s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\left(
\frac{s_{i2}}{s_{i1}s_{12}}+\frac{s_{j2}}{s_{j1}s_{12}}
-\frac{s_{ij}}{s_{i1}s_{j1}}\right)+\Big(1\leftrightarrow 2\Big)\;.
\end{equation}
The first contribution in Eq.~\eqref{eq:psct_soft} can be interpreted as the
eikonal expression for emission of the combined soft-gluon cluster ${12}$
from the hard Wilson lines $i$ and $j$, and the subsequent radiation
of gluon $2$ off the leading-color dipoles spanned by $i1$, $j1$ or the
sub-leading color dipole spanned by $ij$. The second term describes
the same situation with the two gluons interchanged. The last term is
a negative contribution arising from the dipole spanned by $i$ and $j$.
This contribution is sub-leading in the global $1/N_c$ expansion, but it
contributes at leading color in the double-soft limit and must therefore
be included in the parton-shower simulation as the first correction
to leading-color evolution. Partial fractioning Eq.~\eqref{eq:psct_soft}
following the approach in~\cite{Catani:1996vz}, we find
\begin{equation}\label{eq:psct_soft_dec}
\mc{S}_{ij,A}^{\rm(sct)}(1,2)=
\mc{S}_{i,j,A}^{\rm(sct)}(1,2)
+\Big(1\leftrightarrow 2\Big)
+\Big(i\leftrightarrow j\Big)
+\Big(\begin{array}{c}1\leftrightarrow 2\\
i\leftrightarrow j\end{array}\Big)\;,
\end{equation}
where
\begin{equation}\label{eq:psct_soft_tc}
\begin{split}
\mc{S}_{i,j,A}^{\rm(sct)}(1,2)=&\;
\frac{s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\;
\left[\,\frac{1}{s_{12}}\frac{s_{i2}}{s_{i1}+s_{12}}
+\frac{1}{s_{i1}}\left(\frac{s_{i2}}{s_{i1}+s_{12}}-\frac{s_{ij}}{s_{i1}+s_{j1}}\right)
\,\right]\;.
\end{split}
\end{equation}
Equation~\eqref{eq:psct_soft_tc} can be interpreted as the soft enhanced part
of the dipole shower splitting function in the limit where partons $i$, 1 and 2
become triple-collinear, with parton $j$ defining the anti-collinear direction.
Note that in the $i1$-collinear limit, Eq.~\eqref{eq:psct_soft_tc} develops
an integrable singularity that vanishes upon azimuthal integration.
This problem will be discussed in Sec.~\ref{sec:mc}.
The only remaining two-particle singularity is approached as partons $1$ and $2$
become collinear.
The integrals of Eq.~\eqref{eq:psct_soft} have been computed in
Eqs.~\eqref{eq:soft_so_int} and~\eqref{eq:soft_rem}. They combine to give
\begin{equation}\label{eq:pscoll_sum}
\begin{split}
S_{ij,A}^{\rm(sct)}(q)
=\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\bigg[\,&\left(\frac{11}{6\varepsilon}+\frac{67}{18}-\frac{\pi^2}{3}
+\varepsilon\left(\frac{202}{27}-\frac{11}{18}\pi^2-4\zeta_3\right)\right)C_A\\
&\;-\left(\frac{2}{3\varepsilon}+\frac{10}{9}
+\varepsilon\left(\frac{56}{27}-\frac{2}{9}\pi^2\right)\right)T_R n_f
+\mc{O}(\varepsilon^2)\,\bigg]\;.
\end{split}
\end{equation}
Upon defining approximate virtual corrections as
\begin{equation}\label{eq:psvirt}
\begin{split}
\frac{\bar{\alpha}_s^2}{(2\pi)^2}\,\frac{Q^2}{\kappa^2}
\;C_A\left(-\frac{1}{\varepsilon^2}+\frac{\pi^2}{6}-3\,\varepsilon\zeta_3\right)\;,
\end{split}
\end{equation}
we would readily obtain the desired result, Eq.~\eqref{eq:nlo}, at $\mc{O}(\varepsilon)$.
We have verified that the corresponding subtracted real-emission contribution
could be computed directly in four dimensions and cross-checked the finite
term against the difference between Eqs.~\eqref{eq:nlo} and~\eqref{eq:pscoll_sum}.
Nevertheless, $\mc{S}_{ij,A}^{\rm(sct)}$ is not a suitable local subtraction term
for Monte-Carlo simulation, because the difference to the full real-emission
correction contains integrable singularities. In the following section, we will
therefore devise a technique to simulate the complete soft subtraction term,
Eq.~\eqref{eq:psct_soft_sum}, by reweighting the leading-order parton shower.
\subsection{Monte Carlo implementation details}
\label{sec:mc}
We employ the techniques described in~\cite{Hoche:2015sya,Hoche:2017iem}
to generate the final-state momenta, and we evaluate the splitting functions
directly in terms of the kinematic invariants $s_{nm}$ with $n,m\in\{1,2,i,j\}$.
The kinematics mapping in $2\to 3$ branchings is based on~\cite{Catani:1996vz,Catani:2002hc}
and is summarized in App.~A of~\cite{Hoche:2015sya}. The kinematics mapping
and (D-dimensional) phase-space factorization in $2\to 4$ splittings was
derived in~\cite{Hoche:2017iem}, App.~A. Note that in both cases we use the
Lorentz invariant and numerically stable technique of~\cite{Hoche:2017iem}
to construct the transverse components of the momenta.
In order to simulate Eq.~\eqref{eq:psct_soft_tc} in the parton shower, we must
correct the leading-order soft radiation pattern. First we need to account
for the fact that the eikonal generated by the leading-order parton shower
is not identical to $s_{ij}/((s_{i1}+s_{i2})(s_{j1}+s_{j2}))$ if the soft-gluon
emission is followed by a subsequent branching of any of the emerging momenta.
In the transition $(\widetilde{\imath 12},\tilde{\jmath})\to(\tilde{\imath},\widetilde{12},j)$
followed by $(\widetilde{12},\tilde{\imath})\to (1,2,i)$, we obtain instead
the following probability for the emission of the final soft cluster $\widetilde{12}$
\begin{equation}\label{eq:eikonal_from_iterated_ps_12}
\begin{split}
\frac{\tilde{p}_ip_j}{2(\tilde{p}_i\tilde{p}_{12})(\tilde{p}_{12}p_j)}
=&\;\bigg[\,p_ip_j\,\frac{p_ip_1+p_ip_2\pm p_1p_2}{p_ip_1+p_ip_2}\bigg]
\left[2\bigg(p_ip_1+p_ip_2\pm p_1p_2\bigg)
\bigg(p_jp_1+p_jp_2-\frac{p_1p_2}{p_ip_1+p_ip_2}\;p_ip_j\bigg)\right]^{-1}\\
=&\;\frac{s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}\;.
\end{split}
\end{equation}
The similarity of the kinematics mapping in final-state splittings
with a final- and initial-state spectator~\cite{Catani:1996vz,Hoche:2015sya}
implies that Eq.~\eqref{eq:eikonal_from_iterated_ps_12} holds for both final-
and initial-state Wilson lines, $i$ (corresponding to the $\pm$ sign).
Note that the term proportional to $s_{12}$ in the denominator cannot be
neglected in the double-soft limit. We can correct the mismatch between
Eq.~\eqref{eq:eikonal_from_iterated_ps_12} and the target distribution
$s_{ij}/((s_{i1}+s_{i2})(s_{j1}+s_{j2}))$ in Eq.~\eqref{eq:psct_soft}
by applying a reweighting factor in the branching of the soft gluon
$(\widetilde{12},\tilde{\imath})\to (1,2,i)$
\begin{equation}\label{eq:psct_soft_tc_12_weight}
w_{ij}^{12}=1-\frac{s_{ij}s_{12}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})}\;.
\end{equation}
In the transition $(\widetilde{\imath 12},\tilde{\jmath})\to(\widetilde{\imath 1},\tilde{2},j)$
followed by $(\widetilde{\imath 1},\tilde{2})\to (i,1,2)$, with $i$ in the final state,
we obtain the following probability for the emission of the final soft cluster
$\widetilde{12}$
\begin{equation}\label{eq:eikonal_from_iterated_ps_i1}
\begin{split}
\frac{\tilde{p}_{i1}p_j}{2(\tilde{p}_{i1}\tilde{p}_2)(\tilde{p}_2p_j)}
=&\;\bigg[\,p_ip_j+p_1p_j-\frac{p_ip_1}{p_ip_2+p_1p_2}\;p_2p_j\bigg]
\left[2\bigg(p_ip_2+p_1p_2+p_ip_1\bigg)
\bigg(\frac{p_ip_2+p_1p_2+p_ip_1}{p_ip_2+p_1p_2}\;p_2p_j\bigg)\right]^{-1}\\
=&\;\frac{(s_{ij}+s_{j1})(s_{i2}+s_{12})-s_{i1}s_{j2}}{(s_{i1}+s_{i2}+s_{12})^2s_{j2}}\;.
\end{split}
\end{equation}
If the radiator $\widetilde{\imath 1}$ is in the initial state, we obtain instead
Eq.~\eqref{eq:eikonal_from_iterated_ps_12} with $\pm\to -$. The weight factor arising
from Eq.~\eqref{eq:eikonal_from_iterated_ps_i1} generates pseudo-singularities in the
parton shower phase space, which is undesirable in a Monte-Carlo simulation.
We will therefore choose to implement a different strategy in the leading order
parton shower. The kinematics in the soft enhanced part of the $i1$-collinear emission
will be chosen according to the identified particle prescription of~\cite{Catani:1996vz}.
Note that due to our definition of the evolution and splitting variable in final-state
evolution with final-state spectator~\cite{Hoche:2015sya}, the Jacobian factor related
to this modification is unity. Eventually, all kinematical correction factors
are then given by Eq.~\eqref{eq:psct_soft_tc_12_weight}.
Upon including the phase-space correction factors, the collinear terms in the gluon splitting
functions implementing the spin correlations present in Eq.~\eqref{eq:real_ps_coll} read
\begin{equation}\label{eq:psct_soft_tcsfs}
\begin{split}
P_{gg,ij}^{\rm(coll)}(1,2)=&\;2C_A\,z(1-z)\,2\,w_{ij}^{12}\,\cos^2\phi_{12}^{ij}\;,\\
P_{gq,ij}^{\rm(coll)}(1,2)=&\;-T_R\,2\,z(1-z)\,2\,w_{ij}^{12}\,\cos^2\phi_{12}^{ij}\;.
\end{split}
\end{equation}
The remaining phase-space effects leading to $\mc{S}_{i,j,A}^{\rm(sct)}$
are taken into account by multiplying the $1$-soft parts of the $i1$- and
$12$-collinear splitting functions by $w_{ij}^{12}$.
Finally, we need to account for the additional strongly ordered term in
Eq.~\eqref{eq:psct_soft_sum}. This is achieved by means of the identities
\begin{equation}\label{eq:psct_softso_identity}
\begin{split}
\frac{s_{ij}}{s_{i1}s_{12}s_{j2}}=&\;
\frac{s_{i2}}{s_{i1}s_{12}}\left[\frac{s_{ij}}{s_{i2}s_{j2}+s_{i1}s_{j1}}\right]
+\frac{s_{j1}}{s_{j2}s_{12}}\left[\frac{s_{ij}}{s_{i2}s_{j2}+s_{i1}s_{j1}}\right]\;,\\
\frac{s_{ij}^2}{s_{i1}s_{j1}s_{i2}s_{j2}}=&\;
\frac{s_{ij}}{s_{i1}s_{j1}}\left[\frac{s_{ij}}{s_{i1}s_{j1}+s_{i2}s_{j2}}\right]
+\frac{s_{ij}}{s_{i2}s_{j2}}\,\left[\frac{s_{ij}}{s_{i1}s_{j1}+s_{i2}s_{j2}}\right]\;.
\end{split}
\end{equation}
Using Eq.~\eqref{eq:eikonal_from_iterated_ps_12} we can then write
\begin{equation}\label{eq:psct_soft_tc_full}
\begin{split}
\mc{S}_{i,j}^{\rm(sct)}(1,2)=&\;
\frac{s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}
\left[\,\frac{1}{s_{12}}\frac{s_{i2}}{s_{i1}+s_{12}}
+\frac{1}{s_{i1}}\left(\frac{s_{i2}}{s_{i1}+s_{12}}
-\frac{s_{ij}}{s_{i1}+s_{j1}}\right)\,\right]
\frac{w_{ij}^{12}+\bar{w}_{ij}^{12}}{2}\;,
\end{split}
\end{equation}
where we have defined the weight factor
\begin{equation}\label{eq:psct_soft_tc_so_weight}
\bar{w}_{ij}^{12}=\frac{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}{s_{i1}s_{j1}+s_{i2}s_{j2}}\;.
\end{equation}
Note that the negative contribution in Eq.~\eqref{eq:psct_soft_tc_full}
does not have a parton-shower correspondence. At the same time, we have
so far omitted the squared leading-order contribution arising from
Eq.~\eqref{eq:lo_eik_fc}. We can correct for both mismatches by adding
a subleading color contribution to the $i1$-collinear terms of the
splitting function of the Wilson lines. This term reads
\begin{equation}\label{eq:psct_soft_slc_1}
P_{ij,A}^{\rm(slc)}(1,2)=\frac{2\,s_{ij}}{s_{i1}+s_{j1}}\,
\frac{w_{ij}^{12}+\bar{w}_{ij}^{12}}{2}
\Big(\bar{C}_{ij}-C_A\Big)\;
\qquad\text{where}\qquad
\bar{C}_{ij}=\left\{\begin{array}{cc}
2C_F&\text{if $i$ \& $j$ quarks}\\
C_A&\text{else}\end{array}\right.\;.
\end{equation}
The weight factor of the $\bar{C}_{ij}$ term in Eq.~\eqref{eq:psct_soft_slc_1}
was derived by considering its diagrammatic representation, which arises from
the Abelian parts of Figs.~\ref{fig:nlo_box1} and~\subref{fig:nlo_box2}~\cite{Monni:2011gb}.
We may also consider the result of the integration in Sec.~\ref{sec:analytic}
and its Fourier transform in impact parameter space, cf.\ \ App.~\ref{sec:resummation}.
In fact, Eqs.~\eqref{eq:soft_so_3} and~\eqref{eq:soft_rem_3} generate the
exact same result up to $\mc{O}(1)$ as the square of the leading-order term,
Eq.~\eqref{eq:lo_b}, hence proving that Eq.~\eqref{eq:psct_soft_slc_1} is a
valid form in the double-soft region that will allow us to reproduce the
squared leading-order term at the integrated level. In our numerical implementation
we include Eq.~\eqref{eq:psct_soft_slc_1} in the $i1$-collinear sector with spectator
$2$. This means that we mis-identify in principle the related evolution variable,
which should be $\kappa^2$ in the notation of~\cite{Hoche:2015sya}, and hence
proportional to $s_{i1}s_{1j}$ instead of $s_{i1}s_{12}$. We correct for this effect
by reweighting with a ratio of strong couplings, taken at the current vs.\ the correct
evolution variable, and by setting Eq.~\eqref{eq:psct_soft_slc_1} to zero as the
evolution variable falls below the parton-shower cutoff.
A second sub-leading color contribution is given by the difference
\begin{equation}\label{eq:psct_soft_slc_2}
P_{ij,B}^{\rm(slc)}(1,2)=\frac{2\,s_{i2}}{s_{i1}+s_{12}}\,
\frac{w_{ij}^{12}+\bar{w}_{ij}^{12}}{2}\,\Big(C_A-\bar{C}_{ij}\Big)\;.
\end{equation}
It accounts for the fact that the second soft emission off the Wilson lines
occurs with the color charge $C_A$ due to the interference with a color octet.
We can now define the combined sub-leading color contribution to the
parton-shower evolution as
\begin{equation}\label{eq:psct_soft_slc}
P_{ij}^{\rm(slc)}(1,2)=
P_{ij,A}^{\rm(slc)}(1,2)+P_{ij,B}^{\rm(slc)}(1,2)=
\Big(C_A-\bar{C}_{ij}\Big)
\left(\frac{2\,s_{i2}}{s_{i1}+s_{12}}
-\frac{2\,s_{ij}}{s_{i1}+s_{j1}}\right)\,
\frac{w_{ij}^{12}+\bar{w}_{ij}^{12}}{2}\;.
\end{equation}
Note that $P_{ij}^{\rm(slc)}$ vanishes in the $i1$-collinear limit,
such that the correct color factor is recovered in collinear evolution.
The remaining parts of the improved leading-order, fully differential
splitting functions related to the $12$-collinear, $1$-soft final-state
singularities are given by the leading-color expressions
\begin{equation}\label{eq:final_ps_kernels}
\begin{split}
(P_{qq})_2^k(1,i)=&\;C_F\left(\frac{2\,s_{i2}}{s_{i1}+s_{12}}
\frac{w_{2k}^{1i}+\bar{w}_{2k}^{1i}}{2}\right)\;,\\
(P_{gg})_{ij}(1,2)=&\;
C_A\,\left(\frac{2\,s_{i2}}{s_{i1}+s_{12}}\frac{w_{ij}^{12}+\bar{w}_{ij}^{12}}{2}
+w_{ij}^{12}\left(-1+z(1-z)\,2\cos^2\phi_{12}^{ij}\right)\right)\;,\\
(P_{gq})_{ij}(1,2)=&\;T_R\,w_{ij}^{12}\left(1-4z(1-z)\cos^2\phi_{12}^{ij}\right)\;.
\end{split}
\end{equation}
Note that we omitted the collinear parts of the splitting functions related to the
Wilson lines $i$ and $j$, as these are unchanged by the double-soft corrections.
The notation is such that the subscripts indicate the partons which are color-adjacent
to the splitting products, while the superscript indicates one that is color-adjacent
to the adjacent parton. In particular, the color connection in $(P_{qq})_{2}^{k}(1,i)$
would be $2\leftrightarrow 1\leftrightarrow i \leftrightarrow k$.
Note that the ordering of the arguments and lower indices is important. This is apparent
in the $12$-collinear limit, where the gluon-to-gluon kernel receives a second contribution,
$(P_{gg})_{ji}^{\,l}(2,1)$, which is related to a different evolution variable in the
dipole shower approach because it corresponds to the $12$-collinear, $2$-soft
singularity~\cite{Hoche:2015sya}. The leading-order $g\to gg$ splitting function
in the collinear limit is recovered only upon adding $(P_{gg})_{ij}^k(1,2)$ and
$(P_{gg})_{ji}^{\,l}(2,1)$. The pure collinear term in $(P_{qq})_i^k(1,2)$ could
in principle be modified by the weight, Eq.~\eqref{eq:psct_soft_tc_12_weight}, but there
is no indication, based on the double-soft limit, as to whether this would constitute an
improvement of the parton shower or not. We postpone the analysis of this term
to a future publication.
We emphasize that, despite the reweighting of the leading-order parton shower to the
full real-emission pattern of the double-soft limit, a hard correction remains to be
computed using the techniques of~\cite{Hoche:2017iem}. This correction arises
because the leading-order parton shower does not fill the complete two-emission
phase space, see for example~\cite{Fischer:2017yja}. The correction is given by
\begin{equation}\label{eq:subtracted_real}
\begin{split}
&\tilde{\mc{S}}_{ij}^{(gg)}(1,2)=
\frac{2\,s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}\,
\left[\,(P_{gg})_{ij}(1,2)+(P_{ii})_{ij}(1,2)+P_{ii}^{\rm(slc)}(1,2)\,\right]\\
&\qquad\times\left[1-
\Theta\left(\frac{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}{S_{ij,12}}
-\frac{s_{i1}s_{12}}{S_{i,12}}\right)\right]
\Theta\left(\frac{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}{S_{ij,12}}-t_c\right)\\
&\;\qquad\qquad+\Big(1\leftrightarrow 2\Big)+\Big(i\leftrightarrow j\Big)
+\Big(\begin{array}{c}1\leftrightarrow 2\\
i\leftrightarrow j\end{array}\Big)\;,\\
&\tilde{\mc{S}}_{ij}^{(qq)}(1,2)=
\frac{2\,s_{ij}}{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}\;
(P_{gq})_{ij}(1,2)\\
&\qquad\times\left[1-
\Theta\left(\frac{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}{S_{ij,12}}
-\frac{s_{i1}s_{12}}{S_{i,12}}\right)\right]
\Theta\left(\frac{(s_{i1}+s_{i2})(s_{j1}+s_{j2})-s_{ij}s_{12}}{S_{ij,12}}-t_c\right)\\
&\;\qquad\qquad+\Big(1\leftrightarrow 2\Big)+\Big(i\leftrightarrow j\Big)
+\Big(\begin{array}{c}1\leftrightarrow 2\\
i\leftrightarrow j\end{array}\Big)\;,
\end{split}
\end{equation}
where $t_c$ is the infrared cutoff of the parton shower.
The two terms in the $\Theta$-function correspond to the ordering variables in the first
and second emission, respectively. To simplify the notation we have defined $S_{i,12}$,
which is given as $s_{i12}$ in the case of final-state Wilson lines and $2p_i(p_1+p_2)$
in the case of initial-state Wilson lines. Correspondingly, $S_{ij,q}$ is given by
$Q^2$ for two final-state Wilson lines, $2p_ip_j$ for two initial-state Wilson lines,
and $2p_i(p_j+q)$ if $i$ is in the initial, and $j$ is in the final state~\cite{Hoche:2015sya}.
Following the discussion in Sec.~\ref{sec:ps_correspondence}, the $\mc{O}(1)$
remainder in Eqs.~\eqref{eq:lo_nlo_expanded} is implemented as an endpoint
contribution at $s_{12}=0$ and $s_{i1}=0$, $s_{i2}=0$, $s_{j1}=0$, $s_{j2}=0$
for all $\kappa>0$. This allows us to simulate the radiation pattern fully differentially
at the next-to-leading order. The related endpoint terms are given by
\begin{equation}\label{eq:cusp_diff_endpoints}
\begin{split}
\tilde{\mc{S}}_{gq,ij}^{\rm(cusp)}(1,2)=&\;\delta(s_{12})\,
\frac{2\,s_{ij}}{s_{i12}s_{j12}}\,T_R
\Big(2z(1-z)+\big(1-2z(1-z)\big)\ln(z(1-z))\Big)\;,\\
\tilde{\mc{S}}_{gg,ij}^{\rm(cusp)}(1,2)=&\;\delta(s_{12})\,
\frac{2\,s_{ij}}{s_{i12}s_{j12}}\,2C_A\left(\frac{\ln z}{1-z}+\frac{\ln(1-z)}{z}
+\big(-2+z(1-z)\big)\ln(z(1-z))\right)\;,\\
\tilde{\mc{S}}_{wl,ij}^{\rm(cusp)}(1,2)=&\;-\delta(s_{i1})\,
\frac{1}{2}\
\frac{C_A}{2}\,\frac{2\,s_{ij}}{s_{i12}s_{j12}}\,
\left(\frac{\ln z_i}{1-z_i}+\frac{\ln (1-z_i)}{z_i}\right)
+\Big(1\leftrightarrow 2\Big)
+\Big(i\leftrightarrow j\Big)
+\Big(\begin{array}{c}1\leftrightarrow 2\\
i\leftrightarrow j\end{array}\Big)\;.
\end{split}
\end{equation}
The factor $1/2$ in $\tilde{\mc{S}}_{gq,ij}^{\rm(cusp)}$ removes the double counting of
soft-collinear regions when swapping the role of $i$ and $j$. It would in principle
be desirable to work with partial fractions of the eikonals $s_{ij}/(s_{i1}s_{j1})$ and
$s_{ij}/(s_{i2}s_{j2})$. However, these partial fractions cannot be defined
unambiguously in the exact limits $s_{i1}\to 0$, $s_{j1}\to 0$, $s_{i2}\to 0$, $s_{j2}\to 0$.
One possible solution would be to introduce an additional rapidity regulator,
similar to~\cite{Curci:1980uw} or~\cite{Chiu:2011qc,Li:2016axz}.
We leave the investigation of this possibility to future work.
We implement the contributions proportional to the beta function as a
double endpoint which contributes an additional term to the soft enhanced
parts of the leading order splitting functions.~\footnote{We could in principle
implement the terms proportional to $\beta_0$ in the same manner as
Eq.~\eqref{eq:cusp_diff_endpoints} by splitting them into real and virtual
contributions, corresponding to uncanceled infrared and ultraviolet singularities.
When $\mu\approx\kappa$, the impact on the Monte-Carlo predictions will be minor,
and we will therefore leave the investigation of this possibility to future work.}
\begin{equation}
\begin{split}
\tilde{\mc{S}}_{gg,ij}^{\rm(coll)}(q)=&\;\delta(q^2)\,
\frac{2\,s_{ij}}{s_{iq}s_{jq}}\,\ln\frac{\mu^2s_{ij}}{s_{iq}s_{jq}}\;\beta_0\;.
\end{split}
\end{equation}
\section{Numerical results}
\label{sec:results}
In this section we present numerical cross-checks of our algorithm, and we compare the
magnitude of the corrections generated by the double-soft splitting functions to the
leading-order parton shower result in the CMW scheme~\cite{Catani:1990rr}.
We restrict the analysis to pure final-state evolution, but we stress that the
formulae relevant to initial-state evolution have also been presented in Sec.~\ref{sec:mc}.
We have implemented our algorithm into the D\protect\scalebox{0.8}{IRE}\xspace parton showers, which implies
two entirely independent realizations within the general purpose event generation
frameworks P\protect\scalebox{0.8}{YTHIA}\xspace~\cite{Sjostrand:1985xi,Sjostrand:2014zea} and
S\protect\scalebox{0.8}{HERPA}\xspace~\cite{Gleisberg:2003xi,Gleisberg:2008ta} that are cross-checked point by point
and in the full simulation at high statistical precision.
We use the strong coupling according to the CT10nlo PDF set~\cite{Lai:2010vv}.
The process under investigation is $e^+e^-\to$hadrons at LEP I energy (91.2~GeV).
We choose to exemplify the effects of the double-soft corrections using the
$k_T$ jet rates $y_{23}$ and $y_{34}$ in the Durham algorithm~\cite{Catani:1991hj}
and the angle $\alpha_{34}$ between the two softest jets~\cite{Abreu:1990ce}.
\begin{figure}[t]
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig4a1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4a2.pdf}\\[2mm]
\includegraphics[scale=0.5]{Fig4a3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4a4.pdf}
\end{center}
\end{minipage}
\label{fig:psw_validation_gg}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig4b1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4b2.pdf}\\[2mm]
\includegraphics[scale=0.5]{Fig4b3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4b4.pdf}
\end{center}
\end{minipage}
\label{fig:psw_validation_gq}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig4c1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4c2.pdf}\\[2mm]
\includegraphics[scale=0.5]{Fig4c3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig4c4.pdf}
\end{center}
\end{minipage}
\label{fig:psw_validation_qq}}
\caption{The effect of the phase-space weights $w_{ij}^{12}$,
and $\bar{w}_{ij}^{12}$, defined in Eqs.~\eqref{eq:psct_soft_tc_12_weight}
and~\eqref{eq:psct_soft_tc_so_weight}, on the leading-order
parton-shower evolution, limited to two emissions.
We show the differential jet rate $y_{34}$ in the Durham
algorithm~\cite{Catani:1991hj} as a proxy for the rate change,
and the angle defined in Eq.~\eqref{eq:soft_helper_funcs}
as a proxy for the impact on differential distributions.
The process considered is $e^+e^-\to$hadrons at LEP I energies.
\label{fig:psw_validation}}
\end{figure}
\begin{figure}[t]
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig5a1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5a2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5a3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5a4.pdf}
\end{center}
\end{minipage}
\label{fig:sc_validation_gg}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig5b1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5b2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5b3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5b4.pdf}
\end{center}
\end{minipage}
\label{fig:sc_validation_qq}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig5c1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5c2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5c3.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig5c4.pdf}
\end{center}
\end{minipage}
\label{fig:sc_validation_full}}
\caption{The effect of spin correlations in $\mc{S}_{ij,A}^{\rm(coll)}$ compared
to uncorrelated parton-shower evolution, including the phase-space suppression
investigated in Fig.~\ref{fig:psw_validation}. All simulations are limited to two emissions.
The left and middle panels show the impact on the pure $z(1-z)$ contribution
(top panels) and on the complete $g\to gg$ and $g\to q\bar{q}$ splitting function
(bottom panels). In both cases the production of the gluon is described by the
eikonal part of the $q\to qg$ splitting function only. The right panel shows
the effect of spin correlations on the complete two-emission pattern. In order
for the results to be as similar as possible, the weight $w_{ij}^{12}$ from
Eq.~\eqref{eq:psct_soft_tc_12_weight} is included. The process considered is
$e^+e^-\to$hadrons at LEP I energies.
\label{fig:sc_validation}}
\end{figure}
Figure~\ref{fig:psw_validation} shows the impact of the phase-space weights, $w_{ij}^{12}$
and $\bar{w}_{ij}^{12}$, defined in Eqs.~\eqref{eq:psct_soft_tc_12_weight}
and~\eqref{eq:psct_soft_tc_so_weight}. These weights generate a strong suppression
of the radiation probability. The effect is eventually compensated by other corrections
(see Fig~\ref{fig:sv}), such that a fairly good agreement with the leading-order
approximation is obtained. The lower panels in Fig.~\ref{fig:psw_validation}
show a comparison between the results from Pythia against those from Sherpa.
The two predictions agree up to statistical fluctuations, providing a strong
cross-check on the consistency of our implementation.
Figure~\ref{fig:sc_validation} shows the impact of the spin correlations
implemented by the $\cos^2\phi_{12}^{\,ij}$ dependence of
Eq.~\eqref{eq:soft_helper_funcs} compared to a spin averaged simulation.
While the related effects are striking when investigating the $z(1-z)$-dependent
parts of the splitting functions in isolation, they are greatly diminished in
the complete calculation.
Figure~\ref{fig:slc} displays the impact of the generic sub-leading color
corrections in Eq.~\eqref{eq:psct_soft_slc}. The effects are generally
smaller than expected based on a naive estimate (i.e.\ $\mc{O}(1/2N_c)$),
because Eq.~\eqref{eq:psct_soft_slc} is suppressed in the collinear
region, cf. the discussion in Sec.~\ref{sec:mc}.
Figure~\ref{fig:tc} shows the impact of the subtracted real-emission
corrections, Eq.~\eqref{eq:subtracted_real}, and the endpoint terms,
Eq.~\eqref{eq:cusp_diff_endpoints}, on the radiation pattern in
$q\to q(gg)$, $q\to q(q'\bar{q}')$ and $q\to q(q\bar{q})$ splittings,
where the particles in parentheses are the soft emissions.
We have verified that exact agreement between our implementations
is obtained also in the case of $g\to g(gg)$ and $g\to g(q\bar{q})$.
The numerical impact of these corrections is similar to the quark-induced case.
Note that the $3\to 4$ jet rates receive corrections from the
subtracted real emission only, while the $2\to 3$ jet rates are
impacted by both the subtracted real-emission and the endpoint terms.
\begin{figure}[t]
\begin{center}
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig6a1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6a2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6a3.pdf}
\end{center}
\end{minipage}
\label{fig:slc_y_34}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig6b1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6b2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6b3.pdf}
\end{center}
\end{minipage}
\label{fig:slc_cosphi}}\hfill
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig6c1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6c2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig6c3.pdf}
\end{center}
\end{minipage}
\label{fig:slc_alpha34}}
\end{center}
\caption{The impact of the generic sub-leading color corrections,
Eq.~\eqref{eq:psct_soft_slc}, on the radiation pattern in
$e^+e^-\to$hadrons at LEP I energies. The reference result (red)
includes all next-to-leading order effects investigated in
Figs.~\ref{fig:psw_validation} and~\ref{fig:sc_validation}.
All simulations are limited to two emissions. Note that the
simulation results of the sub-leading color corrections alone
(green) and the baseline (red) do not add up to the full result
because of the different Sudakov factors.
\label{fig:slc}}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig7a1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig7a2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig7a3.pdf}
\end{center}
\end{minipage}
\label{fig:tc_gg}}\hskip 5mm
\subfigure{
\begin{minipage}{0.31\textwidth}
\begin{center}
\includegraphics[scale=0.5]{Fig7b1.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig7b2.pdf}\\[-0.5mm]
\includegraphics[scale=0.5]{Fig7b3.pdf}
\end{center}
\end{minipage}
\label{fig:tc_qq}}
\end{center}
\caption{The impact of subtracted real-emission corrections,
Eq.~\eqref{eq:subtracted_real}, and endpoint terms,
Eq.~\eqref{eq:cusp_diff_endpoints} on the radiation pattern in
$e^+e^-\to$hadrons at LEP I energies. We show the contributions
from $q\to qgg$ (left) and $q\to qq'\bar{q}'$ (right) to the
differential $2\to 3$ (red) and $3\to 4$ (right) jet rates
in the Durham algorithm.
\label{fig:tc}}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure{
\includegraphics[scale=0.485]{Fig8a.pdf}
\label{fig:sv_y23}}\hfill
\subfigure{
\includegraphics[scale=0.485]{Fig8b.pdf}
\label{fig:sv_y34}}\hfill
\subfigure{
\includegraphics[scale=0.485]{Fig8c.pdf}
\label{fig:sv_a34}}
\end{center}
\caption{Scale variations in the leading-order and next-to-leading
order (soft) parton shower simulation of $e^+e^-\to$hadrons at
LEP I energies at parton level. We compare to both the plain
leading-order predictions (green) and the result in the CMW
scheme (blue).
\label{fig:sv}}
\end{figure}
Figure~\ref{fig:sv} compares the results from a leading-order simulation
according to~\cite{Hoche:2015sya} to our complete next-to-leading order
prediction. In the leading-order case we present the calculation
with and without the CMW scheme~\cite{Catani:1990rr}.
We observe that the CMW prediction matches the rates of the full
next-to-leading order result for the Durham jet rate $y_{23}$ fairly well
in the intermediate-$y$ region, but there are some discrepancies in the
low and high-$y$ region. In addition, there is a considerable rate change
in $y_{34}$. The angular observable $\alpha_{34}$ shows deviations
between the CMW prediction and the full next-to-leading order result
at very small and very large values. In all cases, the scale uncertainty
is greatly reduced at the next-to-leading order,
and the next-to-leading order predictions lie within the leading order
uncertainty bands. Note in particular that our result presents the first genuine
estimate of the perturbative uncertainty in a parton-shower simulation.
Some earlier attempts, despite generating variations of the same order,
treated the problem in an approximate manner~\cite{Bendavid:2018nar}.
Other techniques~\cite{Badger:2016bpw,Hoche:2017hno} assumed the scale variations
on collinear parts of the splitting functions to be identical to the soft parts,
and therefore generate artificially small uncertainty bands, which may not reflect
the true perturbative precision at the order to which the computation is performed.
\section{Conclusions}
\label{sec:conclusions}
We have presented a calculation of the next-to-leading order corrections
to soft-gluon radiation, differentially in the one-emission phase space.
This is a crucial ingredient in the construction of a next-to-leading order parton shower.
We have demonstrated, for the first time, that the soft next-to-leading order
contribution to the evolution of color dipoles can be obtained in a modified
subtraction scheme, such that both one- and two-emission terms are amenable to
Monte-Carlo integration in four dimensions. The two-loop cusp anomalous dimension
emerges naturally in this method. We observe fair agreement between the results
of the fully differential simulation and the approximate treatment using the CMW
scheme, where the two-loop cusp anomalous dimension is included in an
inclusive manner. The similarity of the results is reassuring, because
the individual higher-order contributions have kinematical dependencies
that can differ strongly from the iterated leading-order result.
Our calculation can be seen as a confirmation that the existing leading-order
parton showers developed over the past decades have been amended by the dominant
effects arising from the higher-order soft corrections, but it also confirms
that the higher-order corrections do have an impact beyond a simple $K$-factor.
We are now in place to compute these effects without the need for approximations,
and to include them in phenomenological studies as well as experimental analyses
at the particle level. This allows in particular to obtain meaningful estimates
of the renormalization scale uncertainty.
\begin{acknowledgments}
\noindent
We thank Lance Dixon and Thomas Gehrmann for comments on the manuscript.
This work was supported by the U.S.\ Department of Energy under contracts
DE--AC02--76SF00515 and DE--AC02--07CH11359.
\end{acknowledgments}
|
2,877,628,089,480 | arxiv | \section{Introduction}
\label{intro}
The purpose of this paper is two-fold. First, we will correct a theorem
in \cite{Top_paper} by the author, and second, study a problem which
arises from the correction. G. Hector and D. Peralta-Salas found out
the error in the theorem in \cite{Top_paper} and informed the author about it.
Moreover, they studied comprehensively the
realization problem which asks if a manifold can be embedded in another
manifold so that it is also a fiber of a submersion to the Euclidean
space (see \cite{Hector-Peralta}).
Before stating the correct theorem, we prepare some notions. We mostly
work in the smooth ($C^{\infty}$) category in this paper. Suppose that
$M$ is an open oriented $3$-manifold and $L$ is an oriented
$n$-component link in $M$. In this paper we say that a manifold is {\em
open} if the boundary is empty and no component is compact. Let $N(L)$
denote a small tubular neighborhood of $L$. A {\em framing} $\nu$ of
$L$ is meant to be an embedding $\nu :\bigsqcup_{j=1}^{n}(S^{1}\times
D^{2})_{j}\rightarrow M$ onto $N(L)$ which maps the cores
$\bigsqcup_{j=1}^{n}(S^{1}\times\{ 0\})_{j}$ onto
$L=\bigsqcup_{j=1}^{n}L_{j}$. Here, $D^{2}$ denotes the unit disk in
$\mathbb{C}$ and $S^{1}=\partial D^{2}$. We assume that any framings and their
restrictions to the cores are orientation-preserving. We note that a
framing of $L$ induces a tangential framing of $TN(L)(=TM|N(L))$ and
vice versa. Here, a tangential framing means a choice of a
trivialization of $TN(L)\cong N(L)\times (\mathbb{R}\times\mathbb{R}^{2})$ with $TN(L)|L
= TL\oplus TL^{\bot}\cong (L\times\mathbb{R})\oplus (L\times\mathbb{R}^{2})$ where
$TL^{\bot}$ denotes a normal bundle to $TL$.
\begin{defun}
Suppose that $L$ represents the null-class in the locally finite
homology group $H^{\infty}_{1}(M;\mathbb{Z})$, i.e., the homology group of
locally finite (possibly) infinite chains. A framing $\nu$
of $L$ is said to be {\em preferred} (or {\em null-homologous}) if the
union of the longitudes $\nu (\bigsqcup_{j=1}^{n}(S^{1}\times\{1 \})_{j})$
represents the null-class in $H_{1}^{\infty}(M\setminus\Int N(L);\mathbb{Z} )$.
We call $\nu (\bigsqcup_{j=1}^{n}(S^{1}\times\{1 \})_{j})$ the {\em preferred
longitudes} of $N(L)$ with respect to the preferred framing $\nu$.
\end{defun}
\begin{remark}
If $L$ represents the null-class in $H_{1}^{\infty}(M;\mathbb{Z} )$, then there
exists a preferred framing of $L$. In fact, there exists an oriented
(possibly non-compact) surface in $M$ bounded by $L$. Choosing such a
surface $S$, we have a framing of $L$ whose longitudes are
$S\cap\partial N(L)$.
\end{remark}
\begin{remark}
Note that a preferred framing is not unique in general. In fact, in
the case of the core circle of the open solid torus, every framing is
preferred. Nevertheless, we invoke the terminology of {\em preferred longitudes} in
\cite{Rolfsen}.
\end{remark}
The correct theorem is the following.
\begin{CrrctThm}\label{correct_thm}
For an oriented link $L$ in an open oriented $3$-manifold $M$, the
following conditions are equivalent:
\begin{enumerate}
\item\label{realization}
there exists a submersion $\varphi : M\rightarrow\mathbb{R}^{2}$ such that up to
isotopy the preimage $\varphi^{-1}(0)$ of the origin is $L$ and
$\varphi$ maps the transverse orientation of $L$ to the standard
orientation of $\mathbb{R}^{2}$, i.e., for any small disk $D$ transverse to
$L$ with the orientation induced from those of $M$ and $L$, the
restriction $\varphi |D$ preserves the orientation, and
\item\label{condition2}
the cycle $L$ represents the null-class in the locally finite homology
group $H_{1}^{\infty}(M;\mathbb{Z} )$ and there exists a preferred
framing of $L$ whose tangential framing is the restriction of some
trivialization of $TM$.
\end{enumerate}
\end{CrrctThm}
\noindent
Theorem \ref{correct_thm} is also a consequence of Theorem 2.4.2 in
\cite{Hector-Peralta}. In the original incorrect theorem (Theorem 1 in
\cite{Top_paper}) the above extension condition of the framing in
(\ref{condition2}) is missing. Here, we explain briefly how it fills the gap in
the original proof. If (\ref{realization}) holds, then the canonical
trivialization of the tangent bundle of $\mathbb{R}^{2}$ is pulled back to a
normal bundle to the fibers. With a trivialization of the tangent
bundle to the fibers, it determines a trivialization of $TM$ which
restricts to a tangential framing of $L$. The projection map from $N(L)$
onto the meridian disk determined by this framing must coincide with
the submersion restricted to $N(L)$. Conversely, by the assumption that
the framing of $L$ is preferred, the projection map
$N(L)\approx\bigsqcup_{j=1}^{n}(S^{1}\times D^{2})_{j}\rightarrow D^{2}$
extends to a map $(M, M\setminus\Int N(L))\rightarrow (\mathbb{R}^{2},
\mathbb{R}^{2}\setminus\Int D^{2})$ and moreover an extension of the
(tangential) framing of $L$ to $M$ ensures that we can take a submersion
$M\rightarrow\mathbb{R}^{2}$ as the extended map. This is an application of the
{\em h-principle}, in this case A. Phillips' submersion classification
theory \cite{Phillips}. In the proof in \cite{Top_paper}, it is only
shown that an extension as a map exists since the framing of $L$ is
preferred. However, in order to apply Phillips' theory to have an
extended submersion, we need the requirement of the simultaneous
extension of the tangential framing of $L$ and the projection map on
$N(L)$ to the whole manifold $M$.
We note that Theorem 2 in \cite{Top_paper} is correct even though the
proof in \cite{Top_paper} is not completed.
\begin{CrrctThm}[Theorem 2 in \cite{Top_paper}]\label{thm_B}
For any link $L$ in an open orientable $3$-manifold, there is a submersion
$\varphi : M\rightarrow\mathbb{R}^{2}$ such that up to isotopy the union of
compact components of $\varphi^{-1}(0)$ is $L$.
\end{CrrctThm}
\noindent
In order to prove this theorem, we have to choose a (tangential) framing
of $L$ which is the restriction of some trivialization over the whole
manifold $M$. This can be always done by twisting a framing once around
the meridional direction if necessary. Note that we need not to require
that the framing is preferred here. This observation is missing in the proof in
\cite{Top_paper}.
Theorem \ref{thm_B} is also proved in Application 2.3.8
in \cite{Hector-Peralta}.
We will give the proof of Theorem \ref{correct_thm} and \ref{thm_B} in
Section \ref{appendix} as an appendix.
As a consequence of the correction, there arises a question to find a
criterion for a link to be a fiber of a submersion to the plane in the words of
well-known invariants. We will answer to this question for the case of a
knot. Suppose that $M$ is an open oriented 3-manifold and $K$ is an
oriented knot in $M$. For the simplicity, we say that $K$ is {\em
realizable} if $K$ satisfies the condition (\ref{realization}) in Theorem
\ref{correct_thm}.
The following is the main theorem of this paper.
\begin{MainThm}\label{main_thm}
Assume that $K$ represents the null-class in $H_{1}^{\infty}(M;\mathbb{Z} )$.
Then, $K$ is realizable if and only if $K$ represents a non-zero class in
$H_{1}(M;\mathbb{Z}_{2} )$, where $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}$.
\end{MainThm}
\noindent
In order to prove the Main Theorem, it suffices to show the following
two claims. Let $\kappa$ denote the homology class $\iota_{\ast}([K])\in
H_{1}(M;\mathbb{Z} )$, where $\iota :K\hookrightarrow M$ is the inclusion map and
$[K]$ denotes the fundamental class of $K$.
Also, let $\kappa_{(2)}$ denote the $\mathbb{Z}_{2}$-reduction of $\kappa$ in
$H_{1}(M;\mathbb{Z}_{2} )$.
\begin{claim}\label{claim_if}
If $\kappa_{(2)}\neq 0$ then $K$ is realizable.
\end{claim}
\begin{claim}\label{claim_onlyif}
If $\kappa_{(2)}=0$ then $K$ is not realizable.
\end{claim}
\begin{remark}
As mentioned earlier, G. Hector and D. Peralta-Salas
\cite{Hector-Peralta} studied this kind of realization problem in the
more general dimensions and setting. As one application of their
theory, they obtained a characterization for a link in $\mathbb{R}^{3}$ to be
realizable and in particular they
showed that no knot in $\mathbb{R}^{3}$ is realizable. One may
consider the Main Theorem generalizes the result.
\end{remark}
\begin{remark}
In the case of links, the argument will be a rather complicated nuisance.
It might be just a technicality, nevertheless we omit here the consideration
in the case of links at all. The complete research including the
general case of links should be done in the sequel.
\end{remark}
In Section \ref{proof_claim_1}, we describe the notion of tangential
framings of oriented knots from the homotopical viewpoint, and prove
Claim \ref{claim_if}. In Section \ref{proof_claim_2}, we study the
properties of framings of oriented knots and prove Claim
\ref{claim_onlyif}. For the reader's convenience, we state a part of
Phillips' theory \cite{Phillips} which we need and give the proofs of
Theorem \ref{correct_thm} and \ref{thm_B} in Section \ref{appendix}, as
an appendix.
\section{Proof of Claim \ref{claim_if}}\label{proof_claim_1}
First, we express the notion of framings of oriented knots
in different words. Suppose that $M$ is an oriented open $3$-manifold
and $K$ is an oriented knot in $M$. We fix a trivialization $\Pi : TM\cong
M\times\mathbb{R}^{3}$ throughout the paper. Suppose any framing $\nu :
S^{1}\times D^{2}\rightarrow N(K)$ is given. Isotoping $\nu$ if
necessary, we may assume that each meridian disk $\nu
(\{\mathrm{exp}t\sqrt{-1}\}\times D^{2})$ is normal to $K$ with respect
to the metric induced by $\Pi$. We will define a map $f_{\nu}:
K\rightarrow\SO(3)$ which is an alternate of $\nu$ under $\Pi$
as follows. For any point $p\in K$, let $(v_{1}(p), v_{2}(p),
v_{3}(p))$ be the orthonormal frame of $T_{p}M=\mathbb{R}^{3}$ determined by the
tangent bundle and the normal bundle to $K$. Precisely, $v_{1}(p)$ is
the unit tangent vector to $K$, $v_{2}(p)$ is the unit normal vector
determined by $\frac{\partial}{\partial x}$, and another unit normal
vector $v_{3}(p)$ is chosen by the orientation of $M$. Here, we write
$z=x+y\sqrt{-1}\in D^{2}$. Thus, with respect to $\Pi$, this orthonormal
frame $(v_{1}(p), v_{2}(p), v_{3}(p))$ can be expressed as a special
orthogonal matrix. We define $f_{\nu}(p):= (v_{1}(p), v_{2}(p),
v_{3}(p))\in\SO(3)$.
\begin{defun}
We call the resulting map $f_{\nu}:K\rightarrow\SO(3)$ a {\em
$\sigma$-framing} of $K$ with respect to $\nu$.
\end{defun}
\begin{remark}
In fact, a $\sigma$-framing is the component of a cross section of the
frame bundle $\mathrm{Fr}(TM)\cong M\times\SO(3)$ associated with $TM$ with the
trivialization induced by $\Pi$.
\end{remark}
It can be easily seen that under the parallelization $\Pi$ any map $K\rightarrow\SO(3)$
determines a framing $S^{1}\times D^{2}\rightarrow N(K)$ up to isotopy.
Thus, to choose a framing (up to isotopy) and to choose a $\sigma$-framing (up
to homotopy) are equivalent. Moreover, since $[S^{1}, \SO
(3)]\cong\mathrm{Hom}(\pi_{1}(S^{1}), \pi_{1}(\SO (3)))\cong\pi_{1}(\SO
(3))\cong H_{1}(\SO(3);\mathbb{Z} )\cong\mathbb{Z}_{2}$ with appropriate choices of base
points, we may identify the homotopy class $[f_{\nu}]$ of a
$\sigma$-framing with its image $(f_{\nu})_{\ast}([K])\in H_{1}(\SO
(3);\mathbb{Z} )\cong\pi_{1}(\SO(3))$. Note that a tangential framing of $K$ is
the restriction of some trivialization of $TM$ if and only if the
corresponding $\sigma$-framing of $K$ extends to $M$ as a map.
Now we have the following criteria for the existence of an extension of a
$\sigma$-framing.
\begin{lemma}\label{extension_lem}
Let $M$ be an open orientable $3$-manifold and $K$ a knot in $M$.
Suppose that a map $f:K\rightarrow\SO (3)$ is given. Then the
following are equivalent:
\begin{enumerate}
\item
the map $f:K\rightarrow\SO (3)$ extends to a map $M\rightarrow\SO (3)$,
\item
the induced homomorphism $f_{\ast}:\pi_{1}(K)\rightarrow\pi_{1}(\SO
(3))$ extends to a homomorphism $\pi_{1}(M)\rightarrow \pi_{1}(\SO
(3))$, and
\item
the homomorphism $f_{\ast}:H_{1}(K;\mathbb{Z} )\rightarrow H_{1}(\SO (3);\mathbb{Z} )$
extends to a homomorphism $H_{1}(M;\mathbb{Z} )\rightarrow H_{1}(\SO (3);\mathbb{Z}
)$.
\end{enumerate}
Moreover, in the implication from $(2)$ or $(3)$ to $(1)$, the resulting
extension map $M\rightarrow\SO(3)$ induces the given extended
homomorphism.
\end{lemma}
\begin{proof}
It is well known that an open orientable $3$-manifold is homotopy
equivalent to a subcomplex of its $2$-skeleton (cf. \cite{Phillips},
\cite{Whitehead} for example). Therefore, by an
elementary obstruction theory, the given map $f:K\rightarrow\SO (3)$ extends to
a map $M\rightarrow\SO (3)$ if and only if the induced homomorphism
$f_{\ast}:\pi_{1}(K)\rightarrow\pi_{1}(\SO (3))$ extends to a
homomorphism $\pi_{1}(M)\rightarrow\pi_{1}(\SO (3))$, i.e., there is a
homomorphism $\Phi:\pi_{1}(M)\rightarrow\pi_{1}(\SO (3))$ such that
$\Phi\circ\iota_{\ast}=f_{\ast}$ where $\iota:K\hookrightarrow M$ is the
inclusion. Moreover, since $\pi_{1}(K)$ and $\pi_{1}(\SO (3))$ are
Abelian it is equivalent to the condition that the homomorphism
$f_{\ast}:H_{1}(K;\mathbb{Z} )\rightarrow H_{1}(\SO (3);\mathbb{Z} )$ extends to a
homomorphism $H_{1}(M;\mathbb{Z} )\rightarrow H_{1}(\SO (3);\mathbb{Z} )$.
\end{proof}
Now, we show Claim \ref{claim_if}.
\begin{proof}[Proof of Claim \ref{claim_if}]
Suppose $f:K\rightarrow\SO(3)$ is a preferred $\sigma$-framing
of $K$, i.e., the $\sigma$-framing associated with a preferred framing
of $K$. By Theorem \ref{correct_thm} and Lemma \ref{extension_lem},
it suffices to show that there exists a homomorphism $\Phi
:H_{1}(M;\mathbb{Z} )\rightarrow H_{1}(\SO(3);\mathbb{Z} )$ which is an
extension of the induced homomorphism $f_{\ast}:H_{1}(K;\mathbb{Z} )\rightarrow
H_{1}(\SO(3);\mathbb{Z} )$. If $f_{\ast}=0$ then the zero
homomorphism is an extension. Hence we assume $f_{\ast}\neq 0$,
which implies $f_{\ast}$ is an epimorphism. On the other hand, since
$\kappa_{(2)}\neq 0$ the composition of the natural homomorphisms
$H_{1}(K;\mathbb{Z} )\rightarrow H_{1}(M;\mathbb{Z} )\rightarrow H_{1}(M;\mathbb{Z}_{2} )$ is
non-trivial. Let $\langle\kappa_{(2)}\rangle$ denote its image. Since
$H_{1}(M;\mathbb{Z}_{2} )$ is a vector space over the field $\mathbb{Z}_{2}$, we have a
projection onto the one-dimensional subspace $H_{1}(M;\mathbb{Z}_{2}
)\rightarrow\langle\kappa_{(2)}\rangle$. Let $\Psi$ denote the
composition of the natural homomorphism $H_{1}(M;\mathbb{Z} )\rightarrow
H_{1}(M;\mathbb{Z}_{2} )$ followed by this projection. Let $\alpha
:\langle\kappa_{(2)}\rangle\cong\mathbb{Z}_{2}$ and $\beta
:H_{1}(\SO(3);\mathbb{Z} )\cong\mathbb{Z}_{2}$ be any isomorphisms. Then the
composition $\Phi =\beta^{-1}\circ\alpha\circ\Psi$ is the desired
extension homomorphism.
\end{proof}
\section{Lemmata and proof of Claim \ref{claim_onlyif}}\label{proof_claim_2}
In this section, we study some properties of $\sigma$-framings and
prove Claim \ref{claim_onlyif}. The following lemma describes a
relation of ($\sigma$-)framings of two oriented knots which are homologous.
\begin{lemma}\label{homologous_framings}
Let $Z_{1}$ and $Z_{2}$ be oriented knots in $M$ and
$\nu_{j}:S^{1}\times D^{2}\rightarrow N(Z_{j})$ their framings $(j=1, 2)$ .
Assume that there is a compact oriented surface $S$ in
$M$ such that $\partial S=Z_{1}\sqcup (-Z_{2})$ and $S\cap\partial
N(Z_{j})=\nu_{j}(S^{1}\times\{1\} )$, where $-Z_{2}$ denotes $Z_{2}$
with the orientation reversed. Then the induced $\sigma$-framings
$f_{\nu_{1}}$ and $f_{\nu_{2}}$ satisfy that
$[f_{\nu_{1}}]=[f_{\nu_{2}}]\in\pi_{1}(\SO (3))$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{homologous_framings}]
We define a map $F:S\rightarrow \SO(3)$ as follows. Choose a unit
tangent vector field $v_{1}:S\rightarrow TS\subset TM|S$ such that
$v_{1}|Z_{1}$ coincides with the unit vector field tangent to $Z_{1}$.
Then we choose another vector field $v_{2}$ so that $(v_{1}, v_{2})$
forms an orthonormal frame field of $S$. Here, $v_{2}$ is chosen to be
inward normal along $Z_{1}$. Picking the normal unit vector field
$v^{\bot}$ to $S$, we have a frame field $F=(v_{1}, v_{2},
v^{\bot}):S\rightarrow\SO(3)$. By the definition,
$F|Z_{1}=f_{\nu_{1}}$. Since the rotation number of $v_{1}|Z_{2}$
along $Z_{2}$ is equal to the Euler characteristic $\chi (S)$ which is
the minus twice of the genus of $S$, we have
$[F|Z_{2}]=[f_{\nu_{2}}]\in\pi_{1}(\SO(3))$. Since $F|Z_{1}$ and
$F|Z_{2}$ are homologous (or bordant) by $F$, we have
$[f_{\nu_{1}}]=[F|Z_{1}]=[F|Z_{2}]=[f_{\nu_{2}}]$.
\end{proof}
Next, we study an oriented knot whose homology class with $\mathbb{Z}_{2}$
coefficient is zero.
First, we consider the ``double'' of a knot and study its framing.
Let $J$ be any oriented knot in $M$ and
$\nu :S^{1}\times D^{2}\rightarrow N(J)$ any framing of $J$. Let
$J_{d}$ be the $(2, 1)$-cable knot in $N(J)$. For the clarity, we define
$J_{d}$ as follows. Define $\tilde{L}$ to be a union of two parallel
lines in $\mathbb{R}\times D^{2}$ as
\[
\tilde{L}:=\{ (t, \pm\tfrac{1}{2}) |\ t\in\mathbb{R}, \}
\]
and $\tau$ to be a self-diffeomorphism on $\mathbb{R}\times D^{2}$ by $\tau (t,
z):=(t, \exp (\pi t\sqrt{-1})z)$. Then the quotient of $(\mathbb{R}\times
D^{2}, \tau (\tilde{L}))$ by the $\mathbb{Z}$-action generated by the translation by $1$
on the $\mathbb{R}$-factor is a manifold pair $(S^{1}\times D^{2}, L)$ such that
$L$ is an oriented knot.
Here, we identify $S^{1}=\mathbb{R} /\mathbb{Z}$.
We define $J_{d}$ to be $\nu (L)$ and call it a
$(2, 1)$-{\em cable knot} of $J$ with respect to $\nu$.
To define a natural framing of a $(2, 1)$-cable knot $J_{d}$ of $J$, we
consider an annulus in $S^{1}\times D^{2}$ defined as follows. Let
$\tilde{A}$ be a union of two strips in $\mathbb{R}\times D^{2}$ defined by
\[
\tilde{A}:=\{ (t, \pm r) |\ t\in\mathbb{R}, \tfrac{1}{2}\leq r\leq 1\}
\]
and set $(S^{1}\times D^{2}, A):=(\mathbb{R}\times D^{2}, \tau (\tilde{A}))/\mathbb{Z}$
as the quotient by the translation. Note that $\partial A-\partial
(S^{1}\times D^{2})=L$. For any small tubular neighborhood $N(L)$ in
$\Int (S^{1}\times D^{2})$, there is a framing $\xi :S^{1}\times
D^{2}\rightarrow N(L)\subset S^{1}\times D^{2}$ of $L$ such that $\xi
(S^{1}\times\{ 1\})=\partial N(L)\cap A$. We call the framing
$\nu\circ\xi :S^{1}\times D^{2}\rightarrow N(J_{d})=\nu (N(L))\subset
N(J)$ a {\em revolution framing} of the $(2, 1)$-cable knot $J_{d}(=\nu
(L))$ of $J$ with respect to $\nu$. See Figure~\ref{fig_RevFraming}.
\begin{figure}[h]
\centering
\includegraphics[height=6cm]{fig_RevFraming.eps}
\caption{A revolution framing of $J_{d}$ at a section}
\label{fig_RevFraming}
\end{figure}
Under the parallelization $\Pi$ of $M$, the revolution framing of
$J_{d}$ induces a {\em revolution $\sigma$-framing}
$J_{d}\rightarrow\SO(3)$.
The following is a key lemma to the proof of Claim \ref{claim_onlyif}.
\begin{lemma}\label{(2,1)-framing}
For any oriented knot $J$ in $M$ and any framing $\nu :S^{1}\times
D^{2}\rightarrow N(J)$, the revolution $\sigma$-framing
$f_{d}:J_{d}\rightarrow\SO(3)$ of the $(2,1)$-cable knot
$J_{d}$ with respect to $\nu$ is not null-homotopic, i.e.,
$[f_{d}]=1\in\mathbb{Z}_{2}\cong\pi_{1}(\SO(3))$.
\end{lemma}
\begin{proof}
Let $f_{\nu}:J\rightarrow\SO(3)$ be the $\sigma$-framing of $J$
with respect to $\nu$. Then along $J_{d}$ the frame $f_{d}(p)$ goes
around twice in the longitudinal direction and rotates once in the
meridional direction. Thus $[f_{d}]=2[f_{\nu}]+1\equiv
1\in\mathbb{Z}_{2}\cong\pi_{1}(\SO(3))$.
\end{proof}
Next, suppose that $K$ is an oriented knot $K$ which represents the
null-class in $H_{1}^{\infty}(M;\mathbb{Z} )$. Recall that $\kappa_{(2)}\in
H_{1}(M;\mathbb{Z}_{2} )$ denotes the $\mathbb{Z}_{2}$-reduction of the homology class $\kappa
=\iota_{\ast}([K])\in H_{1}(M;\mathbb{Z} )$. Let $\lambda$ be the homology class in
$H_{1}(M\setminus\Int N(K);\mathbb{Z} )$ represented
by a preferred longitude of $N(K)$. Then we have the following.
\begin{lemma}\label{longitude_null}
If $\kappa_{(2)}=0$, then the $\mathbb{Z}_{2}$-reduction $\lambda_{(2)}$ of
$\lambda$ is zero.
\end{lemma}
\begin{proof}
Let $E_{M}(K)$ denote the knot exterior $M\setminus\Int N(K)$.
Consider the following diagram.
\[
\begin{array}{cccccccc}
\rightarrow & H_{2}(M, E_{M}(K);\mathbb{Z} ) &
\stackrel{\partial}{\rightarrow} & H_{1}(E_{M}(K);\mathbb{Z} ) &
\stackrel{\iota_{\ast}}{\rightarrow}
& H_{1}(M;\mathbb{Z} ) & \rightarrow & 0\\
& \downarrow & & \downarrow & & \downarrow & & \\
\rightarrow & H_{2}(M, E_{M}(K);\mathbb{Z}_{2} ) &
\stackrel{\partial}{\rightarrow} & H_{1}(E_{M}(K);\mathbb{Z}_{2} ) &
\stackrel{\iota_{\ast}}{\rightarrow} & H_{1}(M;\mathbb{Z}_{2} ) & \rightarrow & 0
\end{array}
\]
Here, the rows are homology exact sequences of the pair $(M, E_{M}(K))$ and
the vertical arrows are natural homomorphisms. Then
$\iota_{\ast}(\lambda)=\kappa , \
\iota_{\ast}(\lambda_{(2)})=\kappa_{(2)}$ and $\lambda ,\kappa$ are mapped down
to $\lambda_{(2)}, \kappa_{(2)}$ respectively. Since $\kappa_{(2)}=0$,
there is $\eta\in H_{2}(M, E_{M}(K);\mathbb{Z}_{2} )$ such that $\partial\eta
=\lambda_{(2)}$. Note that $H_{2}(M, E_{M}(K);\mathbb{Z}_{2} )$ is isomorphic to
$\mathbb{Z}_{2}$ generated by the meridian disk of $N(K)$. Hence $\partial\eta$ is
represented by the meridian loop or equal to zero. However, the
meridian loop intersects exactly once with a (locally finite) relative cycle in
$(E_{M}(K), \partial E_{M}(K))$ bounded by the preferred longitude of
$N(K)$, the representative cycle of $\lambda$. Thus $\partial\eta
=\lambda_{(2)}$ must be zero.
\end{proof}
The following is another key lemma which describes a normal form of the
knot which satisfies the hypothesis of Claim \ref{claim_onlyif}.
\begin{lemma}\label{transportation_lemma}
Suppose that an oriented knot $K$ represents the null-class in
$H_{1}^{\infty}(M;\mathbb{Z} )$ and $\kappa_{(2)}=0$. We fix a preferred
framing $\nu :S^{1}\times D^{2}\rightarrow N(K)$. Then, there
exists an oriented knot $Z\subset M$ such that the $(2, 1)$-cable knot
$Z_{d}$ of $Z$ is homologous to $K$ and
$[f_{d}]=[f_{\nu}]\in\pi_{1}(\SO(3))$, where
$f_{d}:Z_{d}\rightarrow\SO(3)$ is the revolution
$\sigma$-framing of $Z_{d}$ with respect to some framing of $Z$.
\end{lemma}
\begin{proof}
Let $L$ denote the preferred longitude of $N(K)$ with respect to
$\nu$ and $\lambda$ the homology class in $H_{1}(E_{M}(K);\mathbb{Z} )$
represented by $L$. Then we have $\lambda_{(2)}=0$ by Lemma
\ref{longitude_null} with the assumption $\kappa_{(2)}=0$.
Considering the Bockstein homology exact sequence with respect to
$0\rightarrow\mathbb{Z}\stackrel{\times
2}{\rightarrow}\mathbb{Z}\rightarrow\mathbb{Z}_{2}\rightarrow 0$, we have $\lambda =2\zeta$
for some $\zeta\in H_{1}(E_{M}(K);\mathbb{Z} )$. Choose a
representative cycle (an oriented knot) $Z\subset E_{M}(K)$ of $\zeta$ and a
framing of $Z$. Since $\lambda =2\zeta$, there exists an immersed
oriented surface bounded by $L$ and ``twice of $-Z$''. Precisely, there
exists an immersion $h$ of compact oriented surface $S$ into $E_{M}(K)$
which maps $\Int S$ into $\Int E_{M}(K)$ with the
following properties. The boundary $\partial S$ is decomposed
into two parts: $\partial_{+}S\sqcup\partial_{-}S$, where
$h(\partial_{+}S)=L$ and $h(\partial_{-}S)=-Z$ which means
$h|\partial_{-}S$ is orientation-reversing. The immersion $h$ is an
embedding away from $\partial_{-}S$ and $h|\partial_{-}S$ is a
(possibly trivial) two-fold covering onto $Z$. Moreover, $h(S)\cap
N(Z)$ is homeomorphic (in fact diffeomorphic away from $Z$) to the
mapping cylinder of the two-fold covering map $h|\partial_{-}S$. We
may assume that $h(S)$ and $\partial N(Z)$ are transverse to each other
and $h(S)\cap\partial N(Z)$ is a circle or a union of two parallel
circles in $\partial N(Z)$. By choosing another framing of $N(Z)$ if
necessary, we may assume that $h(S)\cap\partial N(Z)$ is the $(2,
1)$-curve or a union of two $(1, 0)$-curves with respect to the chosen
framing of $N(Z)$ restricted to $\partial N(Z)$.
If $h(S)\cap\partial N(Z)$ is the $(2, 1)$-curve, then attaching $A$ to
$h(S)-\Int N(Z)$ along $h(S)\cap\partial N(Z)$ we have an
embedded surface in $E_{M}(K)$ bounded by $L$ and $-Z_{d}$. Here, $A$
is the annulus in $N(Z)$ bounded by a $(2, 1)$-curve in
$\partial N(Z)$ and $Z_{d}$, which is defined in the definition of the
revolution framing of $Z_{d}$, and the orientation of $A$ is determined
by $-Z_{d}$. Since $L$ is isotopic to $K$ in $N(K)$, we have a compact
oriented surface in $M$ bounded by $K$ and $-Z_{d}$. It follows that $K$ and
$Z_{d}$ are homologous and $[f_{\nu}]=[f_{d}]$ by Lemma
\ref{homologous_framings}.
In the case that $h(S)\cap\partial N(Z)$ is two $(1, 0)$-curves, we
modify $h(S)$ in $N(Z)$ as follows. Consider the concentric tubular
neighborhood $N_{1/2}(Z)\subset N(Z)$ where meridian disks are of
radius $\frac{1}{2}$ of the meridian disks of $N(Z)$. We set a $(2,
1)$-cable knot $Z_{d}$ on $\partial N_{1/2}(Z)$ with respect to the
framing of $N(Z)$ and we will construct a compact oriented surface $B$
in $N(Z)$ bounded by $h(S)\cap\partial N(Z)$ and another longitude of
$N(Z_{d})$.
\begin{figure}[h]
\centering
\includegraphics[height=6cm]{fig_DCS.eps}
\caption{Double curve surgery on $(A\cup (-D))-\Int N(Z_{d})$}
\label{fig_DCS}
\end{figure}
First, let $D$ be a meridian disk of $N(Z)$ with the orientation
induced from $S^{1}\times D^{2}$ by the framing and $A$ the annulus in $N(Z)$
which determines the revolution framing of $Z_{d}$ as above. Fix a
small tubular neighborhood $N(Z_{d})$ of $Z_{d}$. Then performing a
double curve surgery on $(A\cup (-D))-\Int N(Z_{d})$, we obtain an
oriented surface $B$ (see Figure~\ref{fig_DCS}). By the construction,
$B\cap\partial N(Z)$ is a union of two $(1, 0)$-curves and
$B\cap\partial N(Z_{d})$ is the longitude of the revolution framing
with two twists corresponding to two intersection points between $D$
and $Z_{d}$. Since on $\partial N(Z)$ two curves $h(S)\cap\partial N(Z)$
and $B\cap\partial N(Z)$ are isotopic, we can attach $B$ to
$h(S)-\Int N(Z)$ along $h(S)\cap\partial N(Z)$. Let $\Sigma$
denote the resulting surface: $\Sigma = (h(S)-\Int N(Z))\cup B$.
Then $\Sigma$ is a compact
oriented proper surface in $E_{M}(K)\setminus\Int N(Z_{d})$ and
$\partial\Sigma = L\sqcup (-B\cap \partial N(Z_{d}))$.
Since $L$ (resp. $B\cap \partial N(Z_{d})$) is isotopic to $K$ (resp. $Z_{d}$)
in $N(K)$ (resp. $N(Z_{d})$), $K$ and $Z_{d}$ are homologous.
On the other hand, the framing
$\xi :S^{1}\times D^{2}\rightarrow N(Z_{d})$ such that
$\xi (S^{1}\times\{ 1\})=-\Sigma\cap\partial N(Z_{d})(=-B\cap\partial
N(Z_{d}))$ determines a
$\sigma$-framing $f_{\xi}:Z_{d}\rightarrow\SO(3)$. By Lemma
\ref{homologous_framings}, we have $[f_{\nu}]=[f_{\xi}]$. Moreover, as
noted above, the longitude with respect to $\xi$ is the longitude of the
revolution framing with two meridional twists. Hence we have
$[f_{\xi}]=[f_{d}]$. Consequently, we have $[f_{d}]=[f_{\nu}]$.
\end{proof}
Now we can prove Claim \ref{claim_onlyif}.
\begin{proof}[Proof of Claim \ref{claim_onlyif}]
Suppose that $f:K\rightarrow\SO(3)$ is any preferred
$\sigma$-framing of $K$. In view of Theorem \ref{correct_thm} and
Lemma \ref{extension_lem}, we only have to show that the induced
homomorphism $f_{\ast}:H_{1}(K;\mathbb{Z} )\rightarrow H_{1}(\SO(3;\mathbb{Z}
))$ never extends to $H_{1}(M;\mathbb{Z} )$. On the contrary to the
conclusion, we assume that there is a homomorphism $\Phi :H_{1}(M;\mathbb{Z}
)\rightarrow H_{1}(\SO(3);\mathbb{Z} )$ such that
$\Phi\circ\iota_{\ast}=f_{\ast}$, where $\iota :K\hookrightarrow M$
denotes the inclusion. Since $\kappa_{(2)}=0$, we have $\Phi (\kappa
)=0$. On the other hand, by Lemma \ref{(2,1)-framing} and
\ref{transportation_lemma}, $f_{\ast}([K])=[f]=1$. Therefore, we have
$0=\Phi (\kappa )=\Phi\circ\iota_{\ast}([K])=f_{\ast}([K])=1$, a
contradiction.
\end{proof}
\section{Appendix: Proofs of Theorem \ref{correct_thm} and \ref{thm_B}}
\label{appendix}
In order to prove Theorem \ref{correct_thm} and \ref{thm_B}, we review
Phillips' submersion classification theory \cite{Phillips}. Let $X$ and
$Y$ be manifolds. We assume that $\mathrm{dim}X\geq\mathrm{dim}Y$ in
the following. The space of all submersions from $X$ to $Y$ is
denoted by $\mathrm{Sbm}(X, Y)$ and the space of all vector bundle
morphisms from $TX$ to $TY$ whose restriction to each fiber has the
maximal rank by $\mathrm{Max}(TX, TY)$. Here, $\mathrm{Sbm}(X, Y)$ and
$\mathrm{Max}(TX, TY)$ are endowed with $C^{1}$-compact-open topology
and $C^{0}$-compact-open topology, respectively. If $X$ has a non-empty
boundary, we impose no other condition on the boundary. The essence of
the Phillips' theory is the following theorem.
\begin{AppendixThm}[Phillips \cite{Phillips}]\label{Phillips_HD_thm}
If $X$ has a handle decomposition with (possibly countably infinitely
many) handles of indices less than $\mathrm{dim}X$, then the
differential map $d:\mathrm{Sbm}(X, Y)\rightarrow\mathrm{Max}(TX, TY)$
is a weak homotopy equivalence.
\end{AppendixThm}
\noindent
Since an open manifold has a handle decomposition with (countably
infinitely many) handles of indices less than the dimension of the
manifold, we have the following theorem as a corollary.
\begin{AppendixThm}[Phillips \cite{Phillips}]\label{Phillips_OM_thm}
If $X$ is an open manifold, then the differential map
$d:\mathrm{Sbm}(X, Y)\rightarrow\mathrm{Max}(TX, TY)$ is a weak
homotopy equivalence.
\end{AppendixThm}
\noindent
In the proof of Theorem \ref{Phillips_HD_thm} (and
\ref{Phillips_OM_thm}), the following are key lemmata.
\begin{AppendixLem}\label{PhillipsLem_D}
The differential map
$d:\mathrm{Sbm}(D, Y)\rightarrow\mathrm{Max}(TD, TY)$ is a weak
homotopy equivalence, where $D$ denotes a disk of dimension
$\mathrm{dim}X$.
\end{AppendixLem}
\begin{AppendixLem}\label{PhillipsLem_H}
Let $V$ be a compact manifold with $\mathrm{dim}V=\mathrm{dim}X$.
Suppose $W$ is obtained by attaching a handle of index less than
$\mathrm{dim}V$. Then, the restriction maps $\rho :\mathrm{Sbm}(W,
Y)\rightarrow\mathrm{Sbm (V, Y)}$ and $\rho :\mathrm{Max}(TW,
TY)\rightarrow\mathrm{Max}(TV, TY)$ are fibrations.
\end{AppendixLem}
\noindent
The proof of Theorem \ref{Phillips_HD_thm} is carried out by starting
with Lemma \ref{PhillipsLem_D}, applying Lemma \ref{PhillipsLem_H}
handle by handle, and an inverse limit argument. We refer
\cite{Phillips} for the detail. Applying the inverse limit argument in
the proof of Theorem \ref{Phillips_HD_thm} and \ref{Phillips_OM_thm},
we have the following.
\begin{AppendixLem}\label{Phillips_byproduct}
Let $W$ be a codimension $0$ compact submanifold of an open manifold
$X$. Then, the restriction maps $\rho :\mathrm{Sbm}(X,
Y)\rightarrow\mathrm{Sbm (W, Y)}$ and $\rho :\mathrm{Max}(TX,
TY)\rightarrow\mathrm{Max}(TW, TY)$ are fibrations.
\end{AppendixLem}
\noindent
To be precise, in the literature an open manifold could have a non-empty
boundary. Thus, the following lemma might be in fact contained in Lemma
\ref{Phillips_byproduct}, however, we give it here as a precise
statement we need in the proof of Theorem \ref{correct_thm}.
\begin{AppendixLem}\label{fibration_lemma}
Suppose that $X$ is a manifold with no compact component and $\partial
X\neq\emptyset$. Let $W$ be a codimension $0$ compact submanifold of
$X$ such that $\partial X\subset\partial W$. Then, the restriction
maps $\rho :\mathrm{Sbm}(X, Y)\rightarrow\mathrm{Sbm (W, Y)}$ and $\rho
:\mathrm{Max}(TX, TY)\rightarrow\mathrm{Max}(TW, TY)$ are fibrations.
\end{AppendixLem}
Now, we can prove Theorem \ref{correct_thm}. We add a correct
consideration on the trivialization of the tangent bundles, however,
we mostly follow the proof in \cite{Top_paper}.
\begin{proof}[Proof of Theorem \ref{correct_thm}]
Assume that (\ref{realization}) holds. Then the preimage by $\varphi$ of
a semiline starting from the origin to the end of $\mathbb{R}^{2}$ is a surface
in $M$ which is bounded by $\varphi^{-1}(0)=L$. By the condition of
$\varphi$ on the transverse orientation to $L$, we may choose the
orientation on the surface so that $L$ represents the null-class in the
locally finite homology group. Moreover, as mentioned in Introduction,
$\varphi$ determines a trivialization of $TM$ which restricts to a
tangential framing of $L$. The projection with respect to the framing
associated with this tangential framing of $L$ coincides with $\varphi$
near $L$. Thus, (\ref{condition2}) holds.
Next, assume that (\ref{condition2}) holds. Choose a framing $\nu
:\bigsqcup_{j=1}^{n}(S^{1}\times D^{2})_{j}\rightarrow N(L)$ which is
preferred and suppose there exists a trivialization of $TM$ which
restricts to the trivialization of $TN(L)$ determined by the tangential
framing induced by $\nu$. For $0\leq r\leq 1$, set $D^{2}(r):=\{ z\in\mathbb{C}\
|\ |z|\leq r\}$ and $N_{r}(L):=\nu (\bigsqcup_{j=1}^{n}(S^{1}\times
D^{2}(r))_{j})$. Define $\pi :N(L)\rightarrow D^{2}\subset\mathbb{C} =\mathbb{R}^{2}$ to
be the composition $\mathrm{pr}\circ\nu^{-1}$, where
$\mathrm{pr}:\bigsqcup_{j=1}^{n}(S^{1}\times D^{2})_{j}\rightarrow
D^{2}$ is the natural projection onto a single disk. Set
$X:=M\setminus\Int N_{1/2}(L)$ and $W:=N(L)\setminus\Int N_{1/2}(L)$.
Note that $\partial X=\partial N_{1/2}(L)$ since $\partial
M=\emptyset$.
We consider the following commutative diagram consisting of the
differential maps $d$ and the restriction maps $\rho$.
\begin{equation}\label{h-principle_diagram}
\begin{array}{ccc}
\mathrm{Sbm}(X, C) & \stackrel{d}{\rightarrow} & \mathrm{Max}(TX,
TC)\\
\rho\downarrow & & \rho\downarrow\\
\mathrm{Sbm}(W, C) & \stackrel{d}{\rightarrow} &
\mathrm{Max}(TW, TC)
\end{array}
\end{equation}
where $C$ denotes $\mathbb{R}^{2}\setminus\Int D^{2}(\frac{1}{2})$. In the diagram
the horizontal arrows are weak homotopy equivalences by Theorem
\ref{Phillips_OM_thm} and \ref{Phillips_HD_thm}, and the vertical
arrows are fibrations by Lemma \ref{fibration_lemma}. The projection
$\pi :N(L)\rightarrow D^{2}$ restricted to $W$, denoted by $\pi |W$,
belongs to $\mathrm{Sbm}(W, C)$. For $d(\pi |W)\in\mathrm{Max}(TW,
TC)$, we have an extension as follows.
\begin{AppendixClaim}
There exists $\Phi\in\mathrm{Max}(TX, TC)$ such that $\rho
(\Phi)=d(\pi |W)$.
\end{AppendixClaim}
\begin{proof}
By the canonical trivialization of $T\mathbb{R}^{2}$, we may consider that
$TC=C\times\mathbb{R}^{2}$. By the assumption, we have a trivialization of
$TX=TM|X$ which restricts to the trivialization of $TW=TN(L)|W$
determined by the framing $\nu$. Thus, $d(\pi |W)$ is represented as
\[
TW\cong W\times\mathbb{R}^{3}\rightarrow C\times\mathbb{R}^{2}\cong TC; (x, (v_{1},
v_{2}, v_{3}))\mapsto (\pi (x), (v_{2}, v_{3}))
\]
Therefore, in
order to obtain an extension of $d(\pi |W)$, we only have to show that
the map $\pi |W:W\rightarrow C$ extends to $X$. For the purpose,
we may consider the problem up to homotopy. Since $W$ (resp. $C$) is homotopy
equivalent to $\partial N_{1/2}(L)=\partial X$ (resp. $S^{1}$), the
projection $\pi |W$ determines a homotopy class $[\pi |W]\in [\partial
X, S^{1}]$. We will give an extension of $[\pi |W]$ in $[X, S^{1}]$.
Since $S^{1}$ is the Eilenberg-MacLane space $K(\mathbb{Z}, 1)$, there are
natural bijections $[X, S^{1}]\rightarrow H^{1}(X;\mathbb{Z} )$ and $[\partial
X, S^{1}]\rightarrow H^{1}(\partial X;\mathbb{Z} )$ which commute the
restriction maps (cf. Spanier \cite{Spanier}). Combining these maps
with Poincar\'{e}-Lefschetz duality (see Massey \cite{Massey} for the
locally finite homology version), we have the following
sign-commutative diagram.
\begin{equation}
\begin{array}{ccccc}
[X, S^{1}] & \rightarrow & [\partial X, S^{1}] & & \\
\parallel & & \parallel & & \\
H^{1}(X;\mathbb{Z} ) & \rightarrow & H^{1}(\partial X;\mathbb{Z} ) &
\stackrel{\delta}{\rightarrow} & H^{2}(X, \partial X;\mathbb{Z}
) \\
\downarrow & & \downarrow & & \downarrow \\
H_{2}^{\infty}(X, \partial X;\mathbb{Z} ) &
\stackrel{\partial}{\rightarrow} & H_{1}(\partial X;\mathbb{Z} ) &
\rightarrow & H_{1}^{\infty}(X;\mathbb{Z} )
\end{array}
\end{equation}
Here, the horizontal rows are cohomology and homology exact sequences and
the vertical arrows are Poincar\'{e}-Lefschetz duality isomorphisms.
As noted above, we may consider that $[\pi |W]$ belongs to $[\partial X,
S^{1}]=H^{1}(\partial X;\mathbb{Z} )$. We claim that $\delta [\pi |W]=0$.
Through the Poincar\'{e}-Lefschetz duality $[\pi |W]$ corresponds to
the homology class represented by the fiber of $\pi $ in
$H_{1}(\partial X;\mathbb{Z} )$. Consequently the class $[\pi |W]$
corresponds to the class represented by the union of longitudes of
$N(L)$. Since the longitudes are preferred, the class vanishes in
$H_{1}^{\infty}(X;\mathbb{Z} )$ which implies that $\delta [\pi |W]=0$.
Hence, by the exactness of the sequence, we have a class in
$H^{1}(X;\mathbb{Z} )=[X, S^{1}]$ which restricts to $[\pi |W]$.
\end{proof}
\begin{AppendixRem}
This extension lemma does not hold under the condition of the original
incorrect theorem in \cite{Top_paper}. In fact, the consideration of
framings of the tangent bundles were necessary.
\end{AppendixRem}
In the diagram (\ref{h-principle_diagram}) the differential map
$d:\mathrm{Sbm}(X, C)\rightarrow\mathrm{Max}(TX, TC)$ is a
weak homotopy equivalence. Therefore, there exists
$\psi\in\mathrm{Sbm}(X, C)$ such that $d\psi$ is homotopic to
$\Phi$ in $\mathrm{Max}(TX, TC)$. Thus $d\rho (\psi)=\rho
(d\psi)$ is homotopic to $d(\pi |W)$ in $\mathrm{Max}(TW, TC)$.
Since the differential map is a weak homotopy equivalence, this implies
that $\rho (\psi)$ and $\pi |W$ are regularly homotopic. Moreover,
the restriction map $\rho :\mathrm{Sbm}(X,
C)\rightarrow\mathrm{Sbm}(W, C)$ is a fibration, the regular
homotopy from $\rho (\psi)$ to $\pi |W$ covered by a regular homotopy
from $\psi$. Hence we conclude that there exists
$\varphi\in\mathrm{Sbm}(X, C)$ whose restriction to $W$ is
$\pi |W$. This completes the proof of Theorem \ref{correct_thm}.
\end{proof}
Finally, we give the proof of Theorem \ref{thm_B}. In the proof below, the
description of cutting open the residual components is improved in comparison
with the proof in \cite{Top_paper}.
\begin{proof}[Proof of Theorem \ref{thm_B}]
Let $L$ be any $n$-component link in $M$. Choose a framing $\nu
:\bigsqcup_{j=1}^{n}(S^{1}\times D^{2})_{j}\rightarrow N(L)$. By
twisting the framing once in the meridional direction, if necessary, we may
assume that there exists a trivialization of $TM$ whose restriction to
$N(L)$ is equal to the trivialization induced by the chosen framing of
$L$. Let $\pi :N(L)\rightarrow D^{2}$ be the projection defined as in the
proof of Theorem \ref{correct_thm}. Now we consider the following
commutative diagram.
\begin{equation}\label{thm_B_diagram}
\begin{array}{ccc}
\mathrm{Sbm}(M, \mathbb{R}^{2}) & \stackrel{d}{\rightarrow} & \mathrm{Max}(TM,
T\mathbb{R}^{2}) \\
\rho\downarrow & & \rho\downarrow \\
\mathrm{Sbm}(N(L), \mathbb{R}^{2}) & \stackrel{d}{\rightarrow} &
\mathrm{Max}(TN(L), T\mathbb{R}^{2})
\end{array}
\end{equation}
Here, the restriction maps $\rho$ are fibrations by Lemma
\ref{Phillips_byproduct} and the differential maps $d$ are weak
homotopy equivalences by Theorem \ref{Phillips_OM_thm} and
\ref{Phillips_HD_thm}. We claim the existence of an extension of $d\pi
$.
\begin{AppendixClaim}
There exists $\Phi\in\mathrm{Max}(TM, T\mathbb{R}^{2})$ such that $\rho
(\Phi)=d\pi $.
\end{AppendixClaim}
\begin{proof}[Proof of Claim]
Since the trivialization of $TM|N(L)$ induced by the chosen framing of
$L$ is the restriction of a trivialization of $TM$, as in the proof of
the claim in the proof of Theorem \ref{correct_thm}, we only have to
show that the map $\pi $ extends to $M$ up to homotopy. However, since
$\mathbb{R}^{2}$ is contractible this is clear.
\end{proof}
\noindent
Now, chasing the diagram (\ref{thm_B_diagram}) in the same way as in
the proof of Theorem~\ref{correct_thm} shows that there exists an
extension $\hat{\varphi}\in\mathrm{Sbm}(M, \mathbb{R}^{2})$ of $\pi $. If the
union of compact components of $\hat{\varphi}^{-1}(0)$ is equal to $L$,
then set $\varphi:=\hat{\varphi}$ and we are done. Otherwise, let $R$
denote the union of compact components of $\hat{\varphi}^{-1}(0)$ which
are not contained in $L$. Note that $R$ has at most countably
infinitely many components. We will cut open these residual circles $R$
by curves tend to ends of $M$. It suffices to consider the case
that there are infinitely many components of $R$. The proof in the
case of only finitely many components is similar and simpler. Note that the
components of $R$ cannot accumulate. Now we fix an increasing
filtration by codimension 0 compact connected submanifolds $N_{k}$ of
$M\ (k\in\mathbb{Z}_{\geq 0})$ such that $\cup_{k=0}^{\infty}N_{k}=M$. (We may
assume that $M$ is connected.) Suppose
that we choose a decreasing filtration by open subsets $U^{e}_{k}$,
each of which is a component of $M\setminus N_{k}$ for $k\in\mathbb{Z}_{\geq
0}$. Then it defines an end $e$ of $M$. Here, we assume that
$\partial N_{k}\cap R=\emptyset$ for any $k\in\mathbb{Z}_{\geq 0}$. Let
$\mathcal{E}$ denote the subset of the end set of $M$ consisting of all
ends $e=\{ U^{e}_{k}\}_{k\in\mathbb{Z}_{\geq 0}}$ such that $U^{e}_{k}\cap
R\neq\emptyset$ for any $k\in\mathbb{Z}_{\geq 0}$. Since $\mathcal{E}$ is at
most a countable set, we index it by natural numbers: $\mathcal{E}=\{
e_{m}\}_{m\in\mathbb{N}}$. Also, since there are at most countably many
components of $R$, we number them as follows. First, number the
components of $R\cap N_{0}$ as $R_{1}\sqcup R_{2}\sqcup\cdots\sqcup
R_{\ell_{0}}$, next $R\cap (N_{1}\setminus\Int
N_{0})=R_{\ell_{0}+1}\sqcup\cdots\sqcup R_{\ell_{1}}$, and inductively
$R\cap (N_{k}\setminus\Int N_{k-1})=R_{\ell_{k-1}+1}\sqcup\cdots\sqcup
R_{\ell_{k}}$ for $k\in\mathbb{N}$.
We then define inductively simple curves $\alpha_{m}:[0,
\infty)\rightarrow M\ (m\in\mathbb{Z}_{\geq 0})$ which cut $R$ open.
First, for the end $e_{1}=\{
U^{1}_{k}\}_{k\in\mathbb{Z}_{\geq 0}}\in\mathcal{E}$, the sequence of the
components of $R\cap(\cup_{k=0}^{\infty}(U^{1}_{k}\cap N_{k+1}))$ is an infinite
subsequence of the components of $R$, which tends to the end $e_{1}$.
Then we choose a simple curve $\alpha_{1}$ in
$\cup_{k=0}^{\infty}(U^{1}_{k}\cap N_{k+1})$ which passes through one
point in each circle of $R\cap(\cup_{k=0}^{\infty}(U^{1}_{k}\cap
N_{k+1}))$ and tends to $e_{1}$. Here, we choose $\alpha_{1}$ so that
it passes through $R_{\ell}$ in order with respect to the indices
$\ell$ of the circle $R_{\ell}$. Set
$R^{(1)}:=R\cap(\cup_{k=0}^{\infty}(U^{1}_{k}\cap N_{k+1}))$.
Inductively, for the end $e_{m}=\{ U^{m}_{k}\}_{k\in\mathbb{Z}_{\geq
0}}\in\mathcal{E}$, the sequence of the components of
$R^{(m)}:=(R\setminus\cup_{i=1}^{m-1}R^{(i)})\cap(\cup_{k=0}^{\infty}(U^{m}_{k}\cap
N_{k+1}))$ is an infinite subsequence of circles of $R$ and we choose a
simple curve $\alpha_{m}$ in $\cup_{k=0}^{\infty}(U^{m}_{k}\cap
N_{k+1})$ which passes through $R^{(m)}$ in order and tends to $e_{m}$.
Moreover, we choose all the curves $\alpha_{m}$ so that they do not
intersect with $L$ and are mutually disjoint. Note that
$R^{(0)}:=R\setminus\cup_{m=1}^{\infty}R^{(m)}$ is compact. Thus, the
components of $R^{(0)}$ are finitely many circles and we can easily
choose a simple curve $\alpha_{0}$ which passes through those circles
and tends to an end of $M$. As is similar to the case of $\alpha_{m}$
above, we take $\alpha_{0}$ so that it does not intersect with $L$ nor
$\alpha_{m}\ (m\in\mathbb{N})$. Now we claim the following.
\begin{AppendixClaim}
$(M\setminus\cup_{m=0}^{\infty}\mathrm{Im}(\alpha_{m}), L)$ is
diffeomorphic to $(M, L)$.
\end{AppendixClaim}
\begin{proof}
Set $P:=D^{2}\times [0, \infty)$ and let $\alpha :[0,
\infty)\rightarrow P$ be the curve defined by $\alpha (t):=(0, t+1)$.
Then we can easily construct a diffeomorphism between $P$ and
$P\setminus\mathrm{Im}(\alpha )$ which is the identity near the
boundary $(D^{2}\times\{ 0\})\cup (\partial D^{2}\times [0, \infty))$.
By the construction of $\alpha_{m}$ the set $\{\alpha_{m}(0)\}$ is
discrete in $M$ and the curves $\{\alpha_{m}\}$ do not accumulate.
Hence, it follows the claim.
\end{proof}
\noindent
Setting $\varphi :=\hat\varphi
|(M\setminus\cup_{m=0}^{\infty}\mathrm{Im}(\alpha_{m}))$, we have the desired
submersion. This completes the proof of Theorem \ref{thm_B}
\end{proof}
\begin{acknowledgements}\label{ackref}
The author would like to express his hearty gratitude to Gilbert Hector
who informed him that his earlier work contains an error. He also thanks
Daniel Peralta-Salas for studying, with G. Hector, the problem which
the author concerned before and is left caused by the author's
misunderstanding. Without their notice the author would never obtain
the correction and new results in the point of the author's view.
\end{acknowledgements}
|
2,877,628,089,481 | arxiv | \section{Introduction}
Strong magnetic fields have been observed in a number of stars,
including classical T Tauri stars (CTTSs) (Basri, Marcy \& Valenti 1992;
Johns-Krull 2007), magnetic white dwarfs (e.g., Warner 1995; Euchner et
al.\ 2002), and various types of neutron stars (e.g., Ghosh \& Lamb
1978; Ghosh 2007). In most stars the structure of the field is unknown.
It is usually suggested that the magnetic fields of gaseous stars are
generated and supported by some type of the dynamo mechanism that
operates in the stellar interior or in the surface layers of the star.
Numerical simulations of uniformly rotating, fully convective stars show
the formation of complex, non-axisymmetric fields that can be
represented only by a superposition of different multipoles (e.g.,
Chabrier \& K\"uker 2006). However, in simulations of differentially
rotating stars, an additional, ordered component appears (e.g., Dobler,
Stix \& Brandenburg 2006).
Measurements of the surface magnetic fields of CTTSs using different
techniques indicate that they have a complex structure (Johns-Krull,
Valenti \& Koresko 1999; Johns-Krull 2007). Measurements of the magnetic
fields of nearby low-mass stars using the Zeeman-Doppler technique show
that their fields are often complex (Donati \& Cameron 1997; Donati et
al.\ 1999; Jardine et al.\ 2002). Recent observations of two CTTSs have
shown that in one star (V2129 Oph) the surface magnetic field associated
with the octupole moment dominates the fields associated with other
moments (Donati et al.\ 2007), whereas in the other star (BP Tau) the
fields associated with the dipole and octupole moments are both
significant and dominate the fields associated with other multipoles
(Donati et al.\ 2008).
Recently, the magnetic field structure of gaseous stars with complex
fields has been studied analytically. It was found that the fraction of
the flux through the stellar surface in open field lines is smaller in
stars with complex fields than in those with purely dipolar fields
(Gregory et al.\ 2008; Mohanty \& Shu 2008). The potential approximation
is usually used to extrapolate the surface magnetic field to larger
distances. Gregory et al.\ (2006) calculated possible gas flows around
stars with magnetic fields constructed from measurements using the
potential approximation. Recently, we were able to compute the flow of
gas around V2129~Oph and BP~Tau in global 3D MHD simulations in which
the magnetic field is not assumed to be given by a potential but is
instead calculated as part of the simulation (Long et al.\ 2009).
Zeeman tomography of magnetic white dwarfs has shown that they also have
complex magnetic fields (Euchner et al.\ 2002). In some, such as
HE~1045$-$0908, the magnetic field associated with the star's quadrupole
moment dominates whereas the fields associated with its dipole and
octupole moments are much weaker (Euchner et al.\ 2005). In others,
representation of the field requires inclusion of misaligned dipole,
quadrupole, octupole, and other multipoles up to $n=4$ or $n=5$ (Euchner
et al.\ 2006; Beuermann et al.\ 2007). These results are in accord with
earlier indications of field complexity, such as asymmetric distribution
of spots on the stellar surface (Meggitt \& Wickramasinghe 1989; Piirola
et al.\ 1987; see Wickramasinghe \& Ferrario 2000 for a review).
Neutron stars also have dynamically important magnetic fields. Soon
after they have formed, the magnetic fields of some may be enhanced by a
dynamo mechanism (e.g., Thompson \& Duncan 1993) and may have a complex
structure. There is evidence that at least some accretion-powered X-ray
pulsars have complex magnetic field structures (Elsner \& Lamb 1976; Gil
et al.\ 2002; Nishimura 2005) or that the dipole moment is off-center
(e.g., Coburn 2001; see also Ruderman 1991; Chen \& Ruderman 1993).
Accreting millisecond X-ray pulsars have relatively weak dipole magnetic
fields (e.g. Psaltis \& Chakrabarty 1999), but the details of their
fields are unknown. They have almost sinusoidal light curves that are
consistent with a dipole magnetic field with a small inclination to the
spin axis (e.g., Gierli\'nski \& Poutanen 2005; Lamb et al.\ 2008a,b).
However, this does not exclude the presence of higher-order multipole
fields near the star. An unusual phenomenon --- rapid, relatively large
shifts in the phase of the light curve --- has been observed in many
millisecond pulsars (e.g., Morgan et al.\ 2003; Markwardt 2004; Burderi
et al.\ 2006; Hartman et al.\ 2009; Patruno et al.\ 2009). Several
mechanisms have been proposed to explain this phenomenon.
Lamb et al.\ (2008a,b) have explained several different properties of
accreting millisecond pulsars, including the phase shifts of their light
curves, with the ``nearly aligned moving spot model". Motion of emitting
areas located close to the spin axis alters both the shape and the
arrival time of the pulse in a complex way. These movements could be
caused by changes in the accretion rate. Earlier 3D MHD simulations of
accretion onto stars with misaligned dipole magnetic fields (Romanova et
al.\ 2003, 2004) have shown that if the dipole is nearly aligned with
the spin axis, the place where the accreting matter impacts the stellar
surface can move prograde or retrograde relative to the surface, which
favors the moving spot idea. In another model, the magnetic field of the
neutron star is assumed to change in time (Burderi et al.\ 2006). Here
we consider stars with complex magnetic fields and investigate both of
these mechanisms for producing phase shifts.
In our earlier studies, we were able to perform global 3D~MHD
simulations of accretion onto stars with a dipole magnetic field
(Koldoba et al.\ 2002; Romanova et al.\ 2003, 2004; Kulkarni \& Romanova
2005) and onto stars with a combination of dipole and quadrupole fields
(Long, Romanova \& Lovelace 2007, 2008). 3D~simulations of accretion
onto stars with multipolar fields are very challenging due to the very
steep gradients of the magnetic field. In this paper, we report on the
first simulations of accretion onto stars with an {\it octupole} field
component. Our methods allow us to investigate the general case of a
magnetic field produced by superposition of dipole, quadrupole, and
octupole fields oriented in different directions. However, even the
simpler case of a magnetic field produced by superposition of misaligned
dipole and quadrupole fields creates very complex field structures
(e.g., Long et al.\ 2008). Consequently, for clarity we omit quadrupole
fields in the present work, restricting consideration to the simpler
case of a superposition of octupole and dipole fields. We investigate
matter flows onto stars with such fields, the shapes of the resulting
hot spots, and the light curves produced by these hots spots. We also
investigate mechanisms for producing phase shifts in the light curves of
stars with complex magnetic fields. Finally, we discuss the validity of
the potential approximation for computing the magnetic field outside
such stars.
Section~2 describes the simulation method and the magnetic field
configurations. Section~3 presents our results for different
combinations of dipole and octupole fields. Section~4 shows examples of
the phase shifts that are possible in the light curves of stars with
complex fields. Section~5 summarizes and discusses our most important
results.
\section{Numerical model and magnetic field configurations}
\subsection{Model}
\begin{figure}
\begin{center}
\includegraphics{fig01_normal_13.jpg}
\caption{\label{grid} The ``cubed sphere" grid of low resolution is shown for demonstration. The left panel shows the grid on the surface of the
inflated cube. Each of the six cube sides has $N^2=13^2$ curvilinear Cartesian grid cells. The grid consists of $N_r=28$ concentric spheres. The
right panel shows two sectors of the grid. Our simulations use a higher grid resolution, up to $N_r\times N^2=200\times 51^2$ in each sector.}
\end{center}
\end{figure}
Our 3D MHD model has been described in a series of papers (Koldoba et al.\ 2002; Romanova et al.\ 2003; 2004; Kulkarni \& Romanova 2005; Long et al.
2007, 2008), where disc accretion onto stars with dipole and more complex magnetic fields has been investigated. Here, we briefly summarize different
aspects of the model.
\subsubsection{Initial Conditions}
We consider a rotating magnetic star surrounded by an accretion disc and a corona. The disc is cold and dense, while corona is hot and rarefied, and
at the reference point ( the inner edge of the disc in the disc plane ), $T_c=100T_d$, $\rho_c=0.01\rho_d$, where the subscripts `d' and `c' denote
the disc and the corona. The disc and corona are initially in rotational hydrodynamic equilibrium, where the sum of the gravitational, centrifugal,
and pressure gradient forces is zero at each point in the simulation region. To avoid an initial magnetic field discontinuity at the disc-corona
boundary, the corona is set to rotate with the Keplerian angular velocity at each cylindrical radius $r$.
An $\alpha$-type viscosity is incorporated into the code and it operates in the disc only (above some density level, $\rho\gtrsim 0.2$), and helps
regulate the accretion rate. We use $\alpha=0.04$ for the octupole-dominated cases, and $\alpha=0.02$ for the dipole-dominated cases. Other
parameters of the disc such as the density distribution and initial structure are fixed in all simulations.
\subsubsection{Boundary Conditions}
At the inner boundary (the surface of the star), most of variables $A$ are set to have free boundary conditions, ${\partial A}/{\partial r}=0$. The
initial magnetic field on the surface of the star is taken to be a superposition of misaligned dipole and octupole fields (see $B_1$ and $B_3$ in
Eqn. 2). As the simulation proceeds, we assume that the normal component of the fields remain unchanged, i.e., the magnetic field is frozen in the
surface of the star. We neglect possible changes of the magnetic field structure inside the star due to, e.g., dynamo processes, because in the most
cases the time scale of such variation is much longer than the length of the simulation time.
At the outer boundary, free conditions are taken for all variables. In addition, matter is not permitted to flow back from the corona into the
region. The simulation region is large enough ($r_{max}\approx 36 R_\star$), and the disc is massive enough, to supply matter
for the entire duration of the simulations.
\subsubsection{The ``Cubed sphere" Grid}
The 3D MHD equations are solved with a Godunov-type code on the ``cubed sphere" grid (Koldoba et al.\ 2002; see also Putman \& Lin 2007). The grid
consists of $N_r$ concentric spheres, where each sphere represents an inflated cube. Fig. \ref{grid} shows that the grid consists of six sectors
corresponding to six sides of the cube with $N\times N$ curvilinear Cartesian grids on each side. The whole grid consists of $6\times N_r\times N^2$
cells. For modeling of octupole fields, a higher radial grid resolution is needed near the star compared with the pure dipole cases. We do this by choosing
the radial size of the grid cells to be 2.5 times smaller than the angular size in the region $r < (5-7)R_\star$ (see \S\ref{ref-units}), while it is
equal to the angular size in the outer region as in all our previous work. The typical grid used in simulations has $6\times N_r\times N^2 =
6\times200\times51^2$ grid cells. Simulations with higher/lower grid resolutions were performed for comparisons.
\subsubsection{Reference Units}\label{ref-units}
The MHD equations are solved using dimensionless variables $\widetilde A$. To obtain the physical dimensional values $A$, the dimensionless values $\widetilde{A}$ should be multiplied by the corresponding
reference units $A_0$; $A=\widetilde{A}A_0$. To choose the reference units, we first choose the stellar mass $M_\star$ and radius $R_\star$. The reference units are then chosen as follows: mass $M_0=M_\star$, distance $R_0=R_\star/0.35$, velocity
$v_0=(GM_0/R_0)^{1/2}$, time scale $P_0=2\pi R_0/v_0$, angular velocity $\Omega_0=v_0/R_0$. The reference
magnetic field $B_0$ can be obtained by choosing a reasonable fiducial value for the surface dipole field strength $B_{1\star}$.
Then $B_0$ is the fiducial dipole field strength at $R_0$; $B_0=B_{1\star
}(R_\star/R_0)^3$. We then define the reference dipole moment $\mu_{1,0}=B_0R_0^3$, quadrupole moment $\mu_{2,0}=B_0R_0^4$, octupole moment
${\mu_{3,0}}=B_0R_0^5$, density $\rho_0=B_0^2/v_0^2$, pressure $p_0=\rho_0v_0^2$, mass accretion rate $\dot{M}_0=\rho_0v_0R_0^2$, angular momentum
flux $\dot{L}_0=\rho_0v_0^2R_0^3$, energy per unit time $\dot{E}_0=\rho_0v_0^3R_0^2$ (the radiation flux $J$ is also in units of $\dot{E}_0$),
temperature $T_0=\mathcal{R}p_0/\rho_0$, where $\mathcal{R}$ is the gas constant, and the effective blackbody temperature $T_{\mathrm{eff,0}} =
(\rho_0 v_0^3/\sigma)^{1/4}$, where $\sigma$ is the Stefan-Boltzmann constant.
Therefore, the dimensionless variables are $\widetilde{r}=r/R_0$, $\widetilde{v}=v/v_0$, $\widetilde{t}=t/P_0$, $\widetilde{B}_n=B_n/B_0$,
$\widetilde{\mu}_n = \mu_n / \mu_{n,0}$ ($n=1,2,3$ for dipole,
quadrupole and octupole components) and so on. In the subsequent
sections, we show dimensionless values for all quantities and drop the tildes ($\sim$). Our dimensionless simulations are applicable to different
astrophysical objects with different scales. We list the reference values for typical CTTSs, cataclysmic variables, and millisecond pulsars in Tab.
\ref{tab:refval}.
\begin{table}
\begin{tabular}{l@{\extracolsep{0.2em}}l@{}lll}
\hline
& & CTTSs & White dwarfs & Neutron stars \\
\hline
\multicolumn{2}{l}{$M_\star(M_\odot)$} & 0.8 & 1 & 1.4 \\
\multicolumn{2}{l}{$R_\star$} & $2R_\odot$ & 5000 km & 10 km \\
\multicolumn{2}{l}{$B_{1\star}$ (G)} & $10^3$ & $10^6$ & $10^9$ \\
\multicolumn{2}{l}{$R_0$ (cm)} & $4\e{11}$ & $1.4\e9$ & $2.9\e6$ \\
\multicolumn{2}{l}{$v_0$ (cm s$^{-1}$)} & $1.6\e7$ & $3\e8$ & $8.1\e9$ \\
\multicolumn{2}{l}{$\Omega_0$ (s$^{-1}$)} & $4\e{-5}$ & 0.2 & $2.8\e3$ \\
\multicolumn{2}{l}{\multirow{2}{*}{$P_0$}} & $1.5\e5$ s & \multirow{2}{*}{29 s} & \multirow{2}{*}{2.2 ms} \\
& & $1.8$ days & & \\
\multicolumn{2}{l}{$B_0$ (G)} & 43 & $4.3\e4$ & $4.3\e7$ \\
\multicolumn{2}{l}{$\rho_0$ (g cm$^{-3}$)} & $7\e{-12}$ & $2\e{-8}$ & $2.8\e{-5}$ \\
\multicolumn{2}{l}{$p_0$ (dy cm$^{-2}$)} & $1.8\e{3}$ & $1.8\e{9}$ & $1.8\e{15}$ \\
\multirow{2}{*}{$\dot M_0$} & (g s$^{-1}$) & $1.8\e{19}$ & $1.2\e{19}$ & $1.9\e{18}$ \\
& ($M_\odot$yr$^{-1}$) & $2.8\e{-7}$ & $1.9\e{-7}$ & $2.9\e{-8}$ \\
\multicolumn{2}{l}{$\dot{L}_0$ (g cm$^2$s$^{-2}$)} & $1.15\e{38}$& $4.9\e{36}$ & $4.5\e{34}$ \\
\multicolumn{2}{l}{$T_0$ (K)} & $1.6\e6$ & $5.6\e8$ & $3.9\e{11}$ \\
\multicolumn{2}{l}{$\dot E_0$ (erg s$^{-1}$)} & $4.8\e{33}$ & $1.2\e{36}$ & $1.2\e{38}$ \\
\multicolumn{2}{l}{$T_{\mathrm{eff},0}$ (K)} & 4800 & $3.2\e5$ & $2.3\e7$ \\
\hline
\end{tabular}
\caption{Sample reference units for typical CTTSs, cataclysmic variables, and millisecond pulsars. Real dimensional values for variables can be
obtained by multiplying the dimensionless values of variables by these reference units.} \label{tab:refval}
\end{table}
\subsection{Magnetic Field Configurations}
When electrical currents outside the star can be neglected, the magnetic field there can be described by a magnetic scalar potential $\varphi
(\mathbf{r})$ and can be represented as a sum of multipoles. In this work we consider intrinsic stellar magnetic fields that can be described outside
the star by a superposition of symmetric dipole, quadrupole, and octupole fields. The total magnetic field can then be written
\begin{equation}
\label{e0} \mathbf{B(r)}=\mathbf{B_1}+\mathbf{B_2}+\mathbf{B_3}\;,
\end{equation}
where
\begin{eqnarray}
\label{e1} \mathbf{B_{1}}&=&\frac{3\mu_1(\hat{\bm\mu_1}\cdot{\hat{\bf r}})}
{r^3}\hat{\bf r}-\frac{\mu_1}{r^3}\hat{\bm\mu_1}\nonumber\\
\mathbf{B_{2}}&=&\frac{3\mu_2}{4r^4}(5(\hat{\bm\mu_2}\cdot\hat{\bf r}) ^2-1)\hat{\bf r}-\frac{3\mu_2}{2r^4}(\hat{\bm\mu_2}\cdot\hat{\bf r})
\hat{\bm\mu_2}\nonumber\\
\mathbf{B_{3}}&=&5\big(\frac{\mu_3}{2r^5}\big)(\hat{\bm\mu_3}\cdot\hat {\bf r})\big[7(\hat{\bm\mu_3}\cdot\hat{\bf r})^2-3\big]\hat{\bf r}
\nonumber\\
&&-3\big(\frac{\mu_3}{2r^5}\big)\big[5(\hat{\bm\mu_3}\cdot\hat{\bf r}) ^2-1\big]\hat{\bm\mu_3}\;.
\end{eqnarray}
Here $\bf B_1$, $\bf B_2$, and $\bf B_3$ are the magnetic fields produced by the symmetric dipole, quadrupole and octupole moments and $ \hat{\bf
r}$, $\hat{\bm\mu_1}$, $\hat{\bm\mu_2}$, and $\hat{\bm\mu_3}$ are the unit vectors describing the position and the symmetric dipole, quadrupole, and
octupole moments, respectively. The magnetic moments may be inclined at angles $\Theta_{1}$, $\Theta_2$, and $\Theta_{3}$ relative to the rotation
axis $\bm{\Omega} $. They may also be in different meridional planes $\bm{\Omega-\mu_2}$ and $\bm{\Omega-\mu_3}$, with at azimuthal angles $\phi_2$
and $\phi_{3}$ relative to the $\bm{\Omega- \mu_1}$ plane defined by the dipole moment and the rotation axis. Fig.~ \ref{config} illustrates the
geometry by an example in which all three magnetic moments have different orientations.
\begin{figure}
\begin{center}
\includegraphics{fig02_config.jpg}
\caption{\label{config} Sketch of the directions of the multipole magnetic moments. The rotation axis $\bm{\Omega}$ is in the $z$ direction. The
magnetic axes of the dipole $\bm{\mu_1}$, quadrupole $\bm{\mu_2}$ and octupole $\bm{\mu_3}$ are inclined at angles of $\Theta_1$, $\Theta_2$ and
$\Theta_3$ from the $z-$ axis. The dipole moment $\bm\mu_1$ is placed in the $xz$ plane. The angles between $\bm{\Omega-\mu_2}$, $\bm{\Omega-\mu_3}$
and $xz$ planes are $\phi_2$ and $\phi_3$ respectively.}
\end{center}
\end{figure}
\subsection{The Alfv\'en Surface and Magnetospheric Radius}
The magnetospheric radius, $r_m$, is a characteristic radius where the inflowing matter is stopped by the magnetosphere. Several expressions for this radius
have been derived since the 1970s for stars with dipole fields. The first estimations were done for spherically-accreting matter. It was suggested
that the radially-falling matter is stopped by the magnetic pressure of the magnetosphere (e.g., Lamb et al.\ 1973):
\begin{equation}\label{e11}
B(r_A)^2/8\pi=\rho(r_A)v(r_A)^2 ,
\end{equation}
\begin{equation}\label{e2}
r_A = kr_A^{(0)}, \ \ r_A^{(0)}=\dot{M}^{-2/7}\mu_1^{4/7}(GM)^{-1/7},
\end{equation}
where $r_A$ is the Alfv\'{e}n radius and $k$ is a dimensionless coefficient of the order of unity
(Elsner \& Lamb 1977; Ghosh et al.\ 1977). For {\it
disc} accretion, a similar criterion can be used but for stresses.
The disk rotates in the azimuthal direction and hence main stresses are connected with the azimuthal
components of the stress tensor:
$T_{\phi\phi}=[p + \rho v_\phi^2] + [B^2/8\pi-B_\phi^2/4\pi]$ (here we neglected the viscous stress which is much
smaller than the matter stress).
The motion of the disk is disturbed by the rotating magnetosphere, when the matter stresses become
comparable with the magnetic stresses, or $p + \rho v_\phi^2 = B^2/8\pi-B_\phi^2/4\pi$.
The dominant component of the dipole magnetic field is the poloidal component, and hence $B_\phi<<B$ and we
obtain the condition for stresses as $p + \rho v_\phi^2 = B^2/8\pi$.
Different criteria were proposed by other authors (Cameron \& Campbell 1993; Armitage
\& Clarke 1996; Matt \& Pudritz 2005). Analysis of these criteria
shows that most of them give a radius similar to that given
by Eqn. \ref{e2}, but with different coefficients $k$ (Bessolaz et al.\ 2008).
A comparison of that formula with simulations gives $k\approx0.5$ (Long, Romanova \& Lovelace
2005), which is very close to the coefficient estimated by Ghosh \& Lamb (1979).
For multipolar fields, we use the generalized formula, such that the $n-$th component of the field is $B_n\sim\mu_n/r^{n+2}$, (see details in
Appendix \ref{rm-mult}), and the magnetospheric radius
\begin{equation}\label{e22}
r_m=k_n r_{m,n}^{(0)},~~~ r_{m,n}^{(0)}=\mu_n^{\frac{4}{4n+3}}\dot{M}^{-\frac{2}{4n+3}}(GM)^{-\frac{1}{4n+3}}.
\end{equation}
\section{Accretion onto Stars with an Octupole and Dipole Field}
\subsection{Pure octupole}
\begin{figure}
\begin{center}
\includegraphics{fig03_oct_bsurf.jpg}
\caption{\label{bsurfoct}
The magnetic field distribution on the star with an octupole field, $\mu_3=0.5$, $\Theta_{3}=10^\circ$,
as seen from the equatorial plane (left panel), the north pole (middle panel) and the south pole (right panel).
}
\end{center}
\end{figure}
First, we consider accretion onto a star with an octupole magnetic field with $\mu_3=0.5$. The magnetic axis of the octupole field is inclined
(misaligned) from the stellar spin axis at a small angle, $\Theta_{3}=10^\circ$.
Fig. \ref{bsurfoct} shows the magnetic field distribution on the stellar surface. There are two polar regions with strong positive (red) and negative
(blue) polarities, as seen in the middle and right panels. There are also two octupolar belts with opposite polarities as seen in the left panel. A
meridian line which goes from the north to the south magnetic poles will pass through the positive northern pole, negative northern belt, positive
southern belt and negative southern pole with zero magnetic field at the magnetic equator. This distribution differs from the dipole field which has
only positive and negative poles.
\begin{figure}
\begin{center}
\includegraphics{fig04_oct_3d.jpg}
\caption{\label{3doct} 3D views of matter flow to a star with an octupolar field with $\mu_3=0.5$, $\Theta_3=10^\circ$
at $t=3.5$. The disc is shown by a constant density surface in green with $\rho=0.15$ in the top panel. Different
density levels in the equatorial plane are shown in the bottom panel. The colors along the field lines represent
different polarities and strengths of the field. The thick cyan and orange lines represent the rotation and octupole
moment axes respectively. }
\end{center}
\end{figure}
Fig. \ref{3doct} (top panel) shows 3D views of matter flow to the star at time $t=3.5$. The magnetic field near the star has an octupolar shape with
three sets of closed loops of field lines connecting regions of different polarities on the star, which we call {\it northern, southern} and {\it
equatorial} sets of loops. Some field lines are closed, while others are dragged by the disc, wrapped around the rotation axis and inflate into the
corona. Matter of the disc is lifted above the equatorial set of loops and flows into the northern magnetic belt on the star. The bottom panel shows
the density distribution in the equatorial plane, where the disc matter is stopped by the equatorial set of loops and forms a low-density gap around
the star.
\begin{figure*}
\begin{center}
\includegraphics{fig05_oct_flow.jpg}
\caption{\label{flowoct} The density distribution in different planes (color background) and 3D magnetic field lines (yellow lines) for an octupole
configuration with $\mu_3=0.5$, $\Theta_{3}=10^\circ$. Panel (a) shows an $xz$ slice at $t=0$; panels (b), (c) and (d) show $xz$, $yz$ and $xy$
slices at $t=3.5$. The red lines show surfaces where $\beta=1$. The thick cyan and orange lines represent the rotation and octupole moment axes
respectively.}
\end{center}
\end{figure*}
Fig. \ref{flowoct} shows the density distribution in different planes and sample 3D magnetic field lines. Panel (a) shows that at $t=0$ the magnetic
field is a pure octupole field, where the three sets of loops of field lines are clearly seen. The equatorial set of loops is expanded to larger
distances than the polar sets, to demonstrate the interaction of the magnetosphere with the disc. Panels (b) and (c) show that the disc matter is
stopped by the magnetosphere, lifted above the disc plane and channeled by the equatorial set of loops, and flows into the octupolar belts on the
stellar surface. Panel (b) shows that matter flow is not axisymmetric: more matter flows in from the direction towards which the octupole is
inclined. Panel (c) shows that there is a low-density magnetospheric gap around the star.
The bold red line represents the magnetospheric surface,
where the matter stresses equal
the magnetic stresses,
\begin{equation}
\label{e-beta}
\beta=\frac{p+\rho v^2}{B^2/8\pi}=1.
\end{equation}
The magnetic field lines stay closed inside the magnetospheric surface, where the magnetic
stresses dominate.
The initial octupolar field shown in the panel (a) is a {\it potential} field, that is, a field that has not been disturbed by the surrounding
plasma. The potential approximation is often used in extrapolation of the observed surface fields to larger distances (e.g., Jardine et al.\ 2002;
Gregory et al.\ 2006; Donati et al.\ 2007). Panels (b)-(d) show that the potential approximation is sufficiently good inside the Alfv\'en surface,
where $\beta < 1$. However, the field strongly departs from the potential one at larger distances.
\begin{figure}
\begin{center}
\includegraphics{fig06_oct_hotspots.jpg}
\caption{\label{hsoct}
The surface density distribution on the star with an octupole magnetic field, $\mu_3=0.5$, $\Theta_3=10^\circ$, as seen
from the equatorial plane (left panel), the north pole (middle panel), and the south pole (right panel).
}
\end{center}
\end{figure}
Fig. \ref{hsoct} shows that the hot spots on the stellar surface represent two rings which are located in the planes parallel to the magnetic
equator. Each ring has a density enhancement on one side, where disc accretion is higher due to the inclination of the magnetic axis. The middle
and right panels show that the density enhancements in the northern and southern rings are antisymmetric relative to the magnetic equatorial plane.
That is why the left panel shows only a part of the southern ring.
\begin{figure}
\begin{center}
\includegraphics{fig07_oct_lc.jpg}
\caption{\label{lcoct} The light curves seen by the observer at different inclination angles $i$ for the pure octupole case with $\mu_3=0.5$ and
$\Theta_3=10^\circ$.}
\end{center}
\end{figure}
For the calculation of the light curves, we use the same approach as Romanova et al.\ (2004). The total energy of the accreting matter is assumed to
be converted into isotropic blackbody radiation on the surface of the star. The specific intensity of radiation from a position $\bm{R}$ on the
stellar surface into a solid angle $\mathrm{d}\Omega$ in the direction $\bm{\hat{k}}$, is $I(\bm{R,\hat{k}})=(1/\pi)F_e(\bm{R})$, where $F_e(\bm{R})$
is the total energy flux of the inflowing matter, $\theta=\arccos{({\bm{\hat R}}\cdot{\bm{\hat{k}}})}$. Therefore we obtain the radiation energy
received per unit time in the direction $\bm{\hat{k}}$, $J=r^2F_{obs}=\int I(\bm{R,\hat{k}})\cos\theta\mathrm{d}S$, where $r$ is the distance between
the star and the observer, $F_{obs}$ is the observed flux, and $\mathrm{d}S$ is the surface area element. Simulations show that the spots usually
``choose" their favorite position on the surface of the star, which does not vary much with time. That is why we fix the spots at some moment of time
and rotate the star to obtain the light curves.
Fig. \ref{lcoct} shows the light curves at $t=3.5$ viewed at different observer inclination angles $i=\arccos{(\hat{\bm\Omega}\cdot\bm{\hat{k}})}$.
They are approximately sinusoidal for relatively small inclination angles, $i=15^\circ$ and $i=30^\circ$, when only the northern ring can be seen.
For larger $i$, the southern ring also contributes to the light curves and produces a second peak per rotation period. The light curve strongly
departs from the sinusoidal one at larger $i$ (see $i=60^\circ$), but two similar peaks per period are observed at $i=90^\circ$.
\subsection{Strong octupole and weak dipole}
Next, we consider accretion onto a star with a superposition of a relatively strong octupole and a much weaker dipole component on the stellar
surface: $\mu_1=0.2$, $\mu_3=0.3$, $\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$. Here, the rotation axis, dipole and octupole moments
are all misaligned.
First, we estimate the radius at which the magnetic fields of the dipole and octupole components are equal. Using Eqn. \ref{e1}, and suggesting for
simplicity that the dipole and octupole moments are aligned, we obtain approximate formulae for the equatorial and polar magnetic fields: (1)
Equatorial field: $B_1=\mu_1/r^3$, $B_3=(3/2)\mu_3/r^5$, and (2) Polar field: $B_1=2\mu_1/r^3$, $B_3=4\mu_3/r^5$. By equating $B_1$ and $B_3$ we
obtain the distances where the dipole and octupole components are equal in the equatorial and polar directions:
\begin{equation}
\label{e-rcrit}
r_{eq}=\bigg({\frac{3}{2}\frac{\mu_3}{\mu_1}}\bigg)^\frac{1}{2}, ~~r_{pole}=\bigg({2 \frac{\mu_3}{\mu_1}}\bigg)^\frac{1}{2}.
\end{equation}
Substituting for $\mu_1$ and $\mu_3$, we obtain $r_{eq}=1.5$ and $r_{pole}=1.4$. At smaller/larger distances the octupole/dipole field dominates. It
is evident that near the star, $r\sim R_\star=0.35$, the octupolar component is much stronger than the dipole component. That is why in Fig.
\ref{bsurfwdip}, the magnetic field distribution, shows similar features to that in pure octupole case (Fig. \ref{bsurfoct}).
\begin{figure}
\begin{center}
\includegraphics{fig08_weakdipole_bsurf.jpg}
\caption{\label{bsurfwdip} The magnetic field distribution on the star with a strong octupole and a weak dipole component, $\mu_1=0.2$, $\mu_3=0.3$,
$\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$, as seen from the equatorial plane (left panel), the north pole (middle panel) and the
south pole (right panel). }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics{fig09_weakdipole_3d.jpg}
\caption{\label{3dwdip} 3D views of matter flow to a star with a strong octupole and weak dipole magnetic field, $\mu_1=0.2$, $\mu_3=0.3$,
$\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$, at $t=5$. The disc is shown by a constant density surface in green with $\rho=0.2$ in
the top panel; different density levels in the equatorial plane are shown in the bottom panel. The colors along the field lines represent different
polarities and strengths of the field. The thick cyan, white and orange lines represent the rotation, dipole and octupole axes respectively.}
\end{center}
\end{figure}
Fig. \ref{3dwdip} shows 3D views of accretion flow onto the star at time $t=5$. The disc matter comes close to the star and mainly interacts with the
octupolar component and accretes onto the octupolar belts on the stellar surface. External field lines are dragged by the disc and inflate. This case
shows similar features to that of the pure octupole, because in both cases the octupole determines matter flow.
\begin{figure*}
\begin{center}
\includegraphics{fig10_weakdipole_flow.jpg}
\caption{\label{flowwdip} Density distribution in different slices (color background) and 3D magnetic field lines (yellow lines) for the case of an
octupole plus weak dipole field with parameters $\mu_1=0.2$, $\mu_3=0.3$, $\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$. Panel (a)
shows an $xz$ slice at $t=0$; panels (b), (c), (d) show $xz$, $yz$ and $xy$ slices at $t=5$. The red lines show the Alfv\'{e}n surface, where
$\beta=1$. The thick cyan, white and orange lines represent the rotation, dipole and octupole axes respectively.}
\end{center}
\end{figure*}
Fig. \ref{flowwdip}(a) shows that initially, at $t=0$, the octupole component dominates near the star, while the dipole dominates at larger
distances, such as the inner edge of the disc, which is different from the pure octupole case. Panels (b)- (d) show that the disc is stopped by the
magnetosphere at $r_m\approx 0.9\approx 2.6 R_\star$. This radius is smaller than $r_{eq}\approx 1.5$, and therefore the octupolar field dominates at
the disc-magnetosphere boundary. The accretion flow is similar to that in the pure octupole case (see Fig. \ref{flowoct}), though here the
misalignment angle $\Theta=20^\circ$ is larger and matter flows more easily above the equatorial set of loops. On the other hand, the octupole is
inclined more strongly (than in the pure octupole case), and part of the funnel stream is located in the equatorial plane, due to which the
magnetospheric gap is not empty (panel d).
Comparison of panel (a) with the other panels shows the difference between the initial potential field (panel a) and the non-potential fields
obtained in the simulations. The magnetic field lines threading the disc and corona inflate, are wrapped around the rotation axis and form some kind
of a magnetic tower (Romanova et al.\ 2009). The field strongly differs from the potential one at $r>r_m$, where the disc and corona matter strongly
disturb the magnetosphere. The potential approximation, however, is reasonably good inside the magnetospheric radius, $r<r_m$, where $\beta<1$
(inside the red line in the figure).
\begin{figure}
\begin{center}
\includegraphics{fig11_weakdipole_hotspots.jpg}
\caption{\label{hswdip}
The surface density distribution on the star with a strong octupole and weak dipole magnetic field,
$\mu_1=0.2$, $\mu_3=0.3$, $\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$, at $t=5$, as seen
from the equatorial plane (left panel), the north pole (middle panel), and the south pole (right panel).
}
\end{center}
\end{figure}
Fig. \ref{hswdip} shows the hot spots on the surface of the star. The spots are similar to those in the pure octupole case, but the rings have a
higher inclination relative to the equatorial plane due to the larger misalignment angle.
\begin{figure}
\begin{center}
\includegraphics{fig12_weakdipole_lc.jpg}
\caption{\label{lcwdip} The light curves in the case of a strong octupole and a weak dipole, $\mu_1=0.2$, $\mu_3=0.3$ and $\Theta_1=30^\circ$,
$\Theta_3=20^\circ$, $\phi_3=70^\circ$, for different inclination angles $i$. }
\end{center}
\end{figure}
Fig. \ref{lcwdip} shows the light curves associated with rotation of the star. The shapes of the light curves strongly depart from sinusoids, in
contrast with the pure octupolar case. However, the reason is not the small dipole component, but instead the fact that the misalignment angle
($\Theta_3=20^\circ$) is higher than that in pure octupolar case ($\Theta_3=10^\circ$), and the southern ring is seen by the observer even at small
inclination angles $i$. At very high $i$, the observer see both rings in similar proportions, and the light curve becomes sinusoidal with two peaks
per period.
\subsection{Strong dipole and weak octupole}
We now consider the case where the dipole component ($\mu_1=2.0$) is much stronger than the octupole component ($\mu_3=0.2$). The misalignment angles
are $\Theta_1=20^\circ$, $\Theta_3=10^\circ$, the phase angle is $\phi_3=180^\circ$.
\begin{figure}
\begin{center}
\includegraphics{fig13_strongdipole_bsurf.jpg}
\caption{\label{bsurfsdip} The surface magnetic field strength of the star with a strong dipole and weak octupole magnetic field, $\mu_1=2.0$,
$\mu_3=0.2$, $\Theta_1=20^\circ$, $\Theta_3=10^\circ$, $\phi_3=180^\circ$, as seen from the equatorial plane (left panel), the north pole (middle
panel) and the south pole (right panel).}
\end{center}
\end{figure}
From Eqn. \ref{e-rcrit}, we find the critical radii where the dipole and octupole components are equal: $r_{eq}=0.39$ and $r_{pole}=0.45$. These
radii are only slightly larger than the radius of the star, $R_\star=0.35$, and hence the dipole determines the dynamics of the flow in the entire
vicinity of the star.
Fig. \ref{bsurfsdip} shows the magnetic field on the surface of the star. The field distribution is complex and different from a dipole or octupole
field. There are two magnetic poles with different polarities near the octupole axis, and two belts of different polarities caused by the octupole
component. The field in the belts, however, is strongly distorted by the dipole component.
\begin{figure}
\begin{center}
\includegraphics{fig14_strongdipole_3d.jpg}
\caption{\label{3dsdip} 3D views of matter flow to a star with a strong dipole and weak octupole magnetic field, $\mu_1=2.0$, $\mu_3=0.2$,
$\Theta_1=20^\circ$, $\Theta_3=10^\circ$, $\phi_3=180^\circ$, at $t=9$. The disc is shown by a constant density surface in green with $\rho=0.25$ in
the top panel; different density levels in the disc plane are shown in the bottom panel. The colors along the field lines represent different
polarities and strengths of the field. The thick cyan, white and orange lines represent the rotation, dipole and octupole axes respectively. }
\end{center}
\end{figure}
Fig. \ref{3dsdip} shows the 3D views of the accretion flow to the star at $t=9$. The magnetic field looks like a dipole at all distances. The matter
is stopped by the dipole component of the magnetosphere and is channeled to the polar regions in two funnel streams which is typical for accretion
onto a star with a dipole field (Romanova et al.\ 2003). In the disc plane (bottom panel), the matter is stopped by the dipole-like field, and no
equatorial accretion is observed.
\begin{figure*}
\begin{center}
\includegraphics{fig15_strongdipole_flow.jpg}
\caption{\label{sdip} Density distribution in different slices (color background) and 3D magnetic field lines (yellow lines) for the case of a strong
dipole and a very weak octupole field with parameters $\mu_1=2.0$, $\mu_3=0.2$, $\Theta_1=20^\circ$, $\Theta_3=10^\circ$, $\phi_3=180^\circ$. Panel
(a) shows an $xz$ slices at $t=0$; panels (b), (c), (d) show $xz$, $yz$ and $xy$ slices at $t=9$. The red lines show the magnetospheric surface,
where $\beta=1$. The thick cyan, white and orange lines represent the rotation, dipole and octupole axes respectively.}
\end{center}
\end{figure*}
Fig. \ref{sdip} (panel a) shows that initially, at $t=0$, the magnetic field has a dipole shape in the whole simulation region excluding the close
vicinity of the star. Panels (b) and (c) show that the disc matter is stopped by the dipole-like magnetosphere and accretes towards the poles in two
funnel streams. Panel (d) shows that a low-density magnetospheric gap forms around the star in the disc plane. Comparison of panel (a) with panels
(b)-(d) shows that the potential approximation of the initially dipole magnetic field shown in panel (a) stays valid at later times only inside the
magnetospheric surface, $r < r_m$, where $\beta<1$. At larger distances, the field lines are dragged by the disc and corona, and the field strongly
departs from the initially potential dipole field.
\begin{figure}
\begin{center}
\includegraphics{fig16_strongdipole_hotspots.jpg}
\caption{\label{hssdip} The hot spots viewed from different angles for the case of $\mu_1=2.0$, $\mu_3=0.2$, $\Theta_1=20^\circ$,
$\Theta_3=10^\circ$, $\phi_3=180^\circ$: from the equatorial plane (left panel); from the north pole (middle panel); and from the south pole (right
panel). The color represents the density.}
\end{center}
\end{figure}
The hot spots are shown in Fig. \ref{hssdip}. As we expect, there are only two polar spots, which is typical for accretion onto a star with a dipole
field. No ring-like spots are observed because the dipole component dominates almost everywhere and matter is not channeled by the octupole-like
field. The hot spots are located near the northern and southern magnetic poles.
\begin{figure}
\begin{center}
\includegraphics{fig17_strongdipole_lc.jpg}
\caption{\label{lcsdip} The light curves in the case of a strong dipole and weak octupole, $\mu_1=2.0$, $\mu_3=0.2$, $\Theta_1=20^\circ$,
$\Theta_3=10^\circ$, $\phi_3=180^\circ$, for different inclination angles $i$.}
\end{center}
\end{figure}
Fig. \ref{lcsdip} shows the light curves associated with the rotation of the star at $t=9$. The shapes of the light curves are quite sinusoidal for
almost all inclination angles, which is also typical for stars with slightly misaligned dipole fields (Romanova et al.\ 2004). At large $i$, two peaks
per period are observed, because the spot near the southern magnetic pole becomes visible and contributes to the light curves.
\subsection{Dipole and octupole of comparable strength}
\begin{figure*}
\begin{center}
\includegraphics{fig18_comp_dip_oct_slice.jpg}
\caption{\label{dip-oct} Close view of the density distribution in different slices (color background) and magnetic field lines (yellow lines) for
the case of a strong dipole and octupole magnetic field with parameters $\mu_1=1.0$, $\mu_3=0.3$, $\Theta_1=30^\circ$, $\Theta_3=20^\circ$,
$\phi_3=70^\circ$. From left to right: $xz$, $yz$ and $xy$ slices at $t=3.5$. The red line shows the magnetospheric surface, where $\beta=1$. }
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics{fig19_comp_dip_oct_hotspots.jpg}
\caption{\label{dip-oct-spot} The hot spots viewed from different angles for the case of $\mu_1=1.0$, $\mu_3=0.3$, $\Theta_1=30^\circ$,
$\Theta_3=20^\circ$, $\phi_3=70^\circ$: from the equatorial plane (left panel); from the north pole (middle panel); and from the south pole (right
panel). The color represents the density. }
\end{center}
\end{figure}
We now consider an interesting case where both dipole and octupole components disrupt the disc and channel matter. The parameters are: $\mu_1=1$,
$\mu_3=0.3$, $\Theta_1=30^\circ$, $\Theta_3=20^\circ$, $\phi_3=70^\circ$. For these parameters we find from Eqn. \ref{e-rcrit}: $r_{eq}=0.67$,
$r_{pole}=0.77$. However, the inclination of the dipole and octupole axes could also influence the result. Fig. \ref{dip-oct} shows a close view of
the matter flow. In the $xz$ plane, the funnel streams are first channeled by the dipole-like field. When the matter flows close to the star, at
about $2 R_\star$, the octupole determines the flow and converts each funnel stream into three small accretion streams between the loops of field
lines. In the $yz$ plane, the disc is stopped by the dipole component, and only a small amount of matter flows along the dipole-like field lines and
is then governed by the octupole. The $xy$ plane shows that matter flow is strongly non-axisymmetric: in some directions the disc matter is stopped
by the dipole, and in other directions matter flows much closer to the star and is stopped by the octupolar component.
Figure \ref{dip-oct-spot} shows the hot spots in this case. Most of the matter is governed by the octupolar field and forms ring-like spots. Some
matter flows into the polar regions located near the dipole magnetic poles.
\section{Phase shifts in light curves of stars with complex fields}
In stars with {\it fixed} complex magnetic fields, and with a {\it constant} accretion rate, the emitting region may have complex shapes, but the
location of these hot spots is approximately {\it fixed} on the stellar surfaces. Rotation of these stars leads to a light curve with a pattern that
repeats every rotation, and no variations like phase shift of the light curves are expected. That is what we see in the light curves shown in Fig.
\ref{lcoct}, \ref{lcwdip} and \ref{lcsdip}, where oscillations occur with the fixed, initial phase. However, the pulse from the emitting region may
change in a complex way and produce the phase shift, if (1) the magnetic field of the star reconstructs, or (2) accretion rate varies with time.
Below we discuss these two possibilities.
\subsection{Phase shift due to field reconstruction}
The magnetic field of a star may vary with time due to internal dynamo processes. Such variations may lead to reconstruction of the field, and
different multipoles may dominate after the reconstruction. Or, the reconstructed field may have the same multipoles, but with different orientations
of their magnetic axes. Here we investigate an example where a predominantly octupolar field changes the orientation of its magnetic axis. We choose
an octupole with $\mu_3=0.3$, which in state (a), has its magnetic axis at $\Theta_3=20^\circ$ and $\phi_3=70^\circ$. Later, in state (b), it has new
orientation angles, $\Theta_3=60^\circ$ and $\phi_3=150^\circ$. In both cases we have a small dipole component: $\mu_1=0.2$, $\Theta_1=30^\circ$,
which does not change.
We observe that in both cases the disc interacts with the octupole-like field close to the star (like in \S 3.1) and the hot spots represent two
octupolar rings. However, the orientation of the rings and their brightness distribution are different due to the different axis directions. The most
important is that the octupole axes have different directions relative to the meridional plane, which leads to changes in hot spots and that
determines the phase shift. Fig. \ref{psreconstrucion} shows that the light curves in states (a) and (b) have different phases with a phase shift of
$\Delta\Phi=\Phi_b-\Phi_a=170^\circ$, where the amplitudes are normalized to the same values. The shapes of the light curves are different because
the angle $\Theta_3$ is different in these states, although this difference may not always lead to changes in the pulse shapes.
In most cases it is not well known how fast the dynamo-driven magnetic field reconstruction is. Observations of the CTTSs type young stars show that
the field may possibly vary on the scales of months or even weeks (Smirnov et al.\ 2004; Donati et al.\ 2007, 2008). It is less clear what the
time-scale of magnetic field variation in neutron stars and white dwarfs is.
We should mention that such phase shifts could occur during reconstruction
of a field with any combination of multipoles, including a purely dipole configuration.
\begin{figure}
\begin{center}
\includegraphics{fig20_ps_reconstruction.jpg}
\caption{\label{psreconstrucion} The phase shift due to the magnetic field reconstruction, where predominantly
octupolar field in the case of $\mu_3=0.3$, $\mu_1=0.2$, $\Theta_1=30^\circ$, changes its orientation. In the state
(a) $\Theta_3=20^\circ$ and $\phi_3=70^\circ$, in the state (b) $\Theta_3=60^\circ$ and $\phi_3=150^\circ$. }
\end{center}
\end{figure}
\subsection{Phase shift due to accretion rate variation}
In another example we consider a star with a fixed complex field where the different multipoles are oriented at different meridional angles, $\phi_n$.
Then, at high accretion rates, the disc comes closer to the star and interacts with the higher order multipoles, while at lower accretion rates, the
truncation distance of the disc moves away from the star and interacts with the lower order multipoles.
To study this case, we need a slight reinterpretation of our reference units defined in \S\ref{ref-units}. So far, we have used the
\textit{dimensionless} parameters $\mu_1$ and $\mu_3$ to control the magnetic moments of the star. However, they can also be used to control the
accretion rate instead, which is more convenient in this subsection; all we need to do is recast the reference units into a different form. In order
to do that, we note that the \textit{dimensional} dipole moment of the star is $\mu_{1\star} = \mu_1\mu_{1,0}$. In the previous sections, we kept the
reference value $\mu_{1,0}$ fixed, which is why varying $\mu_1$ corresponded to varying the dipole moment. Now, however, we keep $\mu_{1\star}$
fixed. It is then convenient to express $\mu_{1,0}$ in terms of $\mu_{1\star}$ as $\mu_{1,0} = \mu_{1\star} / \mu_1$. Then, $\mu_{1,0} = B_0 R_0^3$
gives $B_0 = \mu_{1,0} / R_0^3 = \mu_{1\star} / \mu_1 R_0^3$, and $\mu_{3,0} = B_0 R_0^5$ becomes $\mu_{3,0} = \mu_{1\star} R_0^2/ \mu_1$. The
dimensional octupole moment then is $\mu_{3\star} = \mu_3\mu_{3,0} = (\mu_3/\mu_1) \mu_{1\star} R_0^2$. The reference accretion rate is $\dot M_0 =
\rho_0 v_0 R_0^2 = B_0^2 R_0^{5/2}/\sqrt{GM} = \mu_{1\star}^2 / (\mu_1^2 R_0^{7/2}\sqrt{GM})$. The dimensional accretion rate $\dot M_{dim}$ is then
given by
\begin{equation}
\label{mdot} \dot M_{dim} \approx \left(\frac{\dot M}{\mu_1^2}\right)\frac{\mu_{1\star}^2}{(GM_\star)^{1/2} R_0^{7/2}}.
\end{equation}
It is now clear that changing the \textit{dimensionless} parameter $\mu_1$ has the effect of changing the accretion rate $\dot M_{dim}$, while
keeping $\mu_{1\star}$ fixed as mentioned above. To keep $\mu_{3\star}$ fixed as well, we simply have to change $\mu_3$ such that the ratio
$\mu_{3\star} / \mu_{1\star}$ is fixed. This allows us to change the accretion rate while keeping the stellar magnetic field fixed.
The physical meaning of this recasting is the following. The most natural interpretation of changing $\mu_1$ and $\mu_3$ is changing the stellar
magnetic field, the result of which is a decrease in the magnetospheric radius. However, we can also decrease the magnetospheric radius by fixing the
stellar magnetic field and increasing the accretion rate instead, which is exactly what Eqn. \ref{mdot} shows. The dimensionless parameters $\mu_1$
and $\mu_3$ are therefore best thought of as controlling the {\it magnetospheric size}.
This brings us the convenience of using $\mu_1$ to represent two states of a star with different accretion rates to investigate the phase shifts
between them. As an example, we take a star with a superposition of dipole and octupole fields, where they has parameters, $\Theta_1=30^\circ$,
$\Theta_3=20^\circ$, $\phi_3=70^\circ$ and fixed ratio of $\mu_3/\mu_1$. It is important that the dipole and octupole have different phase angles
$\phi_n$.
We choose two states with the above parameters for the dipole and octupole but different $\mu_1$: state (a) with $\mu_1=2$; and state (b) with
$\mu_1=0.2$. We observe from simulations that in state (a) the disc mainly interacts with the dipole component, while in state (b) it mainly
interacts with the octupole component. Fig. \ref{psmdot} shows a phase shift of peaks $\Delta\Phi=\Phi_b-\Phi_a=120^\circ$ between the states (a) and
(b), where the amplitudes are normalized to the same values. The phase shift occurs because the octupole and dipole axes are located in different
meridional planes and have different angles $\phi$, due to which the matter flows to different places of the star at different accretion rates.
This type of phase shift is expected during periods of accretion rate variation. What is the accretion rate ``jump" between states (a) and (b)? We
can give a simple estimate, $(\dot M)_b/(\dot M)_a=((\mu_1)_a/(\mu_1)_b)^2=(2/0.2)^2=100$. However, we should notice that in state (b) octupole
strongly dominates and Eqn. \ref{mdot} may not be used directly. In addition, the transition may occur at smaller values of $\mu_1$ in state (a). So,
it is clear that this type of phase shift may be produced by accretion rate (luminosity) variation.
Another thing that should
be mentioned is that the phase shifts would be expected to be accompanied by changes
in the light curve pulse profile, because the shape and position of the hot spots is
very different at different field configurations.
\begin{figure}
\begin{center}
\includegraphics{fig21_ps_mdot.jpg}
\caption{\label{psmdot} The phase shift due to the variation of the accretion rate. (a) For low accretion rate ($\mu_1=2$), the dipole component
stops the disc. (b) For high accretion rate ($\mu_1=0.2$), the octupole component stops the disc. The phase shift occurs during the transition of
these two states.}
\end{center}
\end{figure}
\section{Conclusions and discussions}
We performed, for the first time, global 3D MHD simulations of accretion onto stars with predominantly octupolar magnetic fields, and for a combination
of dipole and octupole fields. The calculation become possible due to enhancement of the grid resolution near the star. Simulations have shown that:
\begin{enumerate}
\item
If the disc interacts predominantly with the octupolar field, then matter flows into two octupolar rings on the surface of the star. In the case of
an inclined octupole field, the hot spots are inclined and have a brightness amplification on one side of the ring.
\item
At small misalignment angles of the octupole, $\Theta=10^\circ$, the light curves are quite sinusoidal and ``mimic'' the shape of the light curves of
the pure dipole field, in particular at small inclination angles $i$. However, at higher $\Theta$ (e.g., $\Theta=20^\circ$), the light curves
strongly depart from sinusoidal \textit{even for small $i$}.
\item
If the dipole component strongly dominates, then matter flows in two funnel streams governed by the dipole field, and two round spots form in the
vicinity of the dipole magnetic poles. If the dipole and octupole have comparable strength at the truncation radius, then both components channel
matter to the star. Usually both octupolar rings and polar spots form on the surface of the star.
\item
The potential field approximation is valid only in the region around the star where the
magnetic stresses are higher than the matter stresses. At larger
distances, the field is dragged by the disc and corona and strongly departs from being potential. The external magnetic field usually acquires a
significant azimuthal component and inflates. A number of field lines wrap around the rotation axis,
forming a magnetic tower which may propagate to
larger distances due to magnetic force (e.g., Romanova et al.\ 2009).
\item
Accretion onto stars with multipolar fields may lead to phase shifts in the light curves. This may occur either (a) during dynamo-generated internal
field reconstruction, or (b) during variation of the accretion rate when the complex magnetic field is fixed and has different phase angles
$\phi_n$ for each multipole component, but the disc interacts with multipoles of different orders at different accretion rates.
The phase shifts would be expected to be accompanied by changes
in the light curve pulse profile, because the shape and position of the hot spots is
very different at different field configurations.
\end{enumerate}
These new challenging 3D simulations helped us understand accretion onto stars with octupolar fields. Recent measurements by Donati et al.\ (2007, 2008)
of the two CTTSs V2129 Oph and BP Tau have shown that their fields have a significant octupolar component. In our next paper (Long et al.\ 2009) we
compare our numerical model with observations of these stars.
Magnetic fields in young stars of the CTTS type may have a complex structure and may vary with time due to internal dynamo processes. The recently
measured magnetic fields of the accreting T Tauri stars CV Cha and CR Cha (Hussain et al.\ 2009) show a complex structure consisting of a number of
multipoles. In another CTTS, V2247 Oph, the magnetic field is complex and varies on the very short time-scale of a week (Donati et al.\ 2009).
Frequent phase shifts are expected in this star due to internal field reconstruction, as discussed in this paper. In all these stars the phase
shifts may also occur due to accretion rate variation.
A phase shift of 0.2 was recently observed in the millisecond pulsar SAX J1808.4-3658 on the 14th day of its outburst (Burderi et al.\ 2006). This
phase jump was observed only in the fundamental component of the almost sinusoidal light curve, and much less so in the 1st harmonic. Burderi et al.
(2006) suggested that this phenomenon may be connected with fast spin-down of the millisecond pulsar or with some magnetic field reconstruction
during a stage of enhanced accretion. We suggest that it raises the possibility that the magnetic field of the millisecond pulsar may be more complex
than a dipole, and therefore during enhanced accretion the disc may interact with deeper layers of the magnetosphere where quadrupolar or other
components might possibly influence the flow. We also suggest that the dipole may be slightly off-center (Ruderman 1991; Long et al.\ 2008), and hence
the disc matter accretes to both magnetic poles when the accretion rate is high, or to only one pole if it is low. Superposition of dipoles may also
be a possible reason.
\section*{Acknowledgments}
This research was conducted using the NASA High End Computing Program computing systems, specifically the Columbia,
Discover and Pleiades superclusters. The authors thank A.V. Koldoba and G.V. Ustyugova for the earlier development of
the codes, and A.K. Kulkarni and R.V.E. Lovelace for helpful discussions. The research is supported by NSF grant AST0709015,
and funds of the Fortner Endowed Chair at Illinois. Research of MMR is supported by NASA grant NNX08AH25G and NSF grant
AST-0807129.
|
2,877,628,089,482 | arxiv | \section{introduction}
It is well known that galaxies are not distributed uniformly within (at least) the virialized region, and most of their properties are influenced by their local environments. Early-type galaxies are usually spheroidal, red, star-formation passive, and occupy the inner regions of clusters, whereas late-type galaxies tend to be disky, blue, star-formation active, and prefer low-density environments such as the field \citep[e.g.,][]{dressler1980}. This behavior is known as the ``morphology-density relation"
\citep{postman1984, gomez2003, kauffmann2004, Balogh2004, weinmann2006, weinmann2010, vonderlinden2010, wetzel2012}.
Morphology, colors, and star formation activity are not the only properties of galaxies for which some type of dichotomy has been found. Especially in the last decade, many authors (e.g., \citealt{presotto2012, Balogh2004, Roberts2015, Contini2015, Joshi2016}, just to quote the latest) have focused attention on stellar mass segregation, with the claim that more massive galaxies tend to be distributed closer to the cluster center, usually for dynamical reasons that are often linked to the correlation between a satellite galaxy and its dark matter halo. If more massive or luminous satellites are associated with more massive haloes, as one would expect, dynamical friction would bring more massive haloes (and thus galaxies) more quickly to the innermost regions of the cluster \citep{chandrasekhar1943}.
In the context of dark matter subhaloes, mass segregation has been discussed extensively by several authors \citep[e.g.,][]{delucia2004,contini2012,vandenbosch2016}. \citet{contini2012} showed that haloes that are more massive at the time of infall to the cluster get closer to the center on shorter timescales due to dynamical friction and suffer more significant stripping. On the other hand, the luminosity and the stellar mass are expected to be more strongly related to the mass of the galaxy at the time of infall than to the properties of the present dark matter haloes \citep{gao2004a,vale2006,wang2006}. Consequently, it is natural to expect galaxies to be segregated according to their stellar mass in clusters.
However, there is not yet a general consensus on these issues; both observationally and theoretically different authors either do find segregation \citep[e.g.,][]{lares2004,mcintosh2005,vandenbosch2008,presotto2012,balogh2014,Roberts2015,Contini2015,Joshi2016,vandenbosch2016,nascimento2017} or else they find weak or no segregation \citep[e.g.,][]{ziparo2013,vulcani2013,vonderlinden2010,kafle2016,joshi2017}.
Among observational and theoretical studies in favor of mass segregation, we quote here the most recent ones. \citet{Roberts2015} used galaxy ``group" catalogs derived from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7) and found clear mass segregation, the strength of which depends on both the galactic stellar mass cut (being higher with the inclusion of lower mass galaxies) and on the ``group" halo mass (decreasing with increasing group halo mass). \citet{Contini2015} took advantage of a semi-analytic model of galaxy formation coupled with a high-resolution N-body simulation explicitly to probe mass segregation in groups and clusters. They found mass segregation in both the stellar mass and the dark matter mass out to the virial radius of the main halo. Using galaxy analogues related to N-body simulations \citet{Joshi2016} found a behavior similar to the results shown by \citet{Contini2015} up to 0.5 $R_{\rm vir}$ of the host halo, but with a weaker trend with increasing radial distance. Clear evidence of mass segregation in the dark matter mass has been found also by \citet{vandenbosch2016}. They ran three cosmological N-body simulations and focused on the segregation of 12 properties of subhaloes that depend upon the orbital energy and distance from the halo center, and they found that subhaloes do not show any segregation when their present-day masses are considered. However, when their mass at infall is considered, the segregation in mass is evident. More massive subhaloes fall faster to the center because of stronger dynamical friction, and lose a larger fraction of mass via tidal stripping. As discussed above, these objects are expected to host massive galaxies.
Other authors have found different results. \cite{ziparo2013} studied 52 X-ray selected galaxy groups at $0 < z < 1.6$ and found no clear evidence of mass segregation. Similar results have been found by \cite{vonderlinden2010} at $z < 0.1$ in a sample of SDSS galaxy clusters, using different stellar mass cuts in four redshift ranges. Indirectly, \cite{vulcani2013} worked on the galaxy stellar mass function at intermediate redshift, using bins of distance from the cluster center in different environments, and they found no sign of mass segregation. More recently, \cite{kafle2016} presented a full comparison between the observed Galaxy And Mass Assembly (GAMA) data, a semi-analytic model ({\small GALFORM}), and the EAGLE hydrodynamic simulation data and found negligible mass segregation in galaxy group environments. Similar conclusions have been obtained by \cite{joshi2017}, who found weak mass segregation in the inner regions of groups and clusters.
In this work, we address this topic by taking advantage of a set of zoom-in hydrodynamic simulations and a set of observed galaxy clusters in the local universe. Our goal is to analyze the distributions of galaxies in clusters within 1 $R_{\rm vir}$, by looking at both the stellar and dark matter masses of galaxies and their associated haloes.
This manuscript is structured as follows. In Section 2, we describe the main features of our simulations and the observational data. In Section 3, we present our results, which are discussed in Section 4. We summarize the main conclusions of our analysis in Section 5.
\section{Data}
In this work, we use both an observational and a theoretical approach. We take advantage of a set of observational data of galaxy clusters, named the KASI-Yonsei Deep Imaging Survey of Clusters \citep[KYDISC,][]{Oh2018}, and a set of hydrodynamic zoom-in cluster simulations, named the Yonsei Zoom-in Cluster Simulations \citep[YZiCS,][]{Choi2017}. Their main features are presented below.
\subsection{Observational Data}
KYDISC was performed with the Inamori Magellan Areal Camera and Spectrograph \citep[IMACS,][]{Dressler2006} on the 6.5-meter Magellan Baade telescope at Las Campanas Observatory (LCO) and the MegaCam on the 3.6-meter Canada-France-Hawaii Telescope to target 14 Abell clusters at $0.015 < z < 0.144$.
Then, follow-up spectroscopy was done using the Magellan IMACS, the Wide-field Reimaging CCD Camera (WFCCD) on the du Pont 2.5 m telescope at LCO, and Hydra on the WIYN 3.5 m telescope.
The redshift information of galaxies was attained from the literature (141 galaxies from the NASA/IPAC Extragalactic Database; 619 from the SDSS; and 380 from the Hectospec Cluster Survey; 5 from the 6dF survey) and derived from the observations (112 galaxies from the IMACS; 9 from the du Pont; and 143 from the WIYN).
This results in magnitudes, redshifts, morphologies, bulge-to-total ratios, and local densities for the 1409 cluster galaxies brighter than $-19.8$ in the r-band. The spectroscopic completeness is 0.8 if Abell 1278 is excluded. To derive the stellar mass of cluster member galaxies, we use the equation from \citet{Bell2003}.
Since the formula gives $M_{\rm\ast}$ based on a ``diet" Salpeter initial mass function (IMF), we have multiplied 0.7 to $M_{\rm\ast}$ in order to convert it to a normal Salpeter IMF. We consider the point at which the low mass end of the stellar mass function begins to decline as the mass threshold for our analysis, which is $M_{\rm\ast}$ = $10^{10.3}\ M_{\rm\odot}$ in this case. We use the position of the brightest cluster galaxies (BCGs) as the cluster centers, and we remove the BCGs from our analysis. For both sets of data we used $R_{\rm 200}$ as the virial radius. For more detailed information about the KYDISC data, we refer the reader to \citet{Oh2018}.
\subsection{Hydrodynamic Simulations}
YZiCS is a set of cosmological zoom-in simulations performed with the adaptive mesh refinement code RAMSES \citep{Teyssier2002}. Dark matter only simulation was first run within a cubic volume 200 $\rm h^{-1}$ Mpc on each side. In this parent volume, a total number of 16 dense regions have been selected. The selected volume out to 3 $R_{\rm vir}$ were backtraced until the initial condition, and the hydro zoom-in simulation was conducted for the spherical volume that contains all the particles. The simulation was based on the same baryon physics recipe that was used in the Horizon-AGN simulation \citep{Dubois2012}, including feedback from both active galactic nuclei (AGNs) and supernovae (SNe). Throughout the simulations, the WMAP7 cosmology from \citet{Komatsu2011} was assumed: ${\Omega}_{\rm m}$ = 0.272, ${\Omega}_{\rm\lambda}$ = 0.728, $H_{\rm 0}$ = 70.4 km $\rm s^{-1}\ Mpc^{-1}$, ${\sigma}_{\rm 8}$ = 0.809, and n = 0.963. Objects with more than 200 stellar particles were classified as galaxies, which roughly corresponds to $10^{8}\ M_{\rm\odot}$. We identified a total of 6656 galaxies above the mass threshold $M_{\rm\ast}$ = $10^{9}\ M_{\rm \odot}$ by using the AdaptaHOP halo finder \citep{Aubert2004}. The stellar mass threshold corresponds roughly to the point at which the stellar mass function begins to decline at the low mass end, and we consider it to be the value of the stellar mass to which our galaxy sample is reasonably complete. For more details concerning the properties and features of the simulation, we refer the reader to \citet{Choi2017}.
We have created two catalogs with YZiCS: one is drawn only from the simulations, and the other is adjusted for comparison with the observations. In the first catalog, the distance of a galaxy from the cluster center is measured by projecting into two dimensions (2D) in random directions. To make fair comparisons with the observations, we defined the BCGs as the cluster centers and made 100 random projections into 2D. This results in a total of 665,600 projected properties observed in the simulations. In order to look back in time and inspect various environmental effects, in the second catalog we used the galaxy merger tree constructed by \citet{Lee2018}, which was created using the algorithm given by \citet{jung2014}. To match galaxies to the haloes in which they reside, we made use of the linking length motivated by \citet{Behroozi2013}, where
\begin{equation}
d(h,p) = \left(\frac{{|x_{\rm h}-x_{\rm g}|}^{2}}{{r^{2}_{\rm{dyn,vir}}}}+\frac{{|v_{\rm h}-v_{\rm g}|}^{2}}{\sigma_{\rm v}^{2}}\right)^{1/2}
\end{equation}
is the distance in the 6D phase-space of positions and velocities. The quantities $x_{\rm h}$ and $v_{\rm h}$ are the halo position and velocity, $r_{\rm{dyn,vir}}$ is the virial radius of the halo, $\sigma_{\rm v}$ is the velocity dispersion of the halo, and $x_{\rm g}$ and $v_{\rm g}$ are the galaxy position and velocity. To derive the velocity dispersion of the galaxies included within the linking length, we used the mass-velocity dispersion relation from \citet{munari2013}:
\begin{equation}
\frac{\sigma_{\rm 1D}}{\rm km \ \rm s^{-1}} = A_{\rm 1D} \left[ \frac{h(z)M_{\rm 200}}{10^{15}M_{\rm\odot}} \right]^{\alpha}
\end{equation}
with $A_{\rm 1D}$ = 1244 km\ $\rm s^{-1}$ and $\alpha$ = 1/3.
We sorted the list of galaxies and chose the subhalo with the shortest linking length as the host for each galaxy. We performed this matching process both for galaxies at redshift zero and at their time of infall; i.e., when they crossed 1.5 $R_{\rm vir}$ for the last time. We set the distance at infall to be 1.5 $R_{\rm vir}$ to prevent cluster tides from affecting the galaxies and subhaloes before infall. Although galaxies may cross this baseline many times, we used this criterion in order to remove back-splashed galaxies from `accreted' galaxy samples. This is because galaxies that move out of the cluster are not defined as a part of the accreted portion of the cluster. To make sure that the subhaloes matched at redshift zero and at the time of infall are from the same subhalo tree, we checked manually for sudden changes in mass and position. Unlike the first catalog, in this case the distances of galaxies from their cluster centers are in 3D, and we considered the location of the main haloes with the maximum number of dark matter particles to be the cluster center. Throughout this paper, we removed the BCGs from our analysis. The resulting information of the KYDISC and the YZiCS clusters
is presented in Tables 1 and 2. In Section 3.1 below, we make use of the first catalog, while in Section 3.2, we use the second one described above.
\section{results}
In this section we analyze our observational (KYDISC) and simulation (YZiCS) data to see whether or not they show any mass segregation trend. For fair comparisons, we used the same mass threshold ($10^{10.3}\ M_{\rm \odot}$) to them.
\begin{deluxetable}{lccc}[t!]
\tablecaption{KYDISC Cluster Characteristics}
\tablecolumns{4}
\tablenum{1}
\tablewidth{0pt}
\tablehead{
\colhead{Name} &
\colhead{$M_{\rm vir}$ \tablenotemark{a}} &
\colhead{$R_{\rm vir}$\tablenotemark{b}} &
\colhead{$N_{\rm sat}$\tablenotemark{c}} \\
\colhead{} & \colhead{($10^{14}\ M_{\rm\odot}$)} &
\colhead{(Mpc)} & \colhead{$\geq 10^{10.3}\ M_{\odot}$}
}
\startdata
Abell 116 & 2.25 & 1.26 & 37 \\
Abell 646 & 5.33 & 1.64 & 93 \\
Abell 655 & 6.43 & 1.75 & 166 \\
Abell 667 & 5.12 & 1.61 & 113 \\
Abell 690 & 3.42 & 1.44 & 76 \\
Abell 1126 & 4.99 & 1.63 & 61 \\
Abell 1139 & 1.25 & 1.04 & 40 \\
Abell 1146 & 10.83 & 2.07 & 115 \\
Abell 1278 & 9.54 & 1.99 & 41 \\
Abell 2061 & 11.40 & 2.15 & 221 \\
Abell 2249 & 12.39 & 2.21 & 226 \\
Abell 2589 & 10.40 & 2.11 & 61 \\
Abell 3574 & 1.92 & 1.21 & 8 \\
Abell 3659 & 2.04 & 1.21 & 20 \\
\enddata
\tablenotetext{a}{Virial mass of the clusters.}
\tablenotetext{b}{Virial radius of the clusters.}
\tablenotetext{c}{The number of satellites within 1 $R_{\rm vir}$ of each of the cluster with the mass cut $10^{10.3}\ M_{\rm\odot}$.}
\end{deluxetable}
\begin{deluxetable}{lcccc}[t!]
\tablecaption{YZiCS Cluster Characteristics}
\tablecolumns{6}
\tablenum{2}
\tablewidth{0pt}
\tablehead{
\colhead{Name} &
\colhead{$M_{\rm vir}$ \tablenotemark{a}} &
\colhead{$R_{\rm vir}$\tablenotemark{b}} &
\colhead{$N_{\rm sat}$\tablenotemark{c}} &
\colhead{$N_{\rm sat}$\tablenotemark{d}} \\
\colhead{} & \colhead{($10^{14}\ M_{\rm\odot}$)} &
\colhead{(Mpc)} & \colhead{$\geq 10^{10.3}\ M_{\odot}$} & \colhead{$\geq 10^{9}\ M_{\odot}$}
}
\startdata
C1 & 9.02 & 1.99 & 106 & 489 \\
C2 & 5.12 & 1.64 & 45 & 245 \\
C3 & 2.06 & 1.21 & 27 & 149 \\
C4 & 1.99 & 1.20 & 17 & 110 \\
C5 & 1.98 & 1.20 & 21 & 143 \\
C6 & 1.93 & 1.19 & 19 & 87 \\
C7 & 1.78 & 1.16 & 24 & 159 \\
C8 & 1.53 & 1.10 & 14 & 93 \\
C9 & 1.36 & 1.06 & 17 & 93 \\
C10 & 1.30 & 1.04 & 14 & 91 \\
C11 & 0.68 & 0.84 & 6 & 65 \\
C12 & 0.59 & 0.80 & 8 & 34 \\
C13 & 0.55 & 0.78 & 3 & 40 \\
C14 & 0.50 & 0.76 & 7 & 34 \\
C15 & 0.46 & 0.74 & 3 & 32 \\
C16 & 2.51 & 1.30 & 34 & 164 \\
\enddata
\tablenotetext{a}{Virial mass of the clusters.}
\tablenotetext{b}{Virial radius of the clusters.}
\tablenotetext{c}{The number of satellites within 1 $R_{\rm vir}$ of each of the cluster with the mass cut $10^{10.3}\ M_{\rm\odot}$.}
\tablenotetext{d}{The number of satellites within 1 $R_{\rm vir}$ of each of the cluster with the mass cut $10^{9}\ M_{\rm\odot}$.}
\end{deluxetable}
\subsection{Galaxy Mass Distribution in 2D Projection}
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure1.pdf}
\centering
\caption{The mean stellar mass of the KYDISC and YZiCS galaxies as a function of 2D clustocentric distance. See the text for details. The magenta and gray shades represent the 1${\sigma}$ scatter around the mean values. The mass threshold is $10^{10.3}\ M_{\rm \odot}$ for both the KYDISC and the YZiCS data.}
\end{figure}
Figure 1 shows the mean stellar mass in each radial shell from KYDISC (magenta line) and YZiCS (black line). The region $R_{\rm 2D} \leq 1 \ R_{\rm vir}$ in each cluster was divided into 10 distance bins, which results in 10 concentric shells. The two lines represent the mean values of 16 YZiCS and 14 KYDISC clusters, while the shaded regions represent 1$\sigma$ scatter around the mean. Neither of them clearly shows mass segregation. The lack of mass segregation we find here may be due to several factors. Firstly, to enhance the completeness of our samples we do not include low-mass galaxies which are suspected to reveal the mass segregation trend better \citep{Roberts2015}, and secondly, some of our clusters lie in the highest range of cluster masses compared to those presented in the literature, where mass segregation effect is found to be either weak or null (see Section 3.2). We discuss these two effects in more detail in Section 3.2.
Another complication in the analysis shown in Figure 1 is that it does not take into account the fact that the distribution of galaxy masses depends on the cluster mass. Mixing different galaxy mass distributions of clusters of different masses would hide a trend even though there is any. Besides, Figure 1, which is the format that has been widely used in the literature, shows the mean galaxy mass in each radial bin. In this case, even if more of the massive galaxies are distributed in the inner volume in a more massive cluster, the ``mean'' stellar mass might still be smaller, since there are many more of less massive galaxies in the inner regions.
In order to avoid such effects of cluster masses, we have decided to use a fixed ``fraction'' of more massive galaxies within the sample of each cluster instead of mean masses. For example, if we use 50\% as our cut, we count the top 50\% massive galaxies in each cluster and present their fraction as a function of radial distance. We perform this for all clusters with varying cluster masses. Then, we have the mean radial distribution of the 50\% most massive galaxies in each cluster for all clusters. For reference, the typical mass criterion for the top 50\% is higher than $10^{11}\ \mbox{$\rm M_\odot$}$. In the case of KYDISC (or YZiCS), we have 14 (16) data points as a result in each radial distance bin, and we derive their mean and error through the bootstrapping method. Figure 2 shows the result. The mass segregation trend has become clearer at least in the case of YZiCS. KYDISC shows a less clear trend, and we discuss this in Section 4.
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure2.pdf}
\centering
\caption{The fraction of massive galaxies (the top 50\% of the stellar mass distribution) relative to the total number of galaxies more massive than $10^{10.3}\ M_{\rm\odot}$ as a function of clustocentric distance for the whole sample of KYDISC and YZiCS clusters. The error bars and the shaded region indicate the 1$\sigma$ scatter.}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure3.pdf}
\centering
\caption{Slopes of the linear fits to the data points shown in Figure 2 based on the use of different fractional cuts. Figure 2 shows the case of 50\%. The shaded regions represent the 1$\sigma$ scatter of the distribution. A negative slope indicates the presence of mass segregation. The YZiCS data consistently suggest mass segregation regardless of the choice of the fractional cut. The KYDISC data, on the other hand, suggests a hint of mass segregation for the cut of 20\% or more, because the uncertainty band is consistently below the zero slope.}
\end{figure}
Our choice of the fractional cut of 50\% in mass is arbitrary, but the analysis is stable against the choice as long as a reasonable value is used. We fitted the mass segregation slope of Figure 2 in the range 0 $\leq R_{\rm 2D}/R_{\rm rvir} \leq$ 1 to the linear relation $\alpha$x+$\beta$ using the ${\chi}^{2}$--minimization method, with different percentiles for the high galaxy mass cut, and the slopes `$\alpha$' for different massive galaxy percentiles are shown in Figure 3. Having a negative slope means that mass segregation does exist. Mass segregation appears robust for the YZiCS data regardless of the choice of the fractional mass criterion. It is less clear for the KYDISC data and marginally visible for the range of $\gtrsim 20\%$, because the uncertainty band tends to locate below zero. Considering the limited number of sample galaxies above the mass cut of $10^{10.3} \ \mbox{$\rm M_\odot$}$ (see Tables 1 and 2), a fractional cut of $\gtrsim 20\%$ is recommended.
\subsection{Halo Mass Dependence of Mass Segregation}
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure4.pdf}
\caption{Mass segregation slope of the stellar mass and radial distance relationship as a function of halo mass; the dots represent the slopes and the error bars the associated 1$\sigma$ scatter for each of our 16 YZiCS clusters; the red line is the result of an ${\chi}^{2}$--fit, and its slope and error are shown in the legend. The slopes in this figure are different from that of Figure 3 because all of the galaxy mass over $10^{9}\ M_{\rm\odot}$ were used without clustocentric distance binning. The cluster mass ranges 0.46 $\times 10^{14}\ M_{\odot}$ -- 9.02 $\times 10^{14}\ M_{\odot}$.}
\end{figure}
The mass segregation trend was not visible in the mean stellar mass vs. clustocentric distance diagram (Figure 1). However, mass segregation became visible when we properly normalize the mass functions of galaxies in different clusters based on mass ranking and used the massive galaxy fraction instead of the mean stellar mass (Figure 2).
To give a physical explanation of why mass segregation happens, we use YZiCS to go back in time and monitor the related quantities to the galaxies.
We have decided to use galaxy mass vs. clustocentric distance on individual clusters in this analysis because too few galaxies are in some YZiCS clusters to use the fractional approach.
Figure 4 shows the linear-fitted slope of the stellar mass as a function of clustocentric distance for {\em individual} clusters with different halo masses from YZiCS. The fit was made on the unbinned data.
Because we do not compare simulations against observations any more, we use for better statistics a lower mass cut for the galaxies, that is, $M_* \geq 10^{9}\ M_{\rm\odot}$, which is roughly the completeness limit of the YZiCS simulation.
The haloes of smaller masses show more negative slopes, which means that less massive haloes show more mass segregation.
This is tightly linked with the dynamical friction timescale.
For a given cluster halo mass, a more massive galaxy reaches the cluster center more quickly.
If we simplistically assume that galaxies with different masses arrive at a cluster at the same time, therefore, radial mass segregation is naturally expected after some time.
This may be what happened to low-mass clusters.
It becomes more complicated in massive clusters, however. According to the same dynamical friction formula, a galaxy of a given mass takes longer to reach the center of a more massive cluster. During this longer period, the galaxy is bound to lose more mass due to stellar stripping \citep{Smith2016}.
To consider this cluster mass effect in our analysis, we divide the YZiCS cluster sample into two groups: the most massive two clusters (C1 and C2 in Table 2) in one group and the rest in the other, simply because the mass distribution of the clusters shows a large gap between them.
Figure 5 shows the mean galaxy mass as a function of clustocentric distance for the two groups. Two mean galaxy masses are shown: those at the present epoch marked by the tips of the arrows, and those at the time of infall (when they cross 1.5 $R_{\rm vir}$ for the last time) marked by the bases of the arrows. The arrows show the difference between the galaxy mass at the infall and the current epoch.
The first to note is the mean mass of the galaxies in the central bin of the less massive clusters which is significantly higher than any other points in the diagram. We interpret this as a direct result of dynamical friction.
The infall masses of the galaxies in the central bins of the more massive clusters (three leftmost magenta points), however, are not particularly higher than the values in other bins. This is because they arrived at their clusters relatively earlier, as demonstrated in Figure 6. This figure shows the kernel density estimation of the redshift at infall of galaxies in the two halo mass groups. We consider the galaxies in each cluster as samples from the same population. In reality, clusters are different from each other due to the difference in the detailed history as we are demonstrating in this study. In this sense, our assumption that all the cluster galaxies are from the same parent sample is overly optimistic, and our errors here are upper limits.
Having that in mind, the difference in $z_{\rm inf}$ of the two subsamples appear marginally distinct. Although the scatter do overlap between the two subsamples, more massive clusters have consistently higher values of $z_{\rm inf}$ at $R \lesssim 0.4 \ R_{\rm vir}$ (Figure 6)
Because the galaxies in the inner volume of more massive clusters spend a longer time inside their cluster than those of the less massive clusters, they lose more stellar mass (e.g., magenta arrows at $< 0.3 \ R_{\rm vir}$ in Figure 5). In conclusion, the effect of mass segregation is at work here in the more massive clusters as well, but it is invisible because of the competing effect of arrival time being earlier for the galaxies in the central region.
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure5.pdf}
\caption{Mean stellar mass of each clustocentric shells at the time of infall (bases of the arrows) and present epoch (tips of the arrows), for the 14 less massive clusters and 2 more massive clusters. The inner volume of both subsamples lost more stellar mass than the outskirts, and the galaxies in the inner volume of the more massive clusters lost more mass than the less massive clusters.}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Figure6.pdf}
\caption{Kernel density estimation of redshift at infall ($z_{\rm inf}$) of galaxies in each of the two halo mass groups. The bandwidth is determined by using the Silverman's rule of thumb \citep{Silverman1986}. Both of the subsamples show a decrease of redshift at infall as a function of clustocentric distance and the more massive clusters show higher redshift at infall in general.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=8.5cm]{Figure7.pdf}
\centering
\caption{Subhalo mass at the present epoch ($M_{\rm DM}$) divided by subhalo mass at infall ($M_{\rm DM,inf}$) from YZiCS as a function of clustocentric distance; (a) for groups of different halo mass clusters, (b) for different bins of redshift at infall. The division of the sample has been made to achieve the same subsample size for statistical conveniences. The typical scatter, shown as vertical bars in each panel, is (a) 0.32 for the 14 clusters; 0.34 for the other, (b) 0.33.}
\end{figure}
The dynamical evolution of galaxies inside a cluster, including dynamical friction, is in fact more dominantly determined by the halo properties of galaxies rather than their baryonic properties. Therefore, what has been presented and discussed above should be visible in the dark matter halo properties of galaxies as well. Figure 7 shows the halo properties of galaxies inside clusters. Panel (a) shows the ratio between the present-day halo mass and the halo mass at the time of infall as a function of clustocentric distance. As expected, the galaxies in the inner regions have less of their dark matter haloes remaining due to tidal stripping for a longer time inside the cluster.
Panel (b) indeed shows that the subhaloes that fell in the cluster earlier lose more mass. This is consistent with the earlier studies \citep{rhee2017, Han2018}.
\section{discussion}
Galaxy evolution can be explained either by a `\emph{nature}' or a `\emph{nurture}' scenario. In the context of stellar mass segregation in clusters, a \emph{nature} scenario links the galaxy to its mass at the time it was born, and a \emph{nurture} scenario links mass segregation to what happened from the time of accretion to the present time. Dynamical friction steals the orbital energy of massive galaxies thereby making them spiral inward to the cluster center. This happens because objects orbiting in the cluster slower than the satellite pull the satellites backward, thereby causing a sort of `friction' that results in the loss of the satellite's orbital energy.
Using the dynamical friction formula from \citet{chandrasekhar1943}, we find
\begin{equation}
t_{\rm df} \propto \frac{m_{\rm host}}{m_{\rm sat}}
\end{equation}
where $m_{\rm host}$ is the mass of the host halo, and $m_{\rm sat}$ is the mass of the satellite galaxy it contains.
For a given galaxy mass the dynamical friction timescale is therefore shorter for less massive haloes.
Dynamical friction, however, may not always result in visible mass segregation mainly because galaxy mass is not conserved in some cases. Tidal stripping associated with dynamical friction is effective both for galaxies and their dark haloes.
But the latter are more prone to it because galaxies are in some sense ``protected'' by their host haloes.
It has been suggested that the tidal stripping of dark haloes significantly precedes that of baryonic galaxies \citep{Smith2016}.
But eventually, the galaxies that spent a large amount of time inside their cluster lose some mass.
For stellar mass segregation to be significant, the effect of dynamical friction on the subhalo mass must be larger than the effect of tidal stripping on the galaxy mass.
If this is not the case, mass segregation would not be clearly visible, even though it actually happened.
\begin{figure*}[t!]
\includegraphics[width=18cm]{Figure8.pdf}
\centering
\caption{An example of the galaxies in the (a) less massive cluster C4 ($1.99 \times 10^{14}\ M_{\rm\odot}$) and (b) more massive cluster C2 ($5.12 \times 10^{14}\ M_{\rm\odot}$). The empty stars denote the position of the galaxies at the time of accretion, the filled stars indicate the same at the current epoch, and the color coded dots are the same as the epoch in between the former two. The colorbar indicates the stellar mass at each epoch of the two galaxies. To efficiently demonstrate the orbit and the mass changes of the two galaxies, each of the galaxies was projected onto a plane in time sequence, respectively. The normal vector of the projected plane was derived by using the position vector of the galaxy position at the current epoch and at the time of infall, with their vector origin as the cluster center at the present epoch respectively. Dynamical friction works on both of the clusters, but stellar mass segregation is seen only for (a). (b) shows no mass segregation because of tidal stripping: satellites in more massive clusters lose a greater amount of dark matter.}
\end{figure*}
We showed in Figure 4 a robust trend in the segregation slope according to halo mass. To see and quantify what makes and diminishes segregation, we divided the 16 clusters into two groups using $10^{14.6}\ M_{\rm \odot}$ as the halo mass cut. We found that the stellar mass loss due to tidal stripping is greater for the more massive clusters and in the inner regions of both halo groups. This is due to the longer time since infall for galaxies found in the inner regions and for the more massive clusters, as shown in Figure 6.
This is schematically illustrated in Figure 8 by using the YZiCS data. Panel (a) shows a trajectory of a galaxy that falls into a less massive cluster at $z=0.653$. Its stellar mass, color coded following the key on the right, shows a small decrease through the orbital motion due to tidal stripping. Panel (b) shows another galaxy in a more massive cluster. It was accreted into the cluster much earlier at $z=1.174$ with a similarly large value of mass. But through a long period during its orbital motion, it loses a much larger fraction of mass.
Also, according to Figures 5 and 6, we can infer that less massive galaxies were accreted onto clusters at an earlier epoch in more massive clusters.
These effects combined make the mass segregation effect invisible in massive clusters.
The radial mass segregation of galaxies is linked to the fate of the dark matter haloes that contain them, and to the way they approach the center of their cluster halo. Therefore we use Figure 7, where we show the ratio of the dark matter mass remaining in a subhalo at the present epoch compared to the dark matter mass at infall, when they cross 1.5 $R_{\rm vir}$.
Haloes that have earlier infall times or are closer to the cluster center go through more mass loss via tidal stripping. Therefore, satellites that are more massive at infall get to the cluster center faster and lose their mass more quickly. Also, the satellites that have early infall time and have managed to exist at the present epoch tend to be less massive at the time of infall. They go through a great amount of mass loss due to tidal stripping. On the other hand, the more massive satellites that have higher redshift at the time of infall have a high possibility of being already merged and thus not appearing in the figures. These two effects together may weaken the stellar mass segregation. We minimized this effect in Figure 2 by looking at the ratio of the more massive galaxies in concentric shells in each of the clusters. This greatly increased the segregation trend for YZiCS, while KYDISC showed a slight amount of mass segregation. The segregation trend of KYDISC may show less mass segregation than it actually has due to the following reasons: projecting clustocentric distances from 3D to 2D lessens the segregation trend (Figures 1 and 5); a relatively high mass cut is used ($10^{10.3}\ M_{\rm\odot}$); the average halo mass range is larger than that of YZiCS; some clusters might not be fully relaxed; and finally KYDISC must have a fair level of foreground and background galaxy contamination.
Previous work on this topic can be understood in this same context. For example, it might be that \citet{vonderlinden2010} found no stellar mass segregation because they have a large galaxy mass cut ($10^{9.6}\ M_{\rm\odot}$). \citet{ziparo2013} find weak stellar mass segregation in the inner regions of the clusters. Their segregation trend is weak when $10^{10.3}\ M_{\rm\odot}$ is used as the mass cut, but the trend becomes much stronger when $10^{9}\ M_{\rm\odot}$ is used as the mass cut. \citet{kafle2016} states that they find a lack of stellar mass segregation because the dynamical friction timescale is longer than the crossing timescale. However, their lack of the trend may instead be the result of using a higher redshift range.
\citet{vulcani2013} found no stellar mass segregation, possibly because they used a high mass cut of $10^{10.5}\ M_{\rm\odot}$. \citet{vandenbosch2016} find dark matter mass segregation for the accretion mass and peak mass, but this trend disappears when they reach redshift zero because of tidal disruption and the fact that some subhaloes of larger mass get accreted later.
We assume that the absence of mass segregation in their study is due to a different use of the mass cut, in the sense that they used a ratio of the haloes and subhaloes instead of considering the mass functions of each of the haloes.
\section{summary}
In this paper, we have investigated the existence of mass segregation of galaxies in clusters using both observations (KYDISC, cluster mass range $1.92 \times 10^{14}\ M_{\rm\odot}$ -- $1.24 \times 10^{15}\ M_{\rm\odot}$) and a set of hydrodynamic simulations (YZiCS, cluster mass range $0.46 \times 10^{14}\ M_{\rm\odot}$ -- $9.02 \times 10^{14}\ M_{\rm\odot}$). We found mass segregation depending on the cluster properties, and our main results are the following.
\begin{itemize}
\item When {\em mean stellar mass} is used for the investigation of mass segregation, segregation effect is invisible for both the KYDISC and YZiCS data (Figure 1).
\item The segregation effect became visible when {\em massive galaxy fraction} based on the mass rank in each cluster is used instead. This is a way of normalizing the galaxy mass function in terms of cluster mass (Figures 2 and 3).
\item The stellar mass segregation trend obtained by using all the main haloes of different masses together smears out the trend because individual haloes have trends of different degrees. We find a negative cluster mass dependence of mass segregation effect (Figure 4).
\item Mass segregation is visible in low-mass clusters simply as a result of dynamical friction time being shorter for more massive galaxies (Figure 5).
\item Massive clusters may not show mass segregation mainly because the more centrally located galaxies tend to fall in their clusters relatively earlier and lose more mass during their longer orbital motion. In addition, because of their early arrival, their masses were not particularly large at the time of infall (Figures 5 and 6).
\end{itemize}
Dynamical friction works as a drag force to segregate mass in clusters. It is natural to expect this to result in stellar mass segregation. However, tidal stripping acts as an obstacle and disturbs the mass segregation trend. The galaxies in the inner volume of the more massive clusters seem to have early infall time than those in the less massive clusters. The ratio of the effect of dynamical friction and mass stripping determines the visibility of mass segregation, and the trend is clearly related to the halo mass.
Previous literature in general may have found no mass segregation trend either because they used a high mass cut or a higher range of redshift.
These explanations can account for the discrepancies in the literature concerning the existence of mass segregation trends.
We cannot state that stellar mass segregation exists without knowing the specific conditions in the clusters. What we can conclude is that mass segregation does exist when certain conditions are met in the clusters and satellites. In this context, we conclude that the discrepancies found in the previous literature are not really disagreements after all; they all can be integrated into a single mass segregation scenario.\\
\section*{Acknowledgement}
S.K.Y. acknowledges support from the Korean National Research Foundation (NRF-2020R1A2C3003769).
E. Contini acknowledges the National Key Research and Development Program of China (No. 2017YFA0402703), and the National Natural Science Foundation of China (Key Project No. 11733002). The supercomputing time for numerical simulation was kindly provided by KISTI (KSC-2014-G2- 003), and large data transfer was supported by KREONET, which is managed and operated by KISTI. Parts of this research were conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) through project number CE170100013.
|
2,877,628,089,483 | arxiv | \section{Introduction}
\label{sec:introduction}
The multimedia industry is growing rapidly and consumers are expecting videos of higher quality. On the one hand, video is becoming the main form of information carrier in increasing applications, including remote education, telemedicine, live broadcasting, digital TV, video conference. On the other hand, the demand for video resolution is constantly increasing, from $1080p$ to $2K$, $3K$, $4K$, as well as $8K$. As a result, the extremely large amount of video data needs to be compressed by video compression algorithms, such as H.264/AVC~\cite{wiegand2003overview} and H.265/HEVC~\cite{sullivan2012overview}, to fit the
available storage and network bandwidth.
As the compression ratio increases, the encoder significantly reduces the bit rate while introducing undesirable artifacts that severely degrade the quality of experience (QoE). The introduced artifacts also impair the performance of downstream video-oriented tasks (e.g., action recognition~\cite{sun2022human,zhu2020comprehensive}, object tracking~\cite{xu2019deep,luo2021multiple} and video understanding~\cite{bertasius2021space,wu2021towards,wu2019long}). Therefore, it is vital to enhance the quality of compressed video.
\begin{figure*}[htbp]
\centering\includegraphics[width=18cm]{pic/example.png}
\caption{An illustration showing the quality enhancement performance of our TVQE method, compared with DCN based method and HEVC (Class C, \emph{BasketballDrill}).}
\label{fig:example}
\end{figure*}
Convolutional neural networks (CNNs) have achieved milestones in the task of video quality enhancement (VQE). The CNN-based approaches can generally be classified into two categories: single-frame based methods~\cite{dong2015compression,guo2016building,li2017efficient,tai2017memnet,zhang2017beyond,chen2018dpw} and multi-frame based methods~\cite{yang2018multi,guan2019mfqe,xu2019non,deng2020spatio,zhao2021recursive,yang2021ntire}.
The single-frame based video enhancement method is equivalent to image enhancement, which explores the contextual information within the frame/image by CNNs to reduce compression artifacts and improve the visual quality.
However, the temporal correlations between adjacent frames in the video are ignored, which severely limits the performance.
In multi-frame based methods, the temporal information between adjacent frames are explored. Since there are motions between adjacent frames, the inter-frame information cannot be used directly.
Some works use the optical flow to compensate the motion between frames. For example, \cite{yang2018multi,guan2019mfqe} used dense optical flow for motion compensation. However, the optical flow calculated from compressed video tends to be inaccurate. Thus, some work~\cite{deng2020spatio} utilizes deformable convolution (DCN) to capture the dependencies between multiple adjacent frames and the DCN-based approaches have made great progress in this task. However, deformable convolutional alignment modules are difficult to train, and its instability in training often leads to offset overflow, which ultimately reduces the efficiency of correlation modeling. Thus,~\cite{zhao2021recursive} proposed the Recursive Fusion (RF) module based on~\cite{deng2020spatio}, which saves the temporal information of previously enhanced video frames for correlation modeling, implicitly expanding the temporal range and achieving superior results. However, the RF module consumes huge GPU computing resources and slows down the inference speed.
In order to capture the long-range correlations between frames, we introduce vision transformer into the VQE task for its strong capability to learn long-range dependencies between image patch sequences and adaptability to image content.
Since the computational complexity of the traditional vision transformer~\cite{dosovitskiy2020image} grows quadratically with the increase of image resolution, we use swin transformer~\cite{liu2021swin} in our work, along with the auto-encoder structure. The window-based swin transformer and multi-scale encoder-decoder structure with skip connections can improve the inference efficiency and reduce GPU consumption. Meanwhile, the swin transformer auto-encoder with skip connections facilitates the mining of spatio-temporal information, as well as correlation modeling of temporal information among multiple frames.
As shown in Fig.~\ref{fig:example}, our method achieves better results than other methods. The compressed frame $210$ is enhanced with the information in frames from $207$ to $213$ (only frame $208$ to $212$ are drawn for illustration). It can be seen that the green line on the floor (below the athlete's arm) is totally or partially occluded in frames from $207$ to $209$, and becomes gradually clear from frame $210$ to $213$. In order to recover the green line in frame $210$, proper correlation among pixels should be modeled. The results show that our method achieves better recovery result in this region, which verifies its effective correlation modeling of temporal information among multiple frames.
Besides the spatio-temporal information exploration, we employ \emph{Restormer}~\cite{zamir2022restormer} to calculate the channel attention, which enables efficient fusion of inter-frame information.
The main contributions of this paper are summarized as follows:
\begin{itemize}
\item We propose an compressed video enhancement transformer, which is the first work entirely based on a transformer-based architecture.
\item
Compared to state-of-the-art DCN-based methods, our proposed method has better ability of long-range correlation modelling.
\item Our proposed method calculates both spatio-temporal attention and channel attention, which effectively exploits the spatio-temporal information from multiple reference frames and achieves efficient fusion of the information.
\item We conduct extensive experiments on the JCT-VT test sequence and demonstrate the effectiveness of the proposed method.
\end{itemize}
The rest of this paper is organized as follows. In Section~\ref{sec:related-work}, the vision transformer, deep learning based compressed video enhancement methods are reviewed. Section~\ref{sec:method} describes the task of VQE, the structure of the proposed TVQE method and the training scheme. Section~\ref{sec:experiment} presents the experiments and results. Finally, in section~\ref{sec:con}, we present the conclusions and future work.
\begin{figure*}[htbp]
\centerline{\includegraphics[width=18cm]{pic/frame5.png}}
\caption{The framework of our proposed TVQE method, which consists of the \emph{Swin-AutoEncoder based Spatio-Temporal feature Fusion} (SSTF) Module and the \emph{Channel-wise Attention based Quality Enhancement} (CAQE) Module. The SSTF module is designed to exploit spatio-temporal correlation from multiple frames, where the \emph{Swin-AutoEncoder} (SAE), equipped with skip connections, is used. After SSTF, the information between channels in the feature map is further fused by the CAQE module, and finally generate the enhanced frame.
}
\label{fig:proposed}
\end{figure*}
\section{Related Work}
\label{sec:related-work}
In this section, we first review recent works on deep learning-based quality enhancement of compressed video, including single-frame based methods and multi-frame based methods. Then, a brief overview on vision transformer is provided.
\subsection{Single-frame based video enhancement method}
Single-frame video enhancement methods are considered as image enhancement. Earlier works~\cite{dong2015compression,li2017efficient,zhang2017beyond,chen2018dpw,galteri2017deep,guo2017one,liu2018non,yoo2018image,zhang2019residual} were mainly used for quality enhancement of JPEG compressed images. Specifically, AR-CNN~\cite{dong2015compression} first uses a convolutional neural network for image enhancement, and learns the nonlinear mapping between the original image and the compressed image with four convolutional layers. Subsequently, work such as~\cite{li2017efficient,tai2017memnet,zhang2017beyond,mao2016image,svoboda2016compression,zhang2017learning} proposed deeper networks to further improve the performance. With batch normalization and residual learning proposed, DnCNN~\cite{zhang2017beyond} effectively solves the problem of gradient disappearance in deep image enhancement networks. NLRN~\cite{liu2018non} and RNAN~\cite{zhang2019residual} proposed a residual non-local attention mechanism to capture long-range dependencies between pixels.
In addition to exploiting the information in the image spatial domain, methods such as~\cite{guo2016building,chen2018dpw,yoo2018image} exploited the relevant information in the frequency domain to further improve the subjective visual quality. In particular, \cite{dai2017convolutional,jin2018quality,wang2017novel,yang2018enhancing,yang2017decoder} also utilize the prior knowledge to improve the enhancement performance.
For example, DS-CNN~\cite{yang2017decoder} and QE-CNN~\cite{yang2018enhancing} used different methods to deal with intra-frame coding (e.g., AI) and inter-frame coding (e.g., LDP, LDB, RA). In general, single-frame video enhancement methods ignore the temporal information in the video, thus the performance is severely limited.
\subsection{Multi-frame based video enhancement method}
Multi-frame video enhancement mainly utilizes the spatio-temporal information of multiple adjacent frames. Yang et al.~\cite{yang2018multi} firstly proposed Multi-Frame Quality Enhancement (MFQE $1.0$), which first uses SVM to divide high and low quality frames, and then use two adjacent high quality frames to perform motion compensation through optical flow and enhance the low quality frame. As an enhanced version of MFQE $1.0$, MFQE $2.0$~\cite{guan2019mfqe} proposed an end-to-end quality enhancement network, which pre-trained a bidirectional long short-term memory (BiLSTM) based model to detect peak quality frame (PQF).
The QE-subnet is also advanced by introducing the multi-scale strategy, batch normalization and dense connection. However, the video is compressed, and the compressed video can be severely distorted by various compression artifacts, so the estimated optical flow during motion compensation is often inaccurate and unreliable, resulting in ineffective quality enhancement. To this end, Deng et al. proposed a sliding window based method STDF~\cite{deng2020spatio}, which utilized deformable convolution to avoid explicit calculation of optical flow. This method innovatively proposed to perform feature alignment of moving objects on input multi-frame images through deformable convolution. Based on STDF, RFDA~\cite{zhao2021recursive} proposed the recursive fusion (RF) module, which not only utilized the reference frames within the current time window, but also exploits the temporal information of previously enhanced video frames. By implicitly expanding the time window, RFDA leveraged a larger range of temporal information for better spatio-temporal compensation. However, the computational complexity of RF module is huge.
\subsection{Vision Transformers}
Transformer is a deep neural network based on self-attention mechanism and parallel processing. Transformer~\cite{vaswani2017attention} emerged in the field of NLP. Its proposal solves the problems of recurrent network models, such as Long short term memory (LSTM)~\cite{he2016deep} and Gate recurrent unit (GRU)~\cite{chung2014empirical}. It cannot be trained in parallel and requires a lot of storage resources to memorize the entire sequence information. The successful application of Transformer in the field of NLP has made relevant scholars begin to discuss and try its application in the field of computer vision~\cite{dosovitskiy2020image,carion2020end}.
Image Transformer\cite{parmar2018image} was the first to migrate the Transformer architecture to the field of computer vision. Subsequently, Dosovitskiy et al.~\cite{dosovitskiy2020image} proposed the Visual Transformer (ViT), and ViT completely replaced the Transformer structure with the convolutional structure to deal with the classification task, and achieved results beyond CNN on extremely large-scale datasets~\cite{kolesnikov2020big,mahajan2018exploring,tou2019fixing,xie2020self}. However, the self-attention mechanism calculates the global similarity, and its computational complexity grows quadratically with the expansion of spatial resolution. To improve operational efficiency, an efficient and effective vision transformer called Swin Transformer was proposed in~\cite{liu2021swin}.
Based on the shift window mechanism, Swin Transformer achieves state-of-the-art performance in image classification~\cite{dosovitskiy2020image,wang2021pyramid,liu2021swin}, object detection~\cite{carion2020end,zhu2020deformable}, image segmentation~\cite{xie2021segformer,cheng2022masked}, video understanding~\cite{zhou2018end,zeng2020learning}, image generation~\cite{jiang2021transgan} and point clouds processing~\cite{zhao2021point,guo2021pct}. Zamir et al. proposed Restomer~\cite{zamir2022restormer}, which computes self-attention across channels rather than spatial dimensions, and its complexity grows linearly with image resolution. Thus, Restomer achieves state-of-the-art performance in large image restoration task. In this work, we use the Swin Transformer block as the basic unit to build a \emph{Swin-AutoEncoder} architecture with skip connections for aggregating the temporal information of multiple adjacent video frames. Then, we calculate the channel attention using \emph{Restormer} and efficiently fuse the temporal information to obtain the final result.
\section{Methodology}
\label{sec:method}
Given a compressed video consisting of $T$ frames $V=[X_{1},X_{2}...,X_{t},...,X_{T}]$, where $X_{t} \in \mathbb{R}^{H \times W}$ represents the compressed frame at time $t$, $H$ and $W$ are the height and width of $X_{t}$, the task of compressed video enhancement is to generate an enhanced video $V^{e} = [X^{e}_{1},X^{e}_{2}...,X^{e}_{t},...,X^{e}_{T}]$ from the input compressed video $V$.
The overall framework of the proposed method is shown in Fig.~\ref{fig:proposed}, which consists of two modules: (a) \emph{Swin-AutoEncoder based Spatio-Temporal feature Fusion} (SSTF) module and (b) \emph{Channel-wise Attention based Quality Enhancement} (CAQE) module. The SSTF module explores the spatio-temporal information from multiple frames by modeling the association of these frames with \emph{Swin-AutoEncoder} (SAE). After SSTF, the information between channels in the feature map is further fused by CAQE, and finally generate the residual of the enhanced frame.
For each compressed frame $X_{t}$, its $R$ preceding frames and $R$ succeeding frames are used to exploit correlated temporal information.
With the input $V_{t} = \left\{X_{t-R}, \ldots, X_{t}, \ldots, X_{t+R}\right\}$, the whole process can be expressed as:
\begin{eqnarray}
X_{t}^{m}&=&SA\left(V_{t}; \phi\right),\\
X_{t}^{e}&=&CA\left(X_{t}^{m} ; \varphi\right) + X_{t},
\label{equ:f1}
\end{eqnarray}
where $X_{t}^{e}$ is the output, $SA$ denotes the process of SSTF, and $CA$ denotes the process of CAQE. $\phi$ and $\varphi$ represent the parameters to be learned in the SSTF and CAQE modules, respectively. Finally, residual learning is used to improve the training efficiency.
\begin{figure*}[htbp]
\centerline{\includegraphics[width=18cm]{pic/Channel-attention2.png}}
\caption{Architecture of \emph{Restormer}. It consists of (a) \emph{Multi-Dconv Head Transposed Attention} (MDTA) and (b) \emph{Gated-Dconv Feed-Forward Network} (GDFN). MDTA calculates channel-level attention and GDFN performs feature transformation by GELU to enrich feature representation.}
\label{fig:atten}
\end{figure*}
\subsection{Swin-AutoEncoder based Spatio-Temporal feature Fusion (SSTF)}
The SSTF module consists of the \emph{Patch Partition layer}, \emph{Swin-AutoEncoder} and \emph{Pixel Shuffle layer}. First, the target frame and the adjacent reference frame are partitioned into non-overlapping patches by the \emph{Patch Partition layer}. For the consideration of computing speed, the \emph{Patch Partition layer} downsamples the features and restores it to the original resolution at the final stage by the \emph{Pixel Shuffle layer}. Between the \emph{Patch Partition layer} and the \emph{Pixel Shuffle layer}, the spatio-temporal information is aggregated with the \emph{Swin-AutoEncoder}.
\emph{Swin-AutoEncoder} is a Swin Transformer Block (\emph{Swin-TB}) based auto-encoder structure. In \emph{Swin-AutoEncoder}, each patch after segmentation is treated as a token and then calculated the spatio-temporal attention. In the encoder, \emph{Patch Merging Layer} increases the number of channels while features are downsampled, and \emph{Swin-TB} further enhances the features.
For $V_{t}=\left\{X_{t-R}, \ldots, X_{t}, \ldots, X_{t+R}\right\}$, the whole Encoder process can be expressed as
\begin{eqnarray}
E_{1}&=&Estage1\left(V_{t} \right),\\
E_{2}&=&Estage2\left(E_{1} \right),\\
E_{3}&=&Estage3\left(E_{3} \right),
\label{equ:f2}
\end{eqnarray}
where $Estage$ denotes the combination of \emph{Patch Merging} and \emph{Swin-TB}, and 1, 2, 3 represent each stage of the encoder.
Corresponding to the encoder, the decoder uses a \emph{Patch Expanding} Layer to upsample deep features. For each scale, the low-level features of the encoder are connected with the high-level features of the decoder through skip connections to reduce the loss of spatial information caused by downsampling. The whole Encoder process can be expressed as
\begin{eqnarray}
D_{3}&=&Dstage1\left(E_{3} \right)+E_{3},\\
D_{2}&=&Dstage2\left(D_{3} \right)+E_{2},\\
X_{t}^{m}&=&Dstage3\left(D_{2} \right)+E_{1},
\label{equ:f3}
\end{eqnarray}
where $Dstage$ denotes the combination of \emph{Patch Expanding
} and \emph{Swin-TB}, and 1, 2, 3 represent each stage of the decoder.
By using \emph{Swin-AutoEncoder}, the compressed frames at time $t$ can aggregate the temporal information of the adjacent reference frames and generate a temporal feature map $X_{t}^{m}$.
\subsection{Channel-wise Attention based Quality Enhancement Module (CAQE)}
In order to efficiently fuse the temporal information in each channel and generate residual maps for frames at time $t$, we constructed the CAQE module with efficient channel-level attention. The CAQE module consists of four \emph{Restormer}~\cite{zamir2022restormer} and one \emph{Reconstruct layer}. In which, \emph{Restormer} calculates the channel attention, where temporal information is further fused and enhanced. The final \emph{Reconstruct layer} is a $3\times3$ convolutional layer, which reduces the number of fused feature channels to $1$ to obtain the final residual map. The whole process can be represented as
\begin{eqnarray}
I_{t}&=&Rec\left(Res\left(X_{t}^{m}\right)\right),\\
X_{t}^{e}&=&I_{t}+X_{t},
\label{equ:f4}
\end{eqnarray}
where $Res$ denotes four consecutive stacks of \emph{Restormer} and $Rec$ denotes the final \emph{reconstructed layer}.
As shown in Fig.~\ref{fig:atten}, \emph{Restormer} consists of two parts: \emph{Multi-Dconv Head Transposed Attention} (MDTA) and \emph{Gated-Dconv Feed-Forward Network} (GDFN).
To reduce the computational overhead of the network, MDTA computes the cross-covariance on the channel. Input feature $X_{m}^{t}$, using \emph{Pointwise Convolution} (PW conv) and \emph{Depthwise Convolution} (DW conv) to Generate $\mathbf{Q} \in \mathbb{R}^{H \times W \times C}$, $\mathbf{K} \in \mathbb{R}^{H \times W \times C}$, $\mathbf{V} \in \mathbb{R}^{H \times W \times C}$. Specifically, the PW performs content encoding on the channel and fuses the context information between the channels. DW further encodes spatial context. Then get $\hat{\mathbf{Q}} \in \mathbb{R}^{H W \times C}$, $\hat{\mathbf{K}} \in \mathbb{R}^{H W \times C}$, $\hat{\mathbf{V}} \in \mathbb{R}^{H W \times C}$ through the reshape operation and calculate the dot product of $\hat{\mathbf{Q}}$ and $\hat{\mathbf{K}}$ to generate the channel attention map $\mathbf{M}$ with size $C \times C$. It can be expressed as
\begin{eqnarray}
\mathbf{M}&=&\mathbf{V} \cdot \operatorname{Softmax}(\mathbf{K} \cdot \mathbf{Q})
\label{equ:f5}
\end{eqnarray}
To get more accurate residual information, we utilize GDFN with more complex operations. GDFN adds GELU activation branch and DW on the basis of FN, which can enrich the expression of features and use spatial context information to enhance the recovery of local details.
\subsection{Training Scheme}
For frame $X_{t}$ at time $t$, we use a two-stage training strategy to enhance its quality. In the first stage, we use Charbonnier Loss~\cite{charbonnier1994two} to optimize the parameters of TVQE. In the second stage, we use $\mathcal{L}_{2}$ Loss to further fine-tune the model for a better visual result. Finally, the loss function is defined as
\begin{eqnarray}
\mathcal{L}_{charb}&=&\sqrt{\left({X}^{e}_{t}-{X}^{raw}_{t}\right)^{2} + \epsilon},\\
\mathcal{L}_{mse}&=& \left\|{X}^{e}_{t}-{X}^{raw}_{t}\right\|_{2}^{2},\\
\mathcal{L}&=&\alpha\times \mathcal{L}_{charb}+\beta\times\mathcal{L}_{mse},
\label{equ:loss}
\end{eqnarray}
where ${X}^{e}_{t}$ denotes the enhanced video frame at time $t$, ${X}^{raw}_{t}$ denotes the raw frame, and $\epsilon$ is a constant set to $10^{-6}$. $\alpha$ and $\beta$ are the weights of the loss.
\section{Experiment}
\label{sec:experiment}
\subsection{Datasets}
In this work, we use the MFQE 2.0~\cite{guan2019mfqe} and LDV~\cite{yang2021ntire} datasets for training and JCT-VC~\cite{bossen2013common} dataset for testing.
\subsubsection{MFQE 2.0}
It contains totally $128$ sequences, in which training set contains $108$ sequences. The sequences are acquired from Xiph.org~\cite{montgomery2021xiph} and VQEG~\cite{VGQE}, with resolutions ranging from SIF ($352\times240$) to WQXGA ($2560\times1600$).
\subsubsection{LDV}
It is proposed in NTIRE 2021 challenge~\cite{yang2021ntire1} with $240$ sequences, which consists of training set, validation set and test set. We use $200$ sequences from the training set as training data and $40$ sequences from the validation and test sets for validation, all sequences are $960\times536$ in resolution.
\subsubsection{JCT-VC}
The test set has $18$ sequences, delivered by JCT-VC (Joint Collaborative Team on Video Coding) for evaluating the performance of our model. There are totally five resolutions ranging from $240$p ($416\times240$) to WQXGA ($2560\times1600$), named as Class A to E.
Following~\cite{guan2019mfqe,deng2020spatio}, all sequences were compressed by HEVC HM $16.5$\footnote{https://hevc.hhi.fraunhofer.de/trac/hevc/milestone/HM-16.5} with LDP (Low-Delay-P) configuration. Five QPs (quantization parameters) i.e., $22$, $27$, $32$, $37$,$42$ at different compression bit rates are used for experiments.
\subsection{Implementation Details}
For network structure, the window size is set to $8$ in \emph{Swin-AutoEncoder} from Stage1 to Stage3, the number of \emph{Swin-TB} is [$2$, $2$, $2$], and the attention heads is [$2$, $2$, $2$]. The \emph{Patch embedding} dimension is set to $48$, and the \emph{MLP-ratio} is $1$. The number of \emph{Restormer} in CAQE is set to $4$.
In the training stage, we crop $128\times128$ patches randomly from the compressed video and the corresponding raw video as training samples. Random flips and rotations are also used for data augmentation. The batch size is set to $32$. we train the model using the Adam optimizer ($\beta_{1}$=$0.9$, $\beta_{2}$=$0.999$, $\epsilon$=$10^{-8}$). The learning rate is $10^{-4}$ throughout the training process. In the first stage of training, $\alpha$ is set to $1$ and $\beta$ is set to $0$ in Equ.~(\ref{equ:loss}). In the second stage, $\alpha$ is set to $0$ and $\beta$ is set to $1$. All experiments are performed on the NVIDIA TITAN RTX.
For testing, we evaluate the results using $\Delta$PSNR and $\Delta$SSIM, as well as BD-rate. All tests are performed on the Y-channel in YUV space.
\begin{table*}[htbp]\normalsize
\renewcommand{\arraystretch}{1.2}
\caption{Quantitative results of $\Delta$PSNR (dB) / $\Delta$SSIM ($\times10^{-2}$) on JCT-VC dataset at 5 different QPs. The best and second best performance are bold and underlined, respectively.}
\label{tab:qrtable}
\begin{threeparttable}
\resizebox{\linewidth}{!}{
\begin{tabular}{c|c|l||ccccccc|c}
\toprule[1.5pt]
QP & \multicolumn{2}{c||}{Approach} & \begin{tabular}[c]{@{}c@{}}AR-CNN\\ \cite{dong2015compression}\end{tabular} & \begin{tabular}[c]{@{}c@{}}DnCNN\\ \cite{zhang2017beyond}\end{tabular} & \begin{tabular}[c]{@{}c@{}}RNAN\\ \cite{zhang2019residual}\end{tabular} & \begin{tabular}[c]{@{}c@{}}MFQE 2.0\\ \cite{guan2019mfqe}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Ding et al.\\ \cite{ding2021patch}\end{tabular} & \begin{tabular}[c]{@{}c@{}}STDF-R3L\\ \cite{deng2020spatio}\end{tabular} & \begin{tabular}[c]{@{}c@{}}RFDA\\ \cite{zhao2021recursive}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Ours\\ TVQE\end{tabular} \\ \hline
& \multicolumn{2}{c||}{Metrics} & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM & PSNR / SSIM \\ \hline
\multirow{19}{*}{37} & \multirow{2}{*}{A} & \textit{Traffic} & 0.24 / 0.47 & 0.24 / 0.57 & 0.40 / 0.86 & 0.59 / 1.02 & \textbf{1.08} / \textbf{1.68} & 0.73 / 1.15 & 0.80 / 1.28 & \underline{0.86} / \underline{1.39} \\
& & \textit{PeopleOnStreet} & 0.35 / 0.75 & 0.41 / 0.82 & 0.74 / 1.30 & 0.92 / 1.57 & 0.64 / 1.04 & 1.25 / 1.96 & \underline{1.44} / \underline{2.22} & \textbf{1.46} / \textbf{2.30} \\ \cline{2-11}
& \multirow{5}{*}{B} & \textit{Kimono} & 0.22 / 0.65 & 0.24 / 0.75 & 0.33 / 0.98 & 0.55 / 1.18 & 0.69 / 1.36 & 0.85 / 1.61 & \textbf{1.02} / \textbf{1.86} & \underline{0.99} / \underline{1.82} \\
& & \textit{ParkScene} & 0.14 / 0.38 & 0.14 / 0.50 & 0.20 / 0.77 & 0.46 / 1.23 & 0.49 / 1.21 & 0.59 / 1.47 & \underline{0.64} / \underline{1.58} & \textbf{0.64} / \textbf{1.63} \\
& & \textit{Cactus} & 0.19 / 0.38 & 0.20 / 0.48 & 0.35 / 0.76 & 0.50 / 1.00 & 0.62 / 1.15 & 0.77 / 1.38 & \underline{0.83} / \underline{1.49} & \textbf{0.87} / \textbf{1.59} \\
& & \textit{BQTerrace} & 0.20 / 0.28 & 0.20 / 0.38 & 0.42 / 0.84 & 0.40 / 0.67 & 0.50 / 0.87 & 0.63 / 1.06 & \underline{0.65} / \underline{1.06} & \textbf{0.70} / \textbf{1.22} \\
& & \textit{BasketballDrive} & 0.23 / 0.55 & 0.25 / 0.58 & 0.43 / 0.92 & 0.47 / 0.83 & 0.60 / 1.04 & 0.75 / 1.23 & \textbf{0.87} / \underline{1.40} & \underline{0.82} / \textbf{1.41} \\ \cline{2-11}
& \multirow{4}{*}{C} & \textit{RaceHorses} & 0.22 / 0.43 & 0.25 / 0.65 & 0.39 / 0.99 & 0.39 / 0.80 & 0.40 / 0.88 & 0.55 / 1.35 & \underline{0.48} / \underline{1.23} & \textbf{0.61} / \textbf{1.55} \\
& & \textit{BQMall} & 0.28 / 0.68 & 0.28 / 0.68 & 0.45 / 1.15 & 0.62 / 1.20 & 0.74 / 1.44 & 0.99 / 1.80 & \textbf{1.09} / \textbf{1.97} & \underline{1.02} / \underline{1.95} \\
& & \textit{PartyScene} & 0.11 / 0.38 & 0.13 / 0.48 & 0.30 / 0.98 & 0.36 / 1.18 & 0.51 / 1.46 & 0.68 / 1.94 & \underline{0.66} / \underline{1.88} & \textbf{0.75} / \textbf{2.14} \\
& & \textit{BasketballDrill} & 0.25 / 0.58 & 0.33 / 0.68 & 0.50 / 1.07 & 0.58 / 1.20 & 0.66 / 1.27 & 0.79 / 1.49 & \underline{0.88} / \underline{1.67} & \textbf{0.96} / \textbf{1.91} \\ \cline{2-11}
& \multirow{4}{*}{D} & \textit{RaceHorses} & 0.27 / 0.55 & 0.31 / 0.73 & 0.42 / 1.02 & 0.59 / 1.43 & 0.60 / 1.44 & 0.83 / 2.08 & \underline{0.85} / \underline{2.21} & \textbf{0.87} / \textbf{2.27} \\
& & \textit{BQSquare} & 0.08 / 0.08 & 0.13 / 0.18 & 0.32 / 0.63 & 0.34 / 0.65 & 0.79 / 1.14 & 0.94 / 1.25 & \underline{1.05} / \underline{1.39} & \textbf{1.16} / \textbf{1.62} \\
& & \textit{BlowingBubbles} & 0.16 / 0.35 & 0.18 / 0.58 & 0.31 / 1.08 & 0.53 / 1.70 & 0.62 / 1.95 & 0.74 / 2.26 & \underline{0.78} / \underline{2.40} & \textbf{0.81} / \textbf{2.48} \\
& & \textit{BasketballPass} & 0.26 / 0.58 & 0.31 / 0.75 & 0.46 / 1.08 & 0.73 / 1.55 & 0.85 /1.75 & 1.08 / 2.12 & \textbf{1.12} / \underline{2.23} & \underline{1.07} / \textbf{2.31} \\ \cline{2-11}
& \multirow{3}{*}{E} & \textit{FourPeople} & 0.37 / 0.50 & 0.39 / 0.60 & 0.70 / 0.97 & 0.73 / 0.95 & 0.95 / 1.12 & 0.94 / 1.17 & \underline{1.13} / \underline{1.36} & \textbf{1.20} / \textbf{1.44} \\
& & \textit{Johnny} & 0.25 / 0.10 & 0.32 / 0.40 & 0.56 / 0.88 & 0.60 / 0.68 & 0.75 / 0.85 & 0.81 / 0.88 & \underline{0.90} / \underline{0.94} & \textbf{1.10} / \textbf{1.31} \\
& & \textit{KristenAndSara} & 0.41 / 0.50 & 0.42 / 0.60 & 0.63 / 0.80 & 0.75 / 0.85 & 0.93 / 0.91 & 0.97 / 0.96 & \underline{1.19} / \underline{1.15} &\textbf{1.27} / \textbf{1.21} \\ \cline{2-11}
& \multicolumn{2}{c||}{Average} & 0.23 / 0.45 & 0.26 / 0.58 & 0.44 / 0.95 & 0.56 / 1.09 & 0.69 / 1.25 & 0.83 / 1.51 & \underline{0.91} / \underline{1.62} & \textbf{0.95} / \textbf{1.75} \\\bottomrule[1.5pt]
\toprule[1.5pt]
42 & \multicolumn{2}{c||}{Average} & 0.29 / 0.96 & 0.22 / 0.77 & – / – & 0.59 / 1.65 & 0.69 / 1.86 & – / – & \underline{0.82} / \underline{2.20} & \textbf{0.97} / \textbf{2.58} \\ \hline
32 & \multicolumn{2}{c||}{Average} & 0.18 / 0.19 & 0.26 / 0.35 & 0.41 / 0.62 & 0.52 / 0.68 & 0.67 / 0.83 & 0.86 / 1.04 & \underline{0.87} / \underline{1.07} & \textbf{0.91} / \textbf{1.17} \\\hline
27 & \multicolumn{2}{c||}{Average} & 0.18 / 0.14 & 0.27 / 0.24 & – / – & 0.49 / 0.42 & 0.63 / 0.52 & 0.72 / 0.57 & \underline{0.82} / \underline{0.68} & \textbf{0.85} / \textbf{0.74} \\\hline
22 & \multicolumn{2}{c||}{Average} & 0.14 / 0.08 & 0.29 / 0.18 & – / – & 0.46 / 0.27 & 0.55 / 0.29 & 0.63 / 0.34 & \textbf{0.76} / \underline{0.42} & \underline{0.74} / \textbf{0.42} \\\bottomrule[1.5pt]
\end{tabular}}
\begin{tablenotes}
\footnotesize
\item Video resolution: Class A ($2560 \times 1600$), Class B ($1920 \times 1080$), Class C ($832 \times 480$), Class D ($480 \times 240$), Class E ($1280 \times 720$)
\end{tablenotes}
\end{threeparttable}
\end{table*}
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.2}
\caption{Comparison of inferred speed and GPU consumption between our method and some mainstream methods. For a fair comparison, all methods were retested on the NVIDIA TITAN RTX. Results are reported in frames per second (FPS) and Test Memory (GB). The best and second best performance are bold and underlined, respectively.}
\label{tab:my-table}
\resizebox{\linewidth}{!}{
\begin{tabular}{l||cccc}
\toprule[1pt]
\multicolumn{1}{l||}{\multirow{2}{*}{Method}} & \multicolumn{4}{c}{Test Memory(GB) / FPS} \\ \cline{2-5}
\multicolumn{1}{l||}{} & 240p & 480p & 720p & 1080p \\\hline
STDF-R3L & 1.2 / \textbf{42.5} & 1.9 / \textbf{13.7} & 3.1 / \textbf{6.4} & \;\;5.8 / \textbf{3.1} \\\hline
RFDA & 1.5 / 32.1 & 2.9 / 11.2 & 5.4 / 5.1 & 10.5 / 2.1 \\\hline
TVQE & \textbf{1.1} / \underline{35.9} & \textbf{1.6} / \underline{12.2} & \textbf{2.5} / \underline{5.5} & \;\;\textbf{4.7} / \underline{2.6}\\
\bottomrule[1pt]
\end{tabular}}
\end{table}
\subsection{Comparison with State of the Art Methods}
To demonstrate the effectiveness of our method, we compare the proposed method with seven state-of-the-art methods, including single-frame based methods (AR-CNN~\cite{dong2015compression}, DnCNN~\cite{zhang2017beyond}, RNAN~\cite{zhang2019residual}) and multi-frame based methods (MFQE $2.0$~\cite{guan2019mfqe}, ~\cite{ding2021patch}, STDF-R$3$L~\cite{deng2020spatio} and RFDA~\cite{zhao2021recursive}.)
\subsubsection{Quantitative Results}
Table~\ref{tab:qrtable} presents the quantitative results of our method and seven state-of-the-art methods on $\Delta$PSNR and $\Delta$SSIM. It can be seen that our method TVQE outperforms the seven models in terms of the average $\Delta$PSNR at four QPs and in terms of the average $\Delta$SSIM at all QPs. Meanwhile, the gain of our method TVQE on SSIM is higher than PSNR obviously. Such as QP=$42$, our method TVQE gains $9.7\%$ on $\Delta$PSNR and $11.8\%$ on $\Delta$SSIM over the second best method RFDA, which indicates that our method TVQE provides better visual effects. More specifically, our method outperforms the current state-of-the-art methods on most sequences when QP=$37$.
As for the single-frame based methods, RNAN proposes non-local attention blocks to obtain the remote dependence of the feature map, and finally achieves the best performance among all single-frame based methods. RNAN gains about $69\%$ over DnCNN, which reflects the superiority of the transformer-based method.
However, the single-frame based method cannot use temporal information and has limited performance. Our method computes spatial attention and channel attention over multiple frames, and achieves $\Delta$PSNR of $0.95$, which is about $116\%$ compared to RNAN.
As for the multi-frame based methods, MFQE $2.0$ calculates the explicit optical flow of compressed video and achieves an average $\Delta$PSNR of $0.56$. STDF proposes deformable convolution to align video frames, which solves the problem of inaccurate optical flow estimation of MFQE $2.0$, and achieves an average $\Delta$PSNR of $0.83$. RFDA utilizes the RF (Recursive Fusion) module to exploit temporal information within a longer time range, and obtains $\Delta$PSNR of $0.91$. Our method TVQE utilizes the long-range modeling property of Transformer to exploit the temporal information, which further increases the PSNR with an average $\Delta$PSNR of $0.96$, which demonstrates the effectiveness of our method.
\begin{table*}[htbp]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Results of BD-Bate reduction (\%) at QP = $22, 27, 32, 37$ and $42$ with the HEVC as anchor. The best and second best performance are in bold and underlined, respectively.}
\label{tab:bdrate}
\begin{tabular}{ll|cccccccccc}
\toprule[1.5pt]
\multicolumn{2}{c||}{Sequence} & \begin{tabular}[c]{@{}c@{}}AR-CNN\\\cite{dong2015compression}\end{tabular} & \begin{tabular}[c]{@{}c@{}}DnCNN\\ \cite{zhang2017beyond}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Li et al.\\\cite{li2017efficient}\end{tabular} & \begin{tabular}[c]{@{}c@{}}DCAD\\\cite{wang2017novel}\end{tabular} & \begin{tabular}[c]{@{}c@{}}DS-CNN\\\cite{yang2018enhancing}\end{tabular} & \begin{tabular}[c]{@{}c@{}}MFQE 1.0\\\cite{yang2018multi}\end{tabular} & \begin{tabular}[c]{@{}c@{}}MFQE 2.0\\\cite{guan2019mfqe}\end{tabular} & \begin{tabular}[c]{@{}c@{}}STDF-R3L\\ \cite{deng2020spatio}\end{tabular} & \begin{tabular}[c]{@{}c@{}}RFDA\\\cite{zhao2021recursive}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Ours\\ TVQE\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{A}} & \multicolumn{1}{l||}{\textit{Traffic}} & 7.40 & 8.54 & 10.08 & 9.97 & 9.18 & 14.56 & 16.98 & 21.19 & \underline{22.70} & \textbf{24.00} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{PeopleOnStreet}} & 6.99 & 8.28 & 9.64 & 9.68 & 8.67 & 13.71 & 15.08 & 17.42 & \underline{21.11} & \textbf{22.86} \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{B}} & \multicolumn{1}{l||}{\textit{Kimono}} & 6.07 & 7.33 & 8.51 & 8.44 & 7.81 & 12.60 & 13.34 & 17.96 & \underline{22.32} & \textbf{22.58} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{ParkScene}} & 4.47 & 5.04 & 5.35 & 5.68 & 5.42 & 12.04 & 13.66 & 18.1 & \textbf{19.85} & \underline{19.78} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{Cactus}} & 6.16 & 6.80 & 8.23 & 8.69 & 8.78 & 12.78 & 14.84 & 21.54 & \underline{21.78} & \textbf{23.77} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BQTerrace}} & 6.86 & 7.62 & 8.79 & 9.98 & 8.67 & 10.95 & 14.72 & 24.71 & \underline{24.41} & \textbf{26.89} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BasketballDrive}} & 5.83 & 7.33 & 8.61 & 8.94 & 7.89 & 10.54 & 11.85 & 16.75 & \underline{20.24} & \textbf{20.70} \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{C}} & \multicolumn{1}{l||}{\textit{RaceHorses}} & 5.07 & 6.77 & 7.10 & 7.62 & 7.48 & 8.83 & 9.61 & 15.62 & \textbf{14.29} & \underline{13.67} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BQMall}} & 5.60 & 7.01 & 7.79 & 8.65 & 7.64 & 11.11 & 13.50 & 21.12 & \textbf{21.62} & \underline{20.81} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{PartyScene}} & 1.88 & 4.02 & 3.78 & 4.88 & 4.08 & 6.67 & 11.28 & 22.24 & \underline{21.11} & \textbf{22.28} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BasketballDrill}} & 4.67 & 8.02 & 8.66 & 9.8 & 8.22 & 10.47 & 12.63 & 15.94 & \underline{18.06} & \textbf{20.82} \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{D}} & \multicolumn{1}{l||}{\textit{RaceHorses}} & 5.61 & 7.22 & 7.68 & 8.16 & 7.35 & 10.41 & 11.55 & 15.26 & \textbf{17.57} & \underline{17.03} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BQSquare}} & 0.68 & 4.59 & 3.59 & 6.11 & 3.94 & 2.72 & 11.00 & 33.36 & \underline{31.65} & \textbf{33.50} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BlowingBubbles}} & 3.19 & 5.10 & 5.41 & 6.13 & 5.55 & 10.73 & 15.2 & 23.54 & \underline{22.89} & \textbf{23.62} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{BasketballPass}} & 5.11 & 7.03 & 7.78 & 8.35 & 7.49 & 11.70 & 13.43 & 18.42 & \textbf{20.42} & \underline{20.12} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{E}} & \multicolumn{1}{l||}{\textit{FourPeople}} & 8.42 & 10.12 & 11.46 & 12.21 & 11.13 & 14.89 & 17.50 & 22.91 & \underline{22.84} & \textbf{25.33} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{Johnny}} & 7.66 & 10.91 & 13.05 & 13.71 & 12.19 & 15.94 & 18.57 & 24.55 & \underline{23.87} & \textbf{28.99} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l||}{\textit{KristenAndSara}} & 8.94 & 10.65 & 12.04 & 12.93 & 11.49 & 15.06 & 18.34 & 23.64 & \underline{24.47} & \textbf{27.95} \\ \hline
\multicolumn{2}{c||}{Average} & 5.59 & 7.36 & 8.20 & 8.89 & 7.85 & 11.41 & 14.06 & 20.79 & \underline{21.73} & \textbf{23.04} \\\bottomrule[1.5pt]
\end{tabular}
\end{table*}
\subsubsection{Speed and Cost Comparison}
Table~\ref{tab:my-table} shows the inference speed and GPU consumption of our method, compared to STDF-R$3$L~\cite{ding2021patch} and RFDA~\cite{deng2020spatio}. As can be seen, although our model is based on Transformer, it still has a fast inference speed. At the same time, our method is hardware friendly as it requires less GPU memory. More specifically, comparing to STDF at $1080$p resolution, our method is $16.1\%$ slower at inference speed (from $3.1$ to $2.6$, see Table~\ref{tab:my-table}), but with a $19.0\%$ reduction in memory consumption (from $5.8$ to $4.7$, see Table~\ref{tab:my-table}), as well as with a $14.5\%$ improvement in average $\Delta$PSNR performance at QP$37$ (from $0.83$ to $0.95$, see Table~\ref{tab:qrtable}). RFDA is based on STDF by adding RF module, and thus consumes more GPU resources. Comparing to RFDA, our method outperforms RFDA in terms of inference speed and GPU consumption at all resolutions. Specifically, under $1080$p resolution, the inference speed is improved by $23.8\%$ (from $2.1$ to $2.6$, see Table~\ref{tab:my-table}) and GPU consumption is reduced by $55.2\%$ (from $10.5$ to $4.7$, see Table~\ref{tab:my-table}).
\subsubsection{Quality Fluctuation}
\begin{figure}[htbp]
\centerline{\includegraphics[width=9cm]{pic/basketballdrill.png}}
\centerline{\includegraphics[width=9cm]{pic/bqsquare.png}}
\caption{Illustration of quality fluctuations of two sequences. (Top: Class C, \emph{BasketballDrill}. Bottom: Class D, \emph{BQSquare}.)}
\label{fig:zlbd}
\end{figure}
The PSNR of each frame in two sequences are plotted in Fig.~\ref{fig:zlbd}. It can be observed that the HEVC compressed sequences have severe quality fluctuations (i.e., quality differences between high quality frames and adjacent low quality frames). Compared to both STDF and RFDA, our method provides better PSNR and smaller quality fluctuations, effectively improving the QoE.
\subsubsection{Rate-Distortion performance}
Fig.~\ref{fig:bdrate} presents the rate distortion curves for the four sequences. It can be seen that our method outperforms other methods on both sequences with huge motion (e.g., Class C, \emph{BasketballDrill}) and smooth motion (e.g., Class E, \emph{Johnny}). In addition, we also calculate the BD-rate reduction of PSNR on five QPs (= $22$, $27$, $32$, $37$, $42$). As shown in Table~\ref{tab:bdrate}, our method provides an average BD-rate reduction of $23.04\%$, which is better than the state-of-the-art CNN method RFDA with $21.73\%$. It demonstrates that our method exhibits a better rate distortion performance, which can provide superior visual effects with the same compression rate.
\begin{figure}[htbp]
\centering\includegraphics[width=4.3cm]{pic/bd_basketballdrill.png}
\centering\includegraphics[width=4.4cm]{pic/bd_Johnny.png}\\
\centering\includegraphics[width=4.3cm]{pic/bd_kristenandsara.png}
\centering\includegraphics[width=4.3cm]{pic/bd_peopleonstreet.png}
\caption{Rate-Distortion curves of sequences BasketballDrill, Johnny, KristenAndSara and PeopleOnStreet.}
\label{fig:bdrate}
\end{figure}
\begin{figure*}[htbp]
\centerline{\includegraphics[width=18cm]{pic/sub.png}}
\caption{Qualitative results at QP $37$. Video from top to bottom: \emph{BasketballDrill}, \emph{Racehorses}, \emph{BQSquare}, \emph{BQTerrace}, \emph{BasketballPass}. For a fair comparison, for each method we only enhance on Y component.}
\label{fig:Qualitative}
\end{figure*}
\subsubsection{Qualitative Results}
Fig.~\ref{fig:Qualitative} gives the qualitative results for the five sequences. It can be seen that the single-frame based quality enhancement methods~\cite{dong2015compression,zhang2017beyond} do not make use of temporal information, so the enhanced video frames still have serious compression artifacts (e.g., block effect, ringing effect). With the help of temporal information, CNN-based multi-frame quality enhancement methods~\cite{guan2019mfqe,deng2020spatio} provide better visual effects with the help of reference frames, but the locality of the convolution operation prevents these methods from taking full advantage of the temporal information of reference frames, resulting in enhanced video frames that are too smooth and lack of detailed texture. RFDA~\cite{zhao2021recursive} further implicitly expands the temporal range with RF to better recover details, but the RF module consumes large computational resources and decreases the inference speed. Our proposed method TVQE is based on transformer, which has better remote correlation capability than convolution, thus resulting in better exploration of spatio-temporal information and finer recovery of textures. For example in Fig.~\ref{fig:Qualitative}, the player's fingers in \emph{BaskerballDrill}, the rope on the horse in \emph{Racehorses}, the textures on the railings in \emph{BQSquare} and \emph{BQTerrace}, and the shadow of the sneaker in \emph{BaskerballPass} can be better recovered by our method than other methods.
\subsection{Ablation Study}
In this section, we perform ablation experiments as well as specific analysis of the proposed method. We take STDF-R3L as baseline and replace the modules from different models to analyze their effects. For a fair comparison, all models are retrained by the same training approach as the proposed method. The results of inference speed FPS and GPU consumption are obtained at 1080p resolution, and $\Delta$PSNR / $\Delta$SSIM takes the average results of the test sequences in ClassA-E at QP=$37$.
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.2}
\caption{Ablation study on SSTF and CAQE. The best and second best performance are bold and underlined, respectively.}
\label{tab:withqe}
\resizebox{\linewidth}{!}{
\begin{tabular}{l||ccc}
\toprule[1pt]
Method & $\Delta$PSNR / $\Delta$SSIM & Test Memory(GB) & FPS \\
\hline
STDF+QE & 0.83 / 1.51 & \underline{5.8} & \underline{3.1} \\
STFF+QE & \underline{0.91} / \underline{1.62} & 10.1 & 2.4 \\
SSTF+QE & \textbf{0.91} / \textbf{1.71} & \textbf{5.3} & \textbf{3.4}\\\hline
STDF+CAQE & 0.86 / 1.56 & \underline{5.1} & \underline{2.3} \\
STFF+CAQE & \underline{0.93} / \underline{1.65} & 9.4 & 1.6\\
SSTF+CAQE & \textbf{0.95} / \textbf{1.75} & \textbf{4.7} & \textbf{2.6}\\
\bottomrule[1pt]
\end{tabular}}
\end{table}
\subsubsection{Effectiveness of SSTF}
To illustrate the effectiveness of the SSTF module, we compare the proposed SSTF with the baseline STDF and RFDA. As shown in Table~\ref{tab:withqe}, by replacing the STDF module and the STFF in RFDA with the SSTF (from the first to the third row), the SSTF provides a larger performance improvement while having a faster inference speed and lower GPU consumption. Specifically, compared to the baseline STDF, the Transformer-based SSTF is able to explore global temporal information within a time window, with an improvement of $\Delta$PSNR by $0.08$ (from $0.83$ to $0.91$) and $\Delta$SSIM ($\times10^{-2}$) by $0.2$ (from $1.51$ to $1.71$). Moreover, benefit from the \emph{Swin-AutoEncoder} structure and skip connections, SSTF has a $9.7\%$ speedup (from $3.1$ to $3.4$) inference speed compared to STDF and $8.6\%$ reduction in GPU consumption (from $5.8$ to $5.3$). Compared with STFF, SSTF does not have to utilize additional information outside the time window, resulting in $47.5\%$ lower GPU consumption (from $10.1$ to $5.3$) and $41.6\%$ higher inference express (from $2.4$ to $3.4$), which demonstrates the effectiveness of the SSTF module.
\subsubsection{Effectiveness of CAQE}
To illustrate the effectiveness of the CAQE module, we replace the quality enhancement module in the baseline STDF, RFDA and this method with CAQE (fourth to sixth rows). CAQE calculates the channel attention and effectively fuses the temporal information between channels. In terms of the results of this method (third and sixth lines), CAQE provides a $\Delta$PSNR gain of $0.04$ (from $0.91$ to $0.95$) and a $\Delta$SSIM gain of $0.04$ (from $1.71$ to $1.75$) compared to the QE with CNN structure. Meanwhile, the channel-level attention resulted in a further $11.3\%$ reduction in memory consumption (from $5.3$ to $4.7$) and the inference speed was reduced from $3.4$ to $2.6$, but the overall inference speed was still better than RFDA ($2.1$, see Table~\ref{tab:my-table}), reflecting the effectiveness of the CAQE module.
\section{Conclusion}
\label{sec:con}
In this paper, we propose an end-to-end transformer based network TVQE for compressed video enhancement, which mainly consists of two modules, SSTF module and CAQE module. SSTF module can efficiently explore temporal information within the time window, while CAQE can well fuses the temporal information. The proposed method outperforms the CNN-based methods in terms of performance, inference speed and GPU consumption. The proposed module can also be used in other fields, such as video super-resolution and video interpolation, to explore and fuse temporal information more effectively.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEbib}
|
2,877,628,089,484 | arxiv | \section*{Abstract}
{\bf
Several tensor networks are built of isometric tensors, i.e.\ tensors satisfying $W^{\dagger} W = \mathbbm{1}$.
Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers.
We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g.\ ground states of 1D quantum Hamiltonians.
We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context.
We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes.
We also provide open-source implementations of our algorithms.
}
\vspace{10pt}
\noindent\rule{\textwidth}{1pt}
\tableofcontents\thispagestyle{fancy}
\noindent\rule{\textwidth}{1pt}
\vspace{10pt}
\section{Introduction}%
\label{sec:introduction}
Tensor networks can be used to efficiently represent vectors and operators in very large tensor product spaces, assuming they have a restricted structure of correlations.
This makes them well-suited as ansätze for ground states of local quantum Hamiltonians and other quantum states with limited entanglement~\cite{verstraete_mpsreview_2008,schollwoeck_densitymatrix_2011,vidal_class_2008}; as compact representations of partition functions of large systems in classical statistical mechanics~\cite{levin_trg_2007,evenbly_tnr_2015}; and as representations of tensors of various kinds in other applications~\cite{oseledets2011tensor}, such as machine learning~\cite{cichocki_era_2014,stoudenmire_supervised_2016}.
Many tensor networks have constraints applied to their tensors, most common of them being the requirement of isometricity, i.e.\ the property that $W^{\dagger} W = \mathbbm{1}$ when the tensor $W$ is interpreted as a linear map from the tensor product space associated with a subset of its indices to the space associated with the complementary set of indices.
This constraint arises from removing redundant gauge freedom from the network in the case of canonical forms of matrix product states (MPS)~\cite{schollwoeck_densitymatrix_2011} and tree tensor networks (TTN)~\cite{shi_classical_2006}, but is inherent in the definition of the multiscale entanglement renormalization ansatz (MERA)~\cite{vidal_class_2008}.
Even for projected entangled-pair states (PEPS)~\cite{verstraete_mpsreview_2008,verstraete2004renormalization}, where an isometry constraint does not arise naturally, it might be interesting to consider the restricted set with isometric tensors, as this simplifies certain calculations~\cite{zaletel2020isometric,soejima2020isometric,tepaske2020three}.
Furthermore, tensor networks constructed from isometric tensors are equivalent to quantum circuits that could potentially be implemented on a quantum computer, and have attracted recent attention from this point of view~\cite{peruzzo2014variational,li2017efficient,barratt2020parallel,lin2020real}.
To find a tensor network approximation of an unknown state of interest, e.g.\ a ground state of some local Hamiltonian, the variational principle is invoked, i.e.\ the ground state approximation is identified with the point on the tensor network manifold that minimizes the energy.
The first algorithm for finding such an approximation was the density matrix renormalization group (DMRG)~\cite{white_density_1992}, which optimizes the energy over the set of MPS (although the MPS structure was only implicit in the original formulation of DMRG).
The one-site DMRG algorithm in particular optimizes each tensor in turn, iterating the procedure until convergence, a technique known as alternating least squares optimization.
A similar alternating optimization strategy is also the basis for the standard energy minimization algorithm for MERA~\cite{evenbly_algorithms_2009}, which we refer to as the Evenbly-Vidal algorithm, although the local problem is in this case solved differently in order to respect the isometry condition.
Another paradigm for finding minimal energy tensor networks is based on the idea of imaginary time evolution, using either Trotter decompositions~\cite{vidal2004efficient,orus2008infinite} or the time-dependent variational principle (TDVP)~\cite{hackl2020geometry,haegeman2011time}.
Trotter-based imaginary time evolution has been the prevailing algorithm for the optimization of infinite PEPS until recently~\cite{corboz2016variational,vanderstraeten2016gradient}.
In the context of optimizing unitary or isometric tensor networks, yet another strategy is based on flow equations, as proposed in Ref.~\onlinecite{dawson2008unifying}.
Also in the context of quantum computational tasks, classical optimization of the unitary gates in the quantum circuit with respect to a given cost function is often required, as e.g.\ in Ref.~\onlinecite{lin2020real}.
Well-known gradient-based algorithms for nonlinear optimization have not received a great deal of attention for the optimization of tensor networks, likely due to the astounding efficiency of the DMRG algorithm for the case of MPS\@.
Promising results for using the standard (i.e.\ Euclidean) version of the nonlinear conjugate gradient algorithm were reported for translation-invariant MPS~\cite{milsted2013matrix} and PEPS~\cite{vanderstraeten2016gradient} in the thermodynamic limit.
In this manuscript, we propose to use the well-established Riemannian generalization of the nonlinear conjugate gradient and quasi-Newton algorithms to optimize over manifolds of isometric tensor networks.
We furthermore construct a specific preconditioner for these algorithms, derived from the Hilbert space geometry of the tensor network manifold, and show that the resulting methods can outperform tailor-made optimization algorithms, such as the Evenbly-Vidal algorithm for MERA and the variational uniform MPS (VUMPS) algorithm~\cite{zauner2018variational} for infinite MPS\@.
This manuscript is structured as follows:
Section~\ref{sec:geometry} provides an overview of the Riemannian geometry of complex Grassmann and Stiefel manifolds, the manifolds of isometric matrices and tensors.
In Section~\ref{sec:optimization}, we briefly review the basics of Riemannian extensions of gradient-based optimization methods such as the gradient descent, nonlinear conjugate gradient and quasi-Newton algorithms, and discuss the role of preconditioners in this setting.
In Sections~\ref{sec:mera} and~\ref{sec:mps}, we show how these methods can be applied in the context of MERA and MPS, respectively, and demonstrate how they outperform previous methods in many situations.
Section~\ref{sec:discussion} provides some further discussion and an outlook.
The algorithms presented below are available in open source software packages written in the scientific programming language Julia~\cite{bezanson2017julia}.
The most high-level and user-facing packages are MPSKit.jl~\cite{MPSKit.jl} and MERAKit.jl~\cite{MERAKit.jl}.
The ancillary files in \href{https://arxiv.org/src/2007.03638}{arxiv.org/src/2007.03638} include scripts that use these packages to reproduce all the benchmark results that we show.
\section{Riemannian geometry of isometric tensors}%
\label{sec:geometry}
Throughout this section, we focus on a single isometric matrix $W$ that fulfills $W^{\dagger} W = \mathbbm{1}$.
This could for instance be an isometry or disentangler of a MERA, with its top and bottom indices combined to single matrix indices, or an MPS tensor in left or right canonical form.
In contrast to most literature in numerical optimization, we focus on complex isometric matrices.
Isometric matrices of a given size $n \times p$ form a manifold, called the Stiefel manifold, that can be naturally embedded in the Euclidean vector space $\mathbb{C}^{n\times p}$ of general complex $n\times p$ matrices:
\begin{equation}
\label{eq:stiefel_definition}
\Stiefel{n}{p} = \{ W \in \mathbb{C}^{n \times p} \;|\; W^{\dagger} W = \mathbbm{1} \},
\end{equation}
where we have assumed the necessary condition $n \geq p$.
The case $n = p$ yields the manifold of unitary matrices $\Unitary{n}$, which is thus included as special case.
For instance, for the isometries of a ternary MERA with bond dimension $D$, $n = D^3$ and $p = D$, whereas the corresponding disentanglers have $n = p = D^2$.
In the case of left or right canonical MPS with physical dimension $d$ and bond dimension $D$, $n = dD$ and $p = D$.
The isometry constraint imposes $p^2$ independent real-valued constraints and thus $\Stiefel{n}{p}$ is a real manifold of dimension $(2n - p)p$.
Note that as the isometry constraint is not holomorphic, $\Stiefel{n}{p}$ cannot be understood as a complex manifold, and its tangent space cannot be given the structure of a complex subspace of $\mathbb{C}^{n \times p}$, a point to which we return below.
In many situations, what is of interest is not the exact isometry $W$ itself, but rather the subspace which it defines by the span of its $p$ columns.
In those cases, one should identify $W$ with $WU$, where $U$ can be an arbitrary $p \times p$ unitary, and consider the equivalence class $[W] = \{\, W U \;|\; U \in \Unitary{p} \}$.
In a tensor network, this happens whenever the columns of $W$ correspond to a single virtual index, in which case a gauge transformation $U$ can be applied to it, while $U^\dagger$ can be absorbed into the leg of the tensor to which $W$ is connected.
The manifold of such equivalence classes of isometric tensors $[W]$ is a quotient manifold known as the Grassmann manifold $\Grassmann{n}{p} = \Stiefel{n}{p}/\Unitary{p}$.
While $\Grassmann{n}{p}$ is here defined as the quotient manifold of two manifolds without complex structure, $\Grassmann{n}{p}$ is itself a proper complex manifold with complex dimension $(n - p)p$, or equivalently, real dimension $2(n - p)p$.
This can be understood by noticing that the isometry condition is not necessary to define a subspace, so that $\Grassmann{n}{p}$ can also be defined as $\Grassmann{n}{p} = \GeneralLinear{n} / (\GeneralLinear{p} \times \GeneralLinear{n-p})$, with $\GeneralLinear{n}$ the general linear group of invertible complex $n\times n$ matrices.
In fact, $\Grassmann{n}{p}$ can then be given the structure of a K\"{a}hler manifold, which can be important when studying time evolution~\cite{hackl2020geometry}.
In contrast, optimization of real-valued functions on a manifold is only concerned with the Riemannian structure (and not with possible complex, symplectic, or K\"{a}hler structures), for which only the structure as real manifold is relevant, as we make more explicit below.
Throughout the remainder of this manuscript, we will denote elements from Grassmann manifolds using a single representative $W$ of the corresponding equivalence class $[W]$, and assume that $W$ is isometric.
We briefly review the basic properties of Grassmann and Stiefel manifolds which are required to apply gradient-based optimization methods.
For a more thorough introduction to the properties of Grassmann and Stiefel manifolds, see for instance Refs.~\onlinecite{edelman_geometry_1998,zhu_riemannian_2017}.
Note though, that these references only consider real-valued matrices, whereas we review here the complex case.
\subsection{Tangent vectors}%
\label{sec:geometry_tangents}
For an isometric matrix $W \in \Stiefel{n}{p}$, the tangent space at $W$ consists of all matrices $X$ for which $W^{\dagger} X$ is skew-hermitian.
In other words,
\begin{equation}%
\label{eq:stiefel_tangent}
X = W A + W_\perp B, \;\text{where}\; A = -A^{\dagger}.
\end{equation}
Here $W_\perp$ is a $n \times (n-p)$ isometric matrix such that $WW^{\dagger} + W_\perp W_\perp^{\dagger} = \mathbbm{1}$, i.e.\ it is a unitary completion of $W$ (which is not unique).
$B$ is an arbitrary $(n-p) \times p$ matrix.
The skew-hermiticity condition on $A$ implies that the tangent space only allows for linear combinations with real-valued scalar coefficients, i.e.\ it is a vector space over $\mathbb{R}$, as mentioned above.
Because optimization algorithms are formulated using only real-valued linear combinations of tangent vectors, this does not pose any restriction.
For a point on the Grassmann manifold represented by $W$, we can use the unitary gauge freedom to impose that tangent vectors satisfy the holomorphic condition $W^{\dagger} X = 0$.
This amounts to restricting to tangent vectors with $A=0$, and thus the tangent vectors on a Grassmann manifold can be parameterized as
\begin{equation}
\label{eq:grassmann_tangent}
X = W_\perp B,
\end{equation}
with $B$ again being an arbitrary $(n-p) \times p$ matrix.
Note that the $A=0$ condition is preserved under complex linear combinations, as one would expect given the complex structure of $\Grassmann{n}{p}$.
In both cases, Stiefel and Grassmann, we denote the tangent space at $W$ by $\tangentspace{W}$, to which we can append the manifold if we want to distinguish explicitly between the two cases.
\subsection{Metric}%
\label{sec:geometry_metric}
Implicit in most gradient methods is the idea to use the partial derivatives of the cost function, which constitute a dual vector in the cotangent space, as a direction (i.e.\ a tangent vector) along which to update the state.
This works fine if one assumes to be working in Euclidean space, but otherwise requires a metric.
A natural metric for $\tangentspace{W}$, regardless of whether we are on a Stiefel or Grassmann manifold, is the Euclidean metric $g_W(X, Y) = \Re \Tr [X^{\dagger} Y]$, i.e.\ the real part of the Frobenius inner product in the embedding space $\mathbb{C}^{n \times p}$.
Note that the real part of the inner product of a complex space defines a metric (a real symmetric bilinear), whereas the imaginary part defines a symplectic form.
While a general metric depends on the base point $W$, for $g_W$ this dependence is not explicit.
Another natural metric for the Stiefel manifold is given by what is known as canonical metric, for which we refer to Ref.~\onlinecite{edelman_geometry_1998}.
In this manuscript we use the Euclidean $g_W$, as we found little difference between the two choices in our simulations, and the Euclidean metric is more closely related to the Hilbert space inner product and the preconditioning schemes for the tensor networks that we consider in later sections.
A metric allows one to map cotangent vectors to tangent vectors.
In a case like ours, where the manifold is embedded in a Euclidean space, it more generally allows one to construct an orthogonal projection from the embedding space to the tangent space.
For a given complex matrix $D\in\mathbb{C}^{n \times p}$, we define its orthogonal projection onto $\tangentspace{W}$ as the tangent vector $G$ for which $g_W(G, X) = \Re \Tr [D^{\dagger} X]$, for all $X \in \tangentspace{W}$.
The solution for this projection is
\begin{align}
\label{eq:stiefel_gradient}
G &= D - \frac{1}{2} W(W^{\dagger} D + D^{\dagger} W) & \text{if } W &\in \Stiefel{n}{p},\\%
\label{eq:grassmann_gradient}
G &= D - WW^{\dagger} D & \text{if } W &\in \Grassmann{n}{p}.
\end{align}
$D\mapsto G$ is a complex linear map for the Grassmann manifold, but only real linear for the Stiefel manifold.
Although the names $D$ and $G$ purposefully refer to derivatives and gradients, note that Eqs.~\eqref{eq:stiefel_gradient} and~\eqref{eq:grassmann_gradient} can be used to project any arbitrary matrix from $\mathbb{C}^{n \times p}$ onto the tangent space $\tangentspace{W}$.
\subsection{Gradients, retraction, and transport}%
\label{sec:geometry_gradients_retraction_transport}
For gradient optimization of a cost function $C(W)$, we can first compute the partial derivatives
\begin{align}
D_{ij} = \frac{\partial C}{\partial \Re W_{ij}} + i \frac{\partial C}{\partial \Im W_{ij}} = 2 \frac{\partial C}{\partial W^\ast_{ij}}
\end{align}
without taking the isometry condition into account.
The complex linear combination here is chosen such that
\begin{align}
\left.\frac{\mathrm{d} C(W + \epsilon X)}{\mathrm{d} \epsilon}\right\vert_{\epsilon=0} = \Re \Tr [D^{\dagger} X], \quad \forall X \in \mathbb{C}^{n \times p}
\end{align}
(assuming that the cost-function can meaningfully be extended or continued to non-isometric matrices in such a way that the above derivative is well defined).
Projecting $D$ onto the tangent space with Eq.~\eqref{eq:stiefel_gradient} or~\eqref{eq:grassmann_gradient} yields $G$, which is the tangent vector such that
\begin{align}
\label{eq:stiefel_grassmann_gradient_condition}
g_W(G, X) = \left.\frac{\mathrm{d} C(W + \epsilon X)}{\mathrm{d} \epsilon}\right\vert_{\epsilon=0},\quad \forall X \in \tangentspace{W}.
\end{align}
$G$ will henceforth be referred to as the \emph{gradient} of $C$.
This brings us to the next point, which is that we would often like to change our isometry $W$ by moving in the direction of a tangent vector $X\in \tangentspace{W}$, but $W + \epsilon X$ will only respect the isometry condition up to first order in $\epsilon$.
To travel further in the direction of $X$ while staying on the manifold, Riemannian optimization algorithms employ the concept of \emph{retraction}.
A retraction $\retraction{W}{X}{\alpha}$ is a curve parameterized by $\alpha \in \mathbb{R}$, an initial point $W$ such that $\retraction{W}{X}{0} = W$, and initial direction $X \in \tangentspace{W}$ such that $\frac{\partial}{\partial \alpha} \retraction{W}{X}{\alpha}|_{\alpha=0} = X$, that lies exactly within the manifold for all values of $\alpha$ in some interval containing $\alpha = 0$ (preferably $\alpha \in \mathbb{R}^+$).
For both Stiefel and Grassmann manifolds, several retraction functions exist, even if we impose the requirement that we must be able to numerically compute them efficiently.
One natural choice to consider are geodesics, since the notion of retraction can be seen as a generalization thereof.
Given a tangent vector $X = WA + W_\perp B$, the retraction
\begin{equation}
\label{eq:retraction}
\retraction{W}{X}{\alpha} = e^{\alpha \, Q_X} \, W \, ,
\quad \text{where } Q_X =
\begin{bmatrix}
W & W_\perp
\end{bmatrix}
\begin{bmatrix}
A & -B^{\dagger} \\%
B & 0
\end{bmatrix}
\begin{bmatrix}
W^{\dagger} \\%
W_\perp^{\dagger}
\end{bmatrix},
\end{equation}
is indeed a geodesic for the Grassmann manifold (where $A=0$), but is not a geodesic with respect to the Euclidean metric for the Stiefel manifold (where $A = -A^{\dagger}$).
It is however a geodesic with respect to the canonical metric of the Stiefel manifold, and can certainly be used as viable retraction also in combination with the Euclidean metric.\footnote{%
A closed form expression for the geodesics of the Stiefel manifold with respect to the Euclidean metric is also known, but cannot be written using a unitary applied to $W$; we refer to Ref.~\onlinecite{edelman_geometry_1998} for further details.
}
The retraction in Eq.~\eqref{eq:retraction} requires the matrix exponential of $Q_X$, which can be evaluated with $O(n p^2 + p^3)$ operations (compared to a naive $O(n^3)$ implementation) by exploiting the fact that the maximal rank of $Q_X$ is $2p$.\footnote{%
How this is done depends slightly on the manifold.
In the simpler Grassmann case, the exponential in Eq.~\eqref{eq:retraction} reduces to sines and cosines of singular values of $X$, and we can avoid constructing $W_\perp$ explicitly~\cite{edelman_geometry_1998}.
In the Stiefel case, we need to extract $W_\perp$ from a QR decomposition of $\begin{bmatrix} W & Z \end{bmatrix}$, where $Z = (\mathbbm{1} - WW^{\dagger})X = W_\perp B$, and compute the matrix exponential of $
\begin{bmatrix}
A & -B^{\dagger} \\%
B & 0
\end{bmatrix}
$.
The full details can be found in the source code of the TensorKitManifolds.jl~\cite{TensorKitManifolds.jl} package.
}
Another notable option for retraction is to replace the exponential in Eq.~\eqref{eq:retraction} by a Cayley transform, which can then exploit the reduced rank via the Sherman–Morrison-Woodbury formula, see Ref.~\onlinecite{zhu_riemannian_2017} for details.
While the latter can be somewhat faster, we use the retraction in Eq.~\eqref{eq:retraction} throughout this manuscript.
The above definitions constitute the bare minimum to formulate a Riemannian gradient descent algorithm on a Stiefel or Grassmann manifold.
To exploit information from previous optimization steps, as happens in the conjugate gradient and quasi-Newton algorithms, one more ingredient is needed: a vector transport to transport gradients and other tangent vectors from previous points on the manifold to the current point.
A vector transport generalizes the concept of parallel transport, and needs to be compatible with the chosen retraction.
If $V = \retraction{W}{X}{\alpha}$ is the end point of a retraction, a vector transport maps a tangent vector $Y \in \tangentspace{W}$ at the initial point to a tangent vector $\transport{Y}{W}{X}{\alpha} \in \tangentspace{V}$.
As with the retraction, many choices are possible, but we use the transport
\begin{align}
\label{eq:transport}
\transport{Y}{W}{X}{\alpha} = e^{\alpha \, Q_X} \, Y \, ,
\end{align}
where $Q_X$ is as in Eq.~\eqref{eq:retraction}, both for the Stiefel and the Grassmann case.
This choice can be implemented efficiently, again by exploiting the low-rank property of $Q_X$.
It has the additional benefit that it is a metric connection, which is to say it preserves inner products between tangent vectors, i.e.\ $g_W(Y_1,Y_2) = g_V(\transport{Y_1}{W}{X}{\alpha}, \,\transport{Y_2}{W}{X}{\alpha})$.
This simplifies some steps of the optimization algorithms and guarantees desirable convergence properties~\cite{zhu_riemannian_2017}.
Note that Eq.~\eqref{eq:transport} is not the parallel transport with respect to the Euclidean metric $g$ (nor with respect to the canonical metric), as it corresponds to a metric connection which has torsion, but this does not hinder its usage in optimization algorithms.
Alternatively, one could again replace the exponential in Eq.~\eqref{eq:transport} by a Cayley transform, if this was also done in the retraction.
\subsection{Product manifolds}%
\label{sec:geometry_product_manifolds}
Note, finally, that a function depending on several isometries or unitaries corresponds to a function on the product manifold $\Stiefel{n_1}{p_1} \times\, \Stiefel{n_2}{p_2} \times \ldots$ (with $\times$ being the Cartesian product), where some of the factors could also be Grassmann manifolds instead.
The corresponding tangent space is the Cartesian product of the individual tangent spaces (which corresponds to the direct sum as long as the number of tensors remains finite) and all of the above structures and constructions extend trivially.
\section{Riemannian gradient optimization}%
\label{sec:optimization}
Having established the Riemannian geometry of Grassmann and Stiefel manifolds (and products thereof) in the previous section, we can now discuss how to implement Riemannian versions of some well-known gradient-based optimization algorithms, all of which are described in the literature~\cite{smith_optimization_1994,edelman_geometry_1998,absil2009optimization,ring2012optimization,huang2015broyden,zhu_riemannian_2017}.
We aim to minimize a cost function $C(W)$ defined on our manifold, where we consider a single argument $W$ for notational simplicity.
The simplest approach is the Riemannian formulation of gradient descent, often also referred to as steepest descent.
It is an iterative procedure which at every step computes the gradient of $C$ at the current point on the manifold, and then uses the chosen retraction in the direction of the negative gradient to find the next point.
In steepest descent, the step size $\alpha$ is chosen so as to minimize $C$ along the retraction $\alpha \mapsto \retraction{W}{X}{\alpha}$ with $X = - G$.
Finding $\alpha$ is known as the linesearch, and various algorithms and strategies exist for it.
It is often unnecessary or even prohibitive to determine the minimum accurately; rather an approximate step size $\alpha$ that satisfies the Wolfe conditions~\cite{nocedal2006numerical} is sufficient to guarantee convergence.
If we define $W' = \retraction{W}{X}{\alpha}$ to be the new isometry, $G'$ the gradient at $W'$, and $X' = \mathrm{d} \retraction{W}{X}{\alpha} / \mathrm{d} \alpha$ the local tangent to the retraction, then the Wolfe conditions are
\begin{align}
\label{eq:wolfe_1}
C(W') &< C(W) - c_1 g_W(G, X),\\%
\label{eq:wolfe_2}
g_{W'}(G',X') &> c_2 g_W(G,X),
\end{align}
with $0 < c_1 < c_2 < 1$ being free parameters~\cite{ring2012optimization,huang2015broyden}.
Eq.~\eqref{eq:wolfe_1} states that the cost function should decrease sufficiently, while Eq.~\eqref{eq:wolfe_2} says that its slope (which starts out negative for a descent direction) should increase sufficiently.
Throughout our simulations, we use the linesearch algorithm described in Refs.~\onlinecite{hager2006algorithm,hager_new_2005}, which also takes into account that the descent property of Eq.~\eqref{eq:wolfe_1} (also known as the Armijo rule) cannot be evaluated accurately close to convergence due to finite machine precision, and switches to an approximate but numerically more stable condition when necessary.
In practice, a small number (often two or three) function evaluations suffice to determine a suitable step size $\alpha$.
While (Riemannian) gradient descent with step sizes that satisfy the Wolfe conditions converges in theory, this convergence is only linear and can be prohibitively slow, especially for systems of physical interest exhibiting strong correlations (e.g.\ critical systems)~\footnote{
This can be argued by noting that the Hessian of the corresponding energy function is often related to the dispersion relation of the physical excitations in the system~\cite{haegeman2012variational,haegeman2013post}, and thus has (near)-zero modes for such systems.
}.
An improved algorithm with nearly the same cost is the nonlinear conjugate gradient algorithm, which dates back to the work of Hestenes and Stiefel.
In conjugate gradient the search direction is a linear combination of the (negative) gradient and the previous search direction, a concept known as \enquote{momentum} in the context of optimizers for machine learning.
Various schemes exist for the choice of the $\beta$ coefficient in this linear combination, see Ref.~\onlinecite{hager_survey_2006} and references therein.
All these schemes can be applied in the Riemannian case, although the inner products that need to be computed as part of $\beta$'s definition need to be replaced by the metric $g$.
Furthermore, to build a linear combination between the current gradient and the previous search direction, one needs to invoke the vector transport $\mathcal{T}$ from Sec.~\ref{sec:geometry} for the latter to represent a valid tangent vector at the new base point.
In the simulations below, we use the conjugate gradient scheme of Hager and Zhang~\cite{hager2006algorithm,hager_new_2005}.
From a second order expansion of the cost function around the current point, one arrives at Newton's method, which suggests taking a step of length $1$ in the direction of $-H^{-1}(G)$, where $H$ is the Hessian, i.e.\ the matrix of second derivatives.
While Newton's method has a theoretical quadratic convergence rate close to the minimum, computing $H$ and its inverse is often prohibitively expensive and has various other issues.
The Hessian might not be positive definite far away from the minimum, and furthermore depends on the second order behaviour of the retraction when formulating a Riemannian generalization of Newton's method.
Quasi-Newton methods, on the other hand, construct an approximation to $H^{-1}$ using only gradients, computed at the successive points $W_k$ along the optimization.
The most commonly used is the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm~\cite{nocedal1980updating,nocedal2006numerical}, which keeps a low-rank, positive semi-definite approximation of $H^{-1}$ in memory.
The Riemannian formulation of it also depends on the vector transport and has been well established, see Refs.~\onlinecite{ring2012optimization, huang2015broyden} and references therein.
Both the conjugate gradient and L-BFGS algorithms converge to a local minimum at a rate that is somewhere between the linear convergence of gradient descent and quadratic convergence of Newton's method.
Which one is to be preferred often depends on the application.
The latter requires a few more vector operations, but can use these to scale the inverse Hessian so that step size $\alpha=1$ is typically accepted and no linesearch is needed in most iterations.
Despite the speedup provided by the conjugate gradient and L-BFGS algorithms, it is often beneficial to apply a \emph{preconditioner} to the optimisation.
A preconditioner is a transformation that maps one tangent vector to another, $X \mapsto \tilde{X}$, and that is applied when choosing the search direction.
Using a preconditioner with gradient descent simply means retracting in the direction of the negative \emph{preconditioned} gradient $-\tilde{G}$, instead of $-G$.
Using preconditioners with conjugate gradient and quasi-Newton methods is not much more complicated, and we direct the reader to the numerical optimisation literature~\cite{nocedal2006numerical,nash1985preconditioning,desterck_nonlinearly_2018} for the details.
The choice of the preconditioner $X \mapsto \tilde{X}$ is typically guided by trying to capture some structure of the Hessian.
The inverse Hessian $\tilde{X} = H^{-1}(X)$ would often be an ideal preconditioner, and while it is usually infeasible to implement, using some approximation to it may already help convergence significantly.
A preconditioner (assumed to be positive definite) can also be seen as changing the metric in the problem, hopefully in such a way that the optimisation landscape becomes less singular and hence easier to navigate for the chosen optimisation algorithm.
This geometrical viewpoint is illustrated in Fig.~\ref{fig:preconditioning_geometry}, and is what we will use to justify the preconditioners we use in our tensor network optimisations.
Note that the same effect could be achieved by actually defining a new metric on the relevant Stiefel or Grassmann manifold, and repeating the steps in Sec.~\ref{sec:geometry} again for this metric.
However, we find that using the Euclidean inner product with an additional explicit preconditioning step gives greater flexibility without complicating e.g.\ the metric condition for the vector transport.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.95\linewidth]{preconditioning_geometry.pdf}
\caption{%
On the left, the grey ovals are the contour lines of the cost function in this 2-dimensional optimisation problem.
The purple arrow $X$ is the negative gradient, and the zig-zag line emanating from it is the path that gradient descent takes.
The relatively slow convergence of the gradient descent path is a consequence of the near-singular geometry of the contour lines, where the cost function varies much more along one axis than the other.
The green arrow $\tilde{X}$ would be the optimal choice for the preconditioned search direction, the one that takes us to the optimum in a single retraction.
By changing the geometry (i.e.\ the metric) of the problem, in this case by a simple rescaling of the axes, we can map to the problem on the right, where the geometry of the contour lines has become less singular.
In this new geometry $\tilde{X}$ is in fact the negative gradient.
This suggests that a preconditioner that implements this change of geometry would probably be beneficial for convergence.
While the above is a cartoon example, redefining the metric to make the optimisation landscape less singular can be a useful way to design preconditioners more generally.
}%
\label{fig:preconditioning_geometry}
\end{figure}
In the context of tensor networks, the cost function $C$ will typically be $C(W) = \bra{\psi(W)} H \ket{\psi(W)}$, where $H$ is a local Hamiltonian, and $\ket{\psi(W)}$ is a tensor network state dependent on the isometry $W$.
A tangent vector $X \in \tangentspace{W}$ can then be related to a state $\ket{\Phi_W(X)} = X^i \ket{\partial_i \psi(W)}$ in Hilbert space, which yields an induced inner product $\braket{\Phi_W(X)}{\Phi_W(Y)}$ between tangent vectors $X, Y \in \tangentspace{W}$.
A suitable preconditioner can then be extracted from the explicit expression of $\braket{\Phi_W(X)}{\Phi_W(Y)}$, or some approximation thereof.
As discussed in the applications below, we assume that this inner product can be written as $\braket{\Phi_W(X)}{\Phi_W(Y)} \approx \Tr[X^{\dagger} Y \rho_W]$ for some $W$-dependent, hermitian, positive (semi)-definite $\rho_W$ of size $p \times p$.
We can then implement a preconditioning step $X \mapsto \tilde{X} \in \tangentspace{W}$ such that (henceforth omitting the $W$ dependence)
\begin{equation}%
\label{eq:preconditioner}
\Re \Tr[Y^{\dagger} \tilde{X} \rho] = \Re \Tr[Y^{\dagger} X] \quad \forall \, Y \in \tangentspace{W}.
\end{equation}
In other words, the Euclidean inner product with $X$ equals the more physically motivated inner product with $\tilde{X}$.
If we express $X$ as $X = WA + W_\perp B$, where $A$ is skew-hermitian (Stiefel) or zero (Grassmann), then the solution to Eq.~\eqref{eq:preconditioner} is
\begin{align}%
\label{eq:preconditioner_solution}
& \tilde{X} = W \tilde{A} + W_\perp \tilde{B},\\%
\text{where} \; & \tilde{A} \rho + \rho \tilde{A} = 2 A \;\text{and}\; \tilde{B} = B \rho^{-1}.
\end{align}
The equation for $\tilde{A}$ is a Sylvester equation, that can be solved easily and efficiently using e.g.\ an eigendecomposition of $\rho$ at a cost $O(p^3)$.
The matrix $\rho$ may often be quite ill-conditioned, and in practice we have found the regularized inverse ${\left(\rho^2 + \mathbbm{1} \delta^2\right)}^{-\frac{1}{2}}$ to work well.
We discuss the choice of $\delta$ in the applications below.
Note that this preconditioner accounts for the structure of the physical state space, i.e.\ it corresponds to the induced metric of the variational manifold in Hilbert space.
When implemented exactly, the preconditioned gradient corresponds to the direction in which a state would evolve under imaginary time evolution as implemented by the time-dependent variational principles of Dirac, Frenkel or McLachlan (see Refs.~\onlinecite{hackl2020geometry,Yuan_2019} and references therein) and has been used with MPS as such~\cite{haegeman2011time}.
This choice, or (block)-diagonal approximations thereof, as discussed in the next section for the case of MERA, was recently referred to as the ``quantum natural gradient'' in the context of variational quantum circuits~\cite{Stokes_2020}.
This choice of preconditioner is independent of the Hamiltonian, and it is conceivable that a much bigger speedup can be obtained by explicitly taking the Hamiltonian into account.
Such an improved preconditioner can probably not be implemented efficiently without resorting to an iterative linear solver, such as the linear conjugate gradient method.
Such a scheme would be close in spirit to the set of optimization methods known as truncated Newton algorithms~\cite{nash1985preconditioning,nash2000survey}.
The above preconditioner can then still prove useful to speed up this inner linear problem.
We elaborate on this in the discussion in Section~\ref{sec:discussion}.
\section{Application: MERA}%
\label{sec:mera}
In this section we show how Riemannian optimization methods can be applied to the multiscale entanglement renormalization ansatz (MERA), and demonstrate that the resulting algorithm outperforms the usual Evenbly-Vidal optimization method used for MERA\@.
Specifically, we concentrate on a one-dimensional, infinite, scale invariant, ternary MERA, but the generalization to other types of MERAs is trivial.
A MERA is a tensor network of the form
\begin{equation}
\label{eq:mera}
\includegraphics[scale=1,raise=-1.4em]{mera.pdf}
\; .
\end{equation}
Each tensor in a MERA is isometric in the sense that
\begin{equation}
\label{eq:mera_isometricity}
\includegraphics[scale=1,raise=-0.6em]{mera_isometricity_a.pdf}
\; = \;
\includegraphics[scale=1,raise=-0.6em]{mera_isometricity_b.pdf}
\qquad \text{and} \qquad
\includegraphics[scale=1,raise=-0.6em]{mera_isometricity_c.pdf}
\; = \;
\includegraphics[scale=1,raise=-0.6em]{mera_isometricity_d.pdf}
\;,
\end{equation}
where red borders denote complex conjugation.
The network defines a quantum state $\ket{\text{MERA}}$ living on the lattice at the bottom legs in Eq.~\eqref{eq:mera}.
In the example MERA from Eq.~\eqref{eq:mera}, there are two distinct layers:
There is one transition layer at the bottom, followed by a scale invariant layer, copies of which repeat upwards to infinity.
Each layer $i$ is translation invariant and defined by two tensors, the disentangler $u_i =
\includegraphics[scale=1,raise=-0.2em]{mera_u.pdf}\,$
and the isometry
$w_i =
\includegraphics[scale=1,raise=-0.2em]{mera_w.pdf}\,$.
The cost function we are trying to minimize is $\bra{\text{MERA}} H \ket{\text{MERA}}$, where $H = \sum_i h_i$ is a given local Hamiltonian.
In our benchmark simulations we use the critical Ising Hamiltonian
\begin{align}
h_i = -X_i X_{i+1} - Z_i.
\end{align}
The parameter space in which we are optimising is $\bigtimes_v M_v$, where $\bigtimes_v$ denotes Cartesian product over all the different tensors $v = u_1, w_1, u_2, w_2, \dots$, and $M_v$ is the Stiefel or Grassmann manifold of each tensor $v$.
Any unitary one-site rotation on the top index of an isometry $w_i$ can be absorbed into the disentangler $u_{i+1}$ above it, and hence the natural manifold for $w$'s is the Grassmann manifold: $M_{w_i} = \mathrm{Gr}$.\footnote{%
Note that we are not saying here that any member of the equivalence class $[w_i] = \{ w_i U \;|\; UU^{\dagger} = U^{\dagger} U = \mathbbm{1}\}$ leads to the same MERA\@: This is obviously not the case.
Instead what we are saying is that changes of the form $w_i \mapsto w_i U$ are degenerate from the point of view of our optimization, as they can be cancelled by a corresponding change in one of the disentanglers.
In other words, any tangent directions that correspond to changes of the type $w_i \mapsto w_i U$ are of no interest to us, and can be projected out.
}
The same is not true for the disentanglers, for which similar unitary rotations would entangle the two top indices, and hence we treat them as points on Stiefel manifolds: $M_{u_i} = \mathrm{St}$.\footnote{
Indeed, $\Grassmann{n}{n}$ is the trivial singleton manifold $[\mathbbm{1}]$.
}
We have omitted the dimensions of the manifolds, since they depend on the physical site state space dimension $d$ and the bond dimension $D$ of the upper layers.
As discussed at the end of Section~\ref{sec:geometry}, the tangent space is the Cartesian product of the tangent spaces of the individual tensors, $\bigtimes_v \tangentspace{v}$, which corresponds to a direct sum structure, and the Riemannian geometry and associated operations extend trivially.
The inner product, in particular, is the sum of the inner products on the individual manifolds.
To compute the gradients, we first discuss the partial derivatives.
Hereto, we denote the partial derivative of the state $\ket{\text{MERA}}$ with respect to a tensor $v$ by $\partial_v \ket{\text{MERA}}$.
Since each tensor appears several times in the network, $\partial_v \ket{\text{MERA}}$ has several terms in it, e.g.
\begin{gather}
\label{eq:mera_derivative}
\partial_{w_1} \ket{\text{MERA}} = \;
\includegraphics[scale=1,raise=-1.2em]{mera_derivative_a.pdf}
\; + \;
\includegraphics[scale=1,raise=-1.2em]{mera_derivative_b.pdf}
\; + \;
\includegraphics[scale=1,raise=-1.2em]{mera_derivative_c.pdf}
\; + \; \dots.
\end{gather}
The partial derivative of the cost function is then $D_v = 2 \partial_{v^{\dagger}} \bra{\text{MERA}} H \ket{\text{MERA}}$.
Up to a scalar factor, the same object arises in the context of the usual Evenbly-Vidal optimization algorithm, where it is called the \enquote{environment} of tensor $v$.
These environments can be computed efficiently, and we refer the reader to Ref.~\onlinecite{evenbly_algorithms_2009} for how to do so.
Extra care needs to be taken when dealing with the scale invariant layer, something we discuss in Appendix~\ref{app:mera_scale_invariant_layer}.
The gradient $G_v$ is the projection of the partial derivative $D_v$ onto the tangent space $\tangentspace{v}$, as in Eq.~\eqref{eq:stiefel_gradient} and~\eqref{eq:grassmann_gradient}.
The total gradient $G$ of the whole parameter space is $G = (G_{u_1}, G_{w_1}, G_{u_2}, \dots) \in \bigtimes_v \tangentspace{v}$.
As mentioned above, the inner product between two tangents $X, Y \in \bigtimes_v \tangentspace{v}$ is $\sum_v g_v(X_v, Y_v)$, where $g$ is the Euclidean metric.
However, each $X_v$ is associated with a state in the physical Hilbert space, schematically denoted as $\frac{\partial \ket{\text{MERA}}}{\partial v} X_v$, and we would like to implement a preconditioning that would equate to using instead a metric arising from the physical inner product, namely
\begin{equation}%
\label{eq:mera_full_inner_product}
\sum_{v, v' \in \{u_1, w_1, \dots\}} X_v^{\dagger} \frac{\partial^2 \braket{\text{MERA}}{\text{MERA}}}{\partial v^{\dagger} \partial v'} Y_{v'}
\end{equation}
The cross-terms in this sum are quite expensive to compute, so we settle instead for the diagonal version
\begin{equation}%
\label{eq:mera_diagonal_inner_product}
\sum_{v \in \{u_1, w_1, \dots\}} X_v^{\dagger} \frac{\partial^2 \braket{\text{MERA}}{\text{MERA}}}{\partial v^{\dagger} \partial v} Y_{v}
\;\, = \sum_{v \in \{u_1, w_1, \dots\}} \Tr[X_v^{\dagger} Y_v \rho_v],
\end{equation}
where $\rho_v$ is the reduced density matrix on the top index or indices of $v$.
As discussed at the end of Sec.~\ref{sec:optimization}, preconditioning with this type of metric can be efficiently implemented for both Stiefel and Grassmann tangents.
The regularization parameter $\delta$ used in computing the regularized inverse of $\rho$ (or the equivalent thereof for the Sylvester problem) in the preconditioner can also be allowed to vary.
In particular, using a very small value of $\delta$ can be detrimental to the optimization in the beginning, when we are far from the minimum, and we have found $\delta = \|X_v\|$ to be a good choice.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.8\linewidth]{mera_results.pdf}
\caption{%
A comparison of convergence in optimising a MERA using the Evenbly-Vidal algorithm (solid green lines) and L-BFGS on Riemannian manifolds (dashed blue lines).
Displayed here are the ground state energy error compared to the exact value (top) and the norm of the gradient (bottom).
The benchmark model in question is the critical Ising model.
In all simulations the MERA is a bond dimension $8$ ternary MERA with two transition layers, with the $\mathbb{Z}_2$ symmetry enforced.
For both algorithms three different simulations are shown, corresponding to three different starting points:
One was a MERA initialized with random isometries and identity disentanglers, the two others were MERAs optimized to convergence at a lower bond dimension, $D=3$ and $D=6$, and then expanded to the full bond dimension $D=8$.
This kind of slow ramping up of the bond dimension can be useful for both speed of convergence and for avoiding local minima.
As the energy error plot shows, here, too, some simulations converge to a local minimum instead of the global one.
In all cases the convergence speed of L-BFGS algorithm clearly outperforms the Evenbly-Vidal algorithm.
}%
\label{fig:mera_results}
\end{figure}
To the best of our knowledge, the only algorithm that has systematically been used to minimize $\bra{\text{MERA}}H\ket{\text{MERA}}$ is the Evenbly-Vidal algorithm, described in detail in Ref.~\onlinecite{evenbly_algorithms_2009}.
Fig.~\ref{fig:mera_results} shows benchmark results comparing the Evenbly-Vidal algorithm and an L-BFGS optimization, using the above preconditioning.
The L-BFGS optimization converges significantly faster for all the simulations displayed in the figure.
Note the logarithmic scale of the horizontal axis, which allows to visualize both the initial and final parts of the convergence.
Individual iterations take somewhat longer to run with L-BFGS (though the asymptotic complexity remains the same, $O(D^8)$ for the ternary MERA), typically 1.5--2 times longer in our simulations, but this effect is more than compensated for by the faster rate of convergence~\footnote{%
Note that the speed difference between the Evenbly-Vidal algorithm and gradient methods depends somewhat on the type of MERA\@.
The most costly operations inherent to the gradient methods (retractions, vector transport and applying the preconditioner) scale as $O(D^6)$, whereas the leading-order cost of both algorithms (computing energy and gradients or environments) is $O(D^7)$ for modified binary, $O(D^8)$ for ternary, and $O(D^9)$ for binary MERA\@.
The higher scaling of e.g.\ binary versus ternary MERA is compensated for by ternary MERAs typically needing higher bond dimensions to achieve the same accuracy, which shows as proportionally higher subleading costs.
}.
While our benchmark is the Ising model with a ternary MERA, we find qualitatively similar results for binary MERAs, and for different models such as the XXZ model.
Moreover, we show results for the L-BFGS algorithm as they are slightly better than those of the conjugate gradient method, but the difference is not drastic.
Other small changes, such as treating the isometries as elements of Stiefel manifolds, or using different retractions or the canonical metric, have limited effects on the results.
The use of preconditioning with the Hilbert space inner product, however, is crucial, and thus indicative that further improvements could be made by improving the preconditioner.
Note that MERA optimizations are somewhat prone to getting stuck in local minima, especially at higher bond dimensions, something that affects all optimization methods we have tried.
The strategy of the Evenbly-Vidal algorithm is similar to alternating least-squares algorithms:
At every step a single tensor of the network is updated, while considering the other tensors as independent of it.
The specific update needs to account for the isometry condition and is reviewed in Appendix~\ref{app:ev_steplimit}.
An update like this typically brings down the energy at every step, and the procedure is then iterated over all the different tensors until convergence.
At first this seems entirely different from gradient optimization:
The Evenbly-Vidal algorithm makes discontinuous jumps from one point in the parameter space to another, one tensor at a time, whereas gradient methods perform smooth retractions of all the tensors at once.
However, hidden in the Evenbly-Vidal update is in fact a kind of step size parameter, that is the additive scale of the effective Hamiltonian.
In Appendix~\ref{app:ev_steplimit} we show that there is a particular limit in which the Evenbly-Vidal algorithm reduces to gradient descent preconditioned with the metric from Eq.~\eqref{eq:mera_diagonal_inner_product}.
Although this limit is not necessarily where the algorithm is typically run, this relation to a first-order optimisation method gives some intuition for how a quasi-Newton or conjugate gradient method could outperform it.
\section{Application: MPS}%
\label{sec:mps}
In this section we show how gradient optimization methods on Riemannian manifolds can be applied to optimize a matrix product state (MPS)\@.
The MPS is kept in its left-canonical form, where each tensor is an isometry from its physical index and left virtual index to its right virtual index.
Such an MPS can be depicted as
\begin{equation}
\label{eq:mps}
\includegraphics[scale=1,raise=-1.4em]{mps.pdf}
\; ,
\end{equation}
where
\begin{equation}
\label{eq:mps_isometricity}
\includegraphics[scale=1,raise=-1.4em]{mps_isometricity_a_v2.pdf}
\, = \,
\includegraphics[scale=1,raise=-1.4em]{mps_isometricity_b_v2.pdf}
\;,
\end{equation}
and red borders denote complex conjugation.
Every injective MPS can be gauge-transformed into this form.
For simplicity's sake we concentrate on the case of an infinite MPS with one-site translation symmetry~\cite{zauner2018variational,vanderstraeten2019tangent}.
Such an MPS is defined by a single isometry.
However, the generalization to a finite MPS or to one with a larger unit cell is straightforward.
We consider the tensor
$\,\includegraphics[scale=1,raise=-0.45em]{mps_tensor.pdf}$
defining the MPS as a point on a Grassmann manifold, since unitary rotations on the right virtual index of each tensor are mere gauge transformations, which can be absorbed in the next tensor without changing the physical state.
The inner product between two tangent tensors, as well as retraction and transport functions are as explained in Sec.~\ref{sec:geometry}, but see also Ref.~\onlinecite{haegeman2014geometry} for further details about the Riemannian geometry of MPS manifolds.
The cost function is the expectation value of a Hamiltonian, which we represent as a matrix product operator (MPO)
\begin{equation}
\label{eq:mpo}
\includegraphics[scale=1,raise=-0.85em]{mpo.pdf}
\;\; .
\end{equation}
The partial derivative of the cost function with respect the isometry can be computed as
\begin{equation}
\label{eq:mps_derivative}
2 \cdot\; \includegraphics[scale=1,raise=-1.33em]{mps_derivative.pdf}
\;\; ,
\end{equation}
where $H_l$ and $H_r$ are the left and right energy environments, which can be efficiently be computed as outlined in Refs.~\onlinecite{schollwoeck_densitymatrix_2011,zauner2018variational,vanderstraeten2019tangent}.
The partial derivative can then be projected onto the tangent space of the Grassmann manifold, as in Eq.~\eqref{eq:grassmann_gradient}, to obtain the gradient.
For preconditioning, we want the effective inner product between two tangent vectors for an individual site,
$\,\includegraphics[scale=1,raise=-0.45em]{mps_impurity_a.pdf}$
and
$\,\includegraphics[scale=1,raise=-0.45em]{mps_impurity_b.pdf}$,
to be
\begin{align}
\sum_{n=-\infty}^{\infty} &\includegraphics[scale=1,raise=-1.45em]{mps_inner_a.pdf}
\; = \;
\includegraphics[scale=1,raise=-1.45em]{mps_inner_b.pdf}
\; = \;
\includegraphics[scale=1,raise=-1.45em]{mps_inner_c.pdf}
\;.
\label{eq:mps_inner}
\end{align}
Here $n$ is the separation between the sites, and the first equation follows from the fact that Grassmann tangent vectors are orthogonal to the Grassmann-points they are at, i.e.\ Eq.~\eqref{eq:grassmann_tangent}.
This is known as the left gauge condition for tangent vectors in the context of MPS~\cite{haegeman2014geometry,vanderstraeten2019tangent}.
The tensor at the very right in Eq.~\eqref{eq:mps_inner} is the dominant right eigenvector of the MPS transfer matrix,
\begin{align}
\label{eq:mps_right_transfermatrix}
\includegraphics[scale=1,raise=-1.45em]{mps_with_right_transfermatrix.pdf}
\; = \;
\includegraphics[scale=1,raise=-1.45em]{mps_right_transfermatrix.pdf}
\;,
\end{align}
and plays the role of $\rho$ from Eq.~\eqref{eq:preconditioner}.
In contrast to the MERA case, this expression corresponds to the exact Hilbert space inner product between tangent vectors, without approximations.
Implementing preconditioning with this inner product requires only implementing the map
\begin{align}
\label{eq:mps_preconditioning}
\includegraphics[scale=1,raise=-0.70em]{mps_impurity.pdf}
\; \mapsto \;
\includegraphics[scale=1,raise=-0.70em]{mps_impurity_with_preconditioning.pdf}
\;.
\end{align}
As with MERA, regularising the inverse of the right eigenvector is paramount for performance, especially during the initial iterations of the optimization process.
In the MPS case we use the regularisation $\left(\includegraphics[scale=1,raise=-0.3em]{right_transfermatrix.pdf} + \mathbbm{1} \delta \right)^{-1}$ with $\delta = \left\|\includegraphics[scale=1,raise=-0.5em]{mps_impurity_a.pdf}\right\|^2$.
We would like to note that, with an exact inverse (i.e.\ $\delta = 0$) in Eq.~\eqref{eq:mps_preconditioning}, standard gradient descent in the limit of a small step size $\alpha\to 0$ amounts to imaginary time evolution, implemented using the TDVP~\cite{haegeman2011time}.
This is a consequence of the K\"{a}hler structure of the MPS manifold~\cite{hackl2020geometry,haegeman2014geometry,vanderstraeten2019tangent}.
With the above building blocks, we are ready to use Riemannian gradient methods for a uniform MPS\@.
For benchmarking, we compare against the well-established VUMPS algorithm~\cite{zauner2018variational}.
We are \emph{not} able to consistently outperform VUMPS for all MPS problems, but we are able to do so for some problems.
As an example of a case where gradient optimization performs well, we consider the triangular lattice antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on a cylinder.
The classical analogue of this model is disordered, but quantum fluctuations restore the order again in the infinite 2d plane.
It is an example of order from disorder and has been studied extensively~\cite{chubukov1992,kojima_quantum_2018,zheng_excitation_2006,chernyshev_spin_2009,mourigal_dynamical_2013}.
Considering the cylinder as a 1D system with longer range couplings (\enquote{coiling} around the cylinder), the Hamiltonian can be written as
\begin{equation}%
H = \sum_i (h_{i,i+1} + h_{i, i+c} + h_{i, i+c+1}), \qquad h_{i,j} = X_i X_j + Y_i Y_j + Z_i Z_j,
\end{equation}
where $X$, $Y$, and $Z$ are the spin operators.
Here $c$ is the width of the cylinder, which we fix to $c=6$ for our benchmark.
The appropriate MPS ansatz for this model is a uniform MPS with $c$-site unit cell.
We also enforce the $\mathrm{SU}(2)$ symmetry of the MPS, since continuous symmetry breaking does not take place for finite $c$.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.8\linewidth]{mps_results.pdf}
\caption{%
A comparison of convergence in optimising an infinite MPS using VUMPS (solid green lines), conjugate gradient (dashed blue lines), gradient descent (dotted red lines), and the \enquote{switch} method that combines VUMPS and conjugate gradient (dash-dotted purple lines).
The benchmark model in question is a triangular lattice antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on a cylinder of width $6$.
Results are shown for MPS bond dimensions 1100 (darker colors) and 1900 (lighter colors).
SU(2) symmetry of the tensors is enforced.
VUMPS clearly performs the best at the start of the optimization, but its asymptotic convergence rate is roughly the same as that of gradient descent, whereas conjugate gradient can be seen to converge significantly faster.
A best-of-both-worlds solution is the switch method, which does 30 iterations of VUMPS at the start and then switches over to conjugate gradient.
L-BFGS produces results roughly comparable to those of conjugate gradient, but we do not show them here.
}%
\label{fig:mps_results}
\end{figure}
In Fig.~\ref{fig:mps_results} we show results comparing VUMPS with both gradient descent and conjugate gradient optimizations, with the above preconditioner.
VUMPS does clearly better in the beginning of the optimization, which starts from a randomly initialized MPS\@.
However, its convergence speed after the initial burst is similar to that of gradient descent, whereas conjugate gradient converges at a clearly faster rate.
This is to be expected, as VUMPS was inspired by imaginary time evolution using the TDVP (or thus, Riemannian gradient descent), and should become equivalent to it for small step sizes, i.e.\ when the algorithm is close to convergence.
Note that convergence in Fig.~\ref{fig:mps_results} is shown with respect to number of iterations, not actual running time.
VUMPS iterations, which internally use an iterative eigenvalue solver, take roughly 1.5 times as long as conjugate gradient iterations, thus increasing the gap between the two methods when plotting with respect to running time.
Finally, we have also included results for a method labeled \enquote{switch}, where we use VUMPS for the first few iterations, and then switch over to conjugate gradient, which outperforms both of the individual methods.
As mentioned, the Riemannian optimization methods explained here can be easily applied to a finite MPS as well.
Preliminary benchmarks indicate that for some models, gradient methods can outperform the DMRG algorithm~\cite{white_density_1992}.
The qualitative picture is similar to what we observe with infinite MPS, where variational methods like VUMPS and DMRG are superbly fast at making progress early in the optimization, but if the problem is difficult and a slow convergence sets in, the asymptotic convergence rate of preconditioned conjugate gradient or L-BFGS is often better.
We leave, however, a more detailed study of finite MPS optimization for future work.
\section{Conclusion}%
\label{sec:discussion}
The MERA and MPS results of Secs.~\ref{sec:mera} and~\ref{sec:mps} illustrate that Riemannian gradient-based optimization can be a competitive method for optimising tensor network ansätze.
Partial derivatives of the energy with respect to a given tensor give rise to tensor network diagrams that also appear in current algorithms such as the Evenbly-Vidal algorithm for MERA and the VUMPS algorithm for infinite MPS\@.
Implementing these methods is thus only a matter of computing an actual update direction from the computed gradient using the recipe of the chosen method (gradient descent, conjugate gradient or L-BFGS quasi-Newton), and replacing the update step with a retraction.
Vector transport is subsequently used to bring data from the previous iteration(s), such as former gradients, to the tangent spaces at the current iterate.
This approach is fully compatible with exploiting the sparse structure of tensors arising from symmetries, such as $\mathbb{Z}_2$, $\mathsf{U}_1$ or even non-abelian symmetries such as $\mathsf{SU}_2$.
The isometry condition defines how the tensor should be interpreted as a linear map, so that, when using symmetric tensors, they take a block diagonal form in a basis of fused representations, according to Schur's lemma.
The isometry condition itself, the projection onto the tangent space, the retraction, and the vector transport then all apply at the level of those individual diagonal blocks, and can easily be implemented as such.
Indeed, as mentioned, $\mathbb{Z}_2$ symmetry was used in the MERA results and $\mathsf{SU}_2$ symmetry in the MPS results presented above.
We have demonstrated the usefulness of gradient optimization for MPS and MERA, but there are other tensor network methods that also involve isometric tensors.
Notable cases we have not discussed are tree tensor networks, i.e.\ MERA without the disentanglers, and the tensor network renormalization algorithm~\cite{evenbly_tnr_2015} (TNR), which is closely related to MERA, and for which the usual optimization method is a variant of the Evenbly-Vidal algorithm.
We expect that in both these cases gradient methods could provide similar advantages as they do for MERA\@.
While we focused here on the application of Riemannian gradient-based optimization methods for tensor networks with isometry constraints, even their Euclidean counterparts have not received a great deal of attention as an alternative to the standard recipe of optimizing individual tensors in an alternating sequence using only local information (i.e.\ from the current iteration, not relying on the history of previous iterations).
While the latter can be expected to work extremely well when correlations are relatively short-ranged, there is no particular reason that gradient-based methods which optimize all tensors simultaneously could not replicate this behaviour in this regime, when provided with a suitable preconditioner.
However, gradient-based methods, in particular those that use a history of previous iterations, such as conjugate gradient and quasi-Newton algorithms, have the potential to also work in the regime with long-range and critical correlations.
These conditions typically imply very small eigenvalues in the Hessian, which is detrimental for methods that only use first order information of the current iterate.
A specific example includes situations of low particle density, for which specific multigrid algorithms have been explored~\cite{dolfi2012multigrid}.
It would be interesting to see if gradient-based algorithms would alleviate the problems that plague DMRG in this regime.
Related to this is the case of continuous MPS~\cite{verstraete2010continuous}, where the state is not even a linear or homogeneous function of the matrices containing the variational parameters and DMRG- or VUMPS-like algorithms are unavailable.
In those cases, gradient-based methods are the only alternative~\cite{ganahl2017continuous,tuybens2020variational}.
For all of these applications, a well-considered preconditioner is of paramount importance.
A suitably preconditioned gradient descent can easily outperform a conjugate gradient or quasi-Newton algorithm with ill-chosen parameterization.
In the case of MPS-specific methods such as DMRG or VUMPS, this is implicit in using what is known as the center-gauge.
For gradient methods, the same effect is accomplished by using the reduced density matrix which appears in the physical inner product of these tangent vectors in Hilbert space.
However, it is conceivable that there is plenty of room for improvement by using information of the actual Hamiltonian in constructing a preconditioner, i.e.\ by using its matrix elements with respect to the tangent vectors rather than those of the identity operator.
While the full Hessian needed for Newton's algorithm can be computed for the case of MPS~\cite{haegeman2013post}, this comes with a large cost and would likely be inefficient.
A single application of the Hessian to a given tangent vector requires to solve several non-hermitian linear problems with iterative solvers (e.g.\ the generalized minimal residual algorithm), in order to obtain cubic scaling in the bond dimension.
Hence, Newton's method would amount to three nested levels of iterative algorithms.
A local positive definite approximation of the Hessian which can be applied to a given vector efficiently and directly can be constructed, by (i) ignoring contributions from taking both partial derivatives in the ket or in the bra (somewhat similar to the Gauss-Newton or Levenberg–Marquardt algorithms), as well as (ii) discarding non-local contributions similar to how we ignored off-diagonal contributions in the inner product of MERA tangent vectors.
Such a preconditioner would still need an iterative solver (e.g.\ linear conjugate gradient) to be applied efficiently, but the improvement over the metric or preconditoner constructed here might be sufficiently significant to overcome this overhead.
Indeed, such a scheme is similar in spirit to truncated Newton algorithms~\cite{nash2000survey}, for which dedicated implementations of the inner conjugate gradient method exist, which detect the absence of positive definiteness and produce valid descent directions at every step.
A related strategy might be to directly use the solution of the local problem from DMRG, VUMPS or the Evenbly-Vidal algorithm as some kind of nonlinear preconditioner, as outlined in Ref.~\onlinecite{desterck_nonlinearly_2018}.
These ideas will be explored in a forthcoming paper.
As a final remark, we would like to point out that the techniques explored in this manuscript are relevant beyond the case of tensor network representations of ground states of many body systems.
Various tasks in quantum computation also rely on the classical optimization of the gates in a unitary circuit as a precursory step, and this particular classical task can likewise benefit from the Riemannian optimization methods on which we have reported.
\emph{Note:}
Near completion of this work, the preprint \enquote{Riemannian optimization and automatic differentiation for complex quantum architectures} by Luchnikov, Krechetov, and Filippov~\cite{luchnikov_riemannian_2020} appeared on the arXiv, which also proposes the use Riemannian optimization techniques for applications involving isometric tensor networks, quantum control and state tomography.
In particular, they also consider Stiefel manifolds to perform gradient optimization on a (finite) MERA, although with different gradient-based algorithms inspired by machine learning.
They do not consider the use of preconditioners nor applications to MPS, so that the two articles complement each other and pave the way for a bright future for Riemannian gradient-based optimization of tensor networks.
\section*{Acknowledgements}
We thank Glen Evenbly, Andrew Hallam, Laurens Vanderstraeten, and Frank Verstraete for useful discussions.
We also thank Miles Stoudenmire and an anonymous referee for helpful feedback.
\paragraph{Funding information}
This work has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreements No 715861 (ERQUAF) and 647905 (QUTE)), and from Research Foundation Flanders (FWO) via grant GOE1520N and via a postdoctoral fellowship of MH\@.
\begin{appendix}
\section{Evenbly-Vidal algorithm and its relation to gradient descent}%
\label{app:ev_steplimit}
This appendix summarizes the local update in the Evenbly-Vidal algorithm, illustrates the implicit notion of a step size it contains, and relates it to a preconditioned gradient descent in the limit of small step size.
For a Hamiltonian $H$ the MERA cost function is
\begin{equation}
C(W) = \bra{\text{MERA}(W)} H \ket{\text{MERA}(W)},
\end{equation}
where we have chosen to focus on a single isometry or disentangler $W$ only.
Note that no normalization is necessary, as the state is properly normalized due to the isometry conditions on the tensors.
Because $C$ is a homogeneous function of $W$, $C(W) \propto \Tr[W^{\dagger} D]$, where $D = 2 \partial_{W^\ast} C$ is the partial derivative that we used in the gradient optimization as well, also called the \emph{environment} of $W$.
Given this linear approximation of the cost function, where we assume $D$ to be independent of $W$ (which it in reality is not), the choice of $W$ that extremises it is $W = \pm Q$, where $D = QP$ is the polar decomposition, or as the original paper~\cite{evenbly_algorithms_2009} expresses this, $Q = UV^{\dagger}$ where $D = U S V^{\dagger}$ is the singular value decomposition.
While the sign of $W$ matters for the linearized cost function, it does not for $C$, as $C$ contains only even powers of $W$.
Although the assumption of $D$ being independent of $W$ is clearly false, the update that sets $W = Q$ still works as an iterative step, that in most situations increases $\|C\|$.
This step can then be repeated, and performed in turn for each of the different tensors that make up the MERA, to converge to a local maximum of $\| C \|$.
This algorithm has fixed points where $D = W P$, i.e.\ when $W$ equals the polar factor of $D$.
In that case, it can easily be verified that the gradient $G$ associated to $D$ by orthogonal projection onto the Stiefel tangent space vanishes, which confirms the necessary condition that this scheme converges to local extrema.
However, in order to ensure that maximizing $\| C \|$ amounts to minimizing $C$, the Hamiltonian is redefined as $H_\gamma = H - \gamma \mathbbm{1}$, with $\gamma$ sufficiently large, e.g.\ so as to make $H_\gamma$ negative definite.
In that case, the ground state approximation is indeed the state that maximizes $\| C \|$.
Although $\gamma$ was introduced to shift $H$ by a constant to make it sufficiently negative, it turns out to play the role of an inverse step size.
To see this, first note that
\begin{equation}
C_\gamma \, = \, \bra{\text{MERA}(W)} H_\gamma \ket{\text{MERA}(W)}
\, = \, C - \gamma \Tr[W^{\dagger} W \rho],
\end{equation}
where $\rho$ is the reduced density matrix at the top index or indices of $W$.
Consequently, $D_\gamma = D - \gamma W \rho$.
Now decompose $D$ as $D = W (A + S) + W_\perp B$, where $A$ and $S$ are the skew-hermitian and hermitian parts of $W^{\dagger} D$, and thus
\begin{equation}
D_\gamma \, = \, W(A + S - \gamma \rho) + W_\perp B
\, = \, W(S - \gamma \rho) + G.
\end{equation}
Here $G = W A + W_\perp B$ is the gradient, obtained by projecting $D$ onto the Stiefel tangent space at base point $W$.
As expected, the term in the Hamiltonian $H_\gamma$ that is proportional to the identity operator does not contribute to the Stiefel gradient.
At convergence, $A$ and $B$ will be zero and the role of $\gamma$ is clearly to shift the eigenvalues of $S$ so as to have a fixed sign.
Now consider a small but non-zero $G$, i.e.\ when the algorithm is close to convergence, and treat it as a perturbation to $W(S - \gamma \rho)$.
To see how the Evenbly-Vidal update behaves in this case, we need to understand perturbation theory of the polar decomposition.
If $X = QP$ is the polar decomposition of some arbitrary matrix $X$, and we perturb it as $X + \,\mathrm{d} X$, then an exercise that we omit here shows that
\begin{align}%
\label{eq:polar_perturbation}
X + \,\mathrm{d} X = (Q + \,\mathrm{d} Q) (P + \,\mathrm{d} P)
\end{align}
where $\,\mathrm{d} P$ is some hermitian matrix we do not care about, and
\begin{align}%
\label{eq:polar_perturbation_dQ}
&\,\mathrm{d} Q = Q A_X + Q_\perp B_X\\%
\text{where}\quad & A_X P + P A_X = Q^{\dagger} \,\mathrm{d} X - \,\mathrm{d} X^{\dagger} Q\\%
\text{and}\quad & B_X = Q_\perp^{\dagger} \,\mathrm{d} X P^{-1}.
\end{align}
Matching this up with our case,
\begin{align}%
\label{eq:polar_perturbation_ev}
& D_\gamma = \underbrace{W}_{=Q} \underbrace{(S - \gamma \rho)}_{=P} + \underbrace{W A + W_\perp B}_{= \,\mathrm{d} X},
\end{align}
we obtain
\begin{align}%
& \,\mathrm{d} Q = \,\mathrm{d} W = WA_X + W_\perp B_X\\%
\text{where}\quad & A_X (S - \gamma \rho) + (S - \gamma \rho) A_X = 2 A\\%
\text{and}\quad & B_X = B (S - \gamma \rho)^{-1}.
\end{align}
If we assume that $\gamma$ is sufficiently large so that $S$ is negligible compared to it, this becomes
\begin{align}%
& \,\mathrm{d} W = -\frac{1}{\gamma} (W \tilde{A}_X + W_\perp \tilde{B}_X)\\%
\text{where}\quad & \tilde{A}_X \rho + \rho \tilde{A}_X = 2 A\\%
\text{and}\quad & \tilde{B}_X = B \rho^{-1}.
\end{align}
Comparing this with Eqs.~\eqref{eq:preconditioner_solution} and~\eqref{eq:mera_diagonal_inner_product}, we can identify this with $\,\mathrm{d} W = -\frac{1}{\gamma}\tilde{G}$, where $\gamma^{-1}$ thus plays the role of a step size in the Evenbly-Vidal algorithm, and $\tilde{G}$ is the gradient preconditioned with the same metric that was used in our gradient optimization in Sec.~\ref{sec:mera}.
Indeed, this observation further motivates our specific choice of preconditioner.
Note that in practice, the Evenbly-Vidal algorithm might not satisfy the assumption of large $\gamma$.
The analysis above remains valid up to the final assumption, and might thus give an indication of a better preconditioner for MERA optimization that includes information from the Hamiltonian, yet can still be implemented efficiently.
Instead of $\rho$, we could use $\rho - \gamma^{-1} S$, with $S$ the symmetric part of the $W^{\dagger} D$ and $\gamma$ chosen sufficiently big to ensure positive definiteness.
We leave this proposal for future work.
\section{Efficient computation with the scale invariant layer of a MERA}%
\label{app:mera_scale_invariant_layer}
In the optimization of an infinite MERA, the scale invariant layers at the top need to be treated somewhat differently from the rest.
To discuss this, we first need to lay down some notation.
We denote the local Hamiltonian term ascended to the lowest scale invariant layer by $h$.
We often think of $h$ not as an operator $V \to V$, but as a vector in $V \otimes \bar{V}$, and denote this vector $\bra{h}$.
Similarly, we denote the local scale invariant density matrix $\rho$, and its vectorized version by $\ket{\rho}$.
Finally, we call $A$ the ascending superoperator, thought of as a linear operator $V \otimes \bar{V} \to V \otimes \bar{V}$.
Right-multiplying a vector like $\bra{h}$ by $A$ corresponds to raising it by a layer, and left-multiplying a vector like $\ket{\rho}$ by $A$ corresponds to lowering it by a layer.
There are two problems that need to be solved for $A$ at every iteration of the optimization.
First, to find $\ket{\rho}$, we must solve the eigenvalue equation $A \ket{\rho} = \ket{\rho}$.
Second, when computing the gradient, we need to take the partial derivative $\partial_v \Tr[h \rho] = \partial_v \braket{h}{\rho}$, where $v$ is either the disentangler or the isometry of the scale invariant layer.
Expanding the dependence of $\ket{\rho}$, through $A$, on $v$, one finds
\begin{equation}
\label{eq:scale_invariant_partial}
\partial_v \braket{h}{\rho} = \sum_{i=0}^\infty \bra{h} A^i (\partial_v A) \ket{\rho}.
\end{equation}
To evaluate this we need to find the value of the series $\sum_{i=0}^\infty \bra{h} A^i$.
At face-value this diverges if $\bra{h}$ has overlap with $\bra{\mathbbm{1}}$ (the vectorized version of the identity matrix), since $\bra{\mathbbm{1}} A = \bra{\mathbbm{1}}$.
However, it turns out that any contributions to the partial derivative that are of the form $\bra{\mathbbm{1}}(\partial_v A)\ket{\rho}$ are orthogonal to the Grassmann/Stiefel tangent plane and thus projected out, because they correspond to shifting the cost function by a constant.
Hence we can define $A' = A - \ket{\rho}\bra{\mathbbm{1}}$ and replace the above series by $\sum_{i=0}^\infty \bra{h} A'^i$, which converges like a geometric series, since all eigenvalues of $A'$ are smaller than $1$ in modulus.
Indeed, this can similarly be understood as regular perturbation theory for the eigenvector $\rho$ of the (non-hermitian) operator $A$, whose eigenvalue $1$ does not change under the perturbation.
All of the above is well-known from the original MERA papers~\cite{vidal_class_2008,evenbly_algorithms_2009}, and comes down to solving two relatively simple linear algebra problems.
The reason this is worth mentioning, is that multiplication by $A$ is the leading order cost of the whole MERA optimization, and thus as few such operations should be done as possible.
With the traditional Evenbly-Vidal optimization, approximations have often been used, such as approximating $\bra{h}$ at iteration $i$ as $\bra{h_i} = \bra{h_{i-1}} + \bra{h_{i-1}} A'$, to save computation time~\cite{evenbly_quantum_2011}.
With gradient algorithms like the ones presented here, these kinds of approximations may not be feasible, since the gradient needs to be computed to good accuracy at every step to be able to perform a line search.
We have found, however, that using Krylov subspace methods for the eigenvalue problem $A \ket{\rho} = \ket{\rho}$ and the linear problem of solving $\bra{h}\sum_{i=0}^\infty A'^i$ from $(\bra{h} \sum_{i=0}^\infty A'^i) (\mathbbm{1} - A) = \bra{h}$, with a small Krylov space dimension (e.g.\ $4$) and the solution from the previous iteration as the initial guess, leads to accurate results usually with very few applications of $A$.
This helps make the MERA gradient optimization methods competitive with the Evenbly-Vidal algorithm.
\end{appendix}
|
2,877,628,089,485 | arxiv | \section{Conclusions}
\label{sec:conc}
We have formalized and studied in this paper the problem of
``query-by-explanation", where queries are inferred from example
output tuples and their explanations. We have proposed a generic model, based on the framework of semiring provenance, allowing explanations of varying level of detail and granularity. We have further presented efficient algorithms that infer conjunctive queries from explanations in one of multiple supported semirings. We have theoretically analysed and experimentally demonstrated the effectiveness of the approach in inferring highly complex queries based on a small number of examples. Intriguing directions for future study include further expressive query languages and additional provenance models including in particular the lineage model.
\section{Experiments}
\label{sec:exp}
We have performed an experimental study whose goals were to assess: (1) can users provide meaningful explanations for their examples? (2) once (a small number of) examples and explanations are in place, how effective is the system in inferring queries? (3) how efficient are the algorithms in terms of execution time?
To this end, we have performed two kinds of experiments. The first is a user study based on the IMDB movies database (a part of its schema is depicted in Fig. \ref{scehma}); the second is based on the actual output and provenance of the benchmark queries in \cite{joinQueries}. All experiments were performed on Windows 8, 64-bit, with 8GB of RAM
and Intel Core Duo i7 2.59 GHz processor. We next describe both experiments and their results.
\subsection{User Study}
\label{sec:userstudy}
We have examined the usefulness of the system to non-expert users. To this end, we have loaded the IMDB database to {\tt QPlain}\ (see a partial schema in Figure \ref{scehma}) and have presented each of the tasks in Table \ref{tasks} to 15 users. We have also allowed them to freely choose tasks of the likings, resulting in a total of 120 performed tasks. The intended queries are presented in Table \ref{queries}, where the relation {\em atm} stands for ActorsToMovies.
\begin{figure}[]
\begin{center}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip = true, width=3in]{IMDB_Schema_cropped}
\vspace{-2mm} \caption{Partial IMDB schema} \vspace{-3mm}
\label{scehma}
\end{center}
\end{figure}
\begin{table}[!htb]
\centering \scriptsize
\begin{tabular}{l | p{7.7cm} }
\hline & {\bf Task} \\
\hline
1. & Find all actresses \\
2. & Find all movies that came out in 1994 \\
3. & Find all of Al Pacino's movies \\
4. & Find the entire cast of Pulp Fiction \\
5. & Find all documentary films (by having character ``Himself'') \\
6. & Find the actors who played in all the 3 movies of The Lord Of The Rings \\
7. & Choose an actor and two movies he played in, and find all actors that played with him in these two movies \\
%
%
%
\hline
\end{tabular}
\caption{User Tasks}\label{tasks}
\vspace{-2mm}
\end{table}
\begin{table}[!htb]
\centering \scriptsize
\begin{tabular}{l | p{7.7cm} }
\hline & {\bf Queries} \\
\hline
1. & ans(t) :- actors(mid, t, 'F') \\
2. & ans(t) :- movies(mid, t, '1994') \\
3. & ans(t) :- atm(mid, aid, c), actors(aid, 'Pacino, Al', 'M'), movies(mid, t, y) \\
4. & ans(n) :- atm(mid, aid, c), movies(mid, 'Pulp Fiction', '1994'), actors(aid, n, s) \\
5. & ans(t) :- atm(mid, aid, 'Himself'), actors(aid, n, 'M'), movies(mid, t, y) \\
6. & ans(n) :- atm(mid1, aid, c), atm(mid2, aid, c), atm(mid3, aid, c), actors(aid, n, s), movies(mid1, 'The Fellowship of the Ring', '2001'), movies(mid2, 'The Two Towers', '2002'), movies(mid3, 'The Return of the King', '2003') \\
\hline
\end{tabular}
\caption{Queries (No. 7 depends on user choices, omitted)}\label{queries}
\vspace{-4mm}
\end{table}
In 111 out of the 120 cases, including all cases of freely chosen tasks, users were successful in specifying examples and explanations, and the interface of {\tt QPlain}\ was highly helpful in that.
\begin{example}
Consider task 6 in Table \ref{tasks}. The examples are actor names; for every chosen actor name, {\tt QPlain}\ has proposed as explanations only the tuples that included this actor name as one of its values. In particular, the tuples of $ActorsToMovies$ corresponding to movies in which the actor has played were proposed. Knowing the underlying task, it was natural for users to choose the relevant movies, namely the three ``Lord Of The Rings" movies. Once they did that, the tuples of these movies (and not of any other movie) in the $Movies$ relation has appeared as proposals, allowing their easy selection. A similar case is that of task 3: once a movie is specified as an example, the system proposes its actors as possible explanations. The choice of Al Pacino as an explanation reveals the underlying intention.
\end{example}
In turn, explanations were crucial for allowing the system to focus on the intended query, even with very few examples (in all cases users provided at most 4 examples).
\begin{example}
Re-consider task 6, and now assume that explanations are unavailable. There would be no way of distinguishing the underlying query from, for instance, a much simpler one that looks for the actors of a {\em single} Lord of the rings. More generally, join conditions (or even the fact that a join took place) are in many cases only apparent through the explanations: another example is task 3, where examples are simply movie names that typically have many other characteristics in common in addition to having Al Pacino acting in them. In addition to revealing join conditions, explanations help to learn the query constants: for instance, in task 4, without explanations, the resulting query could have returned all actors who played in a movie with Quentin Tarantino.
\end{example}
Out of the 9 cases where users failed to provide examples and explanations, 5 involved difficulties in providing any example matching the task, and 4 involved errors in providing explanations. Out of the remaining 111 cases, in 98 cases {\tt QPlain}\ has inferred the actual underlying query, and in the remaining 13 cases, it has inferred a ``more specific" query (i.e. with extra constants). This for instance happened when all actors given as examples were males. We next further analyze the accuracy of {\tt QPlain}.
\subsection{Accuracy}
To further examine the effectiveness of the
approach, we have used the system to ``reverse
engineer" multiple complex queries. The queries are Q1--Q6 from \cite{joinQueries} as well as (modified, to drop aggregation and arithmetics) the TPC-H queries TQ2--TQ5, TQ8 and TQ10. The queries have 2--8 atoms, 18--60 variables, and multiple instances of self-joins (We show Q6 for illustration in Figure \ref{fig:query}; the reader is referred to \cite{joinQueries,tpc} for the other queries).
We have evaluated each query using a proprietary provenance-aware
query engine \cite{icde15}, and have then sampled random fragments (of a given size that we vary) of the output database and its provenance (we have tried both $\mathbb{N}[X]$\ and Why(X)), feeding it to our system. In each experiment we have gradually added random
examples until our algorithm has retrieved the original query. This
was repeated 3 times. We report (1) the {\em worst-case} (as
observed in the 3 executions) number of examples needed until the
original query is inferred, and (2) for fewer examples (i.e. before
convergence to the actual query), the differences between the
inferred queries and the actual one (we report the differences
observed in the ``worst-case" run of the experiment).
\begin{table*}[!htb]
\centering\small
\begin{tabular}{| M{1.5cm} | M{3cm} | p{12cm} |}
\hline \textbf{Query} & \textbf{Worst-case number of examples to learn the original query} & \textbf{Difference between original and inferred queries for fewer examples}\\
\hline Q1 (TQ3) & 14 & \noindent\parbox[c]{\hsize}{Inferred Query includes an extra join on a ``status" attribute of two relations. Only 2--3 values are possible for this attribute, and equality often holds.}\\%or 'P' for orderstatus {\em orders} and {\em lineitem} atoms in attributes orderstatus and linestatus since that the\\ two attr. get values that are either 'F', 'O' many times \daniel{Rephrase?}
\hline Q2 & 2 & \noindent\parbox[c]{\hsize}{}\\
\hline Q3 & 5 & \noindent\parbox[c]{\hsize}{For 2 examples, the inferred query contained an extra constant.
For 3 and 4 examples, it included an extra join.}\\% \daniel{actually we never discuss adding a constant! Need to.}
\hline Q4 & 19 & \noindent\parbox[c]{\hsize}{For 2 examples, the inferred query included an extra constant. For 3--18, it included an extra join on a highly skewed ``status" attribute.}\\
\hline Q5 & 11 & \noindent\parbox[c]{\hsize}{The inferred query included an extra join on a ``name" attribute.} \\
\hline Q6 & 3 & \noindent\parbox[c]{\hsize}{The inferred query included an extra constant.}\\
\hline TQ4 & 234 & \noindent\parbox[c]{\hsize}{The inferred query included an extra join on ``orderstatus" and ``linestatus" attributes of two relations (they have two possible values). One of the original join conditions has lead to
occurrence of the same value in these attributes in the vast majority of joined tuples.} \\%Extra join between 9
\hline TQ10 & 4 & \noindent\parbox[c]{\hsize}{The inferred query contained an extra constant.} \\
\hline TQ2 & 3 & \noindent\parbox[c]{\hsize}{The inferred query contained an extra constant.}\\
\hline TQ5 & 3 & \noindent\parbox[c]{\hsize}{The inferred query contained an extra constant.}\\
\hline TQ8 & 18 & \noindent\parbox[c]{\hsize}{For 2 examples, the inferred query contained an extra constant. For 4-17 examples, the query had an extra join between a ``status" attribute of two relations.} \\
\hline
\end{tabular}
\caption{Results for the TPC-H query set and the queries from \protect\cite{joinQueries} with $\mathbb{N}[X]$\ provenance}\label{resultsNx
\end{table*}
\begin{table*}[!htb]
\centering\small
\begin{tabular}{| M{1.5cm} | M{3cm} | p{12cm} |
\hline \textbf{Query} & \textbf{Worst-case number of examples to learn the original query} & \textbf{Difference between original and inferred queries for fewer examples}\\
\hline Q2 & 2 & \\
\hline Q3 & 5 & The inferred query for 2--4 examples did not include self-joins. \\%\begin{tabular}{@{}l@{}}\end{tabular}
\hline Q4 & 19 & \noindent\parbox[c]{\hsize}{For 2--3 examples, the inferred query did not include self-joins. For 4--18 examples, the query had an extra join on a ``status" attribute.}\\%\begin{tabular}{@{}l@{}} \end{tabular}
\hline Q5 & 13 & The inferred query for 2--12 examples did not include self-joins. \\%\begin{tabular}{@{}l@{}} \end{tabular}
\hline Q6 & 3 & The inferred query included an extra constant.\\
\hline TQ8 & 18 & \noindent\parbox[c]{\hsize}{For 2--3 examples, the inferred query contained an extra constant. For 4-17 the query had an extra join between a ``status" attribute of two relations.} \\%{\em nation} and {\em region} atoms \begin{tabular}{@{}l@{}}\end{tabular}
\hline
\end{tabular}
\caption{Results for the TPC-H query set and the queries from \protect\cite{joinQueries} containing self-joins with $Why(X)$ provenance
\label{resultsTrio}
\end{table*}
The results are reported in Table \ref{resultsNx}. Observe that for some queries
the convergence is immediate, and achieved when viewing only 2--5
examples. For other queries, more examples are needed, but with one
exception (TQ4), we converge to the original query after viewing at
most 19 tuples for the different queries. For TQ4 only a very small
fraction of the output tuples reveal that an extra join should not
have appeared, and so we need one of these tuples to appear in the
sample. Furthermore, even for smaller sets of examples, the inferred
query was not ``far" from the actual query. The most commonly
observed difference involved extra constants occurring in the
inferred query (this has typically happened for a small number of
examples, where a constant has co-occurred by chance). Another type
of error was an extra join in the inferred query; this happened
often when two relations involved in the query had a binary or
trinary attribute (such as the ``status" attribute occurring in
multiple variants in TPC-H relations), which is furthermore skewed
(for instance, when other join conditions almost always imply
equality of the relevant attributes). We have also measured the precision and recall of the
output of the inferred query w.r.t. that of the original one. Obviously, when the
original query was obtained, the precision and recall were 100\%. Even when presented with fewer examples, in almost all
cases already with 5 examples, the precision was 100\% and
the recall was above 90\%. The only exception was Q5 with 75\% recall
for 5 examples.
\begin{figure}
\begin{center}
\scriptsize{
\begin{tabular}{|l|}
\hline
\verb"ans(a, b) :- supplier(c, a, add, k, p, d, c1), "\\
\verb"partsupp(h, c, v, j, c2), part(h, i, z, q, t, s, e, rp2, c3), "\\
\verb"partsupp(h, o, x, n, c4), supplier(o, b, y, w, p2, d2, c5), "\\
\verb"nation(k, na1, r, c6), region(r, u, c7), nation(w, na2, r, c8)"\\
\hline
\end{tabular}
} \vspace{-5mm}
\end{center}
\caption{$Q6$} \label{fig:query}
\vspace{-5mm}
\end{figure}
The results for $Why(X)$ are
shown in Table \ref{resultsTrio}. For queries with no self-join, the observed results were naturally the same as
in the $$\mathbb{N}[X]$$ case; we thus report the results only for queries that include
self-joins (some of the queries included multiple self-joins). When presented with a very small number of examples, our algorithm was not always able to detect the
self-joins (see comments in Table 7); but the overall number of
examples required for convergence has only marginally increased with
respect to the $$\mathbb{N}[X]$$ case.
\begin{figure}
\hspace*{-0.8cm}
\centering
\begin{subfigure}[b]{0.26\textwidth}
\centering
\includegraphics[trim=0cm 2cm 0cm 0cm, width=2.0in]{paperQueriesTime2.pdf}
\caption{Queries from \cite{joinQueries}}
\label{timesPaperQueries}
\end{subfigure}%
\begin{subfigure}[b]{0.26\textwidth}
\centering
\includegraphics[trim=0cm 2cm 0cm 0cm, width=2.0in]{TCP-H_QueriesTime2.pdf}
\caption{TPC-H Queries}
\label{timesTCP-H}
\end{subfigure}
\caption{Time of computation in milliseconds as a function of number of examples for $$\mathbb{N}[X]$$} \vspace{-4mm}
\label{rowChoosing}
\end{figure}
\begin{figure}
\hspace*{-0.8cm}
\centering
\begin{subfigure}[b]{0.26\textwidth}
\centering
\includegraphics[trim=0cm 2cm 0cm 0cm, width=2.0in]{paperQueriesTrioTime2.pdf}
\caption{Queries from \cite{joinQueries}}
\label{timesPaperQueriesWhy}
\end{subfigure}%
\begin{subfigure}[b]{0.26\textwidth}
\centering
\includegraphics[trim=0cm 2cm 0cm 0cm, width=2.0in]{TCP-H_QueriesTrioTime2.pdf}
\caption{TPC-H Queries}
\label{timesTCP-HWhy}
\end{subfigure}
\caption{Time of computation in milliseconds as a function of number of examples for $Why(X)$} \vspace{-7mm}
\end{figure}
\subsection{Scalability}
Last, we have examined the scalability of our
solution. To this end, we have increased the number of examples up to 6000, which is well beyond a realistic number of
user-provided examples. The results for $$\mathbb{N}[X]$$ provenance
and Q1--Q6 are presented in Figure \ref{timesPaperQueries}. The
results exhibit good scalability: the computation time for 6000
examples was 1.3 seconds for Q1 (TQ3), 0.4 seconds for Q2, 2.4
seconds for Q3, 2.3 seconds for Q4 and 1.4 and 2 seconds for Q5 and
Q6 respectively. The performance for the TPC-H queries (Figure
\ref{timesTCP-H}) was similarly scalable: for 6000 examples, the
computation time of TQ2 and TQ10 (which are the queries with the maximum number of head attributes: 8 and 7, resp.) was 1.5 and 1.4 seconds
respectively. The number of examples for queries TQ4, TQ5 and TQ8
was limited due to the queries output size: 2500, 15 and
1000 respectively. The running times for these number of
examples were 0.2, 0.2 and 1.7 seconds respectively.
Next, we have repeated the experiment using $Why(X)$ provenance, and the results appear in Figure 12. In general, the computation time was still fast, and only slightly
slower than the $$\mathbb{N}[X]$$ case; this is consistent with our theoretical
complexity analysis.
\begin{figure}
\vspace{-6mm}
\centering
\includegraphics[trim=0cm 2cm 0cm 0cm, width=2in]{matchingQ6.pdf}
\caption{Time of computation in milliseconds of Q6 with a varying number of consistent matchings for $\mathbb{N}[X]$}
\label{matchingQ6}
\vspace{-5mm}
\end{figure}
\paragraph*{Effect of Tuples Choice}
Recall that Algorithm \ref{algo:Efficient} starts by finding queries that are consistent with two example tuples. We have described a heuristic that chooses the two tuples with
the least number of shared values. The effect of this optimization
is demonstrated in Figure \ref{matchingQ6} for Q6: our choice leads
to a single matching in the graph, as oppose to a random choice of tuples that has led to 4 matchings. The average
overhead of making such a random choice,
instead of using our optimization, was 56\%.
\section{Implementation}\label{sec:imp}
\begin{figure}[]
\begin{center}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip = true, width=3.6in]{sysArch.png
\vspace{-3mm} \caption{System architecture} \vspace{-1mm}
\label{systemArchitecture}
\end{center}
\end{figure}
We have implemented our algorithms in an end-to-end system prototype called {\tt QPlain}\ (to be demonstrated in the upcoming ICDE \cite{icde16}), implemented in JAVA with JAVAFX GUI and MS SQL server as its
underlying database management system.
The system architecture is depicted in Figure \ref{systemArchitecture}: users load and view an input database, and then provide examples of output tuples (see Fig. \ref{input}). The users further provide explanations through a dedicated interface (see Fig. \ref{exp}). To form each explanation, users simply drag-and-drop tuples from the input database, that they intuitively view as the cause for the example tuple. Internally, the annotations of tuples chosen for each explanation are combined to form a monomial in the provenance expression.
\begin{figure}[]
\centering
\includegraphics[scale=0.3, trim={0 0 0 1.4cm}]{inputScreenCameraReady_cropped.pdf
\caption{Examples screen}
\vspace{-4mm}
\label{input}
\end{figure}
Importantly, the system assists users in choosing the tuples to form an explanations. To this end, we observe that (unless the intended query is degenerate in that it has only constants in the head) each monomial (explanation) must contain at least one annotation of a tuple that shares at least one value with the example tuple. Consequently, we first only ask the user to choose an explanation tuple out of the input tuples that share at least one value with her example tuple. For instance, in our running example, the user is first asked to choose either the first or last crossing point of the trip between the end-points she has given (see more examples in Section \ref{sec:userstudy}). Once a first explanation tuple is given, the proposals for the following one include again tuples that share values with the example tuple, but now also tuples that share values with the given explanation tuple (this corresponds to a join condition, e.g. a second crossing point in our example), and so on.
\begin{figure}[]
\centering
\includegraphics[height=1.8in]{ExplanationScreenCameraReady_cropped.pdf
\caption{Explanation screen}
\vspace{-2mm}
\label{exp}
\end{figure}
Once the user explanations are in place, the system ``compiles" it into a provenance expression of the corresponding semring. The choice of semiring is done automatically; the system assumes the ``simplest" semiring that can accommodate the user-provided explanations: if there are no repetitions then $Why(X)$ is assumed; if there are repetitions of explanations but not of tuples within an explanation then $Trio(X)$ is assumed, etc.
Finally, the tuples and explanations (in their ``compiled" form) are directed to either of the algorithms presented in the previous sections (according to the identified semiring). The algorithm outputs a query and this query is evaluated with respect to the underlying database, showing the user the full output set, and allowing her to add examples and re-run the inference process as needed.
\section{Introduction}
It has long been acknowledged that writing database queries in a
formal language is a cumbersome task for the non-specialist.
Different solutions have been proposed to assist users in this
respect; a prominent approach (see e.g. \cite{qbo,joinQueries,Shen})
allows users to provide examples of output tuples, based on which
the intended query is automatically inferred. This approach can be
highly effective if the examples provided by the user are plenty and
representative. But coming up with such a set of examples is highly
non-trivial, and unless this is the case, the system would
be unable to distinguish the true intention of the user
from other qualifying queries.
As a simple illustration, consider a user planning to purchase
airline tickets for a trip. She has rather specific requirements:
the trip should include five countries in South America, visiting
each for a week and staying in Bolivia in the third week and in
Argentina in the fourth, in time for meetings she has scheduled.
After viewing a list of border crossings (see Table \ref{relR1}),
she concludes that Argentina and Brazil would serve as good
end-points for the trip, and so would Peru and Paraguay. Since
airfare to these particular destinations is quite expensive, she is
interested in viewing additional recommendations. However, based
only on these two examples of output tuples, there are many
inherently different queries that yield them as a subset of their
results, and there is no reasonable way to distinguish between these
queries. For instance, the trivial query copying the content of
Table \ref{relR1} also yields these two tuples.
Intuitively, if users would provide some form of ``explanations"
for their examples, it could guide the system in identifying
the actual intended query. The explanations should on the one hand
be simple enough so that non-experts are able to specify them (and
in particular their specification should be much easier than query
formulation), and on the other hand be informative enough to
allow inference of the underlying query. Continuing our running
example, an explanation for a pair of end-points involves a
description of actual trips that the user has in mind, and are
compatible with the example end-points. This would in turn limit the
queries of interest to those that not only include the example
output, but rather do so based on criteria that are compatible with
the explanation.
We propose in this paper a novel framework for learning queries
from examples {\em and explanations for these examples}, a problem that we
refer to as {\em query-by-explanation}. The backend of the framework is based on a formal model for explanations,
namely that of {\em provenance semirings} \cite{GKT-pods07}, a formal problem statement that intuitively involves ``reverse-engineering" queries from their provenance, and efficient algorithms for the problem in multiple variants. Importantly, since users can not be expected to understand complex notions of provenance, the framework includes an intuitive Graphical User Interface through which users specify explanations, by essentially dragging-and-dropping relevant input tuples (the system further assists them in this task). The provided explanations are automatically compiled to formal provenance and fed to the algorithms. The effectiveness of the solution is shown through
extensive experiments, including a user study.
Our solution comprises of the following
components.
\paragraph*{A Formal Model for Explanations (Section \ref{sec:prelim})} We first need a formal notion of
explanations to be attached to examples. In this respect, we note
that multiple lines of work have focused on the ``reverse" problem
of the one we consider here, namely that of {\em explaining query
results}. The basic idea in all of these works is to associate with each
output tuple $t$ some description of the input tuples that ``cause" $t$ to appear in the output, i.e. they are used by the
query in a derivation that yields $t$. Different models vary in the granularity of
explanations. For instance, the {\em why-provenance} \cite{why} of
$t$ is a {\em set of sets} of input tuples, where each set includes
the tuples that have been used in a single derivation. The
provenance polynomials model ($\mathbb{N}[X]$\ in \cite{GKT-pods07}) essentially
extends why-provenance to account for multiplicity: each monomial of
a provenance polynomial includes the {\em annotations} (intuitively
identifiers, for our purpose of use) of tuples participating in a
single derivation. Exponents are used to capture that a tuple was
used multiple times, and coefficients capture multiple derivations
based on the same set of variables. Importantly, \cite{Greenicdt09}
has shown that these and other models may be captured through the
{\em provenance semirings} model \cite{GKT-pods07}, via different
choices of semirings. We use here the provenance semirings model as
the underlying model for explanations, and examine the effect of
different semiring choices.
\paragraph*{Query-By-Explanation (Section \ref{sec:model})}
We then formally define the novel problem of learning queries from
examples and their explanations (termed query-by-explanation).
Examples are simply output tuples, and explanations are, formally,
instances of provenance attached to them. We formally define what it
means for a query to be consistent with examples and their
explanations. Intuitively, we want a query that, when evaluated with
respect to the input database does not only yield the specified
example tuples, but also its derivation of these tuples is
consistent with the prescription made by the explanation. This is formalized by
leveraging the inclusion property \cite{Greenicdt09} of relations
annotated with elements of {\em ordered semirings}. Basing our
formal construction on these foundations allows for a ``clean",
generic, problem definition.
We then study the query-by-explanation problem, for Conjunctive Queries (CQs; this is a quite standard choice in this context, see e.g. \cite{Shen,joinQueries,Psallidas}) and for
different semirings used for explanations. As we discuss below (Section \ref{sec:imp}), users do not directly specify explanations in a any semiring model (and in fact do not even need to be aware of these models),
but rather only need to understand the intuitive notion of ``cause", which naturally corresponds to the reasons they had in mind when choosing examples.
Still, the user specification can be of varying levels of details, which we show to correspond to different choices of semirings.
\paragraph*{Learning from Detailed Explanations (Section \ref{sec:NX})} We start by assuming that a detailed form of explanation is given; formally, here we capture explanations as {\em provenance polynomials},
elements of the $$\mathbb{N}[X]$$ semiring. In our example it means that for
each trip the system is aware of all border crossings, including
multiple occurrence of the same crossing, but not of the order in
which they take place (order is abstracted away in
semiring provenance). Also note that not all explanations (trips, in our example) need
to be specified. Technically, a key to generating a consistent query
in this case is to ``align" provenance annotations
appearing in different monomials, eventually mapping them to
constructed query atoms. Indeed, we show that given a permutation
over all annotations appearing in each explanation (formally a
monomial in the provenance polynomial), we can efficiently construct
a corresponding atom for each location of the permutation, or
declare that none exists. However, an algorithm that exhaustively
traverses all such permutations would be prohibitively inefficient
(EXPTIME both in the monomials size and in the number of
examples). Instead, we design an efficient algorithm that is careful
to traverse combinations whose number is only exponential in the
arity of the output relation, and polynomial in the number
of examples and in the provenance size. We further adapt it to find inclusion-minimal queries.
\paragraph*{Relaxing the Level of Detail (Section \ref{sec:trio})} With $$\mathbb{N}[X]$$ provenance, we have assumed complete knowledge of some (but maybe not all) derivations that support the example tuple. There may still be a mismatch between the intuition of explanation that the
users have in mind and that of derivations expressed in the $$\mathbb{N}[X]$$
semiring. This mismatch is reflected in the existence of {\em
multiplicities} in $$\mathbb{N}[X]$$, i.e. multiple occurrences of the same tuple in an
explanation as well as the same explanation occurring multiple
times. Such multiplicities are usually due to the technical
operation of the query, and then they may not be specified in the
user-provided explanation. To this end, we note that the model of
{\em why-provenance} \cite{why} (captured by the $Why(X)$ semiring of \cite{Greenicdt09}) is oblivious to such multiplicities. We show that learning queries from such explanations is more
cumbersome: there may be infinitely many queries that lead to the
same $Why(X)$ provenance (with different multiplicities that are abstracted away). To this end, we prove a small world property, namely
that if a consistent query exists, then there exists such query of
size bounded by some parameters of the input. The bound by itself
does not suffice for an efficient algorithm (trying all $$\mathbb{N}[X]$$
expressions of sizes up to the bound would be inefficient), but we
leverage it in devising an efficient algorithm for the $Why(X)$
case. We then complete the picture by
showing that our solutions may be adapted to explanations specified in other provenance models for which a semiring-based interpretation was given in
\cite{Greenicdt09}.
\paragraph*{Implementation Details (Section \ref{sec:imp})} We have implemented our solution in a prototype
system called {\tt QPlain}, that allows users to specify examples and
explanations through an intuitive GUI. Users formulate explanations
by simply dragging-and-dropping input tuples that serve as support
for their examples. Importantly, our GUI assists users in
identifying those tuples by limiting attention to ``relevant" such
tuples: first, only input tuples that have some values in common with
the provided example tuple are proposed (intuitively these will
correspond to query atoms that contribute to the head); in
subsequent steps, the system will also propose tuples that share common values with
tuples already chosen to appear in the explanation (intuitively
these will e.g. correspond to atoms used as join conditions). The
explanations are automatically compiled to expressions in the
appropriate semiring -- intuitively the least expressive of the
supported semirings whose expressive power suffices to capture the
supplied explanations --- without requiring any user awareness of
semiring model. Finally, the relevant algorithm is invoked to
return a query of interest.
\paragraph*{Experiments (Section \ref{sec:exp})} We have conducted an extensive
experimental study of our system, to asses the feasibility of
forming explanations, the quality of inferred queries (even when very few examples and explanations are provided), and the
computational efficiency of our algorithms. To this end, we have
first conducted a user study, requesting users to provide examples
and explanations both for some complex pre-defined tasks as well as
for tasks of their choice, with respect to the IMDB movies dataset. Users were successful in forming examples and
explanations in the vast majority of the cases, and a small number of examples (up to 4) was typically sufficient for the system to infer the underlying query. Then, we have
further studied the system's ability to ``reverse-engineer" TPC-H
queries as well as highly complex join queries presented as baseline
in \cite{joinQueries}. The queries include multiple joins and
self-joins, with up to 8 atoms and up to 60 variables. In the vast majority of the cases, our algorithms converged to the
underlying query after having viewed only a small number of
examples. Last, further experiments indicate the computational efficiency of
our algorithms.
We survey related work in Section \ref{sec:related} and conclude in
Section \ref{sec:conc}.
\section{Additional Semirings}
\label{sec:more}
So far we have considered the case where full derivations (but
perhaps not all) are given ($$\mathbb{N}[X]$$), and the case where the use of
the same tuple multiple times in a derivation is masked (Trio).
Explaining tuples through the notion of derivations, with every
derivation corresponding to an alternative explanation, is very
natural in cases such as the one of our running example -- where
each explanation is a desired trip. However, there are cases where
the technical notion of derivation does not have such interpretation
that is clear to the non-expert. In these cases, it would be easier
for users to ``explain" an output tuple by detailing a set of
relevant input tuples -- i.e. to provide the tuple {\em lineage}
\cite{lin} as an explanation.
\begin{example}
\daniel{TBD: exemplify}
\end{example}
Interestingly, the freedom in ``breaking" the set of annotations
into monomials allows to show an even better bound for the ``small
world" property, compared to the one shown for Trio:
\begin{proposition}
\label{prop:triobound} For any {Lin(X)} example, if there exists a
consistent query then there exists a consistent query with $k+r$
atoms or less, where $r$ is the number of relations in $D^{in}$.
\end{proposition}
\begin{proof}
\daniel{TBD}
\end{proof}
\IncMargin{1em}
\begin{algorithm}
\SetKwFunction{LinAlgo}{ComputeQueryLin}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\LinesNumbered
\Input{A Lin example Ex}
\Output{A consistent query $q$ or an answer that none exists} \BlankLine
Treat each lineage expression as a single monomial to obtain $\{(t_0,M_0),...,(t_n,M_n)\}$ \;
$Q \gets (t_0,M_0)$ \;
\ForEach{$1 \leq i\leq n$} {
$(V,E) \gets BuildGraph((t_i,M_i),Q)$ \;
$match \gets false$ \;
\ForEach {{\em sub-graph} $E' \subseteq E \text{ s.t. } |E'| \leq k \text{~~and~~} \cup_{e \in E'} label(e) = \{1, \ldots, k\}$}
{
$match \gets true$ \;
\ForEach{conjunct $C \in Q$}
{
Let $t'_1,...,t'_p$ be the tuples whose annotations are
connected by edges in $E'$ to $C$ \;
$Q \gets Q \cup split(C) - \{C\}$ \;
}
\ForEach{{\em relation name} $R$ of a tuple corresponding to an annotation in $M_i$ that is not an end point of an edge in $E'$}
{
$Q \gets Q \cup NewAtom(R)$ \;
}
}
\If{$match = false$}
{
Continue and try next query \;
}
}
\If{a match was found in all steps}
{
return $Q$ \;
}
Output ``No consistent query exists"\;
\caption{FindConsistentQuery (Lin)}\label{algo:lin}
\end{algorithm} \DecMargin{1em}
Similarly to the case of trio, we may use the bound to consider all
options of $$\mathbb{N}[X]$$ provenance expressions that are consistent with the
given lineage. Again similarly to the case of trio, we can do much
better if we gradually consider the different options based on the
matchings we obtain. Algorithm \ref{algo:lin} is similar to
Algorithm \ref{algo:trio}, with subtle differences that we next
highlight. We start by considering the set of annotations comprising
the lineage of each tuple, as if they are all members of a single
monomial. We then proceed as in the trio algorithm, but the main
difference is that we only generate as many atoms as needed to cover
the head, without worrying about covering additional provenance
annotations. The observation is that to use all remaining provenance
atoms it suffices to add a single atom per relation name from which
these annotations were taken. This atom is assigned fresh variables,
so that it may match any tuple of the relation.
\daniel{Not exactly $1$, what if we have multiple relation names}
\paragraph*{Complexity} Since the generated query size is bounded by $k+r$, the complexity is again $O(m^O(k+r) \cdot n)$.
\begin{example}
\daniel{TBD:example}
\end{example}
\paragraph*{Discussion} \daniel{If we keep like that, discuss the disadvantages of obtained queries. One idea: disqualify queries that are not minimal in the standard sense.}
\section{Choosing Best Candidates}
\label{sec:minimization}
\subsection{Query Optimization: Standard Minimization versus p-minimization}
\subsection{Most tight fit}
\section{Query-By-Explanation}
\label{sec:model}
We define in this section the problem of learning queries from
examples and their explanations. We first introduce the notion of
such examples, using provenance as explanations.
\begin{definition}[Examples with explanations]\label{provExample}
Given a semiring $K$, a {\em $K$-example} is a pair $(I,O)$ where $I$ is an abstractly-tagged $K$-database called the {\em input} and $O$ is a $K$-relation called the {\em output}.
\end{definition}
Intuitively, annotations in the input only serve as identifiers, and those in the output serve as explanations -- combinations of annotations of input tuples contributing to the output.
We next define the notion of a query being consistent
with a $K$-example. In the context of query-by-example,
a query is consistent if its evaluation result
includes all example tuples (but maybe others as well). We resort to
\cite{Greenicdt09} for the appropriate generalization to the
provenance-aware settings:
\begin{definition} \cite{Greenicdt09}
Let $(K,+_{K},\cdot_{K},0,1)$ be a semiring and define $a \leq_{K}
b$ iff $\exists c.~ a+_{K} c = b$. If $\leq_{K}$ is a (partial)
order relation then we say that $K$ is naturally ordered.
Given two $K$-relations $R_1,R_2$ we say that $R_1 \subseteq_{K}
R_2$ iff \\
$\forall t. R_1(t) \leq_{K} R_2(t)$.
\end{definition}
Note that if $R_1\subseteq_{K} R_2$ then in particular $supp(R_1) \subseteq supp(R_2)$, so the notion of containment w.r.t.
a semiring is indeed a faithful extension of
``standard" relation containment. In terms of provenance, we note that for $\mathbb{N}[X]$ and Why(X), the natural order corresponds to inclusion of monomials: $p_1 \leq
p_2$ if every monomial in $p_1$ appears in $p_2$. The order relation has different interpretations in other
semirings, still fitting the intuition of a partial explanation.
We are now ready to define the notion of consistency with respect to
a $K$-example, and introduce our problem statement. Intuitively, we
look for a query whose output is contained in the example output,
and for each example tuple, the explanations provided by the user
are ``reflected" in the computation of the tuple by the query.
\begin{definition} [Problem Statement]
\label{def:problem}
Given a {\em $K$-example} (I,O) and a conjunctive query $Q$ we say that $Q$ is consistent with respect to the example if
$O\subseteq_{K} Q(I)$. {\em K-CONSISTENT-QUERY} is the problem of finding a consistent
query for a given $K$-example.
\end{definition}
The above definition allows multiple conjunctive queries
to be consistent with a given $K$-example. This is in line with the
conventional wisdom in query-by-example; further natural desiderata w.r.t. to the query, and are studied in Section 4.3.
\begin{figure}
\begin{center}
\small{
\begin{tabular}{|l|}
\hline
\verb"trip(x, y) :- route(x, z), route(w, y), "\\
\verb"route(t, r), route(k, l)" \\
\hline
\end{tabular}
} \vspace{-5mm}
\end{center}
\caption{$Q_{general}$} \label{fig:qgen}
\vspace{-5mm}
\end{figure}
We next demonstrate the notion of consistent queries with respect to
a given {$K$-example}.
\begin{example}\label{ex:consExact}
Consider Table \ref{outRel}, now treated as an $\mathbb{N}[X]$-example. Each
monomial corresponds to a trip that fits the constraints that the
user has in mind, serving as an explanation that the user has
provided for the trip end-points. Consistent queries must derive the example tuples in the ways specified in the polynomials (and possibly in additional ways). The query $Q_{real}$ from Figure
\ref{fig:qreal} is of course a consistent query with respect to it,
since it generates the example tuples and the provenance of each of
them according to $Q_{real}$ is the same as that provided in the
example. $Q_{real}$ is not the only consistent query; in particular the query $Q_{general}$ presented in Fig. \ref{fig:qgen}
is also consistent (but note that this particular query is not
minimal, see further discussion in Section \ref{sec:NX}).
\end{example}
In the following section we study the complexity of the above
computational problems for the different models of provenance. We
will analyze the complexity with respect to the different facets of
the input, notations for which are provided in Table
\ref{notations}.
\begin{table}[!htb]
\centering
\begin{tabular}{| c | l | c |}
\hline $Ex$ & {$K$-example} \\
\hline $I$ & Input database \\
\hline $O$ & Output relation and its provenance \\
\hline $m$ & Total number of monomials \\
\hline $k$ & Number of attributes of the output relation \\
\hline $n$ & (Maximal) Number of elements in a monomial \\
\hline
\end{tabular}
\vspace{-2mm}
\caption{Notations}\label{notations}
\vspace{-3mm}
\end{table}
\section{Learning from $\mathbb{N}[X]$\ explanations}
\label{sec:NX}
We start our study with the case where the given provenance consists
of $$\mathbb{N}[X]$$ expressions. This is the most informative form of
provenance under the semiring framework. In particular, we note that
given the $$\mathbb{N}[X]$$ provenance, the number of query atoms (and the
relations occurring in them) are trivially identifiable. What
remains is the choice of variables to appear in the query atoms
(body and head). Still, finding a consistent query (or deciding that
there is no such choice) is non-trivial, as we next illustrate.
\subsection{First Try}
\IncMargin{1em}
\begin{algorithm}
\SetKwFunction{goodSettingsAlgo}{ComputeQuery1}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\LinesNumbered
\Input{An N[X] example $Ex = (I, O)$}
\Output{A consistent query $Q$ or an answer that none exists} \BlankLine
Let $(t_1,M_1),...,(t_m,M_m)$ be the tuples and corresponding provenance monomials of $O$ \;
Let $n$ be the size of each monomial \;
$Perms \gets AllPermutations(O)$ \;
\ForEach{$\pi \in Perms$}
{
\If{$\pi$ is inconsistent}{
Continue to the next permutation\;
}
$Cover \gets \emptyset$\;
\ForEach{attribute $A$ of $O$}
{\label{loop:headAttr}
\ForEach{$j<n$}
{\label{loop:monom}
Let $M^{j}_1,...,M^{j}_m$ be the tuples corresponding to the provenance atoms in the $j$'th place of each monomial in
$\pi$ \;\label{line:collect}
Let $\mathcal{R}$ denote the relation name of $M^{j}_1,...,M^{j}_m$\;
\If{$\exists A' \in \mathcal{R} \; \forall 1 \leq i \leq m \;\; t_i.A = M^{j}_i.A'$}
{\label{line:coverA}
$Cover \gets Cover \cup (A,j,A')$ \;
}
}
}
\If{all attributes of $O$ appear in $Cover$}
{\label{line:allCovered}
return $BuildQuery(Cover)$ \;\label{line:buildquery}
}
}
Output ``No consistent query exists"\;
\caption{First (Inefficient) Try}\label{algo:FirstTry}
\end{algorithm} \DecMargin{1em}
We start by describing an inefficient algorithm (Algorithm
\ref{algo:FirstTry}) that retrieves a consistent query for a given
$$\mathbb{N}[X]$$ example. This will serve for explaining some main ideas of the
eventual efficient algorithm, as well as some pitfalls that need to
be avoided.
We first (line 1) ``split" the different monomials so that we obtain pairs $(t_i,M_i)$ where $t_i$ is a tuple and
$M_i$ is a monomial with coefficient $1$. To achieve that, we generate multiple copies of each tuple, one
for each monomial (a monomial with coefficient $C$ is treated as $C$ equal monomials).
Then, the goal of the algorithm is to generate query atoms while mapping the provenance
annotations to generated query atoms, in a way that is consistent
and realizes (``covers") the head attributes. To this end, a first
approach is to consider (Line 3) all possible permutations of the
annotations in every monomial (a single permutation here includes an
permutation of the annotations in $M_1$ {\em and} an permutation of
the annotations in $M_2$, and so on). Note that the need to consider multiple permutations stems from the possibility of multiple occurences of the same relation (self-joins). For each such permutation
(Lines 4-16) we try to compose a corresponding query, as follows. We
first check that the permutation is consistent (Lines 5--6) which
means that (1) for every location $j$, the atoms appearing in
location $j$ of all monomials are all annotations of tuples
appearing in the same input relation (otherwise no query atom can be
the source of all of them); and (2) every two occurrences of the
same monomial are ordered in a different way (otherwise the required
multiplicity will not be achieved). If the permutation is
consistent, we consider the head attributes one by one (Line 8), and
for each such attribute $A$ we try to find a corresponding body atom
and attribute. For that we try every location $j$ in the monomial
ordering (Line 9), and for each such location we ``collect" the
input tuples corresponding to the $j$'th atoms of all monomials
(Line 10). The head variable for $A$ may fit any attribute of the
$j$'th atom, so we need to consider every such attribute $A'$ of the
relation $R$ of the corresponding tuples (Lines 11-13; such a
relation exists due to the consistency of the permutation). This
attribute is a good fit for the head variable if this is the case
for {\em every} example monomial. If such a good fit was found, then the
variable assigned to the head attribute $A$ will appear as the $A'$
attribute of the $j$'th atom, and we continue. If all head
attributes are covered in this fashion, then we generate the
corresponding query (Lines 14-15) assigning a query atom to each
location in the ordering and placing each head variable for
attribute $A$ in the location of the covering attribute $A'$. In
contrast, if after considering all orderings, no such cover is
found, then we conclude that no consistent query exists.
\begin{table}[!htb]
\label{table:exampleperm}
\begin{minipage}{.5\linewidth}
\centering \small
\begin{tabular}{| c | c | c | c | c | c |}
\hline 1 & 2 & 3 & 4 \\
\hline f & e & c & a \\
\hline a & a & c & b \\
\hline d & e & c & h \\
\hline
\end{tabular}
\caption*{First perm.} \label{order1}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\centering \small
\begin{tabular}{| c | c | c | c | c | c |}
\hline 1 & 2 & 3 & 4 \\
\hline f & a & c & e \\
\hline a & a & c & b \\
\hline d & h & c & e \\
\hline
\end{tabular}
\caption*{Second perm.} \label{order2}
\end{minipage}
\caption{Two Permutations in Ex.
\ref{ex:naturalNaive}}\label{table:exampleperm}
\vspace{-5mm}
\end{table}
\begin{example}\label{ex:naturalNaive}
Consider the three monomials in our running example (two of them
belong to the provenance of the same tuple, and are ``split" by the
algorithm). Two of the permutations are depicted in Table
\ref{table:exampleperm}. The first fails in ``covering" all head
attributes: the second output attribute consists of the values $[Brazil,
Brazil,\\Paraguay]$ (in order of the output tuples), but no index
$1\leq j\leq 4$ in the permutation is such that the input tuples
whose annotations appear in the $j$'th column of the permutation
have these values appearing in any attribute $A'$ (so the condition
in line \ref{line:coverA} is not met). In contrast, the second
permutation yields a cover for both head attributes: the first
attribute is covered for $j=1$ (via the first input attribute) and
the second attribute is now covered for $j=2$ (via the second input
attribute). Therefore, the condition in line \ref{line:allCovered}
will hold, and the algorithm will generate the query $Q_{general}$
shown in Figure \ref{fig:qgen}
\end{example}
\vspace{-4mm}
\paragraph*{Pitfalls} There presented algorithm has two pitfalls. The first is that it is prohibitively inefficient: it traverses all $n!^{m}$ possible permutations of monomials. The second pitfall is that the query
generated by the algorithm is a very general one, i.e. it does not
account for possible joins or selection criteria that may appear in
the query. In fact, as exemplified above, the query may include
``redundant" atoms, while an alternative consistent query may be
minimal.
We next address these pitfalls. We start by presenting an efficient variant of the algorithm. Then, we show
how further constraints may be inferred, to obtain a ``tight" fit to
the examples.
\subsection{An Improved Algorithm}\label{subsec:improved}
We present an alternative algorithm that avoids the exponential
dependency on $n$ and $m$. An important observation is that we can
focus on finding atoms that ``cover" the attributes of the output
relation, and the number of required such atoms is at most $k$ (the
arity of the output relation). We may need further atoms so that the
query realizes all provenance tokens (eventually, these atoms will
also be useful in imposing e.g. join constraints), and this is where
care is needed to avoid an exponential blow-up with respect to the
provenance size. To this end, we observe that we may generate a
``most general" part of the query simply by generating atoms with
fresh variables, and without considering all permutations of parts
that do not contribute to the head. This will suffice to guarantee a consistent
query, but may still lead to a generation of a too general
query; this issue will be addressed in Section 4.3.
\IncMargin{1em}
\begin{algorithm}
\SetKwFunction{goodSettingsAlgo}{ComputeQuery2}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\LinesNumbered
\Input{An N[X] example $Ex=(I,O)$}
\Output{A consistent query $q$ or an answer that none exists} \BlankLine
Let $(t_1,M_1),...,(t_m,M_m)$ be the tuples and corresponding provenance monomials of $O$ \;
$(V,E) \gets BuildLabeledGraph((t_1,M_1),(t_2,M_2))$ \; \label{buildGraph}
\ForEach {consistent {\em matchings} $E' \subseteq E \text{ s.t. } |E'| \leq k$}
{\label{EdgeSets}
\If {$\cup_{e \in E'} label(e) = \{1, \ldots, k\}$}
{\label{line:cover}
$Q \gets BuildQueryFromMatching(E',Ex)$ \;\label{line:buildquery2}
\ForEach {$1<j<m$}{
\If{not $consistent(Q,t_j,M_j)$}
{\label{line:notconsistent}
Go to next matching \;
}
}
return $Q$ \;
}
}
Output ``No consistent query exists"\;
\caption{FindConsistentQuery (N[X])}
\label{algo:Efficient}
\end{algorithm} \DecMargin{1em}
We next detail the construction, shown in Algorithm
\ref{algo:Efficient}. Again, we separate monomials similarly to
Algorithm 1, but this time not duplicating monomials with
coefficient greater than $1$ (see a dedicated treatment of
coefficients below). We then start by picking two tuples and
monomials (see below a heuristic for making such a pick) and denote
the tuples by $t_1$ and $t_2$ and their provenance by $M_1=a_1 \cdot
... \cdot a_n$ and $M_2=b_1 \cdot ... \cdot b_n$ respectively. Our
goal is to find all ``matches" of {\em parts} of the monomials so
that all output attributes are covered. To this end, we define (Line
2) a full bipartite graph $G= (V_1 \cup V_2, E)$ where each of $V_1$
and $V_2$ is a set of $n$ nodes labeled by $a_1, \ldots, a_n$ and
$b_1, \ldots, b_n$ respectively. We also define labels on each edge,
with the label of $(a_i,b_j)$ being the set of all attributes that
are {\em covered} by $a_i,b_j$, in the same sense as in lines 12-13
of Algorithm \ref{algo:FirstTry}: an output attribute $A$ is covered
if there is an input attribute $A'$ whose value in the tuple
corresponding to $a_i$ ($b_j$), matches the value of the attribute
$A$ in $t_1$ (respectively $t_2$).
We then (lines 3-4) find all matchings, {\em of size $k$ or less},
that cover all output attributes; namely, that the union of sets
appearing as labels of the matching's edges equals $\{1,...,k\}$. As
part of the matching, we also specify which subset of the edge label
attributes is chosen to be covered by each edge (the number of such
options is exponential in $k$). It is easy to observe that if such a
cover (of any size) exists, then there exists a cover of size $k$ or
less. We further require that the matching is consistent in the
sense that the permutation that it implies is consistent.
For each such matching we generate (line 5) a ``most general query"
$Q$ corresponding to it, as follows. We first consider the matched
pairs $a_i,b_j$ one by one, and generate a query atom for each pair.
This is done in the same manner as in the last step of Algorithm
\ref{algo:FirstTry}, except that the query generation is done here
based on only $k$ pairs of provenance atoms, rather than all $n$
atoms. To this end, we further generate for each provenance token
$a_i$ that was not included in the matching a new query atom with
the relation name of the tuple corresponding to $a_i$, and fresh
variables. Intuitively, we impose minimal additional constraints,
while covering all head attributes and achieving the required query
size of $n$.
Each such query $Q$ is considered as a ``candidate", and its
consistency needs to be verified with respect to the other tuples
of the example (Line \ref{line:notconsistent}). One way of doing so
is simply by evaluating the query with respect to the input, checking that the output tuples are generated, and their provenance
includes those appearing in the example. As a more efficient
solution, we test for consistency of $Q$ with respect to each
example tuple by first assigning the output tuple values to the head
variables, as well as to the occurrences of these variables in the
body of $Q$ (by our construction, they can occur in at most $k$
query atoms). For query atoms corresponding to provenance annotations that
have not participated in the cover, we only need to check that for
each relation name, there is the same number of query atoms and of
provenance annotations relating to it. A subtlety here is in handling
coefficients; for the part of provenance that has participated in the
cover, we can count the number of assignments. This number is
multiplied by the number of ways to order the other atoms (which is
a simple combinatorial computation), and the result should exceed
the provided coefficient.
\paragraph*{Choosing the two tuples} For correctness and worst case complexity guarantees, any choice of
tuples as a starting point for the algorithm (line 1) would suffice.
Naturally, this choice still affects the practical performance, and
we aim at minimizing the number of candidate matchings. A simple but
effective heuristic is to choose two tuples and monomials for which
the number of distinct values (both in the output tuple and in input
tuples participating in the derivations) is maximal.
\begin{figure}[!htb]
\centering
\begin{subfigure}[b]{0.4\columnwidth}
\centering
\begin{tikzpicture}[thick,
every node/.style={draw,circle},
fsnode/.style={fill=myblue,node distance=.2cm},
ssnode/.style={fill=mygreen,node distance=.2cm},
every fit/.style={ellipse,draw,inner sep=-2pt,text width=1.25cm},
-,shorten >= 3pt,shorten <= 3pt
]
\begin{scope}[start chain=going below,node distance=7mm]
\node[fsnode,on chain] (1) [label=left: $f$] {};
\node[fsnode,on chain] (2) [label=left: $e$] {};
\node[fsnode,on chain] (3) [label=left: $c$] {};
\node[fsnode,on chain] (4) [label=left: $a$] {};
\end{scope}
\begin{scope}[xshift=1.8cm,start chain=going below,node distance=7mm
\node[ssnode,on chain] (5) [label=right: $d$] {};
\node[ssnode,on chain] (6) [label=right: $e$] {};
\node[ssnode,on chain] (7) [label=right: $c$] {};
\node[ssnode,on chain] (8) [label=right: $h$] {};
\end{scope}
\node [label=above:$M_1$] {}
\node [xshift=1.8cm,label=above:$M_2$] {}
\draw (1) -- (5)
node[draw=none,fill=none,font=\scriptsize,midway,above,yshift=-.2cm]
{$\{1\}$};
\draw(4)-- (8)
node[draw=none,fill=none,font=\scriptsize,midway,above,yshift=-.2cm]
{$\{2\}$};
\end{tikzpicture}
\caption{Matching A} \label{matching}
\end{subfigure}\hfill%
\begin{subfigure}[b]{0.4\columnwidth}
\centering
\begin{tikzpicture}[thick,
every node/.style={draw,circle},
fsnode/.style={fill=myblue,node distance=.2cm},
ssnode/.style={fill=mygreen,node distance=.2cm},
every fit/.style={ellipse,draw,inner sep=-2pt,text width=1.25cm},
-,shorten >= 3pt,shorten <= 3pt
]
\begin{scope}[start chain=going below,node distance=7mm]
\node[fsnode,on chain] (1) [label=left: $f$] {};
\node[fsnode,on chain] (2) [label=left: $e$] {};
\node[fsnode,on chain] (3) [label=left: $c$] {};
\node[fsnode,on chain] (4) [label=left: $a$] {};
\end{scope}
\begin{scope}[xshift=1.8cm,start chain=going below,node distance=7mm
\node[ssnode,on chain] (5) [label=right: $a$] {};
\node[ssnode,on chain] (6) [label=right: $a$] {};
\node[ssnode,on chain] (7) [label=right: $b$] {};
\node[ssnode,on chain] (8) [label=right: $c$] {};
\end{scope}
\node [label=above:$M_1$] {}
\node [xshift=1.8cm,label=above:$M_2$] {}
\draw (1) -- (5)
node[draw=none,fill=none,font=\scriptsize,midway,above,yshift=-.2cm]
{$\{1\}$};
\draw(4)-- (6)
node[sloped,anchor=south,auto=false,draw=none,fill=none,font=\scriptsize,midway,below,yshift=.2cm]
{$\{1, 2\}$};
\end{tikzpicture}
\caption{Matchings B,C} \label{matching2}
\end{subfigure}
\caption{Matchings for Example \ref{ex:naturalAlgo}} \vspace{-4mm}
\label{matchings}
\end{figure}
\begin{example}\label{ex:naturalAlgo}
Reconsider our running example. Assume that the monomials $f \cdot e
\cdot c \cdot a$ and $d \cdot e \cdot c \cdot h$ were picked for
graph generation. Figure \ref{matching} depicts a matching of size
$2$ where the first attribute of input tuples $f$ and $d$ covers the
first output attribute, and the second attribute of input tuples $a$
and $h$ covers the second output attribute. Generating a query based on this matching results in
the query $Q_{general}$ from Figure \ref{fig:qgen}. The
algorithm now verifies the consistency of $Q_{general}$ with respect to the third
monomial by assigning the output tuple to the head, i.e. assigning
$x$ to $Argentina$ and $y$ to $Brazil$, and returns $Q_{general}$. Two different matchings (one including only the edge (a,a) and another one including (a,a) and (f,a)) are presented in Figure
\ref{matching2}.
\end{example}
\vspace{-4mm}
\paragraph*{Complexity} The algorithm's complexity is $O(n^{2k} \cdot
m)$: at most $n^{k}$
matchings are considered; for each matching, a single query is
generated, and consistency is validated in
$O(n^{k})$ for each of the $m$ example tuples.
We have avoided the exponential factor in $n$ and $m$ and instead the exponential factor only involves $k$, which is much
smaller in practice. Can we do even better? We can show that if $P \neq NP$, there is no algorithm running in time polynomial in $k$.
\begin{proposition}
Deciding the existence of a consistent query with respect to a given
$$\mathbb{N}[X]$$ example is NP-hard in $k$.
\end{proposition}
\subsection{Achieving a tight fit}
\label{subsec:minimize}
We have now developed an efficient algorithm for deciding the
existence of a consistent query, and computing one if exists. Still,
as exemplified above, a downside of the algorithm is that the
generated query is very general. A natural desideratum (employed in
the context of ``query-by-example"), is that the query is
``inclusion-minimal". This notion extends naturally to
$K$-databases.
\begin{definition}
A consistent query $Q$ (w.r.t. a given $K$-example $Ex$) is \emph{inclusion-minimal} if for every query $Q'$ such that $Q'
\subsetneq_{K} Q$ (i.e. for every $K$-database $D$ it holds that
$Q'(D) \subseteq_{K} Q(D)$, but not vice-versa), $Q'$ is not consistent w.r.t. $Ex$.
\end{definition}
To find inclusion-minimal queries, we next refine
Algorithm~\ref{algo:Efficient} as follows. We do not halt when
finding a single consistent query, but instead find all of those
queries obtained for some matching. For each consistent query $Q$,
we examine queries obtained from $Q$ by (i) equating variables and
(ii) replacing variables by constants where possible (i.e. via an
exact containment mapping \cite{AHV}). We refer to both as
\emph{variable equating}. To explore the possible combinations of
variable equatings, we use an algorithm inspired by data mining
techniques (e.g.,~\cite{Gunopulos1997data}): in each iteration, the
algorithm starts from a minimal set of variable equatings that was
not yet determined to be (in)consistent with the example. E.g., in
the first iteration it starts by equating a particular pair of
variables. The algorithm then tries, one-by-one, to add variable
equatings to the set, while applying transitive closure to ensure
the set reflects an equivalence relation. If an additional equating
leads to an inconsistent query, it is discarded. Each equatings set
obtained in this manner corresponds to a homomorphism $h$ over the
variables of the query $Q$, and we use $h(Q)$ to denote the query
resulting from replacing each variable $x$ by $h(x)$.
Importantly, by equating variables or replacing variables by
constants we only impose further constraints and obtain no new
derivations. In particular, the following result holds, as a simple
consequence of Theorem~7.11 in \cite{Greenicdt09} (note that we must
keep the number of atoms intact to be consistent with the
provenance):
\begin{proposition}
Let $Q$ be a CQ over a set of variables
$\mathcal{V}$. Let $h:\mathcal{V} \mapsto \mathcal{V} \cup
\mathcal{C}$ be a homomorphism. For every $$\mathbb{N}[X]$$-example $Ex$, if $Q$
is not consistent with $Ex$, then neither is $h(Q)$.
\end{proposition}
Consequently, the algorithm finds a \emph{maximal set of variable
equatings} that is consistent with the query, by attempting to add
at most $O(k'^2)$ different equatings, where $k'$ is the number of
unique attributes in the body of $Q$. We record every query that was
found to be (in)consistent -- in particular, every subset of a
consistent set of equatings is also consistent -- and use it in the
following iterations (which again find maximal sets of equatings).
\paragraph*{Checking for consistency}
This check may be done very efficiently for query atoms that
contribute to the head, since we only need to check that equality
holds for the provenance annotations assigned to them. For other
atoms we no longer have their consistency as a given and
in the worst case we would need to examine all matchings of these query atoms
atoms to provenance annotations.
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.6]
\node (max) at (0,4) {$\{\{r = k\}, \{w = l\}, \{t = z\}\}$};
\node (a) at (-2,2) {$\{\{r = k\}, \{w = l\}\}$};
\node (d) at (-4,0) {$\{w = l\}$};
\node (e) at (0,0) {$\{r = k\}$};
\node (f) at (3,0) {$\{t = z\}$};
\draw (d) -- (a) -- (max)
(f) -- (max);
\draw[preaction={draw=white, -,line width=6pt}] (a) -- (e);
\end{tikzpicture}
\caption{Part of the lattice in Example \ref{ex:compareVars}}
\label{lattice}
\end{figure}
\begin{example}\label{ex:compareVars}
Reconsider our running example query $Q_{general}$; a part of the
lattice is depicted in Figure \ref{lattice}. The algorithm starts by
considering individually each pair of variables as well as pairs of
variables and constants co-appearing in the two output tuples or in
the tuples used in their provenance. In our example, when
considering the lattice element $\{r=k\}$, the algorithm will find
that the query $Q_{r=k}$ (i.e. $Q_{general}$ after equating $r$ and
$k$), is still consistent. Next, the algorithm will find that
equating $l,w$ in $Q_{r=k}$ also yields a consistent query so it
will proceed with $Q_{r=k,w=l}$, etc. Of course, multiple steps may
yield the same equivalence classes in which case we perform the
computation only once. Three more equalities, namely $t=z$,
$r=`Bolivia'$ and $w=`Argentina'$ may be employed, leading to the
``real" query $Q_{real}$ in Fig. \ref{fig:qreal}. Any further step
with respect to $Q_{real}$ leads to an inconsistent query, and so it
is returned as output.
\end{example}
\vspace{-3mm}
\paragraph*{Choosing a Single Query to Output} For each consistent
query found by Algorithm 2, there may be multiple inclusion-minimal
queries obtained in such a manner (though the number of such queries
observed in practice was not very large, see Section 7). If we wish
to provide a single query as output, we may impose further criteria.
A particularly important consideration here is the ``syntactic"
minimality (in the sense of Chandra and Merlin \cite{ChandraMerlin})
of the inferred query. This is a desirable feature of the inferred
query, but there is a subtlety in this respect when considering
provenance: $$\mathbb{N}[X]$$ provenance is not preserved under syntactic
minimization (in particular, we may get less atoms than specified in
the provenance). We can thus check candidate queries for syntactic
minimality, and prefer those that satisfy it (if any). Testing for
minimality via the algorithm of \cite{ChandraMerlin} is quite costly
(exponential time in the query size), but we run it only on
consistent inclusion-minimal queries whose number is small. Finally,
if multiple inclusion-minimal and syntactically-minimal consistent
queries are obtained, a natural and simple heuristic that we employ
is to prefer a query with the least number of unique variables.
\section{Preliminaries}
\label{sec:prelim}
In this section we give a brief review of the basic notions we use
throughout the paper.
\subsection{Conjunctive Queries}
We will focus in this paper on Conjunctive Queries (see e.g. \cite{hector2002database}). Fix a database schema $\mathcal{S}$ with relation names
$\{\mathcal{R}_1,...,\mathcal{R}_n\}$ over a domain $\mathcal{C}$ of
constants. Further fix a domain $\mathcal{V}$ of variables. A {\em
conjunctive query} $q$ over $\mathcal{S}$ is an expression of the
form $T(\vec{u}^{\,}) :- \mathcal{R}_1(\vec{v_1}^{\,}), \ldots, \mathcal{R}_l(\vec{v_l}^{\,})$
where $T$ is a relation name not in $\mathcal{S}$. For all $1 \leq i
\leq n$, $\vec{v_i}^{\,}$ is a vector of the form $(x_1, \ldots,
x_k)$ where $\forall 1 \leq j \leq k. \; x_j \in \mathcal{V} \cup
\mathcal{C}$. $T(\vec{u}^{\,})$ is the query head, denoted
$head(q)$, and $\mathcal{R}_1(\vec{v_1}^{\,}), \ldots,
\mathcal{R}_l(\vec{v_l}^{\,})$ is the query body and is denoted
$body(q)$. The variables appearing in $\vec{u}^{\,}$ are called the
\textit{head variables} of $q$, and each of them must also appear in
the body. We use $CQ$ to denote the class of all conjunctive
queries, omitting details of the schema when clear from context.
We next define the notion of {\em derivations} for CQs. A derivation
$\alpha$ for a query $q \in CQ$ with respect to a database instance
$D$ is a mapping of the relational atoms of $q$ to tuples in $D$
that respects relation names and induces a mapping over arguments,
i.e. if a relational atom $R(x_1, ..., x_n)$ is mapped to a tuple
$R(a_1, ..., a_n)$ then we say that $x_i$ is mapped to $a_i$
(denoted $\alpha(x_i) = a_i$). We require that a variable $x_i$ will
not be mapped to multiple distinct values, and a constant $x_i$ will
be mapped to itself. We define $\alpha(head(q))$ as the tuple
obtained from $head(q)$ by replacing each occurrence of a variable
$x_i$ by $\alpha(x_i)$.
\begin{figure}
\begin{center}
\small{
\begin{tabular}{|l|}
\hline
\verb"trip(x, w) :- route(x, y), route(y, Bolivia), "\\
\verb"route(Bolivia, Argentina), route(Argentina, w)" \\
\hline
\end{tabular}
} \vspace{-5mm}
\end{center}
\caption{$Q_{real}$} \label{fig:qreal}
\vspace{-5mm}
\end{figure}
\begin{table}[!htb]
\centering \small
\begin{tabular}{| c | c | c |}
\hline prov. & A & B \\
\hline a & Argentina & Brazil \\
\hline b & Brazil & Bolivia \\
\hline c & Bolivia & Argentina \\
\hline d & Peru & Colombia \\
\hline e & Colombia & Bolivia \\
\hline f & Argentina & Colombia \\
\hline g & Peru & Paraguay \\
\hline h & Argentina & Paraguay \\
\hline
\end{tabular}
\caption{Relation $route$}\label{relR1}
\end{table}
\vspace{-4mm}
\begin{example}
\label{ex:prov}
Reconsider our example from the Introduction of a
user planning a trip with specific requirements. Her logic may be
captured by the query $Q_{real}$ presented in Fig. \ref{fig:qreal} ($route$ is depicted in
Table \ref{relR1}). Intuitively, the query outputs all possible endpoints of a trip that
includes visits to five countries, the third and fourth of which being Bolivia and Argentina resp.
Now, consider the result tuple $trip(Argentina,Brazil)$. It is
obtained through two different derivations: one that maps the atoms
to four distinct tuples of $route$ to the four atoms (in order of
the atoms, these are the tuples annotated by {\bf f,e,c,a} in Table
\ref{relR1}), and one that maps the tuple annotated {\bf a} to the
first and last atoms and {\bf b,c} to the second and third
respectively. Each derivation includes the border crossings of a
trip between Argentina and Brazil that satisfies the constraints.
\end{example}
\subsection{Provenance}
The tracking of {\em provenance} to explain query results has been
extensively studied in multiple lines of work, and
\cite{Greenicdt09} has shown that different such models may be
captured using the {\em
semiring approach} \cite{GKT-pods07}. We next overview several aspects of the approach that we will use in our formal framework.
\paragraph*{Commutative Semirings} A {\em commutative monoid} is an algebraic structure $(M,+_{\!_M},0_{\!_M})$ where $+_{\!_M}$ is an associative and
commutative binary operation and $0_{\!_M}$ is an identity for
$+_{\!_M}$. A {\em commutative semiring} is
then a structure $(K,+_{\!_K} ,\cdot_{\!_K},0_{\!_K},1_{\!_K})$
where $(K,+_{\!_K} ,0_{\!_K})$ and $(K,\cdot_{\!_K},1_{\!_K})$ are
commutative monoids, $\cdot_{\!_K}$ is distributive over $+_{\!_K} $,
and $a\cdot_{\!_K}0_{\!_K} = 0\cdot_{\!_K} a = 0_{\!_K}$.
\paragraph*{Annotated Databases}
We will capture provenance through the notion of databases whose tuples are associated (``annotated") with elements of a commutative semiring.
For a schema $\mathcal{S}$ with
relation names $\{\mathcal{R}_1,...,\mathcal{R}_n\}$, denote by
$Tup(\mathcal{R}_i)$ the set of all (possibly infinitely many)
possible tuples of $\mathcal{R}_i$.
\begin{definition} [adapted from \cite{GKT-pods07}]
A {\em $K$-relation} for a relation name $\mathcal{R}_i$ and a
commutative semiring $K$ is a function
$R:Tup(\mathcal{R}_i) \mapsto K$ such that its {\em support} defined
by $\mbox{supp}(R) \equiv \{t \mid R(t)\neq 0\}$ is finite. We say that
$R(t)$ is the annotation of $t$ in $R$. A $K$-database $D$ over a
schema $\{\mathcal{R}_1,...,\mathcal{R}_n\}$ is then a collection of
$K$-relations, over each $\mathcal{R}_i$.
\end{definition}
Intuitively a $K$-relation maps each tuple to its annotation.
We will sometimes use $D(t)$ to denote the
annotation of $t$ in its relation in a database $D$. We furthermore say that a $K$-relation is {\em abstractly tagged}
if each tuple is annotated by a distinct element of $K$ (intuitively, its identifier).
\paragraph*{Provenance-Aware Query Results} We then define Conjunctive
Queries as mappings from $K$-databases to $K$-relations. Intuitively we define the annotation (provenance) of an output tuple as a combination of annotations of input tuples. This combination is based on the query derivations, via the
intuitive association of alternative derivations with the semiring
``$+$" operation, and of joint use of tuples in a derivation with
the ``$\cdot$" operation.
\begin{definition} [adapted from \cite{GKT-pods07}]
\label{def:basicprov} Let $D$ be a $K$-database and let $Q \in CQ$,
with $T$ being the relation name in $head(Q)$. For every tuple $t
\in T$, let $\alpha_{t}$ be the set of derivations of $Q$ w.r.t. $D$
that yield $t$. $Q(D)$ is defined to be a $K$-relation $T$ s.t. for
every $t$, $T(t)=\sum_{\alpha \in \alpha_{t}}\prod_{t' \in
Im(\alpha)}D(t')$. $Im(\alpha)$ is the image of $\alpha$.
\end{definition}
Summation and multiplication in the above definition are done in an
arbitrary semiring $K$. Different semirings give different
interpretations to the operations, as we next illustrate with two important semirings.
\paragraph*{Provenance Polynomials ($\mathbb{N}[X]$)} The most general form of
provenance for positive relational algebra (see \cite{GKT-pods07})
is the {\em semiring of polynomials with natural numbers as coefficients}, namely $($\mathbb{N}[X]$,+,\cdot,0,1)$. The idea
is that given a set of basic annotations $X$ (elements of which may
be assigned to input tuples), the output of a query is represented
by a sum of products as in Def. \ref{def:basicprov}, with only the
basic equivalence laws of commutative semirings in
place. Coefficients serve in a sense as ``shorthand" for multiple
derivations using the same tuples, and exponents as ``shorthand" for
multiple uses of a tuple in a derivation.
Many additional forms of provenance have been proposed in the
literature, varying in their level of abstraction and the details
they reveal on the derivations. We leverage here the work of
\cite{Greenicdt09} that has shown that multiple such provenance
forms may be captured through the semiring model, with the
appropriate choice of semiring. We next show an impotrant such model.
\vspace{-1mm}
\paragraph*{Why(X)} A natural approach to provenance tracking,
referred to as {\em why-}provenance \cite{why}, capturing each
derivation as a {\em set} of the annotations of tuples used in the
derivation. The overall why-provenance is thus a {\em set of such
sets}. As shown (in a slightly different way) in \cite{Greenicdt09},
this corresponds to using provenance polynomials but without ``caring" about exponents and coefficients.
Formally, consider the function $f:$\mathbb{N}[X]$
\mapsto $\mathbb{N}[X]$$ that {\em drops all coefficients and exponents} of its
input polynomial. We then introduce a congruence relation defined by
$P_1 \equiv P_2$ if $f(P_1) = f(P_2)$. $Why(X)$ is then
defined as the quotient semiring of $$\mathbb{N}[X]$$ under this congruence
relation (i.e. two equivalent polynomials are indistinguishable).
\begin{example}
The provenance-aware result (for $\mathbb{N}[X]$\ and Why(X)) of evaluating $Q_{real}$ over the
relation $route$ is shown in Table \ref{outRel}. Reconsider for example
the tuple $trip(Argentina, Brazil)$. Recall its two derivations
shown in Example 2.1. Consequently, its ``exact" ($\mathbb{N}[X]$)
provenance is $f \cdot e \cdot c \cdot a + a^2 \cdot c \cdot b$.
Each summand corresponds to a derivation, and recall that each
derivation stands for an alternative suitable trip that starts at
Argentina and ends at Brazil. Note that the provenance includes a
specification of the bag of tuples used in each derivation, in no
particular order (multiplication is commutative). If we alternatively store Why(X)-provenance, we still have
summands standing for alternative derivations (trips), but we
further lose track of exponents, i.e. the number of times each tuple
was used (as well as multiple identical derivations, if such exist).
The why-provenance here is $f \cdot e \cdot c \cdot a + a \cdot c
\cdot b$. Note that it still specifies the border crossings made
during the trip, but we do not know that a border was crossed twice.
In general, two trips may include the same border crossings, but in
different order (e.g. (Bolivia-Argentina-Bolivia-Brazil-Bolivia) and
(Bolivia-Brazil-Bolivia-Argentina-Bolivia), if the corresponding
tuples are present in the database). In $$\mathbb{N}[X]$$ provenance, the corresponding monomial would have appeared with coefficient $2$; this coefficient would have been omitted in Why(X) provenance.
\end{example}
\begin{table}[!htb]
\vspace{-1mm}
\begin{minipage}{1\linewidth}
\centering \small
\begin{tabular}{| c | c | c | c | c | c |}
\hline A & B & $\mathbb{N}[X]$ & $Why(X)$ \\
\hline Argentina & Brazil & $f \cdot e \cdot c \cdot a+ a^2 \cdot c \cdot b$ & $ f \cdot e \cdot c \cdot a+ a \cdot c \cdot b$ \\
\hline Peru & Paraguay & $d \cdot e \cdot c \cdot h$ & $d \cdot e \cdot c \cdot h$ \\
\hline
\end{tabular}
\caption{Relation $trip$}\label{outRel}
\end{minipage}
\vspace{-6mm}
\end{table}
\section{Related Work}
\label{sec:related}
There is a large body of literature on learning queries from
examples, in different variants. A first axis of these variants
concerns learning a query whose output {\em precisely} matches the
example (e.g. \cite{qbo,joinQueries}), versus one whose output
contains the example tuples and possibly more (e.g.
\cite{Shen,exampler,Psallidas} and the somewhat different problem in
\cite{zloof}). The first is mostly useful e.g. in a use-case where
an actual query was run and its result, but not the query itself, is
available. This may be the case if e.g. the result was exported and
sent. The second, that we adopt here, is geared towards examples
provided manually by a user, who may not be expected to provide a
full account of the output. Another distinguishing factor between
works in this area is the domain of queries that are inferred; due
to the complexity of the problem, it is typical
(e.g. \cite{joinQueries,Psallidas,Bonifati}) to restrict the attention to
join queries, and many works also impose further restrictions on the
join graph \cite{qbo,DasSarma}. We do not impose such restrictions
and are able to infer complex Conjunctive Queries. Last, there is a
prominent line of work on query-by-example in the context of {\em
data exploration} \cite{Kersten,Bonifati,Abouzied,Bonifati:2014}.
Here users typically provide an initial set of examples, leading to
the generation of a consistent query (or multiple such queries); the
queries and/or their results are presented to users, who may in turn
provide feedback used to refine the queries, and so on. In our
settings, the number of examples required for convergence to the
actual intended query was typically small. In cases where more
examples are needed, an interactive approach is expected to be useful in our setting as well.
The fundamental difference between our work and previous work in this area is
the assumed input. Our work is the first, to our knowledge, that
base its inference of queries on explanations that form provenance
information. Leveraging this additional information, we are able to
reach a satisfactory query (1) in a highly complex setting where the
underlying queries includes multiple joins and self-joins, (2) with
no information on the underlying schema beyond relation names and
their number of attributes (in particular no foreign keys are known;
in fact, we do not even need to know the entire input database, but
rather just tuples used in explanations), (3) with only very few
examples (up to 5 were typically sufficient to obtain over 90\%
recall, and less than 20 in all but one case were sufficient to
converge to the actual underlying query), and (4) in split-seconds
for a small number of examples, and in under 3 seconds even with
6000 examples. No previous work, to our knowledge, has exhibited the
combination of these characteristics.
Data Provenance has been extensively studied, for different
formalisms including relational algebra, XML query languages, Nested
Relational Calculus, and functional programs (see e.g.
\cite{trio,GKT-pods07,Userssemiring1,GS13,CheneyProvenance,w3c,ProvenanceBuneman,Olteanu12}).
Many different models have been proposed, and shown useful in a
variety of applications, including program slicing
\cite{slicing}, factorized representation of query results \cite{Olteanu12}, and
``how-to" analysis \cite{Meliou2,Bidoit14query}
We have focused on learning queries from explanations that are either based on the semiring framework or may be expressed using it.
This includes quite a few of the models proposed in the literature, but by no means all of them. Investigating query-by-explanation for
other provenance models is an intriguing direction for future work.
\section{Results and Main Algorithm Ideas}
\daniel{More for ourselves, I think the gist of it will have to go
to the Intro}
\subsection{$\mathbb{N}[X]$}
\section{Learning from Why(X)-Examples}
\label{sec:trio}
We next study the problem of learning queries from $Why(X)$-examples.
Such explanation is often easier for users to provide, but is in
turn more challenging for query inference.
\subsection{Challenges and First try}
A natural approach is to reduce the problem of learning from a
$Why(X)$-example to that of learning from an $$\mathbb{N}[X]$$-example. Recall
that the differences are the lack of coefficients and the lack of
exponents. The former is trivial to address (we simply do not need
to check that coefficients are realized), but the latter means
that we do not know the number of query atoms. Surprisingly, attempting to bound the query size by the size of the
largest monomial fails:
\begin{proposition}
\label{prop:naivefails} There exists a $Why(X)$ example for which
there is no consistent conjunctive query with $n$ atoms
(recall that $n$ is the length of the largest monomial),
but there exists a consistent conjunctive query with more atoms.
\end{proposition}
It is however not difficult to show a ``small world" property, based
on a looser bound.
\begin{proposition}
\label{prop:triobound} For any $Why(X)$ example, if there exists a
consistent query then there exists a consistent query with $k+
r\cdot(n-1)$ atoms or less, where $r$ is the number of distinct
relation names occurring in the provenance monomials.
\end{proposition}
Intuitively, there are at most $k$ atoms contributing to the head.
The worst case is when only one ``duplicated" annotation contributes
to the head, and then for each example there are at most $n-1$
remaining annotations. If the query includes a single relation name
($r=1$), then a query with at most $n-1$ more atoms would be
consistent. Otherwise, as many atoms may be needed for each relation
name. Together with our algorithm for $$\mathbb{N}[X]$$, Proposition
\ref{prop:triobound} dictates a simple algorithm that exhaustively
goes through all $$\mathbb{N}[X]$$ expressions that are compatible with the
$Why(X)$ expressions appearing in the example, and whose sizes are up
to $n+k$. This, however, would be highly inefficient. We next
present a much more efficient algorithm.
\subsection{An Efficient Algorithm}
An efficient algorithm for finding CQs consistent
with a given $Why(X)$ provenance is given in Algorithm
\ref{algo:trioEfficient}. The idea is to traverse the examples one
by one, trying to ``expand'' (by adding atoms) candidate queries
computed thus far to be consistent with the current example. We
start (Line 1), as in the $$\mathbb{N}[X]$$ case, by ``splitting'' monomials if
needed so that each tuple is associated with a single monomial. We
maintain a map $\mathcal{Q}$ whose values are candidate queries, and
keys are the parts of the query that contribute to the head, in a
canonical form (e.g. atoms are lexicographically ordered). This will
allow us to maintain only a single representative for each such
``contributing part'', where the representative is consistent with
all the examples observed so far. For the first step (line
\ref{line:init}) we initialize $\mathcal{Q}$ so that it includes
only $(t_1,M_1)$ (just for the first iteration, we store an example
rather than a query). We then traverse the remaining examples one by
one (line \ref{line:loopRows}). In each iteration $i$, we consider
all queries in $\mathcal{Q}$; for each such query $Q$, we build a
bipartite graph (line \ref{line:buildGraph}) whose one side is the
annotations appearing in $M_i$, and the other side is the {\em atoms
of $Q$}. The label on each edge is the set of head attributes
covered jointly by the two sides: in the first iteration this is
exactly as in the $$\mathbb{N}[X]$$ algorithm, and in subsequent iterations we
keep track of covered attributes by each query atom. Then, instead
of looking for {\em matchings} in the bipartite graph, we find (line
\ref{line:allSubgraphs}) all {\em sub-graphs} whose edges cover all
head attributes (again specifying a choice of attributes subset for
each edge). Intuitively, having $e$ edges adjacent to the same
provenance annotation corresponds to the same annotation
appearing with exponent $e$, so we ``duplicate'' it $e$ times (Lines
\ref{line:expandMonom}-\ref{line:duplicateTuple}). On the other
hand, if multiple edges are adjacent to a single query atom, we also
need to ``split'' (Lines
\ref{line:expandQuery}-\ref{line:duplicateQuery}) each such atom,
i.e. to replace it by multiple atoms (as many as the number of edges
connected to it). Intuitively each copy will contribute to the
generation of a single annotation in the monomial. Now (line \ref{line:buildQuery}), we construct a query $Q'$ based on the
matching and the previous query ``version'' $Q$: the head is
built as in Algorithm 2, and if there were $x$ atoms not
contributing to the head with relation name $R$ in $Q$, then the
number of such atoms in $Q'$ is the maximum of $x$ and the
number of annotations in $M_i$ of tuples in $R$ that were not
matched. Now, we ``combine'' $Q'$ with $Q''$ which is the currently
stored version of a query with the same contributing atoms (lines
\ref{line:getQuery}- \ref{line:putQuery}). Combining means setting
number of atoms for each relation name not contributing to the head
to be the maximum of this number in $Q'$ and $Q''$.
\paragraph*{Complexity} The number of keys in
$\mathcal{Q}$ is exponential only in $k$; the loops thus iterate at
most $m \cdot n^{k} \cdot n^{k} \cdot (n+n^2)$ times, so the overall complexity is $O(n^{O(k)}
\cdot m)$.
\paragraph*{Achieving a tight fit} Algorithm
\ref{algo:trioEfficient} produces a set of candidate queries, which may neither be syntactically minimal nor
inclusion-minimal. To discard atoms that are ``clearly" redundant,
we first try removing atoms not contributing to the head, and test
for consistency. We then perform the process of
inclusion-minimization as in Section \ref{subsec:minimize} (note
that $Why(X)$-inclusion was shown in \cite{Greenicdt09} to be
characterized by onto mappings which is a weaker requirement).
\IncMargin{1em}
\begin{algorithm}
\SetKwFunction{goodSettingsAlgo}{ComputeQuery2}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\LinesNumbered
\Input{A $Why(X)$ example Ex}
\Output{A set of consistent queries (possibly empty, if none exists)} \BlankLine
Let $(t_1,M_1),...,(t_m,M_m)$ be the tuples and corresponding provenance monomials of $Ex$ \;
$\mathcal{Q} \gets \{NULL:(t_1,M_1)\}$ \; \label{line:init}
\ForEach{$2 \leq i\leq m$} {\label{line:loopRows}
\ForEach{$Q \in \mathrm{values}(\mathcal{Q})$} {\label{line:loopQueries}
$\mathcal{Q} \gets \mathcal{Q} - \{Q\}$ \;
$(V_1 \cup V_2, E) \gets BuildGraph(Q, (t_i,M_i))$ \;\label{line:buildGraph}
\ForEach {{\em sub-graph} $E' \subseteq E \text{ s.t. } |E'| \leq k \text{~~and~~} \cup_{e \in E'} label(e) = \{1, \ldots, k\}$} {\label{line:allSubgraphs}
\ForEach{provenance annotation $a$ in $M_i$} {\label{line:expandMonom}
\If{$a$ is an endpoint of more than one edge in $E'$} {
$E' \gets split(E',a)$ \;\label{line:duplicateTuple}
}
}
\ForEach{atom $C \in Q$} {\label{line:expandQuery}
\If{$C$ is an endpoint of more than one edge in $E'$}{
$E' \gets split(E',C)$ \;\label{line:duplicateQuery}
}
}
$Q' \gets BuildQuery(E',Q)$\;\label{line:buildQuery}
$Q'' \gets \mathcal{Q}.get(contribs(Q'))$ \; \label{line:getQuery}
$\mathcal{Q}.put(contribs(Q'),combine(Q',Q''))$
\; \label{line:putQuery}
}
}
}
return $values(\mathcal{Q})$ \;\label{line:returnquery}
\caption{FindConsistentQuery (Why(X))}
\label{algo:trioEfficient}
\end{algorithm} \DecMargin{1em}
\begin{example}\label{ex:algoTrioRunning}
Reconsider our running example, but now with the $Why(X)$ provenance
given in Table \ref{outRel}. If we start from the two monomials of
the tuple $(Argentina,Brazil)$ then we generate a bipartite graph
with $V_1 = \{f,e,c,a\}$ and $V_2 = \{a,b,c\}$, and obtain three
options for covering: $E'_1$ where the edge $(a,a)$ covers
attributes $\{1,2\}$, $E'_2$ where additionally $(f,a)$ covers
$\{1\}$, and $E'_3$ where $(a,a)$ covers $\{2\}$ and $(f,a)$ covers
$\{1\}$ (the latter two are options relevant to the same sub-graph).
When we continue with $E'_{1}$, no duplication is performed, and we
get a query $Q$ with a single $R(x,y)$ atom contributing to the
head, and three most general atoms. Then, we match $Q$ to $d\cdot
e\cdot c\cdot h$, resulting in a sub-graph matching $R(x,y)$ to both
$d,h$. This will lead to a split of the atom to $R(x,z)$ and
$R(w,y)$ that will appear in the final query, together with the
three most general atoms. After variables equating and
removing redundant tuples, we obtain $Q_{real}$. The same query will
be obtained in different ways if choosing $E'_2$ or $E'_3$.
\end{example}
\begin{figure}[!htb]
\vspace{-3mm}
\centering
\begin{subfigure}[b]{0.5\columnwidth}
\centering
\begin{tikzpicture}[thick,
every node/.style={draw,circle},
fsnode/.style={fill=myblue,node distance=.2cm},
ssnode/.style={fill=mygreen,node distance=.2cm},
every fit/.style={ellipse,draw,inner sep=-2pt,text width=1.25cm},
-,shorten >= 3pt,shorten <= 3pt
]
\begin{scope}[start chain=going below,node distance=7mm]
\node[fsnode,on chain] (1) [label=left: $f$] {};
\node[fsnode,on chain] (2) [label=left: $e$] {};
\node[fsnode,on chain] (3) [label=left: $c$] {};
\node[fsnode,on chain] (4) [label=left: $a$] {};
\end{scope}
\begin{scope}[xshift=1.4cm,start chain=going below,node distance=7mm
\node[ssnode,on chain] (5) [label=right: $a$] {};
\node[ssnode,on chain] (6) [label=right: $b$] {};
\node[ssnode,on chain] (7) [label=right: $c$] {};
\end{scope}
\node [label=above:$M_1$] {}
\node [xshift=1.4cm,label=above:$M_2$]{}
\draw(1) -- (5)
node[draw=none,fill=none,font=\scriptsize,midway,above,yshift=-.2cm]
{$\{1\}$};
\draw[red](4)-- (5)
node[sloped,anchor=south,auto=false,draw=none,fill=none,font=\scriptsize,midway,below,yshift=.25cm]
{$\{1,2\}$};
\end{tikzpicture}
\caption{1st iteration in \ref{ex:algoTrioRunning}}
\label{subgraph1}
\end{subfigure}%
\begin{subfigure}[b]{0.5\columnwidth}
\hspace*{-0.8cm} \centering
\begin{tikzpicture}[thick,
every node/.style={draw,circle},
fsnode/.style={fill=myblue,node distance=.2cm},
ssnode/.style={fill=mygreen,node distance=.2cm},
every fit/.style={ellipse,draw,inner sep=-2pt,text width=1.25cm},
-,shorten >= 3pt,shorten <= 3pt
]
\begin{scope}[start chain=going below,node distance=7mm]
\node[fsnode,on chain] (1) [label=left: {$A_1, \{1, 2\}$}] {}
\node[fsnode,on chain] (2) [label=left: {$A_2, \emptyset$}] {}
\node[fsnode,on chain] (3) [label=left: {$A_3, \emptyset$}] {}
\node[fsnode,on chain] (4) [label=left: {$A_4, \emptyset$}] {}
\end{scope}
\begin{scope}[xshift=1.4cm,start chain=going below,node distance=7mm
\node[ssnode,on chain] (5) [label=right: $d$] {};
\node[ssnode,on chain] (6) [label=right: $e$] {};
\node[ssnode,on chain] (7) [label=right: $c$] {};
\node[ssnode,on chain] (8) [label=right: $h$] {};
\end{scope}
\node [label=above:$Q$] {}
\node [xshift=1.4cm,label=above:$M_3$] {}
\draw (1) -- (5)
node[draw=none,fill=none,font=\scriptsize,midway,above,yshift=-.2cm]
{$\{1\}$};
\draw(1) -- (8)
node[sloped,anchor=south,auto=false,draw=none,fill=none,font=\scriptsize,midway,below,yshift=.23cm]
{$\{2\}$};
\end{tikzpicture}
\caption{2nd iteration in \ref{ex:algoTrioRunning}}
\label{subgraph2}
\end{subfigure
\caption{Subgraphs for Example \ref{ex:algoTrioRunning}} \vspace{-4mm}
\label{matchings}
\end{figure}
\vspace{-1mm}
\subsection{Additional Semirings}
To complete the picture, we next show how to adapt our algorithms to explanations taken from semirings (presented in \cite{Greenicdt09}) additional to $$\mathbb{N}[X]$$ and $Why(X)$.
\paragraph*{$Trio(X)$ and $B[X]$} Recall that $$\mathbb{N}[X]$$ keeps both coefficients and exponents, and $Why(X)$ drops both. Other alternatives include the $Trio(X)$ semiring where coefficients are
kept but exponents are not, and the $B[X]$ semiring where exponents
are kept but coefficients are not. For $Trio(X)$ we can employ the same algorithm
designed for $Why(X)$ (Algorithm 3), with a simple modification:
upon checking consistency of a candidate query with a tuple, we further check that there are as many
derivations that use the tuples of the monomial as dictated by the
coefficient (as done in Section \ref{subsec:improved}). For $B[X]$
we adapt the algorithm of $$\mathbb{N}[X]$$, omitting the treatment of
coefficients.
\vspace{-1mm}
\paragraph*{$PosBool[X]$} The semiring of positive boolean
expressions is defined with $+$ and $\cdot$ defined as disjunction
and conjunction respectively. If the expressions are given in DNF,
then our algorithm for $Why(X)$ may be used here as well: the only
difference is the possible absorption of monomials ($a+a \cdot b
\equiv a$), but we already assume that only a subset of the
monomials are given. If the expressions are given in an arbitrary
form there is an additional (exponential time) pre-processing step
of transforming them into DNF.
Last, designing an effective solution for the lineage model \cite{lin} is left as an important task for future work.
\subsection{User Study}
\label{sec:userstudy}
We have examined the usefulness of the system to non-expert users. To this end, we have loaded the IMDB database to {\tt QPlain}\ (see a partial schema in Figure \ref{scehma}) and have presented each of the tasks in Table \ref{tasks} to 15 users. We have also allowed them to freely choose tasks of the likings, resulting in a total of 120 performed tasks. The intended queries are presented in Table \ref{queries}, where the relation {\em atm} stands for ActorsToMovies.
\begin{figure}[]
\begin{center}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip = true, width=3in]{IMDB_Schema_cropped}
\vspace{-2mm} \caption{Partial IMDB schema} \vspace{-3mm}
\label{scehma}
\end{center}
\end{figure}
\begin{table}[!htb]
\centering \scriptsize
\begin{tabular}{l | p{7.7cm} }
\hline & {\bf Task} \\
\hline
1. & Find all actresses \\
2. & Find all movies that came out in 1994 \\
3. & Find all of Al Pacino's movies \\
4. & Find the entire cast of Pulp Fiction \\
5. & Find all documentary films (by having character ``Himself'') \\
6. & Find the actors who played in all the 3 movies of The Lord Of The Rings \\
7. & Choose an actor and two movies he played in, and find all actors that played with him in these two movies \\
%
%
%
\hline
\end{tabular}
\caption{User Tasks}\label{tasks}
\vspace{-2mm}
\end{table}
\begin{table}[!htb]
\centering \scriptsize
\begin{tabular}{l | p{7.7cm} }
\hline & {\bf Queries} \\
\hline
1. & ans(t) :- actors(mid, t, 'F') \\
2. & ans(t) :- movies(mid, t, '1994') \\
3. & ans(t) :- atm(mid, aid, c), actors(aid, 'Pacino, Al', 'M'), movies(mid, t, y) \\
4. & ans(n) :- atm(mid, aid, c), movies(mid, 'Pulp Fiction', '1994'), actors(aid, n, s) \\
5. & ans(t) :- atm(mid, aid, 'Himself'), actors(aid, n, 'M'), movies(mid, t, y) \\
6. & ans(n) :- atm(mid1, aid, c), atm(mid2, aid, c), atm(mid3, aid, c), actors(aid, n, s), movies(mid1, 'The Fellowship of the Ring', '2001'), movies(mid2, 'The Two Towers', '2002'), movies(mid3, 'The Return of the King', '2003') \\
\hline
\end{tabular}
\caption{Queries (No. 7 depends on user choices, omitted)}\label{queries}
\vspace{-4mm}
\end{table}
In 111 out of the 120 cases, including all cases of freely chosen tasks, users were successful in specifying examples and explanations, and the interface of {\tt QPlain}\ was highly helpful in that.
\begin{example}
Consider task 6 in Table \ref{tasks}. The examples are actor names; for every chosen actor name, {\tt QPlain}\ has proposed as explanations only the tuples that included this actor name as one of its values. In particular, the tuples of $ActorsToMovies$ corresponding to movies in which the actor has played were proposed. Knowing the underlying task, it was natural for users to choose the relevant movies, namely the three ``Lord Of The Rings" movies. Once they did that, the tuples of these movies (and not of any other movie) in the $Movies$ relation has appeared as proposals, allowing their easy selection. A similar case is that of task 3: once a movie is specified as an example, the system proposes its actors as possible explanations. The choice of Al Pacino as an explanation reveals the underlying intention.
\end{example}
In turn, explanations were crucial for allowing the system to focus on the intended query, even with very few examples (in all cases users provided at most 4 examples).
\begin{example}
Re-consider task 6, and now assume that explanations are unavailable. There would be no way of distinguishing the underlying query from, for instance, a much simpler one that looks for the actors of a {\em single} Lord of the rings. More generally, join conditions (or even the fact that a join took place) are in many cases only apparent through the explanations: another example is task 3, where examples are simply movie names that typically have many other characteristics in common in addition to having Al Pacino acting in them. In addition to revealing join conditions, explanations help to learn the query constants: for instance, in task 4, without explanations, the resulting query could have returned all actors who played in a movie with Quentin Tarantino.
\end{example}
Out of the 9 cases where users failed to provide examples and explanations, 5 involved difficulties in providing any example matching the task, and 4 involved errors in providing explanations. Out of the remaining 111 cases, in 98 cases {\tt QPlain}\ has inferred the actual underlying query, and in the remaining 13 cases, it has inferred a ``more specific" query (i.e. with extra constants). This for instance happened when all actors given as examples were males. We next further analyze the accuracy of {\tt QPlain}.
|
2,877,628,089,486 | arxiv | \section{Introduction}
\label{sec:1}
In a new series of papers we present the results obtained with
the recent ESC08c-version of the
Extended-Soft-Core (ESC) model \cite{Rij93} for nucleon-nucleon (NN),
hyperon-nucleon (YN), and hyperon-hyperon (YY) interactions with $S=0,-1,-2$.
Moreover, we present predictions for the YY-channels with $S=-3,-4$.
The combined study of all baryon-baryon (BB) interactions,
exploiting all experimental information hitherto available, both on BB-scattering
and (hyper-)nuclear systems, might throw light on the basic mechanisms of
these interactions. The program, which in its original form was formulated in
Refs.~\cite{NRS77,MRS89}, pursuits the aims:
\begin{itemize}
\item To study the assumption of broken $SU(3)$-symmetry. For example we
investigate the properties of the scalar mesons
($\varepsilon(760)$, $f_{0}(975)$, $a_{0}(980)$, $\kappa(800)$).
\item To determine the $F/(F+D)$-ratio's.
\item To study the connection between QCD, the quark-model,
and low energy physics.
\item To extract, in spite of the scarce experimental $YN$- and $YY$-data,
information about scattering lengths, effective
ranges, the existence of resonances and bound states, etc.
\item To provide realistic baryon-baryon potentials, which can be
applied in few-body calculations, nuclear- and hyperonic matter
studies, neutron-stars;
\item To extend the theoretical description to the baryon-baryon channels with
strangeness S=-2. This in particular for the $\Lambda\Lambda$
and $\Xi N$ channels, where some data already exist, and for which
experiments will be realized in the near future.
\item Finally, to extend the theoretical description to all baryon-baryon channels with
strangeness S=-3,-4. These will be parameter free predictions,
and have, like the other BB-channels, relevance for the study of
hyperonic matter and compact stars.
\end{itemize}
With this series of papers this program nears essentially its completion.
As has been amply demonstrated, see Ref.'s \cite{Rij04a,Rij04b,HYPX,PTP185.a},
the ESC-model interactions give excellent simultaneous descriptions of the NN and YN data.
Also it turned out that the
ESC-approach gives great improvements for the NN description as compared to the
One-Boson-Exchange (OBE) models, e.g. \cite{MRS89,RSY99}, and other existing models
in the literature. The ESC08c-model presents the culmination in this respect:
the NN-model has a quality on equal par with the energy-dependent
partial-wave analysis (PWA) \cite{Sto93,Klo93}.
The ESC04-model papers \cite{Rij04a,Rij04b,Rij04c}
contain the first rather extensive exposition of the ESC-approach.
As compared to the earlier versions of the ESC-model, we introduced in ESC04-models
\cite{Rij04a,Rij04b,Rij04c} several innovations:
Firstly, we introduced a zero in the form factor of
the mesons with P-wave quark-antiquark contents, which applies to the scalar and
axial-vector mesons.
Secondly, we exploited the exchange of the axial-vector mesons with $J^{PC}=1^{++}$ and
$J^{PC}=1^{+-}$. Thirdly, we employed some $\Lambda\Lambda, \Xi N$ information.\\
In the ESC08-models on top of these improvements, we introduce
in the ESC-approach for the first time: (i) Odderon-exchange $J^{PC}=1^{-+}$.
Odderon-exchange represents the exchange of an
odd-number of gluons at short-distance. This to complement pomeron-exchange which
stands for the exchange of an even-number of gluons.
(ii) Quark-core effects. The quark-core effects represent
structural effects caused by the occurrence of Pauli-blocked configurations
in two-baryon systems.
These structural effects depend on the BB-channel and cannot be described by
t-channel exchanges.\\
Furthermore, (iii) the
axial-vector ($J^{PC}=1^{++}$) mesons are treated with the most general vertices,
and the
$(\mbox{\boldmath $\sigma$}_1\cdot{\bf q})(\mbox{\boldmath $\sigma$}_2\cdot{\bf q})$-operator
is evaluated in a superior mannner compared to ESC04.
Not included are the potentials from the tensor ($J^{PC}=2^{++}$) mesons.
Attempts including the latter mesons did not lead to substantial potentials from these
mesons or qualitative changes in the other contributions to the potentials.
The first results with the ESC08-model are reported in \cite{HYPX,PTP185.a}.
In this first paper of the series, we display and discuss the NN results of
the simultaneous fit to the NN- and YN-data, including some $\Lambda\Lambda, \Xi N$
and $\Sigma N$ information from hypernuclei, using a single set of parameters.
In the second paper, henceforth referred to as II \cite{RNY10b},
we report on the results for strangeness S=-1 YN-channels,
using the same simultaneous fit of the NN- and YN-data.
This simultaneous fitting procedure was first introduced in \cite{Rij04b}, and its
importance and advantages will be discussed in II.
In the third paper, henceforth referred to as III \cite{RNY10c},
we report on the results and predictions for YY with strangeness $S=-2$.
Finally, in the fourth paper (IV), we describe the predictions for YY
with strangeness $S=-3,-4$.
The contents of this paper are as follows.
In section~\ref{sec:21} a description of the physical background and dynamical
contents of the ESC08-model is given.
In section~\ref{sec:2} the two-body integral equations in momentum space
are discussed. Also, the expansion into Pauli-spinor invariants is reviewed.
In section~\ref{sec:3} the ESC-potentials in momentum and
configuration space for non-strange mesons are discussed in detail.
In particular the new potentials are given.
Section~\ref{sec:4} contains some brief remarks on the ESC-couplings
and the QPC-model.
In section~\ref{sec:5} the simultaneous $NN \oplus YN \oplus YY$ fitting
procedure is reviewed.
Here, also the results for the coupling constants and $F/(F+D)$-ratios
for OBE and MPE are given.
In section~\ref{sec:6} the NN-results for the ESC08c-model are displayed.
In section~\ref{sec:7} a solution for the the nuclear saturation and
neutron star mass is described.
In section~\ref{sec:8} we discuss the results and draw some conclusions.
In appendix~\ref{app:C} the B-field formalism for vector- and axial-vector mesons is
described. The exact treatment of the non-local-tensor operator is explained in
appendix~\ref{app:B}.
In appendix~\ref{app:NTC} the treatment of the non-local tensor potential is reviewed.
In appendix~\ref{app:A} the basic formulas for the
configuration space gaussian-yukawa functions are given.
--------------------------------------------------------------------\\
\twocolumngrid
\section{Physical Content of the ESC-model}
\label{sec:21}
The general physical basis, within the context of QCD, for the Nijmegen
soft-core models has been outlined in the introduction of \cite{Rij04a}.
The description of baryon-interactions at low energies
in terms of baryons and mesons
can be reached through the following stages:
(i) The strongly interacting sector of the standard-model (SM) contains
three families of quarks: (ud), (cs), (tb). (ii) Integrating out the
heavy quarks (c,b,t) leads to a QCD-world with effective interactions
for the (u,d,s) quarks. (iii) This QCD-world is characterized by a phase
transition of the vacuum. Thereby the quarks gets dressed
and become the so-called constituent quarks. The emerging picture
is that of the constituent-quark-model (CQM) \cite{Man84}.
The phase transition has transformed the effective QCD-world into
an complex hadronic-world.
(iv) The strong coupling lattice QCD (SCQCD) seems to be a proper model
to study the low energy meson-baryon and baryon-baryon physics,
see \cite{Mil89} for applications and references.
Here the lattice spacing $a \geq 0.11$ fm provides a momentum scale for
which the QCD coupling $g \geq 1.1$. Emerging is a picture where the
meson-baryon coupling constants get large, and quark-exchange
effects are rather small. The latter is due to the suppression due to the
gluonic overlaps involved. For a similar reason it has been argued \cite{LN87}
that the pomeron is exchanged between the individual quarks of the baryons.
In this picture the Nijmegen soft-core approach
to baryon-baryon interactions has a natural motivation.
(v) For the mesons we restrict ourselves to
mesons with $M \leq 1.5$ GeV$/c^2$, arriving at a so-called
{\it effective field theory} as the arena for our description of the
low energy baryon-baryon scattering.
In view of the success of QCD, pseudo-scalar dominance of the divergence of
the axial-vector current (PCAC) leading to small light ("current") quark masses
\cite{Ynd80,Gas75}, the spectroscopic success of the CQM, where the quarks
have definite color charges, in generating
the masses of the pseudo-scalar and vector nonets, and the masses and
magnetic moments of the baryon octet is rather surprising \cite{Gas81,Pov95}.
The transition from "current" to "constituent" quarks comes from dressing the
quark fields in the original QCD Lagrangian, see e.g.
Ref.~\cite{Pol76,Man84,Lav97}.
In all works of the Nijmegen group on the baryon-baryon models,
(broken) $SU(3)$ flavor-symmetry is explored to connect the $NN$, $YN$, and $YY$
channels, making possible a simultaneous fitting of all
the available BB-data using a single set of model-parameters.
The dynamical basis is the (approximate)
permutation symmetry w.r.t. the constituent (u,d,s)-quarks. This has its roots in the
approximate equality of the quark-masses, and more importantly that the gluons have no
flavor.
This enables the calculation of the baryon-baryon-meson coupling constants
using as parameters the nucleon-nucleon-meson couplings and the $F/(F+D)$-ratio's.
This provides a strong correlation between the (rich) nucleon-nucleon- and the
(scarce) hyperon-nucleon-data.
The obtained coupling constants of the $BBM$-vertices are interpreted studying the
predictions of the constituent quark-model (CQM) in the form of
the quark-antiquark pair creation model (QPC).
It has been argued that the $^3P_0$-mechanism \cite{Mic69,LeY73}
is dominant over the $^3S_1$-mechanism in lattice QCD \cite{Isg85}.
It turned out that the fitted coupling constants in ESC04 and ESC08
indeed follow mainly the pattern of couplings set by the $^3P_0$-model.
Also, all $\alpha=F/(F+D)$-ratios are required to deviate no more than 0.1
from the QPC-model predictions for the $BBM$- and the $BB-Pair$-vertices.
Although it is in principle attractive to study the SU(3)-breaking of the $BBM$-couplings
using the QPC-model, as has been explored in ESC04 \cite{Rij04b}, in ESC08 the couplings
are treated as SU(3)-symmetric.
In the Nijmegen soft-core OBE- and ESC-models the BBM-vertices
are described by coupling constants and gaussian form factors.
Given the fact that in the CQM the quark wave functions for the
baryons are very much like ground state harmonic oscillator
functions, a gaussian behavior of the form factors is most natural.
These form factors guarantee a soft behavior of the potentials
in configuration space at small distances.
The cut-off parameters in the form factors depend only on the type of meson
(pseudoscalar, vector, etc.).
Within a meson SU(3)-multiplet we distinguish between octet and singlet form
factors. Since there is
singlet-octet mixing for the I=0 mesons, we attribute the singlet and octet
cut-off to the dominant singlet or octet particle respectively.
For the considered nonets the singlet and octet cut-off are the same or close.
In this way we have full predictive power for the $S=-2,-3,-4$ baryon-baryon channels,
e.g. $\Lambda\Lambda, \Xi N$-channels which involve the singlet
$\{1\}$-irrep that does not occur in the $N\!N$ and $Y\!N$ channels.
Field theory allows both linear and non-linear realizations of chiral-symmetry (CS)
\cite{Schw67,Wei68,DeAlf73}. At low-energy phenomenologically the non-linear
realization is the most economical and natural.
Therefore, we have chosen the pv-coupling and not the ps-coupling
for the pseudoscalar mesons. This choice affects some $1/M^2$-terms in the
ps-ps-exchange potential, In ESC04 we tested mixtures of the pv- and
ps-coupling, but in ESC08 we use only the pv-coupling. In the non-linear
realization chiral-symmetry for the couplings of the scalar-, vector-,
axial-vector-, etc. mesons is realized through isospin-symmetry SU(2,I)
\cite{Wei68,DeAlf73}.
The potentials of the ESC-model are generated by (i) One-Boson-Exchange (OBE),
(ii) uncorrelated Two-Meson-Exchange (TME), (iii) Meson-Pair-Exchange (MPE),
(iv) Diffractive/Multi-gluon Exchange, (v) Quark-Core Effects (QCE).
\begin{enumerate}
\item [(i)]
The OBE-part of the dynamical contents of the ESC08-models is determined
by the following meson-exchanges:
\begin{enumerate}
\item $J^{PC}=0^{--}$: The pseudoscalar-meson nonet $\pi,\ \eta,\ \eta',\ K$ with the
$\eta-\eta'$ mixing angle $\theta_{P}=-13^{0}$ \cite{KLOE09},
close to the \mbox{Gell-Mann-Okubo} quadratic mass formula \cite{GMO62}.
\item $J^{PC}=1^{--}$: The vector-meson nonet $\rho,\ \phi,\ K^{\star},\ \omega$ with
the $\phi-\omega$ mixing angle $\theta_{V}= 38.70^{0}$ \cite{KLOE09}.
This follows from the quadratic GMO mass-formula, and is close to ideal mixing.
\item $J^{PC}=1^{++}$: The axial-vector-meson nonet $a_1, f_1\ K_1, f_1'$ with
the $f_1-f_1'$ mixing angle $\theta_{A}= 50.0^{0}$ \cite{SR97}.
\item $J^{PC}=0^{++}$: The scalar-meson nonet
$a_0(962)=\delta,f_0(993)=S^{\star},\kappa(800),f_0(760)=\varepsilon$
with the ideal $S^{\star}-\varepsilon$ mixing angle $\theta_{S}=35.26^{0}$.
\item $J^{PC}=1^{+-}$: The axial-vector-meson nonet $b_1, f_1\ K_1, f_1'$ with
the $h_1-h_1'$ ideal mixing angle $\theta_{B}= 35.26^{0}$.
\end{enumerate}
The soft-core approach of the OBE has been given originally for
$NN$ in \cite{NRS78}, and for $YN$ in \cite{MRS89}. With respect to these
OBE-interactions the ESC-models contain the modification of the form factor
by introducing a zero for the mesons being P-wave quark-antiquark states in the
CQM: the scalar- and axial-vector-mesons. Such a zero is natural
in the $^3P_0$-quark-pair-creation (QPC) \cite{Mic69,LeY73} model for the coupling
of the mesonic quark-antiquark ($Q\bar{Q}$) system to baryons.
A consequence of such a zero is that a bound state in $\Lambda p$-scattering is less
likely to occur.
\item [(ii)]
The configuration space soft-core uncorrelated two-meson exchange for $NN$ has been derived
in \cite{Rij91,RS96a}. Similarly to ESC04, also in ESC08 we use these potentials for
ps-ps exchange with a complete $SU(3)_f$-symmetric treatment in NN, YN and YY.
For example, we include double $K$-exchange in $NN$-scattering.
Since this includes two-pion exchange (TPE) the long-range part of the
potentials are represented. Here it is tacitly assumed that other TME potentials,
like ps-vc, ps-sc, etc., are either small due to cancellations, or
can be described adequately by using effective couplings in the OBE-potentials.
When these effective couplings do not deviate from
experimentally determined couplings it may be assumed that the corrections
from these other SU(3) meson-nonets in the TME potentials are small.
This is our working hypothesis for the TME-potentials.
From the point of view of SU(3), since OBE contains only $\{8\}$- and $\{1\}$-exchange,
TME can not be represented completely in terms of OBE. This because TME also has
$\{27\}-, \{10\}$-, and $\{10^*\}$-exchange components.
Therefore, the predictions made by the ESC-models could be sensitive
to this incompleteness of TME in the ESC-models.
At present the BB-data and the hypernuclear-data do not give information at this point.
\item [(iii)]
Meson-pair exchanges (MPE) have been introduced in \cite{Rij93} for $NN$ and described
in detail in \cite{RS96b}. The two-meson-baryon-baryon vertices are the low energy
approximations of (a) the heavy-meson and their two-meson decays, and (b) baryon-resonance
contributions $\Delta_{33}$ etc \cite{RS96b,SR97}.
\item [(iv)]
Diffractive contributions to the soft-core potential have been
introduced from the beginning, cfr. \cite{NRS78}. The pomeron
is thought of being related to an even number of gluon-exchanges.
Here we introduce the Odderon-potential,
which is related to an odd number of gluon exchanges.
\begin{enumerate}
\item $J^{PC}=0^{++}$: The `diffractive' contribution from the pomeron P,
which is a unitary singlet.
These interactions give a repulsive contribution to the
potentials in all channels of a gaussian type.
\item $J^{PC}=1^{--}$: The `diffractive' contribution from the odderon O.
The origin of the odderon is assumed to be purely the exchange of
the color-singlets with an odd number of gluons. Similarly to the
pomeron, the odderon potential is taken to be
an SU(3)$_F$ singlet and of the gaussian form.
\end{enumerate}
\noindent As an explanation of the repulsive character of the pomeron-potential the following:
The $J^{PC}$ is identical to that for the scalar-mesons. Naively, one would expect an
attractive central potential. However, considering the two-gluon model for the
pomeron \cite{Low75,Nus75} the two-gluon parallel and crossed diagram contributions
to the BB-interaction can be shown to cancel adiabatically. The remaining
non-adiabatic contribution is repulsive \cite{Pad89}.
\item [(v)]
Quark-Core-Effects in the soft-core model can supply extra repulsion,
which may be required in some BB-channels.
Baryon-baryon studies with the soft-core OBE and ESC-models
thus far show that it is difficult to achieve a strongly enough repulsive short-range
interactions in (i) the $\Sigma^+ p(I=3/2,^3S_1)$- and (ii) the
$\Sigma N(I=1/2,^1S_0)$-channel. The short-range repulsion in baryon-baryon may
in principle come from: (a) meson- and multi-gluon-exchange \cite{Rij04a,Rij04b},
and/or (b) the occurrence of forbidden six-quark SU(6)-states
by the Pauli-principle \cite{Ots65,Oka00,Fuj07}. In view of the mentioned difficulties,
we have developed a phenomenological method for the ESC-model, which enables us
to incorporate this quark-structural effect. This is an important new ingredient
of the here presented ESC08-model. This structural effect we describe phenomenologically
by gaussian repulsions, similar to the pomeron.
In the ESC08c-model we take the strength of this repulsion proportional to the
weights of the SU(6)-forbidden [51]-configuration in the various BB-channels.
This in contrast to ESC08a,b \cite{HYPX,PTP185.a} where the quark-core effect is
only included in the BB-channels with a dominant occurrence of the [51]-configuration.
\end{enumerate}
The different sources of $SU(3)$-breaking are discussed in paper II of this series.
With this simultaneous treatment of the $N\!N$, $Y\!N$, and $Y\!Y$ channels we have
achieved
a high quality description of the baryon-baryon interactions. The results,
using a single set of meson and quark-core parameters, include:
(a) a description of the NN-data with a $\chi^2_{pdp} = 1.081$
and good low energy parameters for the NN-channels including the binding energy
$E_B$ of the deuteron,
(b) a very good fit to the YN-scattering data.
(c) the fitting parameters with a clear physical significance, like e.g. the
$NN\pi$-, $NN\rho$-couplings etc. and with realistic values of the
$F/(F+D)$-ratio's $\alpha_{PV}$ and $\alpha_{V}^{m}$.
The fitting has been done under the constraints of the G-matrix results for the
ESC08-interactions. These show (i) satisfactory well-depth values for
$U_\Lambda, U_\Sigma > 0$, and $U_\Xi < 0$,
(ii) proper spin-spin ($U_{\sigma\sigma} \geq 1$,
and small spin-orbit interactions for $\Lambda N$.
All these features are in agreement with the Hyperball-data \cite{Has06} and the
NAGARA-event \cite{Tak01}.
As in all Nijmegen models, the Coulomb interaction is included exactly,
for which we solve the multichannel
Schr\"{o}dinger equation on the physical particle basis.
The nuclear potentials are calculated on the isospin basis.
This means that we include only the so-called 'medium strong'
SU(3)-breaking and the charge symmetry breaking (CSB) in the potentials.
\section{Two-Body Integral Equations in Momentum Space}
\label{sec:2}
\subsection{Three-dimensional Two-Body Equations}
We consider the baryon-baryon reactions
\begin{eqnarray}
B(p_{a},s_{a})+B(p_{b},s_{b}) \rightarrow
B(p_{a'},s_{a'})+B(p_{b'},s_{b'}) &&
\label{eq:30.1} \end{eqnarray}
In the following we also refer to a and a' as particles 1 and 1' (or 3),
and to b and b' as particles 2 and 2' (or 4).
The total four-momenta for the initial and the final states are denoted
as $P = p_{a} + p_{b}, P' = p_{a'} + p_{b'}$, and similarly the relative momenta
by $p = \frac{1}{2}(p_{a}-p_{b}), p' = \frac{1}{2}(p_{a'}-p_{b'})$.
In the center-of-mass system (CM-system) for a and b
on-mass-shell one has
$P = ( W , {\bf 0}) \hspace{0.2cm} , \hspace{0.2cm} p = ( 0 , {\bf p})
\hspace{0.2cm} , \hspace{0.2cm} p' = ( 0 , {\bf p}')$.
In the following, the on-mass-shell CM-momenta for the initial
and final states are denoted respectively by ${\bf p}$ and ${\bf p}'$.
So, $p_{a}^{0}=E_{a}({\bf p})=\sqrt{{\bf p}^{2}+M_{a}^{2}}$ and
$p_{a'}^{0}=E_{a'}({\bf p}')=\sqrt{{\bf p'}^{2}+M_{a'}^{2}}$, and
similarly for b(2) and b'(4).
Because of translation-invariance $P=P'$ and
$W=W'=E_{a}({\bf p})+E_{b}({\bf p})=E_{a'}({\bf p}')+E_{b'}({\bf p}')$.
The transition amplitude matrix $M$ is related to the $S$-matrix via
\begin{equation}
\langle f|S|i\rangle = \langle f|i\rangle -i(2\pi)^4\delta^4(P_f-P_i)
\langle f| M | i \rangle.
\label{eq:30.2} \end{equation}
The two-particle states we normalize in the following way
\begin{eqnarray}
\langle {\bf p}_{1}',{\bf p}_{2}'|{\bf p}_{1},{\bf p}_{2}\rangle
&=& (2\pi)^{3}2E({\bf p}_{1}) \delta^{3}({\bf p}_{1}'-{\bf p}_{1})\cdot
\nonumber\\ && \times
(2\pi)^{3}2E({\bf p}_{2}) \delta^{3}({\bf p}_{2}'-{\bf p}_{2}).
\label{eq:30.3} \end{eqnarray}
Three-dimensional integral equations for the amplitudes $\langle f|M|i\rangle$
have been derived in various ways, see e.g. \cite{Log63,Thom70,Ger75,NRS77,Rij85}.
Here, we follow Ref.~\cite{Rij04a} which employs the Macke-Klein procedure \cite{Klein53}.
After redefining the CM-amplitude $M({\bf p}',{\bf p} |W)$ by
\onecolumngrid
\begin{equation}
M({\bf p}',{\bf p} |W) \rightarrow
\sqrt{\frac{M_{a}M_{b}}{E_{a}({\bf p}') E_{b}({\bf p}')} }
M({\bf p}',{\bf p} |W)
\sqrt{\frac{M_{a}M_{b}}{E_{a}({\bf p}') E_{b}({\bf p}')} }
\label{eq:30.4} \end{equation}
one arrives, see for details Ref.~\cite{Rij04a}, at the Thompson equation \cite{Thom70}
\begin{eqnarray}
M({\bf p}',{\bf p}| W) &=& K^{irr}({\bf p}',{\bf p}|W) +
\int\!\frac{d^{3}p^{\prime\prime}}{(2\pi)^3}
K^{irr}({\bf p}',{\bf p}^{\prime\prime}|W)\ E_{2}^{(+)}({\bf p}^{\prime\prime}; W)\
M({\bf p}^{\prime\prime},{\bf p}|W),
\nonumber\\
\label{eq:30.22} \end{eqnarray}
where
$ E_{2}^{(+)}({\bf p}^{\prime\prime}; W)=
\left( W-{\cal W}({\bf p}^{\prime\prime})+i\delta\right)^{-1} $,
and the two-nucleon irreducible kernel is given by
\begin{eqnarray}
K^{{\it irr}}({\bf p}',{\bf p}| W)&=& -\frac{1}{(2\pi)^{2}}
\sqrt{\frac{M_{a}M_{b}}{E_{a}({\bf p}') E_{b}({\bf p}')} }
\sqrt{\frac{M_{a}M_{b}}{E_{a}({\bf p}) E_{b}({\bf p})} }
\left(W-{\cal W}({\bf p}')\right)\left(W-{\cal W}({\bf p})\right)
\nonumber \\[0.2cm] &\times&
\int_{-\infty}^{+\infty} dp_{0}'
\int_{-\infty}^{+\infty} dp_{0} \left[ \vphantom{\frac{A}{A}}
\left\{F_{W}^{(a)}({\bf p}',p_{0}')
F_{W}^{(b)}(-{\bf p}',-p_{0}')\right\}^{-1}
\right.\nonumber \\[0.2cm] &\times& \left.
\left[ I(p_{0}',{\bf p}'; p_{0},{\bf p}) \right]_{++,++}
\left\{F_{W}^{(a)}({\bf p},p_{0})
F_{W}^{(b)}(-{\bf p},-p_{0})\right\}^{-1}
\vphantom{\frac{A}{A}}\right],
\label{Thomp2} \end{eqnarray}
where $F_W({\bf p},p_0)= p_0-E({\bf p})+W/2+i\delta$.
This same expression for the kernel was exploited in \cite{Rij91,RS96a,RS96b}.
In case one does not assume the strong pair-suppression, one must study instead
of equation (\ref{eq:30.22}) a more general equation with couplings between the
positive and negative energy spinorial amplitudes. Also to this more general case
one can apply the described three-dimensional reduction, and we refer the reader to
Ref.~\cite{Klein74} for a treatment of this case.
The $M/E$-factors in (\ref{Thomp2}) are due to the difference
between the relativistic and the non-relativistic normalization of
the two-particle states. In the following we simply put
$M/E({\bf p})=1$ in the kernel $K^{irr}$ Eq.~(\ref{Thomp2}). The corrections
to this approximation would give $(1/M)^{2}$-corrections
to the potentials, which we neglect in this paper. In the same approximation
there is no difference between the Thompson \cite{Thom70}
and the Lippmann-Schwinger equation, when the connection between these
equations is made using multiplication factors. Henceforth, we will not
distinguish between the two.
The contributions to the two-particle irreducible kernel
$K^{{\it irr}}$ up to second order in the meson-exchange
are given in detail in \cite{RS96a,RS96b}.\\
\twocolumngrid
\subsection{Lippmann-Schwinger Equation}
\label{sec:2b}
\begin{figure}[hbt]
\begin{center} \begin{picture}(200,180)(0,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.5}
\ArrowLine(15,10)(15,50)
\Line(15,50)(15,130)
\ArrowLine(15,130)(15,170)
\ArrowLine(85,10)(85,50)
\Line(85,50)(85,130)
\ArrowLine(85,130)(85,170)
\Vertex(15,50){3}
\Vertex(85,130){3}
\DashArrowLine(85,130)(15,50){3}
\PText(15, 00)(0)[b]{p}
\PText(15,175)(0)[b]{p'}
\PText(85, 00)(0)[b]{-p}
\PText(85,175)(0)[b]{-p'}
\PText(50,100)(0)[b]{k}
\PText(50,-5 )(0)[b]{(a)}
\SetOffset(10,0)
\ArrowLine(115,10)(115,50)
\Line(115,50)(115,130)
\ArrowLine(115,130)(115,170)
\ArrowLine(185,10)(185,50)
\Line(185,50)(185,130)
\ArrowLine(185,130)(185,170)
\Vertex(115,130){3}
\Vertex(185, 50){3}
\DashArrowLine(185,50)(115,130){3}
\PText(115, 00)(0)[b]{p}
\PText(115,175)(0)[b]{p'}
\PText(185, 00)(0)[b]{-p}
\PText(185,175)(0)[b]{-p'}
\PText(150,100)(0)[b]{k}
\PText(150,-5 )(0)[b]{(b)}
\end{picture}
\end{center}
\caption{One-boson-exchange graphs:
The dashed lines with momentum ${\bf k}$ refers to the
bosons: pseudo-scalar, vector, axial-vector, or scalar mesons.}
\label{obefig}
\end{figure}
\begin{figure}[hbt]
\begin{center} \begin{picture}(220,350)(0,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.5}
\Vertex(15,220){3}
\Vertex(15,260){3}
\Vertex(85,280){3}
\Vertex(85,320){3}
\ArrowLine(15,190)(15,220)
\ArrowLine(15,220)(15,260)
\Line(15,260)(15,320)
\ArrowLine(15,320)(15,350)
\ArrowLine(85,190)(85,220)
\Line(85,220)(85,280)
\ArrowLine(85,280)(85,320)
\ArrowLine(85,320)(85,350)
\DashArrowLine(85,280)(15,220){3}
\DashArrowLine(85,320)(15,260){3}
\PText( 5,195)(0)[b]{p}
\PText( 5,345)(0)[b]{p'}
\PText(95,195)(0)[b]{-p}
\PText(95,345)(0)[b]{-p'}
\PText( 5,240)(0)[b]{p''}
\PText(95,300)(0)[b]{-p''}
\PText(50,230)(0)[b]{k}
\PText(50,300)(0)[b]{k'}
\PText(50,180)(0)[b]{(a)}
\Vertex(125,220){3}
\Vertex(125,260){3}
\Vertex(195,280){3}
\Vertex(195,320){3}
\ArrowLine(125,190)(125,220)
\ArrowLine(125,220)(125,260)
\Line(125,260)(125,320)
\ArrowLine(125,320)(125,350)
\ArrowLine(195,190)(195,220)
\Line(195,220)(195,280)
\ArrowLine(195,280)(195,320)
\ArrowLine(195,320)(195,350)
\DashArrowLine(195,320)(160,270){3}
\DashLine(160,270)(125,220){3}
\DashArrowLine(195,280)(160,270){3}
\DashLine(160,270)(125,260){3}
\PText(115,195)(0)[b]{p}
\PText(115,345)(0)[b]{p'}
\PText(205,195)(0)[b]{-p}
\PText(205,345)(0)[b]{-p'}
\PText(115,240)(0)[b]{p''}
\PText(205,300)(0)[b]{-p''}
\PText(140,220)(0)[b]{k}
\PText(140,270)(0)[b]{k'}
\PText(160,180)(0)[b]{(b)}
\Vertex(15, 40){3}
\Vertex(85, 60){3}
\Vertex(15,100){3}
\Vertex(85,140){3}
\ArrowLine(15, 10)(15, 40)
\ArrowLine(15, 40)(15,100)
\Line(15,100)(15,140)
\ArrowLine(15,140)(15,170)
\ArrowLine(85, 10)(85, 40)
\Line(85, 40)(85, 60)
\ArrowLine(85, 60)(85,140)
\ArrowLine(85,140)(85,170)
\DashArrowLine(85,140)(15, 40){3}
\DashArrowLine(85, 60)(15,100){3}
\PText( 5, 15)(0)[b]{p}
\PText( 5,160)(0)[b]{p'}
\PText(95, 15)(0)[b]{-p}
\PText(95,160)(0)[b]{-p'}
\PText( 5, 70)(0)[b]{p''}
\PText(95, 90)(0)[b]{-p''}
\PText(40, 50)(0)[b]{k}
\PText(40,100)(0)[b]{k'}
\PText(50, 00)(0)[b]{(c)}
\Vertex(125, 40){3}
\Vertex(195, 80){3}
\Vertex(195,120){3}
\Vertex(125,140){3}
\ArrowLine(125, 10)(125, 40)
\ArrowLine(125, 40)(125,140)
\ArrowLine(125,140)(125,170)
\ArrowLine(195, 10)(195, 40)
\Line(195, 40)(195, 80)
\ArrowLine(195, 80)(195,120)
\Line(195,120)(195,140)
\ArrowLine(195,140)(195,170)
\DashArrowLine(195,120)(125, 40){3}
\DashArrowLine(195, 80)(125,140){3}
\PText(115, 15)(0)[b]{p}
\PText(115,160)(0)[b]{p'}
\PText(205, 15)(0)[b]{-p}
\PText(205,160)(0)[b]{-p'}
\PText(115, 90)(0)[b]{p''}
\PText(205,100)(0)[b]{-p''}
\PText(150, 50)(0)[b]{k}
\PText(150,130)(0)[b]{k'}
\PText(160, 00)(0)[b]{(d)}
\end{picture}
\end{center}
\caption{BW two-meson-exchange graphs: (a) planar and (b)--(d) crossed
box. The dashed line with momentum ${\bf k}_{1}$ refers to the
pion and the dashed line with momentum ${\bf k}_{2}$ refers
to one of the other (vector, scalar, or pseudoscalar) mesons.
To these we have to add the ``mirror'' graphs, and the
graphs where we interchange the two meson lines.}
\label{bwfig}
\end{figure}
\begin{figure}[hbt]
\begin{center} \begin{picture}(220,170)(0,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.5}
\Vertex(15, 40){3}
\Vertex(15,100){3}
\Vertex(85, 70){3}
\Vertex(85,130){3}
\ArrowLine(15, 10)(15, 40)
\ArrowLine(15, 40)(15,100)
\Line(15,100)(15,120)
\ArrowLine(15,120)(15,170)
\ArrowLine(85, 10)(85, 40)
\Line(85, 40)(85, 70)
\ArrowLine(85, 70)(85,130)
\ArrowLine(85,130)(85,170)
\DashArrowLine(85, 70)(15, 40){3}
\DashArrowLine(85,130)(15,100){3}
\PText( 5, 15)(0)[b]{p}
\PText( 5,160)(0)[b]{p'}
\PText(95, 15)(0)[b]{-p}
\PText(95,160)(0)[b]{-p'}
\PText( 5, 70)(0)[b]{p''}
\PText(95, 90)(0)[b]{-p''}
\PText(40, 35)(0)[b]{k}
\PText(40,120)(0)[b]{k'}
\PText(50, 00)(0)[b]{(a)}
\Vertex(125, 40){3}
\Vertex(125,130){3}
\Vertex(195, 70){3}
\Vertex(195,100){3}
\ArrowLine(125, 10)(125, 40)
\ArrowLine(125, 40)(125,130)
\ArrowLine(125,130)(125,170)
\ArrowLine(195, 10)(195, 40)
\Line(195, 40)(195, 70)
\ArrowLine(195, 70)(195,100)
\Line(195,100)(195,130)
\ArrowLine(195,130)(195,170)
\DashArrowLine(195, 70)(125, 40){3}
\DashArrowLine(195,100)(125,130){3}
\PText(115, 15)(0)[b]{p}
\PText(115,160)(0)[b]{p'}
\PText(205, 15)(0)[b]{-p}
\PText(205,160)(0)[b]{-p'}
\PText(115, 85)(0)[b]{p''}
\PText(205, 85)(0)[b]{-p''}
\PText(150, 35)(0)[b]{k}
\PText(150,130)(0)[b]{k'}
\PText(160, 00)(0)[b]{(b)}
\end{picture}
\end{center}
\caption{Planar-box TMO two-meson-exchange graphs.
Same notation as in Fig.~\protect\ref{bwfig}.
To these we have to add the ``mirror'' graphs, and the
graphs where we interchange the two meson lines.}
\label{tmofig}
\end{figure}
\twocolumngrid
The transformation of (\ref{eq:30.22}) to the Lippmann-Schwinger
equation can be effectuated by defining
\begin{eqnarray}
T({\bf p}',{\bf p}) &=& N({\bf p}')\ M({\bf p}',{\bf p}|W)\ N({\bf p}),
\nonumber\\[0.2cm]
V({\bf p}',{\bf p}) &=& N({\bf p}')\ K^{irr}({\bf p}',{\bf p}|W)\ N({\bf p}),
\label{eq:30.24} \end{eqnarray}
where the transformation function is
\begin{equation}
N({\bf p}) = \sqrt{\frac{{\bf p}_i^2-{\bf p}^2}{2M_N(E\left({\bf p}_i)-E({\bf p})\right)}}.
\label{eq:30.25} \end{equation}
Application of this transformation, yields the Lippmann-Schwinger equation
\begin{eqnarray}
T({\bf p}',{\bf p}) &=& V({\bf p}',{\bf p}) +
\int \frac{d^{3}p''}{(2\pi)^3}\
\nonumber\\ && \times
V({\bf p}',{\bf p}'')\ g({\bf p}''; W)\; T({\bf p}'',{\bf p})
\label{eq:30.26} \end{eqnarray}
with the standard Green function
\begin{equation}
g({\bf p};W) =
\frac{M_N}{{\bf p}_i^{2}-{\bf p}^{2}+i\delta}.
\label{eq:30.27} \end{equation}
The corrections to the approximation $E_{2}^{(+)} \approx g({\bf p}; W)$
are of order $1/M^{2}$, which we neglect henceforth.
The transition from Dirac-spinors to
Pauli-spinors, is given in Appendix C of Ref.~\cite{Rij91}, where we write for the
the Lippmann-Schwinger equation in the 4-dimensional Pauli-spinor space
\begin{eqnarray}
{\cal T}({\bf p}',{\bf p})&=&{\cal V}({\bf p}',{\bf p}) + \int \frac{d^{3} p''}{(2\pi)^3}\
\nonumber\\ && \times
{\cal V}({\bf p}',{\bf p}'')\ g({\bf p}''; W)\ {\cal T}({\bf p}'',{\bf p})\ .
\label{eq:30.28} \end{eqnarray}
The ${\cal T}$-operator in Pauli spinor-space is defined by
\begin{eqnarray}
&& \chi^{(a)\dagger}_{\sigma'_{a}}\chi^{(b)\dagger}_{\sigma'_{b}}\;
{\cal T}({\bf p}',{\bf p})\;
\chi^{(a)}_{\sigma_{a}}\chi^{(b)}_{\sigma_{b}} =
\nonumber\\ &&
\bar{u}_{a}({\bf p}',\sigma'_{a})\bar{u}_{b}(-{\bf p}',\sigma'_{b})\
\tilde{T}({\bf p}',{\bf p})\; u_{a}({\bf p},\sigma_{a}) u_{b}(-{\bf p},\sigma_{b}).
\nonumber\\
\label{eq:30.29} \end{eqnarray}
and similarly for the ${\cal V}$-operator.
Like in the derivation of the OBE-potentials \cite{NRS78,NRS77}
we make off-shell and on-shell the approximation,
$ E({\bf p})= M + {\bf p}^{2}/2M $
and $ W = 2\sqrt{{\bf p}_i^{2}+M^{2}} = 2M + {\bf p}_i^{2}/M$ ,
everywhere in the interaction kernels, which, of course,
is fully justified for low energies only.
In contrast to these kinds of approximations, of course the full
${\bf k}^{2}$-dependence of the form factors is kept
throughout the derivation of the TME.
Notice that the gaussian form factors suppress the high momentum
transfers strongly. This means that the contribution to the potentials
from intermediate states which are far off-energy-shell can not
be very large.
Because of rotational invariance and parity conservation, the ${\cal T}$-matrix, which is
a $4\times 4$-matrix in Pauli-spinor space, can be expanded
into the following set of in general 8 spinor invariants, see for example
Ref.~\cite{SNRV71}. Introducing \cite{notation1}
\begin{equation}
{\bf q}=\frac{1}{2}({\bf p}'+{\bf p})\ , \
{\bf k}={\bf p}'-{\bf p}\ , \
{\bf n}={\bf p}\times {\bf p}',
\label{eq:30.30} \end{equation}
with, of course, ${\bf n}={\bf q}\times {\bf k}$,
we choose for the operators $P_{j}$ in spin-space
\begin{eqnarray}
&& P_{1}=1, \hspace{3mm} P_{2}=
\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2,
\nonumber\\[0.0cm]
&& P_{3}=(\mbox{\boldmath $\sigma$}_1\cdot{\bf k})(\mbox{\boldmath $\sigma$}_2\cdot{\bf k})
-\frac{1}{3}(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
{\bf k}^2,
\nonumber\\[0.0cm]
&& P_{4}=\frac{i}{2}(\mbox{\boldmath $\sigma$}_1+
\mbox{\boldmath $\sigma$}_2)\cdot{\bf n}, \hspace{3mm}
P_{5}=(\mbox{\boldmath $\sigma$}_1\cdot{\bf n})(\mbox{\boldmath $\sigma$}_2\cdot{\bf n}),
\nonumber\\[0.0cm]
&& P_{6}=\frac{i}{2}(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2)\cdot{\bf n},
\nonumber\\[0.0cm]
&& P_{7}=(\mbox{\boldmath $\sigma$}_1\cdot{\bf q})(\mbox{\boldmath $\sigma$}_2\cdot{\bf k})
+(\mbox{\boldmath $\sigma$}_1\cdot{\bf k})(\mbox{\boldmath $\sigma$}_2\cdot{\bf q}),
\nonumber\\[0.0cm]
&& P_{8}=(\mbox{\boldmath $\sigma$}_1\cdot{\bf q})(\mbox{\boldmath $\sigma$}_2\cdot{\bf k})
-(\mbox{\boldmath $\sigma$}_1\cdot{\bf k})(\mbox{\boldmath $\sigma$}_2\cdot{\bf q}).
\label{eq:30.31} \end{eqnarray}
Here we follow Ref.~\cite{MRS89}, where in contrast to Ref.~\cite{NRS78},
we have chosen $P_{3}$ to be a purely `tensor-force' operator.
The expansion in spinor-invariants reads
\begin{equation}
{\cal T}({\bf p}',{\bf p}) = \sum_{j=1}^8\ \widetilde{T}_j({\bf p}^{\prime 2},{\bf p}^2,
{\bf p}'\cdot{\bf p})\ P_j({\bf p}',{\bf p})\ .
\label{eq:30.32} \end{equation}
Similarly to (\ref{eq:30.32}) we expand the potentials $V$.
In the case of the axial-vector meson exchange there will occur terms
proportional to
\begin{equation}
P_5'=(\mbox{\boldmath $\sigma$}_1\cdot{\bf q})(\mbox{\boldmath $\sigma$}_2\cdot{\bf q})
-\frac{1}{3}(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2){\bf q}^2.
\label{eq:30.33} \end{equation}
The treatment of such a Pauli-invariant using the Okubo-Marshak identity \cite{Ok58},
see also Ref.~\cite{SNRV71}, is not without problems because it involves the division with
${\bf k}^2$. Therefore, in the ESC04-models \cite{Rij04a,Rij04b} the replacement
$P_5' \rightarrow -P_3$ was chosen. For the ESC08-models a satisfactory treatment
has been developed, which is described in Appendix~\ref{app:B}.
For the treatment of the potentials with $P_8$ we use the identity \cite{BDI70}
\begin{equation}
P_8 = -(1+\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2) P_6.
\label{eq:30.34} \end{equation}
Under time-reversal $P_7 \rightarrow -P_7$ and $P_8 \rightarrow -P_8$.
Therefore for elastic scattering $V_7=V_8=0$.
Anticipating the explicit results for the potentials in section~\ref{sec:3} we
notice the following:
(i) For the general BB-reaction we will find no contribution to $V_7$.
The operators $P_6$ and $P_8$ give spin singlet-triplet transitions.
(ii) In the case of non-strangeness-exchange ($\Delta S=0$), $V_6 \neq 0$
and $V_8$=0. The latter follows from our approximation to neglect the
mass differences among the nucleons, between the $\Lambda$ and $\Sigma$'s, and
among the $\Xi$'s.
(iii) In the case of strangeness-exchange ($\Delta S=\pm 1$), $V_6,V_8 \neq 0$.
The contributions to $V_6$ come from graphs with both spin- and particle-exchange,
i.e. Majorana-type potentials having the $P_f P_\sigma P_6= -P_x P_6$-operator.
Here, $P_f P_\sigma$ reflect our convention for the two-particle wave functions,
see \cite{NRS77}.
The contributions to $V_8$ come from graphs with particle-exchange and
spin-exchange, because $P_8=-P_\sigma P_6$. Therefore, we only have to
apply $P_f$ in order to map the wave functions after such exchange onto
our two-particle wave-functions. So, we have the $P_f P_8= +P_x P_6$-operator.
Here, we used that for BB-systems the allowed physical states satisfy
$P_f P_\sigma P_x=-1$.
\begin{figure}[hbt]
\begin{center} \begin{picture}(220,350)(0,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.5}
\Vertex(15,230){3}
\Vertex(85,270){3}
\Vertex(85,310){3}
\ArrowLine(15,190)(15,230)
\Line(15,230)(15,310)
\ArrowLine(15,310)(15,350)
\ArrowLine(85,190)(85,220)
\Line(85,220)(85,270)
\ArrowLine(85,270)(85,310)
\ArrowLine(85,310)(85,350)
\DashArrowLine(85,270)(15,230){3}
\DashArrowLine(85,310)(15,230){3}
\PText( 5,195)(0)[b]{p}
\PText( 5,345)(0)[b]{p'}
\PText(95,195)(0)[b]{-p}
\PText(95,345)(0)[b]{-p'}
\PText(95,290)(0)[b]{-p''}
\PText(50,230)(0)[b]{k}
\PText(50,280)(0)[b]{k'}
\PText(50,180)(0)[b]{(a)}
\Vertex(125,310){3}
\Vertex(195,230){3}
\Vertex(195,270){3}
\ArrowLine(125,190)(125,230)
\Line(125,230)(125,310)
\ArrowLine(125,310)(125,350)
\ArrowLine(195,190)(195,230)
\ArrowLine(195,230)(195,270)
\Line(195,270)(195,310)
\ArrowLine(195,310)(195,350)
\DashArrowLine(195,230)(125,310){3}
\DashArrowLine(195,270)(125,310){3}
\PText(115,195)(0)[b]{p}
\PText(115,345)(0)[b]{p'}
\PText(205,195)(0)[b]{-p}
\PText(205,345)(0)[b]{-p'}
\PText(205,250)(0)[b]{-p''}
\PText(160,250)(0)[b]{k}
\PText(160,300)(0)[b]{k'}
\PText(160,180)(0)[b]{(b)}
\Vertex(15, 85){3}
\Vertex(85, 40){3}
\Vertex(85,130){3}
\ArrowLine(15, 10)(15, 40)
\Line(15, 40)(15, 85)
\Line(15, 85)(15,130)
\ArrowLine(15,130)(15,170)
\ArrowLine(85, 10)(85, 40)
\ArrowLine(85, 40)(85,130)
\ArrowLine(85,130)(85,170)
\DashArrowLine(85, 40)(15, 85){3}
\DashArrowLine(85,130)(15, 85){3}
\PText( 5, 15)(0)[b]{p}
\PText( 5,160)(0)[b]{p'}
\PText(95, 15)(0)[b]{-p}
\PText(95,160)(0)[b]{-p'}
\PText(95, 85)(0)[b]{-p''}
\PText(40, 55)(0)[b]{k}
\PText(40,110)(0)[b]{k'}
\PText(50, 00)(0)[b]{(c)}
\Vertex(125, 50){3}
\Vertex(195,120){3}
\ArrowLine(125, 10)(125, 50)
\Line(125, 50)(125,120)
\ArrowLine(125,120)(125,170)
\ArrowLine(195, 10)(195, 50)
\Line(195, 50)(195,120)
\ArrowLine(195,120)(195,170)
\DashArrowArc(195,50)(70,90,180){3}
\DashArrowArcn(125,120)(70,00,-90){3}
\PText(115, 15)(0)[b]{p}
\PText(115,160)(0)[b]{p'}
\PText(205, 15)(0)[b]{-p}
\PText(205,160)(0)[b]{-p'}
\PText(150, 40)(0)[b]{k}
\PText(150,110)(0)[b]{k'}
\PText(160, 00)(0)[b]{(d)}
\end{picture}
\end{center}
\caption{One- and Two-Pair exchange graphs.
To these we have to add the ``mirror'' graphs, and the
graphs where we interchange the two meson lines.}
\label{pairfig}
\end{figure}
\section{Extended-Soft-Core Potentials in Momentum Space}
\label{sec:3}
The potential of the ESC-model contains the contributions from
(i) One-boson-exchanges, Fig.~\ref{obefig}, (ii) Uncorrelated
Two-Pseudo-scalar exchange, Fig.~\ref{bwfig} and Fig.~\ref{tmofig},
and (iii) Meson-Pair-exchange, Fig~\ref{pairfig}. In this section we
review the potentials and indicate the changes with respect to
earlier papers on the OBE- and ESC-models.
The spin-1 meson-exchange is an important ingredient for the baryon-baryon force.
In the ESC08-model we treat the vector-mesons and the axial-vector mesons
according to the Proca- \cite{IZ80} and the B-field \cite{Nak72,NO90} formalism
respectively. For details, we refer to Appendix~\ref{app:C}.
\subsection{One-Boson-Exchange Interactions in Momentum Space}
\label{sect.3a}
The OBE-potentials are the
same as given in \cite{NRS78,MRS89}, with the exception of
(i) the zero in the scalar form factor, and
(ii) the axial-vector-meson potentials.
Here, we review the OBE-potentials briefly, and give those potentials
which are not included in the above references.
The local interaction Hamilton densities for the different couplings
are \cite{BD65} \\ \\
a) Pseudoscalar-meson exchange $(J^{PC}=0^{-+})$
\begin{equation}
{\cal H}_{PV}= \frac{f_{P}}{m_{\pi^{+}}}
\bar{\psi}\gamma_{\mu}\gamma_{5}
\psi\partial^{\mu}\phi_{P}. \label{eq:3.1}\end{equation}
This is the pseudovector coupling, and the
relation with the pseudoscalar coupling is
$g_P = 2M_B/m_{\pi^+}$, where $M_B$ is the nucleon or hyperon mass.\\ \\
b) Vector-meson exchange $(J^{PC}=1^{--})$
\begin{equation}
{\cal H}_{V}=g_{V}\bar{\psi}\gamma_{\mu}\psi\phi_{V}^{\mu}
+\frac{f_{V}}{4{\cal M}}\bar{\psi}\sigma_{\mu\nu}
\psi (\partial^{\mu}\phi^{\nu}_{V}-\partial^{\nu}
\phi^{\mu}_{V}), \label{eq:3.2}\end{equation}
where $\sigma_{\mu\nu}= i[\gamma_{\mu},\gamma_{\nu}]/2$,
and the scaling mass ${\cal M}$,
will be taken to be the proton mass.\\ \\
\noindent c)\ Axial-vector-meson exchange ( $J^{PC}=1^{++}$, 1$^{st}$ kind):
\begin{equation}
{\cal H}_A = g_A[\bar{\psi}\gamma_\mu\gamma_5\psi] \phi^\mu_A + \frac{if_A}{{\cal M}}
[\bar{\psi}\gamma_5\psi]\ \partial_\mu\phi_A^\mu.
\label{eq:OBE.1}\end{equation}
In ESC04 the $g_A$-coupling was included, but not the derivative $f_A$-coupling \cite{dercopax}.
Also, in ESC04 we used a local-tensor approximation (LTA) for the
$(\mbox{\boldmath $\sigma$}_1\cdot{\bf q})(\mbox{\boldmath $\sigma$}_2
\cdot{\bf q})$ operator. Here, we improve on that considerably by avoiding such
rather crude approximation. The details of our new treatment are given in
Appendix~\ref{app:B}.
\\[0.2cm]
\noindent d)\ Axial-vector-meson exchange ( $J^{PC}=1^{+-}$, 2$^{nd}$ kind):
\begin{equation}
{\cal H}_B = \frac{if_B}{m_B}
[\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi]\ \partial_\nu\phi_B^\mu\ .
\label{eq:OBE.2}\end{equation}
In ESC04 this coupling was not included. Like for the axial-vector mesons of the
1$^{st}$-kind we include an SU(3)-nonet with members $b_1(1235), h_1(1170), h_1(1380)$.
In the quark-model they are $Q\Bar{Q}(^1P_1)$-states.\\[0.2cm]
\noindent e)\ Scalar-meson exchange ($J^{PC}=0^{++}$):
\begin{equation}
{\cal H}_S = g_S[\bar{\psi}\psi] \phi_S + \frac{f_S}{{\cal M}}
[\bar{\psi}\gamma_\mu\psi]\ \partial^\mu\phi_S,
\label{eq:OBE.3}\end{equation}
which is the most general interaction up to the first derivative.
However, charge conjugation gives
${\cal C}[\bar{\psi}\gamma_\mu \psi] {\cal C}^{-1}=-[\bar{\psi}\gamma_\mu\psi]$,
and therefore $f_S=0$.
\noindent f)\ Pomeron-exchange ($J^{PC}=0^{++}$):
The vertices for this `diffractive'-exchange have the
same Lorentz structure as those for scalar-meson-exchange.\\[0.2cm]
\noindent g)\ Odderon-exchange ($J^{PC}=1^{--}$):
\begin{equation}
{\cal H}_O = g_O[\bar{\psi}\gamma_\mu\psi] \phi^\mu_O + \frac{f_O}{4{\cal M}}
[\bar{\psi}\sigma_{\mu\nu}\psi] (\partial^\mu\phi^\nu_O-\partial^\nu\phi_O^\mu).
\label{eq:OBE.4}\end{equation}
Since the gluons are flavorless, Odderon-exchange is treated as an SU(3)-singlet.
Furthermore, since the Odderon represents a Regge-trajectory with an intercept
equal to that of the Pomeron, and is supposed not to contribute for small ${\bf k}^2$,
we include a factor ${\bf k}^2/{\cal M}^2$ in the coupling.\\[0.2cm]
Including form factors $f({\bf x}'-{\bf x})$ ,
the interaction hamiltonian densities are modified to
\begin{equation}
H_{X}({\bf x})=\int\!d^{3}x'\,
f({\bf x}'-{\bf x}){\cal H}_{X}({\bf x}'),
\end{equation}
for $X= P,\ V,\ A$, and $S$ ($P =$ pseudo-scalar, $V =$ vector,
$A=$ axial-vector, and $S =$ scalar). The
potentials in momentum space are the same as for point interactions,
except that the coupling constants are multiplied by the Fourier
transform of the form factors.
In the derivation of the $V_{i}$ we employ the same approximations as in
\cite{NRS78,MRS89}, i.e.
\begin{enumerate}
\item We expand in $1/M$:
$E(p) = \left[ {\bf k}^{2}/4 +
{\bf q}^{2}+M^{2}\right]^{\frac{1}{2}}$\\
$\approx M+{\bf k}^{2}/8M + {\bf q}^{2}/2M$
and keep only terms up to first order in ${\bf k}^{2}/M$ and
${\bf q}^{2}/M$. This except for the form factors where
the full ${\bf k}^{2}$-dependence is kept throughout
the calculations. Notice that the gaussian form factors
suppress the high ${\bf k}^{2}$-contributions strongly.
\item In the meson propagators
$ (-(p_{1}-p_{3})^{2}+m^{2})
\approx({\bf k}^{2}+m^{2})$ .
\item When two different baryons are involved at a $BBM$-vertex
their average mass is used in the
potentials and the non-zero component of the momentum transfer
is accounted for by using an effective mass in
the meson propagator (for details see \cite{MRS89}).
\end{enumerate}
Due to the approximations we get only a linear dependence on
${\bf q}^{2}$ for $V_{1}$. In the following, separating the local and the
non-local parts, we write
\begin{equation}
V_{i}({\bf k}^{2},{\bf q}^{2})=
V_{i a}({\bf k}^{2})+V_{i b}({\bf k}^{2})({\bf q}^{2}+\frac{1}{4}{\bf k}^2),
\label{vcdec} \end{equation}
where in principle $i=1,8$.
The OBE-potentials are now obtained in the standard way (see e.g.\
\cite{NRS78,MRS89}) by evaluating the $BB$-interaction in Born-approximation.
We write the potentials $V_{i}$ of Eqs.~(\ref{eq:30.33}) and
(\ref{vcdec}) in the form
\begin{equation}
V_{i}({\bf k}\,^{2},{\bf q}\,^{2})=
\sum_{X} \Omega^{(X)}_{i}({\bf k}\,^{2})
\cdot \Delta^{(X)} ({\bf k}^{2},m^{2},\Lambda^{2}).
\label{nrexpv2} \end{equation}
Furthermore for $X=P,V$
\begin{equation}
\Delta^{(X)}({\bf k}^{2},m^{2},\Lambda^{2})= e^{-{\bf k}^{2}/\Lambda^{2}}/
\left({\bf k}^{2}+m^{2}\right),
\label{propm1} \end{equation}
and for $X=S,A$ a zero in the form factor
\begin{equation}
\Delta^{(S)}({\bf k}^{2},m^{2},\Lambda^{2})= \left(1-{\bf k}^2/U^2\right)\
e^{-{\bf k}^{2}/\Lambda^{2}}/
\left({\bf k}^{2}+m^{2}\right),
\label{propm2} \end{equation}
and for $X=D,O$
\begin{equation}
\Delta^{(D)}({\bf k}^{2},m^{2},\Lambda^{2})=\frac{1}{{\cal M}^{2}}
e^{-{\bf k}^{2}/(4m_{P,O}^{2})}.
\label{Eq:difdel}
\end{equation}
In the latter expression ${\cal M}$ is a universal
scaling mass, which is again taken to be the proton mass.
The mass parameter $m_{P}$ controls the ${\bf k}^{2}$-dependence of
the Pomeron-, $f$-, $f'$-, $A_{2}$-, and $K^{\star\star}$-potentials.
Similarly, $m_O$ controls the ${\bf k}^2$-dependence of the Odderon.\\
\noindent {\it In the following we give the OBE-potentials in momentum-space for the
hyperon-nucleon systems. From these those for NN and YY can be deduced easily.
We assign the particles 1 and 3 to be hyperons, and particles 2 and 4 to be
nucleons. Mass differences among the hyperons and among the nucleons will be neglected.}\\
\onecolumngrid
For pseudo-scalar mesons, the graph's of Fig.~\ref{obefig} give for the
potential $ V({\bf k},{\bf q}) \approx K^{(2)}_{PS}({\bf p}',{\bf p}|W) $
\begin{eqnarray}
V_{PS}({\bf k},{\bf q}) & = & -\frac{f_{13}f_{24}}{m_\pi^2}\
\left(1-\frac{({\bf q}^2+{\bf k}^2/4)}{2M_YM_N}\right)\cdot
\left[
\frac{1}{2\omega}\left\{\frac{1}{\omega + a}+\frac{1}{\omega -a}\right\}
(\mbox{\boldmath $\sigma$}_1\cdot{\bf k})(\mbox{\boldmath $\sigma$}_2\cdot{\bf k})
\right.\nonumber\\ && \hspace{-2.0cm} \left.
+\frac{1}{M_Y+M_N}\left\{\frac{1}{\omega + a}-\frac{1}{\omega -a}\right\}
(\mbox{\boldmath $\sigma$}_1\cdot{\bf q}\ \mbox{\boldmath $\sigma$}_2\cdot{\bf k}
-\mbox{\boldmath $\sigma$}_1\cdot{\bf k}\ \mbox{\boldmath $\sigma$}_2\cdot{\bf q})
\right]
\exp\left(-{\bf k}^2/\Lambda^2\right).
\label{eq:psx1}\end{eqnarray}
Here, using the on-energy-shell approximation $E_1+E_2= E_3+E_4$, we have
\begin{eqnarray*}
a &=& E_1+E_4 -W = \frac{1}{2}\left(\vphantom{\frac{A}{A}} E_1+E_4 -E_2-E_3\right) \\
&\approx& \Delta M + \frac{1}{4}\Delta M\left(\frac{1}{M_1M_3}+\frac{1}{M_2M_4}\right)
\left({\bf q}^2+{\bf k}^2/4\right),
\end{eqnarray*}
where $\Delta M = (M_1+M_4-M_3-M_2)/2$,
and we neglected the ${\bf q}\cdot{\bf k}$-term which is of order
$(M_Y-M_N)/2M_YM_N$. Henceforth we neglect the
non-adiabatic effects, i.e. $a \approx \Delta M$, in the OBE-potentials,
except for the $P_8$-terms, where the leading term is proportional to $a$.
One notices that the $P_8$-term in (\ref{eq:psx1}) is only non-zero for K-exchange.
\subsection{Non-strange Meson-exchange}
For the non-strange mesons the mass differences at the vertices are neglected,
we take at the $YYM$- and the $NNM$-vertex the average hyperon and the average
nucleon mass respectively. This implies that we do not include contributions
to the Pauli-invariants $P_7$ and $P_8$.
For vector-, and diffractive OBE-exchange we
refer the reader to Ref.~\cite{MRS89}, where the contributions to the different
$\Omega^{(X)}_{i}$'s for baryon-baryon scattering are given in detail.
\begin{enumerate}
\item[(a)] Pseudoscalar-meson exchange:
\begin{subequations}
\begin{eqnarray}
\Omega^{(P)}_{2a} & = & -g^P_{13}g^P_{24}\left( \frac{{\bf k}^{2}}
{12M_{Y}M_{N}} \right) \ \ ,\ \
\Omega^{(P)}_{3a} = -g^P_{13}g^P_{24}\left( \frac{1}
{4M_{Y}M_{N}} \right), \label{eq1a} \\
\Omega^{(P)}_{2b} & = & +g^P_{13}g^P_{24}\left( \frac{{\bf k}^{2}}
{24M_{Y}^2M_{N}^2} \right) \ \ ,\ \
\Omega^{(P)}_{3b} = +g^P_{13}g^P_{24}\left( \frac{1}
{8M_{Y}^2M_{N}^2} \right). \label{eq1b}
\end{eqnarray}
\end{subequations}
\item[(b)] Vector-meson exchange:
\begin{eqnarray}
\Omega^{(V)}_{1a}&=&
\left\{g^V_{13}g^V_{24}\left( 1-\frac{{\bf k}^{2}}{2M_{Y}M_{N}}\right)
-g^V_{13}f^V_{24}\frac{{\bf k}^{2}}{4{\cal M}M_{N}}
-f^V_{13}g^V_{24}\frac{{\bf k}^{2}}{4{\cal M}M_{Y}}
\vphantom{\frac{A}{A}}\right. \nonumber\\ && \left. \vphantom{\frac{A}{A}}
+f^V_{13}f^V_{24}\frac{{\bf k}^{4}}
{16{\cal M}^{2}M_{Y}M_{N}}\right\},\ \
\Omega^{(V)}_{1b} = g^V_{13}g^V_{24}\left(
\frac{3}{2M_{Y}M_{N}}\right), \nonumber\\
\Omega^{(V)}_{2a} &=& -\frac{2}{3} {\bf k}^{2}\,\Omega^{(V)}_{3a}, \ \
\Omega^{(V)}_{2b} = -\frac{2}{3} {\bf k}^{2}\,\Omega^{(V)}_{3b},
\nonumber\\
\Omega^{(V)}_{3a}&=& \left\{
(g^V_{13}+f^V_{13}\frac{M_{Y}}{{\cal M}})
(g^V_{24}+f^V_{24}\frac{M_{N}}{{\cal M}})
-f^V_{13}f^V_{24}\frac{{\bf k}^{2}}{8{\cal M}^{2}} \right\}
/(4M_{Y}M_{N}), \nonumber\\
\Omega^{(V)}_{3b}&=& -
(g^V_{13}+f^V_{13}\frac{M_{Y}}{{\cal M}})
(g^V_{24}+f^V_{24}\frac{M_{N}}{{\cal M}})
/(8M_{Y}^2M_{N}^2), \nonumber\\
\Omega^{(V)}_{4}&=&-\left\{12g^V_{13}g^V_{24}+8(g^V_{13}f^V_{24}+f^V_{13}g^V_{24})
\frac{\sqrt{M_{Y}M_{N}}}{{\cal M}}
- f^V_{13}f^V_{24}\frac{3{\bf k}^{2}}{{\cal M}^{2}}\right\}
/(8M_{Y}M_{N}) \nonumber\\
\Omega^{(V)}_{5}&=&- \left\{
g^V_{13}g^V_{24}+4(g^V_{13}f^V_{24}+f^V_{13}g^V_{24})
\frac{\sqrt{M_{Y}M_{N}}}{{\cal M}}
+8f^V_{13}f^V_{24}\frac{M_{Y}M_{N}}{{\cal M}^{2}}\right\}
/(16M_{Y}^{2}M_{N}^{2}) \nonumber\\
\Omega^{(V)}_{6}&=&-\left\{(g^V_{13}g^V_{24}
+f^V_{13}f^V_{24}\frac{{\bf k}^{2}}{4{\cal M}^{2}})
\frac{(M_{N}^{2}-M_{Y}^{2})}{4M_{Y}^{2}M_{N}^{2}}
-(g^V_{13}f^V_{24}-f^V_{13}g^V_{24})
\frac{1}{\sqrt{{\cal M}^{2}M_{Y}M_{N}}}\right\}.
\nonumber\\
\label{eq2}\end{eqnarray}
\item[(c)] Scalar-meson exchange: \hspace{2em}
\begin{eqnarray}
\Omega^{(S)}_{1} & = &
-g^S_{13} g^S_{24}
\left( 1+\frac{{\bf k}^{2}}{4M_{Y}M_{N}}
-\frac{{\bf q}^2}{2M_YM_N}\right), \ \
\Omega^{(S)}_{1b} =
+g^S_{13} g^S_{24} \frac{1}{2M_YM_N}
\nonumber\\ &&\nonumber\\
\Omega^{(S)}_{4}&=&
-g^S_{13} g^S_{24} \frac{1}{2M_{Y}M_{N}},\ \
\Omega^{(S)}_{5} = g^S_{13} g^S_{24}
\frac{1}{16M_{Y}^{2}M_{N}^{2} } \nonumber\\
\nonumber\\ &&\nonumber\\
\Omega^{(S)}_{6}&=& -g^S_{13} g^S_{24}
\frac{(M_{N}^{2}-M_{Y}^{2})}{4M_{Y}M_{N}}.
\label{Eq:scal} \end{eqnarray}
\item[(d)] Axial-vector-exchange $J^{PC}=1^{++}$:
\begin{eqnarray}
\Omega^{(A)}_{2a} & = & -g^A_{13}g^A_{24}\left[
1-\frac{2{\bf k}^2}{3M_YM_N}\right]
+\left[\left(g_{13}^A f_{24}^A\frac{M_N}{{\cal M}}
+f_{13}^A g_{24}^A \frac{M_Y}{{\cal M}}\right)
-f_{13}^A f_{24}^A \frac{{\bf k}^2}{2{\cal M}^2}\right]\
\frac{{\bf k}^2}{6M_YM_N}
\nonumber\\ && \nonumber\\
\Omega^{(A)}_{2b} &=&
-g^A_{13}g^A_{24} \left(\frac{3}{2M_{Y}M_{N}}\right)
\nonumber\\ && \nonumber\\
\Omega^{(A)}_{3}&=&
-g^A_{13}g^A_{24} \left[\frac{1}{4M_{Y}M_{N}}\right]
+\left[\left(g_{13}^A f_{24}^A\frac{M_N}{{\cal M}}
+f_{13}^A g_{24}^A \frac{M_Y}{{\cal M}}\right)
-f_{13}^A f_{24}^A \frac{{\bf k}^2}{2{\cal M}^2}\right]\
\frac{1}{2M_YM_N}
\nonumber\\ && \nonumber\\
\Omega^{(A)}_{4} &=&
-g^A_{13}g^A_{24} \left[\frac{1}{2M_{Y}M_{N}}\right]
\ \ ,\ \
\Omega^{(A)}_{6} =
-g^A_{13}g^A_{24} \left[\frac{(M_{N}^{2}-M_{Y}^{2})}{4M_{Y}^2M_{N}^2}\right]
\nonumber\\ && \nonumber\\
\Omega^{(A)'}_{5} & = &
-g^A_{13}g^A_{24} \left[\frac{2}{M_{Y}M_{N}}\right]
\label{eq:axi1} \end{eqnarray}
Here, we used the B-field description with $\alpha_r=1$, see Appendix~\ref{app:C}.
The detailed treatment of the potential proportional to $P_5'$, i.e.
with $\Omega_5^{(A)'}$, is given in Appendix~\ref{app:B}.
\item[(e)] Axial-vector mesons with $J^{PC}=1^{+-}$:
\begin{eqnarray}
\Omega^{(B)}_{2a} & = & +f^B_{13}f^B_{24}\frac{(M_N+M_Y)^2}{m_B^2}
\left(1-\frac{{\bf k}^2}{4M_YM_N}\right)
\left( \frac{{\bf k}^{2}}{12M_{Y}M_{N}} \right),\ \
\Omega^{(B)}_{2b} = +f^B_{13}f^B_{24}\frac{(M_N+M_Y)^2}{m_B^2}
\left( \frac{{\bf k}^{2}}{8M_{Y}^2M_{N}^2} \right)
\nonumber\\
\Omega^{(B)}_{3a} & = & +f^B_{13}f^B_{24}\frac{(M_N+M_Y)^2}{m_B^2}
\left(1-\frac{{\bf k}^2}{4M_YM_N}\right)
\left( \frac{1}{4M_{Y}M_{N}} \right),\ \
\Omega^{(B)}_{3b} = +f^B_{13}f^B_{24}\frac{(M_N+M_Y)^2}{m_B^2}
\left( \frac{3}{8M_{Y}^2M_{N}^2} \right). \nonumber\\
\label{eq:bxi1} \end{eqnarray}
\item[(f)] Diffractive-exchange (pomeron, $f, f', A_{2}$): \\
The $\Omega^{D}_{i}$ are the same as for scalar-meson-exchange
Eq.(\ref{Eq:scal}), but with
$\pm g_{13}^{S}g_{24}^{S}$ replaced by
$\mp g_{13}^{D}g_{24}^{D}$, and except for the zero in the form factor.
\item[(g)] Odderon-exchange:
The $\Omega^{O}_{i}$ are the same as for vector-meson-exchange
Eq.(ref{eq2}), but with
$ g_{13}^{V}\rightarrow g_{13}^{O}$,
$ f_{13}^{V}\rightarrow f_{13}^{O}$ and similarly for the couplings
with the 24-subscript.
\end{enumerate}
As in Ref.~\cite{MRS89} in the derivation of the expressions for $\Omega_i^{(X)}$,
given above, $M_{Y}$ and $M_{N}$ denote the mean hyperon and nucleon
mass, respectively \begin{math} M_{Y}=(M_{1}+M_{3})/2 \end{math}
and \begin{math} M_{N}=(M_{2}+M_{4})/2 \end{math},
and $m$ denotes the mass of the exchanged meson.
Moreover, the approximation
\begin{math}
1/ M^{2}_{N}+1/ M^{2}_{Y}\approx
2/ M_{N}M_{Y},
\end{math}
is used, which is rather good since the mass differences
between the baryons are not large.\\
\noindent {\it The potentials for mesons with strangeness are given in
paper II of this series.}
\subsection{One-Boson-Exchange Interactions in Configuration Space I}
\label{sect.IIIb}
In configuration space the BB-interactions are described by potentials
of the general form
\begin{subequations}
\begin{eqnarray}
V &=& \left\{\vphantom{\frac{A}{A}} V_C(r) + V_\sigma(r)
\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2
+ V_T(r) S_{12} + V_{SO}(r) {\bf L}\cdot{\bf S} + V_Q(r) Q_{12}
\right.\nonumber\\ && \left.
+ V_{ASO}(r)\ \frac{1}{2}(\mbox{\boldmath $\sigma$}_1-
\mbox{\boldmath $\sigma$}_2)\cdot{\bf L}
-\frac{1}{2M_YM_N}\left(\vphantom{\frac{A}{A}}
\mbox{\boldmath $\nabla$}^2 V^{n.l.}(r) + V^{n.l.}(r)
\mbox{\boldmath $\nabla$}^2\right)
\right\}\cdot P, \\
V^{n.l.} &=& \left\{\vphantom{\frac{A}{A}} \varphi_C(r) + \varphi_\sigma(r)
\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2
+ \varphi_T(r) S_{12}\right\},
\label{eq:3b.a}\end{eqnarray}
\end{subequations}
where
\begin{subequations}
\begin{eqnarray}
S_{12} &=& 3 (\mbox{\boldmath $\sigma$}_1\cdot\hat{r})
(\mbox{\boldmath $\sigma$}_2\cdot\hat{r}) -
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2), \\
Q_{12} &=& \frac{1}{2}\left[\vphantom{\frac{A}{A}}
(\mbox{\boldmath $\sigma$}_1\cdot{\bf L})(\mbox{\boldmath $\sigma$}_2\cdot{\bf L})
+(\mbox{\boldmath $\sigma$}_2\cdot{\bf L})(\mbox{\boldmath $\sigma$}_1\cdot{\bf L})
\right], \\
\phi(r) &=& \phi_C(r) + \phi_\sigma(r)
\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2,
\label{eq:3b.b}\end{eqnarray}
\end{subequations}
For the basic functions for the Fourier transforms with gaussian form factors,
we refer to Refs.~\cite{NRS78,MRS89}.
For the details of the Fourier transform for the potentials with $P_5'$, which
occur in the case of the axial-vector mesons with $J^{PC}=1^{++}$, we refer
to Appendix~\ref{app:B}.
\noindent (a)\ Pseudoscalar-meson-exchange:
\begin{subequations}
\begin{eqnarray}
V_{PS}(r) &=& \frac{m}{4\pi}\left[ g^P_{13}g^P_{24}\frac{m^2}{4M_YM_N}
\left(\frac{1}{3}(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)\
\phi_C^1 + S_{12} \phi_T^0\right)\right] P, \\
V_{PS}^{n.l.}(r) &=& \frac{m}{4\pi}\left[ g^P_{13}g^P_{24}\frac{m^2}{4M_YM_N}
\left(\frac{1}{3}(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)\
\phi_C^1 + S_{12} \phi_T^0\right)\right] P.
\label{eq:3b.1}\end{eqnarray}
\end{subequations}
\noindent (b)\ Vector-meson-exchange:
\begin{subequations}
\begin{eqnarray}
&& V_{V}(r) = \frac{m}{4\pi}\left[\left\{ g^V_{13}g^V_{24}\left[ \phi_C^0 +
\frac{m^2}{2M_YM_N} \phi_C^1
\right]
\right.\right.\nonumber\\ && \left.\left. \hspace{0cm}
+\left[g^V_{13}f^V_{24}\frac{m^2}{4{\cal M}M_N}
+f^V_{13}g^V_{24}\frac{m^2}{4{\cal M}M_Y}\right] \phi_C^1 + f^V_{13}f^V_{24}
\frac{m^4}{16{\cal M}^2 M_Y M_N} \phi_C^2\right\}
\right.\nonumber\\ && \left. \hspace{0cm}
+\frac{m^2}{6M_YM_N}\left\{\left[ \left(g^V_{13}+f^V_{13}\frac{M_Y}{{\cal M}}\right)\cdot
\left(g^V_{24}+f^V_{24}\frac{M_N}{{\cal M}}\right)\right] \phi_C^1
+f^V_{13} f^V_{24}\frac{m^2}{8{\cal M}^2} \phi_C^2\right\}
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)\
\right.\nonumber\\ && \left. \hspace{0cm}
-\frac{m^2}{4M_YM_N}\left\{\left[ \left(g^V_{13}+f^V_{13}\frac{M_Y}{{\cal M}}\right)\cdot
\left(g^V_{24}+f^V_{24}\frac{M_N}{{\cal M}}\right)\right] \phi_T^0
+f^V_{13} f^V_{24}\frac{m^2}{8{\cal M}^2} \phi_T^1\right\} S_{12}
\right.\nonumber\\ && \left. \hspace{0cm}
-\frac{m^2}{M_YM_N}\left\{\left[ \frac{3}{2}g^V_{13}g^V_{24}
+\left(g^V_{13}f^V_{24}+f^V_{13}g^V_{24}\right)
\frac{\sqrt{M_YM_N}}{{\cal M}}\right] \phi_{SO}^0
+\frac{3}{8}f^V_{13} f^V_{24}\frac{m^2}{{\cal M}^2} \phi_{SO}^1\right\} {\bf L}\cdot{\bf S}
\right.\nonumber\\ && \left. \hspace{0cm}
+\frac{m^4}{16M_Y^2M_N^2}\left\{\left[ g^V_{13}g^V_{24}
+4\left(g^V_{13}f^V_{24}+f^V_{13}g^V_{24}\right)
\frac{\sqrt{M_YM_N}}{{\cal M}}
+8f^V_{13}f^V_{24}\frac{M_YM_N}{{\cal M}^2}\right]\right\}
\cdot\right.\nonumber\\ && \left. \hspace{0cm} \times
\frac{3}{(mr)^2} \phi_T^0 Q_{12}
-\frac{m^2}{M_YM_N}\left\{\left[
\left(g^V_{13}g^V_{24}-f^V_{13}f^V_{24}\frac{m^2}{{\cal M}^2}\right)
\frac{(M_N^2-M_Y^2)}{4M_YM_N}
\right.\right.\right.\nonumber\\ && \left.\left.\left. \hspace{0cm}
-\left(g^V_{13}f^V_{24}-f^V_{13}g^V_{24}\right)\frac{\sqrt{M_YM_N}}{{\cal M}}\right] \phi_{SO}^0
\right\}\cdot\frac{1}{2}\left(
\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2\right)\cdot{\bf L}\right] P,
\\
&& V_{V}^{n.l.}(r) = \frac{m}{4\pi}\left[ \frac{3}{2} g^V_{13}g^V_{24}\ \phi_C^0
\right.\nonumber\\ && \left. \hspace{0cm}
+\frac{m^2}{6M_YM_N}\left\{\left[ \left(g^V_{13}+f^V_{13}\frac{M_Y}{{\cal M}}\right)\cdot
\left(g^V_{24}+f^V_{24}\frac{M_N}{{\cal M}}\right)\right] \phi_C^1 \right\}
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)\
\right.\nonumber\\ && \left. \hspace{0cm}
-\frac{m^2}{4M_YM_N}\left\{\left[ \left(g^V_{13}+f^V_{13}\frac{M_Y}{{\cal M}}\right)\cdot
\left(g^V_{24}+f^V_{24}\frac{M_N}{{\cal M}}\right)\right] \phi_T^0 \right\} S_{12}
\right].
\label{eq:3b.2}\end{eqnarray}
\end{subequations}
Note: the spin-spin and tensor non-local terms are not included in ESC08c.\\
\noindent (c)\ Scalar-meson-exchange:
\begin{eqnarray}
V_{S}(r) &=& -\frac{m}{4\pi}\left[ \hat{g}^S_{13}\hat{g}^S_{24}\left\{\left[ \phi_C^0
-\frac{m^2}{4M_YM_N} \phi_C^1\right] + \frac{m^2}{2M_YM_N} \phi_{SO}^0\ {\bf L}\cdot{\bf S}
+\frac{m^4}{16M_Y^2M_N^2}
\cdot\right.\right.\nonumber\\ && \left.\left. \times
\frac{3}{(mr)^2} \phi_T^0 Q_{12}
+\frac{m^2}{M_YM_N} \left[\frac{(M_N^2-M_Y^2)}{4M_YM_N}\right] \phi_{SO}^0\cdot
\frac{1}{2}\left(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2\right)\cdot{\bf L}
\right.\right.\nonumber\\ && \left.\left. \hspace{0.0cm}
+\frac{1}{4M_YM_N}\left(\mbox{\boldmath $\nabla$}^2 \phi_C^0
+ \phi_C^0 \mbox{\boldmath $\nabla$}^2\right) \right\}\right] P,
\label{eq:3b.3}\end{eqnarray}
where
\begin{equation}
\hat{g}^S_{13}= g^S_{13}-i\frac{M_3-M_1}{{\cal M}} f^S_{13}\ \ ,\ \
\hat{g}^S_{24}= g^S_{24}-i\frac{M_4-M_2}{{\cal M}} f^S_{24}.
\label{eq:3b.3a}\end{equation}
\noindent (d)\ Axial-vector-meson exchange $J^{PC}=1^{++}$:
\begin{eqnarray}
&& V_{A}(r) = -\frac{m}{4\pi}\left[
\left\{ g^A_{13}g^A_{24}\left(\phi_C^0 +\frac{2m^2}{3M_YM_N} \phi_C^1\right)
+\frac{m^2}{6M_YM_N}\left(g^A_{13}f^A_{24}\frac{M_N}{{\cal M}}
+f^A_{13}g^A_{24}\frac{M_Y}{{\cal M}}\right)\phi_C^1
\right.\right.\nonumber\\ && \left.\left.
+f^A_{13}f^A_{24}\frac{m^4}{12M_YM_N{\cal M}^2}\phi_C^2\right\}
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
-\frac{3}{4M_YM_N} g^A_{13}g^A_{24}\left(\mbox{\boldmath $\nabla$}^2 \phi_C^0
+ \phi_C^0 \mbox{\boldmath $\nabla$}^2\right)
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
\right.\nonumber\\ && \left.
- \frac{m^2}{4M_YM_N}\left\{\left[g^A_{13}g^A_{24}-2\left(g^A_{13}f^A_{24}
\frac{M_N}{{\cal M}}+f^A_{13}g^A_{24}\frac{M_Y}{{\cal M}}\right)\right] \phi_T^0
-f^A_{13}f^A_{24}\frac{m^2}{{\cal M}^2} \phi_T^1\right\} S_{12}
\right.\nonumber\\ && \left.
+\frac{m^2}{2M_YM_N}g^A_{13}g^A_{24} \left\{\phi_{SO}^0\ {\bf L}\cdot{\bf S}
+\frac{m^2}{M_YM_N} \left[\frac{(M_N^2-M_Y^2)}{4M_YM_N}\right] \phi_{SO}^0\cdot
\frac{1}{2}\left(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2\right)\cdot{\bf L}
\right\}\right] P.
\label{eq:3b.4}\end{eqnarray}
\noindent (e)\ Axial-vector-meson exchange $J^{PC}=1^{+-}$:
\begin{subequations}
\begin{eqnarray}
V_{B}(r) &=& -\frac{m}{4\pi}\frac{(M_N+M_Y)^2}{m^2}\left[
f^B_{13}f^B_{24}\left\{\frac{m^2}{12M_YM_N}\left(\phi_C^1+
\frac{m^2}{4M_YM_N} \phi_C^2\right)
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
\right.\right.\nonumber\\ && \left.\left.
-\frac{m^2}{8M_YM_N}\left(\mbox{\boldmath $\nabla$}^2 \phi_C^1
+ \phi_C^1 \mbox{\boldmath $\nabla$}^2\right)
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
+\left[\frac{m^2}{4M_YM_N}\right] \phi^0_T\ S_{12}\right\}\right] P, \\
V_{B}^{n.l.}(r) &=& -\frac{m}{4\pi}\frac{(M_N+M_Y)^2}{m^2}\left[
f^B_{13}f^B_{24}\left\{
\frac{3m^2}{4M_YM_N} \left(\frac{1}{3}
\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2\ \phi_C^1
+ S_{12}\ \phi_T^0\right)\right\}\right] P.
\label{eq:3b.5}\end{eqnarray}
\end{subequations}
\noindent (f)\ Diffractive exchange:
\begin{eqnarray}
&& V_{D}(r) = \frac{m_P}{4\pi}\left[ g^D_{13}g^D_{24}
\frac{4}{\sqrt{\pi}}\frac{m_P^2}{{\cal M}^2}\cdot\left[\left\{
1+\frac{m_P^2}{2M_YM_N}(3-2 m_P^2r^2) + \frac{m_P^2}{M_YM_N} {\bf L}\cdot{\bf S}
\right.\right.\right.\nonumber\\ && \left.\left.\left. \hspace{0.75cm}
+\left(\frac{m_P^2}{2M_YM_N}\right)^2 Q_{12}
+\frac{m_P^2}{M_YM_N} \left[\frac{(M_N^2-M_Y^2)}{4M_YM_N}\right]\cdot
\frac{1}{2}\left(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2\right)\cdot{\bf L}
\right\}\ e^{-m_P^2r^2}
\right.\right.\nonumber\\ && \left.\left. \hspace{1.5cm}
+\frac{1}{4M_YM_N}\left(\mbox{\boldmath $\nabla$}^2 e^{-m_P^2r^2}
+ e^{-m_P^2r^2}\mbox{\boldmath $\nabla$}^2\right) \right]\right] P.
\label{eq:3b.6}\end{eqnarray}
\noindent (g)\ Odderon-exchange:
\begin{subequations}
\begin{eqnarray}
V_{O,C}(r) &=& +\frac{g^O_{13}g^O_{24}}{4\pi}\frac{8}{\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\left[\left(3-2m_O^2r^2\right) \right.\nonumber\\ && \left.
-\frac{m_O^2}{M'M}\left( 15 - 20 m_O^2r^2+4 m_O^4r^4\right)
\right]\exp(-m_O^2 r^2)\ , \\
V_{O,n.l.}(r) &=& -\frac{g^O_{13}g^O_{24}}{4\pi}\frac{8}{\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{3}{4M'M}\left\{\mbox{\boldmath $\nabla$}^2
\left[(3-2m_O^2r^2)\exp(-m_O^2 r^2)\right]+ \right.\nonumber\\
&& \left. + \left[(3-2m_O^2r^2)\exp(-m_O^2 r^2)\right]
\mbox{\boldmath $\nabla$}^2 \right\}\ , \\
V_{O,\sigma}(r) &=&
-\frac{g^O_{13}g^O_{24}}{4\pi}\frac{8}{3\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{m_O^2}{M_YM_N}
\left[15-20 m_O^2r^2+4 m_O^4 r^4\right]\exp(-m_O^2 r^2)\cdot \nonumber\\
&& \times\left(1+\kappa^O_{13}\frac{M_Y}{\cal M}\right)
\left(1+\kappa^O_{24}\frac{M_N}{\cal M}\right)
, \\
V_{O,T}(r) &=& -\frac{g^O_{13}g^O_{24}}{4\pi}\frac{8}{3\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{m_O^2}{M_YM_N}\cdot m_O^2 r^2
\left[7-2 m_O^2r^2\right]\exp(-m_O^2 r^2)\cdot \nonumber\\
&& \times\left(1+\kappa^O_{13}\frac{M_Y}{\cal M}\right)
\left(1+\kappa^O_{24}\frac{M_N}{\cal M}\right)
, \\
V_{O,SO}(r) &=& -\frac{g^O_{13}g^O_{24}}{4\pi}\frac{8}{\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{m_O^2}{M_YM_N} \left[5-2 m_O^2r^2\right]\exp(-m_O^2 r^2)\cdot \nonumber\\
&& \times\left\{3+\left(\kappa^O_{13}+\kappa^O_{24}\right)\frac{\sqrt{M_YM_N}}{\cal M}\right\}
, \\
V_{O,Q}(r) &=& +\frac{g^O_{13}g^O_{24}}{4\pi}\frac{2}{\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{m_O^4}{M_Y^2M_N^2}
\left[7-2 m_O^2r^2\right]\exp(-m_O^2 r^2)\cdot \nonumber\\
&& \times\left\{1+4\left(\kappa^O_{13}+\kappa^O_{24}\right)\frac{\sqrt{M_YM_N}}{\cal M}
+8\kappa_{13}\kappa_{24}\frac{M_YM_N}{{\cal M}^2}\right\}
, \\
V_{O,ASO}(r) &=& -\frac{g^O_{13}g^O_{24}}{4\pi}\frac{4}{\sqrt{\pi}}\frac{m_O^5}{{\cal M}^4}
\frac{m_O^2}{M_YM_N} \left[5-2 m_O^2r^2\right]\exp(-m_O^2 r^2)\cdot \nonumber\\
&& \times\left\{ \frac{M_N^2-M_Y^2}{M_YM_N}
-4\left(\kappa^O_{24}-\kappa^O_{13}\right)
\frac{\sqrt{M_YM_N}}{\cal M} \right\}\ .
\label{eq:3b.7}\end{eqnarray}
\end{subequations}
\subsection{One-Boson-Exchange Interactions in Configuration Space II}
\label{sect.IIIc}
Here we give the extra potentials due to the
zero's in the scalar and axial-vector form factors.
\begin{enumerate}
\item[a)] Again, for $X=V,D$ we refer to the configuration space potentials
in Ref.~\cite{MRS89}. For $X=S$ we give here the additional terms w.r.t. those
in \cite{MRS89}, which are due to the zero in the scalar form factor. They are
\begin{eqnarray}
&& \Delta V_{S}(r) = - \frac{m}{4\pi}\ \frac{m^2}{U^2}\
\left[ g^S_{13} g^S_{24}\left\{
\left[\phi^1_C - \frac{m^2}{4M_YM_N} \phi^2_C\right]
+\frac{m^2}{2M_YM_N}\phi^1_{SO}\ {\bf L}\cdot{\bf S}
\right.\right. \nonumber\\ && \left.\left.
+\frac{m^4}{16M_Y^2M_N^2}\phi^1_T\ Q_{12}
+ \frac{m^2}{4M_YM_N}\frac{M_N^2-M_Y^2}{M_YM_N}\ \phi^{(1)}_{SO}\cdot
\frac{1}{2}(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2)\cdot{\bf L}
\right\}\right]\ . \nonumber\\
\label{eq:3.15}\end{eqnarray}
\item[b)] For the axial-vector mesons, the configuration space potential
corresponding to (\ref{eq:axi1}) is
\begin{eqnarray}
&& V_{A}^{(1)}(r) = - \frac{g_{A}^{2}}{4\pi}\ m \left[
\phi^{0}_{C}\ (\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
-\frac{1}{12M_YM_N}
\left( \nabla^{2} \phi^{0}_{C}+\phi^{0}_{C}\nabla^{2}\right)
(\mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2)
\right. \nonumber \\ && \nonumber \\ & & \left. \hspace*{1.4cm}
+ \frac{3m^{2}}{4M_YM_N}\ \phi^{0}_{T}\ S_{12}
+\frac{m^{2}}{2M_YM_N}\ \phi^{0}_{SO}\ {\bf L}\cdot{\bf S}
\right. \nonumber \\ && \nonumber \\ & & \left. \hspace*{1.4cm}
+ \frac{m^2}{4M_YM_N}\frac{M_N^2-M_Y^2}{M_YM_N}\ \phi^{(0)}_{SO}\cdot
\frac{1}{2}(\mbox{\boldmath $\sigma$}_1-\mbox{\boldmath $\sigma$}_2)\cdot{\bf L}
\right]\ .
\label{eq:3.16}\end{eqnarray}
The extra contribution to the potentials coming from the zero in the axial-vector
meson form factor are obtained from the expression (\ref{eq:3.16}) by making
substitutions as follows
\begin{eqnarray}
\Delta V_{A}^{(1)}(r) &=& V_{A}^{(1)}\left(\phi_C^0 \rightarrow \phi_C^1,
\phi_T^0 \rightarrow \phi_T^1, \phi_{SO}^0 \rightarrow \phi_{SO}^1\right)
\cdot\frac{m^2}{U^2}\ .
\label{eq:3.17}\end{eqnarray}
Note that we do not include the similar $\Delta V_A^{(2)}(r)$ since they involve
${\bf k}^4$-terms in momentum-space.
\end{enumerate}
\subsection{PS-PS-exchange Interactions in Configuration Space}
\label{sect.d}
In Fig.~\ref{bwfig} and Fig.~\ref{tmofig} the included two-meson exchange
graphs are shown schematically. Explicit expressions for
$K^{irr}(BW)$ and $K^{irr}(TMO)$ were derived \cite{Rij91}, where also the
terminology BW and TMO is explained.
The TPS-potentials for nucleon-nucleon have been given in detail in \cite{RS96a,RS96b}
The generalization to baryon-baryon is similar to that for the OBE-potentials.
So, we substitute $M \rightarrow \sqrt{M_YM_N}$, and include all PS-PS
possibilities with coupling constants as in the OBE-potentials.
As compared to nucleon-nucleon in \cite{RS96a,RS96b} here we have in addition
the potentials with double K-exchange. The masses
are the physical pseudo-scalar meson masses. For the intermediate two-baryon
states we take into account of the different thresholds.
We have not included uncorrelated PS-vector, PS-scalar, or PS-diffractive
exchange. This because the range of these potentials is similar to
that of the vector-,scalar-,and axial-vector-potentials. Moreover, for
potentially large potentials, in particularly those with scalar mesons involved,
there will be very strong cancellations between the planar- and crossed-box
contributions.
\subsection{MPE-exchange Interactions}
In Fig.~\ref{pairfig} both the one-pair graphs and the two-pair graphs are shown.
In this work we include only the one-pair graphs. The argument for neglecting
the two-pair graph is to avoid some 'double-counting'. Viewing the pair-vertex
as containing heavy-meson exchange means that the contributions from $\rho(750)$
and $\epsilon=f_0(760)$ to the two-pair graphs is already accounted for by
our treatment of the broad $\rho$ and $\epsilon$ OBE-potential.
For a more complete discussion of the physics behind MPE we refer to our
previous papers \cite{Rij93,RS96a,RS96b}.
The MPE-potentials for nucleon-nucleon have been given in Ref.~\cite{RS96a,RS96b}.
The generalization to baryon-baryon is similar to that for the TPS-potentials.
For the intermediate two-baryon
states we neglect the different two-baryon thresholds. This because,
although in principle possible, it complicates the computation of the
potentials considerably.
For a proper appreciation of the physics it is useful to scale the
phenomenological meson-pair baryon-baryon interaction Hamiltonians different from
the originally used scalings \cite{RS96a,RS96b}. Below we give these Hamiltonians:
\begin{subequations}
\begin{eqnarray}
&& {\cal H}_S = \bar{\psi}\psi \left[g_{(\pi\pi)_0}
\mbox{\boldmath $\pi$}\cdot\mbox{\boldmath $\pi$} +
g_{(\sigma\sigma)}\sigma^2\right]/
{\cal M}, \label{eq:3.17a}\\
&& {\cal H}_V = g_{(\pi\pi)_1}\left[\bar{\psi}\gamma_\mu\mbox{\boldmath $\tau$}\psi\right]\cdot
\left(\mbox{\boldmath $\pi$}\times\partial^\mu
\mbox{\boldmath $\pi$}/m_\pi\right)/{\cal M}
\nonumber\\ && \hspace{1cm}
-\frac{f_{(\pi\pi)_1}}{2M}
\left[\bar{\psi}\sigma_{\mu\nu}\mbox{\boldmath $\tau$}\psi\right]\partial^\nu\cdot\left(
\mbox{\boldmath $\pi$}\times\partial^\mu \mbox{\boldmath $\pi$}/m_\pi\right)/{\cal M},
\label{eq:3.17b}\\
&& {\cal H}_A = g_{(\pi\rho)_1}\left[\bar{\psi}\gamma_5\gamma_\mu
\mbox{\boldmath $\tau$}\psi\right]\cdot
\mbox{\boldmath $\pi$}\times\mbox{\boldmath $\rho$}/{\cal M},\label{eq:3.17c}\\
&& {\cal H}_B = i g_{(\pi\omega)}\left[\bar{\psi}\gamma_5\sigma_{\mu\nu}
\mbox{\boldmath $\tau$}\psi\right]\cdot\partial^\nu\left(
\mbox{\boldmath $\pi$}\phi_\omega^\mu\right)/(m_\pi {\cal M}),\label{eq:3.17d}\\
&& {\cal H}_P = g_{(\pi\sigma)}\left[\bar{\psi}\gamma_5\gamma_\mu
\mbox{\boldmath $\tau$}\psi\right]\cdot
\left(\mbox{\boldmath $\pi$}\partial^\mu\sigma -
\sigma\partial^\mu\mbox{\boldmath $\pi$}\right)/(m_\pi{\cal M}).
\label{eq:3.17e}\end{eqnarray}
\end{subequations}
Here, we systematically scaled the partial derivatives with $m_\pi$.\\
The generalization of the pair-couplings to baryon-baryon is described in
Ref.~\cite{Rij04b}, section III.
Also here in $N\!N$, we have in addition to \cite{RS96a,RS96b}
included the pair-potentials with KK-, KK*-, and K$\kappa$-exchange.
The convention for the MPE coupling constants is the same as in Ref.~\cite{RS96a,RS96b}.
\subsection{The Schr\"{o}dinger equation with Non-local potential}
\label{sect.5}
The non-local potentials are of the central-, spin-spin, and tensor type. The method
of solution of the Schr\"{o}dinger equation for nucleon-nucleon is described in
Ref.~\cite{NRS78}. Here, the non-local tensor is in momentum space
of the form ${\bf q}^2\ \tilde{v}_T({\bf k})$.
\section{ ESC-couplings and the QPC-model}
\label{sec:4}
In the ESC-model for baryon-baryon the meson-baryon couplings are in principle
only restricted by the requirements of relativistic covariance, time-reversal and
parity. However, dynamical input based on e.g. QCD, the QM, chiral-symmetry, and
flavor SU(3), is essential in order to be able to link the NN-, YN-, and YY-systems.
It appeared that in the ESC-model the $^3P_0$ quark-antiquark pair-creation
model \cite{Mic69,LeY73} leads to a scheme for the meson-baryon-baryon couplings which
is very similar to that found in the fits of the ESC-model \cite{Rij04a,Rij04b}.
The couplings found in the ESC08-model fit very well in the $(^3P_0+^3S_1)$-scheme
with a ratio $^3P_0/^3S_1 = 2:1$.
\subsection{ QPC-model Coupling Non-strange Mesons}
\label{sec:4a}
According to the Quark-Pair-Creation (QPC) model, in the $^3P_0$-version
\cite{Mic69,LeY73}, the
baryon-baryon-meson couplings are given in terms of the quark-pair creation
constant $\gamma_M$, and the radii of the (constituent) gaussian quark wave
functions, by \cite{LeY73,LeY75}
\begin{equation}
g_{BBM}(\pm) = \gamma_{q\bar{q}}\ \frac{3}{\sqrt{2}}\ \pi^{-3/4}\
X_M\left(I_M,L_M,S_M,J_M\right)\ F^{(\pm)}_M\ ,
\label{eq:qpc.1}\end{equation}
where $\pm = -(-)^{L_f}$ with $L_f$ is the orbital angular momentum of the final
BM-state, $X_M(\ldots)$ is a isospin, spin etc. recoupling coefficient, and
\begin{eqnarray}
F^{(+)} &=& \frac{3}{2}\ \left(m_MR_M\right)^{+1/2}\ (\Lambda_{QPC}R_M)^{-2},
\nonumber\\
F^{(-)} &=& \frac{3}{2}\ \left(m_MR_M\right)^{-1/2}\ (\Lambda_{QPC}R_M)^{-2}\cdot
3\sqrt{2} (M_M/M_B).
\label{eq:qpc.2}\end{eqnarray}
are coming from the overlap integrals, see Appendix~\ref{app:D}.
Here, the superscripts $\mp$ refer to the parity of the mesons $M$: $(-)$ for
$J^{PC}=0^{+-}, 1^{--}$, and $(+)$ for $J^{PC}= 0^{++}, 1^{++}$. The radii of the
baryons, in this case nucleons, and the mesons are respectively denoted by $R_B$ and $R_M$.
The QPC($^3P_0$)-model gives several interesting relations, such as
$g_\omega = 3 g_\rho, g_\epsilon = 3 g_{a_0}$, and
$g_{a_0} \approx g_\rho, g_\epsilon \approx g_\omega$.
These relations can be seen most easily by applying the Fierz-transformation
to the $³P_0$-pair-creation Hamiltonian, see Appendixref{app:D}.
From $\rho \rightarrow e^+ e^-$, employing the current-field-identities (C.F.I's)
one can derive, see for example \cite{Roy67}, the following relation with the QPC-model
\begin{equation}
f_\rho = \frac{m_\rho^{3/2}}{\sqrt{2}|\psi_\rho(0)|} \Leftrightarrow
\gamma\left(\frac{2}{3\pi}\right)^{1/2}
\frac{m_\rho^{3/2}}{|'\psi_\rho(0)'|}\ ,
\label{eq:gam0}\end{equation}
which, neglecting the difference between the wave functions on the left and
right hand side, gives for the pair creation constant
$\gamma \rightarrow \gamma_0=\frac{1}{2}\sqrt{3\pi} =1.535$. However,
since in the QPC-model gaussian wave functions are used, the $q\bar{q}$-potential
is a harmonic-oscillator one. This does not account for the $1/r$-behavior,
due to one-gluon-exchange (OGE), at short distance. This implies a OG-correction
\cite{LP96} to the wave function, which gives for $\gamma$ \cite{Chai80}
\begin{equation}
\gamma = \gamma_0 \left(1-\frac{16}{3}\frac{\alpha(m_M)}{\pi}\right)^{-1/2}\ .
\label{eq:gam}\end{equation}
In Table~\ref{tab.gam} $\gamma(\mu)$ is shown,
Using from \cite{PDG02} the parameterization
\begin{equation}
\alpha_s(\mu) = 4\pi/\left(\beta_0\ln(\mu^2/\Lambda_{QCD}^2)\right)\ ,
\label{eq:alphas}\end{equation}
with $\Lambda_{QCD} = 100$ MeV and $\beta_0 = 11-\frac{2}{3} n_f$ for $n_f=3$,
\begin{table}[hbt]
\caption{Pair-creation constant $\gamma$ as a function of $\mu$. }
\begin{center}
\begin{tabular}{c|c|c} \hline & \\
$\mu$ [GeV] & $\alpha_s(\mu)$ & $\gamma(\mu)$ \\
& & \\
\hline
& & \\
$\infty$ & 0.00 & 1.535 \\
80.0 & 0.10 & 1.685 \\
35.0 & 0.20 & 1.889 \\
1.05 & 0.30 & 2.191 \\
0.55 & 0.40 & 2.710 \\
0.40 & 0.50 & 3.94 \\
0.35 & 0.55 & 5.96 \\
& & \\
\hline
\end{tabular}
\end{center}
\label{tab.gam}
\end{table}
and taking the typical scale $m_M \approx 1$ GeV, the above formula gives
$\gamma = 2.19$. This value we will use later
when comparing the QPC-model predictions and the ESC08c-model
coupling constants.\\
The formulas (\ref{eq:qpc.2}) are valid for the most simple QPC-model.
For a realistic description of the coupling constants
of the ESC08-model we include two sophistocations: (i) inclusion of both
the $^3P_0$- and the $^3S_1$-mechanism, (ii) inclusion of SU(6)-breaking.
For details, see \cite{THAR11}.
For the latter we use the (\underline{56}) and (\underline 70) SU(6)-irrep
mixing \cite{LeY75}, and a short-distance quark-gluon form factor.
In Table~\ref{tab:cc7} we show the $^3P_0-^3S_1$-model results and
the values obtained in the ESC08c-fit.
In this table we fixed $\gamma_M = 2.19$ for the vector-, scalar-, and
axial-vector-mesons.
From Table~\ref{tab.gam}
one sees that at the scale of $m_M \approx 1$ GeV such a value is reasonable.
Here, one has to realize that the QPC-predictions are kind of "bare" couplings,
which allows vertex corrections from meson-exchange.
For the pseudo-scalar, a different value has to be used, showing indeed
some 'running'-behavior as expected from QCD.
In \cite{Chai80}, for the decays $\rho, \epsilon \rightarrow 2\pi$ etc. it was found
$\gamma =3.33$, which is close to our $\gamma_\pi=4.19$.
For the mesonic decays of the charmonium states $\gamma=1.12$.
One notices the similarity between the QPC($^3P_0$)-model predictions
and the fitted couplings. Here, for $f_1(1420)$
we have to take a larger radius $r_M=1.10$ fm in order to reduce the couplings
in the QPC-model.
Of course, these results are sensitive to the $r_M$ values. We found that for all
solutions with a very good $\chi^2_{NN}$ the $r_M$ values varied by $\pm 0.2$ fm.
\noindent {\it The ESC08c-couplings and the QPC-couplings agree very well.
In particularly,
the SU(6)-breaking is improving the agreement significantly. All this strengthens the
claim that the ESC08c-couplings are realistic ones.}\\
\begin{table}
\caption{SU(6)-breaking in coupling constants, using (\underline{56}) and
(\underline{70})-irrep mixing with angle $\varphi = -22^o$ for the
$^3P_0$- and $^3S_1$-model. Gaussian Quark-gluon cut-off $\Lambda_{QQG}=986.6$ MeV.
Ideal mixing for vector and scalar meson nonets.
For pseudoscalar- and axial-nonets the mixing angles are $-13^0$ and $+50.0^o$ respectively,
imposing the OZI-rule.
Here, $\Lambda_{QPC} = 255.0$ MeV, $\gamma(\alpha_s=0.30)=2.19$ etc.
The weights are A=0.697 and B=0.303 for the $^3P_0$ and $^3S_1$ respectively.
The values in parentheses in the column QPC denote the results for $\varphi=0^o$.
}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l|cc|c|c|c|c} \hline & & & & & & \\
Meson & $r_M[fm]$ & $\gamma_M$
& $^3S_1$ & $^3P_0$ & QPC & ESC08c \\
& & & & & & \\
\hline
& & & & & & \\
$\pi(140)$ & 0.30 & 5.51 & $g=-2.74$ & $g=+6.31$ & 3.57 (3.77) & 3.65 \\
& & & & & & \\
$\eta'(957)$ & 0.70 & 2.22 & $g=-2.49$ & $g=+5.72$ & 3.23 (3.92) & 3.14 \\
& & & & & & \\
$\rho(770)$ & 0.80 & 2.37 & $g=-0.17$ & $g=+0.80$ & 0.63 (0.77) & 0.65 \\
& & & & & & \\
$\omega(783)$ & 0.70 & 2.35 & $g=-0.96$ & $g=+4.43$ & 3.47 (3.43) & 3.46 \\
& & & & & & \\
$a_0(962)$ & 0.90 & 2.22 & $g=+0.19$ & $g=+0.43$ & 0.62 (0.64) & 0.59 \\
& & & & & & \\
$\epsilon(760)$ & 0.70 & 2.37 & $g=+1.26$ & $g=+2.89$ & 4.15 (4.15) & 4.15 \\
& & & & & & \\
$a_1(1270)$ & 0.70 & 2.09 & $g=-0.13$ & $g=-0.58$ & -0.71 (-0.71) & -0.79 \\
& & & & & & \\
$f_1(1420)$ & 1.10 & 2.09 & $g=-0.14$ & $g=-0.66$ & -0.80 (-0.81)& -0.76\\
& & & & & & \\
\hline
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:cc7}
\end{table}
\section{ ESC08-model: Fitting $NN\oplus YN \oplus YY$-data}
\label{sec:5}
In the simultaneous $\chi^2$-fit of the $NN$-, $YN$-, and YY-data a
{\it single set of parameters} was used, which means the same parameters for all
BB-channels.
The input $NN$-data are the same as in Ref.~\cite{Rij04a}, and we refer the reader
to this paper for a description of the employed phase shift analysis \cite{Sto93,Klo93}.
Note that in addition to the NN-phases, including their correlations,
in the ESC08-models also the $NN$-low energy parameters
and the deuteron binding energy are fitted.
The YN-data are those used in Ref.~\cite{Rij04b} with the addition of higher
energy data, see paper II.
Of course, it is to be expected that the accurate and
very numerous $NN$-data essentially fix most of the parameters. Only some of
the parameters, for example certain $F/(F+D)$-ratios, are quite
influenced by the $YN$-data.
In the fitting procedure the following constraints are applied:
(i) A strong restriction imposed on YN-models is the absence of S=-1 bound states.
(ii) During the fitting process sometimes constraints are imposed
in the form of
'pseudo-data' for some YN scattering lengths. These constraints are based
on experiences with Nijmegen YN-models in the past or to impose constraints
from the G-matrix results. In some cases it is necessary
to add some extra weight of the YN-scattering data w.r.t. the NN-data in the
fitting process.
(iii) After obtaining a solution for the
scattering data the corresponding model is tested by checking the corresponding
G-matrix results for the well-depths for $U_\Sigma >0$ and $U_\Xi <0$, and
sufficient s-wave spin splitting in the $U_\Lambda$.
If not satisfactory we refit the scattering data etc. This iterative process
implements the constraints from the G-matrix well-depth's results,
and plays a vital role in obtaining the final results of the combined fit.
(For the G-matrix approach to hyperon-nucleus systems, see e.g. Ref.~\cite{Yam10}.)
The fitting process is discussed more elaborately in paper II.
The $\chi^2$ is a very shallow function of the quark-core parameter.
Accordingly solutions have been obtained using different assumptions about the
quark-core-effects, all with a strength of about 20\% of the
diffractive contribution.
In previous work \cite{PTP185.a}, models ESC08a and ESC08a'', the solutions were obtained
by assuming quark-core effects only for the channels where the [51]-component is dominant:
$\Sigma^+p(^3S_1,I=3/2), \Sigma N(^1S_0,I=1/2)$, and $\Xi N(^1S_0,I=1)$.
The solution ESC08c is obtained by application of the quark-core effects
according to equation~(8.4) in \cite{PTP185.a}, see paper II
for a full description of the Pauli-blocking scheme.
Like in the $NN$-fit, described in Ref.~\cite{Rij04a}, also in the
simultaneous $\chi^2$-fit of the $NN$- and $YN$-data,
it appeared again that the OBE-couplings could be constrained successfully
by the 'naive' predictions of the QPC-model \cite{Mic69,LeY73}. Although these
predictions, see section \ref{sec:4}, are 'bare' ones, we tried to keep during the
searches many OBE-couplings in the neighborhood of the QPC-values.
Also, it appeared that we could either fix the $F/(F+D)$-ratios
to those as suggested by the QPC-model,
or apply the same restraining strategy as for the OBE-couplings.
\subsection{ Fitted BB-parameters}
\label{sec:5a}
The treatment of the broad mesons $\rho$ and $\epsilon$ is similar to that in the
OBE-models \cite{NRS78,MRS89}. For the $\rho$-meson the same parameters are used
as in these references. However, for the $\epsilon=f_0(760)$ assuming
$m_\epsilon=760$ MeV and $\Gamma_\epsilon = 640$ MeV the Bryan-Gersten parameters
\cite{Bry72} are used. For the chosen mass and width they are:
$ m_1=496.39796$ MeV, $m_2=1365.59411$ MeV, and $\beta_1=0.21781, \beta_2=0.78219$.
Other meson masses are given in Table~\ref{table4}.
The sensitivity for the values of the cut-off masses of the $\eta$ and $\eta'$
is very weak.
Therefore we have set the \{1\}-cut-off imass for the pseudoscalar nonet equal to
that for the \{8\}. Likewise, for the two nonets of the axial-vector mesons,
see table~\ref{table5}.
Summarizing the parameters we have for baryon-baryon (BB):\\
(i) NN Meson-couplings: $f_{NN\pi},f_{NN\eta'}$,
$ g_{NN\rho}, g_{NN\omega}$,
$f_{NN\rho},f_{NN\omega}$, $g_{NNa_0},g_{NN\epsilon}$,
$g_{NNa_1}$, $f_{NNa_1}$, $g_{NNf'_1}$, $f_{NNf'_1}$,
$f_{NNb_1}$, $f_{NNh'_1}$\\
(ii) $F/(F+D)$-ratios: $\alpha^{m}_{V}$, $\alpha_{A}$ \\
(iii) NN Pair couplings: $g_{NN(\pi\pi)_1},f_{NN(\pi\pi)_1}$, $g_{NN(\pi\rho)_1}$,
$g_{NN\pi\omega}, g_{NN\pi\eta}, g_{NN\pi\epsilon}$ \\
(iv) Diffractive couplings and masslike parameters $g_{NNP}$, $g_{NNO}$, $f_{NNO}$, $m_P$, $m_O$ \\
(v) Cut-off masses: $\Lambda_{8}^P = \Lambda_{1}^P$, $\Lambda_{8}^V$, $\Lambda_{1}^V$,
$\Lambda_{8}^S$, $\Lambda_{1}^S$, and $\Lambda_{8}^A$ = $\Lambda_{1}^A$.
The pair coupling $g_{NN(\pi\pi)_0}$ was kept fixed at zero.
Note that in the interaction Hamiltonians of the pair-couplings
(\ref{eq:3.17b})-(\ref{eq:3.17e})
the partial derivatives are scaled by $m_\pi$, and there is a scaling mass $M_N$.
The ESC-model described here, is fully consistent with $SU(3)$-symmetry
using a straightforward extension of the NN-model to YN and YY.
This the case for the OBE- and
TPS-potentials, as well as for the Pair-potentials.
For example $g_{(\pi\rho)_1} = g_{A_8VP}$, and
besides $(\pi\rho)$-pairs one sees also that $K K^*(I=1)$- and
$K K^*(I=0)$-pairs contribute to the $NN$ potentials.
All $F/(F+D)$-ratio's are taken as fixed with heavy-meson saturation in mind.
The approximation we have made in this paper is to neglect the baryon mass
differencesi in the TPS-potentials, i.e. we put $m_\Lambda = m_\Sigma = m_N$. This because we
have not yet worked out the formulas for the inclusion of these mass
differences, which is straightforward in principle.
\subsection{ Coupling Constants, $F/(F+D)$ Ratios, and Mixing Angles}
\label{sec:5b}
In Table~\ref{table5} we give the ESC08c meson masses, and the fitted
couplings and cut-off parameters. Note that the axial-vector couplings for the
B-mesons are scaled with $m_{B_1}$.
The mixing for the pseudo-scalar, vector, and scalar mesons, as well as the
handling of the diffractive potentials, has been described elsewhere, see
e.g. Refs.~\cite{MRS89,RSY99}. The mixing scheme of the axial-vector mesons is completely
similar as for the vector etc. mesons, except for the mixing angle.
In the paper II \cite{RNY10b}
the $SU(3)$ singlet and octet couplings are
listed, and also the $F/(F+D)$-ratios and mixing angles.
Also the Pauli-blocking effect parameter $a_{PB}$,
described in \cite{PTP185.a}, section 8, for ESC08c is given.
As mentioned above, we searched for solutions where
all OBE-couplings are compatible with the QPC-predictions. This time the QPC-model
contains a mixture of the $^3P_0$ and $^3S_1$ mechanism, whereas in
Ref.~\cite{Rij04a} only the $^3P_0$-mechanism was considered.
For the pair-couplings all $F/(F+D)$-ratios were fixed to the predictions of
the QPC-model.
\begin{table}
\caption{Meson couplings and parameters employed in the ESC08c-potentials.
Coupling constants are at ${\bf k}^{2}=0$.
An asterisk denotes that the coupling constant is constrained via SU(3).
The masses and $\Lambda$'s are given in MeV.}
\label{table4}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{crccr} \hline\hline
meson & mass & $g/\sqrt{4\pi}$ & $f/\sqrt{4\pi}$ & \multicolumn{1}{c}{$\Lambda$}\\
\hline
$\pi$ & 138.04 & & 0.2687 & 1056.13\ \\
$\eta$ & 547.45 & & \hspace{2mm}0.1265$^\ast$ & ,, \hspace{5mm} \\
$\eta'$ & 957.75 & & 0.2309 & ,, \hspace{5mm} \\ \hline
$\rho$ & 768.10 & 0.6446 & 3.7743 & 695.67\ \\
$\phi$ & 1019.41 &--1.3390$^\ast$ & \hspace{2mm}3.1678$^\ast$ &
,, \hspace{5mm} \\
$\omega$ & 781.95 & 3.4570 & --0.8575 & 758.58\\ \hline
$a_1 $ & 1270.00 &--0.7895 & --0.8192 & 1051.80\ \\
$f_1 $ & 1420.00 & \hspace{3mm}0.7311$^\ast$ &\hspace{2mm} 0.3495$^\ast$ & ,, \hspace{5mm} \\
$f_1'$ & 1285.00 &--0.7613 & \hspace{2mm}--0.4467 & ,, \hspace{5mm} \\ \hline
$b_1 $ & 1235.00 & & --0.2022 & 1056.13 \\
$h_1 $ & 1380.00 & & \hspace{2mm}--0.0621$^\ast$ &
,, \hspace{5mm} \\
$h_1'$ & 1170.00 & & --0.0335 & ,, \hspace{5mm} \\ \hline
$a_{0}$ & 962.00 & 0.5853 & & 994.89\ \\
$f_{0}$ & 993.00 &\hspace{0mm}--1.6898$^\ast$ & &
,, \hspace{5mm} \\
$\varepsilon$ & 760.00 & 4.1461 & & 1113.57 \\ \hline
Pomeron & 220.50 & 3.5815 & & \\
Odderon & 273.35 & 4.6362 & --4.7602 & \\
\hline
\end{tabular}
\end{ruledtabular}
\end{center}
\label{table5}
\end{table}
One notices that all the BBM $\alpha$'s have values rather close to that
which are expected from the QPC-model. In the ESC08c solution $\alpha_A \approx 0.31$,
which is not too far from $\alpha_A \sim 0.4$.
As in previous works, e.g. Ref.~\cite{NRS78}, $\alpha_V^e=1$ is
kept fixed.
Above, we remarked that the axial-nonet parameters may be sensitive to whether
or not the heavy pseudoscalar nonet with the $\pi$(1300) are included.
\begin{table}[hbt]
\caption{Pair-meson coupling constants employed in the ESC08c MPE-potentials.
Coupling constants are at ${\bf k}^{2}=0$.
The F/(F+D)-ratio are QPC-predictions, except that
$\alpha_{(\pi\omega)}=\alpha_{pv}$, which is very close to QPC.}
\label{tab.gpair}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{cclrc} \hline\hline
$J^{PC}$ & $SU(3)$-irrep & $(\alpha\beta)$ &\multicolumn{1}{c}{$g/4\pi$} & $F/(F+D)$ \\
\hline \\
$0^{++}$ & $\{1\}$ & $g(\pi\pi)_{0}$ & --- & --- \\
$0^{++}$ & ,, & $g(\sigma\sigma)$ & --- & --- \\
$0^{++}$ &$\{8\}_s$ & $g(\pi\eta)$ & -1.2371 & 1.000 \\ \hline
$1^{--}$ &$\{8\}_a$ & $g(\pi\pi)_{1}$ & 0.2703 & 1.000 \\
& & $f(\pi\pi)_{1}$ &--1.6592 & 0.400 \\ \hline
$1^{++}$ & ,, & $g(\pi\rho)_{1}$ & 5.1287 & 0.400 \\
$1^{++}$ & ,, & $g(\pi\sigma)$ &--0.2989 & 0.400 \\
$1^{++}$ & ,, & $g(\pi P)$ & --- & --- \\ \hline
$1^{+-}$ &$\{8\}_s$ & $g(\pi\omega)$ &--0.2059 & 0.365 \\
\hline
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
In Table~\ref{tab.gpair} we listed the fitted Pair-couplings for the MPE-potentials.
We recall that only One-pair graphs are included, in order to avoid double
counting, see Ref.~\cite{Rij04a}. The $F/(F+D)$-ratios are all fixed, assuming heavy-boson
domination of the pair-vertices. The ratios are taken from the QPC-model for
$Q\bar{Q}$-systems with the same quantum numbers as the dominating boson.
For example, the $\alpha$-parameter for the axial $(\pi\rho)_1$-pair could fixed
at the quark-model prediction 0.40, see Table~\ref{tab.gpair}.
The $BB$-Pair couplings are calculated, assuming unbroken $SU(3)$-symmetry,
from the $NN$-Pair coupling and the $F/(F+D)$-ratio using $SU(3)$.
Unlike in Ref.~\cite{RS96a,RS96b}, we did not fix pair couplings using
a theoretical model, e.g. based on heavy-meson saturation and chiral-symmetry.
So, in addition to the 14 parameters used in Ref.~\cite{RS96a,RS96b} we now have
6 pair-coupling fit parameters.
In Table~\ref{tab.gpair} the fitted pair-couplings are given.
Note that the $(\pi\pi)_0$-pair coupling gets contributions from the $\{1\}$ and
the $\{8_s\}$ pairs as well, giving in total $g_{(\pi\pi)} \approx 0.10$,
which has the same sign as in \cite{RS96a,RS96b}.
The $f_{(\pi\pi)_1}$-pair coupling has opposite
sign as compared to Ref.~\cite{RS96a,RS96b}.
In a model with a more complex and realistic
meson-dynamics \cite{SR97} this coupling is predicted as found in the present
ESC-fit. The $(\pi\rho)_1$-coupling is large as expected from $A_1$-saturation, see
Ref.~\cite{RS96a,RS96b}. We conclude that the pair-couplings are in general
not well understood quantitatively, and deserve more study.
In Table~\ref{table4} we show the OBE-coupling constants and the
gaussian cut-off's $\Lambda$. The used $\alpha =: F/(F+D)$-ratio's
for the OBE-couplings are:
pseudo-scalar mesons $\alpha_{pv}=0.365$,
vector mesons $\alpha_V^e=1.0, \alpha_V^m=0.472$,
and scalar-mesons $\alpha_S=1.00$, which is calculated using the physical
$S^* =: f_0(993)$ coupling etc..
In Table~\ref{tab.gpair} we show the MPE-coupling constants.
The used $\alpha =: F/(F+D)$-ratio's for the MPE-couplings are:
$(\pi\eta)$ pairs $\alpha(\{8_s\})=1.0$,
$(\pi\pi)_1$ pairs $\alpha_V^e(\{8\}_a)=1.0, \alpha_V^m(\{8\}_a)=0.400$,
and the $(\pi\rho)_1$ pairs $\alpha_A(\{8\}_a)=0.400$.
The $(\pi\omega)$ pairs $\alpha(\{8_s\})$ has been set equal to
$\alpha_{pv}=0.365$.
\begin{figure}
\resizebox{8.cm}{11.43cm}
{\includegraphics[50,50][554,770]{plotnn.aug09/ppi1.ps}}
\caption{Solid line: proton-proton $I=1$ phase shifts for the ESC08c-model.
The dashed line: the m.e. phases of the Nijmegen93 PW-analysis \cite{Sto93}.
The black dots: the s.e. phases of the Nijmegen93 PW-analysis.
The diamonds: Bugg s.e. \cite{Bugg92}.}
\label{ppi1.fig}
\end{figure}
\begin{figure}
\resizebox{8.cm}{11.43cm}
{\includegraphics[50,50][554,770]{plotnn.aug09/ppi1c.ps}}
\caption{Solid line: proton-proton $I=1$ phase shifts for the ESC08c-model.
The dashed line: the m.e. phases of the Nijmegen93 PW-analysis \cite{Sto93}.
The black dots: the s.e. phases of the Nijmegen93 PW-analysis.
The diamonds: Bugg s.e. \cite{Bugg92}.}
\label{ppi1c.fig}
\end{figure}
\begin{figure}
\resizebox{8.cm}{11.43cm}
{\includegraphics[50,50][554,770]{plotnn.aug09/npi0.ps}}
\caption{Solid line: neutron-proton $I=0$, and the I=1 $^1S_0(NP)$
phase shifts for the ESC08c-model.
The dashed line: the m.e. phases of the Nijmegen93 PW-analysis \cite{Sto93}.
The black dots: the s.e. phases of the Nijmegen93 PW-analysis.
The diamonds: Bugg s.e. \cite{Bugg92}.}
\label{npi0.fig}
\end{figure}
\begin{figure}
\resizebox{8.cm}{11.43cm}
{\includegraphics[50,50][554,770]{plotnn.aug09/npi0c.ps}}
\caption{Solid line: neutron-proton $I=0$ phase shifts for the ESC08c-model.
The dashed line: the m.e. phases of the Nijmegen93 PW-analysis \cite{Sto93}.
The black dots: the s.e. phases of the Nijmegen93 PW-analysis.
The diamonds: Bugg s.e. \cite{Bugg92}.}
\label{npi0c.fig}
\end{figure}
\section{ ESC08-model , $N\!N$-Results}
\label{sec:6}
\subsection{ Nucleon-nucleon Fit, Low-energy and Phase Parameters}
\label{sec:6a}
For a more detailed discussion on the NN-fitting we refer to Ref.~\cite{Rij04a}.
Here, we fit to the 1993 Nijmegen representation of the $\chi^2$-hypersurface of the
$NN$ scattering data below $T_{lab}=350$ MeV \cite{Sto93,Klo93},
and also the low-energy parameters are fitted for $pp, np$ and $nn$.
In this simultaneous fit of $NN$ and $YN$, we obtained for ESC08c
for the phase shifts $\chi^2/Ndata =1.081$.
For a comparison with Ref.~\cite{Rij04a}, and for use of this model for the description
of $NN$, we give in Table~\ref{tab.nnphas3} the nuclear-bar
phases for $pp$ in case $I=1$, and for $np$ in the case of $^1S_0(I=1)$
and the $I=0$-phases.
Here, $\Delta\chi^2$ denotes the access in $\chi^2$ of the ESC-model w.r.t. the
phase shift analysis \cite{Sto93,Klo93}.
The deuteron has been included in the fitting procedure, as well as the low-energy parameters.
The fitted binding energy $E_B= 2.224593$ MeV, which
is very close to $E_B(experiment)=2.224644$.
The charge-symmetry breaking is described phenomenologically by having
next to $g_{\rho nn}$ free couplings for $g_{\rho np}$,
and $g_{\rho pp}$. This phenomenological treatment is successful for the
various NN-channels,
especially for the $np(^1S_0,I=1)$-phases, which were included in the NN-fit.
\begin{table}[hbt]
\caption{ ESC08c nuclear-bar $pp$ and $np$ phases in degrees.}
\begin{tabular}{crrrrrrrrrr} \hline\hline & & & & & &&&&&\\
$T_{\rm lab}$ & 0.38& 1 & 5 & 10 & 25 & 50 & 100 & 150 & 215 & 320 \\ \hline
& & & & & &&&&& \\
$\sharp$ data &144 & 68 & 103 & 290& 352 & 572 & 399 & 676 & 756 & 954 \\
& & & & & &&&&&\\
$\Delta \chi^{2}$& 11 & 52 & 11 & 28 & 28 & 75 & 21 & 96 & 140 & 124 \\
& & & & & &&&&&\\ \hline
& & & & & &&&&& \\
$^{1}S_{0}(np)$ & 54.57 & 62.02 & 63.48 & 59.73& 50.49
& 39.82 & 25.40 & 14.99 & 4.37 & --9.02 \\
$^{1}S_{0}$ & 14.61 & 32.62 & 54.75 & 55.17& 48.68
& 38.98 & 25.04 & 14.77 & 4.21 &--9.14 \\
$^{3}S_{1}$ & 159.39 & 147.77& 118.25& 102.73& 80.83
& 63.07 & 43.68 & 31.34 & 19.60 & 5.66 \\
$\epsilon_{1}$ & 0.03 & 0.11 & 0.68 & 1.17 & 1.81
& 2.13 & 2.44 & 2.82 & 3.43 & 4.56 \\
$^{3}P_{0}$ & 0.02 & 0.14 & 1.61 & 3.81 & 8.78
& 11.75 & 9.64 & 4.85 &--1.70 &--11.20 \\
$^{3}P_{1}$ & --0.01 &--0.08 &--0.89 & --2.04 & --4.87
& --8.25 &--13.22 &--17.32 & --21.94 & --28.16 \\
$^{1}P_{1}$ & --0.05 &--0.19 &--1.50 & --3.07 & --6.40
& --9.82 &--14.68 &--18.82 & --23.51 & --29.57 \\
$^{3}P_{2}$ & 0.00 & 0.01 & 0.22 & 0.67 & 2.51
& 5.80 & 10.88 & 14.03 & 16.30 & 17.28 \\
$\epsilon_{2}$ &--0.00 &--0.00 &--0.05 &--0.20 &--0.81
&--1.71 &--2.69 &--2.95 &--2.79 &--2.15 \\
$^{3}D_{1}$ & --0.00 &--0.01 &--0.18 & --0.68 & --2.83
&--6.52 &--12.41 &--16.70 & --20.74 & --25.12 \\
$^{3}D_{2}$ & 0.00 & 0.01 & 0.22 & 0.85 & 3.70
& 8.95 & 17.27 & 22.17 & 24.93 & 24.83 \\
$^{1}D_{2}$ & 0.00 & 0.00 & 0.04 & 0.17 & 0.69
& 1.69 & 3.78 & 5.71 & 7.66 & 9.30 \\
$^{3}D_{3}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.03
& 0.24 & 1.20 & 2.37 & 3.71 & 5.07 \\
$\epsilon_{3}$ & 0.00 & 0.00 & 0.01 & 0.08 & 0.55
& 1.60 & 3.47 & 4.83 & 6.01 & 7.05 \\
$^{3}F_{2}$ & 0.00 & 0.00 & 0.00 & 0.01 & 0.11
& 0.34 & 0.80 & 1.11 & 1.17 & 0.45 \\
$^{3}F_{3}$ & --0.00 & --0.00 &--0.00 & --0.03 & --0.23
&--0.67 &--1.46 &--2.04 & --2.62 & --3.44 \\
$^{1}F_{3}$ & --0.00 & --0.00 &--0.01 & --0.06 & --0.41
&--1.10 &--2.12 &--2.77 & --3.45 & --4.65 \\
$^{3}F_{4}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.02
& 0.12 & 0.51 & 1.05 & 1.82 & 3.00 \\
$\epsilon_{4}$ & --0.00 & --0.00 & --0.00 &--0.00 &--0.05
&--0.19 &--0.53 &--0.83 &--1.13 &--1.45 \\
$^{3}G_{3}$ &--0.00 &--0.00 &--0.00 &--0.00 & --0.05
&--0.26 &--0.93 &--1.73 &--2.77 & --4.17 \\
$^{3}G_{4}$ & 0.00 & 0.00 & 0.00 & 0.01 & 0.17
& 0.71 & 2.12 & 3.53 & 5.18 & 7.33 \\
$^{1}G_{4}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.04
& 0.15 & 0.41 & 0.69 & 1.06 & 1.71 \\
$^{3}G_{5}$ &--0.00 &--0.00 &--0.00 &--0.00 & --0.01
&--0.05 &--0.17 &--0.25 & --0.28 & --0.17 \\
$\epsilon_{5}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.04
& 0.20 & 0.71 & 1.22 & 1.83 & 2.62 \\
& & & & & &&&&& \\
\hline
\end{tabular}
\label{tab.nnphas3}
\end{table}
\begin{table}[hbt]
\caption{ESC08c Low energy parameters: S-wave scattering lengths and
effective ranges, deuteron binding energy $E_B$, and electric
quadrupole $Q_e$.
The asterisk denotes that the low-energy parameters were not searched.}
\begin{center}
\begin{tabular}{ccccc} \hline\hline & & & & \\
& \multicolumn{3}{c}{experimental data}& ESC08c
\\ &&&& \\ \hline
$a_{pp}(^1S_0)$ & --7.823 & $\pm$ & 0.010 & --7.7710\\
$r_{pp}(^1S_0)$ & 2.794 & $\pm$ & 0.015 & 2.7601$^\ast$ \\ \hline
$a_{np}(^1S_0)$ & --23.715 & $\pm$ & 0.015 & --23.7316\\
$r_{np}(^1S_0)$ & 2.760 & $\pm$ & 0.030 & 2.6983$^\ast$ \\ \hline
$a_{nn}(^1S_0)$ & --18.63 & $\pm$ & 0.48 & --17.177\\
$r_{nn}(^1S_0)$ & 2.860 & $\pm$ & 0.15 & 2.8417$^\ast$ \\ \hline
$a_{np}(^3S_1)$ & 5.423 & $\pm$ & 0.005 & 5.4384$^\ast$ \\
$r_{np}(^3S_1)$ & 1.761 & $\pm$ & 0.005 & 1.7481$^\ast$\\ \hline
$E_B$ & --2.224644 & $\pm$ & 0.000046 & --2.224593 \\
$Q_e$ & 0.286 & $\pm$ & 0.002 & 0.2742 \\ \hline\hline
\end{tabular}
\end{center}
\label{tab.lowenergy}
\end{table}
\begin{table}
\caption{ ESC08c $\chi^2$ and $\chi^2$ per datum at the ten energy bins for the
Nijmegen93 Partial-Wave-Analysis. $N_{data}$ lists the number of data
within each energy bin. The bottom line gives the results for the
total $0-350$ MeV interval.
The $\chi^{2}$-access for the ESC model is denoted
by $\Delta\chi^{2}$ and $\Delta\hat{\chi}^{2}$, respectively.}
\begin{ruledtabular}
\begin{tabular}{crrrrrr} & & & & & \\
$T_{\rm lab}$ & $\sharp$ data & $\chi_{0}^{2}$\hspace*{5.5mm}&
$\Delta\chi^{2}$&$\hat{\chi}_{0}^{2}$\hspace*{3mm}&
$\Delta\hat{\chi}^{2}_0$ \\ &&&&& \\ \hline
0.383 & 144 & 137.555 & 14.7 & 0.960 & 0.102 \\
1 & 68 & 38.019 & 60.5 & 0.560 & 0.890 \\
5 & 103 & 82.226 & 8.1 & 0.800 & 0.078 \\
10 & 290 & 257.995 & 28.4 & 1.234 & 0.098 \\
25 & 352 & 272.197 & 33.1 & 0.773 & 0.094 \\
50 & 571 & 538.522 & 37.2 & 0.957 & 0.065 \\
100 & 399 & 382.499 & 19.6 & 0.959 & 0.049 \\
150 & 676 & 673.055 & 72.5 & 0.996 & 0.107 \\
215 & 756 & 754.525 &118.5 & 0.998 & 0.157 \\
320 & 954 & 945.379 &189.5 & 0.991 & 0.199 \\ \hline
& & & & & \\
Total &4313&4081.971& 582.0 &0.948 &0.133 \\
& & & & & \\
\end{tabular}
\end{ruledtabular}
\label{tab.chidistr}
\end{table}
We emphasize that we use the single-energy (s.e.) phases and $\chi^2$-surface
\cite{Klo93} as a means to fit the NN-data.
The multi-energy (m.e.) phases of the PW-analysis \cite{Sto93}
in Fig.~\ref{ppi1.fig}-Fig.~\ref{npi0.fig} are the dashed lines in these figures.
One notices that the central value of the s.e. phases do not correspond
to the m.e. phases in general,
illustrating that there has been a certain amount
of noise fitting in the s.e. PW-analysis, see e.g. $\epsilon_1$ and $^1P_1$
at $T_{lab}=100$ MeV.
The m.e. PW-analysis reaches $\chi^2/N_{data}=0.99$, using
39 phenomenological parameters plus normalization parameters.
The related phenomenological PW-potentials NijmI,II and Reid93 \cite{SKTS94},
with respectively 41, 47, and 50 parameters, turn out all with $\chi^2/Ndata=1.03$.
This should be compared to the ESC-model, which has $\chi^2/N_{data}=1.08$
using for NN 32 parameters. These are 14 QPC-constrained meson-nucleon-nucleon couplings,
6 meson-pair-nucleon-nucleon couplings, 6 gaussian cut-off parameters, 3 diffractive
couplings, and 2 diffractive mass parameters. The 3 remaining fitting parameters
(2 F/(F+D) ratios and the Pauli blocking fraction) are mainly or totally
determined by the YN-fit.
From the figures it is obvious that the ESC-model deviates from the m.e.
PW-analysis in particular at the highest energy.
In Table~\ref{tab.lowenergy} the results for the low energy parameters are given.
In order to discriminate between the $^1S_0$-wave for pp, np, and nn, we introduced
some charge independence breaking by taking $g_{pp\rho} \neq g_{np\rho} \neq g_{nn\rho}$.
With this device we fitted the difference between the $^1S_0(pp)$ and $^1S_0(np)$
phases, and the different scattering lengths and effective ranges as well. We found
$g_{np\rho} = 0.5889,\ g_{pp\rho} = 0.6389$, which are not far from
$g_{nn\rho} = 0.6446$, see Table~\ref{table4}.
The NN low-energy parameters are fitted very well, see Table~\ref{tab.lowenergy}.
For a discussion of the theoretical and experimental situation w.r.t. these low
energy parameters, see \cite{Mil90}.\\
The binding energy of the deuteron is fitted excellently. The electric
quadrupole moment result is typical for models without meson-exchange
current effects. Further properties of the deuteron in this model are:
$P_D=6.07 \%, D/S=0.0257, N_G^2=0.7721$, and $\rho_{-\epsilon,-\epsilon}=1.7273$.
\begin{figure}[hbt]
\resizebox{3.5cm}{!}
{\includegraphics[200,000][400,850]{plotnn.aug09/plot.pom-odd2.ps}}
\caption{Pomeron and Odderon central- and spin-orbit potentials.}
\label{fig:pom-odd2}
\end{figure}
\subsection{ Nucleon-nucleon Potentials}
\label{sec:6c}
The hyperon-nucleon OBE-, TPS-, and Pair-potentials for ESC04 model are shown in
Ref.~\cite{Rij04b}. These potentials are rather similar to those of ESC08c, and
therefore we refer the reader the cited YN-paper. Also, these NN-potentials
are qualitatively rather similar in character.
The odderon potential is a novel feature of ESC08-model. In Fig.~\ref{fig:pom-odd2}
the central and spin-orbit potentials are shown. The spin-spin, tensor, and
quadratic spin-orbit potentials are very small. One notices from this figure that
the pomeron potential is like an 'anti-scalar' potential whereas the odderon
is a normal vector-exchange potential. Note the strong cancellation in the
spin-orbit giving a negligible summed contribution. The upshot is a universal
central repulsion from the pomeron+odderon.\\
\section{Nuclear Saturation and Three-body repulsion}
\label{sec:7}
The lowest-order Brueckner G-matrix calculations
with the continuous (CON) choice for intermediate single particle potentials
were shown to simulate well the results including higher hole-line contributions
up to $3\sim 4$ $\rho_0$~\cite{Baldo98,Baldo02}.
Here, the Brueckner G-matrix theory is considered a good starting point
for studies of many-body systems on the basis of
free-space baryon-baryon interaction models. We study
the properties of high-density nuclear matter
on the basis of the lowest-order G-matrix theory with the CON choice.
As is well known, the experimental nuclear saturation properties, the density $\rho_N$,
the binding energy per nucleon E/A, the compression modulus K, cannot be reproduced
quantitatively with nuclear two-body interactions only, see e.g.~\cite{Lag81}.
Essential for giving the correct energy curve $E(\rho_{N})$ is the inclusion of many-nucleon
interactions. Here, ithe most important seems to be the three-nucleon interaction (TNI),
composed of an attractive (TNA) and a repulsive (TNR) part. Integrating over the third
particle results in a dependence on the nuclear-matter density $\rho_{N}$ of the
'effective' two-nucleon potential (see below).
Since TNA contributes only moderately as a function of $\rho_{N}$,
the saturation curve is not so remarkably changed by the TNA \cite{Lag81}.
Its inclusion is nevertheless important for obtaining the right
nuclear saturation point.
On the other hand, it turns out that the TNR contribution increases rapidly in
the high-density region, giving high values for the incompressibility.
Maximum masses of neutron stars can be reproduced with use of
the stiff equation of state (EoS) realized by the TBR contributions.
The soft-core two-baryon potentials give a too soft EoS.
In particular, ESC08 gives for the mass of the neutron star $1.35 M_{solar}$
\cite{Sch11}, implying for this model the necessity for a TNR contribution.
Therefore, we incorporate the TNR contribution in the ESC-model
together with an additional TNA one,
giving it a key role in stiffening the EoS for symmetric and neutron-star matter.
As will be shown below, this enables to satisfy both the nuclear saturation point
and the observed maximum mass of neutron stars.
At high densities hyperon-mixing occurs in neutron-star matter, which
brings about a significant softening of the EoS canceling the TNR effect
for the maximum mass \cite{BBS00,VPREH00,Nis02}.
To compensate this adverse effect Nishizaki, Takatsuka
and one of the authors (Y.Y.) \cite{Nis02} made the conjecture that
the TNR-type repulsion works universally for $Y\!N\!N$ and $Y\!Y\!N$
as well as for $N\!N\!N$.
They demonstrated that the TNR-stiffening of the EoS
can be recovered clearly by this assumption.
Universal repulsions among three baryons were called
the three-baryon repulsion (TBR).
It is our aim to realize the TBR assumption consistently with the ESC modeling
of the baryon-baryon systems. The presence of three-body forces (3BF)
is a natural possibility in nuclei and hypernuclei,
generating effective two-body forces, which very likely improve the binding
energies and well-depth's. The latter will appear indeed the case for the ESC-model
as shown in the YN-paper \cite{RNY10b} of this series.
\begin{figure}[hbt]
\vspace*{15mm}
\begin{center} \begin{picture}(180,130)(100,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.0}
\ArrowLine(10,120)(60,120)
\Line(60,120)(120,120)
\ArrowLine(120,120)(180,120)
\ArrowLine(10,10)(60,10)
\Line(60,10)(120,10)
\ArrowLine(120,10)(180,10)
\SetColor{Green}
\Gluon(60,120)(90,65){3}{8}
\Gluon(120,120)(90,65){3}{8}
\Gluon(60,10)(90,65){3}{8}
\Gluon(120,10)(90,65){3}{8}
\Gluon(40,95)(70,65){3}{8}
\Gluon(40,35)(70,65){3}{8}
\Gluon(140,95)(110,65){3}{8}
\Gluon(140,35)(110,65){3}{8}
\Vertex(80,100){2}
\Vertex(90,100){2}
\Vertex(100,100){2}
\Vertex(80,30){2}
\Vertex(90,30){2}
\Vertex(100,30){2}
\SetColor{Black}
\GOval(90,65)(15,25)(0.0){0.7}
\PText(90,0)(0)[c]{Pomeron-exchange.}
\SetScale{1.0} \SetWidth{1.0}
\SetOffset(200,0)
\ArrowLine(10,150)(60,150)
\Line(60,150)(120,150)
\ArrowLine(120,150)(180,150)
\ArrowLine(10,120)(60,120)
\Line(60,120)(120,120)
\ArrowLine(120,120)(180,120)
\ArrowLine(10,10)(60,10)
\Line(60,10)(120,10)
\ArrowLine(120,10)(180,10)
\SetColor{Green}
\Gluon(40,150)(90,65){3}{8}
\Gluon(140,150)(90,65){3}{8}
\Gluon(60,10)(90,65){3}{8}
\Gluon(120,10)(90,65){3}{8}
\Gluon(20,120)(70,65){3}{8}
\Gluon(40,35)(70,65){3}{8}
\Gluon(160,120)(110,65){3}{8}
\Gluon(140,35)(110,65){3}{8}
\Text(80,100)[l]{$\bullet \bullet$ }
\Text(80,30)[l]{$\bullet \bullet$ }
\Vertex(80,100){2}
\Vertex(90,100){2}
\Vertex(100,100){2}
\Vertex(80,30){2}
\Vertex(90,30){2}
\Vertex(100,30){2}
\SetColor{Black}
\GOval(90,65)(15,25)(0.0){0.7}
\PText(90,0)(0)[c]{Triple/quartic Pomeron-exchange.}
\end{picture}
\caption{Multi-gluon exchange processes.}
\label{fig.mglue}
\end{center}
\end{figure}
Since in QCD the gluons are flavor blind it is natural to relate the universality
of the TBR repulsion to multi-gluon exchange, see Fig.~\ref{fig.mglue}.
In the Nijmegen soft-core OBE and
ESC models pomeron-exchange can be viewed as due to an even number of
gluon-exchanges contributing a universal repulsion in BB-systems. Like for the
two-baryon systems, in ESC we introduce the multi-gluon three-body forces
with the multi-pomeron exchange potential (MPP) \cite{MPP05,RMPP08,PTP185.a}.
In Fig.~\ref{fig.mpp2} the triple- and quartic-pomeron vertices are illustrated.
We convert the three-body potential into an effective two-body potential by
integrating out the third nucleon.
As demonstrated in \cite{YFYR13}, the MPP gives the stiff
EoS of neutron matter enough to assure the large observed values
of two massive neutron stars with mass $1.97\pm 0.04 M_{solar}$
for PSR J1614-2230 \cite{Demorest10} and $2.01\pm 0.04 M_{solar}$ for
PSR J0348-0432 \cite{Antoniadis13}.
In \cite{Rij04b} the medium effect on the vector masses was assumed as
the dominant mechanism for generating extra repulsion at higher densities.
However, the mass shift of the vector meson masses due to the nuclear medium
has been put in doubt \cite{Mos10}. Therefore, in the ESC08-model, in contrast
to \cite{Rij04b}, we assume that the dominant mechanism is triple and
quartic pomeron exchange \cite{Kai74,Bro77}.
\begin{figure}[hbt]
\begin{center} \begin{picture}(180,150)(100,0)
\SetPFont{Helvetica}{9}
\SetScale{1.0} \SetWidth{1.0}
\SetOffset( 30,0)
\SetColor{Red}
\Gluon( 75, 70.0)(25,125.0){3}{10}
\Gluon( 75, 70.0)(125,125.0){3}{10}
\Gluon( 75, 70.0)( 75,15.0 ){3}{6}
\SetColor{Black}
\CCirc(25,125){15}{White}{White}
\CCirc(125,125){15}{White}{White}
\CCirc( 75,15.0){15}{White}{White}
\ArrowArc(25,125)(15,0,360)
\ArrowArc(125,125)(15,0,360)
\ArrowArc( 75,15)(15,0,360)
\GCirc(75, 70.0){15}{0.8}
\SetOffset(180,0)
\SetColor{Red}
\Gluon( 75, 70.0)(25,125.0){3}{10}
\Gluon( 75, 70.0)(125,125.0){3}{10}
\Gluon( 75, 70.0)(25, 15.0){3}{10}
\Gluon( 75, 70.0)(125, 15.0){3}{10}
\SetColor{Black}
\CCirc(25,125){15}{White}{White}
\CCirc(125,125){15}{White}{White}
\CCirc(25, 15){15}{White}{White}
\CCirc(125, 15){15}{White}{White}
\ArrowArc(25,125)(15,0,360)
\ArrowArc(125,125)(15,0,360)
\ArrowArc(25, 15)(15,0,360)
\ArrowArc(125, 15)(15,0,360)
\GCirc( 75, 70.0){15}{0.8}
\end{picture} \end{center}
\caption{Triple- and quartic-pomeron 3- and 4-body interaction.}
\label{fig.mpp2}
\end{figure}
For the triple pomeron vertex we take the Lagrangian \cite{MPP05,RMPP08}
\begin{eqnarray}
{\cal L}_3 = g_P^{(3)} {\cal M} \sigma_P^3(x)/3! \ .
\label{eq:tbf.1}\end{eqnarray}
Then, the three-body local potential by pomeron exchange is given by
\begin{eqnarray}
&&V({\bf x}_1,{\bf x}_2,{\bf x}_3) = g_P^{(3)} (g_P)^3 \Pi_{i=1}^3
\int \ \frac{d^3k_i}{(2\pi)^3}
\Pi_{i=1}^3
e^{-i{\bf p}_i\cdot{\bf x}}\cdot (2\pi)^3
\delta({\bf k}_1+{\bf k}_2+{\bf k}_3)
\nonumber
\\
&& \times \exp(-{\bf k}_1^2/4m_P^2) \exp(-{\bf k}_2^2/4m_P^2)
\exp(-{\bf k}_3^2/4m_P^2) \cdot {\cal M}^{-5} .
\label{eq:tbf.2}
\end{eqnarray}
Here, the (low-energy) pomeron propagator is the same as used in the
two-body pomeron potential used in all Nijmegen soft-core
OBE and ESC models.
The effective two-body potential in a baryonic medium is obtained
by integrating over the coordinate ${\bf x}_3$.
\begin{eqnarray}
V_{eff}({\bf x}_1,{\bf x}_2)&=& \rho_{NM} \int d^3x_3
V({\bf x}_1,{\bf x}_3,{\bf x}_3)
\nonumber
\\
&=& g_P^{(3)} (g_P)^3 \frac{\rho_{NM}}{{\cal M}} \cdot
\frac{1}{4\pi} \frac{4}{\sqrt{\pi}}
\left(\frac{m_P}{\sqrt{2}}\right)^3
\exp\left(-\frac12 m_P^2 r_{12}^2 \right) \ .
\label{eq:tbf.3}
\end{eqnarray}
In a similar way, one can obtain a four-body interaction
$V({\bf x}_1,{\bf x}_2,{\bf x}_3,{\bf x}_4)$ and a corresponding
effective two-body potential with a quartic pomeron coupling
$g_P^{(4)}$ \cite{RMPP08}. The expressions for the $N$-body interaction and
the effective two-body potential by multiple-pomeron exchange are given
in Ref.~\cite{MPP05,RMPP08,YFYR13}.
Here, we restrict ourselves to the triple and quartic pomeron
couplings where there is information from the ISR pp-data \cite{Kai74}.
Since the pomeron is an SU(3)-singlet, the MPP in nuclear medium
leads to the density-dependent universal repulsion, which can be associated with the
proposal in \cite{Nis02}.
Estimates for $g_P^{(3)}$ and $g_P^{(4)}$ can be obtained from
\cite{Kai74,Bro77}, which shows that $g_P^{(4)} >> g_P^{(3)}$.
The pomeron coupling $g_P$ is fitted to the NN-data etc.,
see Table~\ref{table4}.
In \cite{YFYR13}, the MPP strengths ($g_P^{(3)}$ and $g_P^{(4)}$) were
determined by analyzing the $^{16}$O$+^{16}$O elastic scattering at
$E/A=70$ MeV with use of G-matrix folding potentials, where the TNR
effect appears clearly in the angular distribution. As shown in \cite{FSY},
in such a high scattering energy the frozen-density approximation gives
a good prescription, where G-matrices including TNR at about two times of
normal density contribute to nucleus-nucleus folding potentials.
In addition to MPP, in order to assure the nuclear saturation property
precisely, we introduce also a TNA part phenomenologically as
a density-dependent two-body interaction
\begin{eqnarray*}
V_{TNA}(r;\rho_N)= V^0_{TNA}\, \exp(-(r/2.0)^2)\, \rho_N\,
\exp(-\eta \rho_N)\, (1+P_r)/2 \ ,
\end{eqnarray*}
whose form is similar to the TNA part given in \cite{Lag81}.
$P_r$ is a space-exchange operator. By a $(1+P_r)$ factor, the TNA part
works only in even states, which is needed to reproduce nucleus-nucleus
angular distributions precisely. Then,
$V^0_{TNA}$ and $\eta$ are treated as adjustable parameters.
Strengths of the MPP part ($g_P^{(3)}$ and $g_P^{(4)}$)
and the TNA part ($V_{TNA}$ and $\eta$) are determined so as to reproduce
the $^{16}$O$+^{16}$O angular distribution $E/A=70$ MeV, and
the energy and density at the saturation point of symmetric matter.
The ratio of $g_P^{(3)}$ and $g_P^{(4)}$ unsettled in our analysis was taken
adequately in reference to the results in Ref.~\cite{Kai74,Bro77}.
Quantitatively, however, it is quite uncertain.
In Table~\ref{tabMPP}, we give the three parameter sets of
($g_P^{(3)}$, $g_P^{(4)}$, $V^0_{TNA}$, $\eta$)
named as MPa, MPb and MPc, respectively.
MPa and MPb are very similar to MP1a and MP2a, respectively,
given in Ref.~\cite{YFYR13}. The $g_P^{(3)}$ value of MPc
is the same as that of MPa, but with $g_P^{(4)}=0$.
These sets reproduce equally well the saturation property.
In the case of $^{16}$O$+^{16}$O elastic scattering at $E/A=70$ MeV,
the G-matrix folding potentials derived from the three sets
give the angular distributions similar to the solid and dashed curves
in Fig.1 in Ref.~\cite{YFYR13}, reproducing nicely the experimental data.
\begin{table}
\caption{Parameters of MPP+TNA parts.
$V^0_{TNA}$ and $\eta$ are in MeV and fm$^{-3}$, respectively.}
\label{tabMPP}
\begin{tabular}{|c|cccc|}
\hline
& $g_P^{(3)}$ & $g_P^{(4)}$ & $V^0_{TNA}$ & $\eta$ \\
\hline
MPa & 2.34 & 30.0 & $-32.8$ & 3.5 \\
MPb & 2.94 & 0.0 & $-45.0$ & 5.4 \\
MPc & 2.34 & 0.0 & $-43.0$ & 7.3 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=10cm,height=10cm]{saturation1.eps}
\caption{Energies per nucleon drawn as a function of $\rho_N$
in symmetric nuclear matter (thick curves) and neutron
matter (thin curves). Dotted, solid, dashed and dot-dashed curves are
in cases of ESC08c only, MPa, MPb and MPc,respectively.
The box shows the empirical value.
}
\label{saturation1}
\end{center}
\end{figure}
In Fig.~\ref{saturation1},
we show the energy curves of symmetric nuclear matter (thick curves)
and neutron matter (thin curves), namely binding energy per nucleon
($E/A$) as a function of $\rho_N$. They are obtained from G-matrix
calculations with ESC08c only, and including the MPP+TNA parts.
The box in the figure shows the area where nuclear
saturation is expected to occur empirically.
The dotted curves are obtained only with the two-body interaction ESC08c.
The saturation point in symmetric nuclear matter
is found to deviate substantially from the box. On the other hand,
the solid, dashed and dot-dashed curves are obtained with including
MPa, MPb and MPc contributions, respectively.
As is clearly seen, saturation densities and minimum values of $E/A$ in
these cases are nicely close to the empirical value shown by the box:
For MPa and MPb (MPc), we obtain the value of $\sim -15.8$ ($-15.5$) MeV for
the binding energy per nucleon at the saturation density $\sim 0.16$ fm$^{-3}$.
The incompressibilities $K$ for MPa, MPb and MPc are obtained as
310 MeV, 280 MeV and 260 MeV, respectively, at the saturation densities.
The difference between the $E/A$ curves for neutron matter and
symmetric matter gives the symmetry energy $E_{sym}(\rho)$.
In Fig.~\ref{saturation2}, obtained values of $E_{sym}$ are drawn
as a function of $\rho_N$ in the cases of ESC08c only (dotted) and including
MPa (solid), MPb (dashed) and MPc (dot-dashed).
The values of $E_{sym}$ at the saturation density 0.16 fm$^{-3}$
are 32.2 MeV (ESC08c only), 33.1 MeV (MPa),
33.1 MeV (MPb) and 32.7 MeV (MPc).
The slope parameter is defined by
$L=3\rho_0 \left[\frac{\partial E_{sym}(\rho)}{\partial \rho}\right]_{\rho_0}$.
The values of $L$ at the same density are 69.0 MeV (ESC08c only),
70.4 MeV (MPa), 69.2 MeV (MPb) and 67.1 MeV (MPc).
The above values of $E_{sym}$ and $L$ are in nice agreement to the values
$E_{sym}=32.5\pm0.5$ MeV and $L=70\pm15$ MeV determined
recently on the basis of experimental data~\cite{Yoshida}.
It should be noted the values of $E_{sym}$ and $L$ for the three sets
are similar to the values for ESC08c only, owing to the
isospin-independent nature of the present three-body interaction.
Then, our three sets are specified mainly by the incompressibilities.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=10cm,height=10cm]{saturation2.eps}
\caption{Symmetric energies as a function of $\rho_N$
in cases of ESC08c only and including MPa, MPb and MPc by dotted,
solid, dashed and dot-dashed curves, respectively. The short bar
denotes the experimental value $32.5\pm 0.5$ MeV at normal density.
}
\label{saturation2}
\end{center}
\end{figure}
In the case of pure neutron-matter EOS, the mass-radius relations
of neutron stars can be derived from the present ESC08c+MPP+TNA models
in the same way as those in \cite{YFYR13}.
Calculated values of maximum masses of neutron stars are
$2.5 M_{solar}$ for MPa, $2.2 M_{solar}$ for MPb and
$2.1 M_{solar}$ for MPc,
being larger than the observed value $1.97 M_{solar}$.
The difference between the values for MPa and MPc comes from
the four-body repulsive part included in the former.
When the TNA parts are switched off in the three sets,
we have almost the same values of maximum masses:
The TNA parts contribute very slightly to maximum masses.
Thus, the inclusion of MPP and additional TNA provide a solution
for both the nuclear saturation and the neutron-star mass problem.
It should be noted here that our MPP contributions exist universally
in every baryonic system. It is very interesting to investigate
the relation between the universal MPP repulsions and
the softening effect induced by hyperon mixing to neutron-star matter.
The result will be published in the near future.
Another mechanism for generating extra repulsion at higher densities is
suggested by the relativistic mean-field theory (RMFT), see e.g.\cite{RMFT88}.
Here, at higher densities the scalar field becomes suppressed and
the vector field becomes dominating.
The effect would be similar to that employed in \cite{Rij04b}.
For the Dirac-Brueckner approach to the EoS see \cite{TerHaar87}.
\section{Discussion and Conclusions }
\label{sec:8}
The presentation in this paper reports on the present stage of the ESC-model.
Compared to ESC04 \cite{Rij04a,Rij04b,Rij04c} the model has been developed
further. The new version ESC08 has in addition to meson-exchange also
incorporated quark-core effects. Furthermore, the multi-gluon sector has been
completed by the inclusion of the Odderon. Moreover, the treatment of the
axial-vector mesons is now in a very satisfactory shape by employing the
B-field formalism.
The ESC-approach to the nuclear force is a promising one.
It opens the possibility to make a connection between
the at present available baryon-baryon experimental data on the one hand,
and with the underlying quark structure of the baryons and mesons on the other hand.
Namely, a successful description
of both the $N\!N$- and $Y\!N$-scattering data is obtained with meson-baryon
coupling parameters which all comply with the QPC-model.
We note that by studying the relation between the QPC-processes and the BBM-couplings, we
determined the ratio $\gamma(^3P_0)/\gamma(^3S_1)=2:1$.
In the literature, the $^3P_0$-QPC and the $^3S_1$-QPC in the SCQCD \cite{Mil89}
has been
studied by \cite{KI87} and \cite{KP88} respectively. In this paper we give therefore
an estimation of the relative importance of the QPC processes.
At the same time we comply with the strong constraint of no bound states
in the $S=-1$-systems. Therefore, the ESC-models, ESC04 and ESC08,
are an important step in the determination of
the baryon-baryon interactions for low energy scattering and the description
of hypernuclei in the context of broken SU$(3)$-symmetry.
The values for many parameters, which in previous Nijmegen work were considered to be free to
a large extent, follow now rather well the pattern shown in quark-model predictions.
This is particularly the case for the $F/(F+D)$-ratios of the OBE- and MPE-interactions.
For the nuclear matter description we introduced the multi-pomeron
(multi-gluon) exchange
three-body force potential, achieving three things (i) right nuclear saturation,
(ii) correct neutron star maas, and (iii) better hyperonic well depth's
$U_Y$ for $Y= \Lambda, \Sigma, \Xi$ (see the companion papers II,III).
The combined fit for NN and YN is extremely good in ESC08. It is for the first time
that the quality of the NN-fit does not suffer from the inclusion of the YN-data.
The $\Lambda N$ p-waves seem to be better, which is the result of the truly
simultaneous $NN+YN$-fitting. This is also reflected in the better
Scheerbaum $K_\Lambda$-value \cite{Sch76},
making the well-known small spin-orbit splitting smaller, see Ref.~\cite{Hiya00}.
The G-matrix results showed for ESC04 that basic features of hypernuclear data are reproduced
nicely, improving on the soft-core OBE-models NSC89 \cite{MRS89}
and NSC97 \cite{RSY99}. In spite of this superiority of ESC04 for hypernuclear data,
some problems remained. In particular the well depth $U_\Sigma$ was attractive,
which is very unlikely in view of several other studies e.g.
Ref.'s~\cite{Bat94,Dab99,Nou02,Koh04}
Furthermore, it has been shown \cite{Nis02} that the EOS for nuclear matter is too soft for
the soft-core models.
From this we learn that a good fit to the present scattering data not necessarily
means success in the G-matrix results. To explain this one can think of two reasons:
(i) the G-matrix results are sensitive to the two-body interactions below 1 fm,
whereas the present YN-scattering data are not, (ii) other than two-body forces play an
important role.
Since the problem with $U_\Sigma$ hints at a special feature in
the $\Sigma^+p(^3S_1)$-channel,
it is likely that it is a two-body problem. As we have shown in ESC08 it can be solved by the
inclusion of the quark-core effects. For the softness of the EOS
a natural possibility is the presence of three-body forces (3BF) in nuclear and hyperonic
matter, see Ref.~\cite{Nis02}. This also solves the nuclear saturation problem \cite{Rij04b}.
It is important to stress the role of the information on hypernuclei in our analysis.
We imposed for the ESC08-solution that $U_\Sigma >0$ and $U_\Xi < 0$.
This induced the occurrence of strong tensor-forces with the consequence of a
bound state in the S=-2 systems. Namely, deuteron-like bound states in the
$\Xi N(^3S_1-^3D_1,I=0,1))$-system.
Summarizing the results of the ESC-approach to baryon-baryon interactions, it can be
stated that this is a very successful one. It has been shown that ESC-models are able
to give with single parameter-set's extremely satisfactory descriptions of the
NN$\oplus$YN-data, and at the same time lead to successful G-matrix results.
For the coupling constants (i) flavor SU(3)-symmetry can be maintained, and
(ii) they show rather well the pattern as predicted by the QPC-model.
The tensor-couplings play an important role, especially in the prediction of
a deuteron-like S=-2 bound states. We conclude that these ESC-model predictions,
as well as the applications to the S=-3,-4 systems and hyperonic matter,
have a rather sound physical basis.
We close by remarking that the determination of the MPE-couplings opens the possibility
to compute the 3BF-potentials for baryon-systems where all meson-pair vertices are fixed
by the ESC-model.
\section*{Acknowledgements}
We wish to thank E. Hiyama, T. Motoba and H.-J. Schulze for many
stimulating discussions.
|
2,877,628,089,487 | arxiv | \section{Introduction}
Gain, bandwidth, and noise performance ultimately dictate the quantum efficiency and speed of quantum measurements performed at microwave frequencies in the circuit quantum electrodynamics architecture. Improving these three properties of an amplifier has been the driving force for Josephson junction based amplifier design and characterisation including significant work optimising bandwidth~\cite{Mutus:2014dd, Roy:2015ky}, pump rejection using flux or non-degenerate pumping schemes~\cite{Yamamoto:2008cr, Bergeal:2010iu, Roch:2012gy, Mutus:2013iw, Eichler:2014dk, Frattini:2017ji} and realizing directionality in the amplification process~\cite{Abdo:2013ib,Sliwa:2015vc,Lecocq:2017ge}. Josephson junction based parametric amplifiers utilise the intrinsic nonlinearity of the junction as the basis for parametric wave mixing. Controlling the type and strength of this nonlinearity has been the focus of several amplifier designs since this quantity imposes the input saturation power (characterised as the $\SI{1}{\decibel}$ compression point) of such amplifiers~\cite{Abdo:2011dfa,Eichler:2013cr,Eichler:2014iw,Zhou:2014gt,Eddins:2017ty,Liu:2017bi,Frattini:2018ud}. Moreover, when the strength of the non-linearity reaches a few percent of the operating frequency of the device, higher-order effects lead to imperfect squeezing and non quantum-limited performance~\cite{Yurke:2006cw,Kochetov:2015tp,Boutin:2017dn}. However it is only very recently that the fourth order non-linearity or Kerr non-linearity was identified as the main cause of Josephson parametric amplifiers saturation~\cite{Liu:2017bi}. In their work, \textit{Liu et al.} didn't manage to relate the effective non-linearity of their Josephson junction amplifier to the actual circuit model. This outstanding goal was achieved soon after in the case of the SNAIL Parametric Amplifier, a Josephson device operated in a three-wave mixing mode~\cite{Frattini:2018ud}. In this article we present a parametric amplifier based on four-wave mixing. The subtlety here is that the non-linearity at the root of parametric amplification is the same than the one causing saturation. Our device is formed by a high impedance Josephson meta-material --- an array of $N=80$ SQUIDs --- that forms a $\lambda/4$ non-linear resonator~\cite{YURKE:1996dt, CastellanosBeltran:2008cg, Anonymous:HxLrydHU, Vesterinen:2017ei}. The dispersion relation of this SQUIDs array, obtained via two tone spectroscopy, is fitted using a long range Coulomb interaction or \textit{remote ground} model~\cite{krupko2018kerr} leading to independently inferred values of the circuit components (capacitances and SQUIDs' critical current). We show that the amplifier can be quantitatively described by an effective non-linear $LC$ series resonator with a resonant frequency near the first resonant mode of the array. We report a good quantitative agreement between the saturation power of this JPA and a model without fitting parameters. According to this theory, the 80 SQUID array yields a \SI{15}{\decibel} improvement over a comparable single SQUID device.
This article is organized as follow: In Section II we present an effective model and review the basic description of our device as a single port degenerate Josephson parametric amplifier and in section III we discuss how arrays of SQUIDs can effectively reduce the nonlinearity of the device leading to increased saturation power. In Section IV we present the device and Section V describes its properties in the linear regime. Gain and saturation are reported in Section VI while we discuss the main results in section VII.
\section{Model}
\begin{figure}[h]
\includegraphics[width=\linewidth]{Figure2.pdf}
\caption{\textbf{a} Sketch of the JPA based on an array of 80 SQUIDs. We highlight the capacitance $C_\text{out}$ between the last superconducting pad and the ground and the parasitic shunt capacitance $C_\text{s}$ between the input/output pad and the last superconducting pad. (Zoom-in) SEM image of 7 identical SQUIDs where a single junction has an area of $\SI{10.7}{\micro\meter} \times \SI{0.370}{\micro\meter}$. \textbf{b} Effective LC series nonlinear resonator.}
\label{fig2}
\end{figure}
Our device (See \cref{fig2}) can be modeled as an effective single port, degenerate Josephson parametric amplifier employing a non-linearity of the Kerr type. The circuit can be described by the Hamiltonian of a non-linear resonator:
\begin{equation}
H_\text{JPA}=\hbar\omega_\text{eff}A^\dagger A+\hbar\frac{K_\text{eff}}{2}\left (A^\dagger\right)^2 A^2
\end{equation}
where $A$ is the annihilation operator of the intra-resonator field. It is characterised by an effective resonant frequency $\omega_\text{eff}$, a non-linearity or self-Kerr coefficient $K_\text{eff}$ and a coupling rate to a transmission line $\kappa_\text{eff}$. The link between the circuit model and these effective parameters will be explained in Section V. The JPA is powered by a monochromatic current pump. The physics of such degenerate Josephson parametric amplifiers has been explained in great detail in various articles~\cite{Yurke:2006cw,Vijay:2009wp, Roy:2015ky, Boutin:2017dn}. We recall here the main equations, following the approach of \textit{Eichler and Wallraff}~\cite{Eichler:2014iw}. The dynamics of the circuit is inferred using conventional input-output theory:
\begin{equation}
\dot{A} = -i\omega_\text{eff}A-iK_\text{eff}A^\dagger AA - \frac{\kappa_\text{eff}}{2}A+\sqrt{\kappa_\text{eff}} A_\text{in}
\label{EOM}
\end{equation}
with $A_\text{in}$ the input field coupled with rate $\kappa_\text{eff}$. The boundary conditions of the resonator are taken into account via the equation $A_\text{out} = \sqrt{\kappa_\text{eff}}A - A_\text{in}$, where $A_\text{out}$ is the output field. Next we assume that $A=\alpha+\hat{a}$, where $\alpha$ is a classical part (referring to the strong coherent pump) and $\hat{a}$ is the signal that we treat quantum mechanically. To obtain the gain of the amplifier we follow a two-step procedure: we first solve for the classical field $\alpha$ while setting $\langle\hat{a}\rangle=0$ and then we use a linearisation of the equation of motion for $\hat{a}$ around this working point (see Appendix~\ref{sec:gain} for a detailed derivation). This leads to the standard equation of a parametric amplifier:
\begin{equation}
\hat{a}_\text{out}(\Delta)=g_{S,\Delta}\hat{a}_\text{in}(\Delta)+g_{I,\Delta}\hat{a}_\text{in}^\dagger(-\Delta)
\label{gain}
\end{equation}
The operators $\hat{a}_\text{in}(\Delta)$ and $\hat{a}_\text{out}(\Delta)$ are the Fourier components of the input and outputs signals, where $\Delta$ is the dimensionless frequency detuning $\Delta = (\omega_\text{p} - \omega_\text{signal})/\kappa$ from the pump frequency. \cref{gain} illustrates the link between the output signal and the inputs at signal ($\Delta$) and idler $(-\Delta)$ frequencies. Signal and idler gain ($g_{S,\Delta}$ and $g_{I,\Delta}$, respectively) are expressed as:
\begin{subequations}
\begin{align}
g_{S,\Delta} &= -1 + \frac{i(\delta - 2\xi_\text{$\alpha$} n - \Delta) + \frac{1}{2}}{(i\Delta - \lambda_{-})(i\Delta - \lambda_{+})} \\
g_{I,\Delta} &= \frac{-i\xi_\text{$\alpha$} n e^{2i\phi}}{(i\Delta - \lambda_{-})(i\Delta - \lambda_{+})}
\end{align}
\label{gsgi}
\end{subequations}
with $\lambda_{\pm}=\frac{1}{2} \pm \sqrt{((\xi_\text{$\alpha$} n)^2 - (\delta - 2\xi_\text{$\alpha$}n))}$ (a complete derivation is given in Appendix~\ref{sec:gain}). Both of them encompass the dimensionless detuning $\delta$ between the pump and bare resonator frequencies, $\xi_\text{$\alpha$}$ the product between the non-linearity and the pump power, $n$ the normalised number of pump photons in the cavity, $\phi$ the phase difference between the pump and the signal and $\Delta$. The exact expressions for these parameters are given in Appendix~\ref{sec:gain}. $|g_{S,\Delta}|^2$ is plotted in \cref{fig1}, using the parameters of our amplifier, as a function of the pump power and pump frequency for zero detuning between the pump and the signal ($\Delta = 0$). We define the optimal pump frequency $f_\text{p,opt}$ as the one which maximises the gain for a given pump power, as shown in \cref{gsgi}.
\begin{figure}[h]
\includegraphics[width=\linewidth]{Figure1.pdf}
\caption{
Theoretical maximum signal gain $|g_{S,\Delta}|^2$ versus pump power and pump frequency for signal detuning $\Delta=0$. Larger gain requires larger pump power and lower pump frequency. The inset shows the gain versus frequency for three different pump configurations as indicated by the markers. The maximum expected gain for a given pump power (white square) is reduced when the pump frequency is slightly detuned from this optimal point (circle and cross). }
\label{fig1}
\end{figure}
$\Delta f_p=f_\text{p}-f_\text{p,opt}$ is a frequency shift from optimal pumping conditions. As illustrated on the inset of \cref{fig1}, $\Delta f_p$ as small as $\SI{5}{\mega\hertz}$ leads to a reduction of the gain in excess of $\SI{1}{\decibel}$. This observation is at the heart of JPA saturation~\cite{Liu:2017bi}, since an input power of $n_{s}$ signal photons per second will lead to a shift $\Delta f_p \approx n_{s} \times K_\text{eff}/\kappa_\text{eff}$ Hertz from optimal pumping frequency, thus leading to a drop of the maximum expected gain. This qualitative explanation will be further formalised in Section VI and it can be shown that the input saturation power of JPA increases linearly with the ratio $\kappa_\text{eff}/\abs{K_\text{eff}}$~\cite{Eichler:2014iw}. Maximizing this ratio, and thus minimizing the non-linear self-Kerr term $K_\text{eff}$, is therefore of prime importance.
\section{Decreasing non-linearity using arrays}\label{nonlinearity}
Because reduced $K_\text{eff}$ is important to improve dynamical range, several methods have been introduced to reduce non-linearity in parametric amplifiers. One first option is to use the intrinsic non-linearity of superconductors such as Niobium~\cite{Tholen:2007ea}, NbN~\cite{Chin:ws} or granular Aluminium~\cite{Maleeva:2018uya}, since they often come with very weak non-linearities spanning from $\SI{20}{\milli\hertz}$ to $\SI{30}{\kilo\hertz}$ for resonators in the $\SI{}{\giga\hertz}$ range. However, these non-linearities are so weak that extremely large pump powers are required resulting in experimental challenges. Another option is to dilute the non-linearity of a single Josephson junction into a larger and linear resonator~\cite{Bourassa:2012ej,Zhou:2014gt}. However, in this case the Josephson junction is not purely phase-biased anymore and the usual quartic approximation used to treat the Josephson potential has a limited validity~\cite{Eichler:2014iw,Boutin:2017dn}.
Already in the early days of Josephson Parametric Amplifiers it was recognized that using an array of $N$ Josephson junctions could be beneficial~\cite{YURKE:1996dt}. In this case the total phase drop $\Phi_{tot}$ (or equivalently the voltage drop) occurs across the whole array and not across a single junction. Thus the non-linearity is divided by $N$ since each junction experiences a phase drop $\Phi_{tot}/N$ (See Appendix~\ref{sec:nonlin} for a derivation). This idea can be pushed further by fabricating an array of $N$ Josephson junctions with critical current $N$ times larger; the non-linearity is then divided by $N^2$~\cite{Eichler:2012ixa,Eichler:2014iw,Zhou:2014gt}. However, the approximation that each junction experiences a $\Phi_{tot}/N$ phase drop loses validity when the system becomes very large, reaching a size comparable to the wavelength of the microwave signal. In this case propagating effects should be accounted for properly.
To do so we start by defining the normal modes of the circuit and then treat the non-linearity perturbatively as described in previous works~\cite{Anonymous:2012ek,Weissl:2015do, Roy:2017wp}. Each SQUID is considered as an $LC$ parallel oscillator, described by $C_\text{J}$ and $L_\text{J}$. However, describing the chain as a standard transmission line as it is routinely done, where every $LC$ oscillator is shunted to the ground via a ground capacitance $C_g$ is not the most appropriate description for our system. Given that the distance between the chain and the ground plane is comparable or greater than the modes wavelength (see \cref{fig2}.\textbf{a}), the screening of the charges by the ground plane cannot be considered as local. Capacitive effects between SQUIDs must be accounted for via the long-range part of the Coulomb interaction. We follow the procedure described by \textit{Krupko et al}~\cite{krupko2018kerr} to take this long-range interaction into consideration. This remote ground model gives results closer to the experiment than the standard transmission line model (see Appendix \ref{sec:Remote}). Although this remote ground model is more complex than the standard model, there is still only one parameter describing the screening effect: it is no longer the ground capacitance $C_g$ but a characteristic length of the long range Coulomb interaction preventing from a divergence of the model, called $a_0$. In the description of the capacitive effects in our amplifier, we also consider that the chain is terminated by a metallic pad creating an additional capacitance to ground $C_\text{out}$ and a shunt capacitance $C_{s}$ together with the input transmission line (see figure \cref{fig2}.\textbf{a}). More specifically the system is modelled by considering the Lagrangian $\mathcal{L}$ of the chain, where the fluxes $\Phi_{n}$ between each SQUID are taken as coordinates. This Lagrangian reads:
\begin{equation}
\begin{split}
\mathcal{L} = \sum\limits_{n=0}^{N-1}{\frac{C_J}{2}(\dot{\Phi}_{n+1} - \dot{\Phi}_{n} )^2} -\sum\limits_{n=0}^{N-1}{\frac{1}{2L_J}(\Phi_{n+1} - \Phi_{n})^2} \\
+ \sum\limits_{n=1}^{N-1}{\frac{C_{g,nn}}{2}\dot{\Phi}_n^2} + \sum\limits_{n=1}^{N-1}{\sum\limits_{i\neq n}^{N-1}{\frac{C_{g,ni}}{2}(\dot{\Phi}_n^2 - \dot{\Phi}_i^2}}) \\
+ \frac{C_{out}}{2}\dot{\Phi}_N^2 + \frac{C_s}{2}\dot{\Phi}_N^2
\end{split}
\end{equation}
with $N$ the number of SQUIDs in the chain and $C_{g,ni}$ are the elements of a generalised ground capacitance matrix. We define a new set of variables to describe the system, the charge $Q_n$ and its conjugate $I_n$ at each node $n$
\begin{equation}
\begin{aligned}
Q_n = \frac{\partial{\mathcal{L}}}{\partial{\dot{\Phi}_n } } \\
I_n = \frac{\partial{\mathcal{L}}}{\partial{\Phi_n } }
\end{aligned}
\end{equation}
These new variables lead to capacitance and inductance matrices ($\hat{C}$ and $\hat{L}$ respectively):
\begin{equation}
\begin{aligned}
\vec{Q} = \hat{C}\vec{\dot{\Phi}} \\
\vec{I} = \hat{L}^{-1}\vec{\Phi}.
\end{aligned}
\end{equation}
From these matrices, we can define the angular frequency matrix $\hat{\Omega}$ as :
\begin{equation}
\hat{\Omega}^2=-\hat{C}^{-1}\hat{L}^{-1}
\end{equation}
The eigenvalues $\omega_{n}^2$ and eigenvectors of the matrix $\hat{\Omega}^2$ define respectively the resonant frequency and the wave profile of each mode $n$ of the chain. It allows the definition of an effective capacitance $C_{\text{eff},n}$ and an effective inductance $L_{\text{eff},n}$ for each mode $n$:
\begin{equation}
\begin{aligned}
C_{\text{eff},n} = \vec{\varphi}_{n}^T\hat{C}\vec{\varphi}_{n} \\
L_{\text{eff},n}^{-1} = \vec{\varphi}_{n}^T\hat{L}^{-1}\vec{\varphi}_{n}
\end{aligned}
\end{equation}
With this linear model, we now treat the Kerr non-linearity of the chain. The Josephson non-linearity can be reintroduced as a perturbation of the linear Hamiltonian, by developing the cosine of the Josephson potential up to fourth order~\cite{krupko2018kerr}. By applying the Rotating Wave Approximation (RWA), one can rewrite the full Hamiltonian as:
\begin{equation}
\begin{split}
\hat{H} = \sum\limits_{n}{\hbar\omega_{n}a^{\dagger}_na_n} -\sum\limits_{n}{\frac{\hbar}{2}K_{nn}a^{\dagger}_na_na^{\dagger}_na_n} \\
-\sum\limits_{n,m}{\frac{\hbar}{2}K_{nm}a^{\dagger}_na_na^{\dagger}_ma_m}
\end{split}
\end{equation}
where $K_{nn}$ and $K_{nm}$ are the self and cross Kerr coefficients, respectively:
\begin{equation}
\begin{aligned}
K_{nn} = \frac{2\hbar\pi^4E_\text{J}\eta_{nnnn}}{\Phi_{0}^4C_\text{J}^2\omega_{n}^2} \\
K_{nm} = \frac{4\hbar\pi^4E_\text{J}\eta_{nnmm}}{\Phi_{0}^4C_\text{J}^2\omega_{n}\omega_{m}}
\end{aligned}
\end{equation}
$\eta_{nnmm}$ takes into account the spatial variation of the phase across the chain and $E_\text{J}=\varphi_\text{o}^2/L_\text{J}$ is the Josephson energy of a single SQUID. Given that $\eta_{nnmm}$ depends only on circuit parameters of the chain, the Kerr non-linearities of the modes are fully predictable.
To describe the effect of the transmission line connected to the array and the resulting external quality factor, we model this $\lambda/4$ resonator as an effective non-linear series $LC$ circuit (See \cref{fig2}.\textbf{b}) close to its resonance. From now on, we drop the index $n$ since we only consider the first mode. Using the effective inductance and capacitance defined previously, we can then easily define an effective resonant frequency $\omega_\text{eff}=1/\sqrt{L_\text{eff}C_\text{eff}}$, an effective external quality factor $Q_\text{eff}=\sqrt{L_\text{eff}/C_\text{eff}}/Z_\text{c}$ and an effective coupling rate $\kappa_\text{eff}=\omega_\text{eff}/Q_\text{eff}$, as is very commonly done in microwave engineering~\cite{pozar2009microwave}. The accuracy of this mapping relies on a precise determination of the capacitance and inductance matrices. Using a combination of electromagnetic simulations and two-tone measurements we managed to infer precisely $\hat{C}$ and $\hat{L}$ as will be explained later.
\section{Sample Description}\label{sample}
The JPA presented in this work is made of 80 SQUIDs, obtained using a bridge-free fabrication technique~\cite{Lecocq:2011dk}. It is fabricated on a $\SI{300}{\micro\meter}$ thick, single side polished, intrinsic silicon wafer. The backside of the wafer is metalized using a sandwich of titanium ($\SI{10}{\nano\meter}$) and gold ($\SI{200}{\nano\meter}$). The array is connected galvanically to a $\SI{50}{\ohm}$ microstrip transmission line on one side and to a superconducting pad on the other side (\cref{fig2}.\textbf{a}). Such a design presents two main advantages. It can be fabricated in one single electronic lithography step and since no superconducting ground plane is involved, flux-trapping possibilities and the effect of Meissner currents are strongly reduced.
The associated circuit parameters are $C_\text{J}=\SI{370}{\femto\farad}$, $a_0=\SI{4.3}{\micro\meter}$, $C_\text{out}=\SI{24.7}{\femto\farad}$ $C_\text{s}= \SI{1}{\femto\farad}$. Finally, $L_\text{J}=\SI{165}{\pico\henry}$ at zero magnetic flux, which translates into a critical current of $I_\text{c}=\SI{2}{\micro\ampere}$ for each SQUID. $L_\text{J}$ remains larger than the kinetic inductance of the aluminum wires connecting the SQUIDs. We estimate this stray inductance to be $L_\text{stray}=\SI{30}{\pico\henry}$. Ensuring $L_\text{stray}\ll L_\text{J} $ is important to the validity of our model (\cref{fig2}.\textbf{b}). $C_\text{J}$ is inferred via the size of the junctions and the capacitance density $\SI{45}{\femto\farad\per\square\micro\meter}$~\cite{Fay:2008vi}. The values of $C_\text{out}$ and $C_\text{s}$ are obtained using an electromagnetic simulation software. $L_\text{J}$ and $a_0$ are determined from the dispersion relation of the array, as explained in Section~\ref{linear}.
\section{Characterization in the linear regime}\label{linear}
The device is measured using a conventional cryogenic microwave measurement setup (See Appendix~\ref{sec:setup}). The linear properties of the JPA are inferred by measuring the reflected phase of the microwave signal at zero flux and low power (\cref{fig3}.\textbf{c}). The fit of the phase shift yields $\omega_\text{exp}/2\pi=\SI{7.07}{\giga\hertz}$ and $Q_\text{exp}=19$. The resonant frequency of the JPA can be adjusted over a broad frequency range when flux-biasing the SQUID array (\cref{fig3}.\textbf{a}). We note the smooth behaviour of the device during flux tuning, despite the presence of the SQUID array. We attribute this stability to the absence of superconducting ground plane.
We can go one step further in the characterization of the device and obtain the dispersion relation of the array using two-tone spectroscopy~\cite{Anonymous:2012jo,Weissl:2015do}. The higher-order resonant frequencies of the device are presented in \cref{fig3}.\textbf{b}. Fitting these data using the circuit model presented in Section~\ref{nonlinearity}, we can determine $L_\text{J}$ and $a_0$ independently. Plugging these values in the effective model introduced before, we obtain the values $L_\text{eff}=\SI{21}{\nano\henry}$, $C_\text{eff}=\SI{24}{\femto\farad}$ and $K_\text{eff}=\SI{80}{\kilo\hertz}$. These values translate to an effective resonant frequency $\omega_\text{eff}/2\pi=\SI{7.08}{\giga\hertz}$ and an effective external quality factor $Q_\text{eff}=19$ in very good agreement with the measured values. We note that this external quality factor is much smaller than internal quality factors $Q_\text{int}\sim 10^4$ we measured in devices fabricated using the same procedure~\cite{krupko2018kerr}. This justifies that internal losses can be safely neglected in our model. In the next section, we will use the value of $K_\text{eff}$ to explain the measured gain, bandwidth and $\SI{1}{\decibel}$ compression point of the JPA without any free parameters.
\begin{figure}[h]
\includegraphics[width=\linewidth]{Figure3.pdf}
\caption{
\textbf{a} DC flux modulation of the phase of the reflected signal. \textbf{b} Dispersion relation of the SQUID array for the first modes. Blue stars are the solution of the matrix computation and black diamonds show experimental data. \textbf{c} Cut of \textbf{a} at zero magnetic flux, where the experimental data are fitted by an arctangent.}
\label{fig3}
\end{figure}
\section{Gain and input saturation power}
\begin{figure}[h]
\includegraphics[width=\linewidth]{Figure4.pdf}
\caption{
\textbf{a} Signal power gain versus signal frequency. Experimental (dots) and calculated (solid line) gain at four different pump powers and frequencies. The theoretical pump parameters are chosen as followed: the pump frequency is first set to the one used experimentally. The pump power is then set to maximise the gain at zero detuning ($\Delta$ = 0), as done experimentally. This leads to optimal pump biasing conditions, which can be visualised as the ridge on \cref{fig1}. These optimal conditions are (from low gain to high gain) : ($\SI{-81.65}{dBm}$, $\SI{6.83}{\giga\hertz}$), ($\SI{-81.12}{dBm}$, $\SI{6.80}{\giga\hertz}$), ($\SI{-80.83}{dBm}$, $\SI{6.79}{\giga\hertz}$), ($\SI{-80.57}{dBm}$, $\SI{6.78}{\giga\hertz}$). The bumps on the right tail of the experimental amplification curves are due to the normalization procedure and small losses at zero pump power. \textbf{b} Maximum gain as a function of the input power signal for the four same pump parameters. The pump powers for the theoretical plots have been shifted by up to $\pm\SI{0.03}{dBm}$ from the optimal pump power to account for the fact that a very small variation of pump power translates in a large variation of the gain as explained in the text. Such shifts are compatible with small drifts in the attenuation of the input line over the course of one day.}
\label{fig4}
\end{figure}
In \cref{fig4}.\textbf{a}, we present the gain of the amplifier versus frequency at various pump powers. We measure a $\SI{-3}{\decibel}$ bandwidth of $\Delta f=\SI{45}{\mega\hertz}$ at $\SI{20}{\decibel}$ of gain. All these gain curves can be explained by~\cref{gain} using only the above-mentioned effective parameters. Interestingly it also provides an accurate calibration of the pump power at the JPA level and thus of the attenuation of the input line (see Appendix~\ref{sec:calib}). We note that our JPA can be flux tuned over a band of $\SI{900}{\mega\hertz}$ while reaching $G_\text{max}=\SI{20}{\decibel}$ as shown in Appendix~\ref{sec:flux}. Knowing the attenuation of the input line, the input saturation power of the JPA is quantified by measuring the maximum gain $G_\text{max}$ as a function of input power for different gains (\cref{fig4}.\textbf{b}). More specifically we measure a $\SI{1}{\decibel}$ compression point $P_\text{1dB}=-117\pm 1.4 \text{ dBm}$ at $\SI{20}{\decibel}$ of gain. This point corresponds to the input power at which the amplifier saturates and the gain is compressed by $\SI{1}{\decibel}$ from $G_\text{max}$. Again we show a very good agreement between experiment and theory, without fitting parameters. To describe the saturation of the JPA, the number of signal photons inside the JPA must be taken into account while the pump is on. To do so, we add, in a self-consistent approach ~\cite{Eichler:2014iw}, the terms $2iK\langle a^{\dagger}a \rangle\alpha$ and $iK\langle a^{2} \rangle\alpha^{*}$ to the initial equation of motion of the intra-resonator field (see Appendix~\ref{sec:gain}). This correction to the total number of photons inside the cavity (pump, signal and idler), dependent on the signal power, allows the modelling of the amplifier saturation for a given set of pump frequencies and powers as plotted on \cref{fig4}.\textbf{b}. As will be explained in the next section, this saturation is very sensitive to the pump biasing conditions.
\section{Discussion}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\linewidth]{Figure5.pdf}
\caption{Summary of the amplifier characteristics and agreement between experiment (dots) and theory for optimal pumping condition (line). \textbf{a} Maximum gain and $\SI{-3}{\decibel}$ bandwidth (obtained from a Lorentzian fit of the amplification curve) as a function of the pump frequency. The gain theoretical line follows the highlighted ridge shown on \cref{fig1}.\textbf{b}. \textbf{b} $\SI{1}{\decibel}$ compression point as a function of the initial maximum gain. The shaded area below (above) the theoretical curve shows the effect of a shift of $ + \SI{0.03}{dBm}$ ($ -\SI{0.03}{dBm}$) from the optimal pump power on the $\SI{1}{\decibel}$ compression point. The dashed line shows the $\SI{1}{\decibel}$ compression point of a single-SQUID JPA which would show the same bandwidth and operating frequency. }
\label{fig5}
\end{center}
\end{figure}
To further illustrate the performance of our device and the predictive value of our model, we summarize three important figures of merit of our JPA in~\cref{fig5}. These are the maximum gain, the -3 dB bandwidth, and the 1 dB compression point. The maximum gain $G_\text{max}$ at low signal power and the corresponding $\SI{-3}{\decibel}$ bandwidth $\Delta f$ are measured for different pump powers (panels \textbf{a} and \textbf{b}). The gain-bandwidth product remains equal to $\SI{450}{\mega\hertz}$ over this pump power range, as expected from JPA theory. We now compare the 1 dB compression points measured at various gains to our theoretical predictions. Such a plot should be interpreted with great care since a very small deviation from optimal pump conditions can lead to variations of $P_{1dB}$ as reported previously~\cite{Liu:2017bi}. For example a pump power variation of $0.03\ \text{dBm}$ leads to a change of up to $\SI{3}{dBm}$ in $P_\text{1dB}$, as illustrated by the shaded area of \cref{fig5}.\textbf{c}. To illustrate the advantage of using SQUID arrays, we also plot what would be $P_\text{1dB}$ for a single SQUID JPA as reported in various papers~\cite{Hatridge:2011dh,Mutus:2013iw}. To ensure a meaningful comparison we chose the parameters of this single SQUID JPA so that it displays the same working frequency ($\omega_\text{exp}/2\pi=\SI{7.07}{\giga\hertz}$) and bandwidth ($Q_\text{exp}=19$) as our array JPA. The self-Kerr coefficient of such JPA would be $K_\text{single}/2\pi=\SI{2.4}{\mega\hertz}$ (to be compared to $K_\text{eff}/2\pi=\SI{80}{\kilo\hertz}$). This translates into $P_\text{1dB,single}= -131\ \text{dBm}$ at $G_\text{max}=\SI{20}{\decibel}$ compared to $P_\text{1dB,array}=-116\text{ dBm}$ for our array JPA. This $15 \text{ dBm}$ difference reflects directly the ratio of self-Kerr coefficients since $P_\text{1dB}$ scales as $\kappa_\text{eff}/\abs{K_\text{eff}}$ as explained previously. This illustrates the key advantage of using arrays to fabricate high-performance JPAs. Finally we would like to discuss the data/theory agreement. According to our microscopic model $P_\text{1dB}$ should be $-116\text{ dBm}$, while we measured $-117\pm1.4\text{ dBm}$. This good agreement confirms that adding two terms to the equation of motion of the intra-resonator field is enough to explain the saturation effect observed in our JPA. From a physical point view, the effect of these terms is two-fold. First the bare frequency of the JPA $\omega_\text{eff}$ becomes dependent on the number of signal photons, similarly to the ac-Stark shift effect. Second the number of pump photons inside the JPA depends as well on the number of signal photons; an effect known as pump depletion in parametric amplifiers theory.
\section{Conclusion}
We designed and measured a Josephson parametric amplifier made of 80 SQUIDs. This device relies on a single-step, all-aluminium fabrication process, easily reproducible in a research-grade clean-room. We showed that the number of SQUIDs in the array has a direct and predictable impact on the nonlinearity, which is directly linked to the saturation power of the amplifier. The circuit model we used gives a very good agreement with the experimental data, without fitting parameters. Improvements could be obtained by bringing the Josephson inductance down to $L_\text{J} \approx L_\text{stray}$. Setting $L_\text{J}$ to \SI{40}{\nano\henry}, just above $L_\text{stray}$, adjusting $C_\text{out}$ to $\SI{50}{\femto\farad}$ and the total number of SQUIDs to $N=150$, would lead to a JPA with a bare resonant frequency $f_0=\SI{7.45}{\giga\hertz}$ and external quality factor $Q_\text{e}=9$. According to our model, this JPA would display for \SI{20}{\decibel} signal gain, a bandwidth of \SI{95}{\mega\hertz} and a \SI{1}{\decibel} compression point of $-102\text{dBm}$. A pump power of $-66\ \text{dBm}$ would be necessary to operate a JPA with these figures of merit. This value is comparable to what was reported for Josephson Traveling Wave Parametric Amplifiers (JTWPA)~\cite{macklin2015near,white2015traveling} and, as such, should not be a concern. We would like to stress that these estimates cannot be strictly quantitative since the approximation described in \cref{sample} ($L_\text{stray}\ll L_\text{J}$) does not hold anymore. Theory should be further developed to account for the effect of these stray inductances. Further developments that could be applied to this SQUID array JPA include input impedance engineering to improve the performance of the device~\cite{Mutus:2014dd,Roy:2015ky} or band engineering to bring in new capabilities such as non-degenerate~\cite{Bergeal:2010iu} or multi-mode parametric amplification~\cite{Simoen:2015by}.
\section*{Acknowledgements}
The authors would like to thank W. Wernsdorfer, E. Eyraud,
F. Balestro and T. Meunier for early support with the experimental setup. Very
fruitful discussions with F. W. Hekking and D. Basko and I. Takmakov are strongly
acknowledged. This research was supported in part by the International Centre for Theoretical Sciences (ICTS)
during a visit for participating in the program - Open Quantum Systems (Code: ICTS/Prog-oqs2017/2017/07). This research was supported by the ANR under contracts CLOUD
(project number ANR-16-CE24-0005). J.P.M. acknowledges
support from the Laboratoire d\textquoteright excellence LANEF in Grenoble
(ANR-10-LABX-51-01). R.D. and S.L. acknowledge support from the CFM
foundation.\\
|
2,877,628,089,488 | arxiv | \section{Joint Feature Set Using DTW}
Dynamic time warping (DTW) is an efficient algorithm for handling the matching of non-linearly expanded or contracted signals \cite{Sakoe1978}. The optimal alignment between the two sequences is efficiently calculated using dynamic programming.
Given two time sequences $X=\langle x_1, x_2, ..., x_m \rangle$ and $Y=\langle y_1, y_2, ..., y_n \rangle$, it fills an $m$-by-$n$ matrix representing the distances of the best possible partial path using:
\begin{equation}
D(i,j) = min(D(i,j-1),D(i-1,j),D(i-1,j-1))+d(i,j)
\end{equation}
where $1\leq i\leq m, 1\leq j\leq n, d(i,j)$ represents the distance between $x_i$ and $y_j$, and $D(1,1)$ is initialized to $d(1,1)$.
When a music piece copies another musical piece without proper acknowledgement, even a matching portion of only 2-3 seconds might be treated as a case of plagiarism. Computing a degree of resemblance using the DTW to identify all possible cases of plagiarism is a non-trivial problem. In past cases of musical plagiarism, we observed that, almost in all cases, the two musical pieces differed in rhythm, while retaining some other musical property.
Thus for the rhythm-based features, we use a modified version of the DTW \cite{Muller2009}. We use local weights to prefer an insertion or deletion, to a substitution to better account for the variation in rhythm. The modification is of the form,
\begin{equation}
D(i,j) = min \begin{cases}
D(i-1,j-1)+w_d.d(i,j) \\
D(i,j-1)+w_h.d(i,j) \\
D(i-1,j)+w_v.d(i,j)
\end{cases}
\end{equation}
We keep $w_d$ much larger with respect to $w_h$ and $w_v$. This follows from the fact that, even though the DTW would effectively match two sequences having the same musical property with varying rhythm, the distance between the two sequences would still be large owing to the numerous insertions and deletions required for matching the two time-varying sequences. This would not be representative of the similarity observed between the sequences. Thus, using a lower weight for insertion and deletion would help in bringing out this similarity.
Intuitively, however, a disadvantage of this modification would be that the warping path might get stuck in some position due to a local acceleration or deceleration by a large factor. This is overcome by constraining the slope of the warping path. This ensures that there exists a path between the two sequences if and only if their lengths differ at most by a factor of the upper bound of the allowed slope.
\section{Experimental Results}
\label{sec:experiments}
We used the MIRToolBox in MATLAB \cite{Lar:07} to extract the traditional audio-based features described in Section \ref{subsec:tradfeat}. This was followed by extraction of the NMF features, and the DTW-based distance computation to set up pairwise characterization of tracks, as described in Section \ref{sec:nmf}. These distance features and true class labels for the training data were then used to train a random forest classifier \cite{Breiman2001} with 150 trees.
\subsection{Dataset}
We created a database of 2966 song pairs, comprising of 966 plagiarized (positive data set) and 2000 non-plagiarized (negative data set) song pairs. The positive instances were obtained from music covers and plagiarism lawsuits \cite{Law,ItwoFS} including the Music Copyright Infringement Resource of the UCLA School of Law. The dataset is comprised of music from a wide range of genres and languages. All recordings were resampled to a uniform 16 kHz sampling rate with a frame length of 40 ms. The training and test data were separated randomly to provide a 9:1 train-test split.
\subsection{Results}
We compare performance of 3 systems on the test data. We use the traditional feature sets described in Section \ref{subsec:tradfeat} as the {\it baseline}, and compare it with performance using the {\it NMF-features only}, as well as an {\it enhanced feature set} using baseline features along with NMF-features. First, a comparison of overall classification accuracy on the test set is shown in Table \ref{table:acc}. We find that the NMF-only system significantly outperforms the system using the baseline feature set, while the enhanced system significantly outperforms both of them.
Figure \ref{fig:roc} compares performance of the systems using ROC plots for precision and recall for the plagiarized instances in the test data. Since precision and recall are both metrics of accuracy, the higher the Area Under the Curve (AUC), the better the performance. We note that the NMF-only outperforms the baseline, but the enhanced system significantly outperforms the baseline and NMF-only systems.
\begin{table} [t]
\caption{\label{table:acc} {\it Overall classification accuracy comparison for the various systems on entire test data}}
\vspace{0.1in}
\centerline{
\begin{tabular}{|c|c|c|}
\hline
Baseline & NMF-only & Enhanced\\ \hline
45.1\% & 72.6\% & 78.4\%\\
\hline
\end{tabular}}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[height=48mm, width = 90mm]{images/aucplusroc2.png}
\caption{\it (L): Precision-Recall ROC curves for the various systems on the plagiarized instances in test data; (R) Corresponding Area Under the Curve (AUC) for the various systems}
\label{fig:roc}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[height=48mm, width = 80mm]{genre_classification2.pdf}
\caption{\it Performance across genres (Baseline+NMF features)}
\label{fig:genre}
\end{figure}
Figure \ref{fig:genre} shows the successful and failed detection rates for plagiarized track pairs from different genres. Our method performs the best in the country, jazz and rock categories, and worst in hip-hop. This may be because the presence of rap sequences make the detection of these song pairs difficult.
An inspection of plagiarized sequences shows that plagiarized pieces often contain only a small sequence that is similar to the original, and our feature set does not do anything to explicitly address this. To account for such cases, one may consider deriving features from the alignment trajectories, to detect local occurrences of systematically varied rhythms
While existing systems for the detection of near-duplicate music documents can be used for plagiarism detection \cite{Hanna2007}, we observe that their performance worsens in polyphonic settings, failing in a number of cases where our method proves successful, {\it e.g.} song pairs {\it He's So Fine} and {\it My Sweet Lord}, {\it Oye Mi Canto} and {\it Paginas De Mujer}. Our method proves least effective in cases where the similar portions are in the background, {\it e.g.} a copied guitar riff in the musical pieces involved in the {\it La Cienega Music Co. v. ZZ Top} plagiarism suit.
\section{Introduction}
Music-plagiarism is the use or close imitation of another author's music without proper acknowledgement. Every year, vast numbers of new music tracks are released globally, and questionable similarities exist in some sections of music tracks. Aided by the internet, plagiarism is now noticeable {\em globally}, not just across authors, but also across languages and countries. In 2008 alone, 1.4 billion music tracks were sold internationally. This number has since increased to over 1.8 billion. In 2004, the SACEM, an organization that seeks to protect the rights of the original authors, composers and publishers, was able to manually check only a small percentage of registered pieces for potential copyright violations. With such vast numbers of tracks to monitor, the need for automatic techniques for identification of potential copyright violations and detection of music-plagiarism is clear and paramount.
Current approaches to plagiarism detection use techniques based on musical similarity analysis, which emphasizes finding musical pieces in large databases for retrieval. Various feature sets have been proposed for characterization of musical tracks, including the use of pitch contours, loudness, and cepstral features \cite{Ghias1995, Logan2001}. Approaches to computing similarity given the characterizations of two musical tracks use various approaches including $n$-gram based similarity, geometric distances, and string-matching algorithms \cite{Doraisamy2003, Hanna2007}. There are two main issues with adopting similar techniques for plagiarism detection-- first, these approaches typically ignore the issue of polyphony in the recordings, simply using a monophonic approximation instead \cite{Hanna2007}. Polyphonic music can have multiple overlapping notes and is far more complex from an analysis perspective than monophony. Feature extraction techniques traditionally used for monophonic music cannot be expected to do a good job of representing polyphonic characteristics. As a result, these methods have limited success when applied on polyphonic music. Second, unlike similarity computation for retrieval where a system returns a ranked list, a plagiarism detector needs to decide whether a pair of songs are sufficiently similar that one may have been plagiarized from the other.
In this paper, we present an approach that can effectively deal with these issues. To tackle the problem of polyphony, we present a novel feature set derived from signal separation based on compositional models \cite{Raj2010}. This feature set represents the magnitude spectrum of each frame in a musical data segment as an additive, weighted combination of a set of bases. The bases are not constrained to be physically meaningful in this work, but such constraints may be applied as well; e.g. each base could represent the different notes that are expected to be present. In this framework, the weights assigned to bases to compose each spectral vector can be used as a feature representation for the audio frame.
The second problem of requiring a decision, as opposed to a ranked list in case of retrieval, is tackled by formulating the problem as a discriminative classification task. Given a pair of musical segments and various feature characterizations, we compute an ensemble of distance-based features. These computed distances serve as a representation of the {\it pair} of musical pieces. Each pair is a datapoint with a corresponding label that indicates whether one of the songs is plagiarized from the other or not. We use the labels in conjunction with the distance-based features to train a classification model. Now, given a pair of test musical segments, the system can use this model to decide whether one of the segments may have been plagiarized from the other.
While applications aside from plagiarism detection are beyond the scope of this paper, the techniques described can easily be applied to other related tasks. For instance, for the task of retrieving similar segments, given a musical segment ($M_1$), one can query the classification model using $M_1$ and all other musical segments in the database. Such a system could use the distance from the decision boundary as a score to create a ranked similarity list. The use of signal separation-based embedding for musical segments can also be adopted for other music tasks in polyphonic settings.
The remainder of this paper is organized as follows: in Section \ref{sec:problem}, we present a detailed formulation of the problem of music-plagiarism detection. Section \ref{sec:nmf} discusses the feature representations we use for music tracks as well as to characterize pairs of musical segments as plagiarized or not. In Section \ref{sec:experiments}, we describe the dataset used in our experiments and present our results and our discussion of the same. We conclude the paper in Section \ref{sec:conc}.
\section{Conclusion}
\label{sec:conc}
In this paper, we proposed a novel feature space derived from techniques commonly used in signal separation to account for polyphony in music recordings. We formulated the task of plagiarism detection in a supervised classification framework which using distance features over pairs of music tracks to learn the model. Our approach resulted in a significant improvement in performance over the baseline metrics. It is worth noting that our method does not dispense with the need for agencies that track potential copyright violations. Ideally, it should be used as a filtering mechanism that identifies extremely similar music samples.
This work used exemplar bases obtained from the data for NMF-- alternate learning methods for the bases (such as training of the basis set from the data, using examples of expected notes for initialization) and effects of changing the basis set size are directions that we expect to explore in future work.
The NMF-based feature representation should be useful in other tasks that deal with polyphonic music as well. However, for music retrieval tasks such as query by humming or song-matching tasks, we need to be especially careful in the training/selection of basis vectors, because such retrieval tasks are often applied to databases that include user-generated content, typically of a worse quality than studio recordings due to background noise. For such tasks, we would need to have an additional step for noise removal/reduction or ensure that the basis vectors are not trained from studio-quality recordings only. We continue to actively explore these directions.
\ninept
\bibliographystyle{unsrt}
\section{Feature Representations from compositional models}
\label{sec:nmf}
Our feature-space design is primarily motivated by the fact that most current approaches do not explicitly address the issue of polyphony in recordings, simply using a monophonic approximation instead, while others consider polyphony as a more general multidimensional mathematical issue. While an ideal solution to this problem would involve separating out the multiple notes or voices in the recording into different tracks, this would require information for each track that is not likely to be available.
We introduce a novel feature set that is based on compositional representation of the magnitude spectra. Specifically, we use a non-negative matrix factorization (NMF)-based embedding for the music tracks \cite{Lee1999}, that we expect will account for polyphony much better than the feature sets traditionally used for music representation. These NMF-based features are used in conjunction with traditional feature representations.
In the following, we first describe the NMF-based feature extraction technique for audio, and then briefly describe traditionally used feature sets in Section \ref{subsec:tradfeat}. In Section \ref{subsec:dtw}, we discuss the distance function used to create a characterization for pairs of tracks.
\subsection{NMF features}
\label{subsec:nmffeat}
NMF is a subspace analysis technique which obtains a parts-based representation of data by imposing non-negative constraints \cite{Lee1999}. Given training data, NMF can learn a set of basis vectors so that we can represent any datapoint as a linear weighted non-negative combination of these vectors. We use the magnitude spectra of the audio signals as our data, since they are guaranteed to be non-negative. We can represent $M_t$, a magnitude spectral vector at time $t$ as:
\begin{equation}
M_t = \sum_{i=1}^N \textbf{b}_i w_{i,t}
\end{equation}
where $\textbf{b}_i$ is the $i$-th basis vector and $w_{i,t}$ is the weight of the basis in frame $t$. $N$ is the number of basis vectors.
If we represent the set of basis vectors using matrix $\textbf{B}_N = [ \textbf{b}_1, ..., \textbf{b}_N ]$, the model and the weights using the matrix $[\textbf{W}_N]_{i,t} = w_{i,t}$, we can write the model as,
\begin{equation}
\textbf{M} = \textbf{B} \textbf{W}
\end{equation}
NMF has been applied to various audio tasks, including blind source separation and separation of speech from music \cite{Raj2010,Virtanen2007}. The intuition behind using the NMF formulation for music is that a polyphonic music segment will be composed additively from various notes, and NMF can estimate the contribution of the various notes. Thus, each audio frame can be represented using the NMF technique in the $N$-dimensional basis space, where the weights for the frame correspond to the co-ordinates for the frame in this space. NMF has a significant advantage over dimensionality reduction techniques such as PCA and ICA in that the number of bases used need not be less than the original space, resulting in an overcomplete basis space. For this task, the basis vectors may be thought of as individual notes present in the music.
We use an exemplar-based basis set for our experiments in this paper, where bases are drawn randomly from a collection of spectral vectors for the source (magnitude spectra vectors in the music data, in our case). Such bases, although lacking an intuitive interpretation, have useful theoretical properties \cite{Smaragdis2009}. For alternate tasks, where more information about the notes and instruments is available, one could constrain the basis set to consist of true notes or learn them from audio libraries.
Once the set of bases $\textbf{B}$ is selected, each magnitude spectral vector in the dataset can be represented as a non-negative-weighted combination of the bases. The weights are obtained using an iterative update rule minimizing a generalized Kullback Leibler divergence \cite{Lee2001a} between $\textbf{M}$ and $\textbf{BW}$ as follows:
\begin{equation}
\label{eq7}
\textbf{W} = \textbf{W} \otimes \frac{\textbf{B}^T . [\frac{\textbf{M}}{\textbf{BW}}]}{\textbf{B}^T . \textbf{1}}
\end{equation}
where \textbf{1} is a matrix of ones and the operation $\otimes$ denotes element-wise multiplications. All divisions are element-wise, as well. Weights $\textbf{W}$ are initialized to unity, and we iterate equation \eqref{eq7} to convergence. For each music piece, we now have a sequence of weight vectors $\textbf{W}$ which can be used as a feature representation for the audio in the basis space. These sequences correspond to one {\it feature class}, as described in Section \ref{sec:problem}. We used a 1024-dimensional representation of each audio frame in the data and 64 basis vectors for all experiments reported in this paper.
\subsection{Traditional Representations for Audio}
\label{subsec:tradfeat}
In addition to the features extracted from the NMF, we extract features using traditional means of analysis of the audio content, that describe the temporal and spectral sound structures effectively. We use the F-score for identifying the more discriminative features for the dataset used \cite{Chen2006}. We expect that augmenting these with the NMF-based features will lead to increased efficiency in the detection of similar music documents in polyphonic settings.
In general, perception of structural boundaries in music is mostly influenced by variations in timbre, tonality and rhythm. Timbral features, such as spectral rolloff, which estimates the amount of high frequency of the signal, and zero crossing rate are extracted. Key strength, a tonality feature which indicates the cross-correlation score for each different tonality candidate, is another feature extracted. Other features extracted from the audio include Mel-Frequency Cepstral Coefficients, the relative Shannon entropy, indicating predominant peaks, kurtosis, indicating trends in the audio signal, standard deviation, skewness, and the amplitude envelope. We also use the novelty curve, obtained from convolution along the main diagonal of the similarity matrix using a Gaussian checkerboard. We use this feature set as the baseline for comparison with the enhanced system that also includes the NMF-based features.
\subsection{Distance Features for Pair Characterization}
\label{subsec:dtw}
In the previous subsections, we described the sets of features extracted for each music track. We use these features to compute distances which we use to characterize pairs of tracks as plagiarized or not. Given the feature representations for the two tracks, we use the Dynamic Time Warping (DTW) algorithm to compute distances between the corresponding feature sequences. Dynamic time warping is a dynamic programming algorithm for efficiently computing the optimal alignment of non-linearly expanded or contracted sequences.
Based on an inspection of the dataset, we observed that the music in a pair of plagiarized pieces usually differed in rhythm, while retaining other properties that made them sound similar. We incorporate this intuition into our distance computation by using a modified version of the DTW algorithm for rhythm-based features, with local weights to prefer insertions or deletions to substitutions to better account for the variation in rhythm. The modification is as follows:
\begin{equation}
D(i,j) = min \begin{cases}
D(i-1,j-1)+w_{sub}.d(i,j) \\
D(i,j-1)+w_{del}.d(i,j) \\
D(i-1,j)+w_{ins}.d(i,j)
\end{cases}
\end{equation}
We make $w_{sub}$ much larger than $w_{del}$ and $w_{ins}$. This follows from the fact that, even though the DTW would effectively match two sequences having the same musical property with varying rhythm, the distance between the two sequences would still be large owing to the numerous insertions and deletions required. Thus, using a lower weight for insertion and deletion would help in bringing out this similarity.
However, this modification has a potential disadvantage in that the warping path might begin to prefer axis-parallel trajectories due to the significantly lower costs of insertions and deletions. This is overcome by constraining the slope of the warping path over short windows to be within a pre-specified range.
\subsubsection{Comparison With Existing Systems}
Instances of our method were tested with some existing systems in the detection of near-duplicate music documents. Some of the recent work done in this regard involve the works of Hanna et. al. \cite{Hanna2011} and Robine et. al. \cite{Robine2007}.
The difficulty in the detection of musical plagiarism compared to systems based on the detection of near-duplicate music documents are that, in the latter case, the music pieces, in general are of very similar notes. This is in sharp contrast to many plagiarism cases, where sequences made of very different notes can be musically very similar. Moreover, plagiarised copies may have a very small sequence similar to the original song, while systems which detect near-duplicate music documents consider (and function with) only longer sequences of similarity.
Both the systems achieved good success in detecting successful plagiarism cases using monophonic melody sequences. However, both systems had very limited success for polyphonic melody sequences even for plagiarism cases containing a high level of similarity, such as the Bright Tunes Music v. Harrisongs Music case involving the songs He's So Fine and My Sweet Lord. In contrast, our method was successful in detecting plagiarism for the songs in question.
In another plagiarism suit between Palmieri v. Estefan, the defending song `Oye Mi Canto' had only a 2 second clip similar to `Paginas De Mujer' in the chorus. Though the judge however ruled against the plaintif f due to their inability to prove that the defendant had access to the plaintiff's work, the two songs were substantially similar. Our method was successful in detecting original copies of these two songs as plagiarised copies, showing a significantly better margin for the length by which the two songs need to be similar for successful detection.
Our method, however, failed to detect similarity in some cases where the copied portion was made very faint and in the background. An example would be the similarity in a guitar riff in the musical pieces involved in the La Cienega Music Co. v. ZZ Top plagiarism suit.
\section{Problem Formulation}
\label{sec:problem}
In this paper, we approach the task of detection of music plagiarism in a pairwise discriminative classification framework, where given a pair of musical segments, we wish to predict whether the pair are sufficiently similar so as to be considered plagiarized. Each musical track ($x$) is first represented using a set of feature vectors denoted by $F(x) = \{f_1, f_2, ..., f_n\}$. Each of the individual $f_k$ here represent a {\it class} of features, {\it e.g.} Mel-Frequency Cepstral Coefficient features, pitch contour features.
We then define a set of $n$ distance functions (one for each feature class extracted for a song), $\phi_1, \phi_2,.. \phi_n$, over pairs of such tracks, $x_i$ and $x_j$, where each of these functions $\phi_k$ computes the distance between the feature vectors of the two music tracks for the $k$-th feature class:
\begin{equation}
\phi_k(x_i,x_j) = D(f_k(x_i),f_k(x_j)), \forall k \in [1,2...n]
\end{equation}
where $D$ represents a distance function that is computed over the $k$-th feature classes for each of the music tracks. As we describe in Section \ref{subsec:dtw}, we use an edit-distance measure to compute distance for our task; we note, however, that any other distance metric could be used in this framework, if applied to a different task. The set of distance scores, thus computed, behaves, in effect, as a feature characterization of the degree of difference between the two songs.
\begin{equation}
\mathcal{F}(x_i, x_j) = \{ \phi_1(x_i,x_j), \phi_2(x_i,x_j),.. \phi_n(x_i,x_j)\}
\end{equation}
At training time, given information about whether the pair of songs, $x_i$, $x_j$, represent a positive instance of plagiarism, we can use the distance based feature set in a supervised classification framework to train a model that can predict plagiarism, given {\it a pair} of music tracks. Let $\bar{w}$ represent the set of weights for the features learnt at training time, and $\mathcal{G}$ represent the function that computes a score for the datapoint ({\it e.g.} $\mathcal{G}(\bar{w}, \mathcal{F}(x_i, x_j)) = \sum_{k=1}^{n} w_k \mathcal{F}_k $, for linear regression) . We can then obtain a label $L$ for the pair of music tracks using a thresholded classifier score as follows:
\begin{equation}
H(x_i,x_j) = \mathcal{G}(\bar{w}, \mathcal{F}(x_i, x_j)) - \rho
\end{equation}
\begin{equation}
L = \begin{cases}
+1, & \text{if } H(x_i,x_j)>0, \\
-1, & \text{if } H(x_i,x_j)\leq0.
\end{cases}
\end{equation}
where $L=+1$ denotes plagiarized. Unlike tasks requiring similarity computation between tracks for search-like applications, our task of detecting plagiarism is different in that we cannot simply pick the few most similar songs as potentially plagiarized, since this would result in a large number of songs that need manual examination. The parameter $\rho$ is therefore used as a threshold in this formulation, and we try to find an optimal value for this parameter so that we make the fewest mistakes.
|
2,877,628,089,489 | arxiv | \section{Introduction}\label{Intro}
An essential role in the function theory of the unit disk of the complex plane is played by the property that any function in the Hardy space factorizes (uniquely) as the product of an inner and an outer function. The connections of inner and outer functions are ubiquitous in mathematical analysis, ranging from operator theory to dynamical systems and PDEs (see \cite{CGP} and \cite{Gar}, for instance). One of the main reasons for this is the fact that the closed invariant subspaces (for the shift operator in the Hardy space) can be described via an identification with inner functions, whereas outer functions contain information about approximation properties, and, in fact, coincide with cyclic functions. In the recent paper \cite{ale-giulia}, the first two authors proved an inner-outer factorization theorem for the Hardy space of slice regular functions on the quaternionic unit ball $H^2(\mathbb B)$. Thus, it seems natural to investigate the properties of inner and outer functions in the quaternionic setting more deeply, and the present paper is a first step in this direction. We will see that some properties of holomorphic inner and outer functions are straightforwardly generalized to the quaternionic setting, whereas some other properties are more peculiar of slice regular functions.
The paper is organized as follows. In Section \ref{sec-notation} we fix the notation and we recall some basic definitions and properties of slice regular functions and the quaternionic Hardy space $H^2(\mathbb B)$. We devote Section \ref{sec-inner} to properties of inner functions, whereas in Section \ref{sec-outer} we focus on outer ones. Then, in Section \ref{sec-approx} we investigate cyclicity and properties of optimal approximant polynomials in the quaternionic setting. We conclude formulating some open problems in Section \ref{sec-problems}.
\section{Notation and basic definitions}\label{sec-notation}
In this section we recall a few definitions and properties of slice regular functions and the quaternionic Hardy space $H^2(\mathbb B)$. We do not include any proofs; we refer the reader to the monograph \cite{libroGSS} for the basics on slice regular functions and to \cite{deFGS} for results concerning $H^2(\mathbb B)$.
Let $\HH$ denote the skew field of quaternions, let $\BB=\{q\in\HH:\ |q|<1\}$ be the quaternionic unit ball and
let $\partial \BB$ be its boundary, containing elements of the form $q=e^{tI}=\cos t+\sin t I,\ I\in\SS,\ t\in\RR$, where $\SS=\{q\in \HH : q^2=-1\}$ is the two dimensional sphere of imaginary units in $\HH$. Then,
\[ \HH=\bigcup_{I\in \s}(\rr+\rr I), \hskip 1 cm \rr=\bigcap_{I\in \s}(\rr+\rr I),\]
where the \emph{slice} $L_I:=\rr+\rr I$ can be identified with the complex plane $\C$ for any $I\in\s$.
\medskip
A function $f:\BB\to\HH$ is a \emph{slice regular function} if the restriction $f_I$ of $f$ to $\BB_I:=\BB\cap L_I$
is {holomorphic}, i.e., it has continuous partial derivatives and it is such that
\[\overline{\p}_If_I(x+yI)=\frac{1}{2}\left(\frac{\p}{\p x}+I\frac{\p}{\p y}\right)f_I(x+yI)=0\]
for all $x+yI\in \BB_I$.
The relationship between slice regular functions and holomorphic functions of one complex variable is made clear in the following lemma.
\begin{lem}[Splitting Lemma]\label{splitting-lemma}
If $f$ is a slice regular function on $\BB$, then, for every $I\in\SS$ and for every $J\in\SS$, $J$ orthogonal to $I$, there exist two holomorphic functions $F, G:\BB_I\to L_I$ such that for every $z=x+yI\in \BB_I$,
$$
f_I(z)= F(z)+G(z)J.
$$
\end{lem}
It is a well-known fact that every slice regular function on the unit ball $\mathbb B$ admits a power series expansion of the form
$$
f(q)=\sum_{n\in\N} q^n a_n,
$$
where $\{a_n\}_{n\in\mathbb N}\subseteq\HH$. The \emph{conjugate} of $f$, which we denote by $f^c$, is the function defined by
\begin{equation}\label{regular-conjugate}
f^c(q):=\sum_{n\in\N}q^n\overline{a_n}.
\end{equation}
Morevover, we denote by $\widetilde{f}$ the function
\begin{equation*}\label{function-tilde}
\widetilde f(q):=f(\overline q).
\end{equation*}
The function $\widetilde{f}$ is not slice regular but it is a {\em slice} function.
The class of slice functions was introduced in \cite{ghiloniperotti} in a more general setting than the present one. In this paper we say that a function $f:\B\to \HH$ is a {\em slice function} if for any $I,J\in\SS$,
\begin{equation}\label{repfor2}
\begin{aligned}
f(x+yI)&=\frac{1-IJ}{2}f(x+yJ)+\frac{1+IJ}{2}f(x-yJ).
\end{aligned}
\end{equation}
Slice regular functions are examples of slice functions. Moreover, Formula \eqref{repfor2} furnishes a tool to uniquely extend a holomorphic function defined on the complex disk $\B_J$ to a slice regular function defined on the whole unit ball $\B$ (see \cite{libroGSS}). Given $f_J:\B_J\to \HH$, holomorphic function with respect to the complex variable $x+yJ$, the function $\ext(f_J):\B\to \HH$ defined for any $x+yI\in \B$ as
\[\ext(f_J)(x+yI)=\frac{1-IJ}{2}f_J(x+yJ)+\frac{1+IJ}{2}f_J(x-yJ)\]
is slice regular on $\B$.
\noindent Formula \eqref{repfor2} can also be used to prove the following result concerning the zeros of a slice regular function.
\begin{pro}\label{zerislice}
Let $f$ be a slice regular function on $\B$ such that $f(\B_I)\subseteq L_I$ for some $I\in\s$.
If $f(x+yJ)=0$ for some $J\in\s\setminus \{\pm I\}$, then $f(x+yK)=0$ for any $K\in\s$.
\end{pro}
The structure of the zero set of a slice regular function is completely understood.
\begin{teo
Let $f$ be a slice regular function on $\B$. If $f$ does not vanish identically,
then its zero set consists of the union of isolated points and isolated $2$-spheres of the form $x +y \mathbb{S}$ with
$x,y \in \mathbb{R}$, $y \neq 0$.
\end{teo}
A $2$-dimensional sphere $x+y\s \subseteq \B$ of zeros of $f$ is called a
{\em spherical zero} of $f$.
Any point $x+yI$ of such a sphere is called a {\em generator} of the spherical zero $x+y\s$.
Any zero of $f$ that is not a generator of a spherical zero is called an
{\em isolated zero}
of $f$.
Moreover, on each sphere $x+y\s$ contained in $\B$, the zeros of $f$ are in one-to-one correspondence with the zeros of $f^c$, see \cite[Proposition 3.9]{libroGSS}
In general, the pointwise product of two slice regular functions is not a slice regular function and a suitable product must be considered, namely, the so-called \emph{slice} or $\ast$-\emph{product}. If $f(q)=\sum_{n\in\N} q^n a_n$ and $g(q)=\sum_{n\in\N} q^n b_n$ are two slice regular functions on $\B$, then
\begin{equation}\label{defn-product}
f\ast g(q):=\sum_{n\in\N}q^n\sum_{k\in\N}a_k b_{n-k}.
\end{equation}
This product is related to the pointwise product by the formula
\begin{equation}\label{star-product-pointwise}
f* g(q)=\left\{\begin{array}{l r}
0 & \text{ if $f(q)=0 $}\\
f(q)g(T_{f^c}(q)) & \text{if $f(q)\neq 0 $\,,}
\end{array}\right.
\end{equation}
where $T_{f^c}(q):= f(q)^{-1}q f(q)$.
By means of the $\ast$-product, we can associate to a function $f$ its \emph{symmetrization} $f^s$, that is,
\begin{equation}\label{fs}
f^s(q):= f^c\ast f(q)=f\ast f^c(q).
\end{equation}
We remark here that the symmetrization $f^s$ is a \emph{slice preserving} function, namely $f^s(\mathbb B_I)\subseteq L_I$ for all $I\in\mathbb S$. In particular, this is equivalent to the fact that the coefficients in the power series expansion of $f^s$ are all real numbers.
Finally, we denote by $f^{-\ast}$ the inverse of $f$ with respect to the $\ast$-product, which is given by
\begin{equation*}
f^{-*}(q)=(f^s(q))^{-1}f^c(q).
\end{equation*}
\smallskip
The function $f^{-*}$ is defined on $\{q\in \B \ | \ f^s(q) \neq 0\}$ and $f* f^{-*}=f^{-*}* f =1$. All of $f^{c}, f^s$ and $f^{-\ast}$ are slice regular functions if $f$ is slice regular.
The basic theory of Hardy spaces $H^p(\mathbb B)$ was established in \cite{deFGS}. Here we only recall some facts for the Hilbert case $p=2$ and for the extremal case $p=\infty$ since it is enough for our purposes. Set $\ell^2 := \ell^2(\N,\HH)$. The Hardy space $H^2(\B)$ is the function space defined as
$$
H^2(\B):=\bigg\{f\textrm{ slice regular on $\mathbb B$}: \,f(q)= \sum_{n\in \N}q^n a_n \,,\{a_n\}_{n\in \N} \in \ell^2 \bigg\}.
$$
Each function $f\in H^2(\mathbb B)$ admits a boundary value function, defined in a canonical sense almost everywhere (with respect to a measure $\Sigma$ that will be described later on). We still denote this function by $f$. With this in mind, if $f\in H^2(\B)$, whenever we write $f(q)$ with $|q|=1$, we are implicitly evaluating the boundary value function associated to $f$.
The space $H^2(\BB)$ is a right quaternionic Hilbert space with respect to the inner product \begin{equation}\label{innerproduct}
\Big<\sum_{n\in\N} q^n a_n,\sum_{n\in\N}q^n b_n\Big>:= \sum_{n\in\N}\overline{b_n}a_n.
\end{equation}
This inner product on $H^2(\B)$ admits also an integral representation. Let us endow $\partial\BB$ with the measure
\begin{equation}\label{Sigma}
d\Sigma\left(e^{tI}\right)=d\sigma(I)dt,
\end{equation}
where $dt$ is the Lebesgue measure on $[0,2\pi)$ and $d\sigma$ is the standard area element of $\s$, normalized in such a way that $\sigma(\s)=\Sigma(\p\B)=1$. Then,
\begin{equation}\label{integral inner product}
\left<f,g\right>= \int_{\p\B} \overline{g(q)}f(q)\, d\Sigma(q).
\end{equation}
The measure $\Sigma$, and not the induced Lebesgue measure on $\p\B$, is naturally associated to the Hardy space $H^2(\mathbb B)$, as reasoned in \cite{deFGS, metrica}. An important feature of this inner product is that it can be actually computed by restricting it to any slice $L_I$. In more detail, given any $I\in\SS$ we set
\begin{equation}\label{inner-product-slice2}
\left<f,g\right>_I=\frac{1}{2\pi}\int_{0}^{2\pi}\overline{g(e^{\theta I})}f(e^{\theta I})d\theta,
\end{equation}
where $d\theta$ is the Lebesgue measure on $[0,2\pi)$. For any $I\in\SS$, it holds that
\begin{equation}\label{inner-product-slice}
\left<f,g\right>=\int_{\p\B} \overline{g(q)}f(q)\, d\Sigma(q)=\left<f,g\right>_I.
\end{equation}
We denote by $H^\infty(\B)$ the space of bounded slice regular functions on the unit ball. Notice that $H^\infty(\B) \subseteq H^2(\B)$.
Definitions of inner and outer functions in the quaternionic setting are similar to the classical ones for holomorphic functions and first appeared in \cite{deFGS}.
\begin{defn}\label{defn-inner}
A function $\varphi\in H^\infty(\B)$ is {\em inner} if $|\varphi(q)|\leq 1$ on $\B$ and $|\varphi(q)|=1$ $\Sigma$-almost everywhere on $\p\B$.
\end{defn}
\begin{defn}\label{defn-outer}
A function $g\in H^2(\B)$ is {\em outer} if given any $f\in H^2(\B)$ such that $|g(q)|=|f(q)|$ for $\Sigma$-almost every $q\in\p\B$, then $|g(q)|\geq |f(q)|$ for any $q\in\B$.
\end{defn}
We point out that the definition of inner and outer functions were given in terms of the induced Lebesgue measure $m$ on $\p\B$. It is not difficult to show that $\Sigma$ and $m$ are mutually absolutely continuous.
The following theorem was proved in \cite{ale-giulia} by the first two authors.
\begin{teo}[Inner-outer factorization]\label{thm-factorization}
Let $f\in H^2(\B)$, $f\not\equiv 0$. Then, $f$ has a factorization $f=\varphi*g$ where $\varphi$ is inner and $g$ is outer. Moreover, this factorization is unique up to a unitary constant in the following sense: if $f=\varphi\ast g=\varphi_1\ast g_1$, then $\varphi_1=\varphi\ast\lambda$ and $g_1=\overline{\lambda}\ast g$ for some $\lambda\in\HH$ such that $|\lambda|=1$.
\end{teo}
The proof of this theorem makes use of the concept of cyclicity.
\begin{defn}\label{defn-cyclic}
A function $g$ is {\em cyclic} in $H^2(\BB)$ if
\begin{equation}\label{outer}
[g]:=\overline{\Span\left\{q^n\ast g, n\geq 0\right\}}=H^2(\BB).
\end{equation}
\end{defn}
We stress out that $[g]$ is the smallest closed invariant subspace of $H^2(\B)$ containing $g$. Thus, $g$ is cyclic if the smallest closed subspace containing $g$ is the space $H^2(\mathbb B)$ itself.
In \cite{ale-giulia} it is firstly proved that each function $f\in\ H^2(\BB)$ admits a factorization $f=\varphi\ast g$ with $\varphi$ inner and $g$ cyclic. Afterwards, being cyclic is proved equivalent to being outer in the sense of Definition \ref{defn-outer}.
We remark that we work with right quaternionic Hilbert spaces, therefore the left-hand side of \eqref{outer} denotes the closure in $H^2(\BB)$ of elements of the form
\[\sum_{n=0}^m (q^n\ast g)a_n=\sum_{n=0}^m (g\ast q^n)a_n=g*p_m,\]
where $p_m$ is a quaternionic polynomial with scalar coefficients $\{a_n\}_{n=0}^m\subseteq\mathbb H$.
\section{Inner functions}\label{sec-inner}
Let us now focus on inner functions. To start, we would like to better understand any connection between $f$ being an inner function in $H^2(\mathbb B)$ and the properties of $f_I$ (the restriction of $f$ to the slice $L_I=\mathbb R+\mathbb RI$), or of the splitting components of $f$ (see Lemma \ref{splitting-lemma}).
Some of the results we include in this section are implicit in \cite{ale-giulia}. Here we state them explicitly and we make some remarks.
We first prove a characterization of inner functions in $H^2(\mathbb B)$. In the following statement the $*$-product is the extension of \eqref{defn-product} to the more general setting of slice $L^2$ functions on $\p\B$, that is, the space $L_s^2(\p\B)$ of
functions of the form $q\mapsto \sum_{k\in \Z}q^ka_k$ with $q\in \p\B$ and $\{a_k\}_{k\in\Z}\in \ell^2$, see \cite{ale-giulia}. If $f(q)=\sum_{n\in\Z} q^n a_n$ and $g(q)=\sum_{n\in\Z} q^n b_n$ belong to $L_s^2(\p\B)$, then
\begin{equation}\label{product-slice}
f\ast g(q):=\sum_{n\in\Z}q^n\sum_{k\in\Z}a_k b_{n-k}.
\end{equation}
Clearly, the boundary value function of every $f\in H^2(\B)$ is a slice $L^2$ function.
\begin{pro}\label{thmI}
Let $f\in H^2(\B)$. Then, the following are equivalent:
\begin{enumerate}[(i)]
\item $f$ is inner;
\item $\widetilde f\ast f^c=f^c\ast\widetilde f=1 \quad \Sigma$-almost everywhere on $\p\B$;
\item there exists $I\in\s$ such that $(\widetilde f\ast f^c)_I=(f^c\ast\widetilde f)_I=1$ almost everywhere with respect to the induced Lebesgue measure on $\p\B_I$.
\end{enumerate}
\end{pro}
\vspace{-0.6cm}
\begin{proof}
Recalling Lemma 2.4 in \cite{ale-giulia}, we know that $\widetilde f\ast f^c=f^c\ast\widetilde f=1$ $\Sigma$-almost everywhere on $\p\B$ if and only if $|f|=1$ $\Sigma$-almost everywhere on $\p\B$. Then, if $f$ is inner, we immediately get the first implication of the statement.
Suppose now that $\widetilde f\ast f^c=f^c\ast\widetilde f=1$ $\Sigma$-almost everywhere on $\p\B$, so that $|f|=1$ $\Sigma$-almost everywhere on $\partial \mathbb B$. Let $g\in H^2(\B)$, $g\not \equiv 0$, and denote by $Z_{g^s}=\{q\in\B \, : \, g^s(q)=0\}$. Proposition 2.3 in \cite{ale-giulia} guarantees that $\Sigma(Z_{g^s})=0$, whereas Proposition 5.32 in \cite{libroGSS} guarantees that the map $T_{g}:\p\B\setminus Z_{g^s}\to\p\B\setminus Z_{g^s}$ is a bijection. Therefore, we have
\begin{equation*}
\begin{aligned}
\|f*g\|^2_{H^2}&\!=\!\|g^c*f^c\|^2_{H^2}\!=\!\int_{\p\B\setminus Z_{g^s}}\!\!\!\!|g^c(q)|^2|f^c(T_g(q))|^2d\Sigma(q)=\!\!\!\!\!\int_{\p\B\setminus Z_{g^s}}\!\!\!|g^c(q)|^2d\Sigma(q)=\|g^c\|^2_{H^2}=\|g\|^2_{H^2},
\end{aligned}
\end{equation*}
where we used that $|f|=1$ $\Sigma$-almost everywhere on $\p\B$ if and only if the same holds true for $|f^c|$ (see \cite[Proposition 5]{DRGS}). This implies that $f$ is a multiplier for $H^2(\B)$. Thanks to \cite[Corollary 3.5]{ABCS}
we conclude that $f\in H^{\infty}(\B)$ and, in particular, that $f$ is an inner function. Hence, conditions $(i)$ and $(ii)$ are equivalent.
Clearly $(ii)$ implies $(iii)$. Suppose now that condition $(iii)$ holds. Then, for almost every $t\in[0,\pi)$, we have both
\[1=\widetilde f\ast f^c(e^{tI}) \quad \text{and} \quad 1=\widetilde f\ast f^c(e^{(t+\pi)I})=\widetilde f\ast f^c(e^{-tI}).\]
Using Formula \eqref{repfor2} we obtain that, for any $J\in \s$,
\[\widetilde f\ast f^c(e^{tJ})=\frac{1-JI}{2}+\frac{1+JI}{2}=1.\]
Recalling that $d\Sigma(e^{tI})=dtd\sigma(I)$, see \eqref{Sigma}, we immediately get $(ii)$.
\end{proof}
\begin{oss}\label{remark}
We point out that the previous proof actually showed that condition $(iii)$ implies $(\widetilde f\ast f^c)_I=(f^c\ast\widetilde f)_I=1$ almost everywhere on $\p\B_I$ for \emph{any} $I\in\s$.
\end{oss}
By means of the previous result we obtain another characterization of inner functions in $H^2(\B)$ which is often used as the definition of inner functions in more abstract Hilbert spaces (see, for instance, \cite{secochar}). Recalling the notations from \eqref{innerproduct} and \eqref{inner-product-slice2} and denoting by $\delta_k(j)$ the Kronecker delta, we have the following theorem.
\begin{teo}\label{thm-condition-inner}
Let $f\in H^2(\B)$. Then, the following are equivalent:
\begin{enumerate}[(i)]
\item $f$ is inner;
\item for all $k \in \N$, we have
\[ \big< q^k\ast f,f\big>= \delta_k(0);\]
\item there exists $I\in\SS$ such that, for all $k \in \N$, we have
\[\big< q^k\ast f,f\big>_I= \delta_k(0).\]
\end{enumerate}
\end{teo}
\vspace{-0.6cm}
\begin{proof}
Let $f(q)=\sum_{n\in \N}q^na_n\in H^2(\B)$. Then, for $q\in\p\B$, we get
$$
\big< q^k\ast f,f\big>=\Big< \sum_{n\ge 0}q^{n+k}a_n,\sum_{n\ge 0}q^na_n \Big> = \sum_{n\geq \max\{0,k\}}\overline{a_n}a_{n-k}.
$$
We point out that $q^k\ast f$ has to be interpreted as a $\ast$-product in the setting of slice $L^2$ functions as defined in \eqref{product-slice}. Moreover, for $\Sigma$-almost any $q\in\partial\B$, it holds that
\begin{equation*}
f^c(q)=\sum_{n\geq 0} q^n\overline{a_n}\qquad\text{ and }\qquad \widetilde f(q)=\sum_{n\leq 0}q^n a_{-n}.
\end{equation*}
Hence
\begin{equation*}
f^c\ast\widetilde f(q)=\sum_{k\in \Z}q^k\sum_{n\in\Z} \overline{a_n}a_{n-k}=\sum_{k\in \Z}q^k\sum_{n\geq \max\{0,k\}} \overline{a_n}a_{n-k}.
\end{equation*}
Therefore, for any $k\in\N$, we obtain that $\big< q^k\ast f,f\big>$ is the $k$-th coefficient in the power series expansion of $f^c\ast\widetilde f$. From this fact and Proposition \ref{thmI} is now easy to deduce that $(ii)$ follows from $(i)$. The reverse implication can be proved using the natural extension of the $H^2$ inner product to the bigger space of slice $L^2$ functions, namely,
\[\Big<\sum_{n\in \Z}q^na_n, \sum_{n\in\Z}q^nb_n \Big>_{L^2}=\sum_{n\in\Z}\overline{b_n}a_n.\]
In fact, suppose $(ii)$ holds. Then, the $k$-th coefficient in the power series expansion of $f^c\ast\widetilde f$ vanishes for $k>0$ and equals $1$ for $k=0$. To show that all the coefficients with $k=-n<0$ equal zero consider
\[\langle q^{-n}\ast f,f \rangle_{L^2}=\langle f,q^{n}\ast f \rangle_{L^2}=\overline{\langle q^{n}\ast f,f \rangle}=0,\]
thus the first part of the theorem is proved.
From \eqref{inner-product-slice}, we obtain that $(ii)$ is equivalent to $(iii)$. In fact, the inner product of $H^2(\B)$ can be computed on a single slice and it does not depend on the choice of the slice.
\end{proof}
In general, the restriction $f_I$ of $f$ to the slice $L_I$ is a function of one complex variable, but still quaternion-valued. If $f_I$ were a complex-valued function, then $f_I$ would truly be an inner function of $H^2(\mathbb D)$. Therefore, Theorem \ref{thm-condition-inner} guarantees that the restriction to any slice $L_I$ of a slice regular inner function of $H^2(\B)$ is ``almost'' an inner function of $H^2(\mathbb D)$. At this point it is natural to question about a simple converse. We wonder whether any inner function $F\in H^2(\mathbb D)$ admits a slice regular extension $f:=\ext(F)$ to the unit ball $\B$ such that $f$ is inner for $H^2(\mathbb B)$.
Here we are identifying $\D$ with $\B_i$.
We obtain an answer in the form of the following corollary.
\begin{coro}
Let $F\in H^2(\mathbb D)$ be an inner function. Then, the slice regular extension $f=\ext(F)$ is an inner function of $H^2(\mathbb B)$.
\end{coro}
\begin{proof}
Since the inner product of $H^2(\B)$ can be computed on any slice, it is clear that $f=\ext(F)\in H^2(\mathbb B)$ whenever $F\in H^2(\mathbb D)$. The conclusion now follows from Theorem \ref{thm-condition-inner} since condition $(iii)$ is satisfied for $I=i$.
\end{proof}
We explicitly point out that not all the inner functions of $H^2(\mathbb B)$ trivially arise as the regular extension of some inner function of $H^2(\mathbb D)$.
In fact, if a slice regular function $f$ is the extension of a complex inner function $F$, then it necessarily preserves the slice $L_i$, i.e. $f(\B_i)\subseteq L_i$. Recalling Proposition \ref{zerislice}, we have that if a function preserves a slice, then all its isolated, non-spherical, zeros are contained in that slice. It is enough to take a slice regular Blaschke product which has at least two zeros that are not on the same slice (and neither on the same sphere). See \cite{deFGS} for an explicit construction of such a function.
So far we investigated how $f$ being an inner function in $H^2(\B)$ affects the restriction of $f$ to any slice. We also want to understand how being an inner function affects the splitting components of the function (see Lemma \ref{splitting-lemma}). The following is our best result in this direction.
\begin{teo}\label{thm-inner-components}
Let $f\in H^2(\B)$, $I,J\in\SS$, $J$ orthogonal to $I$, and $F,G:\B_I\to\CC$ be holomorphic functions so that, for any $z\in\B_I$,
\begin{equation}\label{thm-splitting}
f_I(z)=F(z)+G(z)J.
\end{equation}
Then, $f$ is inner if and only if for Lebesgue-almost every $x\in\partial \B_I$, the following conditions hold:\begin{equation}\label{eqn13}
\left\{\begin{array}{l r}
|F(z)|^2+|G(z)|^2=1 \\
\phantom{\footnotesize{sss}}\\
F(z)G(\overline{z})=F(\overline{z})G(z).
\end{array}\right.
\end{equation}
\end{teo}
\begin{proof}
If $F,G$ are the splitting components of $f_I$ as in \eqref{thm-splitting}, then, following \cite[Chapter 1]{libroGSS}, we get
\begin{equation}\label{conditions-system}
f^c_I(z)= \overline{F(\overline{z})}-G(z)J\qquad\text{ and }\qquad \widetilde{f_I}(z)= F(\overline{z})+G(\overline z)J.
\end{equation}
Consider now the power series expansion
\[f(q)=\sum_{n\in \N}q^na_n=\sum_{n\in \N}q^n(\alpha_n+\beta_nJ),\]
where $\alpha_n,\beta_n\in L_I$. Then,
\[F(z)=\sum_{n\in \N}z^n\alpha_n, \quad \overline {F(\overline z)}=\sum_{n\in \N}z^n\overline{\alpha_n},\quad G(z)=\sum_{n\in \N}z^n\beta_n, \quad \overline{G(\overline{z})}=\sum_{n\in \N}z^n \overline{\beta_n}.\]
Hence, for almost every $z\in \p\B_I$,
\begin{equation}\label{splitting-product}
\begin{aligned}
(\widetilde f\ast f^c)_I(z)&= \Big(\sum_{n\le 0}z^{n}(\alpha_{-n}+\beta_{-n}J)\Big)*\Big(\sum_{n\ge 0}z^n(\overline{\alpha_n}-\beta_nJ)\Big)\\
&= \sum_{n\in \Z}z^{n}\sum_{k\leq \min\{0,n\}}(\alpha_{-k}+\beta_{-k}J)(\overline{\alpha_{n-k}}-\beta_{n-k}J)\\
&=\sum_{n\in \Z}z^{n}\sum_{k\leq \min\{0,n\}}(\alpha_{-k}\overline{\alpha_{n-k}}+\beta_{-k}\overline{\beta_{n-k}})+(\beta_{-k}\alpha_{n-k}-\alpha_{-k}\beta_{n-k})J\\
&=F(\overline z)\overline{F(\overline z)}+ G(\overline z)\overline{G(\overline z)}+(G(\overline z)F(z)-F(\overline z)G(z))J.
\end{aligned}
\end{equation}
Combining \eqref{splitting-product} with Remark \ref{remark} we get
$$
F(\overline z)\overline{F(\overline z)}+ G(\overline z)\overline{G(\overline z)}+(G(\overline z)F(z)-F(\overline z)G(z))J=1
$$
for almost every $z\in\p\B_I$. This holds if and only if \eqref{eqn13} is satisfied.
\end{proof}
We conclude this section showing that the characterization of inner functions in \cite{secochar} involving their $H^2$ and $H^\infty$ norms works in the quaternionic setting as well.
\begin{teo}
Let $f\in H^2(\B)$. The following are equivalent:
\begin{enumerate}[(i)]
\item $f$ is inner;
\item $\|f\|_{H^2}=\|f\|_{H^\infty}=1$;
\item $\|f\|_{H^2}=1$ and for all $k \in \N$ and $\lambda \in \HH$ we have
\[ \| (q^k+\lambda) \ast f\|_{H^2} \leq \|q^k + \lambda\|_{H^2}.\]
\end{enumerate}
\end{teo}
\vspace{-0.6cm}
\begin{proof}
Clearly, if $f$ is inner, then its $H^\infty$ norm equals $1$ and
\[\|f\|^2_{H^2}=\int_{\p\B}|f|^2d\Sigma=1,\]
that is, $(i)$ implies $(ii)$. We deduce that $(ii)$ implies $(iii)$ from the fact that the multiplier space of $H^2(\mathbb B)$ can be isometrically identified with $H^\infty(\mathbb B)$ and the fact that $\|f\|_{H^\infty}=1$ implies $\|g\ast f\|_{H^2}\leq \|g\|_{H^2}$ for any $g\in H^2(\mathbb B)$.
To see that $(iii)$ implies $(i)$ we exploit Theorem \ref{thm-condition-inner}. If $\|f\|_{H^2}=1$ and $f$ is not inner, then there must be $k \in \N \backslash \{0\}$ such that \[\big< q^k *f , f \big> \neq 0.\] Choose such $k$ and notice that, for all $\lambda \in \HH$, we have \[\|q^k + \lambda\|_{H^2}^2 = 1 + |\lambda|^2.\] Let us now compute the left-hand side in the condition in $(iii)$ with a $\lambda$ to be chosen later. It holds that
\[ \| (q^k+\lambda) \ast f\|_{H^2}^2= \|q^k \ast f\|_{H^2}^2 + |\lambda|^2 \|f\|_{H^2}^2 + 2 \RRe (\overline{\lambda} \big< q^k *f , f \big>).\]
Since the shift is an isometry over its image on $H^2(\B)$, we see that the first two addends of the right-hand side sum to $\|q^k + \lambda\|_{H^2}^2$. Choosing $\lambda=\big< q^k *f , f \big>$ we contradict the hypothesis in $(iii)$.
\end{proof}
\section{Outer functions}\label{sec-outer}
In \cite{ale-giulia} it was shown that the Definition \ref{defn-outer} of outer functions is equivalent to the concept of cyclicity, extending a celebrated result of Beurling. Recall that a function $g \in H^2(\B)$ is cyclic if $[g]$, the smallest (closed) subspace of $H^2(\B)$ invariant under the action of the shift, is all of $H^2(\B)$. What is currently missing in the quaternionic setting is an analogous of the classical characterization of outer functions on the unit disk in terms of the logarithm. Namely, $f\in H^2(\mathbb D)$ is outer if and only if
\begin{equation}\label{outer-log}
f(z)= \alpha\exp\bigg\{\frac{1}{2\pi}\int_0^{2\pi}\log|f(e^{i\theta})|\frac{e^{i\theta}+z}{e^{i\theta}-z}\, d\theta\bigg\}
\end{equation}
for any $z\in\mathbb D$.
In this section we prove some preliminary results that go in the direction of finding an analogous characterization for quaternionic outer functions. We provide some necessary conditions as well as sufficient ones for a function to be outer. Some of these conditions are in terms of the symmetrization $f^s$ of the function $f$. We will see that, in some cases, $f$ being outer is equivalent to $f^s$ being outer. Since $f^s$ is slice preserving, a logarithm characterization for $f^s$ to be outer is available.
Most of our proofs rely on cyclicity, thus similar proofs could work for function spaces in which outer and cyclic functions do not necessarily coincide.
Our first finding does not come as a surprise, but it will be used in what follows.
\begin{lem}\label{conjucyc}
Let $f \in H^2(\B)$. Then, $f$ is outer if and only if $f^c$ is outer.
\end{lem}
\begin{proof}
Let $f= f_i\ast f_o$ be the inner-outer factorization of $f$ where $f_i$ denotes the inner part of $f$ and $f_o$ denotes the outer part. Assume that $f$ is outer, that is, $f=f_o$, so that $f^c=(f_o)^c$. A priori, we do not know if the conjugate of an outer factor is still an outer factor. Thus, assume for the moment that $f^c$ is not outer, that is, $f^c= (f^c)_i\ast (f^c)_o$.
Then, on the one hand $f=f_o$, on the other hand
$$
f=(f^c)^c= ((f^c)_o)^c\ast ((f^c)_i)^c.
$$
In particular, thanks to \cite[Proposition 2.3]{ale-giulia}, for $\Sigma$-almost every $q\in\partial \mathbb B$,
$$
|f(q)|=|f_o(q)|=|((f^c)_o)^c\ast ((f^c)_i)^c(q)|=|((f^c)_o)^c(q)|.
$$
Since $f$ is outer, we get
$$
|f(q)|=|f_o(q)|\geq |((f^c)_o)^c(q)|
$$
for any $q\in \mathbb B$. However, recall that a function is inner if and only if its conjugate function is inner (\cite[Proposition 2.1]{ale-giulia}), hence $|((f^c)_i)^c(q)|\leq 1$ inside the ball and we also get $$
|f(q)|=|f_o(q)|=|((f^c)_o)^c(q)||((f^c)_i)^c(\widetilde T_{f^c}(q))|\leq |((f^c)_o)^c(q)|,
$$
where $\widetilde T_{f^c}(q)= ((f^c)_o)^c(q)^{-1}q ((f^c)_o)^c(q) $. We remark here that $(f^c)_o$ is never zero in $\mathbb B$ since it is the outer factor of $f^c$, hence $((f^c)_o)^c$ never vanishes as well. As a consequence, $\widetilde T_{f^c}$ is a bijection of $\mathbb B$ to itself; see \cite[Proposition 5.32]{libroGSS}.
Therefore, we obtain that $|((f^c)_i)^c|=1$ in $\mathbb B$, hence, for the maximum modulus principle in the quaternionic setting, we conclude that $((f^c)_i)^c\equiv\alpha$ where $\alpha$ is a quaternion of modulus $1$. Thus,
$$
f^c= \overline \alpha\ast (f^c)_o,
$$
that is, $f^c$ is outer and the proof is concluded.
\end{proof}
Let us now introduce the concept of optimal approximants which will be needed in what follows.
\begin{defn}
Let $n\in\mathbb N$, $f\in H^2(\B)$ and $\mathcal{P}_n:=\{p(q)=\sum_{k=0}^nq^ka_k \ : \ a_k\in \HH \}$. A polynomial $p_n\in\mathcal P_n$ is an {\em optimal approximant} of degree $n$ of $f^{-*}$ if $p_n$ is such that $\|f*p_n-1\|_{H^2}=\min\{\|f*p-1\|_{H^2} \ : \ p \in \mathcal P_n\}$.
\end{defn}
\noindent The existence and uniqueness of such a minimizer is guaranteed by the projection theorem for quaternionic Hilbert spaces, see \cite{Jam}. In particular, the minimizer $f*p_n$ is given by the orthogonal projection of the constant function $1$ on the closed subspace $f*\mathcal P_n \subsetneq H^2(\B)$.
The constant function $1$ plays a special role because it is cyclic. Then, to show the cyclicity of a function $f$ is equivalent to show that its optimal approximants satisfy \begin{equation}\label{1-cyc}
\|f \ast p_n -1\|_{H^2} \rightarrow 0 \quad \text{as} \quad n \rightarrow \infty.
\end{equation}
In fact, if the constant function $1$ satisfies equation \eqref{1-cyc}, then it belongs to $[f]$. The fact that this is a closed and invariant subspace guarantees that $H^2(\B)=[1]\subseteq [f]$, that is, $f$ is cyclic.
The following result states the relationship between the invariant subspace generated by an $H^2$ function $f$ and the inner-outer factorization of $f$.
\begin{lem}\label{leminvariant}
Let $f \in H^2(\B)$ factorizes as $f=f_i \ast f_o$, where $f_i$ is inner and $f_o$ is outer. Then, $[f]=[f_i]$.
\end{lem}
\begin{proof}
Let $\{p_n\}_{n\in \N}$ be a sequence of polynomials such that $\|f_o \ast p_n -1\|_{H^2} \rightarrow 0$ as $n$ tends to $\infty$.
Since $f_i$ is bounded, it is a multiplier, so that
\[ \|f\ast p_n - f_i\|_{H^2} \leq \|f_i\|_{H^\infty} \|f_o\ast p_n -1\|_{H^2}.\]
This shows that $f_i \in [f]$ and hence $[f_i] \subseteq [f]$. The inclusion $[f] \subseteq [f_i]$ follows from the fact that $[f_i]=f_i\ast H^2(\mathbb B)$, and this latter space clearly contains $f$ since $f_o\in H^2(\B)$.
\end{proof}
\begin{lem}\label{prodcyc}
Let $f, g \in H^\infty(\B)$. Then, $f \ast g$ is cyclic if and only if both $f$ and $g$ are cyclic.
\end{lem}
Before proving the lemma, notice that even in the complex case the assumption $f, g \in H^\infty(\D)$ cannot be discarded since there are functions in $H^2(\D)$ whose square is not an element of $H^2(\D)$.
\begin{proof}
To show that $f \ast g$ cyclic implies $g$ cyclic, thanks to Lemma \ref{conjucyc}, it is enough to show that $f \ast g$ cyclic implies $f$ cyclic and then apply the result to $g^c*f^c$. Suppose now that $f*g$ is cyclic and consider the inner-outer factorization of $f=f_i \ast f_o$ with $f_i$ inner and $f_o$ outer. Then, $[f]=f_i*H^2(\B)$. Hence, $f \ast g= f_i *(f_o*g)$ is an element of $[f]$, since $f_o\in H^2(\B)$ and $g\in H^\infty(\B)$ guarantee that $f_o*g\in H^2(\B)$ (see \cite{deFGS}).
Then, $[f \ast g] \subseteq [f]$ but $[f \ast g] = H^2(\B)$. Hence, $f$ is cyclic.
Suppose now that $f$ and $g$ are cyclic, and let $\{p_n\}_{n \in \N}$ and $\{r_m\}_{m \in \N}$ be sequences of polynomials with the property that both the norms $\|f\ast p_n-1\|_{H^2}$ and $\|g\ast r_m-1\|_{H^2}$ tend to zero as $n$ goes to infinity. Then, for each $n, m \in \N$, from the triangle inequality we get
\[\| (f \ast g) \ast (r_m \ast p_n) -1\|_{H^2} \leq \|(f \ast g) \ast (r_m \ast p_n) - f \ast p_n\|_{H^2} + \|f \ast p_n -1\|_{H^2}.\]
The last term on the right-hand side will be arbitrarily small whenever $n$ is large enough. The other one may be estimated using the fact that $f$ and $p_n$ are both multipliers, that is,
\[ \|(f \ast g) \ast (r_m \ast p_n) - f \ast p_n\|_{H^2} \leq \|f\|_{H^\infty} \cdot \| g \ast r_m - 1\|_{H^2} \cdot \|p_n\|_{H^\infty}.\]
Now, whatever the value of $\|f\|_{H^\infty}\|p_n\|_{H^\infty}$ is, it does not depend on $m$. Hence, if we fix $\varepsilon >0$ and $n \in \N$, taking $m$ large enough we obtain \[ \| g \ast r_m - 1\|_{H^2} \leq \varepsilon (\|f\|_{H^\infty}\|p_n\|_{H^\infty})^{-1}.\] Therefore, $f \ast g$ is cyclic.
\end{proof}
The previous result will prove particularly useful when applied to the symmetrization $f^s=f\ast f^c= f^c\ast f$ of a function $f\in H^2(\mathbb B)$.
For each $p \in [1,\infty]$, the function $f^s$ is in $H^p(\B)$ provided that $f$ is in $H^{2p}(\mathbb B)$. In particular, if $f \in H^\infty (\mathbb B)$, then $f^s \in H^\infty (\mathbb B)$, see \cite{deFGS}.
\begin{coro}\label{symmecyc}
Let $f \in H^\infty(\mathbb B)$. Then $f$ is cyclic if and only if $f^s$ is cyclic.
\end{coro}
\begin{proof}
If $f$ is bounded, so is $f^c$ (see \cite{DRGS}) and we can apply both Lemma \ref{prodcyc} and Lemma \ref{conjucyc}.
\end{proof}
As we mentioned, the importance of $f^s$ comes from the fact that it preserves slices, that is, it can be seen as a holomorphic complex-valued function on each slice. This is important because we can transfer the theory from the disk to the quaternionic ball. In the following theorem we denote by $H^2(\mathbb B_I)$, $I\in\SS$, the function space defined as
$$
H^2(\mathbb B_I)=\left\{f\in \mathbb B_I\to L_I: f(z)=\sum_{n=0}^{\infty} z^n \alpha_n, \{\alpha_n\}\subseteq \ell^2(\mathbb N, L_I)\right\}.
$$
It is clear that $H^2(\mathbb B_I)$ can be identified with $H^2(\mathbb D)$.
\begin{teo}\label{thm1out}
Let $f \in H^\infty(\mathbb B)$. The following are equivalent:
\begin{enumerate}[(i)]
\item $f$ is cyclic in $H^2(\mathbb B)$;
\item $f$ is outer in $H^2(\mathbb B)$;
\item $f^s$ is cyclic in $H^2(\mathbb B)$;
\item $f^s$ is outer in $H^2(\mathbb B)$;
\item $f^s_I$ is cyclic in $H^2(\mathbb B_I)$ for all $I \in \SS$;
\item there exists $I\in \SS$ such that $f^s_I$ is cyclic $H^2(\mathbb B_I)$;
\item $f^s_I$ is outer in $H^2(\mathbb B_I)$ for all $I \in \SS$;
\item there exists $I\in \SS$ such that $f^s_I$ is outer in $H^2(\mathbb B_I)$.
\end{enumerate}
\end{teo}
\vspace{-0.6cm}
\begin{proof}
The equivalence between $(i)$, $(ii)$, $(iii)$ and $(iv)$ is guaranteed by Corollary \ref{symmecyc}, and by \cite[Theorem 4.2]{ale-giulia}. Also, since $f^s_I$ is a holomorphic function, the equivalence of $(v)$ and $(vii)$ and the equivalence of $(vi)$ and $(viii)$ are well-known consequences of the Beurling Theorem. \\
Let us now prove that $(iii)$ implies $(v)$. Let $f^s$ be cyclic in $H^2(\B)$ and let $I\in\s$. Then, for any $g_I\in H^2(\B_I)$ there exists a sequence of quaternionic polynomials $\{p_n\}_{n\in\N}$ such that $\|f^s*p_n-\ext g_I\|_{H^2}$ tends to zero as $n$ goes to infinity.
Let now $p_n(z)=P_n(z)+Q_n(z)J$ be the splitting of $p_n$ with respect to $J\in\s$, $J$ orthogonal to $I$. Then, evaluating the $H^2$ norm on the slice $L_I$, we get
\begin{align*}
\|f^s*p_n-\ext g_I\|^2_{H^2}&=\|f_I^s(P_n+Q_nJ)-g_I\|^2_{H^2(\B_I)}=\|f^s_IP_n-g_I\|^2_{H^2(\B_I)}+\|Q_n\|^2_{H^2(\B_I)},
\end{align*}
where the last equality is due to the orthogonality of $I$ and $J$. Therefore, the sequence of complex polynomials $\{P_n\}_{n\in\N}$ in the variable $z\in \B_I$ is such that
$\|f_I^sP_n-g_I\|_{H^2(\B_I)}$ tends to zero as $n$ goes to infinity, that is, $f^s_I$ is cyclic in $H^2(\B_I)$.
To conclude, it suffices to show that $(vi)$ implies $(iii)$, since clearly $(v)$ implies $(vi)$.
Suppose that $f^s_I$ is cyclic in $H^2(\mathbb B_I)$, for some $I\in\s$. Consider $g\in H^2(\B)$ and let $g(z)=F(z)+G(z)J$ be its splitting on $\B_I$ with respect to $J\in\s$, $J$ orthogonal to $I$. By hypothesis, there exist two sequences $\{P_n\}_{n\in\N}$ and $\{Q_n\}_{n\in\N}$ of complex polynomials in $\B_I$ such that $\|f^s_IP_n-F\|_{H^2(\B_I)}$ and $\|f^s_IQ_n-G\|_{H^2(\B_I)}$ tend to zero as $n$ goes to infinity. Then, using again the fact that the $H^2$ norm can be computed on any slice, and the orthogonality of $I$ and $J$, we get
\begin{equation*}
\begin{aligned}
\|f^s*\ext(P_n+Q_nJ)-g\|^2_{H^2}&=\|f_I^s(P_n+Q_nJ)-g_I\|^2_{H^2(\B_I)}=\|f_I^s(P_n+Q_nJ)-(F+GJ)\|^2_{H^2(\B_I)}\\
&=\|f_I^sP_n-F\|^2_{H^2(\B_I)}+\|f_I^sQ_n-G\|^2_{H^2(\B_I)}.
\end{aligned}
\end{equation*}
This latter quantity tends to zero as $n$ goes to infinity, thus we can conclude that $f^s$ is cyclic in $H^2(\B)$.
\end{proof}
Notice that the notion of outer function in $(vii)$ and $(viii)$ is the classical one, hence it may be expressed as in \eqref{outer-log}, that is, in terms of the mean value property of the logarithm. We remark once again that the assumption $f\in H^\infty(\B)$ cannot simply be dropped, not only because we need $f^s$ to be defined as a $H^2(\B)$ function so that we can apply Beurling Theorem to it (and this would be guaranteed if $f\in H^4(\B)$, see \cite{deFGS}), but also because of the applicability of Lemma \ref{prodcyc}, for which we need $f \in H^\infty(\B)$.
The presence in Theorem \ref{thm1out} of the hypothesis $f \in H^\infty(\B)$ is likely unsatisfactory. At the moment, we have not been able to show any characterization in terms of mean value properties for more general $f$. However, sufficient conditions may be shown.
Given $\omega \in \B$ let $\tau_\omega$ denote the slice regular M\"obius transformation of the unit ball taking $0$ to $\omega$, see \cite{moebius},
\[\tau_\omega(q)=(1-q\overline{\omega})^{-*}*(\omega-q),\]
and let ${I_\omega}$ be the imaginary unit identified by $\omega$, that is, ${I_\omega}=\frac{\omega-\RRe \omega}{|\omega-\RRe \omega|}$ if $\omega$ is not real, ${I_\omega}$ is any imaginary unit otherwise. With this notation, it holds that
\[({\tau_\omega})_{{I_\omega}}(z)=(1-z\overline{\omega})^{-1}(\omega-z)\]
for any $z\in \B\cap L_{I_\omega}$.
\begin{pro}\label{thm2out}
Let $f \in H^2(\B)$. Suppose that for all $\omega \in \B$ we have
\begin{equation}\label{eqn401}
\frac{1}{2 \pi} \int_{\p \B_{I_\omega}} \log |f_{I_\omega} \circ \tau_\omega(e^{\theta{I_\omega} })|d\theta = \log |f(\omega)|.
\end{equation}
Then, $f$ is outer.
\end{pro}
\begin{proof}
Suppose that $g$ is a function such that on $\p \B$ we have $|f|=|g|$ $\Sigma$-almost everywhere, and let $\omega \in \B$. Then,
\[ \log |f(\omega)| = \frac{1}{2 \pi} \int_{\p \B_{I_\omega}} \log |f_{I_\omega}\circ \tau_\omega(e^{\theta I_\omega})|d\theta. \]
Since $|f|$ is equal to $|g|$ on the boundary,
the right-hand side is equal to
\[\frac{1}{2 \pi} \int_{\p \B_{I_\omega}} \log |g_{I_\omega} \circ \tau_\omega(e^{\theta I_\omega})|d\theta.\]
Notice that the composition $g_{I_\omega} \circ \tau_\omega$ is well defined on the slice $L_{I_\omega}$ and it is indeed the restriction of the slice regular function $\ext(g_{I_\omega} \circ \tau_\omega)$. Recalling that the logarithm of the modulus of a slice regular function is subharmonic (see \cite{deFGS}), we get
\[\frac{1}{2 \pi} \int_{\p \B_{I_\omega}} \log |g_{I_\omega} \circ \tau_\omega(e^{\theta I_\omega})|d\theta \geq \log |g(\omega)|.\]
All this together yields that $|f(\omega)| \geq |g(\omega)|$. Since $\omega$ was arbitrary, we conclude that $f$ is outer.
\end{proof}
\section{Optimal approximants}\label{sec-approx}
In this section we extend as much as possible the theory of optimal approximants to the quaternionic setting. A good account of the theory of such polynomials in the classical holomorphic setting is given in \cite{daniel-london}.
Recall that, given $n\in\mathbb N$, $\mathcal{P}_n=\{p(q)=\sum_{k=0}^nq^ka_k \ : \ a_k\in \HH \}$.
The reproducing kernel of the subspace $f*\mathcal P_n$ exists since it is a closed subspace of $H^2(\B)$ which is itself a reproducing kernel Hilbert space with kernel function $k(q,w)=(1-q\overline w)^{-*}$.
Let $\{f*\varphi_k\}_{k=0}^n$ be an orthonormal basis of $f*\mathcal P_n$, where $\varphi_k$ is a polynomial of degree $k$ for any $k=0,\ldots,n$.
Then, from the reproducing property we can see that the reproducing kernel of $f*\mathcal P_n$ is given by
\[K_n(q,w)=\sum_{k=0}^{n}f*\varphi_k(q)\langle K_n(q,w), f*\varphi_k \rangle=\sum_{k=0}^{n}f*\varphi_k(q)\overline{f*\varphi_k(w)}.\]
For more information about reproducing kernel Hilbert spaces in the quaternionic setting we refer the reader, for instance, to \cite{QRKHS}.
Notice that, since $f*p_n$ is the orthogonal projection of the constant function $1$,
\[f*p_n(q)=\sum_{k=0}^{n}f*\varphi_k(q)\langle 1, f*\varphi_k \rangle=\sum_{k=0}^{n}f*\varphi_k(q)\overline {f*\varphi_k(0) },\]
i.e.,
\begin{equation}\label{poly-kernel}
f*p_n(q)=K_n(q,0).
\end{equation}
In particular, if $f(0)=0$, then $p_n \equiv 0$ for all $n\in \N$.
\begin{teo}
Let $f\in H^2(\B)$ be such that $f(0)\neq 0$ and let $p_n$ be the optimal approximant of $f^{-*}$ of degree $n\in\mathbb N$. Then, all the zeros of $p_n$ lie outside the closed unit ball $\overline{\B}$.
\end{teo}
\begin{proof}
First, let us show that we can reduce the problem to optimal approximants of degree $1$. Let $\lambda$ be a zero of an optimal approximant $p_n$ for the function $f \in H^2(\B)$. Then, there exists $\hat \lambda$ on the same two dimensional sphere of $\lambda$ such that $p^c_n(q)=(q-\overline{\hat\lambda})*\hat p_n^c$, so that $p_n(q)=\hat{p}_n*(q-\hat{\lambda})$, where $\hat{p}_n$ is a polynomial of degree $n-1$ and $|\hat\lambda|=|\lambda|$. Then, the optimality of $p_n$ guarantees that $(q- \hat{\lambda})$ is the optimal approximant of degree 1 for the function $f*\hat{p}
_n \in H^2(\B)$ which implies that $\hat{\lambda}$ is also a zero of a degree $1$ optimal approximant.
Therefore, in order to understand the possible positions of any such zero, it is enough to understand the same question for $n=1$.
Now, suppose that $p_1(q)=(q-\lambda)c$ is the optimal approximant of degree $1$ of $f^{-*}$. Then, by definition of orthogonal projection, $f*p_1 -1$ must be orthogonal to $f*q$, which translates easily in the equation \[0=\big< f*p_1, f*q \big> =\langle f*qc-f\lambda c, f*q \rangle,\] which implies that
\[
\langle f*q, f*q \rangle c=\langle f, f*q \rangle \lambda c,
\]
that is,
\begin{equation}\label{eqn301} |\lambda| = \frac{\|f*q\|^2}{|\left< f , f*q \right>|}. \end{equation}
Notice that $f*q$ is never a multiple of $f$ unless $f \equiv 0$ (which is against our hypothesis), and hence we can apply Cauchy-Schwarz inequality as a strict inequality to $\langle f, f*q\rangle$ in \eqref{eqn301} to get
\[ |\lambda| > \frac{\|f*q\|}{\|f\|}.\]
Since $f*q=q*f$ and the shift is an isometry, the right-hand side is equal to $1$ and the proof is concluded.
\end{proof}
Notice that all the points outside the closed unit ball are zeros of some optimal approximants. Indeed, if $p_1(q) = q-\lambda$ with $|\lambda| > 1$, then $p_1^{-\ast} \in H^2(\B)$ and $\|p_1^{-\ast} \ast p_1 -1\|=0$. Therefore, $p_1$ must be the only optimal approximant.
We can further understand the relationship between optimal approximants and orthogonal polynomials in the spirit of \cite{daniel-london}.
\begin{teo}
Let $f\in H^2(\B)$ and let $p_n$ be the optimal approximant of degree $n\in\mathbb N$ of $f^{-*}$. Let $\{f\ast\varphi_k\}_{k=0}^n$ be an orthonormal basis of $f\ast\mathcal P_n$, where $\varphi_k \in \mathcal P_k$. Then, the following are equivalent:
\begin{enumerate}[(i)]
\item$f$ is cyclic;
\item $p_n(0)$ converges to $f^{-\ast}(0)$ as $n\to\infty$;
\item $\sum_{k=0}^{\infty}|\varphi_k(0)|^2=|f^{-\ast}(0)|^2$.
\end{enumerate}
\end{teo}
\vspace{-0.6cm}
\begin{proof}
Bearing in mind that $f*p_n$ is the orthogonal projection of the constant function $1$, we can see that \[\|f*p_n -1\|^2 = \big< 1- f*p_n, 1-f*p_n\big>= \big< 1- f*p_n, 1\big>= 1-f(0)*p_n(0).\]
From this equality, the equivalence $(i)-(ii)$ is easily deduced.
Also, from \eqref{poly-kernel} we see that either $(i)$ or $(ii)$ is equivalent to $1-K_n(0,0) \rightarrow 0$ as $n \rightarrow \infty$. However, $K_n(0,0)$ tending to $1$ is equivalent to $(iii)$ and this concludes the proof.
\end{proof}
\section{Some open problems}\label{sec-problems}
We consider that the topic needs more development. We propose a few questions that seem natural from where we stand, beyond the obvious elimination of the boundedness hypothesis in Theorem \ref{thm1out}.
\begin{enumerate}
\item[(A)] If $f\in H^2(\mathbb B)$, let $f= f_i\ast f_o$ and $f^c= (f^c)_i\ast (f^c)_o$ be the inner-outer factorizations of $f$ and of its conjugate function. Then we also have
$f=f_i\ast f_o=[(f^c)_o]^c\ast [(f^c)_i]^c$. Is there any relationship between these two factorizations?
Is there something that can be said about the inner-outer factorization of $f^s$?
\item[(B)] Suppose that the symmetrization $f^s$ of a function $f\in H^2(\B)$ is inner. Is it true that $f$ (or $f^c$) is inner?
\item[(C)] Is the sufficient condition in Proposition \ref{thm2out} necessary for a function to be outer? This can be shown for $f^s$ under the assumption that $f$ is a multiplier.
\item[(D)] The boundary values of slice components of a quaternionic inner function form what is usually called a \emph{Pythagorean pair}, a special situation in which two functions have modulus 1 everywhere when seen as one function in $\T^2$. Such pairs arise in connections with the so-called de Branges-Rovnyak spaces and other areas of mathematics. Can anything else be said about this relation at all?
\end{enumerate}
\bigskip
\small{\noindent\textbf{Acknowledgements.}
The first author is a member of INDAM-GNAMPA and is partially supported by the 2015 PRIN grant
\emph{Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis}
of the Italian Ministry of Education (MIUR).
The second author is partially supported by INDAM-GNSAGA, by the 2014 SIR grant {\em Analytic Aspects in Complex and Hypercomplex Geometry} and by Finanziamento Premiale FOE 2014 {\em Splines for accUrate NumeRics: adaptIve models for Simulation Environments} of the Italian Ministry of Education (MIUR).
The third author is grateful for the financial support by
the Severo Ochoa Programme for Centers of Excellence in R\&D
(SEV-2015-0554) at ICMAT, and by the
Spanish Ministry of Economy and Competitiveness, through grant
MTM2016-77710-P.
\smallskip
Part of this project was carried out during a visit of the first and third author at the University of Firenze, and we wish to thank the Department of Mathematics and Computer Sciences for the financial support and the warm hospitality.
}
|
2,877,628,089,490 | arxiv | \section{Introduction}
In \cite{khovanov1}, Mikhail Khovanov developed a link homology theory categorifying the Jones polynomial. That homology theory utilized a particular rank-2 Frobenius extension to define the boundary operator of its chain complex. Utilizing the well-known correspondence between Frobenius extensions and 2-D Topological Quantum Field Theories (TQFTs), Dror Bar-Natan presented an equivalent development of Khovanov's homology that made use of a category of ``marked cobordisms" (\cite{barnatan}). This marked cobordism category was taken modulo three sets of local relations that recast algebraic properties of Khovanov's Frobenius extension in geometric terms. Khovanov proceeded to develop a second major link homology in \cite{Khovanov2}- an $sl(3)$ link homology that made use of a rank-3 Frobenius extension. In \cite{mackaay}, Marco Mackaay and Pedro Vaz extended his result using a family of ``universal $sl(3)$ Frobenius extensions", and along the way mimicked Bar-Natan's marked cobordism category as part of their category of foams. One intention of this paper is to generalize these marked cobordism constructions to Frobenius extensions of all ranks $n \geq 2$.
In \cite{asaeda}, Marta Asaeda and Charles Frohman introduced the notion of embedding Bar-Natan's marked cobordisms within a 3-manifold, thus producing skein modules. They defined and explored Bar-Natan's original skein module and gave explicit presentations of that skein module for several simple 3-manifolds. Uwe Kaiser made these ideas rigorous in \cite{kaiser}, developing skein modules based on TQFTs from any Frobenius extension. His work gives us a multitude of skein modules to investigate, and in this paper we closely examine a large family of Frobenius extensions that generalize those extensions associated to the $sl(n)$ link homology theories. Our work requires us to prove a number of foundational results about Frobenius extensions that do not seem to appear anywhere else in the literature. In particular, a significantly generalized version of Bar-Natan's original ``neck-cutting relation" is investigated, especially as it relates to the evaluation of closed manifolds in a skein module.
This paper is structured as follows: In Section \ref{sec: towards skein modules} we review the underpinnings of Frobenius extensions and 2-D TQFTs, culminating in a detailed description of Bar-Natan's original category of marked cobordisms. Section \ref{sec: skein modules} generalizes this category to the class of Frobenius extensions in question, while Section \ref{sec: properties of skein modules} is concerned largely with neck-cutting in these extensions and concludes with a major theorem regarding the evaluation of closed compact surfaces. In Section \ref{sec: embedded skein modules} we finally arrive at skein modules, proving several general facts about the entire family of skein modules (with special emphasis on rank $n=2$ Frobenius extensions). We also thoroughly compute an example, in part to demonstrate how complex these skein modules become without certain simplifying assumptions about the underlying Frobenius extensions. The appendix tackles some of the difficult computational challenges revealed in Sections \ref{sec: skein modules} and \ref{sec: properties of skein modules} via linear algebra. Although intended largely as a curiosity, this appendix is interesting in that it betrays a deep indebtedness to the theory of symmetric polynomials.
\section{Towards Skein Modules: Frobenius Extensions \& 2-D TQFTs}
\label{sec: towards skein modules}
\subsection{Frobenius Extensions}
\label{subsec: frobenius extensions}
As in \cite{Khovanov3}, we begin with a ring extension $\iota: R \hookrightarrow A$ of commutative rings with $1$ such that $\iota(1)=1$. $\iota$ endows $A$ with the structure of a $R$-bimodule, allowing for an obvious restriction functor $R: \cat{A-mod} \rightarrow \cat{R-mod}$. By definition, $\iota$ is a Frobenius extension if this functor $R$ has a two-sided adjoint. More specifically, $\iota$ is Frobenius if the induction functor $T: M_R \mapsto (M \otimes_R A)_A$ and the coinduction functor $H: M_R \mapsto (Hom_R(A,M))_A$ are isomorphic as functors $T,H: \cat{R-mod} \rightarrow \cat{A-mod}$. For the remainder of this paper we will treat Frobenius extensions such that $A$ is finitely-generated and projective as an $R$-module.
The functor isomorphism above prompts A-linear isomorphisms $End_R(A) \cong A \otimes_R A$ and $A^* \cong A$ (corresponding, respectively, to $M=A$ and $M=R$). In the finite projective case, the latter of those A-linear isomorphisms leads to the alternative definition of a Frobenius extension as a ring extension such that $A$ is self-dual as a $R$-module. Equivalently, a finite projective Frobenius extension is a ring extension such that $A$ is equipped with an $R$-linear comultiplication map $\Delta : A \rightarrow A \otimes_R A$ that is coassociative, cocommutative, and in possession of an $R$-linear counit map $\varepsilon: A \rightarrow R$. Stated below is yet another equivalent formulation of Frobenius extension that will be the primary definition utilized here. For a detailed proof of the equivalence between these definitions see Kadison \cite{Kadison}, where they deal with the more general case of $R,A$ not necessarily commutative.
\begin{definition}
\label{def: Frobenius extension}
A {\bf Frobenius extension} is a finite projective ring extension $\iota: R \hookrightarrow A$ of commutative rings such that there exists a non-degenerate $R$-linear map $\varepsilon: A \rightarrow R$ and a collection of tuples $(x_i,y_i) \in A \times A$ such that, for all $a \in A$, $a = \displaystyle{\sum_i} x_i \varepsilon(y_i a) = \displaystyle{\sum_i} \varepsilon(a x_i) y_i$.
\end{definition}
$\varepsilon$ is known as the Frobenius map, or trace, and is identified with the counit map mentioned above. For a Frobenius form to be nondegenerate means that there are no (principal) ideals in the nullspace of $\varepsilon$. $(x_i,y_i)$ is referred to as the dual basis, with the duality condition taking the form $\varepsilon(x_i y_j) = \delta_{i,j}$. $(R,A,\varepsilon, (x_i,y_i))$ is collectively referred to as the Frobenius system, and offers a complete description of the Frobenius extension.
One important feature of Frobenius extensions is that the aforementioned self-duality $A \cong A^*$ prompts an R-module isomorphism $A \otimes A \cong A \otimes A^* \cong End(A)$ that is given by $a \otimes b \mapsto a \varepsilon(b \underline{\ \ })$. This map actually extends to an isomorphism of the underlying rings, as long as one defines a multiplication on $A \otimes A$ (known as the $\varepsilon$-multiplication) by $(a \otimes b)(a' \otimes b')=a \varepsilon(ba') \otimes b' = a \otimes \varepsilon(ba')b'$. Note that, via the fundamental property of the Frobenius form in Definition \ref{def: Frobenius extension}, $\sum (x_i \otimes y_i)$ serves as the unit in this multiplication. For full details of this construction, see \cite{Kadison}.
For a fixed base ring $R$, the fundamental notion of equivalence between two Frobenius systems $(R,A,\varepsilon, (x_i,y_i))$ and $(R,\tilde{A},\tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i))$ is known as Frobenius isomorphism. A Frobenius isomorphism is any $R$-linear ring isomorphism $\phi: A \rightarrow \tilde{A}$ such that $\varepsilon = \tilde{\varepsilon} \phi$, with the latter condition implying an isomorphism between the comodule structures of $A$ and $\tilde{A}$. When we also have that $A = \tilde{A}$, Kadison alternatively states that two systems are equivalent iff $\varepsilon = \tilde{\varepsilon}$, which occurs iff $\sum (x_i \otimes y_i) = \sum (\tilde{x}_i \otimes \tilde{y}_i)$. The second iff above follows from the fact that both sums serve as the unit element for the $\varepsilon$-multiplication. This final observation will prove especially significant in Section \ref{sec: properties of skein modules}. In Section \ref{sec: properties of skein modules}, we'll also see how these two distinct notions of equivalence coincide when $A = \tilde{A}$.
\subsection{2-D TQFTs}
\label{subsec: TQFTs and Skein Modules}
There is a well-known correspondence between Frobenius extensions $R \hookrightarrow A$ and 2-dimensional TQFTs over $R$. See \cite{atiyah} or \cite{Kock} for a very detailed discussion of this correspondence in the less general setting of ``Frobenius algebras"- where $R$ is a field. Very briefly, a 2-dimensional TQFT $Z$ is a symmetric monoidal functor from the category of oriented 2-dimensional cobordisms $\cat{2Cob}$ to the category of (left) $R$-modules $\cat{R-mod}$. On the object level, $Z$ sends $S^1$ to an $R$-module $A$ that necessarily acts as a Frobenius extension of $R$. Being a symmetric monoidal functor, $Z$ then sends $n$ disjoint copies of $S^1$ to the tensor product $A^{\otimes_n}$ and the empty 1-manifold to the base ring $R$.
On the morphism level, $Z$ sends a 2-dimensional cobordism $N$ between $X$ and $X'$ to an $R$-linear map $Z(N):Z(X) \rightarrow Z(X')$. In the case of a closed 2-manifold we have $X=X'=\emptyset$ and hence that $Z(N) \in End_R(R)$. We identify this map with an element of $R$ via the image of $1 \in R$, meaning that $Z$ determines an $R$-valued invariant of closed 2-manifolds. This final property will be a central component in the upcoming discussion.
In recent years, Frobenius extensions have seen widespread usage in the construction of link invariants. This dates back to Khovanov's work in \cite{khovanov1}, where he utilized a Frobenius extension to construct the boundary operator in his homology theory that categorified the Jones polynomial. The Frobenius extension used by Khovanov was $R=\mathbb{Z}$, $A=\mathbb{Z}[x]/(x^2)$, with Frobenius form defined on the $R$-module basis $\lbrace 1,x \rbrace$ by $\varepsilon(1)=0,\varepsilon(x)=1$. The resulting dual-basis for this extension was then $\lbrace (1,x),(x,1)\rbrace$.
In \cite{barnatan}, Dror Ban-Natan utilized the correspondence between 2-D TQFTs and Frobenius extensions to give a more geometric interpretation of Khovanov's homology. His primary construction was a category of ``decorated cobordisms", which utilized the associated TQFT to represent algebraic properties of the Frobenius extension via 2-dimensional surfaces. This category $\cat{2Cob}_A$ has the same objects as $\cat{2Cob}$, but its morphisms (2-D cobordisms) may now by ``marked" by elements of $A$ (where a ``marking" appears as an element of $A$ written on the desired component, and an ``unmarked" surface corresponds to a marking by $1 \in A$). Markings are allowed to ``move around" and to be multiplied together (or factored) within a fixed component, but are not allowed to ``jump" across to a distinct components of the same cobordism. The category is also taken to be $R$-linear, and as a convention we always write elements of $R \subseteq A$ ``in front" of cobordisms to emphasize this linear structure.
The morphisms of $\cat{2Cob}_A$ are then taken modulo three sets of local relations $l$ that actually encapsulate the algebraic information about $R \hookrightarrow A$, and it is the resulting quotient category $\cat{2Cob}_A / l$ that is actually of interest. As we will soon be generalizing this construction to more general Frobenius extensions, these local relations are described in detail below. For the original description of these relations, see \cite{barnatan}\\
\\
\underline{Sphere Relations}\\
In the previous subsection we mentioned how $\varepsilon:A \rightarrow R$ is identified with the counit map. In $\cat{2Cob}$ this counit takes the form of the ``cap" surface. Using marked surfaces allows us to graphically depict the value of $\varepsilon$ at a specific value $a \in A$ via precomposition of this ``cap" with a ``cup" decorated by $a$ (this ``cup" is nothing more than the unit map $u:R \rightarrow A$, so we are actually interpreting a marked ``cup" as $a=a*u(1)$).
This all gives rise to Bar Natan's ``sphere relations", which correspond to the function values $\varepsilon(1) = 0$ and $\varepsilon(x) = 1$. They say that we may remove a (disjoint) unmarked sphere from a cobordism and multiply the entire cobordism by $0$, or remove a sphere marked by $x$ and multiply the entire cobordism by $1$. Note that, merely for this sub-section, we adopt Bar Natan's original notation of a dot corresponding to a surface marked by $x$.
\[
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.15}{\includegraphics[angle=270]{spheredot.pdf}}}
\end{picture}
\raisebox{10pt}{\ \ = \ \ 1 \ \ \ \ \ \ }
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.15}{\includegraphics[angle=270]{sphere.pdf}}}
\put(-20,18){\fontsize{9}{10.8}$1$}
\end{picture}
\raisebox{10pt}{\ \ = \ \ 0}
\]
\\
\underline{``Dot Reduction'' Relation}\\
The ``dot reduction" relation follows directly to our choice of $A$, and allows us to re-decorate surfaces by equivalent elements in $A$. As the $R$-module $A$ has a single generating relation in $p(x)=x^2=0$, all such local relations are generated by the one shown below.
\[
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.12}{\includegraphics[angle=270]{sheetdots.pdf}}}
\end{picture}
\raisebox{10pt}{\ \ = \ \ 0}
\]
\\
\underline{Neck-Cutting Relation}\\
The last relation can be applied to any surface $N$ with a compression disk- a copy of $D^2$ such that $\partial D^2 \subset N$ and the interior of $D^2$ is disjoint from $N$. It allows one to ``neck-cut" along the compression disk, and then replace $N$ with a sum of surfaces in which a regular neighborhood of $D^2 \cap N$ has been removed and replaced by two copies of $D^2$ along the two new boundary components. For Bar Natan's category $\cat{2Cob}_A$, this ``neck-cutting" relation is depicted below.
Algebraically, this relation follows from our choice of dual-basis $\lbrace(1,x),(x,1)\rbrace$. In particular, it is a result of the dual basis' non-degeneracy condition that $a = \sum_{i=1}^{n} \varepsilon (a x_i) y_i$ for all $a \in A$, which we interpret as an equality between two endomorphisms of $A$. The left-hand side of the equation sends $a \in A$ to $a$ and hence is identified with the identity map $1_A: A \rightarrow A$: the image of a straight cylinder via our TQFT. The endomorphism on the right applies $\varepsilon (x_i \underline{\ \ })$ to the input (corresponding to the ``cap" decorated by $x_i$) and then outputs $y_i = y_i u(1)$ (corresponding to the ``cup" decorated by $y_i$).
\[
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{leftside.pdf}}}
\ \ = \ \
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{1dots1.pdf}}}
\ + \
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{1dots2.pdf}}}
\]
\\
In practice, the 2-D surfaces of interest will be closed. If we view such surfaces abstractly (i.e.- not embedded within some higher-dimensional space), we can use the local relations above to compress all surfaces down to $R$-linear combinations of spheres that may then be evaluated to a constant in $R$. The nice thing is that, since our local relations were determined by our Frobenius extension, when we restrict ourselves to unmarked surfaces these constants are identical to those of the $R$-valued 2-manifold invariant of the associated 2-D TQFT.
The concerns of this paper will be more general in several respects. Most fundamentally, we will consider skein modules, in which our marked cobordisms of $\cat{2Cob}_A/l$ will actually be embedded within a 3-manifold. This situation was first examined by Frohman and Asaeda in \cite{asaeda}, and complicates the theory in that the topology of the chosen 3-manifold may prevent us from compressing all surfaces down to disjoint spheres. Such embedded skein modules will be the primary focus of Section \ref{sec: embedded skein modules}.
Before considering such skein modules, it will be our primary goal to generalize the category $\cat{2Cob}_A/l$ presented above by considering more general Frobenius extensions $R \hookrightarrow A$. When $A$ is a rank 2 $R$-module, a number of different Frobenius extensions have been used to produce interesting link homologies (\cite{khovanov1},\cite{lee}), and in \cite{Khovanov3} Khovanov has even described a ``universal" rank-2 Frobenius extension that nicely encapsulates all of the distinct theories (although none of this is done within the framework of cobordisms). We are more strongly motivated by the work Mackaay and Vaz in their construction of a universal ``$sl(3)$-link homology" (\cite{mackaay}), in which they introduced a set of local relations similar to those above as part of their category \cat{Foam}.
The ``universal $sl(3)$ Frobenius extension" utilized by Mackaay and Vaz was $R=\mathbb{C}$, $A=\mathbb{C}[x]/(x^3-a x^2 - b x -c)$, with Frobenius form defined on the basis $\lbrace 1,x,x^2 \rbrace$ by $\varepsilon(1)=\varepsilon(x)=0,\varepsilon(x^2)=1$. Notice that this extension is not completely general, in the sense that it doesn't incorporate all possible rank-3 extensions of $\mathbb{C}$ (only ones with a cyclic basis and with a very specific type of Frobenius form). Coming up with a truly ``universal" extension is only a tractable problem in the rank-2 case, where there is always a presentation of the rank-2 $R$-module $A$ of the form $A=R[x]/p(x)$ for some quadratic $p(x) \in R[x]$. Thus in generalizing to arbitrarily high ranks $n \geq 2$, we chose to restrict our attention to a specific family of Frobenius extensions that most closely mimic this ``universal $sl(3)$ extension". These ``$sl(n)$ Frobenius extensions" are the subject of Sections \ref{sec: skein modules} and \ref{sec: properties of skein modules}.
\section{$sl(n)$ Frobenius extensions}
\label{sec: skein modules}
The rank-n Frobenius extensions $R \hookrightarrow A$ that we consider in this paper will be of the form $A=R[x]/p(x)$, where $p(x) \in R[x]$ is a (monic) degree-$n$ polynomial, and we have Frobenius form defined on the standard basis $\lbrace 1,x,...,x^{n-1} \rbrace$ by $\varepsilon(1)=...\varepsilon(x^{n-2})=0,\varepsilon(x^{n-1})=1$. To standardize notation, let $p(x)=x^n - a_1 x^{n-1} - ... - a_n$. From now on we refer to these systems as universal $sl(n)$ Frobenius extensions.
Notice that, in the case of $n=3$ and $R=\mathbb{C}$, this extension coincides with the universal $sl(3)$ extension of Mackaay and Vaz. As opposed to the ``easy choice" of $R=\mathbb{C}$, in the spirit of Khovanov we will choose to work with $R=\mathbb{Z}[a_1,...,a_n]$, a more general setting that will make several of our proofs slightly more involved. Always working with the standard basis $\lbrace 1,x,...,x^{n-1} \rbrace$, we endow our rank-n system with a standard dual-basis as below:
\begin{lemma}
\label{thm: dual basis}
The rank-n Frobenius system defined above has a dual basis given by:
\vspace{-.2 in}
\begin{equation*}
\{ (x^{n-1}, 1), (x^{n-2}, x - a_1), (x^{n-3}, x^2 - a_1 x - a_2), \ldots, (1, x^{n-1} - a_1 x^{n-2} - \ldots - a_{n-2} x - a_{n-1}) \}.
\end{equation*}
\end{lemma}
\begin{proof}
This follows immediately from the fact that the inverse of $\lambda = [[\varepsilon(x^{i+j-2})]]$, after reducing $mod(p(x))$, is of the form
\[
\lambda^{-1}=
\begin{bmatrix}
-a_{n-1} & -a_{n-2} & \ldots & -a_1 & 1 \\
-a_{n-2} & \ldots & \ldots & 1 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
-a_1 & 1 & 0 & \ldots & 0 \\
1 & 0 & \ldots & \ldots & 0
\end{bmatrix}
\]
\end{proof}
Notice that for the universal rank-2 case this gives the familiar dual basis of $\lbrace(x,1),(1,x-a_1)\rbrace$, while for the universal rank-3 case we have $\lbrace(x^2,1),(x,x-a_1),(1,x^2-a_1x-a_2)\rbrace$.
Equipped with a dual basis, we are now ready to determine the local relations for our universal rank-n Frobenius extension. Generalizing the presentation from Subsection \ref{subsec: TQFTs and Skein Modules}, we divide these local relations into the three groups that are outlined in detail below. It is in this general rank $n \geq 2$ case that our distaste for Bar-Natan's ``dot notation" is finally justified, as it obviously becomes unwiedly with increasing rank.\\
\\
\underline{Sphere Relations}\\
Here our Frobenius form is defined by $\varepsilon(x^{n-1})=1$ and $\varepsilon(x^k)=0$ for $0 \leq k \leq n-2$. It follows that the sphere relations for the universal rank-n system are:
\[
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.15}{\includegraphics[angle=270]{sphere.pdf}}}
\put(-25,17){\fontsize{9}{10.8}$x^{n\kern-1pt{-}\kern-1pt{1}}$}
\end{picture}
\raisebox{10pt}{\ \ = \ \ 1 \ \ \ \ \ \ }
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.15}{\includegraphics[angle=270]{sphere.pdf}}}
\put(-20,18){\fontsize{9}{10.8}$1$}
\end{picture}
\raisebox{10pt}{\ \ = \ \ \ldots \ \ = \ \ }
\begin{picture}(40,30)
\raisebox{30pt}{\scalebox{.15}{\includegraphics[angle=270]{sphere.pdf}}}
\put(-25,17){\fontsize{9}{10.8}$x^{n\kern-1pt{-}\kern-1pt{2}}$}
\end{picture}
\raisebox{10pt}{\ \ = \ \ 0}
\]
\\
\underline{``Dot Reduction'' Relation}\\
For $A=R[x]/p(x)$, where $p(x) = x^n - a_1 x^{n-1} - ... - a_n$, we have $x^n = a_1 x^{n-1} + ... + a_n$ and the ``dot reduction" relation allows us to re-mark any fixed component of our surface as:
\[
\begin{picture}(45,30)
\raisebox{20pt}{\scalebox{.05}{\includegraphics[angle=270]{sheet.pdf}}}
\put(-25,10){\fontsize{9}{10.8}$x^n$}
\end{picture}
\raisebox{10pt}{\ \ = \ }
\begin{picture}(80,30)
\raisebox{30pt}{\scalebox{.12}{\includegraphics[angle=270]{longsheet.pdf}}}
\put(-80,10){\fontsize{9}{10.8}$a_1 x^{n-1} + ... + a_n$}
\end{picture}
\]
\\
\underline{Neck-Cutting Relation}\\
In Lemma \ref{thm: dual basis} we saw that the dual basis for the universal rank-n Frobenius system was $\lbrace (x^{n-1}, 1), (x^{n-2}, x - a_1), (x^{n-3}, x^2 - a_1 x - a_2), \ldots, (1, x^{n-1} - a_1 x^{n-2} - \ldots - a_{n-2} x - a_{n-1}) \rbrace$. Grouping terms via the coefficients $a_i$, the corresponding neck-cutting relation then takes the elegant form:
\[
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{leftside.pdf}}}
\ \ = \ \
\begin{picture}(68,0)
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{ndots.pdf}}}
\put(-57,0){\fontsize{16}{19.2}$\sum$}
\put(-58,-12){\fontsize{7}{8.4}$i\kern-.5pt{+}\kern-.5pt{j}=$}
\put(-56,-20){\fontsize{7}{8.4}$n\kern-.5pt{-}\kern-.5pt{1}$}
\put(-22,14){\fontsize{9}{10.8}$x^i$}
\put(-22,-14){\fontsize{9}{10.8}$x^j$}
\end{picture}
\ - \ a_1 \
\begin{picture}(68,0)
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{ndots.pdf}}}
\put(-57,0){\fontsize{16}{19.2}$\sum$}
\put(-58,-12){\fontsize{7}{8.4}$i\kern-.5pt{+}\kern-.5pt{j}=$}
\put(-56,-20){\fontsize{7}{8.4}$n\kern-.5pt{-}\kern-.5pt{2}$}
\put(-22,14){\fontsize{9}{10.8}$x^i$}
\put(-22,-14){\fontsize{9}{10.8}$x^j$}
\end{picture}
\ - \ \ldots \ - \ a_{n-1} \
\raisebox{29pt}{\scalebox{.2}{\includegraphics[angle=270]{0dots.pdf}}}
\]
\section{Properties of $sl(n)$ Frobenius Extensions}
\label{sec: properties of skein modules}
\subsection{Neck-Cutting}
\label{subsec: neck-cutting}
We concluded Section \ref{sec: skein modules} by presenting a neck-cutting relation that was associated with the universal $sl(n)$ Frobenius extension. Here we consider what that relation tells us when the curve that bounds our compression disk is non-separating: when the ``top" and ``bottom" surfaces from the neck-cutting equation are now on the same component. By $R$-linearity and our aforementioned ability to multiply distinct decorations upon a fixed component, in this case neck-cutting amounts to multiplication on the effected component by a ``genus reduction" term of $g = \sum_{i=1}^n x_i y_i \in A$. This value always coincides with $m \circ \Delta (1)=m(\sum_{i=1}^n x_i \otimes y_i)=\sum_{i=1}^n x_i y_i$ via the definition of comultiplication and multiplication in any Frobenius extension, a correspondence that is illustrated in Figure 1 below. Also note that this $g$ is the same as what Kadison and others define to be the $\varepsilon$-index of the ring extension $R \hookrightarrow A$. We choose our slightly unorthodox notion in order to emphasize its geometric importance within the category of marked cobordisms.
\begin{figure}[htb]
\label{decompfig}
\caption{A handle.}
\begin{center}
\scalebox{.4}{\includegraphics[height = 3 in]{composition.pdf}\put(-80,10){\scalebox{2}{$1$}}\put(60,10){\scalebox{2}{$1$}}\put(60,38){\scalebox{2}{$\uparrow$}}\put(45,70){\scalebox{2}{$\Delta(1)$}}\put(60,110){\scalebox{2}{$\uparrow$}}\put(30,150){\scalebox{2}{$m(\Delta(1))$}}
\hspace{2 in}\raisebox{1.5 in}{\scalebox{3}{$=$}\hspace{.2 in}\raisebox{-.4 in}{\includegraphics[height = 1.25 in]{cup.pdf}}}\put(-88,108){\scalebox{1.8}{$m(\Delta(1))$}}}
\end{center}
\end{figure}
In particular, $T^2$ decorated with $1$ is equivalent to $S^2$ decorated with $g$. It follows that the Frobenius form evaluates an unmarked torus to $\varepsilon(g) = n$ in the universal $sl(n)$ Frobenius extension, which is compatible with the fundamental result that, in any 2-D TQFT $Z$, $Z(T^2) \in \mathbb{Z}[a_1,...,a_n]$ equals the rank of the associated Frobenius extension. Similarly, a genus-$i$ closed, compact surface $\Sigma_i$ decorated with $1$ is equivalent to $S^2$ decorated with $g^i$, as shown in Figure 2. It follows that an unmarked $\Sigma_i$ is evaluated by our TQFT $Z$ as $\varepsilon(g^i)$. Note that in this higher genus situation there is some ambiguity in how we choose our compression disks, and to achieve the succinct result above we need to ensure that the $i$ curves bounding those disks are all non-separating (although, naturally, any two ways of compressing down to a incompressible surface must evaluate similarly via $Z$!).
As suggested by its alternative title of $\epsilon$-index, $g$ is dependent upon not only the rings $R,A$ but also upon the choice of Frobenius form $\varepsilon$. What follows are a series of lemmas that hope to characterize how $g$ behaves under changes in Frobenius structure.
\begin{figure}[htb]
\label{cuttingtosphere}
\caption{Cutting down to a sphere.}
\begin{center}
\scalebox{.5}{\raisebox{-1 in}{\includegraphics[height = 1.5 in]{generalsurface.pdf}\put(-100,105){\scalebox{2}{$\dots$}}\put(-100,45){\scalebox{2}{$\dots$}}\put(-128,-15){\scalebox{2}{$i$}}}\put(-163,-2){\scalebox{2}{$1$}} \scalebox{2}{ $=$ } \raisebox{-.5 in}{\includegraphics[height = 1 in]{sphere2.pdf}\put(-40,9){\scalebox{2}{$g^i$}}}}
\end{center}
\end{figure}
We begin by considering the case where our two Frobenius systems $(R,A,\varepsilon, (x_i,y_i))$ and $(R,A,\varepsilon, (\tilde{x}_i,\tilde{y}_i)$ differ only via the choice of dual basis. In Subsection \ref{subsec: frobenius extensions} we noted that $\varepsilon = \tilde{\varepsilon}$ iff $\sum (x_i \otimes y_i) = \sum (\tilde{x}_i \otimes \tilde{y}_i) \in A \otimes A$. Given the latter equality, the well-definedness of the $A$-linear map $A \otimes A \rightarrow A$, $a \otimes b \mapsto ab$ ensures that $g = \sum x_i y_i = \sum \tilde{x}_i \tilde{y}_i = \tilde{g}$. Hence $g$ is independent of our choice of dual basis.
The following well-known result characterizes all possible Frobenius forms over a fixed ring extension $R \hookrightarrow A$. Its proof is readily available in \cite{Kock} or \cite{Kadison} (although \cite{Kock} presents an equivalent argument for the special case of Frobenius algebras with $R=k$ a field):
\begin{lemma}
\label{thm: changing frobenius systems}
Given two Frobenius systems over the same ring extension, $(R,A,\varepsilon, (x_i,y_i))$ and $(R,A,\tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i)$, up to change of dual basis we have $(R,A,\tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i)=(R,A,\varepsilon(d \underline{\ \ }), (x_i,d^{-1} y_i)$ for some invertible $d \in A$. Moreover, there is a bijection between equivalence classes of Frobenius systems over $R \hookrightarrow A$ and invertible $d \in A$.
\end{lemma}
Pause to note that, in light of this lemma, it becomes clear that the ``standard" notion of Frobenius equivalence coincides with Kadison's definition when $R,A$ are fixed. Specifically, the $R$-linear ring automorphism $\phi: A \rightarrow A$ underlying any Frobenius equivalence forces $d=1$ above, thus ensuring $\varepsilon = \tilde{\varepsilon}$. More germane to our discussion is the following corollary:
\begin{corollary}
\label{thm: epsilon dependence of g}
Given two Frobenius systems on the same ring extension, $(R,A,\varepsilon, (x_i,y_i))$ and $(R,A,\tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i)$, their respective genus-reduction terms are related by $\tilde{g}=d^{-1} g$ for some invertible $d \in A$.
\end{corollary}
\begin{proof}
By Lemma \ref{thm: changing frobenius systems}, after an appropriate change of dual basis the second system is of the form $(R,A,\varepsilon(d \_ \_), (x_i,d^{-1} y_i)$ for some invertible $d \in A$. This change of basis leaves $\tilde{g}$ unaffected, so by comparing this modified system to the first system from our theorem we have $\tilde{g}=\sum \tilde{x}_i \tilde{y}_i = \sum x_i d^{-1} y_i = d^{-1} g$.
\end{proof}
This argument obviously extends to show that $\tilde{g}^i = (d^{-1})^i g^i$ for all powers $i \geq 1$. In general, this corollary tells us little about how $\tilde{g}$ or the $\tilde{g}^i$ are actually evaluated by the $R$-linear Frobenius form, as the necessary $d \in A$ need not be in $R$ and hence can't necessarily be ``pulled out" of the argument for $\varepsilon$ and $\tilde{\varepsilon}$. However, do note that if $g^i = 0$ (or $g^i \neq 0$) for any Frobenius system over $R \hookrightarrow A$, then $\tilde{g}^i = 0$ ($\tilde{g}^i \neq 0$) for any Frobenius system over $R \hookrightarrow A$.
One final lemma that we will need is that a general Frobenius equivalence respects genus-reduction terms. When $A = \tilde{A}$, this result follows directly from Lemma \ref{thm: changing frobenius systems} and Corollary \ref{thm: epsilon dependence of g}, but now we allow the case where $A$ and $\tilde{A}$ are merely isomorphic via a Frobenius equivalence.
\begin{lemma}
\label{thm: g under frobenius equivalence}
Assume that $(R,A,\varepsilon, (x_i,y_i))$ and $(R, \tilde{A}, \tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i))$ are Frobenius equivalent via $\phi: A \rightarrow \tilde{A}$. Then $\phi(g) = \tilde{g}$.
\end{lemma}
\begin{proof}
Let $\phi: A \rightarrow \tilde{A}$ be the $R$-linear ring isomorphism such that $\varepsilon = \tilde{\varepsilon} \phi$. We may assume WLOG that $\phi(x_i)=\tilde{x}_i$, as we have already demonstrated that the genus-reduction term is invariant under change of dual basis. Recall that we use the matrix $\lambda = [[\varepsilon(x_i x_j)]]$ to determine the $(y_i)$ half of the dual basis. For our second system above we have $\tilde{\lambda}=[[\tilde{\varepsilon}(\tilde{x}_i \tilde{x}_j)]]=[[\tilde{\varepsilon}(\phi(x_i)\phi(x_j))]]=[[\tilde{\varepsilon}(\phi(x_i x_j))]]=[[\varepsilon(x_i x_j)]]=\lambda$. Thus if $y_i = \sum_m c_{im} x_m$ in our first system ($c_{im} \in R$), we have the same scalars for $\tilde{y}_i = \sum_m c_{im} \tilde{x}_m$. Our genus-reduction terms are then $g= \sum_i x_i y_i = \sum_{i,m} c_{im} x_i x_m$ and $\tilde{g} = \sum_i \tilde{x}_i \tilde{y}_i = \sum_{i,m} c_{im} \tilde{x}_i \tilde{x}_m$, from which it follows that $\phi(g)=\tilde{g}$.
\end{proof}
\subsection{Neck-cutting in $sl(n)$ Frobenius Extensions}
\label{subsec: neck-cutting in universal sln}
Let us now direct our attention towards the specific class of universal $sl(n)$ Frobenius extensions that we introduced at the beginning of Section \ref{sec: skein modules}. With the dual basis that we found in Lemma \ref{thm: dual basis}, we have the genus-reduction term:
\begin{equation}
\label{eq: g}
g = nx^{n-1} - (n-1)a_1x^{n-2} - (n-2)a_2x^{n-3} - ... - a_{n-1}
\end{equation}
Note that this $g$ is merely the derivative $p'(x)$ of the degree-n polynomial from the definition our ring $A=\mathbb{Z}[a_1,...,a_n][x]/(p(x))$. This will greatly simplify some upcoming calculations.
Before continuing on, it will prove useful to fully factor $p(x)$ over $\mathbb{C}$ as $p(x)=x^n-a_1x^{n-1}-...-a_{n-1}x-a_n = \prod_{i=1}^n (x + \alpha_i)$. We may then relate the $a_i$ to the $\alpha_i$ by $a_k = -e_k$, where $e_k$ denotes the k\textsuperscript{th} elementary symmetric polynomial in the $n$ variables $\lbrace \alpha_1, ... \alpha_n \rbrace$. Note that these succinct equations help to motivate our unconventional decision to write the $\alpha_i$ as the negatives of the roots of $p(x)$ as opposed to the roots themselves, as the later choice would have required the introduction of alternating $(-1)^j$ terms in many of our upcoming results. Also note that the roots $\alpha_i$ may not all lie in the ring $R=\mathbb{Z}[a_1,...,a_n]=\mathbb{Z}[e_1,...,e_n]$ (although our original coefficients $a_i$ will always lie in the larger ring $\tilde{R}=\mathbb{Z}[\alpha_1,...,\alpha_n]$). Luckily, despite the fact that a number of our results will depend upon this factorization, and that we will oftentimes need to temporarily pass to the ``more general" Frobenius extension $\tilde{R} \hookrightarrow \tilde{R}[x]/(p(x))$, all of our conclusions will descend back down to our original Frobenius system.
To offer a bit of insight into the general situation, pause to consider the specific case of $n=2$. Here we have $p(x)=x^2-a_1 x-a_2 = (x+\alpha_1)(x+\alpha_2)$ and hence $g = 2x-a_1 = 2x+(\alpha_1 + \alpha_2)$ by our earlier observation. It follows that $g^2 = 4 a_2 +a_1^2 = (\alpha_1 - \alpha_2)^2$, so we have $g^2=0$ (and hence $g^i$ for all $ \geq 2$) in $A$ iff the two roots of $p(x)$ coincide. This result will have a natural extension to higher $n$ that we will address in Theorem \ref{thm: repeated roots}.
At least in the $n=2$ case, the fact that $g^2 \in R$ is a constant also allows us to easily characterize all powers of $g$. In particular, $g^{2i}= (4 a_2 + a_1^2)^{2i}$ and $g^{2i+1}=(4 a_2 + a_1^2)^{2i} (2x-a_1)$ for all $i \geq 0$. We also have have $x*g^{2i}= (4 a_2 + a_1^2)^{2i} x$ and $x* g^{2i+1} = (4 a_2 + a_1^2)^{2i} (2x^2-a_1x)=(4 a_2 + a_1^2)^{2i} (a_1 x + 2 a_2)$. Given our standard Frobenius form $\varepsilon(1)=0$ and $\varepsilon(x)=1$, these results allow us to determine how all (marked) closed genus-$i$ surfaces ($i \geq 1$) evaluate in the Frobenius system. For $\Sigma_k$ a genus-$k$ surface marked with $1$ and $\dot{\Sigma}_k$ a genus-$k$ surface marked with $x$ we have:
\begin{center}
$\Sigma_{2i}=\varepsilon(g^{2i})=0$ \hspace{1.1in} $\Sigma_{2i+1}=\varepsilon(g^{2i+1})= 2(4 a_2 +a_1^2)^{2i}$\\
$\dot{\Sigma}_{2i}=\varepsilon(x*g^{2i})= (4 a_2 + a_1^2)^{2i}$ \hspace{.38in} $\dot{\Sigma}_{2i+1}=\varepsilon(x*g^{2i+1})=(4 a_2 + a_1^2)^{2i} a_1$
\end{center}
Unfortunately, this extremely elegant result does not fully extend to higher $n$, as we don't typically have $g^i \in R$ for any $i \geq 1$. For an attempt at tackling this general problem using linear algebra, see Appendix 1. In order to ensure a relatively simple characterization of the $g^i$ for all $i$, we actually need to impose a condition on our polynomial $p(x)$ akin to what was suggested in the $n=2$ case with $\alpha_1=\alpha_2$. This brings us to the primary theorem of this section, whose converse we briefly delay:
\begin{theorem}
\label{thm: repeated roots}
Consider the universal $sl(n)$ Frobenius extension $R=\mathbb{Z}[a_1,...,a_n] \hookrightarrow A=\mathbb{Z}[a_1,...,a_n][x]/(p(x))$. If every root of $p(x)$ is a repeated root, then $g^2 = 0$ in $A$.
\end{theorem}
\begin{proof}
Let $p(x)=\prod_{i=1}^n(x+\alpha_i)$, and assume that each of the $\alpha_i$ is a repeated root. We temporarily pass to $\tilde{A}=\mathbb{Z}[\alpha_1,...,\alpha_n][x]/(p(x))$ to ensure that $\alpha_i \in \tilde{A}$ for all $i$, and first show that $g^2=0$ in $\tilde{A}$. Recalling that $g(x)=p'(x)$, by the ordinary product rule for derivatives we have $g=\sum_{i=1}^n \frac{p(x)}{(x+\alpha_i)}$ and $g^2= \sum_{i,j=1}^n \frac{p(x)^2}{(x+\alpha_i)(x+\alpha_j)}=p(x) \sum_{i,j=1}^n \frac{p(x)}{(x+\alpha_i)(x+\alpha_j)}$. If every root $\alpha_i$ is repeated, every term in $\sum_{i,j=1}^n \frac{p(x)}{(x+\alpha_i)(x+\alpha_j)}$ can be rewritten with denominator $1$ and we see that $p(x)$ divides $g^2$ in $\mathbb{Z}[\alpha_1,...\alpha_n]$. Thus $g^2 = 0$ in $\tilde{A}$.\\
To prove the stronger statement that $g^2 = 0$ in $A$, we introduce some new notation. Let $e_k^{\alpha_i \alpha_j}$ denote the k\textsuperscript{th} elementary symmetric polynomial in the $n-2$ roots of $p(x)$ that aren't $\alpha_i$ or $\alpha_j$. Expanding the degree $n-2$ polynomial $f(x)=\sum_{i,j=1}^n \frac{p(x)}{(x+\alpha_i)(x+\alpha_j)}$ from above, the coefficient of $x^q$ takes the form $c_q = \sum_{i,j=1}^n e_{n-2-q}^{\alpha_i \alpha_j}$. Each of these $c_q$ is a clearly symmetric polynomial in all of the $\alpha_i$. Thus by the Fundamental Theorem of Symmetric Functions we know that each of the $c_q$ can be generated by the elementary symmetric polynomials in all of the $\alpha_i$, implying that $f(x)$ is actually in $\mathbb{Z}[e_1,...,e_n]=\mathbb{Z}[a_1,...,a_n]$ and that $g^2 = 0$ in $A$.\\
\end{proof}
Apart from allowing us to quickly evaluate all closed compact surfaces of genus $i \geq 2$, this theorem also implies that any surface with a component admitting multiple (non-separating) compressions in the given type of Frobenius system must evaluate to zero. This insight will great aid us in Section \ref{sec: embedded skein modules}, when we attempt to give a presentation of skein modules that are embedded within an arbitrary 3-manifold.
We close this section with the converse of Theorem \ref{thm: repeated roots} and a couple of quick corollaries. This direction of the theorem actually requires a slightly more involved approach, and quite honestly was one that we also could have used above (with a few additional lemmas). The necessity of the distinct approach is due to the fact that the summation from Theorem \ref{thm: repeated roots} can only be easily reduced to $\sum \frac{p(x)^2}{(x+\alpha_i)^2}$, where the sum is over only the non-repeated roots $\alpha_i$, and that it seems rather difficult to demonstrate that this remaining term is necessarily nonzero in $A$.
Whereas we only needed to briefly switch to the larger ring $\tilde{A}=\mathbb{Z}[\alpha_1,...,\alpha_n][x]/p(x)$ in the proof of Theorem \ref{thm: repeated roots}, the proof of its converse requires that we completely pass to the ``more general" Frobenius extension (with equivalent Frobenius form) over $\tilde{R}=\mathbb{Z}[\alpha_1,...,\alpha_n] \hookrightarrow \tilde{A}$. This runs against the tradition, followed by Khovanov and others, of adjoining ``just enough" to $\mathbb{Z}$ when defining Frobenius extensions in their development of associated link homologies. Our departure from convention is justified by the fact that our Frobenius system over $\tilde{R} \hookrightarrow \tilde{A}$ obviously has the same dual basis as the system over $R \hookrightarrow A$, and thus has an identical genus-reduction term $\tilde{g}=g$ that is being reduced mudolo the exact same polynomial $p(x)$.
The following lemma is a straightforward application of the Chinese Remainder Theorem, and explains our reliance upon the ``more general" Frobenius system:
\begin{lemma}
\label{thm: Chinese remainder theorem}
Define $\tilde{R}=\mathbb{Z}[\alpha_1,...\alpha_n]$ as above. Let $p(x)=\prod_{i=1}^n (x+\alpha_i) = \prod_{i=1}^{m} (x+\alpha_i)^{k_i}$, where $n \geq 2$ and in the second product we have fully grouped like roots. The Frobenius system over $\tilde{R} \hookrightarrow \tilde{A}=\tilde{R}[x]/(p(x))$, with Frobenius form $\tilde{\varepsilon}(x^{n-1})=1$, $\tilde{\varepsilon}(x^i)=0$ (for $0 \leq i \leq n-2$), is Frobenius equivalent to the Frobenius system over $\tilde{R} \hookrightarrow \hat{A}=\tilde{R}[x]/(x+\alpha_1)^{k_1} \times ... \times \tilde{R}[x]/(x+\alpha_m)^{k_m}$ if we define a Frobenius form on the direct product by $\hat{\varepsilon}(x^{n-1},...,x^{n-1})=1$, $\hat{\varepsilon}(x^i,...,x^i)=0$ (for $0 \leq i \leq n-2$).
\end{lemma}
\begin{proof}
The underlying ring and $R$-linear isomorphism $\phi: \tilde{A} \rightarrow \hat{A}$ follows from the Chinese Remainder Theorem and is given by $\phi(a)=(a,...,a)$. It is immediate that $\tilde{\varepsilon}=\hat{\varepsilon} \phi$, with the fact that $\tilde{\varepsilon}$ contains no ideals in its nullspace then ensuring the same about $\hat{\varepsilon}$.
\end{proof}
The Frobenius structure that we emplaced on $\tilde{R} \hookrightarrow \hat{A}$ isn't the ``natural" one that brings together the $sl(n)$ systems on each of the coordinates of $\hat{A}$. In particular, we have done none of the prerequisite work towards determing the dual basis (and hence the genus reduction term) of that system. The ``natural" Frobenius structure that we want over $\tilde{R} \hookrightarrow \hat{A}$ is the following:
\begin{itemize}
\item Basis $\lbrace (1,0,...,0),...,(x^{k_1-1},0,...,0),(0,1,0,...,0),... \ , \ ...,(0,...,0,x^{k_m-1}) \rbrace$.
\item Frobenius form on that basis given by $\varepsilon'(x^{k_1-1},0,...,0)=\varepsilon'(0,...,0,x^{k_m-1})=1$ and $\varepsilon'(u)=0$ for every other basis element $u$.
\end{itemize}
The Frobenius matrix $\lambda'$ for this system is then block diagonal, with one block for each distinct root $\alpha_i$. If $\alpha_i$ is a multiplicity one root, its corresponding block is $1 \times 1$ and is the constant matrix $[[1]]$. If $\alpha_i$ is of multiplicity $n \geq 2$, its block is $n \times n$ with entries identical to the Frobenius matrix $\lambda_i$ for the $sl(n)$ Frobenius system over $R_i=\mathbb{Z}[\alpha_i] \hookrightarrow A_i=\mathbb{Z}[\alpha_i][x]/((x+\alpha_i)^{k_i})$. As none of these blocks are zero, $\lambda'$ is invertible and $\varepsilon'$ is in fact a nondegenerate Frobenius form.
$(\lambda')^{-1}$ is also block diagonal, with blocks either $[[1]]$ or $(\lambda_i)^{-1}$. It follows that the dual basis for the above system is $\lbrace ((1,0,...,0),(y_{1,1},0,...,0))$, $...$, $((x^{k_1-1},0,...,0),(y_{1,k_1},0,...,0))$, $((0,1,0,...,0),(0,y_{2,1,0,...,0}))$, $... \ , \ ...$, $((0,...0,x^{k_m-1}),(0,...,0,y_{m,k_m})) \rbrace$, where $y_{i,j}$ is equal to the companion of $x^{j-1}$ in the $sl(n)$ dual basis over $R_i \hookrightarrow A_i$. The associated genus reduction term is then $g'=(g_1,g_2,...,g_m)$, where $g_i$ is the genus reduction term over $R_i \hookrightarrow A_i$ (with $g_i=1*1=1$ in the coordinates corresponding to multiplicity one roots). By Theorem \ref{thm: repeated roots}, $(g_i)^2=0$ whenever $\alpha_i$ is a repeated root, while clearly $(g_i)^2=1$ if $\alpha_i$ is multiplicity one. For this system, it is then obvious that $(g')^2=(g_1^2,g_2^2,...,g_m^2)=0$ iff every root $\alpha_i$ of $p(x)$ is repeated.
We are now ready for the converse of Theorem \ref{thm: repeated roots}:
\begin{theorem}
\label{thm: repeated roots converse}
Let $R= \mathbb{Z}[a_1,...,a_n] \hookrightarrow A= \mathbb{Z}[a_1,...,a_n][x]/(p(x))$ be a $sl(n)$ Frobenius extension ($n \geq 2$), and assume that $p(x)$ has at least one root of multiplicity precisely $1$. Then $g^2 \neq 0$ in $A$
\end{theorem}
\begin{proof}
Let $p(x)= \prod_{i=1}^m (x+\alpha_i)^{k_i}$, where we have completely grouped like roots, and assume WLOG that $k_1=1$. We pass to the ``larger" ring extension $\tilde{R} \hookrightarrow \tilde{A}$ described previously, and show that $\tilde{g}=g \neq 0$ in $\tilde{A}$.\\
By Lemma \ref{thm: Chinese remainder theorem}, $\lbrace \tilde{R}, \tilde{A}, \tilde{\varepsilon}, (\tilde{x}_i,\tilde{y}_i) \rbrace$ is Frobenius equivalent to the ``product" Frobenius system $\lbrace \tilde{R}, \hat{A}, \hat{\varepsilon}, (\hat{x}_i,\hat{y}_i) \rbrace$. By preceding discussion, there exists a Frobenius structure $\lbrace \tilde{R}, \hat{A}, \varepsilon', ({x'}_i, {y'}_i) \rbrace$ over $\tilde{R} \hookrightarrow \hat{A}$ with genus reduction term nonzero. Corollary \ref{thm: epsilon dependence of g} then ensures that $\hat{g} \neq 0$. The aforementioned Frobenius equivalence, combined with Lemma \ref{thm: g under frobenius equivalence}, then gives $\tilde{g} \neq 0$.
With $\tilde{g} \neq 0$ in $\tilde{A}$, there cannot exist $\tilde{f}(x) \in \tilde{R}[x]$ such that $p(x)\tilde{f}(x)= \tilde{g}$. Hence there cannot exist $f(x) \in R[x] \subseteq \tilde{R}[x]$ such that $p(x)f(x) = \tilde{g}=g$, giving $g \neq 0$ in our Frobenius extension over $R \hookrightarrow A$.
\end{proof}
An equivalent argument to Theorem \ref{thm: repeated roots converse} shows that, if $p(x)$ has at least one root of multiplicity $1$, then $g^i \neq 0$ in $A$ for all $i \geq 2$. Combining results then gives the relatively succinct corollary.
\begin{corollary}
\label{thm: behavior of g^i}
Let $R= \mathbb{Z}[a_1,...,a_n] \hookrightarrow A= \mathbb{Z}[a_1,...,a_n][x]/(p(x))$ be a $sl(n)$ Frobenius extension of rank $n \geq 2$. If every root of $p(x)$ if repeated, then $g^i=0$ in $A$ for all $i \geq 2$. Otherwise, $g^i \neq 0$ in $A$ for all $i \geq 2$.
\end{corollary}
This corollary implies that the $sl(n)$ Frobenius extensions associated to $p(x)$ with non-repeated roots have the potential to be extremely complicated, in the sense that they may have closed compact 2-manifolds of arbitrarily high genus that evaluate to nonzero elements of $R$ via $\varepsilon$.
\section{Skein Modules}
\label{sec: embedded skein modules}
In \cite{asaeda}, Asaeda and Frohman explored the free module of isotopy classes of surfaces in a 3-manifold, subject to relations coming from a TQFT over a ring $R$. As in a TQFT, disjoint union behaves like tensor product over $R$. Therefore a surface is viewed as a tensor product of its connected components. The surfaces form a module and this module is an invariant of the 3-manifold the surfaces are embedded in.
The embedded surfaces must be treated slightly differently than the abstract surfaces associated to the TQFT. For instance, it is often the case that the neck-cutting relation cannot be applied as there is no compressing disk present in the 3-manifold. In addition, it is stipulated that the sphere relations only apply to spheres that bound balls. Other than those two considerations, the surfaces are treated as they would be if they are coming from the TQFT. Uwe Kaiser gives a thorough treatment of obtaining skein modules of 3-manifolds from Frobenius extensions in \cite{kaiser}
In the previous sections we have been dealing with the universal $sl(n)$ Frobenius extensions $R \hookrightarrow A \text{ with } A = R[a_1,\dots,a_n]/(p(x))$, where $p(x)=x^n-a_1x^{n-1} - \dots - a_n$. We now define $K_n(M)$ to be the skein module of $M$ where the surfaces are subject to the relations coming from the general $sl(n)$ Frobenius extension.
Often the $a_i$ are simply indeterminates, but sometimes it is interesting or helpful to examine the skein module where the $a_i$ are subject to certain conditions. This will be indicated by $K_n(M)[\{f_j(a_1,\dots a_n)=0\}_j]$, where the $a_i$ satisfy $f_j(a_1,\dots a_n)=0$, for all $j$.
\subsection{3-manifold Preliminaries}
In order to develop and explore the skein modules, we recall some definitions concerning the study of 3-manifolds.
\begin{definition}
A three-manifold is {\textbf{irreducible}} it every two-sphere bounds a three-ball.
\end{definition}
\begin{definition}
A curve on a surface is {\textbf{inessential}} if it bounds a disk on the surface. Otherwise the curve is \textbf{essential}.
\end{definition}
\begin{definition}
Let $S$ be a surface embedded in three-manifold $M$. $S$ is {\textbf{compressible} if $S$ contains an essential curve that bounds a disk, $D$, in $M$ such that $S \cap D = \partial D$. If no such curves exist and $S$ is not a two-sphere that bounds a ball, then $S$ is \textbf{incompressible}}.
\end{definition}
The compressability of a surface is extremely important when dealing with skein modules. For instance, when a surface is compressible the neck-cutting relation can be applied to yield an equivalent surface in the skein module.
\subsection{Linear Independence of Unmarked Surfaces}
In \cite{asaeda}, Asaeda and Frohman showed that under certain conditions in the $n=2$ case the unmarked surfaces are linearly independent. We extend their result to any $n$ below.
\begin{theorem}
\label{linindtheorem}
Let $M$ be an irreducible three-manifold. If every root of p(x) is repeated, then the unmarked incompressible surfaces in $K_n(M)$ are linearly independent over $\mathbb{Z}[a_1,\dots,a_n]$
\end{theorem}
\begin{proof}
Let $p(x) = \prod_{i=1}^r (x+\alpha_i)^{k_i}$, where $k_i > 1$ for all $i$. Note that $-\alpha_i$ is a root of $g_n= p'(x)$ for all $i$. We show that the unmarked incompressible surfaces of $K_n(M)$ are linearly independent over $\mathbb{Z}[\alpha_1,\dots, \alpha_r]$, implying that they are linearly independent over $\mathbb{Z}[a_1,\dots,a_n] \subset \mathbb{Z}[\alpha_1,\dots, \alpha_r]$
Let $F$ be an unmarked incompressible surface in $M$. For each $F$ we define a $\mathbb{Z}[\alpha_1,\dots, \alpha_r]$-linear functional, $\lambda_F$, such that $\lambda_F(F) = 1$ and $\lambda_F(F')=0$ if $F'$ is any other unmarked incompressible surface in $M$. Fix a root $\alpha$ of $p(x)$ (any root will give a suitable family of functionals), and define $\lambda_F$ as follows:
\begin{itemize}
\item $\lambda_F(S) = (-\alpha)^k \prod_\sigma \epsilon(S_\sigma) \prod_\tau \epsilon(T_\tau)$ if $S$ is a disjoint union of $F$, marked with $x^k$, with spheres $S_\sigma$ and compressible tori $T_\tau$.
\item $\lambda_F(S) = \prod_\sigma \epsilon(S_\sigma) \prod_\tau \epsilon(T_\tau)$ if $S$ is a disjoint union of spheres $S_\sigma$ and compressible tori $T_\tau$.
\item $\lambda_F(S) = 0$ otherwise.
\end{itemize}
Note that since $M$ is irreducible all compressible tori compress down to spheres that bound balls, no matter which compressing disk is chosen.
We must show that the functionals respect the relations of the skein module. Thus we must address the neck-cutting relation, the sphere relations and the dot reduction relation.
First we show the functionals respect the neck-cutting relation. The functionals are defined so that all surfaces that are compressible (excluding tori) are sent to zero. Therefore we must show that the result of compressing a surface is also sent to zero by the functionals.
By earlier work we have that
\begin{equation*}
p'(x) = nx^{n-1} - \sum_{j=0}^{n-2} a_{n-1-j} (j+1) x^{j}
\end{equation*}
Consider
\begin{align*}
\lambda_F \left (\sum_{i=0}^{n-1} \raisebox{-.3 in}{\includegraphics[height = .6 in]{cutneck2.pdf}}\put(-18,20){$x^i$}\put(-24,-28){$x^{j-i}$} - \sum_{j=0}^{n-2} a_{n-1-j} \sum_{i=0}^{j} \raisebox{-.3 in}{\includegraphics[height = .6 in]{cutneck2.pdf}}\put(-18,20){$x^i$}\put(-24,-28){$x^{j-i}$} \right ) &=n(-\alpha)^{n-1} - \sum_{j=0}^{n-2} a_{n-1-j} (j+1) (-\alpha)^{j} \\
&= p'(-\alpha) = 0
\end{align*}
It is also necessary to show that if $x^n$ is replaced by $a_1 x^{n-1} + \dots + a_n$ that the functionals respect this.
First note that
\begin{equation*}
0 = p'(-\alpha) = (-\alpha)^n - a_1 (-\alpha)^{n-1} - \dots - a_n,
\end{equation*}
which implies
\begin{equation*}
(-\alpha)^n = a_1(- \alpha)^{n-1} + \dots + a_n.
\end{equation*}
Now we have
\begin{align*}
\lambda( a_1(x^{n-1}) + a_2 (x ^{n-2}) + \dots + a_n(1)) &= a_1 (-\alpha)^{n-1} + a_2 (-\alpha)^{n-2} + \dots + a_n = (-\alpha)^n \\
\\
&= \lambda(x^n).
\end{align*}
Therefore the functionals respect the ring. Now note the functionals respect the sphere relations by how they are defined. By the work above they respect the neck-cutting relation. Since we have defined the appropriate functionals it is apparent that the unmarked incompressible surfaces are linearly independent.
\end{proof}
It is important to observe that while we only utilized one repeated root to define our functionals, we actually needed the fact that all roots were repeated. Otherwise we would have needed to address null-homologous surfaces of genus greater than one in the definition of our functionals. Since all roots are repeated, all higher genus surfaces are equivalent to zero in the skein module, by Corollary \ref{thm: behavior of g^i}.
\subsection{Exploring $K_2(M)$}
In \cite{asaeda}, Asaeda and Frohman required 2 to be invertible in the ring. Therefore, in order to build on their results, we will work over $\mathbb{Q}[a_1,a_2]$, rather than $\mathbb{Z}[a_1,a_2]$, for the rest of this section.
Recall that in the skein modules the surfaces are marked with elements of the ring. In $K_2(M)$ all surfaces can be written in terms of surfaces that are marked with $x$ to the first power, at most. Following the convention set forth by Bar-Natan, among others, we let a dotted surface represent a surface marked with an $x$.
\subsubsection{An Example}
\label{examplesection}
We will determine the skein module $K_2(S^2 \times S^1)[4a_2 + a_1^2 = 0]$. Note that $S^2 \times S^1$ is not irreducible so we cannot apply Theorem \ref{linindtheorem}. In order to concisely do calculations in the skein module we introduce some new notation:
\begin{enumerate}
\item \nd will denote an unmarked sphere in $S^2 \times S^1$ that doesn't bound a ball,
\medskip
\item \od will denote a sphere marked with an $x$ in $S^2 \times S^1$ that also doesn't bound a ball,
\medskip
\item \nd \nd is two parallel unmarked spheres,
\medskip
\item \od \nd is two parallel sphere where one is marked with and $x$ and one is not, etc.
\end{enumerate}
Unless noted otherwise, the spheres are always being viewed locally. That is, there may or may not be other sphere components of the surface in addition to the ones being viewed.
\begin{remark1}
We assume all surfaces are as simple as possible in terms of $x$. If there is an $x^2$ on a component simply replace all of them with $a_1x+a_2$ as follows:
\begin{center}
$\td = a_1 \od + a_2\nd$.
\end{center}
\end{remark1}
\begin{lemma}\label{relationsLemma}
We have the following relations on un-bounding spheres in $S^2 \times S^1$:
\begin{enumerate}
\item \begin{equation*}
\label{relation1}
1 = \od \od + a_2 \nd \nd
\end{equation*}
\item \begin{equation*}
\label{relation2}
\od \nd = - \nd \od + a_1 \nd \nd
\end{equation*}
\item \begin{equation*}
\nd\put(-12,-15){1} \raisebox{.25 em}{\dots} \nd\put(-11,-15){n} = \nd\put(-11,-15){n} \nd\put(-11,-15){1} \raisebox{.25 em}{\dots} \nd\put(-14,-15){n-1}
\end{equation*}
\item \begin{equation*}
\od \od \nd = \nd \od \od
\end{equation*}
\item \begin{equation*}
\od \nd \nd = \nd \nd \od
\end{equation*}
\item \begin{equation*}
\od \nd \dots \nd = \frac{a_1}{2} \nd \nd \dots \nd \text{(exactly one sphere has a dot on left side)}
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Consider two parallel spheres that do not bound a ball. If they are tubed together the new sphere bounds a ball, but now compressing the tube we have just placed yields the two parallel spheres. If the sphere that bounds a ball is marked with a dot we get the desired relation.
\item As in 1, if the new sphere is unmarked we get this relation.
\item This relation comes from the fact that we can cyclically permute the spheres in $S^2 \times S^1$.
\item By repeated applications of relation 2 and relation 5, we have:
\begin{equation*}
\od \od \nd = - \od \nd \od + a_1 \od \nd \nd = \nd \od \od - a_1 \nd \nd \od + a_1 \od \nd \nd = \nd \od \od.
\end{equation*}
\item Again, by repeated applications of relation two we have:
\begin{align*}
\od \nd \nd &= - \nd \od \nd + a \nd \nd \nd = \nd \nd \od - a \nd \nd \nd + a \nd \nd \nd \\
\\
&= \nd \nd \od .
\end{align*}
\item
\begin{equation*}
\od \nd \dots \nd = - \nd \od \dots \nd + a_1 \nd \nd \dots \nd
\end{equation*}
implies that
\begin{equation*}
2 \od \nd \dots \nd = a_1 \nd \nd \dots \nd
\end{equation*}
and since 2 is invertible, we arrive at the desired relation.
\end{enumerate}
\end{proof}
\begin{definition}
An \textbf{odd (even) configuration} of spheres is a surface that consists entirely of an odd (even) number of parallel un-bounding spheres each marked with at most one dot and nothing else.
\end{definition}
\begin{definition}
An even configuration in \textbf{standard position} is in the following form:
\bigskip
\begin{center}
\od \od \dots \od \nd \od \dots \nd \od \nd \nd \dots \nd
\end{center}
An odd configuration in \textbf{standard position} is in the following form:
\bigskip
\begin{center}
\od \od \dots \od \nd \dots \nd
\end{center}
In essence, a configuration in standard position is one where the marked spheres are as close together as possible.
\end{definition}
\begin{lemma}
All configurations can be placed in a unique standard position using relations \ref{relationsLemma}.4 and \ref{relationsLemma}.5.
\end{lemma}
\begin{proof}
First we will consider even configurations. For an even configuration to be in standard position it is necessary to have only one gap between marked spheres of more than one unmarked sphere. If the configuration has two such gaps we can eliminate one by repeated applications of \ref{relationsLemma}.5. If, after eliminating all such gaps, the configuration is still not in standard position it is because there are adjacent dotted spheres surrounded by spheres where every other sphere is dotted. By applying relation \ref{relationsLemma}.4 we can move all the spheres where each sphere is dotted next to each other and now the configuration is in standard position.
Now we must address uniqueness. Since we are dealing with even configuration we can divide the spheres into two sets, where two spheres are in the same set if they are separated by an odd number of spheres. By the fact we only used relations \ref{relationsLemma}.4 and \ref{relationsLemma}.5 the standard position is completely determined by the number of marked spheres in each set.
For an odd configuration to be in standard position it is necessary to have only one gap between marked spheres. Consider if there are two such gaps. If one of the gaps consists of an even number of unmarked spheres then we can eliminate it by \ref{relationsLemma}.5. Otherwise we can move the dot in the opposite direction by using a combination of \ref{relationsLemma}.4 and \ref{relationsLemma}.5. It is then possible to place it next to a dotted sphere since we are in an odd configuration. Uniqueness follows by the fact the standard position is completely determined by the number of marked spheres.
\end{proof}
\begin{lemma}\label{oddzero}
All odd configurations with no marked spheres are equal to 0.
\end{lemma}
\begin{proof}
\begin{align*}
\nd &= \nd \od \od + a_2 \nd \nd \nd & \text{by relation } \ref{relationsLemma}.1 \\
\\
&= -\od \nd \od + a_1 \nd \nd \od + a_2 \nd \nd \nd & \text{by relation } \ref{relationsLemma}.2 \\
\\
&=-(\nd - a_2\nd \nd \nd) + \frac{a_1^2}{2} \nd \nd \nd + a_2 \nd \nd \nd & \text{by relations } \ref{relationsLemma}.2\text{ and }\ref{relationsLemma}.6\\
\\
&= - \nd + \frac{a_1^2}{2} \nd \nd \nd+ 2a_2 \nd \nd \nd\\
\end{align*}
So, $2 \nd = 2a_2 \nd \nd \nd + \frac{a_1^2}{2} \nd \nd \nd$ and thus
\bigskip
$4 \nd = (4a_2+a_1^2) \nd \nd \nd =0$, since we assumed $4a_2+a_1^2=0$. Thus we have that $\nd = 0$.
\end{proof}
\bigskip
We now define an algorithm for reducing the configurations:
\begin{enumerate}
\item Evaluate all trivial tori and spheres.
\item Using the neck-cutting relation remove all handles from non-bounding spheres.
\item Using relations \ref{relationsLemma}.4 and \ref{relationsLemma}.5 put the configuration into standard position.
\item Using relation \ref{relationsLemma}.2 move all of the dots on parallel spheres as close together as possible.
\item Adjacent dotted spheres annihilate each other by relation \ref{relationsLemma}.1.
\item Using relation \ref{relationsLemma}.6 we are able to replace configurations with one dotted sphere with ones with no dotted spheres.
\end{enumerate}
By applying the algorithm and Lemma \ref{oddzero} we can see that the surfaces are spanned by the collection of even unmarked spheres, one dotted sphere and the empty surface. We wish to show that if $4a_2 + a_1^2=0$ then this collection is linearly independent.
\bigskip
The first step to showing linear independence is to define linear functionals on the generators:
$$
\lambda_k(S) =
\left \{
\begin{tabular}{cc}
1 & if $S$ is $2k$ parallel unmarked spheres that don't bound a ball \\
0 & else \\
\end{tabular}
\right .
$$
$$
\lambda_d(S) =
\left \{
\begin{tabular}{cc}
1 & if $S$ is a dotted sphere that doesn't bound a ball \\
0 & else \\
\end{tabular}
\right .
$$
Using the algorithm each linear functional can be extended to a map on any surface in $S^2 \times S^1$. We must show the functionals together with the algorithm are well-defined on the skein module, that is to say that they respect the relations.
By how the functionals are defined it is clear that they respect the sphere relations. Now we must show that both sides of the neck-cutting relation are respected by the functionals. By the definition of the algorithm, the functionals behave well with regards to neck-cutting, with the exception of when a trivial sphere becomes two non-bounding spheres. These situations are related to relations \ref{relationsLemma}.1 and \ref{relationsLemma}.2 and we address them below.
\medskip
Relation 1: Consider the case where one side of the neck-cutting relation is a bounding unmarked sphere and the other side is the result of compressing the sphere to yield:
\begin{center}
$\od \nd + \nd \od - a_1 \nd \nd$
\end{center}
If there is an even number of spheres then note that either \od \nd or \nd \od will need to be moved to put the dots as close together as possible. Without loss of generality we have
\begin{center}
$\od \nd + \nd \od - a_1 \nd \nd = -\nd \od + a_1 \nd \nd + \nd \od - a_1 \nd \nd= 0$
\end{center}
If there is an odd number of spheres then by Lemma \ref{oddzero} all terms are zero.
\begin{center}
$\od \nd + \nd \od - a_1 \nd \nd = 2\nd \od + a_1 \nd \nd=0$
\end{center}
\medskip
Relation 2 : Consider the case where one side of the neck-cutting relation is a bounding marked sphere and the other is the result of compressing the sphere:
\begin{center}
$\od \od + a_2 \nd \nd$
\end{center}
The first summand has one more pair, so at some point the pair is replaced by
\begin{center}
$(1 - a_2\nd \nd) + a_2 \nd \nd = 1= \text{evaluation of marked sphere} $
\end{center}
Thus we have that
\begin{equation*}
K_2(S^2 \times S^1)[4a_2 + a_1^2 = 0] \cong R[x] \oplus Re,
\end{equation*}
where $x^k$ represents $2k$ parallel unmarked spheres and $e$ represents a single marked sphere.
\subsubsection{A Partial Converse}
We were able to prove the linear independence of the unmarked incompressible surfaces of $K_n(M)$ when $M$ is irreducible for any $n$, as long as every root of $p(x)$ is repeated. When $n=2$ we are able to prove a partial converse.
In order to prove the converse we use notation similar to that of Section \ref{examplesection}. We will be dealing with an incompressible surface that fibers over a circle. A vertical line will denote one copy of this surface and multiple lines will denote multiple surfaces. If the lines are decorated with a dot than that particular surface is marked with an $x$.
\begin{theorem}\label{irredclass}
Let $M$ be an irreducible 3-manifold such that some incompressible surface in $M$ fibers over a circle. The unmarked surfaces in $K_2(M)$ are linearly independent if and only if $4a_2+a_1^2 = 0$.
\end{theorem}
\begin{proof}
Right to left is by Theorem \ref{linindtheorem}. We will show the contrapositive of left to right by showing that if $4a_2+a_1^2 \neq 0$, then the unmarked surfaces are linearly dependent.
Let $i$ be the genus of the incompressible surface that fibers over a circle in $M$. Recall the following results from Section \ref{subsec: neck-cutting in universal sln}:
\begin{itemize}
\item
${\Sigma}_{2i} = \text{\{Two parallel genus $i$ surfaces tubed together\}} = \od \nd + \nd \od - a_1 \nd \nd$,
\item
$\dot{\Sigma}_{2i} = \text{\{Two parallel genus $i$ surfaces tubed together, marked with $x$\}} = \od \od + a_2 \nd \nd$.
\end{itemize}
\medskip
Also, note $0 = \Sigma_{2i} = \od \nd + \nd \od - a_1 \nd \nd$, which yields the relation
\begin{equation}\label{convrelation}
a_1 \nd \nd = \od \nd + \nd \od
\end{equation}
By repeated applications of relation \ref{convrelation} above, we have
\begin{equation*}\begin{split}
a_1^2 \nd \nd \nd & = a_1 \nd \nd \od + a_1 \nd \od \nd = 2 a_1 \nd \nd \od = 2 \od \nd \od + 2 \nd \od \od \\
\\
&= 4 \nd \od \od = 4 \dot{\Sigma}_{2i} \nd - 4a_2 \nd \nd \nd,
\end{split}
\end{equation*}
thus, $(4a_2+a_1^2)\nd\nd\nd =4\dot{\Sigma}_{2i} \nd = 4(4 a_2 + a_1^2)^{2i}\nd$, so the unmarked surfaces are linearly dependent.
\end{proof}
Theorem \ref{irredclass} is the only partial converse that we were able to prove for any $n$. Thus, it is an open question as to exactly when the unmarked surfaces are linearly independent for $n > 2$.
\section{Appendix: The Genus Reduction Matrix}
\label{sec: genus reduction matrix}
In Subsection \ref{subsec: neck-cutting in universal sln} we explicitly calculated all powers of our genus reduction term $g$ in the universal $sl(n)$ skein module when $n=2$, and alluded to the fact that this was difficult to do in complete generality for higher $n$. Here we tackle that problem using linear algebra, interpreting $g$ as a $R$-linear operator from $A$ to $A$. Choosing the standard ordered basis $\lbrace 1,x,...,x^{n-1} \rbrace$, we may write $g$ as an $n \times n$ matrix $G_n \in Mat_n(R)$ (where the subscript in $G_n$ corresponds to the rank of the $sl(n)$ extension).
Note that, in terms of our chosen basis, the first column of $G_n$ directly corresponds to our genus-reduction term. The j\textsuperscript{th} column similarly corresponds to $x^{i-1} g_n$, after reducing modulo $p(x)=x^n - a_1 x^{n-1} - ... - a_{n-1} x - a_n$. Also note that what our closed surfaces actually evaluate to via our Frobenius form correspond to the final row of the matrix, so that we can immediately determine the evaluation of a torus decorated by $x^k$ as the $(n,k+1)$ entry of $G$. Our first proposition gives a recursive formula for determining the $(i,j)$ entry of $G_n$, for any $n \geq 2$.
\begin{proposition}
\label{thm: Gn recursive formula}
The entries of $G_n = \left[ \left[ g_{i,j} \right] \right]$, for any $n \geq 2$ are defined recursively as follows:\\
\begin{center}
$g_{i,1} = -i a_{n-i}$ (for $i < n$)\\
$g_{n,1} = n$\\
$g_{i,j} = a_{n-i+1} g_{n,j-1} + g_{i-1,j-1}$ (for $j > 1, i>1$)\\
$g_{i,j} = a_{n-i+1} g_{n,j-1}$ (for $j > 1, i=1$)
\end{center}
\end{proposition}
\begin{proof}
The first two lines follow from the expression for $g$ that we already exhibited at the beginning of Subsection \ref{subsec: neck-cutting in universal sln}. As for the last two lines, we obtain the j\textsuperscript{th} column of $G_n$ from the $(j-1)^{th}$ column via multiplication by $x$. Working modulo $(x^n - a_1 x^{n-1} - ... - a_n)$ we then have:\\
$x*(g_{1,j-1} + g_{2,j-1}x + ... + g_{n-1,j-1}x^{n-2} + g_{n,j-1}x^{n-1}) = g_{1,j-1}x + g_{2,j-1}x^2 + ... + g_{n-1,j-1}x^{n-1} + g_{n,j-1}x^{n} = g_{1,j-1}x + g_{2,j-1}x^2 + ... + g_{n-1,j-1}x^{n-1} + g_{n,j-1}(a_1 x^{n-1} + a_2 x^{n-2} + ... + a_{n-1} x + a_n) = (a_n g_{n,j-1}) + (a_{n-1} g_{n,j-1} + g_{1,j-1}) x + ... + (a_1 g_{2,j-1} + g_{n-1,j-1}) x^{n-1}$
\end{proof}
When our recursive relation it is then easy to produce $G_n$ for small $n$:
\[ G_2 =
\begin{bmatrix}
-a_1 & 2 a_2 \\
2 & a_1
\end{bmatrix}
\]
\[ G_3 =
\begin{bmatrix}
-a_2 & 3 a_3 & a_1 a_3 \\
-2 a_1 & 2 a_2 & a_1 a_2 + 3 a_3 \\
3 & a_1 & a_1^2 + 2 a_2
\end{bmatrix}
\]
\[ G_4 =
\begin{bmatrix}
-a_3 & 4 a_4 & a_1 a_4 & a_1^2 a_4 + 2 a_2 a_4 \\
-2 a_2 & 3 a_3 & a_1 a_3 + 4 a_4 & a_1^2 a_3 + 2 a_2 a_3 + a_1 a_4\\
-3 a_1 & 2 a_2 & a_1 a_2 + 3 a_3 & a_1^2 a_2 + 2 a_2^2 + a_1 a_3 + 4 a_4\\
4 & a_1 & a_1^2 + 2 a_2 & a_1^3 + 3 a_1 a_2 + 3 a_3
\end{bmatrix}
\]
\\
The relatively simple conclusions that we drew about the $n=2$ case in Subsection \ref{subsec: neck-cutting in universal sln} follow directly from the fact that:
\[ (G_2)^2 =
\begin{bmatrix}
a_1^2 + 4 a_2 & 0 \\
0 & a_1^2 + 4 a_2
\end{bmatrix}
=
(a_1^2 + 4 a_2)*E_2
\]
And hence that:
\[
(G_2)^{2k} =
\begin{bmatrix}
(a_1^2 + 4 a_2)^k & 0 \\
0 & (a_1^2 + 4 a_2)^k
\end{bmatrix}
\]
\[
(G_2)^{2k+1} =
\begin{bmatrix}
-a_1 (a_1^2 + 4 a_2)^k & 2 a_2 (a_1^2 + 4 a_2)^k \\
2 (a_1^2 + 4 a_2)^k & a_1 (a_1^2 + 4 a_2)^k
\end{bmatrix}
\]
Now recall our complete factorization of $p(x)$ over $\mathbb{C}$ as $p(x)=x^n-a_a x^{n-1}-...-a_{n-1}x-a_n = \prod_{i=1}^n(x+\alpha_i)$, which provides for the identification of $a_k$ with the (negative of the) k\textsuperscript{th} elementary syymetric polynomial $e_k$ in the $\alpha_i$. When hoping to rewrite our matrices $G_n$ is terms of the $\alpha_i$, we require more general symmetric polynomials than the elementary ones. Hence we introduce the monomial symmetric polynomials, with $m_{(k_1...k_n)}$ standing for the sum of all monomials in the $\alpha_i$ of the form $\alpha_{i_2}^{k_2}...\alpha_{i_n}^{k_n}$. Note that we have as special subcases the elementary symmetric polynomials $e_k = m_{(1^k 0^{n-k})} = m_{(1^k)}$, where the $1^k$ indicates $k$ consecutive 1's and we traditionally drop any trailing 0's for brevity. In this notation we also have the ``power" symmetric polynomials $p_k = m_{(k^1)}$.
We may then quickly rewrite the first several $G_n$ from above:
\[ G_2 =
\begin{bmatrix}
m_{(1^1)} & -2 m_{(1^2)} \\
2 & -m_{(1^1)}
\end{bmatrix}
\]
\[ G_3 =
\begin{bmatrix}
m_{(1^2)} & -3 m_{(1^3)} & m_{(2^1 1^2)} \\
2 m_{(1^1)} & -2 m_{(1^2)} & m_{(2^1 1^1)} \\
3 & -m_{(1^1)} & m_{(2^1)}
\end{bmatrix}
\]
\[ G_4 =
\begin{bmatrix}
m_{(1^3)} & -4 m_{(1^4)} & m_{(2^1 1^3)} & -m_{(3^1 1^3)}\\
2 m_{(1^2)} & -3 m_{(1^3)} & m_{(2^1 1^2)} & -m_{(3^1 1^2)}\\
3 m_{(1^1)} & -2 m_{(1^2)} & m_{(2^1 1^1)} & -m_{(3^1 1^1)}\\
4 & -m_{(1^1)} & m_{(2^1)} & -m_{(3^1)}
\end{bmatrix}
\]
There's an obvious pattern for $G_n$ that begins to emerge here, and that pattern becomes especially simple following the first two columns. To show that this pattern holds for all $n \geq 2$ we require the following basic properties of symmetric polynomials, all of which are directly verifiable:
\begin{lemma}
\label{thm: symmetric polynomial relations}
For $p_a = m_{(a^1)}$ and $e_b = m_{(1^b)}$ in $n$ variables, we have the following relations:
\begin{enumerate}
\item $p_a e_b = m_{((a+1)^1 1^{n-1})}$ (for $b=n$)
\item $p_a e_b = m_{(2^1 1^{b-1})}+(b+1)m_{(1^{b+1})}$ (for $a=1$ and $b<n$)
\item $p_a e_b = m_{((a+1)^1 1^{b-1})}+m_{(a^1 1^b)}$ (for $a >1$ and $b<n$)
\end{enumerate}
\end{lemma}
\begin{proposition}
\label{thm: symmetric polynomial matrix}
For any $n \geq 2$, $G_n$ is of the form:\\
\[ G_n =
\begin{bmatrix}
m_{(1^{n-1})} & -n m_{(1^n)} & m_{(2^1 1^{n-1})} & -m_{(3^1 1^{n-1})} & \ldots & (-1)^{n-1} m_{((n-1)^1 1^{n-1})}\\
2m_{(1^{n-2})} & -(n-1) m_{(1^{n-1})} & m_{(2^1 1^{n-2})} & -m_{(3^1 1^{n-2})} & \ldots & (-1)^{n-1} m_{((n-1)^1 1^{n-2})}\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
(n-1)m_{(1^1)} & -2 m_{(1^2)} & m_{(2^1 1^1)} & -m_{(3^1 1^1)} & \ldots & (-1)^{n-1} m_{((n-1)^1 1^1)}\\
n & -m_{(1^1)} & m_{(2^1)} & -m_{(3^1)} & \ldots & (-1)^{n-1} m_{((n-1)^1)}
\end{bmatrix}
\]
\end{proposition}
\begin{proof}
We use the recursive relations proven in Proposition \ref{thm: Gn recursive formula}. As the pattern stabilizes beginning with the third column, we use those relations to directly verify the entries of columns $j=1$ and $j=2$, and then use induction for columns $j \geq 3$.\\
Column $j=1$:\\
$g_{i,1} = -i a_{n-i}=i e_{n-i} = i m_{(1^{n-i})}$ (for $i<n$)\\
$g_{n,1} = n$\\
Column $j=2$:\\
$g_{1,2} = a_n g_{n,1} = n a_n = -n e_n = -n m_{(1^1)}$\\
$g_{i,2} = a_{n-i+1} g_{n,1} + g_{i-1,1} = -n e_{n-i+1} + (i-1) e_{n-i+1} = -(n-i+1)e_{n-i+1}$ (for $i>1$)\\
Column $j=3$ (inductive base step), noting that $g_{n,2} = -p_1$:\\
$g_{1,3} = a_n g_{n,2} = -e_n (-p_1) = m_{(2^1 1^{n-1})}$ by Lemma \ref{thm: symmetric polynomial relations}(1)\\
$g_{i,3} = a_{n-i+1} g_{n,2} + g_{i-1,2} = -e_{n-i+1} (-p_1 ) + -(n-i+2) e_{n-i+2} = m_{(2^1 1^{n-i})} + (n-i+2) m_{(1^{n-i+2})} - (n-i+2) m_{(1^{n-i+2})} = m_{(2^1 1^{n-i})}$ by Lemma \ref{thm: symmetric polynomial relations}(2) (for $i>1$)\\
Inductive step (assume pattern holds for column $k$), noting that $g_{n,k} = (-1)^{k-1} p_{k-1}$:\\
$g_{1,k+1} = a_n g_{n,k} = -e_n (-1)^{k-1} p_{k-1} = (-1)^k m_{(k^1 1^{n-1})}$ by Lemma \ref{thm: symmetric polynomial relations}(1)\\
$g_{i,k+1} = a_{n-i+1} g_{n,k} + g_{i-1,k} = -e_{n-i+1} (-1)^{k-1} p_{k-1} +(-1)^{k-1} m_{((k-1)^1 1^{n-i+1})} = (-1)^k m_{(k^1 1^{n-i})} + (-1)^k m_{((k-1)^1,1^{n-i+1})} + (-1)^{k-1} m_{((k-1)^1 1^{n-i+1})} = (-1)^k m_{(k^1 1^{n-i})}$ by Lemma \ref{thm: symmetric polynomial relations}(3)\\
\end{proof}
\nocite{*}
\bibliographystyle{plain}
|
2,877,628,089,491 | arxiv | \section{Introduction}
The formation and evolution of massive
($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$) galaxies is one of the
most studied topics in extragalactic astronomy during the last
decade. Early expectations from hierarchical galaxy formation models
considered that star formation began in low mass systems which built
more massive galaxies through sequential merging
\citep{1996MNRAS.283.1361B,2000MNRAS.319..168C}, in a similar process
to the growth of structures in Cold Dark Matter simulations
\citep{2005Natur.435..629S}. In apparent contradiction with
hierarchical assembly, the finding of a substantial population of
massive galaxies at z$>$1
\citep{1988ApJ...331L..77E,1998Natur.394..241H, 2003ApJ...587L..79F,
2004Natur.430..181G}, some of them containing old stellar populations
and evolving passively (according to their red optical colors
--\citealt{2004ApJ...617..746D}, \citealt{2005ApJ...633..748R}--, and
spectra
--\citealt{2006ApJ...649L..71K,2008ApJ...677..219K,2008A&A...482...21C}--),
seems to favor a downsizing formation scenario
\citep{1996AJ....112..839C,2004Natur.428..625H,2005ApJ...619L.135J,
2005ApJ...630...82P,2006ApJ...651..120B}. This population of z$>$1
massive galaxies accounts for a significant fraction of the
local stellar mass density ($\sim$20\% as early as z$\sim$2, and
$\sim$10\% at z$\sim$4:
\citealt{2006A&A...459..745F,2007A&A...476..137A,2008ApJ...675..234P}).
The discovery of such a population reinforced the idea that both stars
and their host galaxies are coeval (resembling a monolithic--like
collapse), and consequently, no expectations of structure evolution in
these galaxies should be expected. For this reason, the recent
observational evidence showing that the most massive galaxies were
much more compact in the past
\citep{2005ApJ...626..680D,2006MNRAS.373L..36T,
2007MNRAS.374..614L,2008A&A...482...21C} has been surprising, and has
again opened the question of how the stellar populations of these
galaxies were assembled into their present shape.
The size evolution of the most massive objects since z$\sim$2 has been
characterized by \citet{2007MNRAS.382..109T}. These authors found
that, at a given stellar mass, disk--like objects at z$\sim$1.5 were a
factor of two smaller than their present-day counterparts. For
spheroid--like objects, the evolution has been even stronger: they
were a factor of four smaller at z$\sim$1.5 than nearby similar mass
ellipticals. In addition, the stellar mass densities of these high--z
galaxies were almost two orders of magnitude higher than objects of
the same mass today. These superdense galaxies have been found at even
higher (z$\sim$2.5) redshifts
\citep{2007ApJ...656...66Z,2007ApJ...671..285T,2008ApJ...677L...5V},
adding more controversy to the debate about the formation and
evolution of massive galaxies.
Two processes have been proposed to allow the superdense high-z
galaxies migrate to the local stellar mass--size relation. The first
process is dissipationless (with absence of star formation) merging.
Given the high metal abundances and old ages of the stellar population
present in local massive elliptical galaxies
(e.g., \citealt{2006MNRAS.370.1106G,2006A&A...457..809S,2007ApJ...669..947J}),
these mergers should preferentially be dry
\citep{2006MNRAS.366..499D}, and occur between z$\sim$1.5 and z$=$0,
the epoch when the red sequence appears
\citep{2007ApJ...665..944L}. In this context, a particular effective
size evolutionary mechanism (r$_e$$\sim$$\mathcal{M}$$^{1.3}$) has
been provided by \citet{2006MNRAS.369.1081B} through head--on mergers
of galaxies. The second possibility is the smooth envelope accretion
scenario \citep{2007ApJ...658..710N}, where accreted stars (mainly
provided by minor mergers) form an envelope whose size increases
smoothly at decreasing redshift.
The goal of this paper is to explore the evolutionary paths followed
by the most massive galaxies and their dependence on the morphology. To
do this, we quantify the growth in stellar mass via star formation
events of massive ($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$)
galaxies as a function of size and brightness profile shape up to
z$\sim$2. We base our discussion on the characterization of the dust
infrared (IR) emission of these systems, which is linked to the amount
of recent star formation and/or the presence of obscured AGN. This
IR-based study is complementary to the more classical approach to the
characterization of the evolution of massive ellipticals based on
rest-frame optical properties.
Throughout this paper, we use a cosmology with $\mathrm
H_{0}=70$~km\,s$^{-1}$\,Mpc$^{-1}$, $\Omega_{\mathrm M}=0.3$ and
$\Lambda=0.7$. All magnitudes refer to the AB system. The results for
stellar masses and SFRs assume a \citet{2003ApJ...586L.133C} initial
mass function (IMF) with 0.1$<$$\mathcal{M}$$<$100~$\mathcal{M}_\sun$.
\section{Data description}
\label{data}
\subsection{The sample}
To analyze the star formation properties of the most massive galaxies
as a function of morphology, we use the catalog of 831 $K$-band
selected massive galaxies
($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$) in the Palomar
Observatory Wide-Field Infrared (POWIR)/DEEP-2 survey
\citep{2003SPIE.4834..161D,2006ApJ...651..120B,2008MNRAS.383.1366C} for which
\citet{2007MNRAS.382..109T} provide redshifts, stellar masses, and
structural parameters (sizes and S\'ersic indices). These data,
jointly with the {\it Spitzer}/MIPS fluxes measured in the observations
carried our by the FIDEL Legacy Program in the Extended Groth Strip
(EGS), allow a detailed analysis of the star formation properties of
the most massive galaxies as a function of morphology up to z$\sim$2.
The sample is described in detail in \citet{2007MNRAS.381..962C} and
\citet{2007MNRAS.382..109T}. Briefly, the $K$-band survey covers
2165~arcmin$^2$ in the EGS and has a depth $K$$=$22.5~mag
(5$\sigma$). Only 710~arcmin$^2$ are covered simultaneously with
HST/ACS $v$- and $i$-band imaging from the All-Wavelength Extended
Groth Strip International Survey \citep[AEGIS,
][]{2007ApJ...660L...1D}, so reliable structural parameters can only
be measured for 831 galaxies within the entire POWIR/DEEP-2 survey in
EGS. Half of those 831 galaxies has spectroscopic redshifts based on
optical data obtained by the DEEP-2 Galaxy Redshift survey
\citep{2003SPIE.4834..161D}. \citet{2007MNRAS.381..962C} estimate
photometric redshifts for the rest of sources with an accuracy $\Delta
z/(1+z)$$=$0.025. The 831 galaxies with
$\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$ in the EGS lie in the
redshift range 0.2$<$z$\lesssim$2. Stellar masses were estimated by
\citet{2005ApJ...625..621B,2006ApJ...651..120B} using the exponential
star formation models of \citet{2003MNRAS.344.1000B} with a
\citet{2003ApJ...586L.133C} IMF and various ages, metallicities and
dust contents included. As shown by \citet{2007MNRAS.381..962C},
typical uncertainties in the stellar masses are a factor of $\sim$2
(typical of any stellar population study; see, e.g.,
\citealt{2003MNRAS.338..525P},
\citealt{2003MNRAS.341...33K},
\citealt{2006ApJ...640...92P}, \citealt{2006A&A...459..745F}, and
\citealt{2008ApJ...675..234P}). As discussed in detail in \citet{2007MNRAS.381..962C},
this factor includes the effects of the photometric redshift
uncertainties, the errors inherent to solution degeneracies, and the
choices of the IMF and the stellar emission library. For example,
using \citet{2005MNRAS.362..799M} models (with an improved treatment
of the TP-AGB stellar evolution phase) would produce a 20\% (at most)
systematic decrease in the mass estimations. Using a
\citet{1955ApJ...121..161S} IMF would increase the stellar masses
by a constant factor of 0.25dex.
\citet{2007MNRAS.382..109T} estimated (circularized) half-light radius
($r_e$) and \citet{1968adga.book.....S} indices ($n$) for all the
galaxies in our sample. They used $i$-band HST/ACS images to fit
surface brightness profiles and divided the sample in disk-like and
spheroid-like galaxies according to the value of the S\'ersic
index. \citet[][see also
\citealt{1995MNRAS.275..874A}]{2004ApJ...604L...9R} demonstrated that
nearby galaxies with $n$$<$2.5 are mostly disks, while spheroids are
characterized by high S\'ersic indices, $n$$>$2.5. They also performed
simulations to check that the S\'ersic index obtained from HST data
can be used as a morphology indicator at
z$>$0. \citet{2007MNRAS.382..109T} extended these simulations to prove
that the structural parameters are robust against the effects
introduced by $K$-corrections, AGN contamination, and surface
brightness dimming.
For our sample, visual inspection of the ACS $i$-band images by one of
the co-authors (I.T.) was used to classify the sample in 4 types:
ellipticals/lenticulars, spirals, irregulars, and mergers. Comparing
this visual classification with the one based on S\'ersic indices, we
find that the visually confirmed spheroids present
$<$$n$$>$$=$4.8$\pm$1.5, and the rest of sources have
$<$$n$$>$$=$1.7$\pm$1.7. There is a 6\% contamination of visually
identified spheroids in the $n$$<$2.5 sample, and a 20\% contamination
of visually identified disks in the $n$$>$2.5 sample (comparable to
the 5\% and 19\% contaminations in \citealt{2004ApJ...604L...9R}). The
fraction of interloper disks decreases to 7\% at $n$$>$4 and 4\% at
$n$$>$5. A 4\% contamination is typical of other works based on visual
or quantitative morphological classifications such as
\citet{2007MNRAS.381..962C} or \citet{2005ApJ...625..621B}.
The visual test shows that the galaxies with $n$$>$4 form a robust
(almost uncontaminated) sample of spheroid-like sources, and $n$$<$2.5
galaxies are mostly disks. Galaxies with 2.5$<$n$<$4 are most probably
spheroid-like galaxies with some contamination of S0 galaxies and
early-type spirals.
\subsection{UV-to-MIR photometric properties of the sample}
\label{merged}
\begin{figure*}
\vspace{-2cm}
\begin{center}
\begin{minipage}{5cm}
\includegraphics[angle=0,width=4.cm]{f1a.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=-90,width=4.cm]{f1b.eps}
\end{minipage}
\begin{minipage}{6.cm}
\includegraphics[angle=-90,width=5.cm]{f1c.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=0,width=4.cm]{f1d.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=-90,width=4.cm]{f1e.eps}
\end{minipage}
\begin{minipage}{6.cm}
\includegraphics[angle=-90,width=5.cm]{f1f.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=0,width=4.cm]{f1g.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=-90,width=4.cm]{f1h.eps}
\end{minipage}
\begin{minipage}{6.cm}
\includegraphics[angle=-90,width=5.cm]{f1i.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=0,width=4.cm]{f1j.eps}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[angle=-90,width=4.cm]{f1k.eps}
\end{minipage}
\begin{minipage}{6.cm}
\includegraphics[angle=-90,width=5.cm]{f1l.eps}
\end{minipage}
\figcaption{\label{postage_stamps} Postage stamps and SEDs for four
typical MIPS-detected galaxies in our sample of massive galaxies. Left
panels show 10$\arcsec$$\times$10$\arcsec$ RGB composite images built
from HST/ACS $v$ and $i$ frames. In the middle column, MIPS 24~$\mu$m\,
images of size 40$\arcsec$$\times$40$\arcsec$ are depicted, with the
red square showing the area covered by the ACS postage stamp. In all
images, North is up and East is left. The right columns show the SEDs
of each galaxy, fitted to stellar population and dust emission models
which are used to estimate photometric redshifts, stellar masses and
SFRs (these parameters are given in each SED plot). The two upper rows
show examples of disk-like galaxies: EGS142126.97$+$531137.4, a galaxy
at z$=$0.67; and EGS142013.18$+$525925.0, lying at z$=$1.65. The two
lower rows show examples of spheroid-like galaxies:
EGS142021.47$+$525543.4, a galaxy at z$=$0.63, and
EGS142125.76$+$531622.8, placed at z$=$1.70.}
\end{center}
\end{figure*}
The positions of the 831 massive galaxies in the EGS were
cross-correlated (using a 1$\arcsec$ search radius) with our own
reduction and catalogs of the {\it Spitzer} IRAC survey of the
EGS. Using the same simulation method described in
\citet{2008ApJ...675..234P}, we found that this catalog is 75\%
complete at $\sim$1.5~$\mu$Jy\, ([3.6]$=$23.5~mag), which corresponds to
8$\sigma$ detections. Our IRAC photometry is consistent with that
published by \citet{2008arXiv0803.0748B} for the same dataset (but
their own reduction and cataloging) within 0.1~mag (typical absolute
uncertainty of IRAC fluxes) for 75\% of the sample, and within
1-$\sigma$ error for virtually all sources.
We found IRAC counterparts down to [3.6]$=$23~mag for all of the 831
galaxies in Trujillo's sample. For 151 sources (18\% of the total),
the IRAC sources were blended with nearby objects, but still resolved
(the separation was larger than 1$\arcsec$). As described in detail in
Appendix A of \citet{2008ApJ...675..234P}, for these sources we
obtained multi-wavelength photometry (including {\it Spitzer}\, fluxes)
using a deblending algorithm based on the deconvolution of the IRAC
and MIPS images. The method takes the known positions of the blended
sources obtained from optical/NIR ground-based images and the PSFs for
the different images and obtain separated fluxes for the blended
sources \citep[see also ][]{2006A&A...449..951G}. The method relies on
the moderate resolution of the IRAC images ($\sim$2$\arcsec$ FWHM, not
that different from an optical ground-based image, but very stable),
which allows the deblending of sources separated by more than
$\sim$1$\arcsec$ (half of the FWHM). For the MIPS 24~$\mu$m\, images, the
resolution is worse ($\sim$6$\arcsec$ FWHM) but the IRAC data can be
used to assign most probable counterparts and help with the
deblending. In any case, the main results in this paper (the MIPS
detection fraction and the statistics of specific SFRs) remain
virtually unchanged (less than 5\% random changes at all redshifts)
when we remove the 151 sources with blending problems.
We measured consistent aperture photometry in several UV, optical,
NIR, and MIR bands with the method described in
\citet{2008ApJ...675..234P}. The multi-wavelength dataset is outlined briefly in
\citet{2008ApJ...677..169V} and will be characterized in detail in
Barro et al. (2008, in preparation). More noticeably, our merged
photometric catalog includes MIPS fluxes at 24~$\mu$m\, obtained from
aperture photometry in the GTO and FIDEL survey (DR2) data in the
AEGIS/EGS field
\citep{2007ApJ...660L...1D,2007ApJ...660L..73S}. Following the same
procedure described in
\citet{2005ApJ...630...82P,2008ApJ...675..234P}, we used the {\sc DAOPHOT}
software package in IRAF\footnote{IRAF is distributed by the National
Optical Astronomy Observatory, which is operated by the Association of
Universities for Research in Astronomy (AURA), Inc., under cooperative
agreement with the National Science Foundation.} to detect sources
(using a 3$\sigma$ detection cut above the local sky noise) in the
MIPS images and measure aperture photometry with a PSF fitting method.
Based on simulations consisting on the creation and recovery of
artificial sources in these images, we found that our 24~$\mu$m\, catalog
of the EGS is 75\% complete at 35~$\mu$Jy\, (consistent with
\citealt{2004ApJS..154...70P} and \citealt{2006ApJ...640..603T}).
$F(24)$$=$35~$\mu$Jy\, corresponds to $\sim$6$\sigma$ detections for the
average sky noise in the FIDEL DR2 images. Our 24~$\mu$m\, catalogs are
cut to 3$\sigma$ detections, which translate to the range
14--17~$\mu$Jy, depending on the location on the sky due to small
differences in exposure time and the effect of confusion (both
presenting small spatial variations throughout the image). As done in
\citet{2005ApJ...630...82P}, we tested the reliability of the MIPS
detections by analyzing the probability of having a counterpart within
the search radius (1$\arcsec$) in other optical/NIR (ground-based and
IRAC) bands for a random position on the sky of the 24~~$\mu$m\,
image. Having a counterpart in 3 different bands within the EGS
dataset has a negligible probability (1.4\%), so we conclude that
(virtually) all the MIPS detections within our sample are not
spurious. Figure~\ref{postage_stamps} shows postage stamps and
spectral energy distributions (SEDs) of two typical examples of
disk-like galaxies and two spheroid-like galaxies detected by MIPS at
24~$\mu$m\, (also one 70~$\mu$m\, detection included).
Using the measured MIPS 24~$\mu$m\, fluxes, the estimated 280~nm
synthetic fluxes inter/extrapolated in the spectral energy
distribution fits, and the spectroscopic and photometric
redshifts published by \citet{2007MNRAS.382..109T}, we obtained total
(unobscured plus obscured) SFRs for each galaxy in the same way
explained in
\citet[][see also \citealt{2005ApJ...625...23B}]{2008ApJ...675..234P}.
Briefly, the MIR fluxes at rest-frame wavelengths longer than 5~$\mu$m\,
are fitted to dust emission models (from several libraries) and the
IR-based SFRs are obtained from integrated total IR (using the
calibration in \citealt{1998ARA&A..36..189K}) and rest-frame 24~$\mu$m\,
(see \citealt{2006ApJ...650..835A}) luminosities from the fits
(averaged through all template libraries). The IR-based (obscured) SFR
is then added to the UV-based (unobscured) SFR to obtain the total
SFR. As discussed in that paper, total SFR estimates should be good
within a factor of 2. The SFRs discussed in the following Sections
were estimated assuming a
\citet{2003ApJ...586L.133C} IMF, obtained by dividing the results
obtained with the calibrations in \citet[][valid for a
\citealt{1955ApJ...121..161S} IMF]{1998ARA&A..36..189K} by a 1.8
factor.
\section{Results and discussion}
\label{results}
\subsection{IR emission of the most massive galaxies at z$\lesssim$2}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=17cm]{f2.eps}
\figcaption{\label{mass_re}Stellar mass--size distribution for different
redshift bins of our massive galaxies separated in disk-like (left
panels) and spheroid-like (right panels) types. Galaxies detected by
MIPS at 24~$\mu$m\, are plotted with filled stars, while open stars show
MIPS non-detections. Red symbols are galaxies whose MIPS emission is
identified with obscured star formation, and green symbols depict
galaxies which harbor an X-ray or/and IR-emitting AGN.}
\end{center}
\end{figure*}
Figure~\ref{mass_re} shows the location of the MIPS detections in a
stellar mass--size diagram for massive galaxies as a function of the
concentration index. Out of the 831 galaxies in Trujillo's sample of
massive galaxies, 485 sources (58\%) were classified as spheroid-like
based on their high ($n$$>$2.5) \citet{1968adga.book.....S} indices,
and 346 (42\%) as disk-like sources ($n$$\leq$2.5). Among the
disk-like sources, 322 (93\% of all disks) are detected by MIPS at
24~$\mu$m\, with a minimum flux $F(24)$$=$15~$\mu$Jy, and 137 galaxies are
detected at 70~$\mu$m\, down to $F(70)$$=$0.5~mJy. Among the spheroids,
297 galaxies (61\% of the total) are detected at 24~$\mu$m\, down to
$F(24)$$=$14~$\mu$Jy, and 84 sources are detected at 70~$\mu$m\, down to
$F(70)$$=$0.7~mJy. If we consider only the MIPS 5$\sigma$ detections
(i.e., more statistically reliable sources, although our simulations
reveal that all our 3$\sigma$ 24~$\mu$m\, sources are reliable to the
99\% level; see Section~\ref{merged}), the MIPS detection fractions
decrease to 92\% for disky systems and 52\% for spheroid-like
sources.
\input{tab1}
Table~\ref{table1} shows the total number of sources and MIPS
detection fractions (for fluxes above the 3$\sigma$ level) as a
function of redshift and morphology. We consider the results based on
the morphological classification using the S\'ersic indices and the
direct visual inspection of the images. The MIPS detection fractions
for $n$$\leq$2.5 galaxies and visually identified disks are almost
identical. Visually confirmed spheroids present less MIPS detections
than the $n$$>$2.5 sources, although the difference is small ($<$10\%)
and consistent with the 20\% contamination of visually identified
disks in the $n$$>$2.5 sample, most of them having 2.5$<$$n$$<$4.0
(see Figure~\ref{re_n_mips}).
Figure~\ref{mass_re} shows that there is basically no difference
between the loci occupied by MIPS detected and undetected galaxies in
the stellar mass--size plane. However, for spheroid-like objects at a
given stellar mass, MIPS non-detections are smaller than IR-bright
sources by a factor of $\sim$1.2 (see also
\citealt{2007ApJ...656...66Z} results at higher z). This suggests that
early (i.e., z$>$2) massive star formation events left even more
compact remnants than starbursts taking place at z$<$2, maybe
reflecting the higher density conditions of the primitive Universe.
Figure~\ref{re_n_mips} shows the MIPS 24~$\mu$m\, detection fractions as
a function of structural parameters. This Figure confirms the bias of
the $n$$\leq$2.5 sample towards galaxies with on-going (possibly
extended through the disk) star formation or harboring an IR-emitting
AGN. Figure~\ref{re_n_mips} demonstrates that virtually all (80-90\%)
the $n$$\leq$2.5 galaxies are detected by MIPS at all redshifts and
(almost) independently of the size of the galaxy. In contrast, the
spheroid sample is biased towards more quiescent systems. There is
still a non-negligible fraction (6\%) of galaxies classified as disky
which fall below the MIPS detection limit or do not present any IR
emission, i.e, they have low-level star formation, no dust, or are
completely quiescent. According to Figure~\ref{re_n_mips}, most of
them lie at z$>$1 (59\% of all disk-like non-detections) and tend to
have comparatively smaller sizes: the MIPS detection fraction
decreases from 90\% for disk-like galaxies with $r_e$$\gtrsim$4~kpc to
70-80\% for $r_e$$\lesssim$1.5~kpc systems. All these sources present
very red SEDs (see Figure~\ref{seds}).
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,width=17.cm]{f3.eps}
\figcaption{\label{re_n_mips} MIPS 24~$\mu$m\, detection fractions as a
function of S\'ersic index (left panels), (circularized) half-light
radius (middle panels), and stellar mass (right panels). The whole
sample has been divided into two redshift intervals (z$<$1 on the
top panels, and z$>$1 on the bottom panels). Galaxies identified as
IR- or X-ray-bright AGN are excluded from the distributions. In all
plots, wide black lines show the results for all the MIPS detections
(with a minimum measured value of $F(24)$$=$15~$\mu$Jy) and gray narrow
lines show the measured fractions for a flux cut
$F(24)$$>$80~$\mu$Jy. On the left panels, the green vertical line shows
the adopted separation between spheroid-like and disk-like
galaxies. On the middle and right panels, red lines show the MIPS
detection fractions for spheroid-like sources, and blue lines for
disk-like sources.}
\end{center}
\end{figure*}
On the contrary, the spheroid-like sample includes at least a 63\% of
''active'' galaxies. Moreover, some more MIPS undetected spheroids may
have some star formation activity or harbor an AGN, since some of the
SEDs in the upper-left panel of Figure~\ref{seds} present a
significant emission in the UV, probably arising from young
stars. Most of these UV-bright galaxies lie at z$>$1, and the MIPS
24~$\mu$m\, flux upper limits\footnote{The flux upper limits at 70~$\mu$m
have been omitted from Figure~\ref{seds} for clarity, given that very
few sources are detected at this wavelength.} are consistent with the
MIR emission from a typical Sc galaxy. Figure~\ref{re_n_mips} shows
that MIPS detections are more common among the largest spheroid-like
galaxies, especially at z$>$1. Interestingly, at z$>$1 the MIPS
detection fraction stays roughly constant at $r_e$$\lesssim$3~kpc. It
may even increase (up to 70\%) for very compact ($r_e$$\lesssim$1~kpc)
spheroid-like galaxies, although there are 3 caveats to this result:
1) the number of sources with $r_e$$\lesssim$1~kpc is small
($\sim$30), so the uncertainties in these bins are of the order of
20-30\%; 2) for z$>$1 and $r_e$$\lesssim$1~kpc we are reaching the
resolution limit in the HST/ACS images, and consequently the size
measurements count with large uncertainties; and 3) these high-z
compact galaxies may be dominated by an obscured AGN (since the
galaxies are detected by MIPS and show no X-ray emission; see also
\citealt{2007MNRAS.382..109T}), which may bias the size measurements.
Our results may be compared with those published by
\citet{2007MNRAS.376..416R}. They found that 20\% of the most securely
identified spheroids\footnote{Note that \citet{2007MNRAS.376..416R}
cut their sample to clear E and E/S0 galaxies as classified visually
by
\citet{2005ApJ...625..621B}, but do not include Bundy's S0 type in
their analysis.} at 0.3$<$z$<$1.0 are detected during phases of
prominent activity. The evidence is the detection at 24~$\mu$m\, above
80~$\mu$Jy\, or at radio wavelengths above 40~$\mu$Jy. Our MIPS data are
deeper than theirs, but if we cut our catalogs to the same flux limit
(see Figure~\ref{re_n_mips}), we obtain an average $\sim$20\%
detection fraction at z$<$1 for the most concentrated objects with
$n$$>$4, probably well correlated with their sample of bona fide
spheroids, although our selection only includes the most massive
galaxies ($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$), and
Rodighiero's selection is not based on stellar mass.
If we consider the high-z (z$>$1) galaxies in our sample, our results
are also consistent with those found in \citet{2006ApJ...640...92P}
for Distant Red Galaxies (DRGs). These galaxies have a typical stellar
mass $\mathcal{M}$$\sim$10$^{11}$~$\mathcal{M}_\sun$ and a mean
redshift z$\sim$2
\citep[see also][]{2007A&A...465..393G,2008ApJ...675..234P}. Given that
our sample has a stellar mass cut
$\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$ and a redshift cut at
z$=$2, the low-redshift, high-mass tail of the general DRG population
must be included in our selection. Indeed, we have 27 DRGs in our
sample, with an average redshift
$<$z$>$$=$1.50$\pm$0.22. \citet{2006ApJ...640...92P} found that
roughly 50\% of DRGs are detected by MIPS at 24~$\mu$m\, down to
80~$\mu$Jy. We find a $\sim$60\% detection fraction for massive galaxies
in the low redshift tail of DRGs (1$<$z$<$2) with deeper data (75\%
completeness level at 35~$\mu$Jy). Among the 27 DRGs in our sample, 25
(93\%) are detected by MIPS. This detection fraction is higher than
the average for 1$<$z$<$2 galaxies, but still consistent with the
results in
\citet{2006ApJ...640...92P}, who argue that z$\lesssim$2 DRGs are mostly
heavily extincted starbursts part of the class of dusty EROs at z$>1$,
and find a $\sim$75\% MIPS detection fraction for their
$\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$ DRGs at z$<$2. Indeed, 18
DRGs in our sample are EROs (16 MIPS detections), and we count with a
total of 305 EROs in our entire sample.
\subsection{Spectral energy distributions}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,width=15.cm]{f4.eps}
\figcaption{\label{seds} Spectral energy distributions (de-redshifted
and normalized to the rest-frame $K$-band flux) of all the galaxies in
the sample of massive galaxies
($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$) in the EGS
\citep{2007MNRAS.382..109T}. The upper and lower panels show the SEDs
for the spheroid-like and disk-like galaxies, respectively. For each
morphological type, the sub-sample has been divided in three groups:
galaxies without MIPS detection (IR-faint, left panels, with arrows
showing an upper limit of the MIPS 24~$\mu$m\, emission corresponding to
15~$\mu$Jy, the minimum flux observed in the sample$^5$), galaxies with
a MIPS counterpart most probably linked to on-going star formation
(IR-bright, middle panels), and galaxies with nuclear activity (AGN,
right panels, see text for details). In each panel, open green circles
show sources at z$<$1, and filled red circles depict galaxies at
1$<$z$<$2. We also show typical templates (from
\citealt{2007ApJ...663...81P}) of an elliptical galaxy (Ell5), a
late-type spiral galaxy (Sc), and a galaxy with an obscured AGN
(I20551, just for the AGN panels on the right). All panels show the
fraction of sources in each type for the spheroid-like and disk-like
samples.}
\end{center}
\end{figure*}
Figure~\ref{seds} shows the SEDs of all the massive galaxies in our
EGS sample divided into morphological and activity types. The two
panels on the right show the SEDs for galaxies with an X-ray detection
(\citealt{2006ApJ...642..126B}, \citealt{2007ApJ...660L..11N}; see
also
\citealt{2007MNRAS.381..962C}) or classified as IRAC
power-law galaxies (PLGs, \citealt{2007ApJ...660..167D}), i.e.,
sources which most probably harbor an X-ray and/or IR emitting
AGN. There are 68 X-ray emitters, 60 of them with MIPS detection, and
5 PLGs in total, all of them with X-ray emission and 4 with MIPS
emission.
The SED distribution of the spheroids present a lower scatter in the
UV/optical, being very similar to a spectral template of an
elliptical. In contrast, there is a very populated tail of disk-like
galaxies with UV/optical fluxes brighter than a template for a typical
Sc galaxy, most probably linked to a recent starburst. The MIR
emission is consistent with the PAH spectrum of a late-type spiral
galaxy (see also Figure~\ref{postage_stamps}), but can be as high as
6-10 times the flux of the Sc template from
\citet{2007ApJ...663...81P}. Spheroid-like galaxies with a
MIPS detection present a lower 24~$\mu$m\, median flux (82~$\mu$Jy, with
the quartiles being 42~$\mu$Jy\, and 188~$\mu$Jy, and the average
160~$\mu$Jy) than MIPS disky sources (190$^{301}_{107}$~$\mu$Jy, and the
average 250~$\mu$Jy). Most of the sources identified as AGN present
relatively bright fluxes at rest-frame wavelengths between 2 and
10~$\mu$m, revealing the presence of very hot dust heated by the central
supermassive black hole and emitting in the NIR and MIR. In several
cases, this NIR/MIR emission hides the 1.6~$\mu$m\, bump, typically seen
in galaxies whose spectrum is dominated by stars rather than dust.
\subsection{Specific Star Formation Rates}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=8.5cm]{f5.eps}
\figcaption{\label{sfrm} Specific SFRs as a function of redshift and morphology
(for galaxies not identified as bright AGN). Galaxies detected at
24~$\mu$m\, are plotted with open (disks) and filled (spheroids) black
circles, while gray symbols show upper limits for sources not detected
by MIPS. Red and blue crosses represent the median and quartiles for
the distribution of specific SFRs in the different redshift ranges
used in \citet[][red widest lines referring to spheroids and blue
narrowest to disky galaxies]{2007MNRAS.382..109T}. Green curves show
the expected positions of galaxies which would multiply their stellar
mass by 5/4, 2, and 4 between their redshift and z$=$0 if they
maintained a constant SFR. Horizontal dashed lines show constant SFR
values for the median stellar mass of our sample
(1.6$\times$10$^{11}$~$\mathcal{M}_\sun$).}
\end{center}
\end{figure}
Figure~\ref{sfrm} shows the specific SFRs of massive galaxies as a
function of redshift and morphology. Table~\ref{table1} gives the
median and quartiles for different redshift ranges and morphological
types (obtained from S\'ersic index and visual classifications). The
median specific SFRs increase by less than 0.1dex and 0.02dex for
spheroids and disks, respectively, when considering MIPS 24~$\mu$m\,
detected above the 5$\sigma$ level. These increments do not affect the
following results significantly.
When segregating the sample based on the S\'ersic indices, we find
that the specific SFRs of spheroid galaxies evolve as
(1+z)$^{5.5\pm0.6}$ from z$=$0 to z$=$2, while the evolution for
disk-like galaxies goes as (1+z)$^{3.6\pm0.3}$. If we consider the
results based on the visual classification, the evolution is more
pronounced for spheroids and almost identical for disks:
(1+z)$^{6.4\pm0.8}$ evolution for the former, and (1+z)$^{3.4\pm0.2}$
for the latter.
The specific SFRs used in Figure~\ref{sfrm} have been estimated by
adding the unobscured SFRs obtained from UV data (at rest-frame
280~nm) and the obscured SFRs from IR data (using the total IR
luminosity) as explained in \citet{2008ApJ...675..234P}. The ratio
between these two quantities allow the estimation of the global
obscuration of the recent star formation in each galaxy (only for
those detected by MIPS). On average, we find that extinctions for MIPS
detected galaxies increase with redshift, ranging from
$<$$A(V)$$>$$=$1.0$\pm$0.5~mag at z$<$0.5 to
$<$$A(V)$$>$$=$1.5$\pm$0.6~mag at z$\sim$1 and
$<$$A(V)$$>$$=$2.0$\pm$0.7~mag at z$\sim$2. These values are
consistent with typical attenuations found for IR-bright galaxies by,
e.g., \citet{2000ApJ...537L..85R}, and the evolution in the extinction
properties of the UV SFR density found by
\citet{2007A&A...472..403T}. According to \citet{2007ApJ...660L..73S},
even larger extinctions (up to a factor of $\sim$100) are needed to
match SFRs obtained from IR or radio data and SFRs obtained from [OII]
spectroscopic observations of IR-bright sources in the EGS.
Figure~\ref{sfrm} shows that below z$=$1.1 spheroid-like galaxies
present very low specific SFRs. On average, they would increase their
stellar mass by less than 25\% at 0$<$z$<$1 if they maintained a
constant SFR. The global mass increase (in the form of newly-formed
stars) for all spheroid-like galaxies is less than 10\% if we take
into account the 38\% of z$<$1 spheroids which are not detected by
MIPS, and even lower ($\sim$5\%) if we only consider the visually
identified spheroids. In contrast, disk-like galaxies could typically
double their mass from z$=$1 to z$=$0 due to newly formed stars if
they maintained a constant SFR, with little change due to the very few
galaxies (less than 5\%) for which we only have SFR upper limits.
In practice, IR-bright intense star-forming bursts are not expected to
last long \citep{1994ApJ...431L...9M}, so a galaxy most probably will
not maintain a high SFR level for several Gyr. The higher SFR values
are expected to be maintained for shorter periods, since gas
exhaustion and supernova winds (and even AGN activity) will help to
suppress star formation. \citet{1996ApJ...464..641M} simulations of
mergers predict the triggering of a very intense and short starburst
event (probably detectable in the MIR by MIPS) lasting a few tens of
Myr (and occurring in late stages of the merger, when the galaxies are
actually joining) for encounters of galaxies with an already formed
bulge. Encounters of disky galaxies would trigger less intense bursts
lasting longer (100-200~Myr) and occurring earlier in the merger
process. For the observed specific SFRs in our sample, those short and
intense starbursts would add up less than 0.01\% (for each merging
event) to the total stellar mass of a typical spheroid-like galaxy at
z$<$1. This very small fraction of young stars would be hidden by the
predominant old stellar population and be undetectable in local
ellipticals. For the disky galaxies at z$<$1, which present specific
SFRs as high as 0.2~Gyr$^{-1}$, the burst strength (ratio of the newly
formed stars to the global stellar mass) could be as high as a few
percent (for each merging event), typical for star-forming galaxies at
low-redshifts \citep{2003MNRAS.341...33K,2003MNRAS.338..525P}.
At z$\gtrsim$1, the specific SFRs of massive galaxies are higher than
0.1~Gyr$^{-1}$, both the active spheroid-like (note that there are
40\% of spheroids which are not detected by MIPS) and disky systems
are forming stars at approximately the same rate, and the number of
quiescent galaxies (those not detected by MIPS) is less than
$\sim$50\% for both types. It is interesting to notice that most
galaxies (disks and spheroids) have significant amounts of dust, since
they are detected at 24~$\mu$m. If some of these galaxies are the
progenitors of nearby ellipticals, that dust should have disappeared
somehow or it is now very cold and may only be detected at very long
wavelengths ($\lambda$$>$100-200~$\mu$m) and low fluxes.
For typical burst durations, and even for star-forming events with a
constant SFR and lasting up to 1~Gyr, the maximum increase in stellar
mass would be $\sim$15\% at 1.1$<$z$<$1.4, $\sim$25\% at
1.4$<$z$<$1.7, and $\sim$50\% at 1.7$<$z$<$2.0, for both spheroids and
disks. This means that a significant fraction (more than 50\%) of the
stellar mass of z$>$1 massive galaxies was assembled at z$>$2
(\citealt{2008ApJ...675..234P}; see also
\citealt{2005ApJ...621L..89B}, \citealt{2005ApJ...633L...9F},
\citealt{2006ApJ...640...92P}). Moreover, we find that about 40\% of spheroids
at z$\sim$1.8 are almost ``dead'' (they present low SFR levels based
on IR and UV data) and evolving passively, or may be experiencing a
quiescent period. Note that most spheroid-like galaxies at z$>$1 would
be qualified as passive based on optical colors alone, but the MIPS
data reveals that $\sim$50\% of them are experiencing dusty starbursts
and 10\% more harbor (also) obscured AGN.
We can estimate how much stellar mass galaxies typically assemble
through star formation from z$=$2 to the present (i.e., the star
formation efficiency to increase the mass of a galaxy in the last
10~Gyr) if a galaxy follows the specific SFR evolution depicted in
Figure~\ref{sfrm}. We assume that the SFRs remain constant within each
redshift interval; since the starbursts probably last 50-200~Myr, as
discussed earlier, the following figures would be an upper
limit. Adding all the mass formed from z$=$2 to z$\sim$0, we estimate
that a disk-like galaxy could increase its stellar mass by up to a
factor of 3.2$\pm$0.5 in the last 10~Gyr: 1.4 times increase at
1.7$<$z$<$2.0 and an almost constant 10-20\% increase in each of our 5
redshift intervals at z$<$1.7. A spheroid-like galaxy could increase
its stellar mass by up to a factor of 1.8$\pm$0.3: 1.2 times increase
at 1.7$<$z$<$2.0, 10-20\% in each of our two intervals at
1.1$<$z$<$1.7 and less than 5\% in each of the three intervals at
z$<$1.1. These figures are almost unchanged ($<$5\% increases) when
considering only the MIPS 24~$\mu$m\, 5$\sigma$ detections. For visually
identified disks and spheroids, the stellar mass increases by up to a
factor of 2.7$\pm$0.4 and 1.8$\pm$0.3, respectively.
Ideally, one would like to disentangle what is the relative
contribution to the size growth of a galaxy of newly-formed stars and
system heating through merger/interactions. However, both processes
are probably linked, since new star formation events are likely
associated to the interactions that could inject energy to the
systems. Consequently, a definitive answer to the problem of how
galaxies grow require the help of elaborate modelling. It is worth
saying, nonetheless, that both the observations at low and high-z show
that galaxies have larger effective radii when observed in bluer
bands. This means that younger stars are preferentially located at
larger galactocentric distances than older populations. In this paper,
we have quantified how much the stellar mass grows through star
formation events only. Once we reach a clear picture of how the
galaxies can increase in size through mergers, our results will
constrain the amount of stellar mass due to dry accretion that is
necessary to migrate the high-z galaxies to the local size--mass
relations.
\section{Summary and conclusions}
We have analyzed the stellar mass growth in the form of newly-born
stars in a sample of 831 $K$-band selected massive galaxies
($\mathcal{M}$$>$10$^{11}$~$\mathcal{M}_\sun$) as a function of
structural parameters (size and concentration). These galaxies lie in
the redshift range between z$=$0.2 and z$\sim$2. Our analysis is based
on the measurement of the specific SFR for each galaxy based on their
UV and IR emission, taking advantage of the deep
{\it Spitzer}\, data obtained by the FIDEL {\it Spitzer}/MIPS Legacy Project in the
Extended Groth Strip.
Our main results follow:
\begin{itemize}
\item[$-$]Most (more than 85\% at any redshift) disk-like galaxies
(identified by small S\'ersic indices, $n$$<$2.5) are detected by MIPS
at 24~$\mu$m\, down to $F(24)$$=$15~$\mu$Jy\, with a median flux
$F(24)$$=$190~$\mu$Jy.
\item[$-$]A significant fraction (more than 55\% at any redshift) of
spheroid-like galaxies is detected at 24~$\mu$m\, down to
$F(24)$$=$14~$\mu$Jy\, with a median flux $F(24)$$=$82~$\mu$Jy.
\item[$-$]The MIPS detection fraction for spheroid-like galaxies is higher
(70--90\%) for larger ($r_e$$\gtrsim$5~kpc) galaxies, especially at
z$>$1, where the detection fraction has a minimum around 30--40\% for
galaxies with $r_e$$\sim$1~kpc. No clear trend is found for disky
galaxies of different sizes.
\item[$-$]There is basically no difference between the loci occupied by
MIPS detected and undetected galaxies in the stellar mass--size
plane. However, for spheroid-like objects at a given stellar mass,
MIPS non-detections are smaller than IR-bright sources by a factor of
$\sim$1.2.
\item[$-$]Most of the galaxies in our sample present spectral energy
distributions which are consistent with an elliptical template in the
UV/optical/NIR spectral range. Some galaxies morphologically
classified as spheroids have UV emission tails which are typical of
star-forming systems, most commonly at z$>$1.
\item[$-$]A $\sim$10\% fraction of the massive galaxies in our sample present X-ray or
power-law-like mid-IR emission which must be linked to the presence of
a bright (unobscured or obscured) AGN.
\item[$-$]Based on the measured specific SFRs, we estimate that spheroid-like
galaxies have doubled (at the most, depending on the burst durations)
their stellar mass due to newly-born stars alone from z$\sim$2 to
z$=$0.2. Most of these mass increase (60\%) occur at z$\gtrsim$1,
where specific SFRs are as high as 0.4~Gyr$^{-1}$.
\item[$-$]Disk-like galaxies have tripled (at the most) their stellar mass by
newly-formed stars at z$<$2, with a more steady growth rate as a
function of redshift.
\end{itemize}
\acknowledgments
We thank an anonymous referee for her/his very constructive
comments. We acknowledge support from the Spanish Programa Nacional de
Astronom\'{\i}a y Astrof\'{\i}sica under grants AYA 2006--02358 and
AYA 2006--15698--C02--02. This work is based in part on observations
made with the {\it Spitzer} Space Telescope, which is operated by the
Jet Propulsion Laboratory, Caltech under NASA contract 1407. PGP-G and
IT acknowledge support from the Ram\'on y Cajal Program financed by
the Spanish Government and the European Union.
\bibliographystyle{apj}
|
2,877,628,089,492 | arxiv | \section{Introduction}
The spatial degrees of freedom offered by a MIMO system with a transmit and receive antenna array can be exploited in the presence of a rich scattering environment. However, in LOS MIMO channels with little or no scattering, the channel responses can become highly correlated, leading to a MIMO channel of rank 1. Nevertheless, with a proper placement of the antennas in the arrays \cite{Driessen99,Gesbert02,Haustein03}, the channel capacity and rank of the LOS MIMO channel can be maximized.
With ULAs at the transmitter and receiver, the best antenna placement is obtained by optimizing the separation between the antennas in the transmit and receive arrays. Although for ULAs there are multiple solutions \cite{Bohagen05,Sarris07} for the optimum antenna separation product, i.e. the product of the antenna separation at the transmit and receive array, the one corresponding to the \emph{smallest} antenna separation product is usually considered, as this leads to the smallest arrays \cite{Bohagen05}.
The previous cited works consider a fixed distance between the transmit and receive array. However, for many applications, a LOS MIMO channel needs to be designed for a \emph{range} of distances between the transmitter and receiver. Since the optimum antenna separation depends on the distance between the transmitter and receiver, there is a performance degradation when the distance is varied for a given antenna placement of the transmit and receive arrays. To reduce the sensitivity to distance variations between the transmitter and receiver, non-uniform linear arrays have been proposed \cite{Torkildson09,Zhou13}. The optimum antenna placement in such cases was found using an exhaustive search, with the
aim of maximizing the range where a minimum condition number or capacity can be guaranteed.
In this paper, we first revisit the derivation of the optimum antenna separation for LOS MIMO systems with ULAs at the transmitter and receiver. In contrast to prior work, we provide the \emph{general} expression for the optimum antenna separation product, which consists of multiple solutions. In addition, we propose to use the multiple solutions for the LOS MIMO design over a range of distances. In particular, we consider the LOS MIMO design for a V2V communication scenario over a range of distances between the transmit and receive vehicle. Although the optimum antenna placement can not be met at all distances, we exploit the fact that some antenna separations are optimum at \emph{several} distances. We show that \emph{larger} antenna separations can be beneficial in certain cases.
This paper is organized as follows. Section~\ref{Sec:ChMod} introduces the LOS MIMO channel model. The optimum antenna separation is derived in Section~\ref{Sec:OptAntSep}. The V2V scenario is described in Section~\ref{Sec:V2V}, where numerical results for the LOS MIMO design are presented. We conclude the paper with Section~\ref{Sec:Con}.
\section{LOS MIMO Channel Model}
\label{Sec:ChMod}
In this paper, we use lower case and capital boldface letters to denote vectors and matrices, respectively. In addition, $(\bullet)^{\operatorname{T}}$ and $(\bullet)^{\operatorname{H}}$ denote the transpose and conjugate transpose, respectively. The cardinality of the set $\mathcal{P}$ is denoted by $|\mathcal{P}|$.
We consider a MIMO channel with a pure LOS between a transmitter and a receiver consisting of a ULA with $N > 1$ and $M > 1$ antennas, respectively. The antenna separation at the \emph{transmit} (Tx) and \emph{receive} (Rx) ULA is $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$, respectively. The distance between the first antenna of the Tx ULA, placed at the origin, and the first antenna of the Rx ULA is given by $R$ as shown in Fig.~\ref{fig:LOSMIMO}. With the Tx array placed on the $xz$-plane, we assume an arbitrary orientation of the arrays given by the angles $\theta_{\text{\tiny Tx}}$, $\theta_{\text{\tiny Rx}}$ and $\phi_{\text{\tiny Rx}}$ as shown in Fig.~\ref{fig:LOSMIMO}, where $0 \le \theta_{\text{\tiny Tx}} \le \frac{\pi}{2}$ and $0 \le \theta_{\text{\tiny Rx}} \le \frac{\pi}{2}$. The carrier frequency and wavelength of the signal are given by $f_{\text{c}}$ and $\lambda$, respectively.
\begin{figure}[!ht]
\begin{center}
%
\begin{tikzpicture}[x=0.5cm,y=0.5cm,z=0.3cm,>=stealth]
\draw[->] (xyz cs:x=0) -- (xyz cs:x=12) node[above] {$x$};
\draw[->] (xyz cs:y=0) -- (xyz cs:y=5) node[right] {$z$};
\draw[->] (xyz cs:z=0) -- (xyz cs:z=-3.5) node[above] {$y$};
\node[fill,circle,inner sep=2.5pt] at (0,0) (Tx1) {};
\node[fill,circle,inner sep=2.5pt] at (-0.5,1) (Tx2) {};
\node[fill,circle,inner sep=2.5pt] at (-1.5,3) (TxN) {};
\draw[-] (Tx1) node[below] {\hspace{8ex} { $(0,0,0)$}} -- (Tx2)
node[below] {\hspace{-3ex} $d_{\text{\tiny Tx}}$};
\draw[dashdotted] (Tx2) -- (TxN) {};
\draw (0,2) arc (90:115:2) node[above] {\hspace{1ex} $\theta_{\text{\tiny Tx}}$};
\node at (-0.2,4) () {\small Tx ULA with $N$ antennas};
\draw[->] (xyz cs:x=8,y=0) -- (xyz cs:x=8,y=5) node[right] {$z^{\prime}$};
\draw[->] (xyz cs:x=8,z=0) -- (xyz cs:x=8,z=-3.5) node[above] {$y^{\prime}$};
\node[fill,circle,inner sep=2.5pt] at (8,0) (Rx1) {};
\node[fill,circle,inner sep=2.5pt] at (8.6,0.8) (Rx2) {};
\node[fill,circle,inner sep=2.5pt] at (9.8,2.4) (RxN) {};
\draw[-] (Rx1) -- (Rx2) node[left] { \hspace{-7ex} $d_{\text{\tiny Rx}}$};
\draw[dashdotted] (Rx2) -- (RxN) {};
\node at (9.1,3.4) () {\small Rx ULA with $M$ antennas };
%
\draw[dashed] (RxN) -- (xyz cs:x=9.8,y=-1.4) ;
\draw[dashed] (Rx1) node[below] {\hspace{-10ex} { $(R,0,0)$}} -- (xyz cs:x=9.8,y=-1.4) ;
\draw[dashed] (xyz cs:x=8,z=-2.3) -- (xyz cs:x=9.8,y=-1.4) ;
\draw[dashed] (xyz cs:x=11) -- (xyz cs:x=9.8,y=-1.4) ;
%
\draw (8,2) arc (90:53:2) node[midway, above] {$\theta_{\text{\tiny Rx}}$};
\draw (9,0) arc (0:-37:1) node[midway,right] {\hspace{-1ex} $\phi_{\text{\tiny Rx}}$};
%
\end{tikzpicture}
\end{center}
\vspace{-1.5ex}
\caption{LOS MIMO channel with a transmit and receive ULA}
\label{fig:LOSMIMO}
\end{figure}
The \emph{normalized} channel matrix for the LOS MIMO system is denoted as
\begin{align}
\m{H} = \left[ \begin{array}{cccc}
\m{h}_{1} & \m{h}_{2} & \cdots & \m{h}_{N}
\end{array} \right] \in \mathbb{C}^{M \times N},
\label{def_H}
\end{align}
where the $n$-th column of $\m{H}$, i.e. $\m{h}_{n}$, corresponds to the channel vector from the $n$-th antenna at the Tx array to the $M$ antennas at the Rx antenna array. With the path loss included in the receive SNR, the \emph{normalized} channel vector $\m{h}_{n} \in \mathbb{C}^{M}$ is determined with ray tracing, i.e. with the spherical wave model instead of the planar wave assumption, and is given as:
\begin{align}
\m{h}_{n} = \left[ \begin{array}{ccc}
\text{exp}\left( \operatorname{j} 2\pi \frac{r_{1,n}}{\lambda}\right), & \cdots, & \text{exp}\left( \operatorname{j} 2\pi \frac{r_{M,n}}{\lambda}\right)
\end{array} \right]^{\operatorname{T}}.
\label{def_hn}
\end{align}
where
$r_{m,n}$ corresponds to the path length between the $n$-th Tx antenna and the $m$-th Rx antenna, for $n=1,\ldots,N$ and $m=1,\ldots,M$, respectively. The path length $r_{m,n}$ can be obtained from the coordinates $(x^{\text{\tiny Tx}}_n,y^{\text{\tiny Tx}}_n,z^{\text{\tiny Tx}}_n)$ of the $n$-th Tx antenna and the coordinates $(x^{\text{\tiny Rx}}_m,y^{\text{\tiny Rx}}_m,z^{\text{\tiny Rx}}_m)$ of the $m$-th Rx antenna, which from Fig.~\ref{fig:LOSMIMO} are given by
\begin{align*}
\begin{array}{ll}
\text{$n$-th Tx ant.}\!: & x^{\text{\tiny Tx}}_n =-(n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}, \\
& y^{\text{\tiny Tx}}_n = 0, \quad \quad z^{\text{\tiny Rx}}_m = (n-1) d_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Tx}} \\
\text{$m$-th Rx ant.}\!: & x^{\text{\tiny Rx}}_m = R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}}, \\
&\hspace{-12ex} y^{\text{\tiny Rx}}_m \!=\! (m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \sin \phi_{\text{\tiny Rx}},
\quad\, z^{\text{\tiny Rx}}_m \!=\!(m\!-\!1) d_{\text{\tiny Rx}} \cos \theta_{\text{\tiny Rx}}.
\end{array}
\end{align*}
With the above coordinates, $r_{m,n}$ can be determined as follows
\begin{align}
r_{m,n} &\boldsymbol{=} \left(\left(x^{\text{\tiny Rx}}_m-x^{\text{\tiny Tx}}_n\right)^{2} + \left(y^{\text{\tiny Rx}}_m-y^{\text{\tiny Tx}}_n\right)^{2} + \left(z^{\text{\tiny Rx}}_m-z^{\text{\tiny Tx}}_n\right)^{2} \right)^{\frac{1}{2}} \nonumber \\
&\hspace{-5ex} \boldsymbol{=}\! \! \left(\!\left(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \! \cos \phi_{\text{\tiny Rx}}\!\! +\! (n\!-\!1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} \!\right)^{2} \!\!\! +\! \big(\!(m\!-\!1) \times \right. \nonumber \\
& \hspace{-5ex} \left. d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \sin \phi_{\text{\tiny Rx}}\big)^{2} \!\! + \!
\left((m\!-\!1) d_{\text{\tiny Rx}} \! \cos \theta_{\text{\tiny Rx}} \! - \!(n\!-\!1) d_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Tx}} \right)^{2} \right)^{\!\frac{1}{2}} \nonumber \\
&\hspace{-5ex} \boldsymbol{=} \Big(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} \Big) \Big( 1 + \nonumber \\
& \hspace{-6ex} \frac{\left( \! (m\!-\!1) d_{\text{\tiny Rx}} \sin \! \theta_{\text{\tiny Rx}} \sin \!\phi_{\text{\tiny Rx}}\right)^{2} \!\!\! + \!
\left(\!(m\!-\!1) d_{\text{\tiny Rx}} \cos\! \theta_{\text{\tiny Rx}}\! \! - \!(n\!-\!1) d_{\text{\tiny Tx}} \!\cos \! \theta_{\text{\tiny Tx}}\!\right)^{\!2}}{\left(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}\right)^{2}} \!\bigg)^{\!\!\!\frac{1}{2}} \nonumber \\
& \hspace{-5ex}\boldsymbol{\approx} R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} + \nonumber \\
& \hspace{-6ex} \frac{\left(\! (m\!-\!1) d_{\text{\tiny Rx}}\! \sin \theta_{\text{\tiny Rx}} \! \sin \phi_{\text{\tiny Rx}}\right)^{2} \!\!\! + \!
\left(\!(m\!-\!1) d_{\text{\tiny Rx}} \!\cos \theta_{\text{\tiny Rx}} \! - \!(n\!-\!1) d_{\text{\tiny Tx}}\! \cos \theta_{\text{\tiny Tx}}\right)^{2}}{2 R}, \label{def_rmn}
\end{align}
where the last step results from the first order approximation of the Taylor series of $\sqrt{1+a}$ with $a \ll 1$, i.e. $\sqrt{1+a}\approx 1 + \frac{a}{2}$, and from $R \approx R+ (m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}$ in the denominator of the argument of the square root, where both approximations hold if the distance $R$ between the transmitter and receiver is much larger than Tx and Rx array dimensions.
\section{Optimum Antenna Separation}
\label{Sec:OptAntSep}
Consider the case when $N \le M$, such that\footnote{The case $N > M$ can be derived in a similar manner by simply interchanging the tranmsitter and the receiver.} $\text{rank}\left(\m{H}\right) \le N$. As discussed in \cite{Bohagen05}, the capacity of the LOS MIMO system at high SNR is maximized if $\m{H}^{\text{H}}\m{H} = M \m{1}_{N}$, i.e. if the columns of $\m{H}$ are orthogonal. For this case, $\m{H}$ achieves the maximum rank of $N$ and the $N$ eigenvalues of $\m{H}^{\text{H}}\m{H}$ are all equal to $M$, as $\text{tr}\left(\m{H}^{\text{H}}\m{H}\right)=MN$.
\subsection{Solution of the Orthogonality Condition}
In order to design the channel matrix $\m{H}$ of the LOS MIMO system to have orthogonal columns, from (\ref{def_H}) we need to have
\begin{align}
\m{h}_{k}^{\operatorname{H}} \m{h}_{l} = 0, \quad \quad \text{for} \quad k \ne l; \quad k,l=1,\cdots,N.
\label{orth_cond}
\end{align}
Using (\ref{def_hn}), we can write
\begin{align}
\m{h}_{k}^{\operatorname{H}} \m{h}_{l}
&= \sum_{m=1}^{M} \text{exp}\left(\operatorname{j} 2\pi \frac{r_{m,l}-r_{m,k}}{\lambda} \right) \nonumber \\
&\!\!\!\!\overset{\text{(a)}}{\approx} \! \sum_{m=1}^{M} \! \text{exp}\left(\! \operatorname{j} 2\pi \! \left(\!\frac{\gamma}{\lambda}\! -\!\frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (l\!-\!k) (m\!-\!1) \!\right) \!\right) \nonumber \\
&\!\!\!\!\overset{\text{(b)}}{=} \Gamma \cdot \sum_{m^{\prime}=0}^{M-1} \!\! \text{exp}\left(\operatorname{j} 2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) m^{\prime} \right) \nonumber \\
&\!\!\!\!\overset{\text{(c)}}{=} \Gamma \cdot
\frac{1-\text{exp}\left(\operatorname{j} 2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} M (k-l) \right)}{1-\text{exp}\left(\operatorname{j} 2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) \right)},
\label{inner_prod}
\end{align}
where step (a) results from
\begin{align}
r_{m,l}-r_{m,k} \approx &
\, \gamma - \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos \theta_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Rx}}}{R} (l\!-\!k) (m\!-\!1),
\end{align}
which follows from using the approximation (\ref{def_rmn}) for $r_{m,n}$, and where
$\gamma \!=\! (l\!-\!k) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} - \frac{ \left((l-1)^{2}\! - (k-1)^{2} \right)d^{2}_{\text{\tiny Tx}} \cos^{2} \theta_{\text{\tiny Tx}}}{2 R}$. For step (b), we use the substitutions $m^{\prime}=m\!-\!1$ and $\Gamma\! \!= \!\text{exp}\left(\operatorname{j} 2\pi \! \frac{\gamma}{\lambda}\right)$, with $\Gamma$ being independent of $m^{\prime}$. For step (c), we employ the expression for the finite sum of a geometric series for $w\ne 1$:
\begin{align}
\sum_{m^{\prime}=0}^{M-1} w^{m^{\prime}} = \frac{1 - w^M}{1-w} ,
\end{align}
with $w = \text{exp}\left(\operatorname{j} 2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) \right) $.
Given that $\m{h}_{k}^{\operatorname{H}} \m{h}_{l}$ depends on $(k-l)$, as observed from (\ref{inner_prod}), and that $|\m{h}_{k}^{\operatorname{H}} \m{h}_{l}|=|\m{h}_{l}^{\operatorname{H}} \m{h}_{k}|$, the conditions given in (\ref{orth_cond}) required to have orthogonal columns of $\m{H}$ are equivalent to
\begin{align}
\m{h}_{k}^{\operatorname{H}} \m{h}_{l} = 0, \quad \quad \text{for} \quad (k -l)=1,\cdots,N-1.
\label{orth_cond_0}
\end{align}
From (\ref{inner_prod}) and as $\Gamma \ne 0$, the {equivalent} orthogonality conditions in \eqref{orth_cond_0} are fulfilled\footnote{Due to the approximation (\ref{def_rmn}) for $r_{m,n}$, (\ref{orth_cond_0}) can only be fulfilled approximately with (\ref{ortho_cond_1}). As the error introduced with (\ref{def_rmn}) is negligible for practical systems \cite{Sarris07}, we assume in the following that (\ref{ortho_cond_1}) can be met with equality.} if
\begin{align}
\frac{1-\text{e}^{\operatorname{j} 2\pi \delta M q}}{1-\text{e}^{\operatorname{j} 2\pi \delta q }} =0, \quad \forall \, q\in\{1, 2,\! \cdots\!, N\!-\!1\},
\label{ortho_cond_1}
\end{align}
where we introduce $q=k-l$ and define
\begin{align}
\delta \overset{\Delta}{=}
\frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R}.
\label{delta}
\end{align}
Solving \eqref{ortho_cond_1} with respect to $\delta$, allows us to determine the optimum antenna separations $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$ of the Tx and Rx ULAs, which lead to a channel matrix $\m{H}$ that maximizes the capacity of the LOS MIMO system.
To satisfy (\ref{ortho_cond_1}), the numerator of the expression in (\ref{ortho_cond_1}) needs to be zero while the denominator is non-zero, i.e.
\begin{align}
\text{e}^{\operatorname{j} 2\pi \delta M q}=1, \quad \forall \, q \in \{1, 2,\! \cdots\!, N\!-\!1\},
\label{num_zero_0}
\end{align}
while
\begin{align}
\text{e}^{\operatorname{j} 2\pi \delta q} \ne 1, \quad \forall \, q \in\{1, 2,\! \cdots\!, N\!-\!1\}.
\label{den_zero_0}
\end{align}
As the solution of (\ref{num_zero_0}) for $q=1$ is also a solution of (\ref{num_zero_0}) for $q=2,\cdots,N-1$, the solution of (\ref{num_zero_0}) for all $q$ results from $\text{e}^{\operatorname{j} 2\pi \delta M }=1$, i.e. the solution of (\ref{num_zero_0}) is
\begin{align}
\delta = \frac{p}{M}, \quad \forall \,\, p \in \mathbb{Z}_{+}, \label{num_zero}
\end{align}
where $\mathbb{Z}_{+}$ represents the set of positive integers. The set of negative integers is excluded from the solution since all the terms in $\delta$ are positive, as can be seen in \eqref{delta}.
On the other hand, to avoid the denominator of the expression in \eqref{ortho_cond_1} being zero for any value of $q$, from \eqref{den_zero_0} we get
\begin{align}
&\delta \ne \frac{p_1}{q}, \quad \, \, \forall \, p_1 \in \mathbb{Z}_{+}, \,\, q\in\{1, 2,\! \cdots\!, N\!-\!1\}.
\label{den_zero}
\end{align}
Thus, given \eqref{num_zero} and \eqref{den_zero}, we have that (\ref{ortho_cond_1}) is fulfilled if $\delta = \frac{p}{M}$ for $p \in \mathbb{Z}_{+}$ but excluding the integers $p$ for which $\frac{p}{M} = \frac{p_1}{q}$ for $q= 1, 2,\cdots, N\!-\!1$, i.e. when
\begin{align}
\delta = \frac{p}{M}, \quad \forall \, p \in \mathbb{Z}_{+} \setminus \left\{p^{\prime}: p^{\prime} \!=\! \frac{p_1 M}{q}, \!\!\! \begin{array}{c} p_1 \in \mathbb{Z}_{+} , p^{\prime} \in \mathbb{Z}_{+}, \\
q\in\{1, 2,\! \cdots\!, N\!-\!1\}
\end{array}
\!\!\!\! \right\} \!\! . \label{ortho_sol_0}
\end{align}
Writing $q$ as the product of \emph{any} two (positive integer) factors, i.e. $q=q_1 q_2$, $p^{\prime}=\frac{p_1 M}{q}$ is an integer if $\frac{p_1}{q_1}$ and $\frac{M}{q_2}$ are both integers. As there is always a $p_1\in \mathbb{Z}_{+}$ such that $\frac{p_1}{q_1} \in \mathbb{Z}_{+}$, we only need to consider when $\frac{M}{q_2}$ is an integer for any factor $q_2$ of $q$. Given that $q_2 \le q \le N-1$, $\forall q$, we have that $p^{\prime}=\frac{p_1 M}{q}$ is an integer if $\frac{M}{q}$ is an integer for $q=1, \ldots, N-1$.
The possible values, in ascending order, of $\frac{M}{q}$ for $q=1, \ldots, N-1$, are $\frac{M}{N-1}$, $\frac{M}{N-2}$, \ldots $\frac{M}{2}$, $M$, out of which those that are integers (recall that $N \le M$), correspond to the \emph{divisors} of $M$ which are larger than or equal to $\frac{M}{N-1}$. Let us denote the set of divisors of $M$ which satisfy this condition as $\mathcal{D}_M(N)$, i.e.
\begin{align}
\mathcal{D}_{M}(N)= \left\{\nu: \nu \,\, | \, \, M, \,\,\nu \ge \frac{M}{N-1} \right\} ,
\label{div_set}
\end{align}
where $a \,\, | \, \, b$ means that $a$ is a divisor of $b$. Given \eqref{div_set}, we can rewrite the solution \eqref{ortho_sol_0} for the orthogonality conditions as
\begin{align}
\delta = \frac{p}{M}, \quad \forall \, p \in \mathbb{Z}_{+} \setminus \, \left\{
p^{\prime} \, \nu, \, \, p^{\prime} \in \mathbb{Z}_{+}, \, \nu \in \mathcal{D}_M(N) \right\},
\label{ortho_sol}
\end{align}
i.e. \eqref{ortho_cond_1} is fulfilled if $\delta = \frac{p}{M}$ for the set of positive integers $p$ \emph{excluding} the multiples of divisors of $M$ which are larger than or equal to $\frac{M}{N-1}$.
Prior solutions of (\ref{ortho_cond_1}) provided in the literature, e.g. as in \cite{Sarris07}, include only a subset of the possible integers $p$ given in (\ref{ortho_sol}). In addition, in contrast to prior work, our derived expression (\ref{ortho_sol}) shows the dependency on $N$, which corresponds to the number of Tx antennas and the $\text{rank}(\m{H})$. We discuss this dependency with two examples: $N\!=\!2$ and $N\!=\!M$. For $N\!=\!2$, $\frac{M}{N-1}=M$ such that from \eqref{div_set}, $\mathcal{D}_M(2)=\{M\}$. On the other hand, for $N\!=\!M$, $\frac{M}{N-1}=1+\frac{1}{M-1}$ such that $\mathcal{D}_M(M) = \left\{\nu: \nu \,\, | \, \, M, \,\,\nu > 1 \right\}$, i.e. $\mathcal{D}_M(M)$ consists of all the divisors\footnote{If $M$ is prime, $\mathcal{D}_M(M)=\{M\}$ and hence, (\ref{ortho_sol}) is independent of $N$.} of $M$ \emph{except} $1$. As $|\mathcal{D}_M(M)| \ge |\mathcal{D}_M(2)|$, we see that in general a larger set of positive integers $p$ are excluded in (\ref{ortho_sol}) when $N=M>2$ compared to when $N=2$. This is a consequence of the fact that the orthogonality conditions in (\ref{ortho_cond_1}) becomes more stringent with increasing $N$: for $N=2$, only two channel vectors need to be orthogonal, whereas for $N=M$, $M$ orthogonal channel vectors need to be designed.
\subsection{Design of LOS MIMO Systems}
Using \eqref{delta}, we rewrite (\ref{ortho_sol}) in terms of the \emph{antenna separation product} (ASP) \cite{Bohagen05}, i.e. in terms of the product of the antenna separation at the transmitter and receiver
\begin{align}
&\quad d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} = p \cdot \frac{\lambda R}{M \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}, \label{ortho_sol_asp}
\\ &\forall \, p \in \mathbb{Z}_{+} \setminus \, \left\{
p^{\prime} \, \nu, \, \, p^{\prime} \in \mathbb{Z}_{+}, \, \nu \in \mathcal{D}_M(N) \right\},
\nonumber
\end{align}
for $N \le M$. For $N > M$, the optimum solution for the ASP results from exchanging $N$ with $M$ in the expression above.
By setting the antenna separations $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$ of the Tx and Rx ULAs according to (\ref{ortho_sol_asp}), the channel matrix $\m{H}$ of the LOS MIMO system can be designed to have orthogonal columns, for a given distance $R$ between the arrays and a given orientation of the arrays. Although multiple solutions for the ASP exist\footnote{Despite infinite solutions, not all solutions fulfill (\ref{orth_cond_0}) in practice. As $p\! \rightarrow \!\infty$, the length of the arrays increase such that the assumption that the distance between the arrays is much larger than the array dimensions becomes invalid.}, only the first solution of \eqref{ortho_sol_asp}, i.e. $p=1$, is usually considered in the literature as this leads to the smallest antenna separations and hence, to the smallest arrays \cite{Bohagen05,Sarris07}.
However, for certain applications, \emph{other} solutions for the ASP, i.e. $p>1$, might be of interest. Take for instance the LOS MIMO design over a \emph{range} of distances between the transmitter and receiver, which is relevant for many applications. As observed in (\ref{ortho_sol_asp}), the optimum antenna separations at the Tx and Rx arrays depends on the \emph{fixed} distance $R$ between the arrays. Thus, varying the distance between the transmitter and receiver with a given optimum antenna separation, leads to a capacity reduction, i.e. reduced $\text{rank}(\m{H})$ or non-equal eigenvalues of $\m{H}^{\text{H}}\m{H}$. To reduce the sensitivity to distance variations, non-uniform linear arrays have been proposed \cite{Torkildson09,Zhou13}, where the optimum antenna placement is found via an exhaustive search, in order to maximize the range for which a certain metric can be guaranteed. In this paper, we propose the use of ULAs for the LOS MIMO design over a set of distances between the transmitter and receiver, by exploiting the multiple solutions for the ASP given in (\ref{ortho_sol_asp}). In particular, we consider the LOS MIMO design for a V2V link as discussed next.
\section{LOS MIMO Design for V2V}
\label{Sec:V2V}
Due to the importance of V2V communication in future wireless networks, e.g. 5G, we consider the LOS MIMO design for a V2V link between two vehicles located in the same lane, where the front car (Tx car) is communicating with a rear car (Rx car) separated by a longitudinal distance $D$ as shown in Fig.~\ref{fig:V2V}. The Tx car is equipped in the \emph{rear} bumper with a Tx ULA consisting of $N$ antennas, whereas the Rx car is equipped in the \emph{front} bumper with a Rx ULA consisting of $M$ antennas. The \emph{maximum} length of the Tx and the Rx ULA is assumed to be $L_{\text{\tiny Tx}}$ and $L_{\text{\tiny Rx}}$, respectively. From Fig.~\ref{fig:V2V}, we can see that the Tx ULA and the Rx ULA are always parallel and hence, the orientation of both arrays are the same, i.e. $\theta_{\text{\tiny Tx}}=\theta_{\text{\tiny Rx}}$ (c.f. Fig.~\ref{fig:LOSMIMO}). We assume a pure LOS channel between the Tx ULA and the Rx ULA, as well as the same speed for the Tx and Rx car. We assume a carrier frequency of $f_{\text{c}}=28$ GHz ($\lambda \approx 10.7$ mm) and a normalized LOS channel as discussed in Section~\ref{Sec:ChMod}, with a fixed receive $\text{SNR}=$ 13 dB for the considered distances $D$, i.e. with perfect sidelink power control.
\begin{figure}[!ht]
\vspace{-1ex}
\begin{center}
\begin{tikzpicture}[x=0.5cm,y=0.5cm,>=stealth,scale=0.65, every node/.style={transform shape}]
\draw[-,dash pattern=on 8pt off 8pt,line width=1.5pt] (0,-2.2) -- (0,20.2) {};
\draw[-,line width=1.75pt] (5,-2.5) -- (5,20.5) {};
\draw[-,line width=1.75pt] (-5,-2.5) -- (-5,20.5) {};
\draw[<->,loosely dashed] (0,-2.7) -- (5,-2.7) node[align=center, midway,below=5pt] {{ Lane Width} \\ { 3.5 m}};
\draw[-,dashed] (-1.1,4.4) -- (7.5,4.4) {};
\draw[-,dashed] (-1.1,13.6) -- (6,13.6) {};
\draw[<->,loosely dashed] (-0.6,4.4) -- (-0.6,13.6) node[align=center,midway,left] {\large{Longitudinal \, Distance $D$} \\ {\large between \, cars \hspace{2ex}}};
\node[] (RxCar) at (2.9,1.1) {\Fcar{}};
\node[] (TxCar) at (2,16.9) {\Rcar{}};
\draw[-,dashed] (2.225,2) -- (0.875,16) {};
\draw[<->] (2,4.4) -- (1.1,13.6) node[midway,right] {\large $R$};
\draw[-,dashed] (0,13.5) -- (6,14.0784) {};
\draw[-,dashed] (1,4.3) -- (7.5,4.8784) {};
\draw (7,4.4) arc (0:5:5) node[midway,xshift=9pt] {\hspace{3ex} \large $\theta_{\text{\tiny Rx}}$};
\draw (5.5,13.6) arc (0:5:5) node[midway,xshift=9pt] {\hspace{3ex} \large $\theta_{\text{\tiny Tx}}$};
\draw[->] (-6.5,1.5) node[left,align=center] { {\large Rx Car with} \\ {\large Rx ULA in} \\ {\large front bumper} } -- (2.9,0.5);
\draw[->] (-6.5,17) node[left,align=center] { {\large Tx Car with} \\ {\large Tx ULA in} \\ {\large rear bumper} } -- (2,16) ;
\draw[decoration={brace,raise=5pt},decorate] (1.85,4.5) -- node[above=10pt] {\large \hspace{2ex} $\le L_{\text{\tiny Rx}}$} (3.9,4.5);
\draw[decoration={brace,mirror,raise=5pt},decorate] (.95,13.5) -- node[below=10pt] {\large \hspace{2ex} $\le L_{\text{\tiny Tx}}$} (3.05,13.5);
\draw[->] (6,11) node[right,align=center] { {\large Tx ULA with} \\ {\large $N$ antennas} } -- (3.05,13.5);
\draw[->] (6,7) node[right,align=center] { {\large Rx ULA with} \\ {\large $M$ antennas} } -- (3.9,4.5);
\end{tikzpicture}
\end{center}
\vspace{-3ex}
\caption{V2V Scenario with two vehicles within a lane}
\label{fig:V2V}
\end{figure}
We consider the LOS MIMO design over a range of distances $D$ between the two cars with $10 \le D \le 100$. Due to lack of space, we do not consider the horizontal displacement of the two cars within the lane (of width equal to $3.5$ m), which leads to slightly different orientation angles of the arrays. We assume the cars are facing each other, such that $\theta_{\text{\tiny Tx}}=\theta_{\text{\tiny Rx}}=0$ and $R=D$. Furthermore, we assume the same number of antennas in the Tx and Rx array and set it to $3$, i.e. $N=M=3$, as well as the same antenna separation $d$ at both the Tx and Rx array, i.e. $d=d_{\text{\tiny Tx}}^{}=d_{\text{\tiny Rx}}^{}$. The maximum length of the arrays is assumed to be equal and set to $1.8$ m, in order to fit in the bumpers of a standard car, i.e. $L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$ m. Note that for the considered distances, $D \gg L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$.
From (\ref{ortho_sol_asp}) with $d_{\text{\tiny Tx}}^{}=d_{\text{\tiny Rx}}^{}=d$, $R=D$, and $N=M=3$, the optimum antenna separation for both arrays is given by
\begin{align}
&d_{\text{}}^{} = \sqrt{ p \cdot \frac{\lambda D}{M}} \quad \quad \text{for} \quad p \in \left\{1,2,4,5,7,8,\cdots \right\}, \label{ortho_sol_ex}
\end{align}
where only the multiples of $M=3$ are excluded from the set of positive integers for the possible values of $p$ in \eqref{ortho_sol_asp}. To observe the multiple solutions for the optimum antenna separation $d$ which maximize the capacity, i.e. which result in an orthogonal LOS MIMO channel with $3$ equally strong eigenmodes, we plot $d$ given in (\ref{ortho_sol_ex}) as a function of the longitudinal distance $D$ between the cars for the first eight values of $p$. As mentioned before, only the solution corresponding to $p=1$ is usually considered in the literature, as this corresponds to the smallest optimum antenna separation which then results in the shortest Tx and Rx arrays.
\begin{figure}[!th]
\begin{center}
\begin{tikzpicture
\begin{axis}[ylabel= Opt. Ant. Separation $d$ (meters),
xlabel=Distance $D$ between the cars (meters),
grid,
xmin=10,
xmax=100,
ymin=0,
ymax=0.9,
xtick={10,20,...,100},
legend pos= south east,
]
,
\foreach \F/\p/\c/\M in {1/1/myblue/o,2/2/myorange/square,3/4/myyellow/+,4/5/mypurple/diamond,5/7/mygreen/triangle,6/8/myred/asterisk,7/10/mylightblue/pentagon,8/11/magenta/x}
{
\edef\temp{\noexpand\addplot[color=\c,line width=1pt, mark=\M,] table[x index=0, y index=\F] {Ant_Spacing_vs_Distance_R_M3.txt};}
\temp
\edef\legendentry{\noexpand\addlegendentry{$p =\noexpand\pgfmathprintnumber[fixed]{\p}$}};
\legendentry
}
\draw[thick, black, dashed] (axis cs: 0,0.5976) -- (axis cs: 100,0.5976);
\node at (axis cs:70,0.6) [anchor=south west] {\small $d=0.5976$};
\end{axis}
\end{tikzpicture}
\end{center}
\vspace{-3ex}
\caption{Optimum Antenna Separations for the V2V link}
\label{fig:res1}
\end{figure}
However, the curves for $p>1$ result in larger antenna separations which also maximize the channel capacity. For a given distance $D$ between the arrays, the optimum antenna separation increases with $\sqrt{p}$ as can be shown in (\ref{ortho_sol_ex}). This results in an increasing length of the arrays with $p$, given by $(M-1) \sqrt{ p \cdot \frac{\lambda D}{M}}$. Due to the maximum length of the Tx and Rx arrays (car bumpers) in our V2V scenario given by $L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$ m, we consider only those solutions for $d$ which are less than or equal to $\frac{1.8}{M-1}$, i.e. with $M=3$, we consider only the optimum antenna separations which fulfill
\begin{align}
d \le 0.9 .
\end{align}
With this constraint, we observe from Fig.~\ref{fig:res1} there are at least two possible antenna separations which guarantee a $3 \times 3$ orthogonal LOS MIMO channel for each distance $D$ in the considered range of distances up to $100$ m.
More interestingly we observe in Fig.~\ref{fig:res1} that some antenna separations are optimum at \emph{several} distances! For example, $d=0.5976$ is an optimum antenna separation at $D~=~10,\, 12.5,\, 14.2857,\, 20,\, 25,\, 50$ and $100$ m, which can be obtained from (\ref{ortho_sol_ex}). At these distances, the LOS MIMO channel matrix with $d=0.5976$ is orthogonal with three equally strong eigenmodes as shown in Fig.~\ref{fig:res2}, where the eigenvalues of $\m{H}(D)\m{H}^{\text{H}}(D)$ are depicted for the considered range of distances between the cars. As $\text{tr}\left(\m{H}(D)\m{H}^{\text{H}}(D)\right)=MN=9$, the capacity with $\m{H}(D)$ is maximized when the three eigenvalues of $\m{H}(D)\m{H}^{\text{H}}(D)$ are equal to $3$. The channel matrix $\m{H}(D) \in \mathbb{C}^{3 \times 3}$ corresponds to a LOS MIMO system given by (\ref{def_H}), (\ref{def_hn}) and (\ref{def_rmn}) with a Tx and Rx ULA consisting of $3$ antennas with an antenna separation of $0.5976$ and a distance $D$ between the Tx and Rx arrays. $\m{H}(D)$ is given as a function of $D$ to highlight its dependency on the distance $R=D$ between the arrays via (\ref{def_rmn}). From Fig.~\ref{fig:res2}, we also see that at certain distances some eigenvalues go to zero and hence, the LOS channel $\m{H}(D)$ becomes rank deficient, e.g. at $D=34$ and $D=68$ the channel rank is $1$ and $2$, respectively.
\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture
\begin{axis}[ylabel= Eigenvalues of $\, \m{H}(D)\m{H}^{\text{H}}(D)$,
xlabel=Distance $D$ between the cars (meters),
grid,
xmin=10,
xmax=100,
ymin=0,
ymax=10,
ytick={0,3,6,9},
xtick={10,20,...,100},
legend pos= north east,
]
,
\addplot[color=mygreen,line width=1pt,dashdotted] table[x index=0, y index=1] {Eigs_vs_Distance_R_M3sol_2_D50.txt};
\addplot[color=mylightblue,line width=1pt] table[x index=0, y index=2] {Eigs_vs_Distance_R_M3sol_2_D50.txt};
\addplot[color=myred,line width=1pt,dashed] table[x index=0, y index=3] {Eigs_vs_Distance_R_M3sol_2_D50.txt};
\node at (axis cs:50,7) [anchor=south west,align=left] {\footnotesize{$3$ eigenvalues for} \\ \footnotesize{the case $d=0.5976$}};
\end{axis}
\end{tikzpicture}
\end{center}
\vspace{-3ex}
\caption{Eigenvalues of $\, \m{H}(D)\m{H}^{\text{H}}(D)$ with $d=0.5976$}
\label{fig:res2}
\end{figure}
To elaborate further on the performance over the considered range of distances, we depict in Fig.~\ref{fig:res3} the capacity of the LOS MIMO channel for the described V2V link for three different antenna separations
$d=0.5,\,0.5976,\,0.7$ for the Tx and Rx ULAs. In this case, the maximum capacity with an SNR of $13$ dB is $13.18$ bps/Hz, whereas the capacity is $10.72$ and $7.50$ bps/Hz when one or two eigenmodes go to zero, respectively. As can be seen in Fig.~\ref{fig:res3}, the maximum capacity with $d=0.5976$ is achieved for the set of distances mentioned previously. For $d=0.5$ and $d=0.7$, the maximum capacity is achieved at other sets of distances. In fact, we observe a \emph{stretching} and \emph{shift} to the right of the capacity curve as the antenna separation $d$ increases, which can be explained as follows. Given that (\ref{ortho_sol_ex}) can be rewritten as $\frac{d^{2}}{D}=p\frac{\lambda}{M}$, we can find other pairs of antenna separation $d^{\prime}$ and distance $D^{\prime}$ which achieve the same value $p \cdot\frac{\lambda}{M}$, i.e.
\begin{align}
\frac{d^{\prime,2}}{D^{\prime}} = \frac{d^{2}}{D}, \quad \quad \text{such that} \quad \quad
D^{\prime} = D \, \frac{d^{\prime,2}}{d^{2}}.
\label{Ddexpression}
\end{align}
For instance, from Fig.~\ref{fig:res1} the optimum antenna separation for $p=2$ at $D=50$ is $d=0.5976$. From (\ref{Ddexpression}), the distance $D^{\prime}$ which achieves the same value $p \cdot\frac{\lambda}{M}$ as the previous setting but with $d^{\prime}=0.7$ is given by $D^{\prime}=50\cdot\frac{0.7^2}{0.5976^2}=68.8$ m. Thus, in Fig.~\ref{fig:res3} the point on the capacity curve for $d=0.5976$ at $D=50$ is shifted to the right by a factor of $\frac{0.7^2}{0.5976^2} \approx 1.37$ when the antenna separation $d=0.7$ is employed. The stretching of the capacity curve can also be explained in a similar manner.
\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture
\begin{axis}[ylabel= Capacity (bps/Hz),
xlabel=Distance $D$ between the cars (meters),
grid,
xmin=10,
xmax=100,
ymin=5,
ymax=15,
ytick={1,2,...,15},
xtick={10,20,...,100},
legend pos= south east,
]
,
\addplot[color=mygreen, densely dotted, thick, line width=1pt] table[x index=0, y index=2] {Cap_vs_Distance_R_M3d05.txt};
\addlegendentry{Max. Capacity}
\addplot[color=myblue,line width=1pt,smooth,] table[x index=0, y index=1] {Cap_vs_Distance_R_M3d05.txt};
\addlegendentry{$d=0.5$}
\addplot[color=myred,line width=1pt,smooth, dashed] table[x index=0, y index=1] {Cap_vs_Distance_R_M3sol_2_D50.txt};
\addlegendentry{$d=0.5976$}
\addplot[color=myyellow,line width=1pt,smooth,] table[x index=0, y index=1] {Cap_vs_Distance_R_M3d07.txt};
\addlegendentry{$d=0.7$}
\draw[->] (axis cs: 35,6) node[right,,align=center] {{\footnotesize $2$ eigenvalues} \\ {\footnotesize go to zero}} -- (axis cs: 34,7.39) ;
\draw[->] (axis cs: 70,9.1) node[right,align=center] {{\footnotesize $1$ eigenvalue} \\ {\footnotesize goes to zero}} -- (axis cs: 68,10.6) ;
\node[draw,shape=circle,fill=myred,inner sep=1.5pt] at (axis cs:50,13.18) () {};
\node[draw,shape=circle,fill=myyellow,inner sep=1.5pt] at (axis cs:68.8,13.18) () {};
\draw[->,thick, black] (axis cs: 50,13.8) -- (axis cs: 68.8,13.8) node[above,midway,black] { \footnotesize{Shift by $1.37$}};
\end{axis}
\end{tikzpicture}
\end{center}
\vspace{-3ex}
\caption{Capacity of the LOS MIMO channel for different antenna separations}
\label{fig:res3}
\end{figure}
\section{Conclusion
\label{Sec:Con}
We have derived the general expression for the optimum antenna separation product for maximizing the capacity of a LOS MIMO channel with a Tx and Rx ULA. The expression leads to multiple solutions of the optimum antenna separation product which depend on the number of Tx and Rx antennas. We have proposed to exploit the multiple solutions for the LOS MIMO design over a range of distances between the transmitter and receiver, such as for V2V. We have shown that larger antenna separations can be beneficial and that some antenna separations are optimum at several distances. The provided results can serve as guidelines for the LOS MIMO design for V2V. Future work includes considering non-uniform linear arrays as well as the ground reflection in the V2V link.
\section*{Acknowledgment}
{The authors would like to acknowledge support of this work
under the 5GPPP European Project 5GCAR (grant agreement
number 761510).}
|
2,877,628,089,493 | arxiv | \section{Introduction}
\label{sec:intro}
The hot ($T\sim10^6$\,K), X-ray emitting plasma observed in the disk of our
Galaxy is an important factor when considering the energy balance and enrichment
of the interstellar medium (ISM). Energised by the winds of young massive stars and
supernovae, its properties relate to the supernova rate, the evolution of
supernova remnants (SNRs) and the impact of young star clusters on their environment.
In normal spiral galaxies like the Milky Way, hot gas produced by multiple
supernova explosions may escape from the disk via galactic fountains and/or chimneys
(\citealt{shapiro76}; \citealt{bregman80}; \citealt{norman89})
resulting in a dynamic corona that is bound to the galaxy. This gas then cools,
condenses and falls back to the disk, possibly as high-velocity clouds
\citep{bregman80}. In contrast, the extremely high star-formation rates in
starburst galaxies give rise to hotter ($T\ga10^7$\,K) large-scale metal-rich
outflows from the starburst regions in the form of ``superwinds''
({\it e.g.,} \citealt*{heckman90}). Such superwinds are energetic enough to fully escape
the influence of the galaxy and are potentially important for the enrichment
and heating of the intergalactic medium ({\it e.g.,} \citealt*{martin02}).
Unfortunately, the opacity of the galactic plane to soft X-rays greatly
complicates the study of the extent, physical properties and filling factor of
the X-ray emitting plasma in our own galaxy. However, observations of nearby
spiral galaxies with
configurations ranging from face-on to edge-on can circumvent many of the
problems inherent in studies of the Milky Way. The {\it Einstein~\/} observatory provided
the first detection of ultra-hot out-flows from the disks of edge-on starburst
galaxies (\citealt{watson84}), although only upper limits were obtained on
the diffuse emission produced in the disks of normal spiral galaxies
(\citealt{bregman82};
\citealt{mccammon84}). The improved throughput and spatial resolution
of subsequent {\it ROSAT~\/} PSPC/HRI observations revealed unambiguous evidence for
truly diffuse emission in spiral galaxies (\citealt{cui96};
\citealt{read97}). More recently, diffuse structures have been observed with the
{\it XMM-Newton~\/} and {\it Chandra~\/} observatories in samples of both edge-on
(\citealt{pietsch01}; \citealt*{stevens03}; \citealt{strickland04a};
\citealt{strickland04b}) and face-on spiral galaxies (\citealt{kuntz03};
\citealt{tyler04}). In the face-on systems, diffuse X-rays were found to
correlate with the nuclear regions, trace recent star formation within the
spiral arms and correlate spatially with H$\alpha$ and mid-infrared emission
\citep{tyler04}. In the survey of edge-on systems by Strickland et
al. (2004a,b), extra-planar diffuse emission was detected in all starburst
galaxies and one normal spiral galaxy, the extent of which correlated with
estimates of the rate of star formation within the disks.
\begin{table*}
\caption{Details of the three {\it XMM-Newton~\/} observations of M101.}
\centering
\begin{tabular}{lcccccccccc}
\hline
\# & Observation ID & Date & & Filter & & \multicolumn{2}{c}{Pointing co-ordinates} & & \multicolumn{2}{c}{Useful exposure (ks)} \\
& & (yyyy-mm-dd) & & & & RA (J2000) & Dec (J2000) & & pn & MOS \\
\hline
1 & 0104260101 & 2002-06-04 & & Medium & & $14^h03^m10.0^s$ & $+54\deg20\arcm24\hbox{$^{\prime\prime}$~\/}$ & & 25.7 & 36.7 \\
2 & 0164560701 & 2004-07-23 & & Medium & & $14^h03^m32.3^s$ & $+54\deg21\arcm03\hbox{$^{\prime\prime}$~\/}$ & & 26.2 & 28.9 \\
3 & 0212480201 & 2005-01-08 & & Thin & & $14^h03^m32.7^s$ & $+54\deg21\arcm02\hbox{$^{\prime\prime}$~\/}$ & & 27.7 & 25.8 \\
\hline
\end{tabular}
\label{table:obs}
\end{table*}
Clearly, if the purpose is to image the X-ray emission over a wide bandwidth,
one should preferentially select galaxies in
directions where the line-of-sight Galactic hydrogen column density is low. In
this context, the Scd supergiant spiral M101 (NGC~5457), for which
$N_H = 1.1\times10^{20} {\rm ~cm^{-2}}$ \citep{dickey90}, is an ideal candidate for a
face-on study. A diffuse X-ray component was first hinted at in the central
regions of M101 with {\it Einstein~\/} at energies in the range 0.2--1.5\,keV
(\citealt{mccammon84}; \citealt{trinchieri90}). Subsequent {\it ROSAT~\/} PSPC
observations \citep{snowden95} revealed extensive diffuse X-ray emission in the
0.1--1\,keV band, corresponding to thermal emission in the temperature
range 1--3$\times10^6$\,K. The
dominant soft component was spatially peaked towards the
centre of the galaxy but could also be traced out to a radius of at least
7\hbox{$^\prime$~\/} (15\,kpc at a distance of 7.2\,Mpc). It was argued that
this soft emission could not originate solely in the disk as this would require
a filling factor greater than unity. After correcting for absorption intrinsic
to M101, the total diffuse luminosity was estimated to be at least
$10^{40} {\rm ~erg~s^{-1}}$.
In the {\it ROSAT~\/} HRI study of \citet{wangetal99}, low-surface-brightness X-ray
emission was detected both in the central region of M101 and in the vicinity
of bright H{\small II}$~$ regions in the spiral arms of the galaxy. The improved
spatial resolution of the HRI
($\sim5$\hbox{$^{\prime\prime}$~\/}) allowed the detection and subtraction of point sources down
to a luminosity of $4\times10^{37} {\rm ~erg~s^{-1}}$, after which the total
0.5--2\,keV luminosity of the residual emission within a radius of 12\hbox{$^\prime$~\/}
was estimated to be $9\times10^{39} {\rm ~erg~s^{-1}}$. These authors agreed with
\citet{snowden95} that faint discrete X-ray sources were likely to account
for only a fraction of the observed flux, implying that the bulk of the soft
emission is truly diffuse in nature. The X-ray spectra
of the emission in the central region from both the {\it ROSAT~\/} PSPC and {\it ASCA~\/} GIS
were fitted simultaneously, yielding a soft thermal component with a temperature
of $kT\sim0.2$\,keV, plus a hard power-law continuum most plausibly explained
as the integrated emission of the population of hard X-ray binary sources (XRBs)
in M101.
Further progress was made in the {\it Chandra~\/} study of \citet{kuntz03}, using data
from the 8\hbox{$^\prime$~\/} square ACIS-S3 chip from the first $\sim$100\,ks
{\it Chandra~\/} observation of M101. The superb spatial resolution of {\it Chandra~\/}
allowed the removal of point sources down to luminosities of $\sim10^{36}
{\rm ~erg~s^{-1}}$, leaving residual emission of luminosity
2.3$\times10^{39} {\rm ~erg~s^{-1}}$ in the 0.45--1.0\,keV band, 16\% of which
was estimated to come from unresolved XRBs and dwarf stars. The bulk of the soft
emission was found to trace the spiral arms and was also spatially correlated
with H$\alpha$ and far ultra-violet (UV) emission, physically linking the soft
X-ray emission to regions of on-going star formation. The X-ray spectrum of the
diffuse emission was well-described by a two-temperature thermal plasma with
$kT$=0.2/0.75\,keV, with a large covering factor implying that a significant
fraction of the softest component originates in the halo of M101.
This is the third in a series of papers reporting the results of {\it XMM-Newton~\/} observations
of M101. In \citet{jenkins04a}~(hereafter Paper~I), the spectral and timing
properties of the brightest X-ray sources were investigated,
whereas in \citet{jenkins05b}~(hereafter Paper~II) we reported on the
full catalogues of X-ray source detected in M101 by {\it XMM-Newton}.
In the current
paper we focus on the morphology and spectral properties of the
``diffuse'' X-ray component in this galaxy.
The structure of this paper is as follows.
First we describe the {\it XMM-Newton~\/} observations and the methods used to
construct images and spectra of the residual emission after the exclusion of bright
discrete sources (section \ref{sec:obs}). Next we investigate the likely
composition, spectral properties and spatial morphology of this residual
emission (section \ref{sec:res}). We then go on to discuss the implications of our
results (section \ref{sec:disc}) and, finally, provide a brief summary of our conclusions
(section \ref{sec:conc}). Throughout this paper we assume a distance to
M101 of 7.2 Mpc (\citealt*{stetson98}), implying that an angular scale of 1\hbox{$^\prime$~\/}
corresponds to a linear extent of 2.09 kpc in M101.
\section{Observations \& Data Reduction}
\label{sec:obs}
Three {\it XMM-Newton~\/} EPIC observations have been made of M101 as summarised in
Table~\ref{table:obs}. Our earlier analysis of the point source population
in M101, presented in Papers~I and II of this series, utilised only the first
observation performed in 2002. However, for this paper, we have
taken advantage of two additional observations carried out by
{\it XMM-Newton~\/} in 2004/5 as a Target of Opportunity (TOO) programme to follow up the
outburst of the ultraluminous supersoft X-ray source J140332.3+542103,
(see \citealt*{kong04}; \citealt*{mukai05}). The first {\it XMM-Newton~\/} observation was
targeted at the nucleus of the galaxy with the result that the
30\hbox{$^\prime$~\/} diameter field-of-view of the EPIC
cameras encompassed the full 28.8\hbox{$^\prime$~\/} (D$_{25}$) extent of M101
(\citealt{devaucouleurs91}). In the second and third observations the source
J140332.3+542103 was placed on axis, resulting in a $3.3'$ eastward offset
of the field-of-view with respect to the first observation.
All datasets were screened for periods of high background by
accumulating full-field 10--15\,keV light curves. MOS data were
excluded during periods when the 10-15 keV count rate exceeded
0.3 $\rm ct~s^{-1}$, and the pn data were similarly screened
when the pn count rate exceeded 3 $\rm ct~s^{-1}$ (except in
observation~1, which was more heavily contaminated by
flaring, where the pn threshold was set at 0.9 $\rm ct~s^{-1}$).
For the subsequent image and spectral analysis, single and double pixel
events were selected for the pn (pattern 0--4), whereas single to quadruple
events (pattern 0--12) were utilised in the case of the MOS data.
\subsection{Image processing and point source subtraction}
To study the morphology of the diffuse component, pn and MOS images and associated
exposure maps were created for each observation in three energy bands:
soft (0.3--1\,keV), medium (1--2\,keV) and hard (2--6\,keV).
In each case the pixel size was set at $4.35'' \times 4.35''$.
The pn, MOS 1 and MOS 2 images in each of the three bands were first flat-fielded
by subtracting a constant (non-vignetted) background particle rate (estimated
from the count rates recorded in the corners of the CCD detectors
not exposed to the sky) and then dividing by the appropriate exposure
map. In the same process bad pixels, hot columns and spurious data along chip
gaps were excised. The resulting data from the three observations were then
mosaiced into flat-fielded pn and MOS (MOS 1 plus MOS 2) images for each band.
Since the medium and hard band images showed no evidence for diffuse emission,
here we focus primarily on the images obtained in the soft (0.3-1 keV) band.
Figure~\ref{fig:im_big}(a) shows an adaptively-smoothed soft-band image,
obtained from the combined pn and MOS datasets, encompassing the central
20\hbox{$^\prime$~\/} diameter region centred on the optical nucleus
of M101 (RA $\rm 14^h~3^m~12.55^s$, Dec $+54\hbox{$^\circ$~\/}~20\hbox{$^\prime$~\/}~56.5\hbox{$^{\prime\prime}$~\/}$) . The amplitude
scaling is logarithmic and adjusted so as to highlight the extended low surface
brightness emission in M101.
\begin{figure*}
\centering
\caption{ {\it (a)} {\it XMM-Newton~\/} soft-band (0.3-1 keV) image of the central $20'$ of M101. This
flat-fielded image is constructed using pn and MOS data from the three {\it XMM-Newton~\/}
observations. The amplitude scaling is logarithmic with a dynamic range (maximum
to minimum in the colour table) of 50. The position of the optical nucleus of
M101 is marked with a cross. {\it (b)} A ``model'' image constructed from a set of
count-rate scaled PSF sub-images. A source mask, defined by the black contours,
was produced by applying a cut at a surface brightness threshold of 0.01 pn+MOS $\rm
ct~ks^{-1}~pixel^{-1}$. {\it (c)} The same X-ray image as shown in panel (a)
with the regions confused by bright sources masked out. The amplitude scaling is
logarithmic with a dynamic range of 10. {\it (d)} The {\it GALEX} near-UV
(NUV:2310 \AA) band image of M101 on the same spatial scale
as the X-ray data. The scaling is logarithmic with a dynamic range of 10.
In all four panels the two large circles have
radii of $5'$ and $10'$. {\tt Figures submitted to arXiv as jpgs.}
}
\label{fig:im_big}
\end{figure*}
To facilitate the investigation of the contribution of bright sources to the
total X-ray emission in M101, a master sourcelist, applicable to the three
observations, was created. To this end, each of the new observations was analysed in turn
using the source detection procedure described in Paper II (applied to
the soft, medium and hard band data). Then, by comparing the resulting source lists,
the 108 sources detected in observation 1 (Paper II) were supplemented by 21 new sources
from observation 2 and an additional 9 new sources from observation 3, to
give a combined source catalogue of 138 sources (see the Appendix for details of the
new sources).
The next step involved the construction of a spatial mask for the suppression of the
bright X-ray sources in the central field of M101.
Using the pn+MOS flat-fielded soft-band image, a soft-band source count
rate, net of the local background, was determined at each source position; in practice,
we extracted the count rate within a cell of $16''$ radius centred on the source
position, subtracted the local background, and lastly applied a correction for the extent
of the point spread function (PSF) beyond the source cell. We then selected the sources
within 11\hbox{$^\prime$~\/} radius of the nucleus of M101 with count rates above
1.5 pn+MOS $\rm ct~ks^{-1}$ as a ``bright
source sample'' to be excised from the the soft-band image. Here our approach was to
construct a model image in which each source in our bright source list was
represented by a PSF sub-image
centred on the source position and an intensity scaling commensurate
with the measured source count rate. Then by applying a cut at a surface brightness
threshold of 0.01 pn+MOS $\rm ct~ks^{-1}~pixel^{-1}$, we were able to construct
a bright source mask - see Fig.~\ref{fig:im_big}(b). A total of 65 bright soft-band
sources were removed within a radius of 11\hbox{$^\prime$~\/} of the nucleus of M101 (25 sources
within 5\hbox{$^\prime$~\/}).
Figure \ref{fig:im_big}(c) shows the pn+MOS soft-band image of M101 after applying the
bright source mask
with the colour table adjusted to highlight the distribution
of the residual emission\footnote{Hereafter in this paper we use the word ``residual''
to describe the X-ray emission of M101 after masking out the bright
discrete X-ray sources. This residual component will include the truly
diffuse emission, the integrated emission of discrete sources with count rates
below our source-exclusion threshold and also some low-level contamination from
bright sources due to the spillover beyond the masked regions of their extended
PSFs - see Fig.~\ref{fig:im_big}(b).}.
The residual component is most prominent within the the central 5\hbox{$^\prime$~\/} radius region,
although it can be traced out to 10\hbox{$^\prime$~\/} in the radial profile distribution
shown in Fig.~\ref{fig:radial}. There is a clear correlation with the spiral structure
of the galaxy, as is evident from a comparison with the {\it GALEX} near-UV
(NUV; 2310 \AA) band image of M101 (\citealt{bianchi05}; \citealt{popescu05})
shown in Fig.~\ref{fig:im_big}(d).
\begin{figure*}
\rotatebox{270}{\scalebox{0.4}{\includegraphics{fig_2.ps}}}
\caption{The radial distribution of the soft-band (0.3-1 keV) pn+MOS count
rate. The sky background level has been subtracted. The solid line shows an
exponential fall-off in the surface brightness with a scale length of
2.6\hbox{$^\prime$~\/} (5.4 kpc), as determined from optical B data (\citealt{okamura76}).}
\label{fig:radial}
\end{figure*}
\subsection{Spectral extraction of the diffuse component}
\label{sec:diff_extr}
The images in Fig.~\ref{fig:im_big} demonstrate the existence of an extensive
residual component in the soft X-ray emission of M101. In order to investigate
the spectral properties of this component we have extracted spectral data for
the 5\hbox{$^\prime$~\/}\ radius region, centred on the position of
the nucleus. A total of nine individual diffuse emission spectra were
derived from the appropriate
event files (for the three EPIC instruments, over the three observations). In each
case, regions around bright point sources were excluded using the mask described
earlier (in practice this involved
the use of circular exclusion cells of radius $12'' - 53''$, as an approximation
to the source mask used in the image analysis).
Given the extended nature of the diffuse emission, the background subtraction
is more complicated than is usual with EPIC data. The use of other {\it XMM-Newton~\/} data
sets deriving from ``blank-field'' observations is, in our view, not a good option
due to the variation from field to field in the soft X-ray background from the sky
- clearly an important consideration when attempting to measure a soft diffuse
signal. Instead local background (LBG) spectra
were extracted from each event file for the annular region from
5\hbox{$^\prime$~\/} to 10\hbox{$^\prime$~\/} radius - see Fig.~\ref{fig:im_big}(a) -
again with the regions around bright sources excluded. We also extracted particle
background (PBG) spectra from the out-of-field-of-view corner regions in each EPIC
camera. The true background
beneath the residual M101 emission in the central source region will be made up
primarily of (vignetted) photons from the X-ray sky
and (non-vignetted) particles, which comprise the instrumental background. The
LBG spectra will also have photon and particle components, but, as the LBG region
is further
off-axis than the source region, these components are not in the correct ratio
for accurate background subtraction (since the signal reduction due to the vignetting
is larger in the LBG region). However, a very good model of the true background
can be made from a combination of the LBG and PBG spectra, provided the
contributions of the two are appropriately scaled\footnote{The scale
factors applied to the LBG and PBG spectra were respectively
$(A_{c}/A_{LBG})\times(V_{c}/V_{LBG})$ and $(A_{c}/A_{PBG})\times(1-V_{c}/V_{LBG})$. Here
$A_{c}$, $A_{LBG}$ and $A_{PBG}$ refer to the actual areas on the detector
of the central, LBG and PBG extraction regions. Similarly $V_{c}$ and $V_{LBG}$
are the average vignetting factors within the central and LBG regions.}
to allow for the vignetting-corrected and actual areas of the central, LBG and
PBG extraction regions.
Using this approach, an appropriate composite background was determined for each
central region spectrum together with the corresponding ARF and RMF files.
Finally for each dataset, the spectral channels were binned together to give a
minimum of 20 counts per bin.
Given the radial profile in Fig.~\ref{fig:radial}, the above amounts to a differential
measurement of the galaxy spectrum, which in effect compares the integrated signal
in the central 5\hbox{$^\prime$~\/} region with the equivalent signal measured in the
5\hbox{$^\prime$~\/} - 10\hbox{$^\prime$~\/} annular region.
\section{Results}
\label{sec:res}
\subsection{Luminosity and composition of the residual emission}
\label{sec:cont}
After the exclusion of the bright source regions and subtracting the mean sky surface
brightness measured in the 5\hbox{$^\prime$~\/}--10\arcmin annulus, the count rate in the
central 5\hbox{$^\prime$~\/} region is 0.28 pn+MOS $\rm ct~s^{-1}$ (0.3--1 keV).
However, we need to apply three correction factors to obtain the ``true'' central
5\hbox{$^\prime$~\/} count rate. The first accounts for the loss of ``diffuse'' signal due to the
source mask which blocks 12\% of the central area. The second factor allows for the
differential nature of the measurement ({\it i.e.,} the fact that we extract the
background from the 5\hbox{$^\prime$~\/}-10\hbox{$^\prime$~\/} annulus where the galaxy is still bright); on
the basis of the radial profile determined earlier we estimate that this
effect suppresses the central count rate by $\approx 19\%$. Finally, the spillage
of the bright source signal beyond the masked region (due to the
extended PSF) contributes about 9\% of the measured residual count rate
(as determined from the source model image shown in Fig.~\ref{fig:im_big}(b)). The
correction of all three effects translates to a 28\% upward scaling of the residual
count rate. If we adopt the best-fitting two-temperature model described
in section \ref{sec:spec}, the count rate to flux conversion factor is
1 pn+MOS $\rm ct~s^{-1}$ = $0.97 \times 10^{-12} \rm~erg~cm^{-2}~s^{-1}$
for the 0.5--2.0 keV band (fluxes
corrected for Galactic absorption). The equivalent factors for the 0.3--1 keV and
0.45--1 keV bands are $1.16$ and $0.91 \times 10^{-12}
\rm~erg~cm^{-2}~s^{-1}~(ct~s^{-1})^{-1}$ respectively.
The implied luminosity of the central emission of M101 after excluding
bright sources is $2.1 \times 10^{39} \rm erg~s^{-1}$ in the 0.5--2 keV band.
However, we must take into account the fact that our source exclusion threshold is
relatively high (roughly a factor 10 higher than that employed in the equivalent
{\it Chandra~\/} study - \citealt*{kuntz03}). {\it Chandra~\/} has shown that the log N - log S
relation
for relatively faint sources in the central region of M101 (after correction for
background AGNs) is quite flat with a slope of $-0.80\pm 0.05$ in the integral
counts \citep{pence01}, implying that bright sources provide the dominant
contribution to the discrete source luminosity of the galaxy. Using the count
rate to flux conversion factor quoted earlier, our source exclusion threshold
of 1.5 pn+MOS $\rm ct~ks^{-1}$ (0.3--1 keV) corresponds to a source luminosity
of $9 \times 10^{36} {\rm ~erg~s^{-1}} $ (0.5--2 keV), although this could
be an underestimate for the harder spectral forms more typical of bright X-ray
binaries. In fact, a source exclusion threshold of $1 \times 10^{37} {\rm ~erg~s^{-1}} $
appears to be a reasonable estimate based on the average count rate to flux conversion
obtained when we compare our measurements with those quoted in \citet{pence01}
for the sub-set of common sources.
As a check, at the latter luminosity threshold the {\it Chandra~\/} source counts
predict $\approx 25$ sources within a central 5\hbox{$^\prime$~\/} region, which is very consistent
with the number of sources actually excluded in our study. Using the {\it Chandra~\/}
source counts, we find the integrated luminosity in discrete sources from
our exclusion cut-off down to a factor 10 deeper (in effect the {\it Chandra~\/} limit)
amounts to about 20\% of the residual signal measured by {\it XMM-Newton}.
In other words, our estimate
of the ``diffuse'' luminosity of M101 (after allowing for the contribution of
sources brighter than $\sim 10^{36} {\rm ~erg~s^{-1}} $) is $1.7 \times 10^{39} {\rm ~erg~s^{-1}}$
(0.5--2 keV) in good agreement with the value
reported by \citet{kuntz03} based on {\it Chandra~\/} measurements
of the same region.
\subsection{Spectral properties of the residual emission}
\label{sec:spec}
The nine epic spectra were fitted simultaneously using a variety of
spectral models, with identical model parameters applied to each
of the data sets. Visual inspection of the spectra
indicated almost no emission above $\approx 1.5$\,keV. No acceptable
fit could be obtained using just a single hot plasma component.
However, a good fit (reduced $\chi^{2}$=0.87 for 2986 degrees of freedom)
was obtained with a model incorporating two thermal (solar-abundance) Mekal
components, with absorption attributable to the hydrogen column density of
our Galaxy in the direction of M101 (1.16$\times10^{20}$\,cm$^{-2}$;
\citealt{dickey90}).
The derived temperatures for the two components were 0.20$^{+0.01}_{-0.01}$ keV and
0.68$^{+0.06}_{-0.04}$ keV (90\% confidence limits assuming 1 interesting parameter).
The data plus best fitting two-temperature
model are shown in Fig~\ref{fig:spec}. (In this figure we show the averages
of the three pn and the six MOS spectra for clarity of
display; however, the spectral fitting results quoted here pertain to the fits
to the nine individual spectra).
The absorption-corrected flux in the 0.3--1 keV band derived from the spectral fits
was 3.0 $\times10^{-13}$\,erg~cm$^{-2}$~s$^{-1}$ of which the hotter component
contributes $18$\%. The corresponding values for the 0.5--2.0 keV
band are 2.5 $\times10^{-13}$\,erg~cm$^{-2}$~s$^{-1}$ with the hotter component
contributing $27$\%.
When we allowed the metallicity of the two plasmas to vary from
unity ({\it i.e.,} solar values) we obtained a best fit of 0.044
(0.038-0.055) solar for the lower-temperature component and 0.17
(0.10-0.50) solar for the hotter component (with a reduced $\chi^{2}$ for the fit
of 0.81). However, this determination of strongly sub-solar abundances
is most probably an artifact of our attempt to fit relatively low spectral
resolution data pertaining to a complex multiphase plasma with a
simplistic spectral model, a problem which has in fact been well documented
in the literature ({\it e.g.,} \citealt{strickland00}; \citealt{wang01}). As noted by
\citet{kuntz03}, the abundances in the hot plasma in the central region of M101
are more likely to be near to solar rather than grossly sub-solar.
We know that the measured {\it XMM-Newton~\/} spectrum includes some residual contamination
by the wings of the bright sources (estimated to be 9\% of the
0.3--1 keV count rate) and also includes a contribution from the underlying
unresolved source population (estimated to contribute $20\%$ of the
0.5--2 keV flux). Since the sources may have somewhat harder spectra than
the diffuse emission, presumably they contribute preferentially to the
higher temperature spectral component. However, our derived temperatures
of $\approx 0.2$ and $\approx 0.7$ keV
are in very good agreement with the results of \citet{kuntz03} based on
{\it Chandra~\/} observations, suggesting that some fraction of the harder
emission does have a diffuse origin. In the following analysis we assume
50\% of the measured hard emission is attributable to diffuse emission.
We also scale the component normalisations obtained from the spectral
fitting so as to correct for both the area of the source mask and
the differential nature of the spectral measurement (as detailed in \S 3.1).
\begin{figure*}
\centering
\rotatebox{270}{\scalebox{0.5}{\includegraphics{fig_3a.ps}}}
\rotatebox{270}{\scalebox{0.5}{\includegraphics{fig_3b.ps}}}
\caption{The EPIC spectra of the residual X-ray emission component in M101
after the exclusion of bright sources. {\it (a)} The pn spectrum averaged
over the three observations. {\it (b)} The combined MOS spectrum averaged
over the three observations. In both cases the solid line corresponds to the
best-fitting 2-component thermal spectrum with the dashed lines showing the
individual contributions of the 0.2-keV and 0.7-keV components - see text.
The $\chi^{2}$ residuals with respect to the best-fitting model are also
shown. }
\label{fig:spec}
\end{figure*}
One can infer mean physical properties of the hot diffuse gas once
some assumptions have been made regarding the geometry of the diffuse
emission. Here we assume that the gas is contained within a cylindrical
region of radius of 10.5 kpc in the plane of the galaxy (matching a
5\hbox{$^\prime$~\/} angular extent) with a half-width perpendicular to the plane
of 0.5 kpc (representing the extent of a putative shallow halo).
Using this volume, the derived emission measure $\eta n^{2}_{e} V$
(where $\eta$ is the `filling factor' - the fraction of the total volume $V$
which is occupied by the emitting gas) can be used to infer the mean electron
density $n_{e}$. For the 0.2 keV plasma we find $n_{e} \approx 0.003$
$\eta^{-1/2}$\,cm$^{-3}$ compared to a value of $\approx 0.001$
$\eta^{-1/2}$\,cm$^{-3}$ for the 0.7 keV plasma, implying that these
two components are in rough pressure balance. The thermal energy
residing in each of the plasma components is comparable and totals
$E_{th} \approx 6\times10^{55}$$\eta^{1/2}$\,erg. The cooling is dominated by the
line emission with radiative cooling timescales of
$t_{\mbox{\small soft}} \approx 1.8 \times 10^{8} \eta^{1/2}$\,yr
and $t_{\mbox{\small hard}} \approx 1.5 \eta^{1/2}$\,Gyr for the
cooler and hotter components respectively.
\subsection{Morphology of the residual emission}
\label{sec:morph}
As previously noted, the spatial distribution of the residual soft X-ray emission in M101
shows a good correlation with the optical/UV emission of the galaxy, at least
over the central 5\hbox{$^\prime$~\/} region. We have investigated this correlation using the
FUV (1530 \AA) and NUV (2310 \AA) images of M101 from the {\it GALEX} pipeline
(\citealt{morrissey05}), together with images recorded by the {\it XMM-Newton~\/}
Optical Monitor ({\it OM}) in the UVW1 ($\approx$ 2800 \AA), U, B and V filters
(\citealt{mason01}).
The latter were derived from data taken during observation~1 (UVW1) and observation 3
(U,B,V) and subject to the standard SAS pipeline processing.
To aid the comparison, the soft X-ray image of Fig.~\ref{fig:im_big}(c) was compressed
in both coordinates by a factor 4 (to give $17.4\hbox{$^{\prime\prime}$~\/} \times 17.4\hbox{$^{\prime\prime}$~\/}$ pixels)
and then lightly smoothed. The UV and optical images were first rebinned into the same
image format as the original soft X-ray data, blurred by a mask representative of
the on-axis {\it XMM-Newton~\/} soft X-ray PSF and then further compressed in an identical way to
the soft X-ray data.
Fig.~\ref{fig:xuv} shows a detailed comparison of the 0.3--1 keV surface brightness
measured in the central 5\hbox{$^\prime$~\/} radius region of M101 with the corresponding
NUV, U and V band images. A striking similarity between the soft X-ray
and U band images is immediately obvious. In particular, the ``S-feature'' which
delineates the
inner spiral structure of M101 is very prominent in both cases.
The fall-off in the surface brightness towards the edges of the 5\hbox{$^\prime$~\/} field is also
very similar in both wavebands, with the north-south asymmetry evident at the
level of the lowest soft X-ray contour level also present in the U band image.
Interestingly, the correspondence becomes slightly less strong when the soft
X-ray image is compared to both the NUV and V band images.
The NUV image is clearly totally dominated by bright HII regions
and young star associations which together delineate the
spiral arms of the galaxy, whereas in the soft X-ray image the general rise in
surface brightness towards the nucleus has more relative impact.
It would seem highly unlikely that these morphological trends could
be induced by the increase in extinction as we move from the U band into
the NUV. In contrast, comparison with the V band data veers in the other
direction with the soft X-ray image showing more structure outside the central
2\hbox{$^\prime$~\/} zone than the optical image.
\begin{figure*}
\centering
\caption{{\it (a)} The pn+MOS soft-band (0.3--1 keV) image of
the central 5\hbox{$^\prime$~\/} radius region of M101. The scaling is linear.
The contour levels increase
in steps of 0.3 pn+MOS $\rm ct~ks^{-1}~pixel^{-1}$ from a
starting level of 0.9 pn+MOS $\rm ct~ks^{-1}~pixel^{-1}$.
{\it (b)} The {\it GALEX} NUV image of the same region
overlaid with the soft X-ray contours. The scaling is linear
in $\rm count~pixel^{-1}$ units.
{\it (c)} The {\it XMM-Newton~\/} {\it OM} U image of the same region
overlaid with the soft X-ray contours.The scaling is linear
in $\rm count~pixel^{-1}$ units.
{\it (d)} The {\it XMM-Newton~\/} {\it OM} V image of the same region
overlaid with the soft X-ray contours. The scaling is linear
in $\rm count~pixel^{-1}$ units.
{\tt Figures submitted to arXiv as jpgs.}
}
\label{fig:xuv}
\end{figure*}
The above trends are further illustrated in Fig.~\ref{fig:cor}
which shows the pixel-by-pixel correlation between the soft X-ray surface
brightness and the optical/UV surface brightness in the various wavebands.
(here we use unsmoothed data with a pixel size of $34.8\hbox{$^{\prime\prime}$~\/} \times 34.8\hbox{$^{\prime\prime}$~\/}$,
{\it i.e.} a factor of 8 compression of the original images).
The correlation, as measured by the Pearson product moment correlation
coefficient, R, is at a maximum in the U band.
\begin{figure*}
\centering
\rotatebox{270}{\scalebox{0.8}{\includegraphics{fig_5.ps}}}
\caption{The correlation of the surface brightness measured in the soft X-ray band
with that measured in other wavebands from the FUV through to V. The units in all
cases are effectively counts per pixel. The numerical values quoted in each panel
are the Pearson product moment correlation coefficient.
}
\label{fig:cor}
\end{figure*}
\section{Discussion}
\label{sec:disc}
After excluding bright point sources (with $L_X > 10^{37} {\rm ~erg~s^{-1}}$)
we measure the residual soft X-ray luminosity of the inner (10.5 kpc radius)
disk of M101 to be $L_X \approx 2.1 \times 10^{39} {\rm ~erg~s^{-1}}$ (0.5--2 keV).
Up to 20\% of this flux may originate in relatively bright point X-ray sources
(with $L_X$ down to $10^{36} {\rm ~erg~s^{-1}}$), which remain unresolved in
the {\it XMM-Newton~\/} observations. A further 5\% of the measured luminosity may
be attributable to intermediate luminosity sources (with $L_X$ down to
$10^{34}{\rm ~erg~s^{-1}}$) and, at very faint levels, the integrated emission of
dwarf stars may account for another 10\% contribution (\citealt{kuntz03}).
Nevertheless, it seems that the bulk of observed X-ray emission is likely
to be diffuse in origin.
The EPIC spectrum of the residual emission can be modelled as the sum of
two thermal plasma components at temperatures of $\approx 0.2$ keV and
$\approx 0.7$ keV with the harder component contributing $27\%$ of the flux in the
0.5--2 keV band ($18\%$ in the 0.3--1 keV band). These spectral
characteristics appear to be quite typical of the diffuse components seen
in normal and starburst galaxies ({\it e.g.,} \citealt{fraternali02};
\citealt{jenkins04b}; \citealt {jenkins05a}) and,
albeit with some caveats, consistent
with spectral measurements for M101 reported in the literature
(\citealt{snowden95}; \citealt{wang99}; \citealt{kuntz03}).
Earlier studies have demonstrated that for normal spiral galaxies, the
total soft X-ray luminosity correlates with the total infrared luminosity
(\citealt{read01}; \citealt{strickland04b}), the latter serving as a direct
indicator of the current star formation rate.
For the diffuse X-ray components, this is in line with the mechanism
whereby the mechanical energy input from supernovae and the winds from
young massive stars serves to heat the galactic ISM
to million-degree temperatures. In the case of M101 the required rate
of energy input is readily supplied by two supernovae per century
(\citealt{matonick97}), if the efficiency of the conversion of the
mechnical energy of the supernova explosion into X-rays
is $10^{51}/E~\%$, where $E$ is the energy of the
supernova in $\rm erg$.
The linkage of X-ray emission to recent star formation
is further strengthened by the finding that in a number of galaxies
the X-ray morphology in the spiral arms matches that seen in the
mid-infrared or H${\alpha}$ (\citealt{tyler04}).
A recent {\it Chandra~\/} study has demonstrated that, in M101, the
diffuse X-ray emission traces the spiral arms and is correlated with both the
H${\alpha}$ and far-UV emission (\citealt{kuntz03}). In the same study
there was no evidence for any significant trend of X-ray spectral hardness
with radius\footnote{We have subsequently checked the {\it XMM-Newton~\/} data for
spectral trends by extracting images in the 0.3--0.8 keV
and 0.8--1.2 keV sub-bands. The radial profiles of the residual X-ray emission
in these sub-bands are consistent with the results obtained for the broader
0.3--1.0 keV band (see Fig.~\ref{fig:radial}) and within the errors, exhibit a
constant ratio for radii in the range 1'-10'. The {\it XMM-Newton~\/} observations thus confirm
that there is no significant radial dependence of spectral hardness in this Galaxy.}
implying that the 0.2- and 0.7-keV spectral components are closely linked with
one another and are both associated with star formation.
In the present paper we
have investigated this correlation in more detail by contrasting the X-ray
morphology with that observed in far-UV through to V band.
We find that the correlation with the X-ray morphology is quite strong for all
bands considered (FUV, NUV, UVW1, U, B and V), although the best match is
obtained with the U band data. One interpretation of this result is that,
in broad terms, there are two underlying components present in the
X-ray surface brightness distribution, namely a smooth rise towards the
centre of the galaxy as reflected in the radial profile of the emission
(Fig.~\ref{fig:radial}) plus a superimposed X-ray spiral arm structure.
In moving from the FUV through to the V band, clearly the balance shifts between
the extreme population I components (distributed in the spiral arms) and
a more intermediate population (associated with the galactic disk) and
it is in the U band that this balance happens to best match the
X-ray morphology.
In the disks of spiral galaxies star-formation is thought to be triggered
by the passage of a spiral density wave through the ISM.
The formation of massive young stars results in a large UV/far-UV flux
with reprocessing in the immediate environment subsequently giving rise to associated
H${\alpha}$ and mid-IR emission. All of these serve
as tracers of the spiral arms of the galaxy. Population synthesis models
predict that following a star-formation burst, the peak in the UV/H${\alpha}$
production will last $\sim 3 \times 10^{6}$ yr, whilst the most massive stars
complete their life cycle. The rate of energy deposition into the ISM
from the subsequent supernovae rises at this time and remains fairly
constant for $\sim 3 \times 10^{7}$ yr (\citealt{leitherer95}; \citealt{cervino02}).
Given the lag between the peak in UV/H${\alpha}$ emission and the
X-ray heating one might predict a spatial offset between the diffuse X-ray emission
and the other spiral tracers, although as yet there is no clear observational
evidence for such an effect (\citealt{tyler04}).
In the case of M101, we assume a flat rotation curve beyond a few kpc of the galaxy
centre and a rotational velocity $v_{rot}\approx200~\rm~km~s^{-1}$ (\citealt{bosma81}).
We further assume
that the spiral pattern corotates with the disk material at 3 radial scale lengths,
{\it i.e.} at $r\approx15$ kpc. Then at $ r\approx7.5 $ kpc a
delay of $3 \times 10^{7}$ yr
translates to a rotational lag of $24^{\circ}$ or about 3 kpc in a direction perpendicular
to the spiral arm. The effective pixel size in our multi-waveband correlation is $35\hbox{$^{\prime\prime}$~\/}$
or 1.2 kpc, implying that a delay in the X-ray production following the passage of a
spiral density wave might just be measurable in the current data. Unfortunately the
restricted signal to noise and limited clarity of the spiral arm features
in the X-ray image mitigates against the detection of such an effect.
Earlier we estimated a radiative cooling timescale for the more prominent 0.2-keV
plasma component to be $t_{\mbox{\small soft}} \approx 1.8 \times 10^{8}$\,yr
for a filling factor $\eta \sim 1$. Clearly this timescale is inconsistent
with the presence of reasonably narrow spiral features in the soft X-ray image.
A very small filling factor, {\it e.g.} $\eta \sim 0.01$, is required to match
the properties of X-ray spiral arms in M101, suggesting a very clumpy thin-disk
distribution. In fact a reasonable scenario is that the spiral arms in X-rays
are delineated by a combination of truly diffuse emission, possibly in the form
of hot gas bubbles and superbubbles, plus contributions from individual SNRs and
concentrations of unresolved discrete sources.
As noted by \citet{kuntz03}, the available
constraints suggest that superbubbles in the disk of M101 may have similar
properties to the Galactic Loop I superbubble (\citealt{egger95}),
although they are not individually resolvable even in {\it Chandra~\/} data.
Where there is no confinement by chimneys or similar structures in the ISM,
energy losses arising from adiabatic expansion of the hot gas in the disk
into the lower halo of M101 may also help localise the spiral arm component
({\it nb.} in M101 the sound speed of the soft component is comparable to the escape
velocity from the disk - \citealt{kuntz03}).
The second spatial component of the X-ray emission considered above broadly
follows the distribution of the optical light attributable to the intermediate
disk stellar population. The integrated X-ray emission of dwarf stars will
presumably follow a disk distribution but this is unlikely to represent a
substantial contribution (10\% overall in the 0.5--2 keV band but, given the
typical $kT \sim 0.8$ keV temperature of dwarf-star coronae,
falling to $\approx 5\%$ in the softer 0.3--1 keV band; \citealt{kuntz03}).
Other contributions to this smoother distribution might be expected from supernovae
in the interarm regions associated with disk population stars and X-ray emitting gas
which has accumulated in a shallow halo in M101 as a result of galactic chimney/fountain
activity. In the case of the latter, gas which has cooled substantially due to adiabatic
expansion may, as a result of a frozen-in non-ionization equilibrium,
exhibit an emission spectrum dominated by oxygen lines, which mimics that expected
from a $0.2$ keV plasma (\citealt{breitschwerdt99}). The fact that the X-ray
surface brightness increases towards the centre of M101 is most naturally
explained in terms of an increase in activity per unit disk area, with the filling
factor of the extended z-height component possibly approaching unity near the
centre of the galaxy. With the latter assumption, the implied electron density and
pressure in the inner halo of M101 is comparable to that inferred for the centre of
Galactic Loop I and the Local Hot Bubble.
\section{Conclusions}
\label{sec:conc}
This is the third and final paper in a series presenting the results of {\it XMM-Newton~\/} observations
of the nearby face-on Scd spiral galaxy M101. Here we focus on the spatial and spectral
properties of the galaxy, when bright X-ray sources with $L_X > 10^{37} {\rm ~erg~s^{-1}}$ are
removed using an appropriate point source mask.
The residual soft X-ray luminosity of the central (10.5 kpc radius) region of M101 was
measured as $L_X \approx 2.1 \times 10^{39} {\rm ~erg~s^{-1}}$ (0.5--2 keV), the bulk of which
appears to originate as diffuse emission. We find a two-temperature model best fits
the spectral data with the derived temperatures of 0.20$\pm$0.01 keV and
0.68$^{+0.06}_{-0.04}$ keV, typical of the diffuse components seen
in other normal and starburst galaxies.
In line with earlier studies, we find that the observed X-ray surface brightness
distribution is well correlated with images recorded in optical/UV wavebands.
In particular the detection of spiral arms in X-rays establishes a close link between
the X-ray emission and recent starformation. Closer investigation suggests that the
X-ray morphology may comprise both a spiral arm component and a smoother disk component.
In spiral galaxies, star formation is thought to be triggered by the passage of a
spiral density wave through the ISM. In principle one might observe a lag between more
immediate star-formation indicators such as the UV/far-UV flux or H${\alpha}$ and
the heating of the interstellar medium to X-ray temperatures as result of
subsequent supernova. The {\it XMM-Newton~\/} data show no evidence for such an effect
but it may be observable in future high resolution and high signal-to-noise data.
Whereas the spiral arm component, on the basis of the radiative cooling
timescale of the 0.2-keV plasma, may be deduced to have a clumpy, low-z distribution,
the smoother disk component may represent longer-lived X-ray gas with a relatively
large filling factor, which has been transported to the lower halo of M101 via the
galactic chimney/fountain mechanism.
Future progress in understanding the X-ray properties of normal spiral
galaxies will no doubt follow from very intensive studies of individual
galaxies ({\it e.g.,} the recent 1 Ms observation of M101 with {\it Chandra~\/} - Kuntz et al.
in preparation) and by applying appropriately optimised analysis procedures to
samples of nearby galaxies, as is now possible using archival datasets.
\section*{Acknowledgments}
LPJ, TPR and RAO acknowledge PPARC support at various junctures during
this project. We should also like to thank the anonymous referee for
some helpful comments and suggestions.
\label{lastpage}
|
2,877,628,089,494 | arxiv | \section{Introduction}
\input{introduction}
\section{Methods}
\input{methods}
\section{Experiments}
\input{experiments}
\section{Discussion}
\input{discussion}
\vfill
\pagebreak
\subsubsection*{Acknowledgments}
This work was supported by the ERC-StG NeuroLang ID:757672.
We would like to warmly thank Dr. Thomas Yeo and Dr. Ru Kong (CBIG) who made pre-processed HCP functional connectivity data available to us.
We also would like to thank Dr. Majd Abdallah (Inria) for his insights and perspectives regarding our functional connectivity results.
\subsubsection*{Reproducibility Statement}
All experiments were performed on a computational cluster with 16 Intel(R) Xeon(R) CPU E5-2660 v2 @ 2.20GHz (256Mb RAM), 16 AMD EPYC 7742 64-Core Processor (512Mb RAM) CPUs and 1 NVIDIA Quadro RTX 6000 (22Gb), 1 Tesla V100 (32Gb) GPUs.
All methods were implemented in Python. We implemented most methods using \textit{Tensorflow Probability} \citep{dillon_tensorflow_2017}, and SBI methods using the SBI Python library \citep{tejero-cantero_sbi_2020}.
As part of our submission we release the code associated to our experiments. Our supplemental material furthermore contains an entire section dedicated to the implementation details of the baseline methods presented as part of our experiments. For our neuromimaging experiment, we also provide a section dedicated to our pre-processing and post-processing steps
\bibliographystyle{iclr2022_conference}
\section{Complements to the methods: model descriptors for automatic variational family derivation}
This section is a complement to \cref{sec:methods}. We formalize explicitly the \textit{descriptors} of the generative HBM needed for our method to derive its dual architecture. This information is of experimental value, since those descriptors need to be available in any API designed to implement our method.
If we denote $\operatorname{plates}(\theta)$ the plates the RV $\theta$ belongs to, then the following HBM descriptors are the needed input to derive automatically our ADAVI dual architecture:
\begin{equation}
\label{eq:descriptors}
\begin{aligned}
\mathcal{V} &= \{ \theta_i \}_{i=0 \hdots L} \\
\mathtt{P} &= \{ \mathcal{P}_p \}_{p=0 \hdots P} \\
\operatorname{Card} &= \{ \mathcal{P}_p \rightarrow \# \mathcal{P}_p \}_{p=0 \hdots P} \\
\operatorname{Hier} &= \{ \theta_i \mapsto h_i = \min_p \{ p\ :\ \mathcal{P}_p \in \operatorname{plates}(\theta_i) \} \}_{i=0 \hdots L} \\
\operatorname{Shape} &= \{ \theta_i \mapsto \mathcal S^{\text{event}}_{\theta_i} \}_{i=0 \hdots L} \\
\operatorname{Link} &= \{ \theta_i \mapsto (l_i:S_{\theta_i}\rightarrow S^{\text{event}}_{\theta_i}) \}_{i=0 \hdots L}
\end{aligned}
\end{equation}
Where:
\begin{itemize}
\item $\mathcal{V}$ lists the RVs in the HBM (vertices in the HBM's corresponding graph template);
\item $\mathtt{P}$ lists the plates in the HBM's graph template;
\item $\operatorname{Card}$ maps a plate $\mathcal{P}$ to its \textit{cardinality}, that is to say the number of independent draws from a common conditional density it corresponds to;
\item $\operatorname{Hier}$ maps a RV $\theta$ to its \textit{hierarchy}, that is to say the level of the pyramid it is placed at, or equivalently the smallest rank for the plates it belongs to;
\item $\operatorname{Shape}$ maps a RV to its \textit{event shape} $S^{\text{event}}_{\theta_i}$. Consider the plate-enriched graph template representing the HBM. A single graph template RV belonging to plates corresponds to multiple similar RVs when grounding this graph template. $S^{\text{event}}_{\theta_i}$ is the potentially high-order shape for any of those multiple ground RVs.
\item $\operatorname{Link}$ maps a RV $\theta$ to its \textit{link function} $l$. The \textit{Link function} projects the latent space for the RV $\theta$ onto the event space in which $\theta$ lives. For instance, if $\theta$ is a variance parameter, the link function would map $\mathbb{R}$ onto $\mathbb{R}^{+*}$ ($l=\operatorname{Exp}$ as an example). Note that the latent space of shape $S_\theta$ is necessary an order 1 unbounded real space. $l$ therefore potentially implies a reshaping to the high-order shape $S^{\text{event}}_{\theta_i}$.
\end{itemize}
Those descriptors can be readily obtained from a static analysis of a generative model, especially when the latter is expressed in a modern probabilistic programming framework \citep{dillon_tensorflow_2017, bingham2019pyro}.
\section{Complements to our discussion}
\subsection{Amortization}
Contrary to traditional VI, we aim at deriving a variational family $\mathcal{Q}$ in the context of amortized inference \citep{rezende_variational_2016, cranmer_frontier_2020}. This means that, once an initial training overhead has been ``paid for'', our technique can readily be applied to a new data point. Amortized inference is an active area of research in the context of Variational Auto Encoders (VAE) \citep{kingma_auto-encoding_2014, wu_meta-amortized_2019, shu_amortized_nodate, iakovleva_meta-learning_nodate}. It is also a the original setup of normalizing flows (NF) \citep{rezende_variational_2016, radev_bayesflow_2020}, our technology of choice. From this amortized starting point, \citet{cranmer_frontier_2020, papamakarios_sequential_2019, thomas_likelihood-free_2020, greenberg_automatic_2019} have notably developed \textit{sequential} techniques, refining a posterior -and losing amortization- across several rounds of simulation. To streamline our contribution, we chose not build upon that research, and rather focus on the amortized implementation of normalizing flows. But we argue that our contribution is actually rather orthogonal to those: similar to \citet{pmlr-v139-ambrogioni21a} we propose a principled and automated way to combine several density estimators in a hierarchical structure. As such, our methods could be applied to a different class of estimators such as VAEs \citep{kingma_auto-encoding_2014}. We could leverage the SBI techniques and extend our work into a sequential version through the reparameterization of our conditional estimators $q_i$ (see \cref{sec:ADVI}). Ultimately, our method is not meant as an alternative to SBI, but a complement to it for the pyramidal class of problems described in \cref{sec:pyramidal}.
We choose to posit ourselves as an amortized technique. Yet, in our target experiment from \citet{kong_spatial_2018} (see \cref{sec:exp-yeo}), the inference is performed on a specific data point. An amortized method could therefore appear as a more natural option. What's more, it is generally admitted that amortized inference implies an \textit{amortization gap} from the true posterior, which accumulates on top of the \textit{approximation gap} that depends on the expressivity of the considered variational family. This amortization gap further reduces the quality of the approximate posterior for a given data point.
Our experimental experience on the example in \cref{exp:GM} however makes us put forth the value that can be obtained from sharing learning across multiple examples, as amortization entitles \citep{cranmer_frontier_2020}. Specifically, we encountered less issues related to local minima of the loss, a canonical issue for MF-VI \citep{blei_variational_2017} that is for instance illustrated in our supplemental material. We would therefore argue against the intuition that a (locally) amortized technique is necessarily wasteful in the context of a single data point. However, as the results in \cref{tab:GRE_scaling} and \cref{tab:NC&GM} underline, there is much work to be done for amortized technique to reach the performance consistency and training time of amortized techniques, especially in high dimension, where exponentially more training examples can be necessary to estimate densities properly \citep{donohoHighdimensionalDataAnalysis2000}.
Specializing for a local parameter regime -as sequential method entitles \citep{cranmer_frontier_2020}- could therefore make us benefit from amortization without too steep an upfront training cost.
\subsection{Extension to a broader class of simulators}
The presence of exchangeability in a problem's data structure is not tied to the explicit modelling of a problem as a HBM: \citet{zaheer_deep_2018} rather describe this property as an permutation invariance present in the studied data. As a consequence, though our derivation is based on HBMs, we believe that the working principle of our method could be applied to a broader class of simulators featuring exchangeability. Our reliance on HBMs is in fact only tied to our usage of the reverse KL loss (see \cref{sec:training}), a readily modifiable implementation detail.
In this work, we restrict ourselves to the pyramidal class of Bayesian networks (see \cref{sec:pyramidal}). Going further, this class of models could be extended to cover more and more use-cases. This bottom-up approach stands at opposite ends from the generic approach of SBI techniques \citep{cranmer_frontier_2020}. But, as our target experiment in \ref{sec:exp-yeo} demonstrates, we argue that in the long run this bottom-up approach could result in more scalable and efficient architectures, applicable to challenging setups such as neuroimaging.
\subsection{\textcolor{black}{Relevance of likelihood-free methods in the presence of a likelihood}}
\textcolor{black}{As part of our benchmark, we made the choice to include likelihood-free methods (NPE-C and SNPE-C), based on a forward KL loss. In our supplemental material (\cref{sec:all_losses_derivation}) we also study the implementation of our method using a forward KL loss.}
\textcolor{black}{There is a general belief in the research community that likelihood-free methods are not intended to be as competitive as likelihood-based methods in the presence of a likelihood \citep{cranmer_frontier_2020}. In this manuscript, we tried to provide quantitative results to nourish this debate. We would argue that likelihood-free methods generally scaled poorly to high dimensions (\cref{exp:scaling}). The result of the Gaussian Mixture experiment also shows poorer performance in a multi-modal case, but we would argue that the performance drop of likelihood-free methods is actually largely due to the label switching problem (see \cref{sec:label-switching}). On the other hand, likelihood-free methods are dramatically faster to train and can perform on par with likelihood-based methods in examples such as the Gaussian Random Effects for $G=3$ (see \cref{tab:GRE_scaling}). Depending on the problem at hand, it is therefore not straightforward to systematically disregard likelihood-free methods.}
\textcolor{black}{As an opening, there maybe is more at the intersection between likelihood-free and likelihood-based methods than meets the eye. The symmetric loss introduced by \citet{weilbach_structured_nodate} stands as a fruitful example of that connection.}
\subsection{\textcolor{black}{Inference over a subset of the latent parameters}}
\textcolor{black}{Depending on the downstream tasks, out of all the parameters $\theta$, an experimenter could only be interested in the inference of a subset $\Theta_1$. Decomposing $\theta = \Theta_1 \cup \Theta_2$, the goal would be to derive a variational distribution $q_1(\Theta_1)$ instead of the distribution $q(\Theta_1, \Theta_2)$.}
\textcolor{black}{\paragraph{Reverse KL setup}
We first consider the reverse KL setup. The original ELBO maximized as part of inference is equal to:
\begin{equation}
\begin{aligned}
\text{ELBO}(q) &= \log p(X) - \operatorname{KL}[q(\Theta_1, \Theta_2) || p(\Theta_1, \Theta_2 | X)] \\
&= \mathbb{E}_q[\log p(X, \Theta_1, \Theta_2) - \log q(\Theta_1, \Theta_2)]
\end{aligned}
\end{equation}
To keep working with normalized distributions, we get a similar expression for the inference of $\Theta_1$ only via:
\begin{equation}
\begin{aligned}
\text{ELBO}(q_1) &= \mathbb{E}_{q_1}[\log p(X, \Theta_1) - \log q_1(\Theta_1)]
\end{aligned}
\end{equation}
In this expression, $p(X, \Theta_1)$ is unknown: it results from the marginalization of $\Theta_2$ in $p$, which is non-trivial to obtain, even via a Monte Carlo scheme. As a consequence, working with the reverse KL does not allow for the inference over a subset of latent parameters.}
\textcolor{black}{\paragraph{Forward KL setup}
Contrary to reverse KL, in the forward KL setup the evaluation of $p$ is not required. Instead, the variational family is trained using samples $(\theta, X)$ from the joint distribution $p$. In this setup, inference can be directly restricted over the parameter subset $\Theta_1$. Effectively, one wouldn't have to construct density estimators for the parameters $\Theta_2$, and the latter would be marginalized in the obtained variational distribution $q(\Theta_1)$. However, as our experiments point out (\cref{exp:scaling}, \cref{exp:GM}), likelihood-free training can be less competitive in large data regimes or complex inference problems. As a consequence, even if this permits inference over only the parameters of interest, switching to a forward KL loss can be inconvenient.}
\subsection{\textcolor{black}{Embedding size for the Hierarchical Encoder}}
\textcolor{black}{An important hyper-parameter in our architecture is the embedding size for the \textit{Set Transformer} ($\operatorname{ST}$) architecture \citep{pmlr-v97-lee19d}. The impact of the embedding size for a single $\operatorname{ST}$ has already been studied in \citet{pmlr-v97-lee19d}, as a consequence we didn't devote any experiments to the study of the impact of this hyper-parameter.}
\textcolor{black}{However, our architecture stacks multiple $\operatorname{ST}$ networks, and the evolution of the embedding size with the hierarchy could be an interesting subject:
\begin{itemize}
\item it is our understanding that the embedding size for the encoding ${\tens{E}}_h$ should be increasing with:
\begin{itemize}
\item the number of latent RVs $\theta$ whose inference depends on ${\tens{E}}_h$, i.e. the latent RVs of hierarchy $h$
\item the dimensionality of the latent RVs $\theta$ of hierarchy $h$
\item the complexity of the inference problem at hand, for instance how many statistical moments need to be computed from i.i.d data points
\end{itemize}
\item experimentally, we kept the embedding size constant across hierarchies, and fixed this constant value based on the aforementioned criteria (see \cref{sec:implem}). This approach is probably conservative and drives up the number of weights in $\operatorname{HE}$
\item higher-hierarchy encodings are constructed from sets of lower-hierarchy encodings. Should the embedding size vary, it would be important not to "bottleneck" the information collected at low hierarchies, even if the aforementioned criteria would argue for a low embedding size.
\end{itemize}
There would be probably experimental interest in deriving algorithms estimating the optimal embedding size at different hierarchies. We leave this to future work.}
\subsection{\textcolor{black}{Bounds for ADAVI's inference performance}}
\textcolor{black}{When considering an amortized variational family, the non-amortized family with the same parametric form can be considered as an upper bound for the inference performance -as measured by the ELBO. Indeed, considering the fixed parametric family $q(\theta ; \Psi)$, for a given data point $X^1$ the best performance can be obtained by freely setting up the $\Psi^1$ parameters. Instead setting $\Psi^1 = f(X^1)$ -amortizing the inference- can only result in worst performance. On the other hand the parameters for another data point $X^2$ can then readily be obtained via $\Psi^2 = f(X^2)$ \citep{cremer_inference_2018}.}
\textcolor{black}{In a similar fashion, it can be useful to look for upper bounds for ADAVI's performance. This is notably useful to compare ADAVI to traditional MF-VI \citep{blei_variational_2017}:
\begin{enumerate}
\item \textbf{Base scenario: traditional MF-VI} In traditional MF-VI, the variational distribution is $q^{\text{MF-VI}} = \prod_i q^{\text{Prior's parametric form}}_i(\theta_i)$:
\begin{itemize}
\item $q^{\text{Prior's parametric form}}_i$ can for instance be a Gaussian with parametric mean and variance;
\item in non-conjugate cases, using the prior's parametric form can result in poor performance due to an approximation gap, as seen in \cref{exp:NC};
\item due to the difference in expressivity introduced by normalizing flows, except in conjugate cases, $q^{\text{MF-VI}}$ is \textbf{not} an upper bound for ADAVI's performance.
\end{itemize}
\item \textbf{Superior upper limit scenario: normalizing flows using the Mean Field approximation} A family more expressive then $q^{\text{MF-VI}}$ can be obtained via a collection of normalizing flows combined using the mean field approximation: $q^{\text{MF-NF}} = \prod_i q^{\text{Normalizing flow}}_i(\theta_i)$:
\begin{itemize}
\item every individual $q^{\text{Normalizing flow}}_i$ is more expressive than the corresponding $q^{\text{Prior's parametric form}}_i$: in a non-conjugate case it would provide better performance \citep{papamakarios_normalizing_2019};
\item since the mean field approximation treats the inference over each $\theta_i$ as a separate problem, the resulting distribution $q^{\text{MF-NF}}$ is more expressive than $q^{\text{MF-VI}}$;
\item consider a plate-enriched DAG \citep{koller_probabilistic_2009}, a \textit{template} RV $\theta_i$, and $\theta_i^j$ with $j=1\hdots \operatorname{Card}(\mathcal{P})$ the corresponding \textit{ground} RVs. In $q^{\text{MF-NF}}$, every $\theta_i^j$ would be associated to a separate normalizing flow;
\item consequently, the parameterization of $q^{\text{MF-NF}}$ is linear with respect to $\operatorname{Card}(\mathcal{P})$. This is less than the quadratic scaling of TLSF or NPE-C -as explained in \cref{sec:ADVI} and \cref{sec:implem}. But this scaling still makes $q^{\text{MF-NF}}$ not adapted to large plate cardinalities, all the more since the added number of weights -corresponding to a full normalizing flow- per $\theta_i^j$ is high;
\item this scaling is similar to the one of Cascading Flows \citep{pmlr-v139-ambrogioni21a}: CF can be considered as the improvement of $q^{\text{MF-NF}}$ with statistical dependencies between the $q_i$;
\item as far as we know, the literature doesn't feature instances of the $q^{\text{MF-NF}}$ architecture. Though straightforward, the introduction of normalizing flows in a variational family is non-trivial, and for instance marks the main difference between CF and its predecessor ASVI \citep{ambrogioni_automatic_nodate}.
\end{itemize}
\item \textbf{Inferior upper limit scenario: non-amortized ADAVI} At this point, it is useful to consider the non-existent architecture $q^{\text{ADAVI-NA}}$:
\begin{itemize}
\item compared to $q^{\text{MF-NF}}$, considering the \textit{ground} RVs $\theta_i^j$ corresponding to the \textit{template} RV $\theta_i$, each $\theta_i^j$ would no longer correspond to a different normalizing flow, but to the same conditional normalizing flow;
\item each $\theta_i^j$ would then be associated to a separate independent encoding vector. There wouldn't be a need for our Hierarchical Encoder anymore -as referenced to in \cref{sec:ADVI};
\item as for $q^{\text{MF-NF}}$, the parameterization of $q^{\text{ADAVI-NA}}$ would scale linearly with $\operatorname{Card}(\mathcal{P})$. Each new $\theta_i^j$ would only necessitate an additional embedding vector, which would make $q^{\text{ADAVI-NA}}$ more adapted to high plate cardinalities than $q^{\text{MF-NF}}$ or CF;
\item using separate flows for the $\theta_i^j$ instead of a shared conditional flow, $q^{\text{MF-NF}}$ can be considered as an upper bound for $q^{\text{ADAVI-NA}}$'s performance;
\item due to the amortization gap, $q^{\text{ADAVI-NA}}$ can be considered as an upper bound for ADAVI's performance. By transitivity, $q^{\text{MF-NF}}$ is then an even higher bound for ADAVI's performance.
\end{itemize}
\end{enumerate}}
\textcolor{black}{It is to be noted that amongst the architectures presented above, ADAVI is the only architecture with a parameterization invariant to the plate cardinalities. This brings the advantage to theoretically being able to use ADAVI on plates of any cardinality, as seen in \cref{eq:parametrization}. In that sense, our main claim is tied to the amortization of our variational family, though the linear scaling of $q^{\text{ADAVI-NA}}$ could probably be acceptable for reasonable plate cardinalities.}
\section{Complements to the Gaussian Random Effects experiment: hyperparameter analysis}
This section features additional results on the experiment described in \cref{eq:gaussian_re_model} with $G=3$ groups. We present results of practical value, mostly related to hyperparameters.
\subsection{Descriptors, inputs to ADAVI}
We can analyse the model described in \cref{eq:gaussian_re_model} using the descriptors defined in \cref{eq:descriptors}. Those descriptors constitute the inputs our methodology needs to automatically derive the \textit{dual} architecture from the generative HBM:
\begin{equation}
\begin{aligned}
\mathcal{V} &= \{ \mu, M^G, X \} \\
\mathtt{P} &= \{ \mathcal{P}_0, \mathcal{P}_1 \} \\
\operatorname{Card} &= \{ \mathcal{P}_0 \mapsto N, \mathcal{P}_1 \mapsto G \} \\
\operatorname{Hier} &= \{ \mu \mapsto 2, M^G \mapsto 1, X \mapsto 0 \} \\
\operatorname{Shape} &= \{ \mu \mapsto (D,), M^G \mapsto (D,), X \mapsto (D,) \} \\
\operatorname{Link} &= \{ \mu \mapsto \operatorname{Identity}, M^G \mapsto \operatorname{Identity}, X \mapsto \operatorname{Identity} \}
\end{aligned}
\end{equation}
\subsection{Tabular results for the scaling experiment}
A tabular representation of the results presented in \cref{fig:scaling} can be seen in \cref{tab:GRE_scaling}.
\begin{table}[ht]
\centering
\begin{tabular}{rr|l|l|r|r|r}
\toprule
G & Type & Method & ELBO ($10^2$) & \# weights & \textcolor{black}{Inf. (s)} & \textcolor{black}{Amo. (s)} \\
\midrule
3 & \textcolor{black}{Grd truth proxy} & MF-VI & \ 2.45 ($\pm$ 0.15) & 10 & 5 & - \\
\rule{0pt}{3ex} & Non amortized & SNPE-C & \ 2.17 ($\pm$ 33) & 45,000 & 53,000 & - \\
& & TLSF-NA & \ 2.33 ($\pm$ 0.20) & 18,000 & 80 & - \\
& & CF-NA & \ 2.12 ($\pm$ 0.15) & 15,000 & 190 & - \\
\rule{0pt}{3ex} & Amortized & NPE-C & \ 2.33 ($\pm$ 0.15) & 12,000 & - & 920 \\
& & TLSF-A & \ 2.37 ($\pm$ 0.072) & 12,000 & - & 9,400 \\
& & CF-A & \ 2.36 ($\pm$ 0.029) & 16,000 & - & 7,400 \\
& & \textbf{ADAVI} & \ 2.25 ($\pm$ 0.14) & 12,000 & - & 11,000 \\
\hline
30 & \textcolor{black}{Grd truth proxy} & MF-VI & \ 24.4 ($\pm$ 0.41) & 64 & 18 & - \\
\rule{0pt}{3ex} & Non amortized & SNPE-C & -187 ($\pm$ 110) & 140,000 & 320,000 & - \\
& & TLSF-NA & \ 24.0 ($\pm$ 0.49) & 63,000 & 400 & - \\
& & CF-NA & \ 21,2 ($\pm$ 0.40) & 150,000 & 1,800 & - \\
\rule{0pt}{3ex} & Amortized & NPE-C & \ 23.6 ($\pm$ 50) & 68,000 & - & 6,000 \\
& & TLSF-A & \ 22.7 ($\pm$ 13) & 68,000 & - & 130,000 \\
& & CF-A & \ 23.8 ($\pm$ 0.06) & 490,000 & - & 68,000 \\
& & \textbf{ADAVI} & \ 23.2 ($\pm$ 0.89) & 12,000 & - & 140,000 \\
\hline
300 & \textcolor{black}{Grd truth proxy} & MF-VI & \ 244 ($\pm$ 1.3) & 600 & 240 & - \\
\rule{0pt}{3ex} & Non amortized & SNPE-C & -9,630 ($\pm$ 3,500) & 1,100,000 & 3,100,000 & - \\
& & TLSF-NA & \ 243 ($\pm$ 1.8) & 960,000 & 5,300 & - \\
& & CF-NA & \ 212 ($\pm$ 1.5) & 1,500,000 & 30,000 & - \\
\rule{0pt}{3ex} & Amortized & NPE-C & \ 195 ($\pm 3\times10^6$)\textsuperscript{1} & 3,200,000 & - & 72,000 \\
& & TLSF-A & \ 202 ($\pm$ 120) & 3,200,000 & - & 2,800,000 \\
& & CF-A & \ 238 ($\pm$ 0.1) & 4,900,000 & - & 580,000 \\
& & \textbf{ADAVI} & \ 224 ($\pm$ 9.4) & 12,000 & - & 1,300,000 \\
\bottomrule
\end{tabular}
\caption{Scaling comparison on the Gaussian random effects example (see \cref{fig:exp-graphs}-GRE). Methods are ran over 20 random seeds (Except for SNPE-C and TLSF: to limit computational resources usage, those non-amortized computationally intensive methods were only ran on 5 seeds per sample, for a number of effective runs of 100).
Are compared: from left to right ELBO median (higher is better) and standard deviation (ELBO for all techniques except for Cascading Flows, for which ELBO is the numerically comparable \textit{augmented ELBO} \citep{ranganath_hierarchical_2016}); number of trainable parameters (weights) in the model; \textcolor{black}{for non-amortized techniques: CPU inference time for one example (seconds); for amortized techniques: CPU amortization time (seconds).}
\textsuperscript{1}- Results for NPE-C are extremely unstable, with multiple NaN results: the median value is rather random and not necessarily indicative of a good performance}
\label{tab:GRE_scaling}
\end{table}
\subsection{Derivation of an analytic posterior}
To have a ground truth to which we can compare our methods results, we derive the following analytic posterior distributions.
Assuming we know $\sigma_\mu,\ \sigma_g,\ \sigma_x$:
\begin{subequations}
\begin{align}
\label{eq:analytical_GRE}
\hat \mu^g &= \frac{1}{N}\sum_{n=1}^N x_n^g \\
\tilde \mu^g | \hat \mu^g &\sim \mathcal{N}\left( \hat \mu^g, \frac{\sigma_x^2}{N}\operatorname{Id}_D \right) \label{eq:mu_g_approx}\\
\hat \mu &= \frac{1}{G}\sum_{g=1}^G \hat \mu^g \\
\tilde \mu | \hat \mu &\sim \mathcal{N}\left( \frac{\frac{G}{\sigma_g^2}\hat \mu}{\frac{1}{\sigma_\mu^2} + \frac{G}{\sigma_g^2}}, \frac{1}{\frac{1}{\sigma_\mu^2} + \frac{G}{\sigma_g^2}}\operatorname{Id}_D \right)
\end{align}
\end{subequations}
Where in equation \ref{eq:mu_g_approx} we neglect the influence of the prior (against the evidence) on the posterior in light of the large number of points drawn from the distribution. We note that this analytical posterior is \textit{conjugate}, as argued in \cref{exp:NC}.
\subsection{Training losses full derivation and comparison}
\label{sec:all_losses_derivation}
\paragraph{Full formal derivation}
Following the nomenclature introduced in \citet{papamakarios_normalizing_2019}, there are 2 different ways in which we could train our variational distribution:
\begin{itemize}
\item using a \textit{forward} KL divergence, benefiting from the fact that we can sample from our generative model to produce a dataset $\{ (\theta^m, X^m) \}_{m=1\hdots M}, \theta^m \sim p(\theta), X^m \sim p(X|\theta)$. This is the loss used in most of the SBI literature \citep{cranmer_frontier_2020}, as those are based around the possibility to be \textit{likelihood-free}, and have a target density $p$ only implicitly defined by a simulator:
\begin{equation}
\begin{aligned}
\Psi^\star &= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\operatorname{KL}(p(\theta | X) || q_\Psi(\theta | X)] \\
&= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\mathbb{E}_{\theta \sim p(\theta | X)}[\log p(\theta | X) - \log q_\Psi(\theta | X)]] \\
&= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\mathbb{E}_{\theta \sim p(\theta | X)}[- \log q_\Psi(\theta | X)]] \\
&= \arg \min_\Psi \int p(X) \big[ \int - p(\theta | X) \log q_\Psi(\theta | X) d\theta \big] dX \\
&= \arg \min_\Psi \int \int - p(X, \theta) \log q_\Psi(\theta | X) d\theta dX \\
&\approx \arg \min_\Psi \frac{1}{M} \ \times \ \sum_{m=1}^M -\log q_\Psi(\theta^m | X^m) \\
&\text{where } \theta^m \sim p(\theta), X^m \sim p(X|\theta)
\end{aligned}
\end{equation}
\item using a \textit{reverse} KL divergence, benefiting from the access to a target joint density $p(X, \theta)$. The reverse KL loss is an amortized version of the classical ELBO expression \citep{blei_variational_2017}. For training, one only needs to have access to a dataset $\{ X^m \}_{m=1\hdots M}, X^m\sim p(X)$ of points drawn from the generative HBM of interest. Indeed, the $\theta^m$ points are sampled from the variational distribution:
\begin{equation}
\begin{aligned}
\Psi^\star &= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\operatorname{KL}(q_\Psi(\theta | X) || p(\theta | X)] \\
&= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\mathbb{E}_{\theta \sim q_\Psi(\theta | X)}[\log q_\Psi(\theta | X) - \log p(\theta | X)]] \\
&= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\mathbb{E}_{\theta \sim q_\Psi(\theta | X)}[\log q_\Psi(\theta | X) - \log p(X, \theta) + \log p(X)]] \\
&= \arg \min_\Psi \mathbb{E}_{X \sim p(X)}[\mathbb{E}_{\theta \sim q_\Psi(\theta | X)}[\log q_\Psi(\theta | X) - \log p(X, \theta)]] \\
&= \arg \min_\Psi \int p(X) \big[ \int q_\Psi(\theta | X) [\log q_\Psi(\theta | X) - \log p(X, \theta) ] d\theta \big] dX \\
&= \arg \min_\Psi \int \int p(X) q_\Psi(\theta|X) [\log q_\Psi(\theta | X) - \log p(X, \theta) ] d\theta dX \\
&\approx \arg \min_\Psi \frac{1}{M} \ \times \ \sum_{m=1}^M \log q_\Psi(\theta^m | X^m) - \log p(X^m, \theta^m) \\
&\text{where } X^m \sim p(X), \theta^m \sim q_\Psi(\theta | X)
\end{aligned}
\end{equation}
\end{itemize}
As it more uniquely fits our setup and provided better results experimentally, we chose to focus on the usage of the \textit{reverse} KL divergence. During our experiments, we also tested the usage of the \textit{unregularized ELBO} loss:
\begin{equation}
\begin{aligned}
\Psi^\star &= \arg \min_\Psi \frac{1}{M} \ \times \ \sum_{m=1}^M - \log p(X^m, \theta^m) \\
&\text{where } X^m \sim p(X), \theta^m \sim q_\Psi(\theta | X)
\end{aligned}
\end{equation}
This formula differs from the one of the reverse KL loss by the absence of the term $q_\Psi(\theta^m | X^m)$, and is a converse formula to the one of the forward KL (in the sense that it permutes the roles of $q$ and $p$).
Intuitively, it posits our architecture as a pure sampling distribution that aims at producing points $\theta^m$ in regions of high joint density $p$. In that sense, it acts as a first moment approximation for the target posterior distribution (akin to MAP parameter regression). Experimentally, the usage of the \textit{unregularized ELBO} loss provided fast convergence to a mode of the posterior distribution, with very low variance for the variational approximation.
We argue the possibility to use the \textit{unregularized ELBO} loss as a \textit{warm-up} before switching to the reverse KL loss, with the latter considered here as a regularization of the former. We introduce this training strategy as an example of the modularity of our approach, where one could transfer the rapid learning from one task (amortized mode finding) to another task (amortized posterior estimation).
\paragraph{Graphical comparison}
In Figure \ref{fig:losses_comparison} we analyse the influence of these 3 different losses on the training of our posterior distribution, compared to the analytical ground truth. This example is typical of the relative behaviors induced on the variational distributions by each loss:
\begin{itemize}
\item The forward KL provides very erratic training, and results after several dozen epochs (several minutes) with a careful early stopping in posteriors with too large variance.
\item The unregularized ELBO loss converges in less then 3 epochs (a couple dozen seconds), and provides posteriors with very low variance, concentrated on the MAP estimates of their respective parameters.
\item The reverse KL converges in less 10 epochs (less than 3 minutes) and provides relevant variance.
\end{itemize}
\begin{figure}
\includegraphics[width=\textwidth]{FigC1.pdf}
\caption{Graphical results on the Gaussian random effects example for our architecture trained using 3 different losses. Rows represent 3 different data points. Left column represents data, with colors representing 3 different groups. Other columns represent posterior samples for $\mu$ (black) and ${\color{blue}\mu^1}, {\color{orange}\mu^2}, {\color{green}\mu^3}$. Visually, posterior samples ${\color{blue}\mu^1}, {\color{orange}\mu^2}, {\color{green}\mu^3}$ should be concentrated around the mean of the data points with the same color, and the black points $\mu$ should be repartitioned around the mean of the 3 group means (with a shift towards 0 due to the prior). Associated with the posterior samples are analytical solutions (thin black circles), centered on the analytical MAP point, and whose radius correspond to 2 times the standard deviation of the analytical posterior: 95 \% of the draws from a posterior should fall within the corresponding circle.
}
\label{fig:losses_comparison}
\end{figure}
\paragraph{Losses convergence speed}
We analyse the relative convergence speed of our variational posterior to the analytical one (derived in \cref{eq:analytical_GRE}) when using the 3 aforementioned losses for training. To measure the convergence, we compute analytically the KL divergence between the variational posterior and the analytical one (every distribution being a Gaussian), summed for every distribution, and averaged over a validation dataset of size 2000.
We use a training dataset of size 2000, and for each loss repeated the training 20 times (batch size 10, 10 $\theta^m$ samples per $X^m$) for 10 epochs, resulting in 200 optimizer calls. This voluntary low number allows us to asses how close is the variational posterior to the analytical posterior after only a brief training. Results are visible in \cref{tab:loss_convergence}, showing a faster convergence for the \textit{unregularized ELBO}. After 800 more optimizer calls, the tendency gets inverted and the \textit{reverse KL} loss appears as the superior loss (though we still notice a larger variance).
The large variance in the results may point towards the need for adapted training strategies involving Learning rate decay and/or scheduling \citep{kucukelbir_automatic_2016}, an extension that we leave for future work.
\begin{table}
\label{tab:loss_convergence}
\centering
\begin{tabular}{r c c c c c}
\ & \ & \multicolumn{4}{c}{Mean of analytical KL divergences} \\
\ & \ & \multicolumn{4}{c}{from the theoretical posterior (low is good)} \\
\cmidrule(r){3-4} \cmidrule(r){5-6}
\ & \ & \multicolumn{2}{c}{Early stopping} & \multicolumn{2}{c}{After convergence} \\
\cmidrule(r){3-4} \cmidrule(r){5-6}
Loss & NaN runs & Mean & Std & Mean & Std \\
\cmidrule(r){1-1} \cmidrule(r){2-2} \cmidrule(r){3-4} \cmidrule(r){5-6}
forward KL & 0 & 3847.7 & 5210.4 & 2855.3 & 4248.1 \\
unregularized ELBO & 0 & 6.6 & 0.7 & 6.2 & 0.9 \\
reverse KL & 2 & 12.3 & 19.8 & 3.0 & 4.1
\end{tabular}
\caption{Convergence of the variational posterior to the analytical posterior over an early stopped training (200 batches) and after convergence (1000 batches) for the Gaussian random effects example}
\end{table}
\subsection{Monte Carlo approximation for the gradients and computational budget comparison}
\label{sec:MC_budget}
In section \ref{sec:training}, for the reverse KL loss, we approximate expectations using Monte Carlo integration. We further train our architecture using minibatch gradient descent, as opposed to stochastic gradient descent as proposed by \citet{kucukelbir_automatic_2016}. An interesting hyper-parametrization of our system resides in the effective batch size of our training, that depends upon:
\begin{itemize}
\item the size of the mini batches, determining the number of $X^m$ points considered in parallel
\item the number of $\theta^m$ draws per $X^m$ point, that we use to approximate the gradient in the ELBO
\end{itemize}
More formally, we define a computational budget as the relative allocation of a constant effective batch size $\operatorname{batch\ size} \times \ \theta \ \operatorname{draws\ per\ X}$ between $\operatorname{batch\ size}$ and $\theta \ \operatorname{draws\ per\ X}$.
To analyse the effect of the computational budget on training, we use a dataset of size 1000, and run experiment 20 times over the same number of optimizer calls with the same effective batch size per call. Results can be seen in \cref{fig:budget}. From this experiment we can draw the following conclusions:
\begin{itemize}
\item we didn't witness massive difference in the global convergence speed across computational budgets
\item the bigger the budget we allocate to the sampling of multiple $\theta^m$ per point $X^m$ (effectively going towards a stochastic training in terms of the points $X^m$), the more erratic is the loss evolution
\item the bigger the budget we allocate to the $X^m$ batch size, the more stable is the loss evolution, but our interpretation is that the resulting reduced number of $\theta^m$ draws per $X^m$ augments the risk of an instability resulting in a NaN run
\end{itemize}
Experimentally, we obtained the best results by evenly allocating our budget to the $X^m$ batch size and the number of $\theta^m$ draws per $X^m$ point (typically, 32 and 32 respectively for an effective batch size of 1024). Overall, in the amortized setup, our experiment stand as a counterpoint to those of \citet{kucukelbir_automatic_2016} who pointed towards the case of a single $\theta^m$ draw per point $X^m$ as their preferred hyper-parametrization.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{FigC2_a.pdf}
\includegraphics[width=0.5\textwidth]{FigC2_b.pdf}
\caption{Loss evolution across batches for different computational budgets. All experiments are designed so that to have the same number of optimizer calls (meaning that $\operatorname{batch\ size} \times \operatorname{epochs} = 1000$) and the same effective batch size (meaning that $\operatorname{batch\ size} \times \ \theta \ \operatorname{draws\ per\ X} = 1000$). Every experiment is run 20 times, error bands showing the standard deviation of the loss at the given time point. Note that the {\color{blue} blue} line (batch size 1, 1000 $\theta$ draws per $X$) is more erratic than the other ones (even after a large number of batches). On the other hand, the {\color{red} red} line (batch size 1000, 1 $\theta$ draws per $X$) is more stable, but 19 out of 20 runs ultimately resulted in an instability}
\label{fig:budget}
\end{figure}
\section{Complements to the Gaussian mixture with random effects experiment: further posterior analysis}
This section is a complement to the experiment described in \cref{exp:GM}, we thus consider the model described in \cref{eq:mixture_of_gaussians}. We explore the complexity of the theoretical posterior for this experiment.
\subsection{Descriptors, inputs to ADAVI}
We can analyse the model described in \cref{eq:mixture_of_gaussians} using the descriptors defined in \cref{eq:descriptors}. Those descriptors constitute the inputs our methodology needs to automatically derive the \textit{dual} architecture from the generative HBM:
\begin{equation}
\begin{aligned}
\mathcal{V} &= \{ M^L, M^{L, G}, \Pi^G, X \} \\
\mathtt{P} &= \{ \mathcal{P}_0, \mathcal{P}_1 \} \\
\operatorname{Card} &= \{ \mathcal{P}_0 \mapsto N, \mathcal{P}_1 \mapsto G \} \\
\operatorname{Hier} &= \{ M^L \mapsto 2, M^{L, G} \mapsto 1, \Pi^G \mapsto 1, X \mapsto 0 \} \\
\operatorname{Shape} &= \{ M^L \mapsto (L, D), M^{L,G} \mapsto (L, D), \Pi^G \mapsto (L,), X \mapsto (D,) \} \\
\operatorname{Link} &= \{ \\
M^L &\mapsto \operatorname{Reshape}((LD,) \rightarrow (L, D)),\\
M^{L,G} &\mapsto \operatorname{Reshape}((LD,) \rightarrow (L, D)),\\
\Pi^G &\mapsto \operatorname{SoftmaxCentered}((L - 1,) \rightarrow (L,)),\\
X &\mapsto \operatorname{Identity}\\
\} \ \ \ \ \ \ \ \ \ \
\end{aligned}
\end{equation}
For the definition of the $\operatorname{SoftmaxCentered}$ link function, see \citet{dillon_tensorflow_2017}.
\subsection{Theoretical posterior recovery in the Gaussian mixture random effects model}
\label{sec:label-switching}
We further analyse the complexity of model described in \cref{exp:GM}.
Due to the label switching problem \citep{jasra_markov_2005}, the relative position of the L mixture components in the D space is arbitrary. Consider a non-degenerate example like the one in \cref{fig:multimodality}, where the data points are well separated in 3 blobs (likely corresponding to the $L=3$ mixture components). Since there is no deterministic way to assign component $l=1$ unequivocally to a blob of points, the marginalized posterior distribution for the position of the component $l=1$ should be multi-modal, with -roughly- a mode placed at each one of the 3 blobs of points. This posterior would be the same for the components $l=2$ and $l=3$. In truth, the posterior is even more complex than this 3-mode simplification, especially when the mixture components are closer to each other in 2D (and the grouping of points into draws from a common component is less evident).
In \cref{fig:multimodality}, we note that our technique doesn't recover this multi-modality in its posterior, and instead assigns different posterior components to different blobs of points. Indeed, when plotting only the posterior samples for the first recovered component $l=1$, all points are concentrated around the bottom-most blob, and not around each blob like the theoretical posterior would entail (see \cref{fig:multimodality} second row).
This behavior most likely represents a local minimum in the reverse KL loss that is common to many inference techniques \citep[for instance consider multiple non-mixing chains for MCMC in][]{jasra_markov_2005}. We note that training in forward KL wouldn't provide such a flexibility in that setup, as it would enforce the multi-modality of the posterior, even at the cost of an overall worst result (as it is the case for NPE-C and SNPE-C in \cref{tab:NC&GM}. Indeed, let's imagine that our training dataset features $M'$ draws similar to the one in \cref{fig:multimodality}. Out of randomness, the labelling $l$ of the 3 blobs of points would be permuted across those $M'$ examples. A forward-KL-trained density estimator would then most likely attempt to model a multi-modal posterior.
Though it is not similar to the theoretical result, we argue that our result is of experimental value, and close to the intuition one forms of the problem: using our results one can readily estimate the original components for the mixture. Indeed, for argument's sake, say we would recover the theoretical, roughly trimodal posterior. To recover the original mixture components, one would need to split the 3 modes of the posterior and arbitrarily assign a label $l$ to each one of the modes. In that sense, our posterior naturally features this splitting, and can be used directly to estimate the $L=3$ mixture components.
\begin{figure}
\includegraphics[width=\textwidth, trim={3cm 3cm 3cm 3cm}, clip]{FigD3.pdf}
\caption{Graphical comparison for various methods on the Gaussian mixture with random effects example. First column represents a non-degenerate data point, with colored points corresponding to ${\color{blue} [x^{1, 1},...,x^{1, N}]}$, ${\color{orange} [x^{2, 1},...,x^{2, N}]}$, ${\color{green} [x^{3, 1},...,x^{3, N}]}$. Note the distribution of the points around in 3 multi-colored groups (population components), and 3 colored sub-groups per group (group components). All other columns represent the posterior samples for population mixture components $\mu_1,\ \hdots \ ,\mu_3$ (black) and group mixture components ${\color{blue}\mu_1^1,\ \hdots \ ,\mu_3^1}, {\color{orange}\mu_1^2,\ \hdots \ ,\mu_3^2}, {\color{green}\mu_1^3,\ \hdots \ ,\mu_3^3}$. Second column represents the results of a non-amortized MF-VI (best ELBO score across all random seeds): results are typical of a local minimum for the loss. Third column represents the result of an amortized CF-A (best amortized ELBO). Last column represents our amortized ADAVI technique (best amortized ELBO). First row represents the full posterior samples. Second and third row only represents the first mixture component samples (third row zooms in on the data).
Notice how neither technique recovers the actual multi-modality of the theoretical posterior.
We obtain results of good experimental value, usable to estimate likely population mixture components.
}
\label{fig:multimodality}
\end{figure}
\section{Complements to the neuroimaging experiment}
This section is a complement to the experiment described in \cref{sec:exp-yeo}, we thus consider the model described in \cref{eq:mshbm_model} and \cref{eq:mshbm_x}. We present a toy dimension version of our experiment, useful to build an intuition of the problem. We also present implementation details for our experiment, and additional neuroimaging results.
\subsection{Neuroimaging context}
The main goal of \citet{kong_spatial_2018} is to address the classical problem in neuroscience of estimating population commonalities along with individual characteristics. In our experiment, we are interested in parcelling the region of left inferior frontal gyrus (IFG). Anatomically, the IFG is decomposed in 2 parts: pars opercularis and triangularis. Our aim is to reproduce this binary split from a functional connectivity point of view, an open problem in neuroscience~\citep[see e.g.][]{heimEffectiveConnectivityLeft2009}.
As \citet{kong_spatial_2018}, we consider a population of S=30 subjects, each with $T=4$ acquisition sessions, from the Human Connectome Project dataset~\citep{pmid22366334}. The fMRI connectivity between a cortical point and the rest of the brain, split in $D=1,483$ regions, is represented as a vector of length D with each component quantifying the temporal correlation of blood-oxygenation between the point and a region. A main hypothesis of \citet{kong_spatial_2018}, and the fMRI field, is that the fMRI connectivity of points belonging to the same parcel share a similar connectivity pattern or correlation vector.
Following \citet{kong_spatial_2018}, we represent D-dimensional correlation vectors as RVs on the positive quadrant of the $D$-dimensional unit-sphere. We do this efficiently assuming they have a $\mathcal{L}$-normal distribution, or Gaussian under the transformation of the link function $\mathcal L(x)=\sqrt{\operatorname{SoftmaxCentered}(x)}$~\citep{dillon_tensorflow_2017}:
\textcolor{black}{
\begin{subequations}
\begin{equation}
\label{eq:mshbm_model}
\begin{aligned}
\pi,\ s^-,\ s^+ &= 2,\ -10,\ 8 &\
\\
\mathcal{L}^{-1}(\mu^g_l) &\bm{\sim} \mathcal{N}(\vec 0_{D-1}, \Sigma_g) &\
M^L &= [\mu^g_l]^{l=1\hdots L} \\
\log(\epsilon_l) &\bm{\sim} \mathcal{U}(s^-, s^+) &\
E^L &= [\epsilon_l]^{l=1\hdots L} \\
\mathcal{L}^{-1}(\mu^s_l) \boldsymbol{\mid} \mu^g_l, \ \epsilon_l &\bm{\sim} \mathcal{N}(\mathcal{L}^{-1}(\mu_l^g), \epsilon_l^2) &\
M^{L,S} &= [\mu_l^s]^{\substack{l=1\hdots L \\ s=1\hdots S}} \\
\log(\sigma_l) &\bm{\sim} \mathcal{U}(s^-, s^+) &\
\Sigma^L &= [\sigma_l]^{l=1\hdots L} \\
\mathcal{L}^{-1}(\mu^{s, t}_l) \boldsymbol{\mid} \mu^s_l, \ \sigma_l &\bm{\sim} \mathcal{N}(\mathcal{L}^{-1}(\mu_l^s), \sigma_l^2) &\
M^{L,S,T} &= [\mu_l^{s,t}]{\substack{l=1\hdots L \\ s=1\hdots S \\ t=1\hdots T}} \\
\log(\kappa) &\bm{\sim} \mathcal{U}(s^-, s^+) &\
\\
\Pi &\bm{\sim} \operatorname{Dir}([\pi] \times L) &\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\mathcal{L}^{-1}(X_n^{s, t}) \boldsymbol{\mid} [\mu_1^{s, t},\ \hdots \ ,\mu_L^{s, t}], \kappa, \Pi &\bm{\sim}\operatorname{Mix}(\Pi, [\textit{N}(\mathcal{L}^{-1}(\mu_1^{s, t}), \kappa^2), \ \hdots \ ,\textit{N}(\mathcal{L}^{-1}(\mu_L^{s, t}), \kappa^2)]) \\
X &= [X_n^{s,t}]{\substack{s=1\hdots S \\ t=1\hdots T \\ n=1\hdots N}}
\end{aligned}
\label{eq:mshbm_x}
\end{equation}
\end{subequations}
}
Our aim is therefore to identify $L=2$ functional networks that would produce a functional parcellation of the studied IFG section. In this setting, the parameters $\theta$ of interest are the networks $\mu$. Instead of the complex EM computation derived in \citet{kong_spatial_2018}, we perform full-posterior inference for those parameters using our automatically derived architecture.
\subsection{Descriptors, inputs to ADAVI}
We can analyse the model described in \cref{eq:mshbm_model} and \cref{eq:mshbm_x} using the descriptors defined in \cref{eq:descriptors}. Those descriptors constitute the inputs our methodology needs to automatically derive the \textit{dual} architecture from the generative HBM:
\begin{equation}
\begin{aligned}
\mathcal{V} &= \{ M^L, E^L, M^{L,S}, \Sigma^L, M^{L,S,T}, \kappa, \Pi, X \} \\
\mathtt{P} &= \{ \mathcal{P}_0, \mathcal{P}_1, \mathcal{P}_2 \} \\
\operatorname{Card} &= \{ \mathcal{P}_0 \mapsto N, \mathcal{P}_1 \mapsto T, \mathcal{P}_2 \mapsto S \} \\
\operatorname{Hier} &= \{ M^L \mapsto 3, E^L \mapsto 3, M^{L,S} \mapsto 2, \Sigma^L \mapsto 3, M^{L,S,T} \mapsto 1, \kappa \mapsto 3, \Pi \mapsto 3, X \mapsto 0 \} \\
\operatorname{Shape} &=\{ \\
M^L &\mapsto (L, D),\\
E^L &\mapsto (L,),\\
M^{L,S} &\mapsto (L, D),\\
\Sigma^L &\mapsto (L,),\\
M^{L,S,T} &\mapsto (L, D),\\
\kappa &\mapsto (1,),\\
\Pi &\mapsto (L,),\\
X &\mapsto (D,)\\
\} \ \ \ \ \ \ \ \ \ \
\\
\operatorname{Link} &=\{ \\
M^L &\mapsto \mathcal{L} \circ \operatorname{Reshape}((LD,) \rightarrow (L, D)),\\
E^L &\mapsto \operatorname{Exp}, \\
M^{L,S} &\mapsto \mathcal{L} \circ \operatorname{Reshape}((LD,) \rightarrow (L, D)),\\
\Sigma^L &\mapsto \operatorname{Exp},\\
M^{L,S,T} &\mapsto \mathcal{L} \circ \operatorname{Reshape}((LD,) \rightarrow (L, D)),\\
\kappa &\mapsto \operatorname{Exp},\\
\Pi &\mapsto \operatorname{SoftmaxCentered}((L - 1,) \rightarrow (L,)),\\
X &\mapsto \mathcal{L}\\
\} \ \ \ \ \ \ \ \ \ \
\end{aligned}
\end{equation}
\subsection{Experiment on MS-HBM model on toy dimensions}
\label{sec:MSHBM_toy}
To get an intuition of the behavior of our architecture on the MS-HBM model, we consider the following toy dimensions reproduction of the model:
\begin{equation}
\begin{aligned}
N, T, S, D, L &= 50, 2, 2, 2, 2 \\
g^-, g^+ &= -4, 4 \\
\kappa^-,\ \kappa^+,\ \sigma^-,\ \sigma^+,\ \epsilon^-,\ \epsilon^+ &= -4,\ -4,\ -3,\ -3,\ -2,\ -2,\ -1 \\
\pi &= 2 \\
\mathcal{L}^{-1}(\mu^g_l) &\sim \mathcal{U}(-g^-, g^+) \\
\log(\epsilon_l) &\sim \mathcal{U}(\epsilon^-, \epsilon^+) \\
\mathcal{L}^{-1}(\mu^s_l) | \mu^g_l, \ \epsilon_l &\sim \mathcal{N}(\mathcal{L}^{-1}(\mu_l^g), \epsilon_l^2) \\
\log(\sigma_l) &\sim \mathcal{U}(\sigma^-, \sigma^+) \\
\mathcal{L}^{-1}(\mu^{s, t}_l) | \mu^s_l, \ \sigma_l &\sim \mathcal{N}(\mathcal{L}^{-1}(\mu_l^s), \sigma_l^2) \\
\log(\kappa) &\sim \mathcal{U}(\kappa^-, \kappa^+) \\
\Pi &\sim \operatorname{Dir}([\pi] \times L) \\
\mathcal{L}^{-1}(X_n^{s, t}) | [\mu_1^{s, t},\ \hdots \ ,\mu_L^{s, t}], \kappa, \Pi &\sim \operatorname{Mix}(\Pi, [\mathcal{N}(\mathcal{L}^{-1}(\mu_1^{s, t}), \kappa^2), \ \hdots \ ,\mathcal{L}(\mathcal{L}^{-1}(\mu_L^{s, t}), \kappa^2)])
\end{aligned}
\end{equation}
The results can be visualized on \cref{fig:synthetic_yeo}. This experiment shows the expressivity we gain from the usage of link functions.
\begin{figure}
\centering
\includegraphics[height=0.8\textheight]{FigE4.pdf}
\caption{Visual representation of our results on a synthetic MS-HBM example. Data is represented as colored points on the unit positive quadrant, each color corresponding to a subject $\times$ session. Samples from posterior distributions are represented as concentric colored markings. Just below the data points are $\mu^{s, t}$ samples. Then samples of $\mu^s$. Then samples of $\mu^g$ (black lines). Notice how the $\mu$ posteriors are distributed around the angle bisector of the arc covered by the points at the subsequent plate.}
\label{fig:synthetic_yeo}
\end{figure}
\subsection{Implementation details for the neuroimaging MS-HBM example}
\label{sec:yeo_implem}
\paragraph{Main implementation differences with the original MS-HBM model}
Our implementation of the MS-HBM (\cref{eq:mshbm_model} and \cref{eq:mshbm_x}) contains several notable differences with the original one from \citet{kong_spatial_2018}:
\begin{itemize}
\item we model $\mu$ distributions as Gaussians linked to the positive quadrant of the unit sphere via the function $\mathcal{L}$. In the orignal model, RVs are modelled using \textit{Von Mises Fisher} distributions. Our choice allows us to express the entirety of the connectivity vectors (that only lie on a portion of the unit sphere). However, we also acknowledge that the densities incurred by the 2 distributions on the positive quadrant of the unit sphere are different.
\item we forgo any spatial regularization, and also the assumption that the parcellation of a given subject $s$ should be constant across sessions $t$. This is to streamline our implementation. Adding components to the loss optimized at training could inject those constraints back into the model, but this was not the subject of our experiment, so we left those for future work.
\end{itemize}
\paragraph{Data pre-processing and dimensionality reduction}
Our model was able to run on the full dimensionality of the connectivity, $D^0=1483$. However, we obtained better results experimentally when further pre-processing the used data down to the dimension $D^1=141$. The displayed results in \cref{fig:kong_experiment} are the ones resulting from this dimensionality reduction:
\begin{enumerate}
\item we projected the $(S, T, N, D^0)$ $X$ connectome (lying on the $D^0$ unit sphere) to the unbounded $\mathbb{R}^{D^0 - 1}$ space using the function $\mathcal{L}$
\item in this $\mathbb{R}^{D^0 - 1}$ space, we performed a Principal Component Analysis (PCA) to bring us down to $D^1 - 1 = 140$ dimensions responsible for 80\% of the explained data variance
\item in the resulting $\mathbb{R}^{D^1 - 1}$ space, we calculated the mean of all the connectivity points, and their standard deviation, and used both to whiten the data
\item from the whitened data, we calculated the Ledoit-Wolf regularised covariance \citep{LEDOIT2004365}, that we used to construct the $\Sigma_g$ matrix used in \cref{eq:mshbm_model}
\item finally, we projected the whitened data onto the unit sphere in $D^1=141$ dimensions via the function $\mathcal{L}$
\end{enumerate}
To project our results back to the original $D^0$ space, we simply ran back all the aforementioned steps. Our prior for $\mu^g$ has been carefully designed so has to sample connectivity points in the vicinity of the data point of interest. Our implementation is therefore in spirit close to SBI \citep{cranmer_frontier_2020, papamakarios_sequential_2019, greenberg_automatic_2019, thomas_likelihood-free_2020} that aims at obtaining an amortized posterior only in the relevant data regime.
\paragraph{Mutli-step training strategy}
\label{sec:multi_step_training}
In \cref{sec:implem} we describe our conditional density estimators as the stacking of a MAF \citep{papamakarios_masked_2018} on top of a diagonal-scale affine block. To accelerate the training of our architecture and minimize numerical instability (resulting in NaN evaluations of the loss) we used the following 3-step training strategy:
\begin{enumerate}
\item we only trained the \textit{shift} part of our affine block into a Maximum A Posteriori regression setup. This can be viewed as the amortized fitting of the first moment of our posterior distribution
\item we trained both the \textit{shift} and \textit{scale} of our affine block using an \textit{unregularized ELBO} loss. This is to rapidly bring the variance of our posterior to relevant values
\item we then trained our full posterior (\textit{shift} and \textit{scale} of our affine block, in addition to our MAF block) using the \textit{reverse KL} loss.
\end{enumerate}
This training strategy shows the modularity of our approach and the transfer learning capabilities already introduced in \cref{sec:all_losses_derivation}. Loss evolution can be seen in \cref{fig:3fold_loss}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{FigE5.pdf}
\caption{3-step loss evolution across epochs for the MS-HBM ADAVI training. Losses switch are visible at epochs 1000 and 2000. Training was run for a longer period after epoch 3000, with no significant results difference.}
\label{fig:3fold_loss}
\end{figure}
\paragraph{Soft labelling}
In \cref{eq:mshbm_model} and \cref{eq:mshbm_x} we define $\mu$ variables as following Gaussian distributions in the latent space $\mathbb{R}^{D^1 - 1}$. This means that, considering a vertex $X_n^{s, t}$ and a session network $\mu_k^{s, t}$, the squared Euclidean distance between $\mathcal{L}^{-1}(X_n^{s, t})$ and $\mathcal{L}^{-1}(\mu_k^{s, t})$ in the latent space $\mathbb{R}^{D^1 - 1}$ is proportional to the log-likelihood of the point $\mathcal{L}^{-1}(X_n^{s, t})$ for the mixture component $k$:
\begin{equation}
\label{eq:x_likelihood}
\begin{aligned}
\lVert \mathcal{L}^{-1}(X_n^{s, t}) - \mathcal{L}^{-1}(\mu_k^{s, t}) \rVert^2 &= \log p(X_n^{s, t} | l=k) + C(\kappa)
\end{aligned}
\end{equation}
Note that $\kappa$ is the same for both networks. Additionally, considering Bayes theorem:
\begin{equation}
\label{eq:likelihood_ratio}
\begin{aligned}
\log p(l=k | X_n^{s, t}) &= \log p(X_n^{s, t} | l=k) + \log p(l=k) - \log p(X_n^{s, t}) \\
\log \frac{p(l=0 | X_n^{s, t})}{p(l=1 | X_n^{s, t})} &= \log p(X_n^{s, t} | l=0) - \log p(X_n^{s, t} | l=1) + \log \frac{p(l=0)}{p(l=1)}
\end{aligned}
\end{equation}
Where $\log p(X_n^{s, t} | l=k)$ can be obtained through \cref{eq:x_likelihood} and $\log p(l=k)$ via draws from the posterior of $\Pi$ (see \cref{eq:mshbm_model}). To integrate those equations, we used a Monte Carlo procedure.
\subsection{Additional neuroimaging results}
\subsubsection{Subject-level parcellation}
As pointed out in \cref{sec:exp-yeo}, the MS-HBM model aims at representing the functional connectivity of the brain at different levels, allowing for estimates of population characteristics and individual variability \citep{kong_spatial_2018}. It is of experimental value to compare the parcellation for a given subject, that is to say the soft label we give to a vertex $X_n^{s, t}$, and how this subject parcellation can deviate from the population parcellation. Those differences underline how an individual brain can have a unique local organization. Similarly, we can obtain the subject networks $\mu^s$ and observe how those can deviate from the population networks $\mu^g$. Those results underline how a given subject can have his own connectivity, or, very roughly, his own "wiring" between different areas of the brain. Results can be seen in \cref{fig:subject_parcellation}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{FigE6.jpg}
\caption{Subject-level parcellations and networks. For 3 different HCP subjects, we display the individual parcellation (on the bottom) and the individual $\mu^s$ networks (on the top). Note how the individual parcellations, though showing the same general split between the \textit{pars opercularis} and \textit{pars triangularis}, slightly differ from each other and from the population parcellation (\cref{fig:kong_experiment}). Similarly, networks $\mu^s$ differ from each other and from the population networks $\mu^g$ (\cref{fig:kong_experiment}) but keep their general association to semantic/phonologic processing (0, in blue) and language production (1, in red) \citep{heimEffectiveConnectivityLeft2009, zhangConnectingConceptsBrain2020}. To be able to model and display such variability is one of the interests of models like the MS-HBM \citep{kong_spatial_2018}.}
\label{fig:subject_parcellation}
\end{figure}
\subsubsection{\textcolor{black}{Comparison of labelling with the MF-VI results}}
\textcolor{black}{We can compare the subject-level parcellation resulting from the latent networks recovered using the ADAVI vs the MF-VI method. The result, for the same subjects as the previous section, can be seen in \cref{fig:neuro-diff-mfvi}, where the difference in ELBO presented in our main text translates into marginal differences for our downstream task of interest.}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{FigE7.jpg}
\caption{\textcolor{black}{Gap in labelling between ADAVI and MF-VI results. Following \cref{eq:likelihood_ratio}, we compute the difference in latent space between the odds for the ADAVI and MF-VI techniques, before applying a sigmoid function. Differences are marginal, and interestingly located at the edges between networks, where the labelling is less certain.}}
\label{fig:neuro-diff-mfvi}
\end{figure}
\section{Baselines for experiments}
\subsection{Baseline choice}
\label{sec:baselines}
In this section we justify further the choice of architectures presented as baselines in the our experiments:
\begin{itemize}
\item \textit{Mean Field VI} (MF-VI) \citep{blei_variational_2017}. This methods stands as a common-practice non-amortized baseline, is fast to compute, and due to our choice of conjugate examples (see \cref{exp:NC}) can be considered as a proxy to the ground truth posterior. We implemented MF-VI in its usual setup, fitting to the posterior a distribution of the prior's parametric form;
\item \textit{(Sequential) Neural Posterior Estimation} (SNPE-C) architecture \citep{greenberg_automatic_2019}. NPE-C is an typical example from the SBI literature \citep{cranmer_frontier_2020}, and functions as a \textit{likelihood-free}, \textit{black box} method. Indeed, NPE-C is trained using forward KL (samples from the latent parameters), and is not made "aware" of any structure in the problem. NPE-C fits a single normalizing flow over the entirety of the latent parameter space, and its number of weights scales quadratically with the parameter space size. When ran over several simulation rounds, the method becomes \textit{sequential} (SNPE-C), specializing for a certain parameter regime to improve performance, but loosing amortization in the process;
\item \textit{Total Latent Space Flow} (TLSF) architecture \citep{rezende_variational_2016}. Following the original normalizing flow implementation from \citet{rezende_variational_2016}, we posit TLSF as a counterpoint to SNPE-C. Like SNPE-C, TLSF fits a single normalizing flow over the entirety of the latent parameter space, and is not made "aware" of the structure of the model. But contrary to SNPE-C, TLSF is trained using reverse KL and benefits from the presence of a likelihood function. We can use TLSF in a non-amortized setup (TLSF-NA), or in an amortized setup (TLSF-A) trough an observed data encoder conditioning the single normalizing flow;
\item \textit{Cascading Flows} (CF) \citep{pmlr-v139-ambrogioni21a}. CF is an example of a structure-aware, prior-aware VI method, trained using reverse KL. By design, its number of weights scales linearly with the plate's cardinalities. CF can be ran both in a non-amortized (CF-NA) and amortized (CF-A) setup, with the introduction of amortization through observed data encoders in the auxiliary graph. As a structure-aware amortized architecture, CF-A is our main point of comparison in this section;
\end{itemize}
\subsection{Implementation details}
\label{sec:implem}
In this section, we describe with precision and per experiment the implementation details for the architectures described in \cref{sec:baselines}. We implemented algorithms in Python, using the Tensorflow probability \citep[TFP, ][]{dillon_tensorflow_2017} and Simulation Based Inference \citep[SBI, ][]{tejero-cantero_sbi_2020} libraries. For all experiments in TFP, we used the Adam optimizer \citep{adam_kingma2014}. For normalizing flows, we leveraged Masked Autoregressive Flow \citep[MAF, ][]{papamakarios_fast_2018}.
For all experiments:
\begin{itemize}
\item \textit{Mean Field VI} (MF-VI) \citep{blei_variational_2017}. We implemented MF-VI in TFP. The precise form of the variational family is described below for each experiment.
\item \textit{Sequential Neural Posterior Estimation} (SNPE-C) architecture \citep{greenberg_automatic_2019}. We implemented SNPE-C with the SBI library, using the default parameters proposed by the API. Simulations were ran over $5$ rounds, to ensure maximal performance. We acknowledge that this choice probably results in an overestimate of the runtime for the algorithm. To condition the density estimation based on the observed data, we designed an encoder that is a variation of our Hierarchical Encoder (see \cref{sec:ADVI}). Its architecture is the same as $\operatorname{HE}$ -the hierarchical stacking of 2 Set Transformers \citep{pmlr-v97-lee19d}- but the encoder's output is the concatenation of the G per-group encodings with the population encodings. This encoder is therefore parsimoniously parametrized, and adapted to the structure of the problem.
\item \textit{Neural Posterior Estimation} (NPE-C). Though we acknowledge that NPE-C can be implemented easily using the SBI library, we preferred to use our own implementation, notably to have more control over the runtime of the algorithm. We implemented the algorithm using TFP. We used the same encoder architecture as for SNPE-C.
\item \textit{Total Latent Space Flow} (TLSF). We implemented TLSF using TFP. Our API is actually the same as for NPE-C, since TLSF-A and NPE-C only differ by their training loss.
\item \textit{Cascading Flows} (CF) \citep{pmlr-v139-ambrogioni21a}. We implemented our own version of Cascading Flows, using TFP, and having consulted with the authors. An important implementation detail that is not specified explicitly in \citet{pmlr-v139-ambrogioni21a} (whose notations we follow here) is the implementation of the target distribution over the auxiliary variables $r$, notably in the amortized setup. Following the authors specifications during our discussion, we implemented $r$ as the Mean Field distribution $r = \prod_j p_j(\epsilon_j)$.
\item ADAVI (ours). We implemented ADAVI using TFP.
\end{itemize}
Regarding the training data:
\begin{itemize}
\item All amortized methods were trained over a dataset of $20,000$ samples
\item All non-amortized methods except SNPE-C were trained on a single data point (separately for $20$ different data points)
\item SNPE-C was trained over 5 rounds of simulations, with $1000$ samples per round, for an effective dataset size of $5000$
\end{itemize}
For the non conjugate experiment (see \cref{exp:NC}):
\begin{itemize}
\item MF-VI: variational distribution is
\begin{align*}
q &=\textit{Gamma}(a; \text{concentration=}V_{(D,)}, \text{rate=}\operatorname{Softplus}(V_{(1,)}))
\end{align*}
We used the Adam optimizer with a learning rate of $10^{-2}$. The optimization was ran for $20,000$ steps, with a sample size of $32$.
\item CF: auxiliary size 8, observed data encoders with 8 hidden units. Minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-2}$), 40 epochs using a reverse KL loss.
\item ADAVI: NF with $1$ Affine block with triangular scale, followed by $1$ MAF with $[32, 32, 32]$ units. $\operatorname{HE}$ with embedding size $8$, $2$ modules with $2$ ISABs ($2$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 40 epochs using a reverse KL loss.
\end{itemize}
For the Gaussian random effects experiment (see \cref{exp:scaling}):
\begin{itemize}
\item MF-VI: variational distribution is
\begin{align*}
q &=\mathcal{N}(\mu; \text{mean=}V_{(D,)}, \text{std=}\operatorname{Softplus}(V_{(1,)})) \\ &\times \mathcal{N}([\mu^1,...,\mu^G] ; \text{mean=}V_{(G, D)}, \text{std=}\operatorname{Softplus}(V_{(1,)}))
\end{align*}
We used the Adam optimizer with a learning rate of $10^{-2}$. The optimization was ran for $10,000$ steps, with a sample size of $32$.
\item SNPE-C: $5$ MAF blocks with $50$ units each. Encoder with embedding size $8$, $2$ modules with $2$ SABs ($4$ heads) and $1$ PMA (seed size $1$) each, and $1$ linear unit. See SBI for optimization details.
\item NPE-C: 1 MAF with $[32, 32, 32]$ units. Encoder with embedding size $8$, $2$ modules with $2$ ISABs ($2$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 15 epochs with Adam ($10^{-3}$) using a forward KL loss.
\item TLSF: same architecture as NPE-C. TLSF-A: minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), 15 epochs with Adam ($10^{-3}$) using a reverse KL loss. TLSF-NA: minibatch size 1, 32 theta draws per X point (see \cref{sec:MC_budget}), 1500 epochs with Adam ($10^{-3}$) using a reverse KL loss.
\item CF: auxiliary size 16, observed data encoders with 16 hidden units. CF-A: minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 40 epochs using a reverse KL loss. CF-NA: minibatch size 1, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-2}$), 1500 epochs using a reverse KL loss.
\item ADAVI: NF with $1$ Affine block with triangular scale, followed by $1$ MAF with $[32, 32, 32]$ units. $\operatorname{HE}$ with embedding size $8$, $2$ modules with $2$ ISABs ($2$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 10 epochs using an unregularized ELBO loss, followed by 10 epochs using a reverse KL loss.
\end{itemize}
For the Gaussian mixture with random effects experiment (see \cref{exp:GM}):
\begin{itemize}
\item MF-VI: variational distribution is
\begin{align*}
q &=\mathcal{N}([\mu_1,\hdots,\mu_L]; \text{mean=}V_{(L, D,)}, \text{std=}\operatorname{Softplus}(V_{(1,)})) \\ &\times \mathcal{N}([[\mu_1^1,\hdots,\mu_L^1],...,[\mu_1^G,...,\mu_L^G]] ; \text{mean=}V_{(G, L, D)}, \text{std=}\operatorname{Softplus}(V_{(1,)})) \\ &\times \operatorname{Dir}(\text{concentration=}\operatorname{Softplus}(V_{(G, L)}))
\end{align*}
We used the Adam optimizer with a learning rate of $10^{-2}$. The optimization was ran for $10,000$ steps, with a sample size of $32$.
\item SNPE-C: $5$ MAF blocks with $50$ units each. Encoder with embedding size $8$, $2$ modules with $2$ SABs ($4$ heads) and $1$ PMA (seed size $1$) each, and $1$ linear unit. See SBI for optimization details.
\item NPE-C: 1 MAF with $[32, 32, 32]$ units. Encoder with embedding size $8$, $2$ modules with $2$ ISABs ($2$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 20 epochs with Adam ($10^{-3}$) using a forward KL loss.
\item TLSF-A: same architecture as NPE-C, minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), $250$ epochs with Adam ($10^{-3}$) using a reverse KL loss.
\item TLSF-NA: same NF architecture as NPE-C. Encoder with embedding size $16$, $2$ modules with $2$ ISABs ($2$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 1, 32 theta draws per X point (see \cref{sec:MC_budget}), $1000$ epochs with Adam ($10^{-3}$) using a reverse KL loss.
\item CF: auxiliary size 8, observed data encoders with 8 hidden units. CF-A: minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 200 epochs using a reverse KL loss. CF-NA: minibatch size 1, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-2}$), 1500 epochs using a reverse KL loss.
\item ADAVI: NF with $1$ Affine block with diagonal scale, followed by $1$ MAF with $[32]$ units. $\operatorname{HE}$ with embedding size $16$, $2$ modules with $2$ ISABs ($4$ heads, 8 inducing points), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 50 epochs using a MAP loss on the affine blocks, followed by 2 epochs using an unregularized ELBO loss on the affine blocks, followed by 50 epochs of reverse KL loss (see \cref{sec:multi_step_training} for the training strategy, total 102 epochs).
\end{itemize}
For the MSHBM example in toy dimensions (see \cref{sec:MSHBM_toy}):
\begin{itemize}
\item ADAVI: NF with $1$ Affine block with diagonal scale, followed by $1$ MAF with $[32]$ units. $\operatorname{HE}$ with embedding size $32$, $2$ modules with $2$ SABs ($4$ heads), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 32, 32 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 5 epochs using a MAP loss on the affine blocks, followed by 1 epoch using an unregularized ELBO loss on the affine blocks, followed by 5 epochs of reverse KL loss (see \cref{sec:multi_step_training} for the training strategy, total 11 epochs).
\end{itemize}
For the MSHBM example in (see \cref{sec:exp-yeo}):
\begin{itemize}
\item \textcolor{black}{MF-VI: variational distribution is
\begin{align*}
q &=\mathcal{L} \circ \mathcal{N}([\mu^g_1,\hdots,\mu^g_L]; \text{mean=}V_{(L, D - 1,)}, \text{std=}\operatorname{Exp}(V_{(L,)}) \\
&\times \operatorname{Exp} \circ \mathcal{N}([\epsilon_1,\hdots,\epsilon_L]; \text{mean=}V_{(L,)}, \text{std=}\operatorname{Exp}(V_{(L,)}) \\
&\times \mathcal{L} \circ \mathcal{N}([\mu^s_1,\hdots,\mu^s_L]; \text{mean=}V_{(S, L, D - 1,)}, \text{std=}\operatorname{Exp}(V_{(L,)}) \\
&\times \operatorname{Exp} \circ \mathcal{N}([\sigma_1,\hdots,\sigma_L]; \text{mean=}V_{(L,)}, \text{std=}\operatorname{Exp}(V_{(L,)}) \\
&\times \mathcal{L} \circ \mathcal{N}([\mu^{st}_1,\hdots,\mu^{st}_L]; \text{mean=}V_{(S, T, L, D - 1,)}, \text{std=}\operatorname{Exp}(V_{(L,)}) \\
&\times \operatorname{Exp} \circ \mathcal{N}(\kappa; \text{mean=}V_{(1,)}, \text{std=}\operatorname{Exp}(V_{(1,)}) \\
&\times \operatorname{Dirichlet}(\Pi; \text{concentration=}V_{(L,)})
\end{align*}
We used the Adam optimizer with a learning rate of $10^{-2}$. The optimization was ran for $10,000$ steps, with a sample size of $32$.}
\item ADAVI: NF with $1$ Affine block with diagonal scale, followed by $1$ MAF with $[1024]$ units. $\operatorname{HE}$ with embedding size $1024$, $3$ modules with $1$ SABs ($4$ heads), $1$ PMA (seed size $1$), $1$ SAB and $1$ linear unit each. Minibatch size 1, 4 theta draws per X point (see \cref{sec:MC_budget}), Adam ($10^{-3}$), 1000 epochs using a MAP loss on the affine blocks, followed by 1000 epochs using an unregularized ELBO loss on the affine blocks, followed by 1000 epochs of reverse KL loss (see \cref{sec:multi_step_training} for the training strategy, total 3000 epochs).
\end{itemize}
\subsection{Inference amortization}
\label{exp:amo}
In this experiment we illustrate the trade-off between amortized versus non-amortized techniques \citep{cranmer_frontier_2020}. For this, we define the following Gaussian random effects HBM \citep{gelmanbda04} (see \cref{fig:exp-graphs}-GRE):
\begin{equation}
\label{eq:gaussian_re_model}
\begin{aligned}
D,\ N &= 2,\ 50\ &\
\mu &\sim \mathcal{N}(\vec 0_D, \sigma_\mu^2) &\
&\ \\
G &= 3 &\
\mu^g | \mu &\sim \mathcal{N}(\mu, \sigma_g^2) &\
M^G &= [\mu^g]^{g=1\hdots G} \\
\sigma_\mu,\ \sigma_g,\ \sigma_x &= 1.0,\ 0.2,\ 0.05 &\
x^{g,n} |\mu^g &\sim \mathcal{N}(\mu^g, \sigma_x^2) &\
X &= [x^{g,n}]^{\substack{g=1\hdots G \\ n=1\hdots N}}
\end{aligned}
\end{equation}
In \cref{fig:amo_vs_non_amo} we compare the cumulative time to perform inference upon a batch of examples drawn from this generative HBM. For a single example, a non-amortized technique can be faster -and deliver a posterior closer to the ground truth- than an amortized technique. This is because the non-amortized technique fits a solution for this specific example, and can tune it extensively. \textcolor{black}{In terms of ELBO, on top of an \textit{approximation gap} an amortized technique will add an \textit{amortization gap} \citep{cremer_inference_2018}.} On the other hand, when presented with a new example, the amortized technique can infer directly whereas the optimization of the non-amortized technique has to be repeated. As the number of examples rises, an amortized technique becomes more and more attractive. This result puts in perspective the quantitative comparison later on performed between amortized and non-amortized techniques, that are qualitatively distinct.
\subsection{Expressivity in a non-conjugate case}
\label{exp:NC}
In this experiment, we underline the superior expressivity gained from using normalizing flows -used by ADAVI or CF- instead of distributions of fixed parametric form -used by MF-VI. For this we consider the following HBM (see \cref{fig:exp-graphs}-NC):
\begin{equation}
\begin{aligned}
N,\ D &= 10,\ 2 &\
r_a,\ \sigma_b &= 0.5,\ 0.3 &\
&\ \\
a &\sim \textit{Gamma}(\vec 1_D, r_a) &\
b^n |a &\sim \textit{Laplace}(a, \sigma_b) &\
B &= [b^n]^{n=1\hdots N}
\end{aligned}
\end{equation}
This example is voluntarily non-canonical: we place ourselves in a setup where the posterior distribution of $a$ given an observed value from $B$ has no known parametric form, and in particular is not of the same parametric form as the prior. Such an example is called \textit{non-conjugate} in \citet{gelmanbda04}. Results are visible in \cref{tab:NC&GM}-NC: MF-VI is limited in its ability to approximate the correct distribution as it attempts to fit to the posterior a distribution of the same parametric form as the prior. As a consequence, contrary to the experiments in \cref{exp:scaling} and \cref{exp:GM} -where MF-VI stands as a strong baseline- here both ADAVI and CF-A are able to surpass its performance.
\textcolor{black}{\textbf{Proxy to the ground truth posterior} MF-VI plays the role of an ELBO upper bound in our experiments GRE (\cref{exp:scaling}), GM (\cref{exp:GM}) and MS-HBM (\cref{sec:exp-yeo}). We crafted those examples to be conjugate: MF-VI thus doesn't feature any approximation gap, meaning $\operatorname{KL}(q(\theta) || p(\theta | X)) \simeq 0$. As such, its $ELBO(q) = \log p(X) - \operatorname{KL}(q(\theta) || p(\theta | X))$ is approximately equal to the evidence of the observed data. As a consequence, any inference method with the same ELBO value -calculated over the same examples- as MF-VI would yield an approximate posterior with low KL divergence to the true posterior. Our main focus in this work are flow-based methods, whose performance would be maintained in non-conjugate cases, contrary to MF-VI \citep{papamakarios_normalizing_2019}. We further focus on amortized methods, providing faster inference for a multiplicity of problem instances, see e.g.\ \cref{exp:amo}. MF-VI is therefore not to be taken as part of a benchmark but as a proxy to the unknown ground truth posterior.}
\begin{table*}[t]
\centering
\caption{Expressivity comparison on the non-conjugate (NC) and Gaussian mixture (GM) examples.
\textbf{NC}: both CF-A and ADAVI show higher ELBO than MF-VI. \textbf{GM}: TLSF-A and ADAVI show higher ELBO than CF-A, but do not reach the ELBO levels of MF-VI, TLSF-NA and CF-NA.
Are compared from left to right: ELBO median (larger is better) and standard deviation; \textcolor{black}{for non-amortized techniques: CPU inference time for one example (seconds); for amortized techniques: CPU amortization time (seconds).}
Methods are ran over 20 random seeds, except for SNPE-C and TLSF-NA who were ran on 5 seeds per sample, for a number of effective runs of 100.
For CF, the ELBO designates the numerically comparable \textit{augmented ELBO} \citep{ranganath_hierarchical_2016}.
}
\begin{tabular}{r|rr|l|l|l}
\toprule
HBM & Type & Method & ELBO & \textcolor{black}{Inf. (s)} & \textcolor{black}{Amo. (s)} \\
\midrule
NC & \textcolor{black}{Fixed param. form } & MF-VI & -21.0 ($\pm$ 0.2) & 17 & - \\
\rule{0pt}{3ex} (\cref{exp:NC})& \textcolor{black}{Flow-based} & CF-A & -17.5 ($\pm$ 0.1) & - & 220 \\
&& \textbf{ADAVI} & -17.6 ($\pm$ 0.3) &- & 1,000 \\
\midrule
GM & \textcolor{black}{Ground truth proxy} & MF-VI & \ 171 ($\pm$ 970) & 23 & - \\
\rule{0pt}{3ex} (\cref{exp:GM}) & Non amortized & SNPE-C & -14,800 ($\pm$ 15,000) & 70,000 & - \\
&& TLSF-NA & \ 181 ($\pm$ 680) & 330 & - \\
&& CF-NA & \ 191 ($\pm$ 390) & 240 &- \\
\rule{0pt}{3ex} & Amortized & NPE-C & -27,000 ($\pm$ 19,000) & - &1300 \\
&& TLSF-A & -530 ($\pm$ 980) & - & 360,000 \\
&& CF-A & -7,000 ($\pm$ 640) & - & 23,000 \\
&& \textbf{ADAVI} & -494 ($\pm$ 430) & - & 150,000 \\
\bottomrule
\end{tabular}
\label{tab:NC&GM}
\end{table*}
\subsection{Performance scaling with respect to plate cardinality}
\label{exp:scaling}
In this experiment, we illustrate our plate cardinality independent parameterization defined in \cref{sec:ADVI}.
We consider 3 instances of the Gaussian random effects model presented in \cref{eq:gaussian_re_model}, increasing the number of groups from $G=3$ to $G=30$ and $G=300$. In doing so, we augment the total size of the latent parametric space from $8$ to $62$ to $602$ parameters, and the observed data size from $300$ to $3,000$ to $30,000$ values.
Results for this experiment are visible in \cref{fig:scaling} (see also \cref{tab:GRE_scaling}). \textcolor{black}{On this example we note that amortized techniques only feature a small amortization gap \citep{cremer_inference_2018},} reaching the performance of non-amortized techniques -as measured by the ELBO, using MF-VI as an upper bound- at the cost of large \textcolor{black}{amortization times}. We note that the performance of (S)NPE-C quickly degrades as the plate dimensionality augments, while TLSF's performance is maintained, hinting towards the advantages of using the likelihood function when available. \textcolor{black}{As the HBM's plate cardinality augments, we match the performance and amortization time of state-of-the-art methods, but we do so maintaining a constant parameterization.}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{Fig3.pdf}
\caption{Scaling comparison on the Gaussian random effects example.
ADAVI -in red- maintains constant parameterization as the plates cardinality goes up (first panel); it does so while maintaining its inference quality (second panel) and a comparable amortization time (third panel).
Are compared from left to right: number of weights in the model; \textcolor{black}{closeness of the approximate posterior to the ground truth via the $\frac{ELBO}{G}$ median} -that allows for a comparable numerical range as G augments; \textcolor{black}{CPU amortization + inference time (s) for a single example -this metric advantages non-amortized methods}.
\textit{Non-amortized} techniques are represented using dashed lines, and \textit{amortized} techniques using plain lines.
\textcolor{black}{\textit{MF-VI}, in dotted lines, plays the role of the upper bound for the ELBO.}
Results for SNPE-C and NPE-C have to be put in perspective, as from $G=30$ and $G=300$ respectively both methods reach data regimes in which the inference quality is very degraded (see \cref{tab:GRE_scaling}).
Implementation details are shared with \cref{tab:NC&GM}.
}
\label{fig:scaling}
\end{figure}
\subsection{Expressivity in a challenging setup}
\label{exp:GM}
In this experiment, we test our architecture on a challenging setup in inference: a mixture model. Mixture models notably suffer from the \textit{label switching} issue and from a loss landscape with multiple strong local minima \citep{jasra_markov_2005}. We consider the following mixture HBM (see \cref{fig:exp-graphs}-GM):
\begin{subequations}
\begin{equation}
\label{eq:mixture_of_gaussians}
\begin{aligned}
\kappa,\ \sigma_\mu,\ \sigma_g,\ \sigma_x &= 1,\ 1.0,\ 0.2,\ 0.05 &\
G,L,D,N &= 3,3,2,50 \\
\mu_l &\sim \mathcal{N}(\vec 0_D, \sigma_\mu^2) &\
M^L &= [\mu_l]^{l=1\hdots L} \\
\mu_l^g | \mu_l &\sim \mathcal{N}(\mu_l, \sigma_g^2) &\
M^{L,G} &= [\mu_l^g]^{\substack{l=1\hdots L \\ g=1\hdots G}} \\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\pi^g \in [0,1]^L &\sim \operatorname{Dir}([\kappa] \times L) &\
\Pi^G &= [\pi^g]^{g=1\hdots G} \\
x^{g,n} | \pi^g, [\mu_1^g,\hdots,\mu_L^g] &\sim \operatorname{Mix}(\pi^g, [\mathcal{N}(\mu_1^g, \sigma_x^2)\hdots\mathcal{N}(\mu_L^g, \sigma_x^2)]) &\
X &= [x^{g,n}]^{\substack{g=1\hdots G \\ n=1\hdots N}}
\end{aligned}
\end{equation}
\end{subequations}
\textcolor{black}{Where $\operatorname{Mix}(\pi, [p_1, \hdots ,p_N])$ denotes the finite mixture of the densities $[p_1, \hdots ,p_N]$ with $\pi$ the mixture weights.} The results are visible in \cref{tab:NC&GM}-GM. In this complex example, similar to TLSF-A we obtain significantly higher ELBO than CF-A, but we do feature an amortization gap, not reaching the ELBO level of non-amortized techniques. We also note that despite our efforts (S)NPE-C failed to reach the ELBO level of other techniques. We interpret this result as the consequence of a forward-KL-based training taking the full blunt of the \textit{label switching} problem, as seen in \cref{sec:label-switching}. \cref{fig:multimodality} shows how our higher ELBO translates into results of greater experimental value.
\subsection{Neuroimaging: modelling multi-scale variability in Broca's area functional parcellation}
\label{sec:exp-yeo}
To show the practicality of our method in a high-dimensional context, we consider the model proposed by \citet{kong_spatial_2018}.
We apply this HBM to parcel the human brain's Inferior Frontal Gyrus in $2$ functional MRI (fMRI)-based connectivity networks.
Data is extracted from the Human Connectome Project dataset \citep{pmid22366334}.
\textcolor{black}{The HBM models a population study with 30 subjects and 4 large fMRI measures per subject, as seen in \cref{fig:exp-graphs}-MSHBM: this nested structure creates a large latent space of $\simeq 0.4$ million parameters and an even larger observed data size of $\simeq 50$ million values. Due to our parsimonious parameterization, described in \cref{eq:parametrization}, we can nonetheless tackle this parameter range without a memory blow-up, contrary to all other presented flow-based methods -CF, TLSF, NPE-C. Resulting population connectivity profiles can be seen in \cref{fig:kong_experiment}.}
\textcolor{black}{We are in addition interested in the stability of the recovered population connectivity considering subsets of the population. For this we are to sample without replacement hundreds of sub-populations of 5 subjects from our population. On GPU, the inference wall time for MF-VI is $160$ seconds per sub-population, for a mean $\log (-\text{ELBO})$ of $28.6 (\pm0.2)$ (across 20 examples, 5 seeds per example). MF-VI can again be considered as an ELBO upper bound. Indeed the MSHBM can be considered as a 3-level (subject, session, vertex) Gaussian mixture with random effects, and therefore features conjugacy. For multiple sub-populations, the total inference time for MF-VI reaches several hours. On the contrary, ADAVI is an amortized technique, and as such features an amortization time of $550$ seconds, after which it can infer on any number of sub-populations in a few seconds. The posterior quality is similar: a mean $\log (-\text{ELBO})$ of $29.0 (\pm0.01)$. As shown in our supplemental material -as a more meaningful comparison- the resulting difference in the downstream parcellation task is marginal (\cref{fig:neuro-diff-mfvi}).}
\textcolor{black}{We therefore bring the expressivity of flow-based methods and the speed of amortized techniques to parameter ranges previously unreachable. This is due to our plate cardinality-independent parameterization. What's more, our automatic method only necessitates a practitioner to declare the generative HBM, therefore reducing the analytical barrier to entry there exists in fields such as neuroimaging for large-scale Bayesian analysis.
Details about this experiment, along with subject-level results, can be found in our supplemental material.}
\begin{figure}[t]s
\centering
\includegraphics[height=.16\textheight]{Fig4.jpg}
\caption{Results for our neuroimaging experiment. On the left, networks show the top 1\% connected components. Network 0 (in blue) agrees with current knowledge in semantic/phonologic processing while network 1 (in red) agrees with current networks known in language production~\citep{heimEffectiveConnectivityLeft2009,zhangConnectingConceptsBrain2020}. Our soft parcellation, where coloring lightens as the cortical point is less probably associated with one of the networks, also agrees with current knowledge where more posterior parts are involved in language production while more anterior ones in semantic/phonological processing~\citep{heimEffectiveConnectivityLeft2009,zhangConnectingConceptsBrain2020}.}
\label{fig:kong_experiment}
\end{figure}
\subsection{Pyramidal Bayesian Models}
\label{sec:pyramidal}
We are interested in experimental setups modelled using plate-enriched Hierarchical Bayesian Models (HBMs) \citep{kong_spatial_2018, 10.1093/braincomms/fcab110}. These models feature independent sampling from a common conditional distribution at multiple levels, translating the graphical notion of \textit{plates} \citep{gilks_language_1994}. \textcolor{black}{This nested structure, combined with large measurements -such as the ones in fMRI- can result in massive latent parameter spaces}. For instance the population study in \citet{kong_spatial_2018} features multiple subjects, with multiple measures per subject, and multiple brain vertices per measure, \textcolor{black}{for a latent space of around $0.4$ million parameters}. Our method aims at performing inference in the context of those large plate-enriched HBMs.
Such HBMs can be represented with Directed Acyclic Graphs (DAG) templates \citep{koller_probabilistic_2009} with vertices -corresponding to RVs- $\{ \theta_i \}_{i=0 \hdots L}$ and plates $\{ \mathcal{P}_p \}_{p=0 \hdots P}$. We denote as $\operatorname{Card}(\mathcal{P})$ the -fixed- \textit{cardinality} of the plate $\mathcal{P}$, i.e. the number of independent draws from a common conditional distribution it corresponds to. In a template DAG, a given RV $\theta$ can belong to multiple plates $\mathcal{P}_h, \hdots \mathcal{P}_P$. When \textit{grounding} the template DAG into a ground graph -instantiating the repeated structure symbolized by the plates $\mathcal{P}$- $\theta$ would correspond to multiple RVs of similar parametric form \textcolor{black}{$\{\theta^{i_h,\hdots,i_P}\}$, with $i_h=1\hdots \operatorname{Card}(\mathcal{P}_h), \ \hdots \ , i_P=1\hdots \operatorname{Card}(\mathcal{P}_P)$}. This equivalence visible on the left on \cref{fig:ADAVI}, \textcolor{black}{where the template RV $\mathbf{\Gamma}$ corresponds to the ground RVs $[\gamma^1, \gamma^2]$.} We wish to exploit this plate-induced \textcolor{black}{exchangeability}.
We define the sub-class of models we specialize upon as \textit{pyramidal} models, which are plate-enriched DAG templates with the 2 following differentiating properties. First, we consider a single stack of the plates $\mathcal{P}_0, \hdots , \mathcal{P}_P$. This means that any RV $\theta$ belonging to plate $\mathcal{P}_p$ also belongs to plates $\{ \mathcal{P}_q \}_{q > p}$. We thus don't treat in this work the case of \textit{colliding} plates \citep{koller_probabilistic_2009}. Second, we consider a single observed RV $\theta_0$, with observed value $X$, belonging to the plate $\mathcal{P}_0$ (with no other -latent- RV belonging to $\mathcal{P}_0$). The obtained graph follows a typical pyramidal structure, with the observed RV at the basis of the pyramid, as seen in \cref{fig:ADAVI}. This figure features 2 plates $\mathcal{P}_0$ and $\mathcal{P}_1$, the observed RV is $\mathbf{X}$, at the basis of the pyramid, and latent RVs are $\mathbf{\Gamma}$, $\lambda$ and $\kappa$ at upper levels of the pyramid. \textcolor{black}{Pyramidal HBMs delineate models that typically arise as part of population studies -for instance in neuroimaging- featuring a nested group structure and data observed at the subject level only \citep{kong_spatial_2018, 10.1093/braincomms/fcab110}.}
The fact that we consider a single pyramid of plates allows us to define the \textit{hierarchy} of an RV $\theta_i$ denoted $\operatorname{Hier}(\theta_i)$. An RV's \textit{hierarchy} is the level of the pyramid it is placed at. Due to our pyramidal structure, the observed RV will systematically be at hierarchy $0$ and latent RVs at hierarchies $>0$. For instance, in the example in \cref{fig:ADAVI} the observed RV $\mathbf{X}$ is at hierarchy 0, $\mathbf{\Gamma}$ is at hierarchy 1 and both $\lambda$ and $\kappa$ are at hierarchy 2.
Our methodology is designed to process generative models whose dependency structure follows a pyramidal graph, and to scale favorably when the plate cardinality in such models augments. Given the observed data $X$, we wish to obtain the posterior density for latent parameters $\theta_1,\ \hdots \ ,\theta_L$, exploiting the exchangeability induced by the plates $\mathcal{P}_0,\hdots ,\mathcal{P}_P$.
\subsection{Automatic derivation of a dual amortized variational family}
\label{sec:ADVI}
In this section, we derive our main methodological contribution. We aim at obtaining posterior distributions for a generative model of pyramidal structure. For this purpose, we construct a family of variational distributions $\mathcal{Q}$ \textit{dual} to the model. This architecture consists in the combination of 2 items. First, a Hierarchical Encoder ($\operatorname{HE}$) that aggregates summary statistics from the data. Second, a set of conditional density estimators.
\textbf{Tensor functions} We first introduce the notations for tensor functions which we define in the spirit of \citet{magnus99}. We leverage tensor functions throughout our entire architecture to reduce its parameterization. Consider a function $f:F \rightarrow G$, and a tensor ${\tens{T}}_F \in F^{\mathcal{B}}$ of shape $\mathcal{B}$. We denote the tensor ${\tens{T}}_G \in G^{\mathcal{B}}$ resulting from the element-wise application of $f$ over ${\tens{T}}_F$ as ${\tens{T}}_G = \overrightarrow{f}^{(\mathcal{B})}({\tens{T}}_F)$ (in reference to the programming notion of \textit{vectorization} in \citet{numpy2020}). In \cref{fig:ADAVI}, $\overrightarrow{\operatorname{ST}}_0^{(\mathcal{B}_1)}$ and $\overrightarrow{l_\gamma \circ \mathcal{F}_\gamma}^{(\mathcal{B}_1)}$ are examples of tensor functions. At multiple points in our architecture, we will translate the repeated structure in the HBM induced by plates into the repeated usage of functions across plates.
\textbf{Hierarchical Encoder} For our encoder, our goal is to learn a function $\operatorname{HE}$ that takes as input the observed data $X$ and successively exploits the permutation invariance across plates $\mathcal{P}_0, \hdots ,\mathcal{P}_P$. In doing so, $\operatorname{HE}$ produces encodings ${\tens{E}}$ at different hierarchy levels. Through those encodings, our goal is to learn summary statistics from the observed data, that will condition our amortized inference. For instance in \cref{fig:ADAVI}, the application of $\operatorname{HE}$ over $X$ produces the encodings ${\tens{E}}_1$ and ${\tens{E}}_2$.
To build $\operatorname{HE}$, we need at multiple hierarchies to collect summary statistics across i.i.d samples from a common distribution. To this end we leverage \textit{SetTransformers} \citep{pmlr-v97-lee19d}: an attention-based, permutation-invariant architecture. We use \textit{SetTransformers} to derive encodings across a given plate, repeating their usage for all larger-rank plates.
We cast the observed data $X$ as the encoding ${\tens{E}}_0$. Then, recursively for every hierarchy $h=1\ldots P + 1$, we define the encoding ${\tens{E}}_h$ as the application to the encoding ${\tens{E}}_{h-1}$ of the tensor function corresponding to the set transformer $\operatorname{ST}_{h-1}$. $\operatorname{HE}(X)$ then corresponds to the set of encodings $\{ {\tens{E}}_1,\hdots,{\tens{E}}_{P+1} \}$ obtained from the successive application of $\{\operatorname{ST}_h\}_{h=0,\hdots, P}$. If we denote the batch shape $\mathcal{B}_h = \operatorname{Card}(\mathcal{P}_h) \times \hdots \times \operatorname{Card}(\mathcal{P}_P)$:
\begin{equation}
\begin{aligned}
{\tens{E}}_h &= \overrightarrow{\operatorname{ST}}_{h-1}^{(\mathcal{B}_h)}({\tens{E}}_{h-1})
&\
\operatorname{HE}(X) = \{ {\tens{E}}_1,\hdots,{\tens{E}}_{P+1} \}
\end{aligned}
\end{equation}
\textcolor{black}{In collecting summary statistics across the i.i.d. samples in plate $\mathcal{P}_{h-1}$, we decrease the order of the encoding tensor ${\tens{E}}_{h-1}$.} We \textit{repeat} this operation in parallel on every plate of larger rank than the rank of the contracted plate. We consequently produce an encoding tensor ${\tens{E}}_h$ with the batch shape $\mathcal{B}_h$, which is the batch shape of every RV of hierarchy $h$. In that line, successively summarizing plates $\mathcal{P}_0, \ ,\hdots \ ,\mathcal{P}_P$, of increasing rank results in encoding tensors ${\tens{E}}_1, \ \hdots \ ,{\tens{E}}_{P+1}$ of decreasing order. In \cref{fig:ADAVI}, there are 2 plates $\mathcal{P}_0$ and $\mathcal{P}_1$, hence 2 encodings ${\tens{E}}_1 = \overrightarrow{\operatorname{ST}}_0^{(\mathcal{B}_1)}(X)$ and ${\tens{E}}_2 = \operatorname{ST}_1({\tens{E}}_1)$. ${\tens{E}}_1$ is an order 2 tensor: it has a batch shape of $\mathcal{B}_1 = \operatorname{Card}(\mathcal{P}_1)$ -similar to $\mathbf{\Gamma}$- whereas ${\tens{E}}_2$ is an order 1 tensor. We can decompose ${\tens{E}}_1 = [e_1^1, e_1^2] = [\operatorname{ST}_0([X^{1, 1}, X^{1, 2}]), \operatorname{ST}_0([X^{2, 1}, X^{2, 2}])]$.
\textbf{Conditional density estimators} We now will use the encodings ${\tens{E}}$, gathering hierarchical summary statistics on the data $X$, to condition the inference on the parameters $\theta$. The encodings $\{ {\tens{E}}_h \}_{h = 1 \hdots P+1}$ will respectively condition the density estimators for the posterior distribution of parameters sharing their hierarchy $\{ \{ \theta_i : \operatorname{Hier}(\theta_i) = h \} \}_{h = 1 \hdots P+1}$.
Consider a latent RV $\theta_i$ of hierarchy $h_i = \operatorname{Hier}(\theta_i)$. Due to the plate structure of the graph, $\theta_i$ can be decomposed in a batch of shape $\mathcal{B}_{h_i} = \operatorname{Card}(\mathcal{P}_{h_i}) \times \hdots \times \operatorname{Card}(\mathcal{P}_P)$ of multiple similar, conditionally independent RVs of individual size $S_{\theta_i}$. This decomposition is akin to the grounding of the considered graph template \citep{koller_probabilistic_2009}.
A conditional density estimator is a 2-step diffeomorphism from a latent space onto the event space in which the RV $\theta_i$ lives. We initially parameterize every variational density as a standard normal distribution in the latent space $\mathbb{R}^{S_{\theta_i}}$. First, this latent distribution is reparameterized by a conditional \textit{normalizing flow} $\mathcal{F}_i$ \citep{rezende_variational_2016, papamakarios_normalizing_2019} into a distribution of more complex density in the space $\mathbb{R}^{S_{\theta_i}}$. The flow $\mathcal{F}_i$ is a diffeomorphism in the space $\mathbb{R}^{S_{\theta_i}}$ conditioned by the encoding ${\tens{E}}_{h_i}$. Second, the obtained latent distribution is projected onto the event space in which $\theta_i$ lives by the application of a \textit{link function} diffeomorphism $l_i$. For instance, if $\theta_i$ is a variance parameter, the link function would map $\mathbb{R}$ onto $\mathbb{R}^{+*}$ ($l_i=\operatorname{Exp}$ as an example).
The usage of $\mathcal{F}_i$ and the link function $l_i$ is \textit{repeated} on plates of larger rank than the hierarchy $h_i$ of $\theta_i$. The resulting conditional density estimator $q_i$ for the posterior distribution $p(\theta_i | X)$ is given by:
\begin{equation}
\label{eq:density_estimators}
\begin{aligned}
u_i &\sim \mathcal{N}\left(\overrightarrow{0}_{\mathcal{B}_{h_i} \times S_{\theta_i}}, {\bm{I}}_{\mathcal{B}_{h_i} \times S_{\theta_i}}\right)
&\
\tilde \theta_i = \overrightarrow{l_i \circ \mathcal{F}_i}^{(\mathcal{B}_{h_i})}(u_i; {\tens{E}}_{h_i}) &\sim q_i(\theta_i; {\tens{E}}_{h_i})
\end{aligned}
\end{equation}
In \cref{fig:ADAVI} $\mathbf{\Gamma}=[\gamma^1, \gamma^2]$ is associated to the diffeomorphism $\overrightarrow{l_\gamma \circ \mathcal{F}_\gamma}^{(\mathcal{B}_1)}$. This diffeomorphism is conditioned by the encoding ${\tens{E}}_1$. Both $\mathbf{\Gamma}$ and ${\tens{E}}_1$ share the batch shape $\mathcal{B}_1 = \operatorname{Card}(\mathcal{P}_1)$. Decomposing the encoding ${\tens{E}}_1=[e_1^1, e_1^2]$, $e_1^1$ is used to condition the inference on $\gamma^1$, and $e_1^2$ for $\gamma^2$. $\lambda$ is associated to the diffeomorphism $l_\lambda \circ \mathcal{F}_\lambda$, and $\kappa$ to $l_\kappa \circ \mathcal{F}_\kappa$, both conditioned by ${\tens{E}}_2$.
\textbf{Parsimonious parameterization} Our approach produces a parameterization effectively independent from plate cardinalities.
Consider the latent RVs $\theta_1,\hdots,\theta_L$. Normalizing flow-based density estimators have a parameterization quadratic with respect to the size of the space they are applied to \citep[e.g.][]{papamakarios_masked_2018}. Applying a single normalizing flow to the total event space of $\theta_1,\hdots,\theta_L$ would thus result in $\mathcal{O}([\sum_{i=1}^L S_{\theta_i} \prod_{p=h_i}^P \operatorname{Card}(\mathcal{P}_p)]^2)$ weights. But since we instead apply multiple flows on the spaces of size $S_{\theta_i}$ and repeat their usage across all plates $\mathcal{P}_{h_i},\ \hdots \ ,\mathcal{P}_P$, we effectively reduce this parameterization to:
\begin{equation}
\text{\# weights}_{\text{ADAVI}} = \mathcal{O}\left(\sum_{i=1}^L S_{\theta_i}^2\right)
\end{equation}
As a consequence, our method can be applied to HBMs featuring large plate cardinalities without scaling up its parameterization to impractical ranges, \textcolor{black}{preventing a computer memory blow-up.}
\subsection{Variational distribution and training}
\label{sec:training}
Given the encodings ${\tens{E}}_p$ provided by $\operatorname{HE}$, and the conditioned density estimators $q_i$, we define our parametric amortized variational distribution as a \textit{mean field approximation} \citep{blei_variational_2017}:
\begin{equation}
\label{eq:parametrization}
\begin{aligned}
q_{\chi, \Phi}(\theta | X) = q_\Phi(\theta ; \operatorname{HE}_\chi(X))
&= \prod_{i=1 \hdots L} q_i(\theta_i ; {\tens{E}}_{h_i}, \Phi)
\end{aligned}
\end{equation}
In \cref{fig:ADAVI}, we factorize $q(\mathbf{\Gamma}, \kappa, \lambda | X) = q_\gamma(\mathbf{\Gamma}; {\tens{E}}_1) \times q_\lambda(\lambda ; {\tens{E}}_2) \times q_\kappa(\kappa ; {\tens{E}}_2)$. Grouping parameters as $\Psi= (\chi, \Phi)$, our objective is to have $q_{\Psi}(\theta | X) \approx p(\theta | X)$.
Our loss is an amortized version of the classical ELBO expression \citep{blei_variational_2017, rezende_variational_2016}:
\begin{equation}
\begin{aligned}
\Psi^\star &= \arg \min_\Psi \frac{1}{M} \sum_{m=1}^M \log q_\Psi(\theta^m | X^m) - \log p(X^m, \theta^m), \ \ \ X^m\sim p(X), \theta^m\sim q_\Psi(\theta | X)
\end{aligned}
\end{equation}
Where we denote $z \sim p(z)$ the sampling of $z$ according to the distribution p(z). \textcolor{black}{We jointly train $\operatorname{HE}$ and $q_i, i=1\hdots L$ to minimize the amortized ELBO. The resulting architecture performs amortized inference on latent parameters. Furthermore, since our parameterization is invariant to plate cardinalities, our architecture is suited for population studies with large-dimensional feature space.}
|
2,877,628,089,495 | arxiv | \section{Introduction}
\label{sec:int}
Dark matter, dark energy, and the theory of gravitation dictate the evolution of large-scale structure in the Universe. The physical conditions
allowing for the formation of galaxies, ultimately lead to a bias between the distribution of galaxies and the underlying mass density. However,
large-scale motions of galaxies are most certainly locked to the peculiar velocity field associated with the gravitational tug of the total underlying
mass fluctuations. This assumes that gravity is the only relevant large-scale force and neglects the contribution from decaying linear modes. Despite
the bias between the distribution of galaxies and the underlying matter field, the clustering properties of galaxies have been the main tool for
testing cosmological models. Currently planned galaxy surveys will allow us to quantify the clustering of galaxies on hundreds of comoving Mpcs and to
even measure coherent distortions of galaxy images which arise from gravitational lensing by the foreground matter.
On the other hand, the peculiar motions of galaxies have traditionally been less successful as a cosmological tool, and there are several reasons
for that. Neglecting other potentially important effects (see section \ref{method} for details), peculiar velocities are approximately equal to the redshifts less the corresponding
Hubble expansion recession velocities. The latter require direct measurements of galaxy distances which are available for only a small fraction
of galaxies. Presently, the number of galaxies with measured distances is several orders of magnitude below that of galaxies in redshift surveys used
for clustering studies. Furthermore, although the underlying peculiar velocity is an honest tracer of the general matter flow, the inference of
peculiar velocities from observational data is plagued with observational biases \cite{lyn88}. Traditional peculiar velocity catalogs are expected
to improve within the next few years, but it remains questionable how well observational biases will be controlled, especially at large distances.
An alternative probe of the peculiar velocity field may be astrometric measurements of galaxies by the Gaia space mission \cite{perryman01,nbdgaia}.
This probe is essentially free of the classic biases contaminating traditional peculiar velocity measurements, but it is also limited to nearby galaxies
within $\sim 100h^{-1}\,{\rm Mpc}$.
Here we describe a method for deriving strong constraints on the power spectrum of the galaxies' peculiar velocity field independent of
conventional direct distance measurements, which are prone to systematic errors, and any biasing relation between galaxies and mass. The
method is an extension of the approaches we have recently proposed \citep{NBDL,ND11a,BDN12}, and it relies on using the observed fluxes of galaxies as a proxy for their cosmological distance \cite{TYS1979}. Although
this most basic distance indicator is very noisy, the large number of galaxies available in future surveys will allow one to beat down this noise
to a sufficiently low level. Planned galaxy redshift surveys such as Euclid \cite{euclidL,EuclidRB} will probe the structure of the Universe over
thousands of (comoving) Mpcs, comprising $\lesssim 10^8$--$10^9$ galaxies at $z\sim 1$ and beyond. These observations will provide redshifts and fluxes,
$z$ and $f$, respectively.
The observed redshifts deviate from the cosmological redshifts which would be observed in a purely homogeneous universe. The
large number of galaxies and the large sky coverage of these surveys can be used to derive a mean global relation between the mean redshift of a
galaxy and its apparent magnitude $m=-2.5 \log f + {\rm const}$. We interpret this relation as yielding the cosmological redshift $z_{_{\rm cos}}(m)$
for a given apparent magnitude $m$. Angular power spectra of the difference $z_i- z_{_{\rm cos}}(m_i)$ between the observed redshift $z_i$ of a
galaxy and its expected cosmological redshift $z_{_{\rm cos}}$ should contain valuable information, mainly on the peculiar velocity field which
is the main cosmological source for $z_i- z_{_{\rm cos}}(m_i)$. In this work, we will show that the velocity power spectrum on large scales of a
few 100 Mpcs could be constrained with significant signal-to-noise ratio ($S/N$) at the effective depth of the survey. Another contribution to
$z_i- z_{_{\rm cos}}(m_i)$ results from the time evolution of the gravitational potential along the photon path, but is significantly smaller than
that induced by peculiar velocities as we will show below.
There are two additional, indirect effects which modify the relation $z_{_{\rm cos}}(m)$ along a given line of sight. The first effect is related
to the environmental dependence between galaxy luminosities and the large-scale structure in which they reside. Since this dependence is closely
connected to the underlying density field, it can be self-consistently removed in our analysis. The second effect is caused by gravitational lensing
magnification which changes the apparent magnitudes of galaxies in a given direction. This latter contribution could actually be very rewarding since
gravitational lensing provides a direct probe of the underlying mass distribution. Considering the analysis presented below, we will therefore treat
it as part of the sought signal.
The paper is structured as follows: We begin with a detailed description of the method and its application to galaxy redshift surveys in section
\ref{method}. In section \ref{sec:tpk}, we consider predictions for the standard $\Lambda$CDM model and discuss the method's viability as well as
its expected performance. Finally, we present our conclusions in section \ref{sec:cnl}. For clarity, some of the technical material is given separately
in an appendix. In the following, we adopt the standard notation. The matter density and the cosmological constant in units of the critical density
are denoted by $\Omega$ and $\Lambda$, respectively. The scale factor $a$ is normalized to unity at the present time ($t=t_0$), and the Hubble
function is defined as $H=\dot{a}/a$. Further, $r=c \int_{t}^{t_0} dt'/a(t')$ will be the comoving distance to an object and $z$ its corresponding
redshift, assuming a homogeneous and isotropic cosmological background. Throughout the paper, the subscript ``$0$'' will refer to quantities given
at $t=t_0$, and a dot symbol denotes partial derivatives with respect to time $t$, i.e. $\dot{A}\equiv\partial A/\partial t$.
\section{Methodology}
\label{method}
In an inhomogeneous universe, the observed redshift $z$ of a given object differs from its cosmological redshift $z_{_{\rm cos}}$ (defined for the
unperturbed background). The relative difference between these redshifts, $\Theta \equiv (z-z_{_{\rm cos}})/(1+z_{_{\rm cos}})$, can be expressed as
\cite{SW}
\begin{equation}
\label{eq:Theta}
\Theta=\frac{V(t,r)}{c} - \frac{\Phi(t,r)}{c^2} - \frac{2}{c^2}\int_{t(r)}^{t_0}dt \frac{\partial \Phi\left\lbrack\hat {\vr} r(t),t\right\rbrack}{\partial t},
\end{equation}
where $\hat {\vr} $ is a unit vector along the line-of-sight to the object. Here the radial peculiar velocity $V$ and the usual metric potential $\Phi$ are
assumed as relative to their present-day values at $r=0$ ($t=t_{0}$). The first and second terms on the right-hand side of eq. \eqref{eq:Theta} are
the Doppler and gravitational shifts, respectively, while the third term describes the energy change of light as it passes through a time-varying
gravitational potential. Note that this third term is equivalent to the late-time integrated Sachs-Wolfe effect experienced by photons of the cosmic
microwave background (CMB). In what follows, we will denote the three terms as $\Theta{^{\rm V}}$, $ \Theta{^{\rm \Phi}}$ and $ \Theta{^{\rm { \dot \Phi}}}$, respectively. Also, we will consider
angular power spectra (equivalent to angular correlations) of $\Theta$ on large scales where the corresponding signal is significant only relative to
the expected error. Throughout this paper, we therefore rely on linear theory in a $\Lambda\rm CDM$ model where perturbations on all scales grow at
the same rate. Introducing $D(t)$ as the growth rate of the underlying mass density contrast $\delta=\rho/\bar \rho-1$, linear theory yields the
well-known relations
\begin{align}
\delta(t,\pmb{$r$}) &= D(t) \delta_0(\pmb{$r$}),\\
\Phi(\pmb{$r$},t) &= \frac{D(t)}{a}\Phi_0(\pmb{$r$}),\\
\label{eq:linv} V(t,\pmb{$r$}) &= -\frac{2}{3}\frac{a \dot D(t)}{\Omega_0 H_0^2}\frac{\partial \Phi_0}{\partial r},
\end{align}
where $D(t_0)=1$. The second relation is obtained from the first using Poisson's equation, i.e. $\pmb{$\nabla$}_r^2\Phi = 3 H_0^2\Omega_0\delta/2a $. For the
special case $\Omega_0=1$, we have $D=a$ and fluctuations in the gravitational potential remain constant with time.
\subsection{Cosmological redshift versus apparent magnitude}
\label{methodsub1}
We aim to derive the field $\Theta$ sampled at the positions of all galaxies in a flux limited redshift survey covering a significant region of
the sky together with a large number of galaxies. As an example, we consider the planned Euclid redshift survey \cite{euclidL,EuclidRB}. In order
to obtain an estimate of $\Theta_{i}$ for each galaxy, we shall use apparent galaxy magnitudes (or equivalently fluxes) as a proxy to the cosmological
redshift or distance. Although this most trivial distance indicator is very noisy, we will see that the high number of available galaxies allows
one to beat down its scatter.
The mean cosmological redshift $z_{_{\rm cos}}(m)$ corresponding to a given apparent magnitude $m$ is
\begin{equation}
z_{_{\rm cos}}(m)=\bar z_{_{\rm cos}}=\int_{z_1}^{z_2}z_{_{\rm cos}} P(z_{_{\rm cos}}|m) {\rm d} z_{_{\rm cos}} ,
\end{equation}
and the root mean square (rms) scatter around this relation is
\begin{equation}
\sigma_z(m)=\int_{z_1}^{z_2} (z_{_{\rm cos}} -\bar z_{_{\rm cos}} )^2P(z_{_{\rm cos}}|m){\rm d} z_{_{\rm cos}} ,
\end{equation}
where $z_1$ and $z_2$ are the limiting redshifts in the survey, and $P(z_{_{\rm cos}}|m)$ denotes the probability that a galaxy with measured apparent
magnitude $m$ has a cosmological redshift $z_{_{\rm cos}}$. The $z_{_{\rm cos}}(m)$ relation can be linked to certain characteristics of the galaxy survey. The
luminosity of a galaxy at $z_{_{\rm cos}}$ is $L=4\pi f d_L^2$ where $d_L(z_{_{\rm cos}})$ is the luminosity distance to the galaxy, and its absolute magnitude
is defined as $M=-2.5 \log L +{\rm const}=m-5 \log d_L$. Let $\Phi(M)$ be the underlying luminosity function such that $\Phi {\rm d} M $ is the
number density of galaxies within the magnitude interval $[M,M+{\rm d} M]$. Generally, the function $\Phi(M)$ depends on cosmic time, but in favor
of a simplified description, we will assume for the moment that it varies little throughout the depth of the survey considered. Note that it
is trivial to include a time-dependent evolutionary term in $\Phi(M)$. The probability $P(z_{_{\rm cos}}|m)$ satisfies
\begin{equation}
\label{eq:pcos}
P(z_{_{\rm cos}}|m)\propto P(m|z_{_{\rm cos}})n(z_{_{\rm cos}}),
\end{equation}
where $n(z_{_{\rm cos}}) $ is the underlying mean number density of galaxies at $z_{_{\rm cos}}$. In the absence of galaxy population evolution, we have
\begin{equation}
n(z_{_{\rm cos}}) \propto d_c(z_{_{\rm cos}})^2 \frac{{\rm d} d_c(z_{_{\rm cos}})}{{\rm d} z_{_{\rm cos}}},
\end{equation}
where $d_c$ is the comoving distance from the observer to $z_{_{\rm cos}}$. The relation in eq. \eqref{eq:pcos} can be easily derived using Bayes'
theorem which yields
\begin{equation}
P(z_{_{\rm cos}} |m)P(m) = P(m|z_{_{\rm cos}} )P(z_{_{\rm cos}}),
\end{equation}
where $P(m)$ can be directly estimated from observations, $P(z_{_{\rm cos}})$ is proportional to $n(z_{_{\rm cos}})$, and $P(m|z_{_{\rm cos}})$ is related to $\Phi(M)$ through
\begin{equation}
P(M|z_{_{\rm cos}}) = \frac{\Phi(M)}{\int_{-\infty}^{M_l(z_{_{\rm cos}})}\Phi(M){\rm d} M}.
\end{equation}
Here $M_l = m_l-5\log d_L(z_{_{\rm cos}})$ is the absolute magnitude which corresponds to the limiting apparent magnitude $m_l$ of the galaxy survey.
Note that the scatter of $z_{_{\rm cos}}$ about the mean relation is not Gaussian. However, the associated $1\sigma$ error in the derived angular
power spectra depends only on the rms quantity $\sigma_z$ (see section \ref{methodsub2} for a detailed discussion of systematic errors).
Moreover, the central limit theorem implies that the errors in the power spectra tend to be Gaussian. Considering the actual observations,
the two quantities $z_{_{\rm cos}}(m)$ and $\sigma_z(m)$ can be estimated from the full survey by dividing the data in magnitude bins without
actually computing $P(z_{_{\rm cos}}|m)$. In this case, $\sigma_z(m)$ will include a positive contribution from the cosmological deviations, but
this is overwhelmed by both the intrinsic scatter in the $z_{_{\rm cos}}(m)$ relation and the uncertainties in the photometric redshifts.
The underlying assumption is that the sought cosmological deviations between observed and cosmological redshifts cancel out when all
galaxies of the entire survey are used. Clearly, a global constant mode should persist in this procedure, but we shall neglect it in this
paper, assuming that it does not affect modes in the power spectra on smaller scales. Once we obtain $z_{_{\rm cos}}(m)$, we compute
\begin{equation}
\Theta_{i} = \frac{z_{i} - z_{_{\rm cos}}(m_{i})}{1 + z_{i}}
\end{equation}
for all galaxies in the survey. Note that in the denominator of the above, we have used $z_{i}$ instead of $z_{_{\rm cos}}(m_{i})$. This
substitution is consistent at linear order and motivated by the fact that $z_{i}$ is actually a better estimate of the true $z_{_{\rm cos}} $
than $z_{_{\rm cos}}(m_{i})$ which additionally includes deviations from the actual value of $z_{_{\rm cos}}$ due to large random errors with an rms of
$\sigma_z$.
Environmental dependences of the luminosity distribution on the large-scale density field may systematically shift all $\Theta_i$ for galaxies
lying in the direction of a certain line of sight. However, we will show below that the resulting signal contamination is small and in any case,
it can be removed from the correlations since information on the underlying density field at the relevant scales will be directly available from
the observations (see section \ref{sec:env}).
\subsection{Expectations for Euclid}
\label{methodsub2}
As a test case for a future survey, we consider Euclid which aims at measuring photometric and spectroscopic redshifts of galaxies with
$z_{_{\rm cos}} \sim 1$ over $15,000$ $\rm deg^2$. More details on the Euclid mission can be found in \cite{EuclidRB}. Here we focus on the
photometric redshift survey since we aim at good statistics rather than precise redshift measurements. Photometric redshifts will be
measured with an error of $\sigma_{\rm phot}\le 0.05(1+z_{\rm phot})$ for $\sim 30$ galaxies per arcmin$^2$, with $z_{_{\rm cos}} \ge 0.7$ and
magnitudes in the broad R+I+Z band (550--920 nm) RIZ$_{\rm AB} \le 24.5$. Estimates of the photometric redshifts will rely on three
near-infrared (NIR) bands (Y, J, and H in the range 0.92--2.0 $\mu$m) for objects with Y$_{\rm AB}\le 24$, J$_{\rm AB}\le 24$, and
H$_{\rm AB}\le 24$. These will further be complemented by ground-based photometry in the visible bands derived from public data or
through engaged collaborations.
The expected number density of Euclid galaxies with measured photometric redshift can be parametrized as follows \citep{theoryWG}:
\begin{equation}
\label{eq:nz}
n(z_{\rm phot})\propto z_{\rm phot}^2 {\rm e}^{-(z_{\rm phot}/z_0)^{3/2}},
\end{equation}
where $z_{0} = z_{\rm mean}/1.412$ is the peak of the redshift distribution and $z_{\rm mean}$ the median. Here we assume that
$z_{\rm mean} =0.9$. We have compared the expected distribution $n(z)$ of eq. \eqref{eq:nz} to that measured from galaxies with
H$_{\rm AB}\le 24$ and RIZ$_{\rm AB} \le 24.5$ in the zCOSMOS catalog \cite{Ilbert09,Mck10,Bielby11}. Indeed, we have found that eq.
\eqref{eq:nz} does provide a good fit to the data in the range $0.7 < z_{\rm phot} < 2.0$ which we will consider in our analysis.
The observed $z_{\rm phot}$-H relation of zCOSMOS galaxies has been used to derive $\bar z_{\rm phot}(\rm H)$ and $\sigma_{z_{\rm phot}}(\rm H)$
in different H-magnitude bins. To match Euclid constraints, we have only considered galaxies with RIZ$_{\rm AB} \le 24.5$,
H$_{\rm AB}\le 24$, and $0.7 < z_{\rm phot} < 2.0$. The additional constraints Y$_{\rm AB}\le 24$ and J$_{\rm AB}\le 24$ do not
significantly modify our results and have therefore not been enforced. Our results for the zCOSMOS data are shown in figure \ref{fig:one}.
The (blue) solid curve and the (red) dashed curve represent the expected $z_{_{\rm cos}}(H)$ and $\sigma_z(H)$ for Euclid galaxies, respectively.
As can be directly read off the figure, the expected scatter in $z_{_{\rm cos}}(H)$ is $\sigma_z(H)\lesssim 0.3$.
Errors on the measured redshift, i.e. $\sigma_{\rm phot}$, dominate the scatter for H$<20$. They will be added in quadrature to
determine the effective scatter in the $z_{_{\rm cos}}$-H relation.
\begin{figure}
\centering
\includegraphics[scale=0.7]{fig1.ps}
\caption{The mean relation $z_{_{\rm cos}}(m)$ and the corresponding rms scatter $\sigma_z(m)/(1+z)$ for Euclid galaxies with
RIZ$_{\rm AB} \le 24.5$, H$_{\rm AB}\le 24$, and photometric redshifts in the range $0.7 < z_{\rm phot} < 2.0$: The shown results
are based on zCOSMOS data.}
\label{fig:one}
\end{figure}
Since our aim is to estimate the angular correlation properties of $\Theta_{i}$, we are concerned with all potential sources of systematic
errors that are coherent over large angular scales. Random errors in the H-band photometry and $z_{\rm phot}$ can induce systematic errors
as a result of the galaxies' non-uniform distribution in the $z_{\rm phot}$-H plane. However, the resulting offsets bear no angular
coherence and can be safely ignored in our analysis. The lack of angular correlations also characterizes systematic errors induced by a
gross misestimate of $z_{\rm phot}$ which are commonly known as ``catastrophic errors".
Angular-dependent systematic errors may arise when calibrating the photometry across a large area of the sky. The angular structure of
these errors generally depends on the survey strategy and, considering ground-based observations, its interplay with the atmospheric
conditions at the telescope's site. In current surveys such as SDSS, the relative photometric errors are already on the order of 1\%
(10 mmag) and have only little angular structure \cite{Padma2008}. Clearly, these will be further reduced in next-generation surveys, especially in those that will be based in space. Systematic errors of this kind may propagate into zeropoint offsets affecting the
estimate of $z_{\rm phot}$. To assess the significance of this effect for our analysis, we have performed a number of simulations in
which we have computed $z_{\rm phot}$ after introducing photometric offsets of about $1$\% in different bands, using various templates
of spectral energy distributions for different galaxy types. As a reference case, we have considered a set of $1000$ galaxies at $z=1$
observed with the Euclid filters RIZ, Y, J, and H. A small, but sizable zeropoint offset $\Delta z_{\rm phot} = 0.01$ can be obtained
if the photometric offset runs smoothly from $-5$ mmag in RIZ to $5$ mmag in the H band. It is rather unlikely that such a
configuration will ever occur, but even so, there are plenty of reasons to ignore these systematic errors. First of all, the amplitude
and probability of $\Delta z_{\rm phot}$ decreases dramatically with the number of used filters. For instance, if additional photometry
in the visible bands $g$, $r$, $i$, and $z$ is considered (which is expected to be the case for Euclid), then $\Delta z_{\rm phot}$ drops
by a factor of approximately $10$. Second, we have found that the offset's amplitude can be further reduced by selecting homogeneous
subsamples of objects. Third, $\Delta z_{\rm phot}$ turns out proportional to the photometry offset, and will decrease when sub-percent
calibration accuracy is achieved. Finally, $z_{\rm phot}$ can be independently calibrated in different areas of the sky with the help of
spectroscopic redshift information, which may be used to reveal a possible angular correlation among errors.
\subsection{Spherical harmonics decomposition for discrete noisy data}
\label{methodsub3}
Much of the analysis presented below is very similar to previous work done in the lensing community, and before that, in the context of the
CMB \cite{Knox1995, Hobson1996}. As usual, angular power spectra are defined in terms of the spherical harmonics $Y_{lm}(\hat {\vr})$. For an
all-sky continuous field $f(\hat {\vr})$, the decomposition is
\begin{equation}
f_{lm} = \int d\Omega f(\hat {\vr})Y_{lm}(\hat {\vr}),\quad f({\hat {\vr}}) = \sum_{l=0}^\infty\sum_{m=-l}^{+l} f_{lm}Y^*_{lm}(\hat {\vr}),
\end{equation}
In our case, however, we have to deal with partial sky coverage (around $30\%$ of the sky for Euclid) and consider that the field is sampled
at discrete points given by the galaxy positions. The limited sky coverage could formally be described by an appropriate masking of the sphere
\cite{Peeb73,Peeb80}. However, the description in terms of masks will unnecessarily complicate the notation and somewhat obscure the physical
interpretation of the results. We shall therefore resort to a simplified description and assume that we are provided with a survey covering
$4\pif_{_{\rm sky}} $ steradians of the sky. For each degree $l$, we assume that there are $(2l+1)f_{_{\rm sky}}$ independent modes, instead of $2l+1$ for full
sky coverage. Considering modes with angular resolution much smaller than the extent of the survey, we then have
\begin{equation}
\label{eq:rules}
\begin{split}
&\int d\Omega Y_{lm}(\hat {\vr}) Y^*_{l^{\prime}m^{\prime}}(\hat {\vr}) = f_{_{\rm sky}}\delta^K_{ll^{\prime}}\delta^K_{mm^{\prime}},\\
&\sum_m Y_{lm}(\hat {\vr}) Y^*_{lm}(\hat {\vr}^{\prime}) = \frac{(2l+1)f_{_{\rm sky}}}{4\pi}P_l(\hat {\vr}\cdot \hat {\vr}^{\prime}),
\end{split}
\end{equation}
where $P_l$ is the Legendre polynomial of degree $l$. Throughout the paper, the angular integration is carried out only over the observed part
of the sky, and the number of terms in the sum of the second relation is $(2l+1)f_{_{\rm sky}}$. As is obvious, these relations should be understood to
hold in the approximate sense.\footnote{We emphasize that a future analysis of the real data should properly account for the lack of a full sky
coverage.}
Considering a function $f(\hat {\vr})$ with limited sky coverage in the continuous limit, we define
\begin{equation}
\label{eq:flm}
f_{lm} = \frac{1}{ f_{_{\rm sky}}^{1/2}} \int d \Omega f(\hat {\vr}) Y_{lm}(\hat {\vr}),
\end{equation}
where the integration is again taken over the observed region only. The angular power spectrum $C_{l}$ is defined as the variance of the
$f_{lm}$'s and given by
\begin{equation}
C_{l} = \langle\left\vert f_{lm} \right\vert^{2}\rangle_{_{\rm ens}},
\end{equation}
where the average is taken over many different realizations of a field with the same power spectrum as $f$. For brevity, we will use the symbol
$\langle\cdot\rangle$ without any subscript to refer to this kind of averaging. Up to cosmic variance, the values of $C_{l}$ obtained from the ensemble average
$\langle\left\vert f_{lm} \right\vert^{2}\rangle$ should equal those computed by averaging over the $(2l+1)f_{_{\rm sky}}$ modes for each $l$. Thus, the angular power
spectrum may be estimated as
\begin{equation}
C_l= \langle\left\vert f_{lm} \right\vert^{2}\rangle_{_{m}}\equiv \frac{1}{(2l+1)f_{_{\rm sky}}}\sum_m \left\vert f_{lm} \right\vert^2 .
\end{equation}
In the following, we shall interchange between these different kinds of averaging whenever appropriate. The factor $1/f_{_{\rm sky}}^{1/2}$ in eq.
\eqref{eq:flm} and the rules in eq. \eqref{eq:rules} guarantee that
\begin{equation}
\label{eq:CLC}
C(\cos\theta) = \sum\limits_l C_l \frac{2l+1}{4\pi} P_l(\cos\theta),\quad C_{l} = \int d\Omega C(\cos\theta) P_l(\cos\theta),
\end{equation}
which can be shown by decomposing $P_{l}$ into $Y_{lm} $ according to the second relation in eq. \eqref{eq:rules} and assuming that
$C(\theta) $ is negligible for angular separations larger than the extent of the survey (see appendix \ref{sec:proof}).
For a discrete sampling of $f$ at the positions of $N$ galaxies distributed over the observed part of the sky, we write \cite{afsh04}
\begin{equation}
\sum_i^N f(\hat {\vr}_i) \approx \int d\Omega n(\hat {\vr})f(\hat {\vr}) \approx \bar n \int d\Omega f(\hat {\vr}),
\end{equation}
where $n(\hat {\vr})$ is the projected number density of objects and $\bar n=N/(f_{_{\rm sky}} 4\pi)$ is the corresponding mean number density over
the observed part of the sky. Here we have assumed that $f$ itself depends on the density contrast $\delta=n/\bar n-1$, so that the
last step applies to linear order in the fluctuations. Again, the angular integration is carried out only over the observed part of
the sky. In analogy to eq. \eqref{eq:flm}, we further define
\begin{equation}
\label{eq:fdef}
f_{lm} = \frac{1}{\bar n f_{_{\rm sky}}^{1/2}} \sum_{i=1}^N f(\hat {\vr}_i) Y_{lm}(\hat {\vr}_i) .
\end{equation}
In this case, one can show that the angular power spectrum takes the form (see appendix \ref{sec:shotn})
\begin{equation}
C_{l} = \langle\left\vert f_{lm} \right\vert^{2}\rangle - \frac{\sigma_f^2}{\bar n},
\end{equation}
where the second term represents the contribution of shot noise due to the discrete sampling of $f$ and $\sigma_f^2=\sum_i f(\hat {\vr}_i)^2/N$.
To estimate the values of $C_{l}$, we therefore use
\begin{equation}
\label{eq:CLf}
C_{l} = \langle\left| f_{lm}\right|^{2}\rangle_{_{m}} - \frac{\sigma_f^2}{\bar n}.
\end{equation}
If $f_{i} = S(\hat {\vr}_i) + \epsilon_{i}$ where $S$ is an underlying cosmological signal and $\epsilon_{i}$ is an uncorrelated random
error, this simply reflects the fact that $C_{l} = \langle\left| S_{lm} \right|^{2}\rangle$. The expected error in this estimate of $C_l$ is given
by (again, see appendix \ref{sec:shotn})
\begin{equation}
\label{eq:SIGMA}
\Sigma^{2} = \frac{2}{(2l+1)f_{_{\rm sky}}}\left(\frac{\sigma_f^2}{\bar n}+C_{l}\right)^{2}
\end{equation}
which includes contributions from both shot noise $\Sigma_{\rm sn}$ and cosmic variance $\Sigma_{\rm cv}$,
\begin{equation}
\Sigma_{\rm sn} = \sqrt{\frac{2}{(2l+1)f_{_{\rm sky}}}}\frac{\sigma_f^2}{\bar{n}},\quad \Sigma_{\rm cv} = \sqrt{\frac{2}{(2l+1)f_{_{\rm sky}}}}C_l
\end{equation}
The variance of the scatter $\sigma^2_i$,
\begin{equation}
\label{eq:sigi_def}
\sigma_{i}^{2} = \langle\epsilon_{i}^{2}\rangle = \frac{\sigma_{\rm phot}^{2}(z_i) + \sigma_{z}^{2}(m_i)}{(1+z_{i})^{2}},
\end{equation}
depends on both magnitude and redshift. Note that the factor of $(1+z_{i})$ arises from the definition of $\Theta$ in eq. \eqref{eq:Theta}.
Considering the application to real data, it is prudent to weight each galaxy according to the $\sigma_{i}$ in the sum of eq. \eqref{eq:fdef}.
To minimize the effects of shot noise, we weight each galaxy by a factor of $w_{i}$ which is given by
\begin{equation}
\label{eq:w}
w_{i}^{2} = \frac{N \sigma_{i}^{-2}}{\sum_{j} \sigma_{j}^{-2}}.
\end{equation}
This particular weighting scheme yields
\begin{equation}
\label{eq:sigf}
\sigma_{f}^{2} = \frac{N}{\sum_{j}\sigma_j^{-2}},
\end{equation}
where we have assumed that that the underlying signal makes a negligible contribution to $\sigma_{f}^{2}$. The weighting does not affect the
ensemble average of $\lvert f_{lm}\rvert^{2}$, and its net effect is that $\sigma_{f}$ should be computed as given by eq. \eqref{eq:sigf},
i.e. it merely reduces shot noise errors (for additional details, see appendix \ref{sec:shotn}). In principle, one may use any weighting scheme.
\subsection{Environmental dependences in the luminosity functions and magnification by gravitational lensing}
\label{sec:env}
So far, we have assumed that the systematic shifts in the $z_{_{\rm cos}}(m)$ relation for galaxies in a given direction are solely due to the terms appearing
on the right-hand side of eq. \eqref{eq:Theta}. However, additional shifts may arise from changes in the mean magnitudes due to large-scale density
fluctuations in a given direction, i.e. environmental dependences and evolution in the luminosity function, and the magnification effect caused by
gravitational lensing. In what follows, we shall denote their contribution as $\Theta^{\rm env}$ and $\Theta^{\rm lens}$, respectively. Both effects
result in a magnitude shift which translates into additional correlated deviations of the estimated $z_{_{\rm cos}}$ from its mean relation. To model the
contribution of these effects in the correlation function of $\Theta$, we need to translate a magnitude shift $\Delta m$ into a corresponding shift
$\Delta z$. This can directly be read off the (blue) solid curve $z_{_{\rm cos}}(m)$ shown in figure \ref{fig:one} in which a shift $\Delta m=0.2$ leads to mean
shift $\Deltaz_{_{\rm cos}}\approx 0.017$.\footnote{Note that although more complicated schemes are possible, we adopt this mean relation for simplicity.}
To quantify the impact of environmental dependences, we assume that the systematic shift in galaxy magnitude depends on the density contrast $\delta$
in regions where they reside, irrespective of the smoothing scale \cite{Falt2010,moyang04}. Here we adopt the linear relation $\Delta m=0.2 \delta$
which is observationally inferred from the visible band \cite{crotL05}. The actual dependence is a function of the photometric band, and it is
expected to be much weaker in Euclid's H-band as indicated by the weak environmental dependence of the Schechter parameter $M_*$ fitted to NIR
luminosities \cite{mercurio12}. The remaining Schechter parameter, usually dubbed $\alpha$, exhibits a stronger dependence, but since this parameter
fixes the shape at the faint end, it will have only a small impact in deep surveys like Euclid. Nonetheless, it will be possible to remove most of the
contamination caused by this effect using the observed distribution of galaxies in the Euclid survey.
In the weak-field limit, the magnification induced by gravitational lensing is proportional to $1+2\kappa$, where $\kappa$ is the effective convergence
field, i.e. an integral over the (weighted) density contrast along the line of sight \cite{BS01}. For a flat $\Lambda$CDM model and a fixed source
redshift corresponding to a comoving source distance $r$, one finds
\begin{equation}
\kappa(\bm{\theta},r) = \frac{3H_{0}^{2}\Omega_{0}}{2c^{2}}\int_{0}^{r}dr^{\prime}\frac{r^{\prime}(r - r^{\prime})}
{a(r^{\prime})r}\delta(r^{\prime}\bm{\theta},r^{\prime}),
\end{equation}
where the two-dimensional angular vector $\bm{\theta}$ is perpendicular to the line of sight.
Therefore, it is straightforward to model the magnification and its effect in our analysis. The magnification field contains valuable information since
it probes the growth of the angular derivatives of the gravitational potential \cite[e.g.][]{broad95,BS01}. It can also be used to constrain the
gravitational slip which arises in certain modifications of the general theory of relativity \cite[e.g.][]{zhang07,bert11}. Given a theory of gravity,
much of the contribution to the power spectra from gravitational lensing can, in principle, be removed using the underlying large-scale density field
which is inferred from the foreground galaxy distribution. Considering the following analysis, however, we will treat the effects of lensing
magnification as part of the signal. As we will see below, its contribution is not negligible, and can be constrained together with the power spectrum
of the velocity field.
\section{Theoretical angular power spectra}
\label{sec:tpk}
In the following, we will present predictions for the angular power spectra corresponding to the various terms in eq. \eqref{eq:Theta}. Since the observed
galaxies cover a redshift range $z_{1} < z < z_{2}$, we do not consider the angular power spectra of a quantity $f$ defined at a specific redshift, but
instead we use
\begin{equation}
\tilde{f}(\hat {\vr}) = \int_{r_{1}}^{r_{2}}f\left\lbrack\hat {\vr} r, t(r)\right\rbrack p(r) dr,
\end{equation}
where $r_{1}$ and $r_{2}$ are the comoving distances at the survey's limiting redshifts, $z_1$ and $z_2$, respectively, and $p(r)dr$ is the probability of
observing a galaxy within the interval $[r,r+dr]$. Note that the function $f$ represents any of the terms on the right-hand side of eq. \eqref{eq:Theta}, i.e.
$\Theta{^{\rm V}}$, $\Theta{^{\rm \Phi}}$, and $\Theta{^{\rm { \dot \Phi}}}$. Beginning with $\Theta{^{\rm \Phi}}$, we write
\begin{equation}
{\tilde \Theta}{^{\rm \Phi}}_{lm} = \int d \Omega {\tilde \Theta}{^{\rm \Phi}} Y_{lm}(\hat {\vr}) = \frac{1}{c^{2}}\int d\Omega Y_{lm}(\hat {\vr}) \int_{r_1}^{r_2} W_{ \phi}(r) \Phi_0(\hat {\vr} r) dr ,
\end{equation}
where we have used the linear relation $\Phi(\pmb{$r$},t)=(D/a)\Phi_0(\pmb{$r$},t_0)$ and defined $W_{ \phi} = Dp(r)/a$, with $D(t)$ and $a(t)$ evaluated at $t=t(r)$.
Expanding $\Phi_0(\pmb{$r$})$ in Fourier space,
\begin{equation}
\Phi_0(\pmb{$r$}) = \frac{1}{(2\pi)^3}\int d^3 k \Phi_{\pmb{$k$}}{\rm e}^{{\rm i}\pmb{$k$} \cdot{ \pmb{$r$}}},
\end{equation}
and using
\begin{equation}
\label{eq:ekr}
{\rm e}^{{\rm i}\pmb{$k$} \cdot{\pmb{$r$}}} = 4\pi \sum\limits_{l,m}{\rm i}^lj_l(kr) Y^*_{lm}(\hat {\pmb{$n$}})Y_{lm}(\hat {\pmb{$k$}}),
\end{equation}
where $j_{l}$ is the usual first-kind spherical Bessel function of degree $l$, we get
\begin{equation}
{\tilde \Theta}{^{\rm \Phi}}_{lm} = \frac{{\rm i}^l}{2\pi^2 c^2}\int_{r_{1}}^{r_{2}} dr W_{ \phi} \int d^3 k \Phi_{\pmb{$k$}}j_l(kr) Y_{lm}(\hat {\pmb{$k$}}).
\end{equation}
Therefore, we finally arrive at
\begin{equation}
\label{eq:clphi}
C_{l}^{\Phi} = \langle |{\tilde \Theta}{^{\rm \Phi}} _{lm}|^{2}\rangle = \frac{2}{\pi c^4}\int dk k^2 P_\Phi(k) \left\vert \int_{r_1}^{r_2} d r W_{ \phi} j_l(kr)\right\vert^{2},
\end{equation}
where we have used $\langle\Phi_{\pmb{$k$}}\Phi_{\pmb{$k$}'}\rangle = (2\pi)^3 \delta^D(\pmb{$k$}-\pmb{$k$}')P_\Phi(k)$.
Similarly, using the linear relation in eq. \eqref{eq:linv}, we obtain
\begin{equation}
C_l^{\rm V} = \frac{2}{\pi c^2}\int dk k^2 P_\Phi(k) \left\vert \int_{r_{1}}^{r_{2}} dr W_{ \rm V} \left(\frac{lj_l}{r}-kj_{l+1}\right)\right\vert^{2},
\end{equation}
where $W_{ \rm V} = 2a\dot{D}p(r)/3\Omega_0 H_{0}^{2}$. As for $\Theta{^{\rm { \dot \Phi}}}$, the last term appearing in eq. \eqref{eq:Theta}, we will assume that the signal
is mostly caused by the large-scale structure between the observed high redshift galaxy sample and the observer. Therefore, $\Theta{^{\rm { \dot \Phi}}}$ is
approximately the same for all galaxies along a common line-of-sight such that ${\tilde \Theta}{^{\rm { \dot \Phi}}}\approx \Theta{^{\rm { \dot \Phi}}}$. The angular power spectrum then reads
\begin{equation}
\label{eq:isw}
C_{l}^{\dot{\Phi}} = \langle |\Theta{^{\rm { \dot \Phi}}}_{lm}|^{2}\rangle = \frac{8}{\pi c^6}\int dk k^2 P_\Phi(k) \left\vert \int_{0}^r dr^{\prime} W_{{\dot \phi}} j_l(kr^{\prime})\right\vert^{2},
\end{equation}
where $W_{{\dot \phi}}(t)/a = (d/dt)[D/a]$. The integration over $r^{\prime}$ is taken from $r=0$ out to a distance beyond which $\Phi$ becomes nearly
constant with time. For simplicity, we will assume that most of the contribution to the integral comes from the inner edge of the considered
galaxy sample, and adopt $z = 1$ in eq. \eqref{eq:isw} for all galaxies in the survey. For Euclid, this seems to be the case since the mean
redshift is expected to be $z\sim 1$. Note that the same assumptions are used when calculating the angular power spectrum $C^{\rm lens}$.
Although not explicitly given, similar expressions may be obtained for the corresponding cross-correlations of the above contributions.
\subsection{Predictions of the $\Lambda$CDM scenario}
Having derived the relevant expressions above, we are now ready to make predictions for the framework of $\Lambda$CDM. Here we adopt a spatially
flat model with best-fit parameters based on the CMB anisotropies measured by the Wilkinson Microwave Anisotropy Probe (WMAP) \cite{wmap7}. In
this case, the total mass density parameter is $\Omega_m=0.266$, the baryonic density parameter $\Omega_b=0.0449$, the Hubble constant $h=0.71$
in units of $100\ {\rm km\,s^{-1}} {\rm Mpc}^{-1}$, the scalar spectral index $n_{s}=0.963$, and $\sigma_{8} = 0.80$ for the rms of linear density fluctuations
within spheres of $8h^{-1}\,{\rm Mpc}$. We work with a parametric form of the power spectrum taken from ref. \cite{EH98} (see eqs. 29--31 in their paper). For
the calculation of angular power spectra, we use the expressions from section \ref{sec:tpk} together with $p(r)$ corresponding to the redshift
distribution of galaxies appropriate for Euclid which is given by eq. \eqref{eq:nz}.
The angular power spectra and associated errors are plotted in figure \ref{fig:two}. The shot-noise error $\Sigma_{\rm sn}$ is
computed using eqs. \eqref{eq:sigi_def} and \eqref{eq:sigf}, which gives $\sigma_{f} = 0.17$ as inferred from the (red) dashed curve in
figure \ref{fig:one}. For completeness, we also show the power spectrum $C^{\rm env}$ which results from environmental dependences in the luminosity function. As explained in section \ref{sec:env}, the
contamination arising from this effect can be removed given the observed distribution of galaxies in the survey. The accuracy to which this can be
achieved for Euclid is represented by $\Sigma_{\rm env}$, the $1\sigma$ error within which $C^{\rm env}$ can be estimated explicitly from the data
(see appendix \ref{sec:shotn} for details). As is clear from $C^{\rm lens}$ in the figure, magnification caused by gravitational lensing introduces
significant angular correlations in $(z_i-z_{_{\rm cos}})/(1+z_i)$.
The quantity $C^{\rm tot}$ is the power spectrum of the sum of the three signals $\Theta^{\rm lens}$, $\Theta^{\rm V}$, and $\Theta^\Phi$,
corresponding to the lensing magnification, the Doppler shift, and the gravitational shift, respectively. Note that the calculation of
$C^{\rm tot}$ does include covariance between the three individual signals. The contribution from $\Theta^{\dot \Phi}$ is negligible as is
indicated by $C^{\dot \Phi}$, and we do not include it in $C^{\rm tot}$. At $l\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$, $C^{\rm V}$ is the dominant
contribution, but the lensing term $C^{\rm lens}$ takes over at $l\gtrsim 10$--$15$, roughly until $l\sim 60$ where it drops below the shot-noise
level. The shaded area represents the $1\sigma$ error of $C^{\rm tot}$ due to cosmic variance. For practical purposes, it is therefore possible
to provide measurements of $C^{\rm V}$ and $C^{\rm lens}$ by an appropriate fitting procedure of the two corresponding curves to the measured
$C^{\rm tot}$. Unfortunately, at low $l$, cosmic variance is so large that accurate constraints on $C^\Phi$ do not seem possible.
\begin{figure}
\centering
\includegraphics[scale=0.7]{fig2.ps}
\caption{Angular power spectra in the $\Lambda$CDM model as explained in the text: The shaded area represents the cosmic variance uncertainty
$\Sigma_{\rm cv}$ on $C^{\rm tot}$, and $\Sigma_{\rm sn}$ is the $1\sigma$ shot-noise error for the Euclid survey. The desired signal is
contaminated by $C^{\rm env}$ which results from environmental dependences in the luminosity function and can be removed from the actual
Euclid data with high precision represented by $\Sigma_{\rm env}$. Only contributions (including cross-correlations) from $\Theta^{\rm V}$
(peculiar velocity), $\Theta^{\Phi}$ (gravitational shift), and $\Theta^{\rm lens}$ (lensing magnification) are included in $C^{\rm tot}$.
At $l\sim 60$ (not shown), $\Sigma_{\rm sn}$ and $C^{\rm tot}$ become comparable.}
\label{fig:two}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.7]{fig3.ps}
\caption{Confidence levels for measuring the amplitudes of the velocity and lensing power spectra by fitting
$a_{\rm v}^2 C^{\rm V} + a_{\rm lens}^2 C^{\rm lens}$ to random realizations of $C^{\rm tot}$ which include both shot noise
and cosmic variance: The fitting procedure ignores physical cross-correlations between the used quantities. Contours are shown
for $\Delta\chi^{2} = 2.3$, $6.17$, and $9.2$, corresponding to confidence levels of 68\%, 95.4\% and 99\%, respectively.}
\label{fig:three}
\end{figure}
\subsection{Signal-to-noise and expected error on model normalization}
Suppose that observations yield an estimate $C^{\rm obs}_{l}$ for the total power spectrum $C^{\rm tot}_{l}$. Let $C_{l}^{\rm H_{0}} = 0$
and $C^{\rm H_{1}}_{l}$ be the expected total power spectra for the null hypothesis with no correlations ($H_{0}$) and of the $\Lambda$CDM
model ($H_{1}$), respectively. The ratio of probabilities for $ H_{1}$ and $H_{0}$ is given by
\begin{equation}
-2{\rm ln}\frac{P(H_0)}{P(H_1)} = \sum_l\left[ \frac{\left(C^{\rm obs}_l-C^{\rm H_0}_l\right)^2}{\Sigma_{\rm sn}^2}- \frac{\left(C^{\rm obs}_l-C^{\rm H_1}_l\right)^2}{\Sigma^2}
+2{\rm ln} \left(\frac{\Sigma_{\rm sn}}{\Sigma}\right)\right].
\end{equation}
If $H_1$ holds, it follows that $C_l^{\rm H_1}$ is equal to the $C_l^{\rm tot}$ shown in figure \ref{fig:two}, and $C^{\rm obs}_l-C_l^{\rm H_1}$
is a random variable with variance $\Sigma^2$ given by eq. \eqref{eq:SIGMA}. Therefore, the signal-to-noise ratio ($S/N$) for rejecting $H_0$
is
\begin{equation}
\left(\frac{S}{N}\Big\vert_{0}\right)^2 = -2{\rm ln}\frac{P(H_0)}{P(H_1)} = \sum_l \left[\frac{(2l+1)f_{_{\rm sky}} C_l^2} {2\sigma_f^4/\bar n^2}
+\left(1+\frac{\bar nC_l}{\sigma_f^2}\right)^2-2{\rm ln}\left(1+\frac{\bar nC_l}{\sigma_f^2}\right) \right] ,
\end{equation}
where $C_l=C_l^{\rm H_1}=C_l^{\rm tot}$. Substituting the relevant quantities into the above, we obtain $(S/N)\vert_{0}=101$, where the above
sum rapidly converges by $l = 30$. On the other hand, if $H_0$ is true, then the $S/N$ for rejecting $H_1$ is given by
\begin{equation}
\begin{split}
\left(\frac{S}{N}\Big\vert_{1}\right)^2 &= -2{\rm ln}\frac{P(H_1)}{P(H_0)}\\
{ } &= \sum_l \left[\frac{(2l+1)f_{_{\rm sky}} C_l^2} {2(\sigma_f^2/\bar n+C_l)^2}
+\left(1+\frac{\bar nC_l}{\sigma_f^2}\right)^{-2}+2{\rm ln}\left(1+\frac{\bar nC_l}{\sigma_f^2}\right) \right ],
\end{split}
\end{equation}
which quickly converges at $l\sim 50$ and yields $(S/N)\vert_{1} = 14.7$. The fact that $(S/N)\vert_{1}\ll (S/N)\vert_{0}$ is a result of
cosmic variance which is zero for the $H_{0}$ hypothesis, but significant in the $\Lambda$CDM scenario, i.e.
$\Sigma_{\rm cv}^{2} = 2C_{l}^{2}/(2l+1)f_{_{\rm sky}}$.
In addition to $S/N$ considerations, we can assess how well a measurement of $C^{\rm tot}$ would constrain the normalization of the
$\Lambda$CDM power spectrum in terms of $\sigma_8$. To this end, we write the model's total power spectrum as
$C^{\rm tot,m} = (\sigma_8/0.8)^2C^{\rm tot}$, where $C^{\rm tot}$ as illustrated in figure \ref{fig:two} is obtained for $\sigma_{8}=0.8$. The
expected $1\sigma$ error on $\sigma_{8}$ is then $(-\partial^2 {\rm ln} P(H_1)/\partial \sigma_{8}^{2})^{-1}$, where $P(H_1)$ is now expressed as
\begin{equation}
\label{eq:phone}
{\rm ln} P(H_{1}) = -\sum_l\frac{\left(C_l^{\rm obs}- C_l^{\rm tot, m}\right)^2 }{2\Sigma_l^2}-\sum_l{\rm ln} \Sigma_{l},
\end{equation}
and $\Sigma_l$ is given by eq. \eqref{eq:SIGMA} with $C_l=C_l^{\rm tot,m}$. Using a normal probability distribution of the form given in eq.
\eqref{eq:phone} with fixed $\sigma_{8} = 0.8$, we have generated $1000$ random realizations of $C^{\rm obs}$. For each of these realizations,
we have maximized $P(H_{1})$ given by eq. \eqref{eq:phone} with respect to $\sigma_8$. As a result, we find that the true value $\sigma_{8}=0.8$
is recovered within a relative $1\sigma$ error of less that $4\%$, without any statistically significant bias.
We have also inspected the possibility of constraining the velocity and lensing signal amplitudes with the help of the two-parameter model
\begin{equation}
C^{\rm tot, m} = a_{\rm V}^2 C^{\rm V} + a_{\rm lens}^2 C^{\rm lens}.
\end{equation}
This model for $C^{\rm tot, m}$ neglects the contribution of $\Theta^{\Phi}$ to $C^{\rm obs}$ as well as any covariance between the remaining
signals, $\Theta^{\rm V}$ and lensing magnification $\Theta^{\rm lens}$. Using the above expression, we have repeated the procedure described
above for constraining $\sigma_8$ with the $1000$ random realizations of $C^{\rm obs}$. The result is presented in figure \ref{fig:three} which
shows contours of $\Delta \chi^2=-2{\rm ln} P+2{\rm ln} \max(P)$ as a function of $a_{\rm V}$ and $a_{\rm lens}$.
The contours have been computed for one of the random realizations, but the values of the best-fit parameters (giving the lowest $\Delta\chi^2$)
obtained for this particular realization have been shifted to their underlying value, i.e. unity for both. This is reasonable since the average
best-fit parameters of $1000$ realizations are essentially unbiased. As seen from the figure, $a_{\rm lens} $ is constrained with better
accuracy than the velocity amplitude $a_{\rm V}$. This is consistent with figure \ref{fig:one} which shows that $C^{\rm lens}$ dominates the total
signal over a large range of $l$ while $C^{\rm V}$ is significant only for low $l$ where cosmic variance becomes increasingly important. Still,
both parameters are constrained with good accuracy.
\section{Discussion}
\label{sec:cnl}
In this paper, we have presented a novel method for deriving direct constraints on the peculiar velocity and gravitational potential power
spectra from currently planned galaxy redshift surveys. The large number of galaxies with photometric redshifts in these surveys allows
one to exploit apparent galaxy magnitudes as a proxy for their cosmological redshifts since it beats down the large scatter in the $z_{_{\rm cos}}(m)$
relation and the uncertainty in the photometric redshifts. The method aims at directly constraining power spectra of the underlying
fluctuation fields independent of the way galaxies trace mass. Other methods for extracting cosmological information from redshift surveys
rely on accurate measurements of the galaxy power spectrum in redshift space (since galaxy distances remain unknown). The power spectrum and
other statistical measures based on the distribution of galaxies have been successful at probing the nature of dark matter and placing
important constraints on neutrino masses \cite{tegmark04,deputter12,sanchez12}. Having said that, however, they depend on a very accurate
knowledge of the relation between galaxies and the full underlying matter distribution.
The method we have considered here is less precise, but it is completely independent of the galaxy formation process and offers a much more
sensitive assessment of the underlying physical mechanism driving cosmic acceleration and structure formation. This approach is particularly
worthwhile if such constraints on the velocity field and the gravitational potential are contrasted with local constraints obtained from
data at low redshifts. For example, peculiar motions of galaxies within a distance of $\sim 100h^{-1}\,{\rm Mpc}$ can be measured using tight relations
between intrinsic observables of galaxies \cite[e.g.][]{TF77,springsixdf}, and also using astrometric observations of the Gaia space mission
which is currently scheduled for launch in $2013$. These peculiar motions of galaxies have been useful for constraining cosmological
parameters \citep{DN10} as well as the amplitude of the velocity field in the nearby Universe \citep{ND11a,bilicki11}.
Although we have presented predictions for the Euclid survey, the science proposed in this paper will not have to wait for this space
mission. In fact, several ground-based photometric surveys in the optical and near-infrared bands, which will constitute the backbone of
Euclid's photometry, will provide photometric redshift catalogs that can be used for our purposes well before the launch of the satellite.
On a shorter timescale, the VLT Survey Telescope (VST) will be used to carry out the Kilo Degree Survey (KiDS), one of the ESO public
surveys. It will cover 1,500 deg$^2$ to $u=24$, $g=24.6$, $r=24.4$, and $i=23.1$, and will probably contain $\sim 10^{8}$ galaxies with
measured photometric redshifts.\footnote{\protect\url{http://www.eso.org/public/teles-instr/surveytelescopes/vst.html}} Also, the Dark
Energy Survey (DES) will start its operations soon, and it will cover 5000 deg$^2$ of the Southern sky within 5 years, reaching magnitudes
up to $\sim 24$ in SDSS $griz$ filters, comparable to the limiting magnitudes of Euclid and with a redshift distribution $dN/dz$ similar
to that of Euclid galaxies. DES will measure photometric redshifts of $\sim 3\times 10^8 $ galaxies with $\sigma_{\rm photo}\sim 0.12$ at
$z\sim 1$.\footnote{\protect\url{http://www.darkenergysurvey.org/reports/proposal-standalone.ps}} Furthermore, the first of the four
planned Pan-STARRS telescopes has been operational since May 2010.\footnote{\protect\url{http://pan-starrs.ifa.hawaii.edu/public/home.html}}
The planned 3$\pi$ area of the sky will be considerably shallower, detecting galaxies below a limiting
magnitude of $\sim 24$ in the $griz$ bands. A deeper survey involving the PS1 and PS2 telescopes is currently being planned. The survey
should cover $\sim 7,500 \rm deg^2$ with limiting fluxes $g = 24.7$, $r = 24.3$, $i = 24.1$, and $z = 23.6$. Photometric redshifts will
then be measured for $\sim 4.5 \times 10^8 $ galaxies with similar errors.
Finally, on the long run, the Large Synoptic Survey Telescope (LSST) is expected to start operations in 2020. Its main deep-wide-fast survey
is expected to observe $\sim 20,000$ deg$^2$ in the $ugrizy$ bands. After about 10 years of operation, it will reach much deeper depths
(down to a co-added magnitude $r=27$), detecting about $3 \times 10^9$ galaxies \cite{ivezic08}.
\acknowledgments
We are grateful to Micol Bolzonella for computing all redshift-magnitude relations used in this paper from the zCOSMOS data
and for running simulations to assess the amplitude of zeropoint errors on the measured photometric redshifts.
We also thank Henry McCracken for providing us with the H-band magnitudes of zCOSMOS galaxies. E.B. thanks Gianni Zamorani and
Massimo Meneghetti for useful discussions and suggestions. This work was supported by THE ISRAEL SCIENCE FOUNDATION (grant No.203/09),
the German-Israeli Foundation for Research and Development, the Asher Space Research Institute and the WINNIPEG RESEARCH FUND.
E.B. acknowledges the support provided by MIUR PRIN 2008 ``Dark energy and cosmology with large galaxy surveys'' and by Agenzia
Spaziale Italiana (ASI-Uni Bologna-Astronomy Dept. 'Euclid-NIS' I/039/10/0). M.F. is supported in part at the Technion by the
Lady Davis Foundation.
|
2,877,628,089,496 | arxiv | \section{Introduction}
The current consensus is that the
Compact Steep Spectrum (CSS) and Giga-hertz Peaked Spectrum (GPS)
sources are young radio sources, evolving in the
Inter-Stellar Medium (ISM) of the host galaxy \citep[cf.][]{fanti.etal95, carvalho98, odea98,
murgia.etal99, sjk.etal05}.
These classes of radio sources provide the opportunity to study
the evolution of the radio sources
in their youth and the interaction of the radio jets and lobes with the
material in the ISM. The GPSs are selected based on the turnover in the spectrum.
Although the CSSs are not selected based on such a turnover,
the spectra of most of the CSSs show either a turnover or flattening
at very low frequencies \citep{fanti.etal90}.
The processes that can produce turnover in the spectrum are
synchrotron self-absorption (SSA), free-free absorption (FFA) and
Induced Compton Scattering (ICS) \citep{odea&baum97,bicknell.etal97, kuncic.etal98}.
Studies of a few individual sources report evidence of FFA in them
\citep{bicknell.etal97, xie.etal05, kameno.etal03a, kameno.etal03b}.
\citet{muto02} find that the polarisation properties are not
according to the trend expected from SSA. However using a sample of
GPS galaxies \citet{snellen.etal00} find that the turnover is
consistent with SSA.
The intrinsic turnover frequency (\nupeak) is found to be anti-correlated
with the linear size (LS) of the source, in the complete samples of CSS
and GPS sources (O'Dea \& Baum 1997, hereafter OB97; Fanti et al. 1990).
It has been shown that FFA by the material ionised by the fast moving cocoon
of the young radio source can explain this trend \citep{bicknell.etal97}.
Alternatively, using a model for the evolution of the source
\citet{odea&baum97} could predict such an anti-correlation using
a simple homogeneous SSA model. However the observed trend is
flatter than that predicted by SSA. This could be due
to the evolution of the luminosity of the radio source resulting
in the observed trend \citep{odea&baum97} or due to the picture
of SSA by a homogeneous medium being too simple.
So, in this work a 3D model is used to study the effects of the evolution
of the radio source on the observed trend in the \nupeak\ --~LS plane,
in the context of SSA.
New samples of GPS and CSS sources have become
available in recent years which have been used to extend
the \nupeak\ --~LS plane. In addition, the luminosity evolution of the
sources is also studied.
The sample of sources is described in Section~\ref{sec:sample}.
The SSA model and the model results are described in
Sections~\ref{sec:model} and \ref{sec:results} respectively.
A summary of the results is presented in Section~\ref{sec:conclusions}.
\section{The sample of sources}
\label{sec:sample}
The sample of sources is derived from the complete samples
of CSS and GPS sources available in the literature.
The GPS sources have well defined turnover
in the spectrum by selection. The CSS sources are selected based
on the spectral index and is not necessary to have a turnover in the spectrum.
Since the aim here is to study the turnover in the spectrum,
only those CSS sources with a clear turnover or flattening of
the spectrum at low frequencies are considered here.
In the case of flattening of the spectrum at low frequencies,
the measurement at the lowest available frequency is chosen as
the turnover frequency. Those CSSs, mostly from the B3-VLA CSS sample,
which show a curvature at low frequencies without a clear flattening
are also not considered here \citep{fanti.etal01}.
Variability studies of the GPS sources have shown that most of the
quasars identified as GPSs may not be genuine peakers, but
flaring blazars \citep{tinti.etal05, tinti&zotti05, tornikoski.etal01}.
To avoid such sources affecting the results,
only those GPS catalogues for which structural information is available
for most of the sources are considered.
However galaxies from the GPS catalogues are considered
to study the dependence of the \nupeak\ with \lpeak\ (luminosity at the
peak frequency) even if the structural information is not available
\citep[cf.][]{tinti.etal05}. The samples of CSS and GPS sources used here
are described below.
\noindent
(C) The \citet{fanti.etal90} sample of
CSSs from the 3C catalogue. The sources in this sample which do not follow
the above criteria are, 3C43, 3C186, 3C190,
3C303.1, 3C305.1 and 3C455. These sources have been removed from
the current sample.
\noindent
(S) The \citet{stanghellini.etal98} sample of GPSs.
These two samples are well studied. The sources from these samples
are listed in \citet{odea&baum97}.
\noindent
(F) The B3-VLA sample of CSS sources \citep{fanti.etal01}.
Information on the structure of the sources is available for most of
the sources (Dallacasa et al. 2002a,b).
\nocite{dallacasa.etal02a, dallacasa.etal02b}
\noindent
(W) The sample of GPS galaxies from the WENSS catalogue \citep{snellen.etal00}.
\noindent
(A) The sample of CSS and GPS sources in the southern sky
\citep{edwards&tingay04}. From this sample only those galaxies
listed in \citet{tinti&zotti05} are considered.
\noindent
(P) The sample of confirmed CSOs \citep{peck&taylor00, gugliucci.etal05}
are also used for this study irrespective of the liberal selection
criteria on the spectral index,
since the CSOs are known to be young radio sources
\citep{readhead.etal96a,owsianik&conway98, gugliucci.etal05}.
In this sample, some of the sources show flat spectra (see below)
making it difficult to estimate the turnover frequencies.
Such sources have been removed from the current sample.
\noindent
(D) The galaxies from the sample of High Frequency Peakers
\citep{dallacasa.etal00} listed in \citet{tinti&zotti05}.
\noindent
(H) The sample of GPS galaxies from the Parkes half-Jy catalogue
\citep{snellen.etal02}.
\noindent
(B) The galaxies from the \citet{bolton.etal04} sample of GPS sources listed in
\citet{tinti&zotti05}.
\noindent
(L) The GPSs from the CORALZ sample \citep{snellen.etal04}.
For the sources in the samples (C), (P) and (F) the flux density
measurements available in the literature (from NED and the CATS database)
have been used to fit the spectrum as described in \citet{steppe.etal95}.
The values of \nupeak\ and \speak\ (flux density at the peak frequency)
estimated from this fit are listed
in the tables.
\begin{table}
\begin{minipage}{8.5cm}
\caption{The sample of GPS galaxies}
\label{table:samplegps}
\begin{tabular}{l c l l l c }
\hline
Source & Sample$^{a}$ & z$^{b}$ & \nupeak$^c$\ & \speak\ & log(\lpeak) \\
& & & MHz & Jy & W/Hz/Sr \\
\hline
\input{table_gps_nup.tex}
\hline
\end{tabular}
\medskip
$^{a}$~See text for the meaning of the labels in the sample column.
$^{b}$~The * denotes a photometric redshift.
$^{c}$~Here \nupeak\ refers to the observed value\\
\end{minipage}
\end{table}
Of these samples (H), (B) and (L) have been used to study
only the dependence of \nupeak\ with \lpeak, since
information on the source structure is not available for the majority of the
sources. The sources from these three samples are listed in
Table~\ref{table:samplegps}. The columns are arranged as follows:
column 1: source name; column 2: master sample; column 3: redshift ;
column 4: observed \nupeak\ in MHz ; column 5: \speak\ in Jy; column 6:
log of \lpeak\ in W/Hz/Sr.
The sources from all other samples are listed in Table~\ref{table:samplecss}.
The columns are arranged as follows:
column 1: source name; column 2: master sample; column 3: optical id;
column 4: redshift ; column 5: largest angular size (LAS) in arc seconds;
column 6: observed \nupeak\ in MHz ; column 7: \speak\ in Jy; column 8:
log of \lpeak\ in W/Hz/Sr ; column 9: references to LAS, \nupeak\ and \speak.
Sources appearing on many samples are counted only once.
A redshift of 1.5 is assumed for those sources without any redshift
measurement. The cosmological constants used here are, $H_0=75$~kms$^{-1}$Mpc$^{-1}$ and $q_0$=0.0.
Other samples of CSS and GPS sources available in the literature
are, the CSS sources from the FIRST catalogue \citep{kunert.etal02},
the weak CSSs \citep{tschager.etal03}, the B2 sample of CSSs
\citep{saikia.etal02}, the sample of CSSs
from the S4 sample \citep{saikia.etal01} and the sample of GPSs from
the JVAS sample \citep{marecki.etal99}.
Since the optical identification or structural information
is less complete in these catalogues, they are not considered here.
The combined sample consists of a total of 203 sources of which
150 are CSS/GPS sources and 53 are GPS galaxies. The sample size used
in the present study is about three times larger than OB97 sample of sources.
The upper ranges of redshift, linear size and peak luminosity
are similar in both the samples. There are only three sources with large rest frame
peak frequency in the current sample. The lowest redshift in this sample is 0.004 as
compared to 0.08 in the OB97 sample. However the present sample extends the range of
linear sizes by an order of magnitude lower to about 0.65~pc and the peak luminosity
by 2.5 orders of magnitude lower than the OB97 sample.
\begin{table*}
\begin{minipage}{14cm}
\caption{The sample of CSS \& GPS sources$^\dagger$}
\label{table:samplecss}
\begin{tabular}{l c l l l l l l c l}
\hline
Source & Sample$^{a}$ & ID & z$^{b}$ & LAS & \nupeak$^{c}$\ & \speak\ & LS & log(\lpeak) & Ref$^{d}$\\
& & & & '' & MHz & Jy & kpc & W/Hz/Sr & \\
\hline
\input{table_css_nup.tex}
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{table*}
\setcounter{table}{1}
\begin{minipage}{14cm}
\caption{continued }
\begin{tabular}{l c l l l l l l c l}
\hline
Source & Sample$^{a}$ & ID & z$^{b}$ & LAS & \nupeak$^{c}$\ & \speak\ & LS & log(\lpeak) & Ref$^{d}$\\
& & & & '' & MHz & Jy & kpc & W/Hz/Sr & \\
\hline
\input{table_css_nup1.tex}
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{table*}
\setcounter{table}{1}
\begin{minipage}{14cm}
\caption{continued }
\begin{tabular}{l c l l l l l l c l}
\hline
Source & Sample$^{a}$ & ID & z$^{b}$ & LAS & \nupeak$^{c}$\ & \speak\ & LS & log(\lpeak) & Ref$^{d}$\\
& & & & '' & MHz & Jy & kpc & W/Hz/Sr & \\
\hline
\input{table_css_nup2.tex}
\hline
\end{tabular}
\medskip
$^\dagger$ A machine readable version of this table is available in CDS.
$^a$~See text for the meaning of the labels in the sample column.
$^b$~The * denotes a photometric redshift.
$^c$~ Here \nupeak\ refers to the observed value.
$^d$~The references are: (1) \citet{odea98};
(2) \citet{taylor&vermeulen97};
(3) \citet{fanti.etal01}; (4) \citet{peck&taylor00};
(5) \citet{gugliucci.etal05}; (6) \citet{xu.etal95};
(7) \citet{taylor.etal94}; (8) \citet{taylor.etal98};
(9) \citet{snellen.etal00}; (10) \citet{edwards&tingay04};
(11) \citet{henstock.etal95}; (12) \citet{kameno.etal01};
(13) \citet{tingay.etal03}; (14) \citet{ojha.etal04} ;
(15) \citet{xiang.etal02};
(16) \citet{stanghellini.etal99};
(17) \citet{polatidis.etal95};
(18) \citet{kellermann.etal98};
(19) \citet{dallacasa.etal00};
(20) \citet{stanghellini.etal98};
(21) Dallacasa et al. (2002a); \nocite{dallacasa.etal02a}
(22) Dallacasa et al. (2002b); \nocite{dallacasa.etal02b}
(23) \citet{tinti.etal05};
\end{minipage}
\end{table*}
\section{Model for spectral turnover}
\label{sec:model}
It is well known that SSA can produce
turnover in the spectrum. The \nupeak\ occurs at a frequency
where the SSA optical depth becomes unity.
The SSA optical depth, $\tau_{SSA}=\alpha_\nu R$,
where R is the path length through the source and
$\alpha_\nu$ is the synchrotron self absorption coefficient which is
given in Eqn.~\ref{eqn:alphanu}.
The $\tau_{SSA}$ depends on the electron number density, the magnetic field and
the path length, which all vary as the source evolves.
Thus a model for the evolution of the radio source is required to
understand the evolution of \nupeak\ with LS.
In the recent years, semi-analytical models
of dynamical and spectral evolution of classical radio sources
have been constructed (Kaiser \& Alexander 1997, hereafter KA;
Kaiser, Dennett-Thorpe \& Alexander 1997, hereafter KDA; Kaiser 2000, hereafter
K00; Blundell et al. 1999; Manolakou \& Kirk 2002),
based on self-similarity \citep{falle91}.
\nocite{kaiser&alexander97, kaiser.etal97, kaiser2000,blundell.etal99,manolakou&kirk02}
These models provide a very useful tool to study the evolution of radio
emission from the radio sources evolving in a power-law ambient medium.
Of these models, K00 provide a method of constructing
the 3-dimensional emissivity of the cocoon \citep[see also][]{chyzy97}.
Here, 3-dimensional synchrotron self-absorption coefficient,
$\alpha_\nu$, has been constructed using the approach of K00.
This has been used to evolve the radio spectrum with the age of the source,
including synchrotron self absorption. The method employed here and
the parameters involved are described below.
The radio source with jet power, $Q_0$ evolves in an ambient medium of
density $\rho(L_j) = \rho_0a_0^\beta L_j^{-\beta}$, where $L_j$ is the length
of the jet. The explicit expressions for the length $L_j$ is given by KA.
The synchrotron emissivity of a small volume element
which was injected into the cocoon at time $t_i$ and has evolved to the
present time $t$, can be written as
\begin{eqnarray}
\epsilon(\nu,t, t_i) = { c\sigma_T \over 6\pi\nu} \gamma(t)^3 u_b(t,t_i) n(\gamma,t, t_i)
\end{eqnarray}
where $u_b(t,t_i)$ is the energy density of the magnetic field and
$n(\gamma, t, t_i)$ is the electron energy spectrum.
The synchrotron self absorption coefficient is,
\begin{eqnarray}
\label{eqn:alphanu}
\alpha_\nu = K_0 n_0 B^{(p+2)/2} \nu^{-(p+4)/2}
\end{eqnarray}
where $K_0$ is a constant \citep{shu}.
The scaling factor $n_0 = n(\gamma,t,t_i)/\gamma^{-p}$, where
$p$ is the power-law energy index of the electrons.
The lower and upper bound of the $\gamma$ factor at injection are
$\gamma_{min}$ and $\gamma_{max}$ respectively.
The equations for quantities $u_b$, $n(\gamma)$ injected at time $t_i$
and their evolution to the present time $t$ are given in KDA.
In order to construct a 3D model, K00 uses a specialised
geometry for the cocoon and assume that the volume element injected at
time $t_i$ is located at $l$ (in units of L$_j$) as a thin cylindrical
volume of thickness $\delta l$ and radius $r_c(l, \alpha_1, \alpha_2)$
(equation~12 of K00).
The location of the cylindrical volume element $l$ is given by $(t_i/t)^{\alpha_3}$.
Since all the $\delta l$ have to add up to the length of the jet,
the constant $\alpha_3$ can be calculated self consistently
for the assumed geometry parameters $\alpha_1$ and $\alpha_2$,
using the equation 15 of K00.
Using the above approach, $\epsilon_\nu$ and $\alpha_\nu$ are obtained
at the position $(l,y)$, where $y$ is the perpendicular distance
from the jet axis. These quantities are assumed to be axi-symmetric
with respect to the jet axis.
If the source is viewed edge-on, the path length for a ray passing
through the position $(l,y)$ is, $R(l,y)=2\sqrt{(r_c(l)^2 - y^2}$
and $r_c(l)$ is the cocoon radius at $l$.
The surface brightness, including the optical depth effects,
along this ray can be written as,
\begin{eqnarray}
S_\nu(l,y) = {\epsilon(\nu,l,y)\over \alpha(\nu,l,y)} \left\{ 1- exp\left[-\alpha(\nu,l,y)R(l,y)\right]\right\}
\end{eqnarray}
The total emission from the cocoon, obtained by integrating over the
surface, is given by
\begin{eqnarray}
\label{eqn:pnu}
P_\nu = 4 \int_{l_{low}}^1 L_j(t) dl \int_0^{r_c(l)} S_\nu(l,y) dy
\end{eqnarray}
Here $l_{low}$ depends on whether that part of the cocoon can have
electrons with $\gamma$ corresponding to the given frequency, had
that value at injection been $\gamma_{max}$.
In the model of KA, the jet length has to be large enough to allow
for the pressure balance between the cocoon and the jet that passed through
the re-confinement shock. For a source size of 10~pc and $\beta=1.9$
the ratio of the location of the re-confinement shock to the
length of the jet, $R_{conf}$, is about 0.1.
Thus the jet length is much larger than the location of the
re-confinement shock and the model results can be applied to such sources.
However, the density at the inner region of the host galaxy is not expected
to vary as steeply as in the outer regions. A realistic situation
is a King type profile with a constant density core.
The models with shallower density profile $\beta \sim 0.0$
are a good representation for the smaller sources.
For $R_{conf}$ of about 0.3 and $\beta$ of 0.0 and 0.5 the
size of the source is about 150 and 50~pc respectively.
Therefore at smaller sizes the model results are to be treated as extrapolation
of the trend at larger sizes.
\begin{figure}
\includegraphics[width=8cm,clip=]{specb1.9_1_2_10.eps}
\caption{
The model spectra for radio sources of sizes 1 (solid),
2 (dashed) and 10 (dotted) kpc are plotted.
The parameters used for the model calculations are
$Q_0=0.13\times10^{40}$, $a_0= 2$~kpc and $\beta=1.9$.
} \label{fig:spec}
\end{figure}
\begin{figure}
\includegraphics[width=8cm,clip=]{speccross.eps}
\caption{
The model spectra for a source with linear size of 4.88~kpc.
The solid line represents the model
with $\beta=0$ and the dotted line represents $\beta=1.9$.
The parameters used for the model calculations are
as in Fig.~\ref{fig:spec}.
} \label{fig:speccross}
\end{figure}
\citet{alexander00} has presented a method of switching from
one power-law model to another that can apply
to evolution from smaller to larger sources and has shown that the
luminosity from different models match very close to the point of switching.
It turns out that the location where the pressure from different
models match is same as the location where the analytical
luminosities of \citet{alexander00} match.
So model results are also considered by switching from
$\beta=0$ model to $\beta=\beta_{outer}$
at the distance where the pressure of the cocoon of different
power-law models match.
For $\beta_{outer}=1.9$, the pressures match at $L_j = 1.22~a_0$.
Although, all the quantities involved in
estimating the luminosity of the source are related to the pressure,
the continuity of the evolutionary tracks in the
\nupeak\ --~\lpeak\ or the \nupeak\ --~LS plane is not guaranteed. In addition,
the evolutionary history of the energy of the electrons of the two
power-law models are different.
However it can be seen from Fig.~\ref{fig:speccross}
that indeed the luminosity and the \nupeak\ are very close to each other at
the switching point, but not exactly the same. Results obtained using
this approach are also presented in the next section.
\begin{figure}
\includegraphics[width=8.5cm]{ifreqlsizemodel.ps}
\caption{The \nupeak, is plotted against
the linear size of the radio source. The symbols representing the samples
are:
unfilled square $-$ C;
filled squre $-$ S;
unfilled triangle $-$ P;
filled triangle $-$ F;
unfilled circle $-$ H ;
filled circle $-$ W;
unfilled star L;
filled star $-$ D;
plus $-$ B ;
cross $-$ A.
The quasars are plotted with symbol sizes smaller
than that of the galaxies.
The thick solid line represents the
least squares fit to the data.
The thin lines from the bottom to the top of the figure represent
models with $Q_0$$=$$1.3\times10^{37}$, $1.3\times10^{38}$,
$1.3\times10^{39}$, $1.3\times10^{40}$ and $5\times10^{50}$~W respectively.
The model with $\beta=0$ at the inner core and $\beta=1.9$ at the
outer regions is plotted with a line of intermediate thickness.
\label{fig:nupeakls} }
\end{figure}
\section{Results}
\label{sec:results}
The model spectrum is obtained by numerically integrating Eqn.~\ref{eqn:pnu}.
The geometry parameters used in the model calculations are,
$\alpha_1=2$ and $\alpha_2=1/3$. A very high value is chosen
for $\gamma_{max}$ and $\gamma_{min}=1.0$. In addition, the value
of the hotspot to the cocoon pressure is taken as $4R_T^2$, where $R_T = 2$.
Other parameters assumed for the model calculations correspond to
the case-2 of KDA.
The model spectrum is presented in Fig.~\ref{fig:spec},
for a radio source of size 1, 2 and 10 kpc for $\beta=1.9$.
The figure shows that the \nupeak\ shifts to
low frequencies as the source grows in size. The spectral index
at the optically thick side of the spectrum is about 5/2.
The point at which the pressure of the cocooon of the models
with $\beta=0$ and 1.9 match, corresponds to a source size of 4.88~kpc.
The model spectrum for a source with this size is presented for
$\beta=0$ and 1.9 in Fig.~\ref{fig:speccross}. This figure shows
that the spectra from the two different power-law models are
indeed close to each other.
\subsection{Dependence of \nupeak\ with LS}
The value of \nupeak\ is plotted against the
LS in Fig.~\ref{fig:nupeakls},
for the quasars and the galaxies from the combined sample listed in
Table~\ref{table:samplecss}. The meaning of the different
symbols are explained in the caption.
A linear fit to all the data gives a slope of -0.51$\pm$0.03, which
is shown as thick solid line in the figure.
A linear fit to only the galaxies give a slope of -0.50$\pm$0.04.
The slope obtained with the new sample is somewhat flatter than
the slope of -0.65$\pm$0.05 obtained by \citet{odea&baum97}.
The same dependence calculated using the current model is
also plotted in Fig.~\ref{fig:nupeakls} for
jet powers of, $1\times10^{37}$, $1.3\times10^{38}$, $1.3\times10^{39}$,
$1.3\times10^{40}$ and $5\times10^{40}$~W. These curves fit the data well.
The slope of the model curves for $\beta=1.9$, is about -0.85.
For $\beta=0$ the slope of the model curve is -0.56 which is close to the
linear fit to the observed data.
These results suggest that SSA can explain the turnover in the spectrum
and the observed trend in \nupeak\ --~LS plane can be explained by SSA alone.
The sources evolve from GPS to CSS in the \nupeak\ --~LS plane and do not
leave this plane during the evolution.
However large CSS sources with low jet power may not show turnover in the observable
frequency range and may leave this plane during the later stages of evolution.
Using the radio source model of \citet{begelman96} and assumption
on the variation of the magnetic field, B,
\citet{odea&baum97} predict a slope from -1.65 to -1.8 depending on
their model parameters. The is steeper than that given by this
model. Interestingly the slope predicted by
\citep{bicknell.etal97} for the case of FFA, for $\beta=1.9$
is about -0.91. This is very similar to the
slope predicted by the current model with $\beta=1.9$.
For $\beta=0.0$, the current model prediction fits the observed trend well.
It is not possible to favour any one mechanism from
this slope alone. However, if both mechanisms are at work,
then the spectral index on the optically thick side is expected to be
steeper than that predicted by either of the mechanisms.
Although in general this is not the case, it is interesting observationally.
Detailed modelling of the spectrum of high resolution observations
of a few GPS sources suggest that both mechanisms could be at work
\citep{xie.etal05, kameno.etal03a, kameno.etal03b}.
\subsection{Luminosity evolution}
\begin{figure}
\includegraphics[width=8.5cm]{lumlsizemodel.ps}
\caption{The \lpeak\ is plotted against the LS for the galaxies from
Tables~\ref{table:samplegps} and ~\ref{table:samplecss}.
The meaning of the symbols and the line types are as in Fig.~\ref{fig:nupeakls}.
\label{fig:lsizelum} }
\end{figure}
\begin{figure}
\includegraphics[width=8.5cm]{lumifreqmodel.ps}
\caption{The \lpeak\ is plotted against the \nupeak\ for the galaxies from
Tables~\ref{table:samplecss} and ~\ref{table:samplegps}.
The meaning of the symbols and the line types are as in Fig.~\ref{fig:nupeakls}.
\label{fig:nupeaklum} }
\end{figure}
The vaule of \lpeak\ is plotted against the LS in Fig.~\ref{fig:lsizelum}.
for sources from Table~\ref{table:samplegps}.
Since the luminosities of the quasars are affected by the beaming effects,
only galaxies are plotted in the above diagram.
The figure shows that the \lpeak\ increases with the LS for the sources with
sizes less than about 1~kpc. Beyond this size the \lpeak\ does not
vary with LS. A parabolic fit to the data (not shown here)
clearly shows a flattening at large sizes. A linear least squares fit to
the data for log(LS) smaller than -0.5 is shown as thick solid line.
The slope of the fit is 1.8$\pm$0.2.
The results from the model are also plotted in the figure for the
same parameters used in the above section. The slope of
the model curves at smaller linear sizes for
$Q_0=1.3\times10^{39}$~W and $\beta$ of 0 and 1.9 are 0.94 and 1.41
respectively. The slope is higher for higher jet powers.
This trend is also consistent with the prediction of \citet{snellen.etal00}.
Although the trend shown by the model curves are in agreement
with the observed trend, the observations suggest a steeper evolution
than the model prediction.
It is possible to study this luminosity evolution in the \lpeak\ --~\nupeak\ plane,
where all the GPSs without angular size estimates can be used. This will
increase the number of sources with very high frequencies or correspondingly
small linear sizes. The dependence of \lpeak\ with \nupeak\ is shown
in Fig.~\ref{fig:nupeaklum} for all the galaxies.
The figure shows a trend of \lpeak\ decreasing with \nupeak.
The relations \nupeak\ $\propto$~LS$^{-0.51}$ and \lpeak\ $\propto$~LS$^{1.8}$
obtained from the least squares fits above can be translated to
\lpeak\ $\propto$~\nupeak$^{-3.5}$. This trend is shown as a thick solid line.
Such a trend is expected for a power-law electron energy spectrum
since the $\gamma$ of the electrons
corresponding to the \nupeak\ is higher and the electron density is lower
for higher $\gamma$ values. However the slope expected from this
argument is the spectral index $\alpha$ which is smaller than the least
squares fit.
The curves of the model results are plotted for the same
parameters used for the model
curves in Fig.~\ref{fig:nupeakls}.
The model curves show that the peak luminosity decreases with
increasing \nupeak\ for values of \nupeak$~> \nu_{cut}$. The
value of $\nu_{cut}$ is the turnover frequency expected for a source size of
about 1 kpc. For smaller \nupeak\ values corresponding to larger sources the \lpeak\
is almost constant.
The slope of the model curves
for $\beta=1.9$ and 0 is -1.58 and -1.67 respectively. This reflects
the same trend seen in the \lpeak\ --~LS plane.
Most of the sources in this diagram can be bound by curves corresponding
to jet powers between $1.3\times10^{37}$ and $5\times10^{40}$~W.
The observed trend and the model curves suggest that
the luminosity increases as the source evolves from GPS to CSS
and it is unlikely to observationally miss the CSS sources due to evolution.
However it is possible to miss the smaller GPSs since they will appear
more fainter at the frequencies away from the \nupeak.
These missing sources do not affect the trend seen in Fig.~\ref{fig:nupeakls}.
\section{Conclusions}
\label{sec:conclusions}
The dependence of \nupeak\ with linear size
is presented for the CSS and GPS sources derived from complete
samples available in the literature.
The \nupeak\ is anti-correlated with the linear size,
as \nupeak\ $\propto$~LS$^{-0.51}$ for this sample.
The sources evolve in luminosity as they grow in linear size.
The \lpeak\ increases with the LS of the source for sizes smaller
than about 0.3~kpc, as \lpeak\ $\propto$~LS$^{1.8}$.
Beyond this size the \lpeak\ is almost constant.
This luminosity evolution is seen in the \lpeak\ -- \nupeak\ plane also.
Optical depth effect has been included to the 3D model of K00.
Using this model the observed dependence of the \nupeak\ with linear size
and \lpeak\ can be explained. These results suggest that
synchrotron self absorption can explain the turnover
in the young radio sources. The luminosity evolution does not affect
the trend seen in the \nupeak\ --~LS plane.
\section*{Acknowledgments}
Critical comments and suggestions made by the anonymous referee,
which have helped in refining the interpretation and also improving the presentation are acknowledged.
SJ thanks Paul J. Wiita for useful comments on the paper.
This research has made use of the 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. The authors made use of the database, CATS
\citep{verkhodanov.etal97} of the Special Astrophysical Observatory.
This work was partially supported by PROMEP/103-5/07/2462 and Conacyt CB-2009-01/130523 grants.
\bibliographystyle{mn2e}
|
2,877,628,089,497 | arxiv | \section{Introduction}
Let $F$ be a field and let $k$ be a Galois extension of $F$ of degree $n$ with cyclic group generated by $\sigma$. For $a\in F^{\times}$, we let $(k,\sigma, a)$ denote the cyclic $F$-algebra generated over $k$ by a single element $y$ with defining relation $ycy^{-1}=\sigma(c)$ for $c\in k$ and $y^n=a$. If $F$ contains a primitive $n$-th root of unity $\zeta$, it follows from Kummer theory that one may write $k$ in the form $k=F(\sqrt[n]{b})$ for some $b\in F^{\times}$. The algebra $(k, \sigma, a)$ is then isomorphic to the \emph{symbol} algebra $(a,b)_n$ over $F$, that is, a central simple $F$-algebra generated by two elements $i$ and $j$ satisfying $i^n=a$, $j^n = b$ and $ij=\zeta ji$ (see for instance \cite[\S 15. 4]{Pie82}). In the case $n=2, \zeta=-1$, one gets a quaternion algebra over $F$ that will be denoted $(a, b)$.
A central division algebra decomposes into a tensor product of symbol algebras (of degree $2$ in \cite{ART79} and of degree an arbitrary prime $p$ in \cite{Row82}) if and only if it contains a set whose elements satisfy some commuting properties: \emph{$q$-generating set} in \cite{ART79}, \emph{$p$-central set} in \cite{Row82}, and a set of representatives of an \emph{armature} in \cite{Tig82}. Our approach is based on these notions.
The main goal of this paper is to further investigate the decomposability of central simple algebras; the study of power central-elements is a constant tool. Let $A$ be a central simple algebra over $F$ and let $k_1,\ldots, k_r$ be cyclic extensions of $F$, of respective degree $n_1,\ldots, n_r$, contained in $A$ such that $k_1\otimes_F \cdots \otimes_F k_r$ is a field. It is natural to ask: \emph{when does there exist a decomposition of $A$ into a tensor product of symbol algebras in which each $k_i$ is in a symbol $F$-subalgebra factor of $A$ of degree $n_i$? } Starting from $A$, we construct a division algebra $E$ whose center is the iterated Laurent series with $r$ indeterminates over $F$ and show that $A$ admits such a decomposition if and only if $E$ is a tensor product of symbol algebras (see Theorem \ref{thm3.1} and Corollary \ref{thm3.2} for details). Note that Corollary \ref{thm3.2} is very close to a result of Tignol \cite[Prop. 2.10]{Tig87}. In contrast with Tignol's result (which is stated in terms of Brauer equivalence and only for prime exponent), Corollary \ref{thm3.2} is stated in terms of isomorphism classes and is valid for any exponent. Moreover, our approach is completely different. We will give an example, pointed out by Merkurjev, of a division algebra which is a tensor product of three quaternion algebras, and containing a quadratic field extension which is in no quaternion subalgebra (Corollary \ref{cor3.1}). Using valuation theory, we give a general method for constructing tensor products of quaternion algebras containing a quadratic field extension which is in no quaternion subalgebra. As an application, let $A$ be a central simple algebra of degree $8$ and exponent $2$ over $F$, and containing a quadratic field extension which is in no quaternion subalgebra. We use Corollary \ref{thm3.2} to associate with $A$ an example of an indecomposable algebra of degree $8$ and exponent $2$ over a field of rational functions in one variable over a field of $2$-cohomological dimension $3$ (see Theorem \ref{application}). This latter field is obtained by an inductive process pioneered by Merkurjev \cite{Mer92}.
We next recall some results related to our main question: let $A$ be a $2$-power dimensional central simple algebra over $F$ and let $F(\sqrt{d_1}, \sqrt{d_2})\subset A$ be a biquadratic field extension of $F$. If $A$ is a biquaternion algebra, it follows from a result of Albert that $A\simeq (d_1, d_1')\otimes_F (d_2, d_2')$ where $d_1', d_2'\in F^{\times}$ (see for instance \cite{Rac74}). As observed above, this is not true anymore in higher degree. More generally, if $A$ is of degree $8$ and exponent $2$ and $F(\sqrt{d})\subset A$ is a quadratic field extension, there exists a cohomological criterion associated with the centralizer of $F(\sqrt{d})$ in $A$ which determines whether $F(\sqrt{d})$ lies in a quaternion $F$-subalgebra of $A$ (see \cite[Prop. 4.4]{Bar}). In the particular case where the $2$-cohomological dimension of $F$ is $2$ and $A$ is a division algebra of exponent $2$ over $F$, the situation is more favorable: it is shown in \cite[Thm. 3.3]{Bar} that there exits a decomposition of $A$ into a tensor product of quaternion $F$-algebras in which each $F(\sqrt{d_i})$ (for $i=1,2$) is in a quaternion $F$-subalgebra.
An outline of this article is the following: in Section \ref{sect2} we collect from \cite{Tig82} and \cite{TW87} some results on armatures of algebras that will be used in the proofs of the main results. Section \ref{sect3} is devoted to the statements and the proofs of the main results. The particular case of exponent $2$ is analyzed (in more details) in Section \ref{sect4}.
All algebras considered in this paper are associative and finite-dimensional over their center. A central simple algebra $A$ over a field $F$ is \emph{decomposable} if $A \simeq A_1 \otimes_F A_2$ for two central simple $F$-algebras $A_1$ and $A_2$ both non isomorphic to $F$; otherwise A is called \emph{indecomposable}.
Throughout this article, we shall use freely the standard terminology and notation from the theory of finite-dimensional algebras and the theory of valuations on division algebras. For these, as well as background information, we refer the reader to Pierce's book \cite{Pie82}.
\section{Armatures of algebras}\label{sect2}
Armatures in central simple algebras are a major tool for the next section. The goal of this section is to recall the notion of an armature and gather some preliminary results that will be used in the sequel.
We write $|H|$ for the cardinality of a set $H$. Let $A$ be a central simple $F$-algebra. For $a\in A^{\times}/F^{\times}$, we fix an element $x_a$ of $A$ whose image in $A^{\times}/F^{\times}$ is $a$, that is, $a=x_aF^{\times}$. For a finite subgroup $\mathcal A$ of $A^{\times}/F^{\times}$,
\[
F[\mathcal A]= \Big\{\sum_{a\in\mathcal A}c_ax_a\,|\, c_a\in F \Big\}
\]
denotes the $F$-subspace of $A$ generated by $\{x_a\,|\, a\in \mathcal A\}$. Note that this subspace is independent of the choice of representatives $x_a$ for $a\in\mathcal A$. Since $\mathcal A$ is a group, $F[\mathcal A]$ is the subalgebra of $A$ generated by $\{x_a\,|\, a\in \mathcal A\}$. As was observed in \cite{Tig82}, if $\mathcal A$ is a finite abelian subgroup of $A^{\times}/F^{\times}$ there is an associated pairing $\langle\,,\,\rangle$ on $\mathcal A\times\mathcal A$ defined by
\[
\langle a,b\rangle=x_ax_bx^{-1}_ax_b^{-1}.
\]
This definition is independent of the choice of representatives $x_a, x_b$ for $a,b$ and $\langle a,b\rangle$ belongs to $F^{\times}$ as $\mathcal A$ is abelian. Hence, $\langle a,b\rangle$ is central in $A$, and it follows that the pairing $\langle\,,\,\rangle$ is bimultiplicative. It is also alternating, obviously. Thus, as $\mathcal A$ is finite, the image of $\langle\,,\,\rangle$ is a finite subset of $\mu(F)$ (where $\mu(F)$ denotes the group of roots of unity of $F$). For any subgroup $\mathcal H$ of $\mathcal A$ let
\[
\mathcal H^{\perp}=\{a\in \mathcal A\,|\,\, \langle a,h\rangle =1 \,\text{ for all } h\in \mathcal H\},
\]
a subgroup of $\mathcal A$. The subgroup $\mathcal H^{\perp}$ is called the \emph{orthogonal} of $\mathcal H$ with respect to $\langle\,,\,\rangle$. The \emph{radical}\index{radical} of $\mathcal A$, rad$(\mathcal A)$, is defined to be $\mathcal A^{\perp}$. The pairing $\langle\,,\,\rangle$ is called \emph{nondegenerate}\index{nondegenerate} on $\mathcal A$ if $\text{rad}(\mathcal A)=\{1_{\mathcal A}\}$.
For $g\in \mathcal A$, we denote by $(g)$ the cyclic subgroup of $\mathcal A$ generated by $g$ . The set $\{g_1,\ldots,g_r\}$ is called a base of $\mathcal A$ if $\mathcal A$ is the internal direct product
\[
\mathcal A=(g_1)\times\cdots\times (g_r).
\]
If $\langle\,,\,\rangle$ is nondegenerate then $\mathcal A$ has a \emph{symplectic base}\index{symplectic base} with respect to $\langle\,,\,\rangle$, i.e, a base $\{g_1,h_1,\ldots, g_n,h_n\}$ such that for all $i,j$
\[
\langle g_i,h_i\rangle= c_i,\text{ where } \operatorname{ord}(g_i)=\operatorname{ord}(h_i)=\operatorname{ord}(c_i)
\]
\[
\langle g_i,g_j\rangle=\langle h_i,h_j\rangle=1 \text{ and, if } i\ne j,\,\, \langle g_i,h_j\rangle=1
\]
(see \cite[1.8]{Tig82}).
\begin{dfn}
For any finite-dimensional $F$-algebra $A$, a subgroup $\mathcal A$ of $A^{\times}/F^{\times}$ is an \emph{armature}\index{armature} of $A$ if $\mathcal A$ is abelian, $|\mathcal A|=\dim_F A$, and $F[\mathcal A]=A$.
\end{dfn}
If $\mathcal A=\{a_1,\ldots, a_n\}$ is an armature of $A$, the above definition shows that the set $\{x_{a_1},\ldots,x_{a_n}\}$ is an $F$-base of $A$. The notion of an armature was introduced by Tignol in \cite{Tig82} for division algebras. The definition given here, slightly different from that given in \cite{Tig82}, comes from \cite{TW87}. This definition allows armatures in algebras other than division algebras. The following examples will be used repeatedly in the next section.
\begin{examples}\label{ex3.1}
(a) (\cite{Tig82}) Let $A=A_1\otimes_F\cdots\otimes_F A_r$ be a tensor product of symbol $F$-algebras where $A_k$ is a symbol subalgebra of degree $n_k$. Suppose $F$ contains a primitive $n_k$-th root $\zeta_k$ of unity for $k=1,\ldots,r$. So, $A_k$ is isomorphic to a symbol algebra $(a_k,b_k)_{\zeta_k}$ of degree $n_k$ for some $a_k, b_k\in F^\times$. For each $k$, let $i_k,j_k$ be a symbol generator of $(a_k,b_k)_{\zeta_k}$. The image $\mathcal A$ in $A^{\times}/F^{\times}$ of the set
\[
\{i_1^{\alpha_1}j_1^{\beta_1}\ldots i_r^{\alpha_r}j_r^{\beta_r}\,|\,\, 0\le \alpha_k,\beta_k\le n_k-1\}
\]
is an armature of $A$ isomorphic to $(\mathbb{Z}/n_1\mathbb{Z})^2\times\cdots\times (\mathbb{Z}/n_r\mathbb{Z})^2$. We observe that for all $1\ne a\in\mathcal A$ there exists $b\in \mathcal A$ such that $\langle a,b\rangle\ne 1$; that is the pairing $\langle\,,\,\rangle$ is nondegenerate on $\mathcal A$. Furthermore, $\{i_1F^\times, j_1F^\times,\ldots, i_rF^\times, j_rF^\times\}$ is a symplectic base of $\mathcal A$.
\vspace{0.2cm}
(b) Let $M$ be a finite abelian extension of a field $F$ and let $G$ be the Galois group of $M$ over $F$. Let $\ell$ be the exponent of $G$. If $F$ contains an $\ell$-th primitive root of unity, the extension $M/F$ is called a \emph{Kummer extension}\index{Kummer extension}. Let
\[
S=\{x\in M^{\times}\,\,|\,\, x^\ell\in F^{\times}\} \quad \text{and} \quad \text{\sf Kum}(M/F)= S/F^{\times}.
\]
It follows from Kummer theory (see for instance \cite[p.119-123]{Jac64}) that $\text{\sf Kum}(M/F)$ is a subgroup of $M^{\times}/F^{\times}$ and is dual to $G$ by the nondegenerate Kummer pairing $G\times \text{\sf Kum}(M/F)\to \mu(F) $ given by $(\sigma, b)=\sigma(x_b)x_b^{-1}$, for $\sigma\in G$ and $b\in \text{\sf Kum}(M/F)$. Whence, $\text{\sf Kum}(M/F)$ is isomorphic (not canonically in general) to $G$. As observed in \cite[Ex. 2.4]{TW87}, the subgroup $\text{\sf Kum}(M/F)$ is the only armature of $M$ with exponent dividing $\ell$.
\end{examples}
Let $\mathcal A$ be an armature of a central simple $F$-algebra $A$ and let $\{a_1,b_1,\ldots, a_n,b_n\}$ be a symplectic base of $\mathcal A$ with respect to $\langle\,,\,\rangle$. We shall denote by $ F[(a_k)\times(b_k)]$ the subalgebra of $A$ generated by the representatives $x_{a_k}$ and $x_{b_k}$ of $a_k$ and $b_k$. It is clear that $F[(a_k)\times(b_k)]$ is a symbol subalgebra of $A$ of degree $n_k=\operatorname{ord}(a_k)=\operatorname{ord}(b_k)$ and generated by $x_{a_k}$ and $x_{b_k}$. It is shown in \cite[Lemma 2.5]{TW87} that $A\simeq F[(a_1)\times(b_1)]\otimes_F\cdots\otimes_F F[(a_n)\times(b_n)]$.
\vspace{0.2cm}
Actually, the notion of an armature is a generalization and refinement of the notion of a quaternion generating set ($q$-generating set) introduced in \cite{ART79}. Indeed, a central simple algebra $A$ over $F$ has an armature if and only if $A$ is isomorphic to a tensor product of symbol algebras over $F$ (see \cite[Prop. 2.7]{TW87}).
\vspace{0.2cm}
Note that if the exponent of $\mathcal A$ is a prime $p$, we may consider $\mathcal A$ as a vector space over the field with $p$ elements $\mathbb F_p$. Identifying the group of $p$-th roots of unity with $\mathbb F_p$ by a choice of a primitive $p$-th root of unity, we may suppose the pairing has values in $\mathbb F_p$. So, two elements $a,b\in\mathcal A$ are orthogonal if and only if $\langle a,b\rangle=0$. We need the following proposition:
\begin{proposition}\label{prop3.1}
Let $V$ be a vector space over $\mathbb F_p$ of dimension $2n$ and let $\langle\,,\,\rangle$ be a nondegenerate alternating pairing on $V$. Let $\{e_1,\ldots,e_r\}$ be a base of a totally isotropic subspace of $V$ with respect to $\langle\,,\,\rangle$. There are $f_1, \ldots,f_r,e_{r+1},f_{r+1}, \ldots,e_n, f_n$ in $V$ such that $\{e_1,f_1,\ldots,e_n,f_n\}$ is a symplectic base of $V$.
\end{proposition}
\begin{proof}
We argue by induction on the dimension of the totally isotropic subspace spanned by $e_1,\ldots,e_r$. If $r=1$, since the pairing is nondegenerate, there is $f_1\in V$ such that $\langle e_1,f_1\rangle\ne 0$ . Denote by $U=\text{span}(e_1,f_1)$ the subspace spanned by $e_1,f_1$. We have $V=U\perp U^{\perp}$ since the restriction of $\langle\,,\,\rangle$ to $U$ is nondegenerate. We take for $\{e_2, f_2,\ldots, e_n, f_n\}$ a symplectic base of $U^{\perp}$.
Assume the statement for a totally isotropic subspace of dimension $r-1$. Let $W=\text{span}(e_2,\ldots,e_r)$; we have $W\subset W^{\perp}$. First, we find $f_1\in W^{\perp}$ such that $\langle e_1,f_1\rangle\ne 0$. For this, consider the induced pairing, also denoted by $\langle\,,\,\rangle$, on $W^{\perp}/W$ defined by $\langle x+W,y+W\rangle=\langle x,y\rangle$ for $x,y\in W^{\perp}$. It is well-defined, and nondegenerate since $(W^{\perp})^{\perp}=W$. The element $e_1+W$ being non-zero in $W^{\perp}/W$, there is $f_1+W\in W^{\perp}/W$ such that $0\ne\langle e_1,f_1\rangle =\langle e_1+W,f_1+W\rangle$. Letting $U=\text{span}(e_1,f_1)$, we have $V=U\perp U^{\perp}$ and $e_2,\ldots,e_r\in U^{\perp}$. Induction yields $f_2,\ldots, f_r, e_{r+1}, f_{r+1},\ldots, e_n, f_n\in U^{\perp}$ such that $\{e_2, f_2,\ldots, e_n, f_n\}$ is a symplectic base of $U^{\perp}$. Then $\{e_1, f_1,\ldots, e_n, f_n\}$ is a symplectic base of $V$.
\end{proof}
\section{Decomposability}\label{decomposability}\label{sect3}
Let $A$ be a central simple algebra over $F$ and let $t_1,\ldots,t_r$ be independent indeterminates over $F$. For $i=1,\ldots,r$, let $k_i$ be a cyclic extension of $F$ of degree $n_i$ contained in $A$. We assume that $M=k_1\otimes_F\cdots \otimes_F k_r$ is a field and denote by $G$ the Galois group of $M$ over $F$. So $[M:F]=n_1\ldots n_r$ and $G=\langle \sigma_1\rangle\times\cdots \times\langle \sigma_r\rangle$, where $\langle \sigma_i\rangle$ is the Galois group of $k_i$ over $F$, and the order of $G$ is $n_1\ldots n_r$. Every element $\sigma\in G$ can be expressed as $\sigma=\sigma_1^{m_1}\ldots \sigma_r^{m_r} $ ($0\le m_i< n_i$). We shall denote by $C=C_AM$ the centralizer of $M$ in $A$. Let $t_1,\ldots, t_r$ be independent indeterminates over $F$. Consider the fields
\[
L'=F(t_1, \ldots, t_r) \quad \text{ and }\quad L= F((t_1))\ldots((t_r))
\]
and the following central simple algebras over $L'$ and $L$ respectively
\[
N'=(k_1\otimes_FL', \sigma_1\otimes\text{id},t_1)\otimes_{L'}\cdots\otimes_{L'}(k_r\otimes_FL', \sigma_r\otimes\text{id},t_r)
\]
and
\[
N=(k_1\otimes_FL, \sigma_1\otimes\text{id},t_1)\otimes_L\cdots\otimes_L(k_r\otimes_FL, \sigma_r\otimes\text{id},t_r).
\]
We let
\[
R'=A\otimes_F N'\quad \text{ and }\quad R=A\otimes_F N.
\]
In this section our goal is to prove the following results:
\begin{theorem}\label{thm3.1}
Let $\mathcal M$ be a finite group with a nondegenerate alternating pairing $\mathcal M\times \mathcal M \to \mu(F)$. Suppose $C$ is a division subalgebra of $A$, and $F$ contains a primitive $\exp(\mathcal M)$-th root of unity. Then, the following are equivalent:
\begin{enumerate}
\item[(i)] The division algebra Brauer equivalent to $R'$ (respectively $R$) has an armature isomorphic to $\mathcal M$.
\item[(ii)] The algebra $A$ has an armature isomorphic to $\mathcal M$ and containing $\text{\sf Kum}(M/F)$ as a totally isotropic subgroup.
\end{enumerate}
\end{theorem}
\vspace{0.3cm}
In the particular case where $A$ is of degree $p^n$ and exponent $p$ (for a prime number $p$) and each $k_i$ is a cyclic extension of $F$ of degree $p$, we have:
\begin{corollary}\label{thm3.2}
Assume that $A$ is of degree $p^n$ and exponent $p$. Suppose $C$ is a division subalgebra of $A$, $F$ contains a primitive $p$-th root of unity, and $n_i=p$ for $i=1,\ldots, r$. Then, the following are equivalent:
\begin{enumerate}
\item[(i)] The division algebra Brauer equivalent to $R'$ (respectively $R$) is decomposable into a tensor product of symbol algebras of degree $p$.
\item[(ii)] The algebra $A$ decomposes as
\[
A\simeq (k_1,\sigma_1, \delta_1)\otimes_F\cdots\otimes_F(k_r,\sigma_r, \delta_r)\otimes_F A_{r+1}\otimes_F\cdots \otimes_F A_n
\]
for some $\delta_1,\ldots,\delta_r\in F^\times$ and some symbol algebras $A_{r+1},\ldots, A_n$ of degree $p$.
\end{enumerate}
Moreover, if these conditions are satisfied then the division algebras Brauer equivalent to $R'$ and $R$ decompose respectively as
\[
(k_1\otimes_FL', \sigma_1\otimes\text{id},\delta_1t_1)\otimes_{L'}\cdots\otimes_{L'}(k_r\otimes_FL', \sigma_r\otimes\text{id},\delta_rt_r)\otimes_F A_{r+1}\otimes_F\cdots \otimes_F A_n
\]
and
\[
(k_1\otimes_FL, \sigma_1\otimes\text{id},\delta_1t_1)\otimes_{L}\cdots\otimes_{L}(k_r\otimes_FL, \sigma_r\otimes\text{id},\delta_rt_r)\otimes_F A_{r+1}\otimes_F\cdots \otimes_F A_n.
\]
\end{corollary}
\vspace{0.3cm}
As opposed to part (i), it is not enough in part (ii) of Theorem \ref{thm3.1} to assume simply that $A$ has an armature to get an armature in the division algebra Brauer equivalent to $R'$ or $R$. The following example shows that the existence of an armature and the existence of an armature containing $\text{\sf Kum}(M/F)$ are different.
\begin{example}
Denote by $\mathbb{Q}_2$ the field of $2$-adic numbers, and let $A$ be a division algebra of degree $4$ and exponent $4$ over $F=\mathbb{Q}_2(\sqrt{-1})$. Such an algebra is a symbol (see \cite[Th., p. 338]{Pie82}). Note that $M=F(\sqrt{2},\sqrt{5})$ is a field since the set $\{-1, 2, 5\}$ forms a $\mathbb{Z}/2\mathbb{Z}$-basis of $\mathbb{Q}_2^\times/\mathbb{Q}_2^{\times 2}$ (see for instance \cite[Lemma 2.24, p. 163]{Lam05}). It also follows by \cite[Prop., p. 339]{Pie82} that $A\otimes M$ is split. Hence, since $[M:F]$ divides $\deg(A)$, we deduce that $M\subset A$. Moreover, it is clear that the centralizer of $M$ in $A$ is M. The algebra $A$ being a symbol, it has an armature $\mathcal A$ isomorphic to $(\mathbb{Z}/4\mathbb{Z})^2$ (see Example \ref{ex3.1}). Assume that $\mathcal A$ contains $\text{\sf Kum}(M/F)\simeq (\mathbb{Z}/2\mathbb{Z})^2$. Since $\mathcal A^2=\{a^2\,|\,\, a\in \mathcal A\}=\text{\sf Kum}(M/F)$, the algebra $A$ is a symbol of the form $A=(c,d)_4$ with $c.F^{\times 2}$, $d.F^{\times 2}\in \{2.F^{\times 2}, \,5.F^{\times 2},\, 2.5.F^{\times 2}\}$. It follows that $A\otimes A$ is either Brauer equivalent to the quaternion algebra $(2,5)$ or $(2, 2.5)$ or $(5, 2.5)$. Since $(2,5)\simeq (-1,-1)$ over $\mathbb{Q}_2$, the algebra $(2,5)$ is split over $F$; that is $A\otimes A$ is split. Therefore the exponent of $A$ must be $2$; impossible. Therefore $A$ has no armature containing $\text{\sf Kum}(M/F)$.
Now, let $E$ be the division algebra Brauer equivalent to $A\otimes (2,t_1)\otimes (5,t_2)$. As we will see soon (Lemma \ref{lm3.1} and Lemma \ref{lm3.2}), $\deg(E)=\exp(E)=4$. But $E$ has no armature, that is, $E$ is not a symbol. Indeed, suppose $E$ has an armature $\mathcal B$. If $\mathcal B\simeq (\mathbb{Z}/2\mathbb{Z})^4$ then $E$ is a biquaternion algebra, so $\exp(E)=2$; contradiction. Therefore $\mathcal B$ is isomorphic to $(\mathbb{Z}/4\mathbb{Z})^2$. It follows then by Theorem \ref{thm3.1} that $A$ has an armature containing the amature $\text{\sf Kum}(M/F)$ of $M$. This is impossible as we showed above; therefore $E$ has no armature.
\end{example}
\subsection{Brauer classes of $R'$ and $R$}
For the proof of the results above, we need an explicit description of the division algebras Brauer equivalent to $R'$ and $R$. First, we fix some notation: recall that we denoted by $G=\langle\sigma_1\rangle\times\cdots\times \langle\sigma_r\rangle$ the Galois group of $M$ over $F$. By the Skolem-Noether Theorem, for each $\sigma_i$ there exists $z_i\in A^\times$ such that $\sigma_i(b)=z_ibz_i^{-1}$ for all $b\in M$. Notice that $z_i^{n_i}=:c_i\in C$ and $z_iz_jz_i^{-1} z_j^{-1}=:u_{ij}\in C$ for all $i,j$. For all $\sigma=\sigma_1^{m_1}\ldots\sigma_r^{m_r}\in G$, we set $z_{\sigma}=z_1^{m_1}\ldots z_r^{m_r}$ ($0\le m_i<n_i$). Setting $c(\sigma,\tau)=z_\sigma z_\tau (z_{\sigma\tau})^{-1}$, a simple observation shows for all $\sigma,\tau$
\begin{equation}\label{eq3.1}
c(\sigma,\tau)\in C \quad \text{ and }\quad z_\sigma b=\sigma(b)z_\sigma \text{ for all } b\in M.
\end{equation}
In fact, $c(\sigma,\tau)$ can be calculated from the elements $u_{ij}$ and $c_i$.
Let $y_i$ be a generator of $(k_i\otimes L', \sigma_i\otimes 1, t_i)$, that is an element satisfying the relations $y_i(d\otimes l)y_i^{^{-1}} = \sigma_i(d)\otimes l$ for all $d\in k_i$ and $l\in L'$ and $y_i^{n_i}=t_i$. For $\sigma=\sigma_1^{m_1}\ldots \sigma_r^{m_r}\in G$ with $0\le m_i< n_i$, we set $y_\sigma=y_1^{m_1}\ldots y_r^{m_r}$.
The algebras $N'$ and $N$ being crossed products, we may write
\[
N'=\bigoplus_{\sigma\in G}(M\otimes L')y_\sigma \quad \text{ and } \quad N=\bigoplus_{\sigma\in G}(M\otimes L)y_\sigma
\]
and the $y_\sigma$ satisfy
\begin{equation}\label{eq3.2}
y_\sigma y_\tau(y_{\sigma\tau})^{-1} \in M\otimes L \text{ and } y_\sigma(b\otimes l)=(\sigma(b)\otimes l)y_\sigma
\end{equation}
for all $b\in M$ and $l\in L$. Set $f(\sigma,\tau)= y_\sigma y_\tau(y_{\sigma\tau})^{-1}$. The elements $f(\sigma,\tau)$ are in fact in $L'$. Indeed, let $y_\sigma= y_1^{\alpha_1}\ldots y_r^{\alpha_r}$ and $y_\tau= y_1^{\beta_1}\ldots y_r^{\beta_r}$ with $0\le \alpha_i, \beta_i < n_i$ . We get
\begin{equation}\label{eq3.2.0}
f(\sigma,\tau)=y_\sigma y_\tau(y_{\sigma\tau})^{-1}=t_1^{\varepsilon_1}\ldots t_r^{\varepsilon_r}
\end{equation}
where $\varepsilon_i=0$ if $\alpha_i+\beta_i< \operatorname{ord}(\sigma_i)= n_i$ and $\varepsilon_i=1$ if $\alpha_i+\beta_i\geq \operatorname{ord}(\sigma_i)$. Note that $f(\sigma,\tau)=f(\tau,\sigma)$ for all $\sigma,\tau\in G$.
Now, let $e$ be the \emph{separability idempotent} of $M$, that is, the idempotent $e\in M\otimes_F M$ determined uniquely by the conditions that
\begin{equation}\label{eq3.3}
e\cdot(x\otimes 1)=e\cdot(1\otimes x) \quad \text{for all } x\in M
\end{equation}
and the multiplication map $M\otimes_F M\,\to\, M$ carries $e$ to $1$ (see for instance \cite[\S 14. 3]{Pie82}). For $\sigma\in G$, let
\[
e_\sigma=(\text{id}\otimes \sigma)(e)\in M\otimes_F M.
\]
The elements $(e_\sigma)_{\sigma\in G}$ form a family of orthogonal primitive idempotents of $ M\otimes_F M$ (see \cite[\S 14. 3]{Pie82} ) and it follows, by applying $\text{id}\otimes\sigma$ to each side of (\ref{eq3.3}), that
\begin{equation}\label{eq3.4}
e_\sigma\cdot(x\otimes 1)=e_\sigma\cdot(1\otimes \sigma(x)) \quad \text{for } x\in M.
\end{equation}
We need the following lemma:
\begin{lemma}\label{lm3.1}
Notations are as above.
\begin{enumerate}
\item The elements $z_{\sigma}\otimes y_\sigma$ are subject to the following rules
\begin{itemize}
\item[(i)] For all $\sigma,\tau\in G$, there exists $u\in C_L$ such that
\[
(z_\sigma\otimes y_\sigma)(z_\tau\otimes y_\tau)=(u\otimes 1)(z_{\sigma\tau}\otimes y_{\sigma\tau}).
\]
\item[(ii)] $(z_\sigma\otimes y_\sigma)(c\otimes 1)=((z_{\sigma}cz^{-1}_{\sigma})\otimes 1)(z_\sigma\otimes y_\sigma)$ for all $c\in C$. Moreover, $z_{\sigma}cz^{-1}_{\sigma}\in C$ for all $\sigma\in G$.
\end{itemize}
\item The sums
\[
E'=\sum_{\sigma\in G}C_{L'}z_\sigma\otimes y_\sigma \quad\text{and} \quad E=\sum_{\sigma\in G}C_{L}z_\sigma\otimes y_\sigma
\]
are direct and are central simple subalgebras of $R'$ and $R$ respectively. Moreover, the algebras $R'$ and $R$ are Brauer equivalent to $E'$ and $E$ respectively, and $\deg E'=\deg E=\deg A$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The statement of (i) follows from relations (\ref{eq3.1}), (\ref{eq3.2}) and the fact that $f(\sigma,\tau)\in L'$. Since the elements $1\otimes y_\sigma$ centralize $C_{L}\otimes 1$, the statement of (ii) is clear.
(2) Suppose $\sum_{\sigma\in G}c_\sigma z_\sigma\otimes y_\sigma=0$ with $c_\sigma\in C_{L'}$ (or $C_{L}$). Pick such a sum with a minimal number of non-zero terms. There are at least two non-zero elements, say $c_\rho z_\rho\otimes y_\rho$, $c_\tau z_\tau \otimes y_\tau$, in the sum. Let $b\in M$ be such that $\rho(b)\ne b$ and $\tau(b)=b$. One has
\[
(b\otimes 1)\Big(\sum_{\sigma\in G}c_\sigma z_\sigma\otimes y_\sigma\Big)(b\otimes 1)^{-1}-\sum_{\sigma\in G}c_\sigma z_\sigma\otimes y_\sigma=0
\]
and the number of non-zero terms is nontrivial and strictly smaller; contradiction. Whence we have the direct sums $E'=\bigoplus_{\sigma\in G}C_{L'}z_\sigma\otimes y_\sigma$ and $E=\bigoplus_{\sigma\in G}C_{L}z_\sigma\otimes y_\sigma$. It is clear that $E'\subset R'$ and $E\subset R$. On the other hand, the computation rules of the part (1) show that $E'$ and $E$ are generalized crossed products (see \cite[Th. 11.11]{Alb61} or \cite[\S 1.4]{Jac96}). Hence, the same arguments as for the usual crossed products show that $E'$ and $E$ are central simple algebras over $L'$ and $L$ respectively. Moreover, since $\dim C = \frac{1}{n_1\ldots n_r}\dim A$, we have $\deg E'=\deg E=\deg A$.
Now, it remains to show that $R'$ and $R$ are respectively Brauer equivalent to $E'$ and $E$. For this we work over $L$; the same arguments apply over $L'$. For $z_\sigma\otimes y_\sigma$ as above, consider the inner automorphism
\[
\text{Int}(z_\sigma\otimes y_\sigma)\,:\, R\, \longrightarrow\, R.
\]
Notice that Int$(z_\sigma\otimes y_\sigma)(e_\tau)$ is in $M\otimes M$ and is a primitive idempotent for all $\tau\in G$ (since $e_\tau$ is primitive). Moreover, for $x\in M$,
\begin{eqnarray*}
\text{Int}(z_\sigma\otimes y_\sigma)(e_\tau)\cdot(x\otimes 1) &=& (z_\sigma\otimes y_\sigma)e_\tau (z_\sigma^{-1}\otimes y_\sigma^{-1})\cdot(x\otimes 1)\\
& = & (z_\sigma\otimes y_\sigma)e_\tau \cdot (\sigma^{-1}(x)\otimes 1)(z_\sigma^{-1}\otimes
y_\sigma^{-1})\\
& = & (z_\sigma\otimes y_\sigma)e_\tau \cdot (1 \otimes \tau\sigma^{-1}(x) )(z_\sigma^{-1}\otimes
y_\sigma^{-1})\\
& = & (1 \otimes \tau(x) )\cdot\text{Int}(z_\sigma\otimes y_\sigma)(e_\tau).
\end{eqnarray*}
Therefore $\text{Int}(z_\sigma\otimes y_\sigma)(e_\tau)=e_\tau$ by comparing with the definition of $e$ and the relation (\ref{eq3.4}). Hence, each $e_\tau$ centralizes $E$ in $R$. On the other hand, since the degree of the centralizer $C_RE$ of $E$ in $R$ is $n_1\ldots n_r$ and $(e_\tau)_{\tau\in G}\subset C_RE$, the algebra $C_RE$ is split. So $R$ is Brauer equivalent to $E$.
\end{proof}
\begin{lemma}\label{lm3.2}
Notations are as in Lemma \ref{lm3.1}. If $C$ is a division algebra then there exists a unique valuation on $E$ which extends the $(t_1,\ldots, t_r)$-adic valuation on $L$. Consequently $E'$ and $E$ are division algebras.
\end{lemma}
\begin{proof}
The $(t_1,\ldots, t_r)$-adic valuation on $L$ being Henselian, it extends to a unique valuation to each division algebra over $L$ (see for instance \cite{Wad86}). It follows that there is a valuation on $C_L$ extending the $(t_1,\ldots, t_r)$-adic valuation on $L$. More precisely this valuation is constructed as follows: writing $C_L$ as
\[
C_L=\Bigg\{\sum_{i_1\in\mathbb{Z}}\cdots \sum_{i_r\in\mathbb{Z}}c_{i_1\ldots i_r}t_1^{i_1}\ldots t_r^{i_r}
\left| \begin{array}{ll} c_{i_1\ldots i_r} \in C \text{ and }\\
\{ (i_1,\ldots, i_r)\, |\, c_{i_1\ldots i_r}\ne 0 \text{ is well-ordered}\\
\text{for the right-to-left lexicographic ordering}\}
\end{array}
\right.
\Bigg\},
\]
computations show that the map $v: C_L^{\times} \to \mathbb{Z}^r$ defined by
\[
v\Big(\sum_{i_1\in\mathbb{Z}}\cdots \sum_{i_r\in\mathbb{Z}}c_{i_1\ldots i_r}t_1^{i_1}\ldots t_r^{i_r}\Big)= \min \{(i_1,\ldots, i_r) \,|\, c_{i_1\ldots i_r}\ne 0 \}
\]
is a valuation. Clearly $v$ extends the $(t_1,\ldots, t_r)$-adic valuation on $L$. Recall that $N$ is a division algebra over $L$ (see e.g. \cite[Ex. 3.6]{Wad02}). We also denote by $v$ the unique extension of the $(t_1,\ldots, t_r)$-valuation to $N$. Since $y_i^{n_i}=t_i$, we have
\[
v(y_i)=(0,\ldots,0,\frac{1}{n_i},0,\ldots, 0)\in \frac{1}{n_1}\mathbb{Z}\times\ldots \times\frac{1}{n_r}\mathbb{Z} .
\]
Hence, for $y_\sigma = y_1^{m_i}\ldots y_r^{m_r}$ where $\sigma= \sigma_1^{m_1}\ldots \sigma_r^{m_r}$ (with $1\le m_i < n_i$), we have $v(y_\sigma)=(\frac{m_1}{n_1},\ldots, \frac{m_r}{n_r})$. It then follows that
\begin{equation}\label{lmeq1}
v(y_\sigma)\not\equiv v(y_\tau)\bmod \mathbb{Z}^r \quad \text{ if }\quad \sigma\ne \tau.
\end{equation}
Now, define a map $w\,:\, E^\times \to \frac{1}{n_1}\mathbb{Z}\times\ldots \times\frac{1}{n_r}\mathbb{Z}$ as follows: for any $\sigma\in G$ and any $c\in C_L^\times$, set
\[
w(cz_\sigma\otimes y_\sigma)= v(c)+v(y_\sigma).
\]
For any $s\in E^\times$, $s$ has a unique representation $s=\sum_{\sigma\in G}c_\sigma z_\sigma\otimes y_\sigma$ with the $c_\sigma\in C_L$ and some $c_\sigma\ne 0$. Define
\[
w(s)= \min_{\sigma\in G}\{w(c_\sigma z_\sigma\otimes y_\sigma)\, |\, c_\sigma \ne 0\}.
\]
It follows by (\ref{lmeq1}) that $w(c_\sigma z_\sigma\otimes y_\sigma)\ne w(c_\tau z_\tau\otimes y_\tau)$ for $\sigma\ne \tau$. Thus, there is a unique summand $c_\iota z_\iota\otimes y_\iota$ of $s$ such that $w(s)=w(c_\iota z_\iota\otimes y_\iota)$; this $ c_\iota z_\iota\otimes y_\iota$ is called the \emph{leading term} of $s$.
We are going to show that $w$ is a valuation on $E$. Let $s'=\sum_{\sigma\in G}d_\sigma z_\sigma\otimes y_\sigma\in E^\times$ with $d_\sigma\in C_L$ and $s+s'\ne 0$. Let $(c_\rho+ d_\rho)z_\rho\otimes y_\rho$ be the leading term of $s+s'$. If $ c_\rho\ne 0$ and $ d_\rho\ne 0$, we have
\begin{eqnarray*}
w (( c_\rho + d_\rho)z_\rho\otimes y_\rho) & = & v(c_\rho+d_\rho)+v(y_\rho)\\
& \geq & \min (v(c_\rho)+ v(y_\rho), v(d_\rho)+v(y_\rho))\\
& = & \min (w(c_\rho z_\rho\otimes y_\rho), w(d_\rho z_\rho\otimes y_\rho))\\
& \geq & \min (w(s), w(s')).
\end{eqnarray*}
Thus, $w(s+s')= w (( c_\rho + d_\rho)z_\rho\otimes y_\rho))\geq \min (w(s), w(s'))$. This inequality still holds if $c_\rho =0$ or $d_\rho=0$.
By the usual argument, we also check that
\begin{equation}\label{lmeq2}
\text{if } w(s)\ne w(s') \text{ then } w(s+s')=\min (w(s), w(s')).
\end{equation}
It remains to show that $w(ss')= w(s)+w(s')$. For $\sigma,\tau\in G$, recall that
\[
(z_\sigma\otimes y_\sigma)(z_\tau\otimes y_\tau)= c(\sigma,\tau)f(\sigma,\tau)(z_{\sigma\tau}\otimes y_{\sigma\tau})
\]
for some $c(\sigma,\tau)\in C^\times$ and some $f(\sigma,\tau)\in L^\times$. It follows that, for $c_\sigma, d_\tau\in C^\times_L$,
\begin{eqnarray} \label{lmeq3}
w((c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau)) &=& w(c_\sigma(z_\sigma d_\tau z_\sigma^{-1})(z_\sigma\otimes y_\sigma)(z_\tau\otimes y_\tau)) \quad \text{(Lemma \ref{lm3.1})} \nonumber\\
& = & w(c_\sigma(z_\sigma d_\tau z_\sigma^{-1}) c(\sigma,\tau)f(\sigma,\tau)(z_{\sigma\tau}\otimes y_{\sigma\tau}))\nonumber\\
& = & v(c_\sigma)+ v(d_\tau)+ v(f(\sigma,\tau)y_{\sigma\tau})\nonumber\\
& = & v(c_\sigma)+ v(y_\sigma)+ v(d_\tau)+ v(y_\tau) \nonumber\\
& = & v(c_\sigma z_\sigma\otimes y_\sigma) + v(d_\tau z_\tau\otimes y_\tau).
\end{eqnarray}
Hence, we have
\begin{eqnarray} \label{lmeq4}
w(ss') &=& w\Big(\sum_{\sigma,\tau}(c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau)\Big)\nonumber\\
& \geq & \min_{\sigma,\tau}\{w((c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau))\,|\, c_\sigma, d_\tau \ne 0\}\nonumber\\
& = & \min_{\sigma,\tau} \{w(c_\sigma z_\sigma\otimes y_\sigma)+ w(d_\tau z_\tau\otimes y_\tau)\,|\, c_\sigma, d_\tau \ne 0\}\nonumber\\
& \geq & w(s)+ w(s').
\end{eqnarray}
Let $c_\rho z_\rho\otimes y_\rho$ and $d_\iota z_\iota \otimes y_\iota$ be the leading terms of $s$ and $s'$ respectively. Set $s_1=s-c_\rho z_\rho\otimes y_\rho$ and $s'_1= s'-d_\iota z_\iota \otimes y_\iota$. So,
\[
w(s)=w(c_\rho z_\rho\otimes y_\rho)< w(s_1) \quad \text{and}\quad w(s')= w(d_\iota z_\iota \otimes y_\iota)< w(s'_1).
\]
Writing
\[
ss'=(c_\rho z_\rho\otimes y_\rho)(d_\iota z_\iota \otimes y_\iota)+ s_1(d_\iota z_\iota \otimes y_\iota)+(c_\rho z_\rho\otimes y_\rho)s_1' + s_1s_1',
\]
it follows by (\ref{lmeq3}) and (\ref{lmeq4}) that the first summand in the right side of the above equality has valuation strictly smaller than the other three. Hence, by (\ref{lmeq2}) and (\ref{lmeq3}), one has $ss'\ne 0$ and
\[
w(ss')= w((c_\rho z_\rho\otimes y_\rho)(d_\iota z_\iota \otimes y_\iota))= w(s)+w(s').
\]
Therefore $w$ is a valuation on $E$. Since $E=E'\otimes_{L'}L$, the restriction of $w$ to $E'$ is also a valuation. The uniqueness of $w$ follows from its existence by \cite{Wad86}.
\end{proof}
Let $D$ be a division algebra with a valuation. The \emph{residue} division algebra of $D$ is denoted by $\overline{D}$.
We keep the notations above. Now, suppose $C$ is a division subalgebra of $A$ and denote by $\Gamma_E$ and $\Gamma_{L}$ the corresponding value groups of $E$ and $L$ respectively. Furthermore, assume that $E$ has an armature $\mathcal A$. The diagram
\[
\xymatrix {
1 \ar[r] & L^\times \ar[r] \ar[d]^v & E^\times \ar[r] \ar[d]^w & E^\times/L^\times \ar[r] \ar@{.>}[d] & 1 \\
0 \ar[r] & \Gamma_L \ar[r] & \Gamma_E \ar[r] & \Gamma_E/\Gamma_L \ar[r] & 0}
\]
induces a homomorphism
\[
w'\,:\,\mathcal A\subset E^{\times}/L^\times\,\longrightarrow\, \Gamma_E/\Gamma_L.
\]
Put $\mathcal A_0=\ker w'$ and let $a\in \mathcal A_0$. The above diagram shows that there exists a representative $x_a$ of $a$ such that $w(x_a)=0$. Define
\[
\bar\,\,\,:\,\mathcal A_0\,\longrightarrow\,\overline{E}^\times/\overline L^\times=C^\times/F^\times
\]
by
\[
a=x_aL^{\times}\,\longmapsto\, \bar a=\bar x_aF^{\times}
\]
where $x_a$ is such that $w(x_a)=0$ and $\bar x_a$ is the residue of $x_a$. If $y_a$ is another representative of $a$ such that $w(y_a)=0$, there is $h\in L^\times$ with $w(h)=0$ such that $y_a=x_ah$. Hence $\bar y_a=\bar x_a\bar h$, that is $\bar y_a=\bar x_aF^\times$, so $\bar\,\,$ is well defined. We have:
\begin{proposition}\label{prop3.3}
Assume that $C$ is a division algebra and $\mathcal A$ is an armature of $E$ as above. Then
\begin{itemize}
\item[(1)] The map $\bar\,\, \,:\, \mathcal A_0\,\longrightarrow\,C^\times/F^\times$ is an injective homomorphism.
\item[(2)] The image $\overline{\mathcal A_0}$ of $\mathcal A_0$ is an armature of $C$ over $F$.
\end{itemize}
\end{proposition}
\begin{proof}
(1) Let $a=x_aL^\times$ and $b=x_bL^\times$ be such that $w(x_a)=w(x_b)=0$. Choosing $x_{ab}=x_ax_b$, we have $w(x_{ab})=0$. Therefore $\bar x_{ab}=\bar x_a\bar x_b$; this shows that $\bar\,\,$ is a homomorphism.
Let $c\in \ker\bar\,\,$ with $c\ne 1$ and let $x_c\in E^{\times}$ be a representative of $c$ such that $w(x_c)=0$. Let $\bar x_c=\alpha \in F^{\times}$; then $x_c=\alpha+ x_c'$ for some $x_c'\in E$ with $w(x_c')>0$. The pairing $\langle \,,\,\rangle$ being nondegenerate on $\mathcal A$, there exists $d \in \mathcal A$ such that $\langle d,c\rangle=\zeta$ for some $1 \ne\zeta\in \mu(F)$. Thus,
\[
\overline{x_{d}x_cx_{d}^{-1}}=\zeta\bar x_c=\zeta\alpha.
\]
On the other hand,
\[
x_{d}x_cx_{d}^{-1}=x_{d}\alpha x_{d}^{-1}+x_d x_c' x_{d}^{-1}=\alpha+x_{d} x_c' x_{d}^{-1}.
\]
Hence, we get
\[
\overline{x_{d}x_cx_{d}^{-1}}=\alpha \,\,\,\text{ since } \,\,\, w(x_{d}x_c' x_{d}^{-1})>0; \,\, \text{contradiction}.
\]
Therefore the map $\bar\,\,$ is injective. Consequently, we have $|\overline{\mathcal A_0}|=|\mathcal A_0|$.
(2) We first show that $\vert \mathcal A_0\vert\le \dim_F C$: since $C=\overline E$, it suffices to prove that $(\bar x_a)_{a\in\mathcal A_0}$ are linearly independent over $F$. Let $\sum_{a\in \mathcal A_0}\lambda_a\bar x_a=0$,
with $\lambda_a\in F$, be a zero linear combination such that the set
\[
S=\{a\in \mathcal A_0\,\,|\,\,\lambda_a \ne 0\}
\]
is not empty and of least cardinality. For $s\in S$, let $x_s$ be a representative of $s$ in $E^{\times}$ such that $w(x_s)=0$. We have
\[
\bar x_s\Big(\sum_{a\in \mathcal A_0}\lambda_a\bar x_a\Big)\bar x_s^{-1}=\sum_{a\in \mathcal A_0}\langle a,s\rangle \lambda_a\bar x_a=0=\sum_{a\in \mathcal A_0}\lambda_a\bar x_a.
\]
Then the linear combination $\sum_{a\in \mathcal A_0}(1-\langle a,s\rangle) \lambda_a\bar x_a$ is zero and the number of non-zero coefficients is less than the cardinality of $S$ because $\langle s,s\rangle=1$. Therefore, $\langle a,s\rangle=1$ for all $a,s\in S$; this implies that $\bar x_s$ and $\bar x_{s'}$ commute for all $s, s'\in S$. It follows from \cite[Lemma 1.5]{Tig82} that the elements $\bar x_s$, for $s\in S$, are linearly independent; contradicting the fact that $S$ is not empty. Combining with the part (1), we get
\[
\vert\overline{\mathcal A_0}\vert=\vert\mathcal A_0\vert\le\dim C=\frac{1}{n_1\ldots n_r}\vert\mathcal A\vert.
\]
On the other hand, since $\vert w'(\mathcal A)\vert=\frac{\vert\mathcal A\vert}{\vert\mathcal A_0\vert}$, we have $\vert w'(\mathcal A)\vert\geq n_1\ldots n_r$. We already know that $\vert w'(\mathcal A)\vert\le n_1\ldots n_r$ because $w'(\mathcal A)\subset \Gamma_E/\Gamma_L$ and $|\Gamma_E/\Gamma_L|=n_1\ldots n_r$. It follows that $\vert w'(\mathcal A)\vert= n_1\ldots n_r$ and $\vert\overline{\mathcal A_0}\vert=\dim_F C$. Since we showed that $(\bar x_a)_{a\in \mathcal A_0}$ are linearly independent, the subgroup $\overline{\mathcal A_0}$ is an armature of $C$ over $F$.
\end{proof}
\subsection{Proof of the main result}
Recall that the division algebras Brauer equivalent to $R'$ and $R$ are respectively $E'$ and $E$ by Lemma \ref{lm3.1}.
\begin{proof}[Proof of Theorem \ref{thm3.1}]
(i) $\Rightarrow$ (ii): we give the proof for $E$; the proof for $E'$ follows because if $E'$ has an armature then $E = E' \otimes_{L'} L$ has an isomorphic armature. Assume that $E$ has an armature $\mathcal A$. Let $c\in \mathcal A$ and let $c_\rho z_\rho\otimes y_\rho$ be the leading term of a representative $x_c$ of $c$ in $E$. That is, $x_c=c_\rho z_\rho\otimes y_\rho+ x'_c$ with $w(x_c)= w(c_\rho z_\rho\otimes y_\rho)$ and $w(x'_c)> w(x_c)$. Define the map
\[
\nu\,:\,\mathcal A\,\longrightarrow\, A^{\times}/F^{\times}
\]
by
\[
c=x_c.L^\times\,\longmapsto\, c_\rho z_\rho.F^{\times}.
\]
If $y_c$ is another representative of $c$, we have $y_c=\ell x_c$ for some $\ell\in L^{\times}$. Note that the leading term of $y_c$ is the leading term of $\ell$ multiplied by $c_\rho z_\rho\otimes y_\rho$ (see the proof of Lemma \ref{lm3.2}); moreover, the leading term of $\ell$ lies in $F^{\times}$. One deduces that $\nu(x_c.L^{\times})= \nu(y_c.L^{\times})$. So $\nu$ is well-defined.
We show that $\nu(\mathcal A)$ is an armature of $A$: first, we claim that $\nu$ is an injective homomorphism. Indeed, let $a,b\in\mathcal A$ with respective representatives $x_a$ and $x_b$. Let $c_\sigma z_\sigma\otimes y_\sigma$ and $d_\tau z_\tau\otimes y_\tau$ be the leading terms of $x_a$ and $x_b$ respectively. As showed in the proof of Lemma \ref{lm3.2}, the leading term of $x_ax_b$ is $(c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau)$. On the other hand, it follows by (\ref{eq3.1}), (\ref{eq3.2}), (\ref{eq3.2.0}) and Lemma \ref{lm3.1} that
\[
(c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau)=c_\sigma (z_\sigma d_\tau z_\sigma^{-1}) f(\sigma,\tau)c(\sigma,\tau)z_{\sigma\tau}\otimes y_{\sigma\tau}
\]
for some $f(\sigma,\tau)\in L^\times$ and some $c(\sigma,\tau)\in C^\times$, and $z_\sigma d_\tau z_\sigma^{-1}\in C$.
Since $x_{ab}=x_ax_b\bmod L^\times$ (because $a,b\in\mathcal A$), we may take $x_ax_b$ as a representative of $ab$, so
\[
x_{ab}= x_ax_b= c_\sigma (z_\sigma d_\tau z_\sigma^{-1}) c(\sigma,\tau)z_{\sigma\tau}\otimes y_{\sigma\tau}+ x_{ab}' \quad \text{ with }\quad w(x_{ab}') >w(x_{ab}).
\]
Hence, it follows by the definition of $\nu$ that
\begin{eqnarray*}
\nu(ab) & = &c_\sigma (z_\sigma d_\tau z_\sigma^{-1}) c(\sigma,\tau)z_{\sigma\tau} \bmod F^\times\\
& = & (c_\sigma z_\sigma)(d_\tau z_\tau) \bmod F^\times\\
& = & \nu(a)\nu(b).
\end{eqnarray*}
Therefore, $\nu$ is a homomorphism.
For the injectivity, we start out by proving that the pairing $\langle\,,\,\rangle$ is an isometry for $\nu$: by definition
\[
\langle a,b \rangle=x_ax_bx_a^{-1}x_b^{-1}=\zeta \quad \text{ for some }\quad \zeta\in \mu(F).
\]
Hence
\[
x_ax_b= (c_\sigma z_\sigma\otimes y_\sigma)(d_\tau z_\tau\otimes y_\tau)+(x_ax_b)' =\zeta x_b x_a=\zeta (d_\tau z_\tau\otimes y_\tau)(c_\sigma z_\sigma\otimes y_\sigma)+\zeta(x_bx_a)',
\]
with $w((x_ax_b)')=w((x_bx_a)')> w(x_ax_b)$. Since $y_\sigma$ and $y_\tau$ commute, it follows that
\[
(c_\sigma z_\sigma)(d_\tau z_\tau)=\zeta (d_\tau z_\tau)(c_\sigma z_\sigma).
\]
Therefore
\[
\langle \nu(a),\nu(b) \rangle=(c_\sigma z_\sigma)(d_\tau z_\tau)(c_\sigma z_\sigma)^{-1}(d_\tau z_\tau)^{-1}=\zeta.
\]
The pairing $\langle\, ,\,\rangle$ is then an isometry for $\nu$. Now, to see that $\nu$ is an injection, let $a\in\mathcal A$ be such that $\nu(a)=1$. Since $\langle\, ,\,\rangle$ is an isometry for $\nu$, for all $b\in\mathcal A$, one has
\[
1= \langle \nu(a),\nu(b) \rangle= \langle a,b \rangle.
\]
We infer that $a=1$ because the pairing is nondegenerate on $\mathcal A$. It follows that $\nu(\mathcal A)$ is an abelian subgroup of $A^{\times}/F^{\times}$ isomorphic to $\mathcal A$. Consequently, $\nu(\mathcal A)$ is an armature of $A^{\times}/F^{\times}$ isomorphic to $\mathcal M$ ($\simeq \mathcal A$ by hypothesis) since $\deg E=\deg A$ by Lemma \ref{lm3.1}.
Now, we prove that $\text{\sf Kum}(M/F)\subset \nu(\mathcal A)$. Since $\overline{\mathcal A_0}$ is an armature of $C$ by Proposition \ref{prop3.3}, it follows by \cite[Lemma 2.5]{TW87} that rad$(\overline{\mathcal A_0})$ is an armature of the center of $C$ which is $M$. The extension $M/F$ being a Kummer extension, Examples \ref{ex3.1} (b) indicates that $\text{rad}(\overline{\mathcal A_0})=\text{\sf Kum}(M/F)$. Therefore, we have $\text{\sf Kum}(M/F)\subset \nu(\mathcal A) $ since $\nu$ is the identity on $\mathcal A_0$.
(ii) $\Rightarrow$ (i): let $\mathcal B$ be an armature of $A$ isomorphic to $\mathcal M$ and containing $\text{\sf Kum}(M/F)$ as a totally isotropic subgroup. We construct an isomorphic armature in $E'$. Note that if $E'$ has an armature, then $E$ also has an armature. For each $\sigma\in G$, set
\[
\mathcal B_\sigma=\{a\in \mathcal B\, | \, \langle a,b\rangle=\sigma(x_b)x_b^{-1} \text{ for all } b\in \text{\sf Kum}(M/F)\}.
\]
One easily checks that $\mathcal B_{\mathrm{id}}= \text{\sf Kum}(M/F)^\perp$ and for $a,c\in \mathcal B_\sigma$, $ac^{-1}\in \mathcal B_{\mathrm{id}}$. The sets $\mathcal B_\sigma$ are the cosets of $\mathcal B_{\mathrm{id}}$ in $\mathcal B$. So, we have the disjoint union $ \mathcal B=\bigsqcup_{\sigma\in G} \mathcal B_\sigma$. On the other hand, for $a\in \mathcal B_\sigma$ and $b\in \text{\sf Kum}(M/F)$, comparing the equality $\langle a,b\rangle=\sigma(x_b)x_b^{-1}$ and the definition $\langle a,b\rangle=x_ax_bx_a^{-1} x_b^{-1}$ we get $x_ax_b=\sigma(x_b)x_a^{-1}$; this implies $\mathcal B_\sigma\subset C^{\times} z_\sigma/F^{\times}$. Now, let us denote
\[
\mathcal B_\sigma'=\{(x_{a_\sigma} \otimes y_\sigma).L^{' \times}\,\,|\,\, a_\sigma\in \mathcal B_\sigma \} \quad \text{and}\quad \mathcal B'=\bigsqcup_{\sigma\in G}\mathcal B'_\sigma.
\]
Note that $\mathcal B_\sigma'\subset C_{L'}^{\times}(z_\sigma\otimes y_\sigma)/L^{' \times}$ and we readily check that $\mathcal B'$ is a subgroup of $E^{' \times}/L^{' \times}$. We claim that $\mathcal B'$ is an armature of $E'$: since $\deg A=\deg E'$, it follows by the definition of $\mathcal B'$ that $|\mathcal B'|=\dim E'$. Moreover, as in the part (2) of the proof of Proposition \ref{prop3.3}, one verifies that the representatives of the elements of $\mathcal B'$ in $E'$ are linearly independent over $L'$. It remains to show that $\mathcal B'$ is commutative. Let $a_\sigma\in \mathcal B_\sigma$ and $d_\tau\in \mathcal B_\tau$ with $\sigma,\tau\in G$. By (\ref{eq3.2.0}), $y_\sigma y_\tau = y_\tau y_\sigma$ because $f(\sigma,\tau)=f(\tau,\sigma)$. Furthermore, taking $x_{a_\sigma}x_{d_\tau}$ as a representative of $a_\sigma d_\tau$ (since $\mathcal B$ is an armature), we have $ x_{a_\sigma}x_{d_\tau}= x_{a_\sigma d_\tau} =x_{d_\tau}x_{a_\sigma}$. The commutativity of $\mathcal B'$ follows; and therefore $\mathcal B'$ is an armature of $E'$.
Using the same arguments as above, we see that the map $\mathcal B' \to \mathcal B$ that carries $(x_{a_\sigma}\otimes y_\sigma).L^{' \times}$ to $x_{a_\sigma}.F^{\times}$ is an isomorphism. Consequently, the armature $\mathcal B'$ is also isomorphic to $\mathcal M$. This concludes the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{thm3.2}]
(i) $\Rightarrow$ (ii): as in the proof of Theorem \ref{thm3.1}, it is enough to give the proof for $E$. Assume that $E$ decomposes into a tensor product of symbol algebras of degree $p$. Recall that if $E$ decomposes into a tensor product of symbol algebras of degree $p$ then $E$ has an armature of exponent $p$ (see \cite[Prop. 2.7]{TW87}). It follows by Theorem \ref{thm3.1} that $A$ decomposes into a tensor product of symbol algebras of degree $p$. More precisely, if $\mathcal A$ is an armature of $E$ of exponent $p$, we showed that $\nu(\mathcal A)$ is an armature of $A$ isomorphic to $\mathcal A$ and $\text{\sf Kum}(M/F)\subset \nu(\mathcal A)$. Now, let $x_1,\ldots, x_r\in A$ be such that $k_i= F(x_i)$. The subgroup generated by $(x_i F^{\times})$ for $i=1,\ldots, r$ is $\text{\sf Kum}(M/F)$.
The exponent of $\nu(\mathcal A)$ being $p$, we may view $\nu(A)$ as a vector space over the field with $p$ elements. Since $M$ is a field, the elements $e_1:=x_1F^\times,\ldots, e_r:= x_rF^\times$ are linearly independent in $\nu(\mathcal A)$. On the other hand, $M$ being commutative, the subspace spanned by $e_1, \ldots, e_r$ is totally isotropic with respect to $\langle\,,\,\rangle$. It follows then by Proposition \ref{prop3.1} that there are $f_1,\ldots, f_r, e_{r+1}, f_{r+1},\ldots, e_n, f_n$ in $\nu(\mathcal A)$ such that $\{e_1, f_1,\ldots, e_n, f_n \}$ is a symplectic base of $\nu(\mathcal A)$. Expressing $f_i= y_iF^\times$ for $i=1,\ldots,r$ and $\nu(\mathcal A)= \mathcal A_1\times\ldots\times\mathcal A_n$, where $\mathcal A_i=(e_i)\times(f_i)$ for $i=1,\ldots,n$, we get
\[
A \simeq (k_1,\sigma_1, \delta_1)\otimes_F\cdots\otimes_F(k_r,\sigma_r, \delta_r)\otimes_F F[\mathcal A_{r+1}]\otimes_F\cdots\otimes_F F[\mathcal A_n]
\]
with $\delta_i=y_i^p$ for $i=1,\ldots,r$.
(ii) $\Rightarrow$ (i): assume that $A$ decomposes as
\[
A\simeq (k_1,\sigma_1, \delta_1)\otimes_F\cdots\otimes_F(k_r,\sigma_r, \delta_r)\otimes_F A_{r+1}\otimes_F\cdots \otimes_F A_n
\]
for some $\delta_1,\ldots,\delta_r\in F^\times$ and some symbols subalgebras $A_{r+1},\ldots, A_n$ of $A$. We give the proof for $R$. The same argument is valid for $R'$. We have
\begin{multline*}
R= A\otimes_F (k_1\otimes_FL, \sigma_1\otimes\text{id},t_1)\otimes_L\cdots\otimes_L(k_r\otimes_FL, \sigma_r\otimes\text{id},t_r)\sim \\
(k_1\otimes_FL, \sigma_1\otimes\text{id},\delta_1t_1)\otimes_L\cdots\otimes_L(k_r\otimes_FL, \sigma_r\otimes\text{id},\delta_rt_r)\otimes_F A_{r+1}\otimes_F\cdots \otimes_F A_n
\end{multline*}
(see for instance \cite[\S 10]{Dra83}). Since this latter algebra has the same degree as $A$ and $\deg(A)=\deg(E)$ by Lemma \ref{lm3.1}, it is isomorphic to $E$. The proof is complete.
\end{proof}
\section{Square-central elements}\label{square-central}\label{sect4}
Let $A$ be a central simple $F$-algebra of exponent $2$ and let $g\in A^{\times}-F$ be a square-central element. The purpose of this section is to investigate conditions for $g$ to be in a quaternion subalgebra of $A$ and to give examples of tensor products of quaternion algebras containing a square-central element which is in no quaternion subalgebra.
\subsection{The algebra $A$ is not a division algebra}
Here we distinguish two cases, according to whether $g^2\in F^{\times 2}$ or $g^2\notin F^{\times 2}$. Actually, we will not need to mention in the following proposition that $A$ is not a division algebra because this is encoded by the fact that $g\in A^{\times}-F^{\times}$ and $g^2\in F^{\times 2}$. Indeed, if $g^2=\lambda^2$ with $\lambda\in F^{\times}$ then $(g-\lambda)(g+\lambda)=0$; this means that $A$ is not division.
\begin{proposition}\label{prop1.1}
Let $A$ be a central simple $F$-algebra and let $g\in A^{\times}-F^{\times}$ be such that $g^2=\lambda^2$, $\lambda\in F^{\times}$. The element $g$ is in a quaternion subalgebra of $A$ if and only if $\dim(g-\lambda)A=\dim(g+\lambda)A$. If the characteristic of $F$ is $0$, this condition holds if and only if the reduced trace $\operatorname{Trd}_A(g)$ of $g$ is zero.
\end{proposition}
\begin{proof}
We can write $A\simeq \text{End}_D(V)$ where $D$ is a division algebra Brauer equivalent to $A$ and $V$ is some right $D$-vector space. Suppose there is a quaternion $F$-subalgebra $Q$ of $A$ such that $g\in Q$. Then $A=Q\otimes C_AQ$, where $C_AQ$ is the centralizer of $Q$ in $A$. Since $g^2=\lambda^2$ with $\lambda\in F^{\times}$, we may identify $Q$ with $M_2(F)$ in such a way that $g$ is the diagonal matrix diag$(\lambda,-\lambda)$. Computations show that $\dim(g-\lambda)A=\dim(g+\lambda)A = 2\dim C_AQ$.
Conversely, suppose $\dim(g-\lambda)A=\dim(g+\lambda)A$. Let $V_+$ and $V_-$ be the $\lambda$-eigenspace and $-\lambda$-eigenspace of $g$ respectively. For all $u\in V$, we have $u=\frac{1}{2}(u+\lambda^{-1}g(u))+\frac{1}{2}(u-\lambda^{-1}g(u))\in V_+ +V_-$ and $V_+ \cap V_-=\{0\}$. Hence $V=V_+ \oplus V_-$. Denote by $r$ and $s$ the dimensions of $V_+$ and $V_-$ respectively. Since $g^2=\lambda^2$, $g$ is represented in $A\simeq \text{End}_D(V)$ by the diagonal matrix $\text{diag}(\lambda,\ldots,\lambda,-\lambda,\ldots,-\lambda)$ where the number of $\lambda$ is $r$ and the number of $-\lambda$ is $s$. Computations show that $\dim(g-\lambda)A=s\dim D$ and $\dim(g+\lambda)A=r\dim D$; therefore the hypothesis yields $r=s$. The endomorphism $f$ whose matrix is the block matrix
$
f=\begin{pmatrix}
0 & 1\\ 1 & 0
\end{pmatrix},
$
where each block is an $r\times r$ matrix, anticommutes with $g$ and is square-central. It follows that $g$ lies in the split quaternion subalgebra of $A$ generated by $g$ and $f$.
Now suppose the characteristic of $F$ is $0$. The element $g$ is represented by diag$(\lambda,\ldots,\lambda,-\lambda,\ldots,-\lambda)$. So $\operatorname{Trd}_A(g)=(r-s)\lambda \deg D$ and $\dim(g-\lambda)A=s\dim D$ and $\dim(g+\lambda)A=r\dim D$. Therefore $\operatorname{Trd}_A(g)= 0$ if and only if $\dim(g-\lambda)A=\dim(g+\lambda)A$. The proof is complete.
\end{proof}
If the characteristic of $F$ is positive, the hypothesis on the trace does not suffice as we observe in the following counterexample:
\begin{contrexample}\label{cex3.1}
Assume that $F=\mathbb F_3$, the field with three elements, and take $A=M_8(F)$. The diagonal matrix $g= \text{diag}(1,\ldots, 1, -1)$ is such that $g^2=1$ and the trace of $g$ is $0$. But Proposition \ref{prop1.1} shows that $g$ is not in a quaternion subalgebra of $A$ since $\dim(g+1)A=56\ne\dim(g-1)A=8$.
\end{contrexample}
\begin{proposition}\label{prop1.2}
Let $A$ be a central simple $F$-algebra and let $g\in A^{\times}-F^{\times}$ be such that $g^2=a\in F^{\times}-F^{\times 2}$. The element $g$ lies in a split quaternion subalgebra of $A$ if and only if $\frac{\deg A}{\operatorname{ind}(A)}$ is even.
\end{proposition}
\begin{proof}
If $g\in M_2(F)\subset A$, then $A=M_2(F)\otimes C$ where $C$ is the centralizer of $M_2(F)$ in $A$. Hence, $\frac{\deg A}{\operatorname{ind}(A)}$ is even. Conversely, assume $\frac{\deg A}{\operatorname{ind}(A)}$ is even. So, we may write $A\simeq M_2(F)\otimes A'$ for some algebra $A'$ Brauer equivalent to $A$. Set
$
g'=\begin{pmatrix}
0 & 1\\
a & 0
\end{pmatrix}
$; we have $g'\in M_2(F)\subset A$ and $g'^2=a$. By the Skolem-Noether Theorem, $g$ and $g'$ are conjugated. It follows that $g$ is in a split quaternion subalgebra of $A$ since $g'\in M_2(F)$.
\end{proof}
\subsection{The algebra $A$ is a division algebra}
Let $A$ be a division algebra and let $x\in A^{\times}-F^{\times}$. Recall that, $A$ being a division algebra, we have necessarily $x^2\in F^{\times}-F^{\times 2}$. Here we argue on the degree of the division algebra.
\subsection*{Degree $4$}
The following result is due to Albert and many proofs exist in the literature (see for instance \cite{Rac74}, \cite[Prop. 5.2]{LLT93}, \cite[Thm. 4.1]{Bec04}). We propose the following proof for the reader's convenience.
\begin{proposition}\label{prop1.3}
Suppose $A$ is a central simple algebra over $F$ of degree $4$ and exponent $2$. Let $x\in A^{\times}-F^{\times}$ be a square-central element with $x^2\not\in F^{\times 2}$. Then, $x$ is in a quaternion $F$-subalgebra of $A$.
\end{proposition}
\begin{proof}
Note that $F(x)$ is isomorphic to a quadratic extension of $F$ since $x^2\in F^{\times}-F^{\times 2}$. If $A$ is not a division algebra, the result follows by Proposition \ref{prop1.2}. We assume that $A$ is a division algebra. The centralizer $C_A(x)$ of $x$ in $A$ is a quaternion algebra over $F(x)$. The algebra $C_A(x)$ is Brauer equivalent to $A_{F(x)}$ (see for instance \cite[\S 13.3]{Pie82}). Since
\[
\text{cor}_{F(x)/F}[C_A(x)]=\operatorname{cor}_{F(x)/F}(\operatorname{res}_{F(x)/F}[A])=2[A]=0
\]
in Br$(F)$ (see for instance \cite[(3.13)]{KMRT98}), it follows from a result of Albert (see \cite[(2.22)]{KMRT98}) that there is a quaternion algebra $Q$ over $F$ such that $C_A(x)=Q\otimes_F F(x)$. Then, $A=Q\otimes C_A(Q)$ and the centralizer $C_A(Q)$ of $Q$ is a quaternion $F$-subalgebra of $A$ containing $x$.
\end{proof}
\subsection*{Degree $8$}
Here we give an example of a tensor product of three quaternion algebras containing a square-central element which is in no quaternion subalgebra. This example is a private communication from Merkurjev to Tignol based on the following result:
\begin{lemma}[Tignol]\label{lm1.1}
Let $A$ be a division algebra over $F$ of degree $8$ and exponent $2$. Let $x\in A^{\times}-F^{\times}$ be such that $x^2=a\in F^{\times}$. Then, there exists quaternion algebras $Q_1, Q_2, Q_3$ such that $M_2(A)\simeq Q_1\otimes Q_2\otimes Q_3\otimes(a,y)$ for some $y\in F$.
\end{lemma}
\begin{proof}
It is shown in \cite[Thm. 5.6.38]{Jac96} that $M_2(A)$ is a tensor product of four quaternion algebras. The proof shows that one of these quaternion
algebras can be chosen to contain $x$.
\end{proof}
\begin{corollary}[Merkurjev]\label{cor3.1}
There exists a decomposable $F$-algebra of degree $8$ and exponent $2$ containing a square-central element which is in no quaternion subalgebra.
\end{corollary}
\begin{proof}
Let $A$ be an indecomposable $F$-algebra of degree $8$ and exponent $2$ and let $x\in A$ be such that $x^2=a\in F^{\times}$ with $x\notin F$. Such an algebra $A$ exists by \cite{ART79} and the existence of such an element $x$ follows from a result of Rowen \cite[Thm. 5.6.10]{Jac96}. Lemma \ref{lm1.1} indicates that $M_2(A)\simeq Q_1\otimes Q_2\otimes Q_3\otimes(a,y)$ for some $y\in F$. Set $D = Q_1\otimes Q_2\otimes Q_3$. We claim that $D$ is a division algebra. Indeed, if $D$ is not a division algebra then $D\simeq M_2(D')$ where $D'$ is an algebra of degree $4$ and exponent $2$. Since an exponent $2$ and degree $4$ central simple algebra is always decomposable by a well-known result of Albert (see for instance \cite{Rac74}), we deduce that $A$ is isomorphic to a product of quaternion algebras; this contradicts our hypothesis. Hence $D$ is a division algebra. Since the algebras $D_{F(\sqrt{a})}$ and $A_{F(\sqrt{a})}$ are isomorphic and $A_{F(\sqrt{a})}$ is not a division algebra, $D_{F(\sqrt{a})}$ is not a division algebra. Then, by \cite[Thm. 4.22]{Alb61} the algebra $D$ contains an element $\alpha$ such that $\alpha^2=a$ with $\alpha \notin F$. Assume that $D$ contains a quaternion subalgebra containing $\alpha$, say $(a,b)$ for some $b\in F$. The centralizer of $(a,b)$ in $D$ is an algebra of exponent $2$ and degree $4$. Thus, we have $D\simeq H_1\otimes H_2\otimes (a,b)$ where $H_i$ are quaternion algebras. It follows that
$
M_2(A)\simeq H_1\otimes H_2\otimes (a,b)\otimes (a,y)\simeq M_2(H_1\otimes H_2\otimes (a,yb)).
$
Whence $A\simeq H_1\otimes H_2\otimes (a,yb)$; contradiction. The algebra $D$ satisfies the required conditions.
\end{proof}
\subsection*{Degree $2^n$, $n>3$}
In this part, we generalize Corollary \ref{cor3.1}: we are going to construct a tensor product of $n$ (with $n>3$) quaternion algebras containing a square central element which is not in a quaternion subalgebra. To do this, we use valuation theory.
Let $L=F((t_1))((t_2))$ be the iterated Laurent power series field where $t_1,t_2$ are independent indeterminates over $F$ and let $D$ be a division $F$-algebra. Set $ D'=D\otimes (t_1,t_2)_L$ and let $i,j\in D'$ be such that
$i^2=t_1,\quad j^2=t_2 \quad \text{ and } \quad ij=-ji$. Since $i^2=t_1$ and $j^2=t_2$, every element $f\in D'$ can be written as an iterated Laurent series in $i$ and $j$ with coefficients in $D$:
\[
f=\sum_{\begin{subarray}{l}
\beta\geq n
\end{subarray}}\sum_{\alpha\geq m_\beta}d_{\alpha,\beta}i^{\alpha}j^{\beta} \quad \text{ with } d_{\alpha,\beta}\in D \text{ and } n, m_\beta\in\mathbb{Z}.
\]
Define $v\, :\, D'^{\times}\, \longrightarrow\, (\frac{1}{2}\mathbb{Z})^2$ (where $(\frac{1}{2}\mathbb{Z})^2$ is ordered lexicographically from right-to-left) by
\[
v(f)=\inf\Big\{(\frac{\alpha}{2},\frac{\beta}{2})\,|\,\, d_{\alpha,\beta}\ne 0\Big\}.
\]
Computations show that $v$ is a valuation on $D'$. Actually, $v$ is the unique extension of the $(t_1,t_2)$-adic valuation on $L$ (which is Henselian). As in the previous section, for $f\in D'^{\times}$, the leading term of $f$ is defined to be
\[
\ell(f)=d_{m,n}i^mj^n \quad \text{ where }\quad (m,n)= v(f).
\]
Straightforward computations show that
\begin{itemize}
\item[(i)] $\ell(fg)=\ell(f)\ell(g)$, for $f,g\in D'^{\times}$;
\item[(ii)] $\ell(d)=d$, for $d\in D^{\times}$;
\item[(iii)] $\ell(z)\in L $, for $z\in L^{\times}$.
\end{itemize}
We have the following generalization of Corollary \ref{cor3.1}:
\begin{proposition}\label{prop1.4}
Let $D$ be a division algebra over $F$. Let $x\in D^{\times}-F$ be a square-central element which is in no quaternion subalgebra of $D$. Then $D\otimes (t_1,t_2)_L$ has no quaternion subalgebra containing $x$.
\end{proposition}
\begin{proof} Suppose there is $y\in D\otimes (t_1,t_2)_L$ such that $y^2\in L^\times$ and $xy=-yx$. Let $\ell(y)= d i^\alpha j^\beta$ with $d\in D^\times$ and $\alpha,\beta\in\mathbb{Z}$. We have $\ell(y)i^{-\alpha}j^{-\beta}=d$ and $d^2\in F^{\times}$. Since $xy=-yx$ we have $\ell(x)\ell(y)=-\ell(y)\ell(x)$, that is, $x\ell(y)=-\ell(y)x$. Hence, $d$ anticommutes with $x$; contradiction with the choice of $x$.
\end{proof}
\subsection{An application}
Corollary \ref{cor3.1} implies that if $F$ is the center of an indecomposable algebra of degree $8$ and exponent $2$, then there exist a decomposable division algebra of degree $8$ and exponent $2$ containing a square-central element which is not in a quaternion subalgebra. Conversely, let $D$ be a division algebra of degree $8$ and exponent $2$ over $F$ and let $F(\sqrt{a})\subset D$ be a quadratic field extension of $F$ such that $F(\sqrt{a})$ is not in a quaternion subalgebra of $D$ (the algebra $D$ could be decomposable). Theorem \ref{thm3.2} shows that the division algebra Brauer equivalent to $D\otimes_F(a,t)_{F(t)}$ is an indecomposable algebra of degree $8$ and exponent $2$ over $F(t)$.
As an application, we are going now to give an example of indecomposable algebra of degree $8$ and exponent $2$ over a field of $2$-cohomological dimension $4$. Let $F$ be a field of characteristic different from $2$ and let us denote $K=F(\sqrt{a})$. Let $B$ be a biquaternion algebra over $K$ with trivial corestriction, $\operatorname{cor}_{K/F}(B)=0$. In \cite{Bar}, it is associated with $B$ a degree three cohomological invariant $\delta_{K/F}(B)$ with value in $H^3(F,\mu_2)/\operatorname{cor}_{K/F}((K^\times)\cdot [B])$ where $H^3(F,\mu_2)$ is the third Galois cohomology group of $F$ with coefficients in $\mu_2=\{\pm 1\}$. It is also shown in \cite{Bar} that $B$ has a descent to $F$ (that is, $B=B_0\otimes_F K$ for some biquaternion algebra $B_0$ defined over $F$) if and only if $\delta_{K/F}(B)=0$.
Now, let $D$ be a central simple algebra of degree $8$ and exponent $2$ over $F$ containing $K$ such that $K$ is not in a quaternion subalgebra of $D$. That also means $\delta_{K/F}(C_DK)\ne 0$ where $C_DK$ denotes the centralizer of $K$ in $D$. Consider the following \emph{Merkurjev extension} $\mathbb{M}$ of $F$:
\[
F=F_0\subset F_1\subset\cdots \subset F_{\infty}=\bigcup_iF_i =:\mathbb{M}
\]
where the field $F_{2i+1}$ is the maximal odd degree extension of $F_{2i}$; the field $F_{2i+2}$ is the composite of all the function fields $F_{2i+1}(\pi)$, where $\pi$ ranges over all $4$-fold Pfister forms over $F_{2i+1}$. The arguments used by Merkurjev in \cite{Mer92} show that the $2$-cohomological dimension $cd_2(\mathbb{M})\le 3$. We have the following result:
\begin{theorem}\label{application}
The algebra $D$ and the field $\mathbb{M}$ are as above. The division algebra Brauer equivalent to
\[
D_{\mathbb M}\otimes_{\mathbb M}(a,t)_{\mathbb M(t)}
\]
is indecomposable of degree $8$ and exponent $2$ over $\mathbb M(t)$, where $t$ is an indeterminate.
\end{theorem}
\begin{proof}
Put $B=C_DK$. As observed in the proof of \cite[Thm. 1.3]{Bar} the $2$-cohomological dimension of $\mathbb{M}$ is exactly $3$. It follows from \cite[Prop. 4.7]{Bar} and a result of Merkurjev (see Theorem A.9 of \cite{Bar}) that the scalar extension map
\[
\frac{H^3(F,\mu_2)}{\operatorname{cor}_{K/F}((K^\times)\cdot [B])}\longrightarrow \frac{H^3(\mathbb M,\mu_2)}{\operatorname{cor}_{\mathbb M(\sqrt{a})/\mathbb M}(\mathbb M(\sqrt{a})^\times\cdot [B_{\mathbb M(\sqrt{a})}])}
\]
is an injection. So, $\delta_{\mathbb{M}(\sqrt{a})/\mathbb{M}}(B)\ne 0$ since $\delta_{K/F}(B)\ne 0$. Hence the extension $\mathbb M(\sqrt{a})$ is not in a quaternion subalgebra of $D_{\mathbb M}$. Therefore the division algebra Brauer equivalent to $D_{\mathbb M}\otimes_{\mathbb M}(a,t)_{\mathbb M(t)}$ is an indecomposable algebra of degree $8$ and exponent $2$ over $\mathbb M(t)$ by Corollary \ref{thm3.2}; as desired.
\end{proof}
\bigskip
\subsection*{Acknowledgements}
This work is a generalization of some results of my PhD thesis. I gracefully thank my advisors A. Qu\'eguiner-Mathieu and J.-P. Tignol for their support, ideas and for sharing their knowledge. I thank A. S. Merkurjev for providing Corollary \ref{cor3.1} (a private communication to J.-P. Tignol).
|
2,877,628,089,498 | arxiv | \section{Introduction}\label{intro}
The so-called Plummer model was introduced by \citet{p11} as a
description of the stellar density distribution in Galactic globular
clusters \citep{p11}. Subsequently, \citet{e16} showed that this
spherically symmetric density profile could be derived from a
phase-space distribution of the form
\begin{equation}
F(\epsilon) \propto (-\epsilon)^{7/2},
\end{equation}
with $\epsilon = \psi+v^2/2 $ the Newtonian specific energy of a star
and $\psi$ the Newtonian gravitational potential of the stellar
cluster \citep{de87}. This distribution function self-consistently
generates a mass distribution with a gravitational potential \begin{equation}
\psi(r) = -\frac{GM}{a} \frac{1}{\sqrt{1+\left( \frac{r}{a}
\right)^2}} \end{equation} and density profile \begin{equation} \rho(r) =
\frac{3}{4\pi} \left( 1+\left( \frac{r}{a}
\right)^2\right)^{-\frac{5}{2}}\frac{M}{a^3} = \frac{3}{4\pi} \left(
-\frac{a}{GM} \psi \right)^5\frac{M}{a^3}. \end{equation} Here, $M$ is the
total mass of the cluster and $a$ a scale length.
Certain general relativistic (GR) extensions of the Plummer model can
already be found in the literature and we give an overview here.
For instance, in the case of a spherically symmetric cluster, the
density, the potential (or some generalization thereof), and the
distribution function are all functions of one argument, so it makes
sense to construct the metric around a single, unknown function of the
radius, usually denoted simply by $f(r)$. An example, inspired by the
Schwarzschild metric is \begin{equation} ds^2 = \left( \frac{1-f}{1+f} \right)^2
c^2 dt^2 - (1+f)^4( dr^2 + r^2 d\Omega^2 ). \end{equation} In the Newtonian
limit, $f(r)$ reduces to $-\psi/(2c^2)$.
One approach is to choose $f(r)$ such that it produces a meaningful
cluster model in the Newtonian limit, for instance by equating it to
the gravitational potential of the Newtonian cluster. \citet{nl13}
show how this technique can be used to recover GR extensions of the
hypervirial models of which the Plummer model is a special
case. Solving the time-time-component of the field equations yields a
density that together with $f$, by construction, correctly reduces to
the corresponding Newtonian potential-density pair. However, as these
authors note, the pressure does not reduce to the expected Newtonian
limit. This is because the underlying distribution function does not
reduce to the proper Newtonian limit.
Another possibility is to equate the radial and tangential field
equations, thus enforcing isotropy, and to solve the resulting
equation for the metric. This solution can then be plugged in the
time-time-component of the field equations to yield the density
profile. \citet{bu64} has used this procedure to produce a cluster
model with an equation of state analogous to that of the Plummer
model, i.e. a polytrope with index $n=5$. \citet{fa71}, using a metric
of the form \begin{equation} ds^2 = e^{\nu(r)} c^2 dt^2 -e^{\lambda(r)} dr^2 -
r^2 d\Omega^2, \end{equation} subsequently derived a rather unwieldy analytical
expression for the distribution function of this model and showed
that, unless the central value of the potential satisfies
$\exp(\nu(0)) > 0.413$, it can show a ``temperature inversion'' in
the sense that it is not a monotonically decreasing function of
energy. Alternatively, one can impose the polytropic equation of state
on the field equations, which yields a generalization of the
Lane-Emden equation, and thus solve for the unknown function in the
metric \citep{to64,ka67}.
Both techniques avoid an explit calculation of the distribution
function. Using a generalization of Eddington's integral equation, it
can, however, be determined from the density profile
\citep{fa68,pk96}. Unfortunately, this distribution function is not
guaranteed to be positive everywhere in phase space, although
necessary conditions for positivity have been derived \citep{su77}.
Since employing the Eddington integral equation can lead to rather
cumbersome expressions for the distribution function and, moreover,
the latter's positivity is not guaranteed from the outset, we here
advocate another approach. We first write down a mathematically simple
distribution function that is everywhere positive and that reduces to
a well-defined Newtonian limit. From this distribution function, the
density and pressure profiles can be calculated. By construction, all
moments of the distribution function will reduce to the proper
Newtonian limit. Solving the field equations finally yields the
metric. While such generalizations of Newtonian cluster models may not
have the same equation of state as in the Newtonian limit, they have
the benefit of having a mathematically simple, strictly non-negative
distribution function with a properly defined, meaningful Newtonian
limit. Our goal is to produce general relativistic cluster models with
isotropic, polytropic distribution functions, to study their dynamical
properties, and to investigate their Newtonian limits. In particular,
we wish to study how the Newtonian polytropes, of which the well-known
Plummer model is a special case, fit in this more general scheme of
models.
In section \ref{isodyn}, we develop the dynamical theory of general
relativistic stellar cluster models and calculate the properties of
models with isotropic, polytropic distribution functions. In section
\ref{solve}, we present our method for solving the field equations for
such models. We end with a discussion of the models in section
\ref{discussion} and conclude in section \ref{conc}.
\section{Isotropic dynamical models for general relativistic stellar clusters}
\label{isodyn}
\subsection{The internal dynamics of isotropic clusters}
In general relativistic dynamics, the distribution function (DF)
$F(x^\mu,p^i) d^3x d^3p$ counts the number of occupied world lines
that intersect a 6-dimensional submanifold of the 8-dimensional phase
space. This 6-dimensional submanifold consists of a 3-dimensional
spatial hypersurface and its future mass hyperboloid. In the absence
of particle creation/annihilation or collisions, the Lie derivative of
the DF is zero, or \begin{equation} \left[ p^\mu \frac{\partial}{\partial x^\mu}
- \Gamma^\mu_{\alpha\beta}p^\alpha p^\beta \frac{\partial}{\partial
p^\mu} \right] F(x,\vec{p}) = 0. \end{equation}
Let $\hat{p}^\mu$ be the components of the momentum 4-vector in a
local orthonormal frame at rest, such that $\hat{p}_0^2 - \sum_i
\hat{p}_i^2 = (mc)^2$, with $m$ the rest mass of a single star. In
such a local orthonormal frame, tensor quantities of the form \begin{equation}
T_{\mu \nu \ldots \kappa}(x) = \int \frac{\hat{p}_\mu \hat{p}_\nu \ldots
\hat{p}_\kappa}{\hat{p}_0} F(x,\vec{p}) d\hat{p}_1 d\hat{p}_2
d\hat{p}_3 \end{equation} can be defined. If the Lie derivative of the DF
disappears, then all these quantities have zero covariant
divergence. The most well-known such tensor quantities are those with
one index (the stream density vector) and two indices (the
energy-momentum tensor).
In an isotropic cluster, the DF depends only on $p_0$, the 0-component
of the momentum 4-vector, which is a constant in a time-independent
gravitating system (see below). Obviously, what matters in the above
definition of the momentum moments of the DF is the number of
instances of each momentum component. We therefore re-write these
momentum moments as \begin{equation} \mu_{k,2m,2n,2l}(x) = \int
{\hat{p}_0}^k \hat{p}_1^{2m} \hat{p}_2^{2n}
\hat{p}_3^{2l} F(p_0) \frac{d\hat{p}_1 d\hat{p}_2
d\hat{p}_3}{\hat{p}_0}. \end{equation} Using the parameterization
\begin{align}
\hat{p}_0 &= \sqrt{ (mc)^2+p^2 } \nonumber \\
\hat{p}_1 &= p \cos \vartheta \nonumber \\
\hat{p}_2 &= p \sin \vartheta \cos \varphi \nonumber \\
\hat{p}_3 &= p \sin \vartheta \sin \varphi
\end{align}
this reduces to
\begin{align} \mu_{k,2m,2n,2l}(x) &= \frac{1}{2\pi} \frac{
\Gamma\left(m+\frac{1}{2}\right) \Gamma\left(n+\frac{1}{2}\right)
\Gamma\left(l+\frac{1}{2}\right)}{
\Gamma\left(m+n+l+\frac{3}{2}\right) } \times \nonumber
\\ & \hspace*{2em} 4\pi \int F(p_0) \hat{p}_0^{k-1} p^{2(m+n+l)+2} dp.
\end{align}
Let $E$ be the energy of a star as measured by an obsever at rest at
infinity, where the geometry of spacetime is essentially flat. The
energy measured by a local observer at rest, denoted by $E_{\sf
local}$, is linked to $E$ via \begin{equation} E = \sqrt{g_{00}} E_{\sf local}
= e^{\phi/2} E_{\sf local} = cp_0. \end{equation} The 0-momentum in the local
orthonormal frame, $\hat{p}_0$, is related to the energy at infinity
as \begin{equation} c \hat{p}_0 = \frac{c p_0 }{\sqrt{g_{00}}} =
\frac{E}{\sqrt{g_{00}}}.
\end{equation}
Therefore,
\begin{equation}
E^2 - m^2 c^4 g_{00} = g_{00} p^2 c^2
\end{equation}
and
\begin{equation}
p dp = \frac{E dE}{c^2 g_{00}}.
\end{equation}
Then
\begin{align}
\mu_{k,2m,2n,2l}(x) &= \frac{1}{2\pi} \frac{
\Gamma\left(m+\frac{1}{2}\right) \Gamma\left(n+\frac{1}{2}\right)
\Gamma\left(l+\frac{1}{2}\right)}{
\Gamma\left(m+n+l+\frac{3}{2}\right) } \times
\nonumber \\ & \hspace*{2em} \mu_{k,2(m+n+l)}(x)
\end{align}
which defines the set of isotropic $k$-moments \begin{align}
\mu_{k,2q} &= 4 \pi \int F(p_0) \hat{p}_0^k \left(
\hat{p}_0^2-m^2c^2 \right)^{q+\frac{1}{2}} d\hat{p}_0 \nonumber
\\ &= \frac{(mc)^{2q+k+2}}{2^{q+\frac{3}{2}} E_0^{2q+k+2}}
\tilde{\mu}_{k,2q}(E_0^2) \label{mu2q}
\end{align}
with \begin{align} \tilde{\mu}_{k,2q}(E_0^2) &= \nonumber
\\ & \hspace*{-2em} 2^{q+\frac{5}{2}}\pi \int_{E_0^2} F(E^2) \left(
E^2 \right)^{\frac{k-1}{2}} (E^2-E_0^2)^{q+\frac{1}{2}}
dE^2 \label{mut2q} \end{align} and $E_0^2 = (mc^2)^2
g_{00}$. Deriving this equation $q$ times with respect to $E_0^2$
leads to \begin{equation} \tilde{\mu}_{k,0}(E_0^2) = \frac{(-1)^q}{(2q+1)!!}
D^q_{E_0^2} \tilde{\mu}_{k,2q}(E_0^2). \end{equation} Here, \begin{equation} (2q+1)!! =
(2q+1)(2q-1) \ldots 1 \end{equation} indicates the double factorial. This
equation is formally identical to equation (1.3.7) in \citet{de86}. We
can therefore simply invoke equation (1.3.8) from that same work to
invert the above expression and to write all higher order $k$-moments
of the DF in terms of the zeroth-order $k$-moment: \begin{equation}
\tilde{\mu}_{k,2q}(E_0^2) = \frac{(2q+1)!!}{(q-1)!!} \int_{E_0^2}
(E^2-E_0^2)^{q-1} \tilde{\mu}_{k,0}(E^2) dE^2. \end{equation}
With the aid of equation (1.3.12) from \citet{de86}, (\ref{mut2q}) can
be inverted as
\begin{align}
E^{k-1} F(E) & \nonumber \\
&\hspace*{-4em}= \frac{1}{2^{q+\frac{5}{2}} \pi^{\frac{3}{2}} \Gamma\left(
q+\frac{3}{2}\right)} D^{q+2}_{E^2} \int_{E^2}
\frac{\tilde{\mu}_{k,2q}({E}_0^2)}{\sqrt{ {E}_0^2-E^2}} d{E}_0^2 \nonumber \\
&\hspace*{-4em}= \frac{(mc)^{-2q-k-2}}{2 \pi^{\frac{3}{2}} \Gamma\left(
q+\frac{3}{2}\right) } D^{q+2}_{E^2} \int_{E^2}
\frac{E_0^{2q+k+2}\mu_{k,2q}({E}_0^2)}{\sqrt{ {E}_0^2-E^2}} d{E}_0^2.
\end{align}
In particular, for $(q=0,k=2)$ and for $(q=1,k=0)$ the above inversion
relation reduces to the two important special cases
\begin{align} E F(E) &=
\frac{1}{\pi^2 m^4c^3} D^2_{E^2} \int_{E^2} \frac{E_0^4\rho(E_0^2)}{\sqrt{
E_0^2-E^2}} dE_0^2 \nonumber \\ \frac{1}{E}F(E) &= \frac{2}{\pi^2
m^4 c^5} D^3_{E^2} \int_{E^2} \frac{E_0^4P(E_0^2)}{\sqrt{
E_0^2-E^2}} dE_0^2
\end{align}
with $\rho$ the mass density and $P$ the pressure. These are none
other than the inversion relations derived by \citet{fa68} and
\citet{pk96}. Here, we made use of the fact that
\begin{align}
\mu_{2,0} &= \frac{4\pi}{g_{00}^2c^4} \int F(E) E^2 \sqrt{ E^2-E_0^2 } dE = \rho c \nonumber \\
\mu_{0,2} &= \frac{4\pi}{g_{00}^2c^4} \int F(E) \left( E^2-E_0^2\right)^{\frac{3}{2}} dE = \frac{3P}{c}
\end{align}
\citep{zp65,os74}. Hence, we have shown that these two relations
linking the density and pressure to the isotropic DF are simply
specific cases of a more general link between the DF and any of its
moments $\mu_{k,2q}$.
\subsection{A generalized polytropic distribution function}
For a static, spherically symmetric gravitational system, the metric
can always be brought in the form \begin{equation} ds^2 = e^{\phi(r)} c^2 dt^2 -
\left( 1-\frac{2GM(r)}{c^2r} \right)^{-1} dr^2 - r^2 d\Omega^2, \end{equation}
with $M(r)$ the total gravitating mass interior to the areal radius
$r$ and $\phi$ a potential function that, in the Newtonian limit,
reduces to $2\psi/c^2$. We propose a distribution function of the form
\begin{equation} F(E)= f_0 \left( \frac{mc^2}{E} \right)^{2\beta}\left( \frac{
m^2c^4 e^{\Phi}-E^2}{m^2c^2} \right)^\alpha, \label{propdf} \end{equation} with $\alpha$ and
$\beta$ positive real numbers, $f_0$ a constant forefactor, and $\Phi
= \phi(R)$, the value of the potential at the outer edge of the
cluster at radius $r=R$.
For the isotropic distribution function given above, the energy
density is given by
\begin{align}
\rho c^2 &= \frac{4\pi}{c^3} e^{-2\phi}
\int_{mc^2e^{\phi/2}}^{mc^2e^{\Phi/2}} F(E) E^2 \sqrt{ E^2 - m^2c^4
e^\phi} dE \nonumber \\ &= \pi^{3/2}
\frac{\Gamma(\alpha+1)}{\Gamma\left(\alpha+\frac{5}{2}\right)} f_0 m^4
c^{3+2\alpha} e^{-2\phi} \left( e^\Phi-e^\phi \right)^{\alpha+\frac{3}{2}}
\times \nonumber \\ & e^{\left(\frac{1}{2}-\beta \right)\Phi}
{_2F_1}\left(\beta-\frac{1}{2},\alpha+1;\alpha+\frac{5}{2}; \frac{e^\Phi-e^\phi}{e^\Phi} \right)
\end{align}
\citep{zp65,os74}. Here, $\Gamma(x)$ is Euler's gamma-function and
${_2F_1}(a,b;c;z)$ is the Gaussian hypergeometric function
\begin{align}
{_2F_1}(a,b;c;z) &= \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 t^{b-1}
(1-t)^{c-b-1}\frac{dt}{(1-zt)^a} \nonumber \\
&= \sum_{n\ge 0} \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!}
\end{align}
with $(q)_n$ the Pocchammer symbol, defined as $(q)_n =
\Gamma(q+n)/\Gamma(q)$. We can choose a scale-length $a$ and denote
the scaled radius by $x = r/a$. With the choice of a mass scale $M$,
we can introduce the dimensionless parameter \begin{equation} {\cal A}=
\frac{c^2a}{2GM}. \label{Qparam} \end{equation} If the mass scale $M$ is taken
to coincide the model's total mass, then $\cal A$ is simply the ratio
of the scale-length $a$ to the model's Schwarzschild radius.
We can then take \begin{equation} f_0 =
\frac{3}{4\pi^{\frac{5}{2}}c^{3+2\alpha}}
\frac{\Gamma\left(\alpha+\frac{5}{2}\right)}{\Gamma(\alpha+1)}
\frac{M{\cal A}^{\alpha+\frac{3}{2}}}{m^4 a^3}. \end{equation} With this choice for
the forefactor $f_0$, we find the following expression for the density
\begin{align}
\rho(\phi) &= \frac{3}{4\pi} {\cal A}^{\alpha+\frac{3}{2}}
e^{-2\phi} \left( e^\Phi-e^\phi \right)^{\alpha+\frac{3}{2}} \times \nonumber
\\ & e^{\left(\frac{1}{2}-\beta \right)\Phi}
{_2F_1}\left(\beta-\frac{1}{2},\alpha+1;\alpha+\frac{5}{2};
\frac{e^\Phi-e^\phi}{e^\Phi} \right)\frac{M}{a^3}.
\end{align}
Clearly, the choice $\beta=0.5$ yields the ``simplest'' mass density
profile since in that case the hypergeometric function is identically
one and
\begin{equation}
\rho(\phi) = \frac{3}{4\pi} {\cal A}^{\alpha+\frac{3}{2}}
e^{-2\phi} \left( e^\Phi-e^\phi \right)^{\alpha+\frac{3}{2}}\frac{M}{a^3}.
\end{equation}
The expression for the pressure follows
from
\begin{align}
P &= \frac{4\pi}{3 c^3 } e^{-2\phi}
\int_{mc^2e^{\phi/2}}^{mc^2e^{\Phi/2}} F(E) \left( E^2 - (mc^2)^2
e^\phi \right)^{\frac{3}{2}} dE \nonumber \\ &= \frac{3}{4\pi}
\frac{{\cal A}^{\alpha+\frac{3}{2}}}{(2\alpha+5)}
e^{-\left(\beta+\frac{1}{2}\right)\Phi} e^{-2\phi} \left(
e^\Phi-e^\phi \right)^{\alpha+\frac{5}{2}} \nonumber \\ & \hspace{3em} \times
{_2F_1}\left( \beta+\frac{1}{2}, \alpha+1; \alpha+\frac{7}{2}; \frac{e^\Phi-e^\phi}{e^\Phi}
\right)\frac{Mc^2}{a^3}.
\end{align}
The proper mass density is given by
\begin{align}
nm &= \frac{4\pi m}{c^3} e^{-3\phi/2}
\int_{mc^2e^{\phi/2}}^{mc^2e^{\Phi/2}} F(E) E \sqrt{ E^2 - m^2c^4
e^\phi} dE \nonumber \\
&= \frac{3}{4\pi} {\cal A}^{\alpha+\frac{3}{2}} e^{-\beta\Phi} e^{-3\phi/2}
\left(
e^\Phi-e^\phi \right)^{\alpha+\frac{3}{2}} \nonumber \\ & \hspace{3em} \times
{_2F_1}\left( \beta , \alpha+1; \alpha+\frac{5}{2};\frac{e^\Phi-e^\phi}{e^\Phi}
\right)\frac{M}{a^3},
\end{align}
with $n$ the stellar proper number density. The proper mass of the
cluster is then \begin{equation} M_p(r) = 4\pi \int_0^r \frac{n(r)m r^2 dr}{\sqrt{
1-\frac{2GM}{c^2r}}}. \end{equation} The difference between the total
proper mass $M_p(R)$ and the total gravitating mass $M(r)$ can be
interpreted as the gravitational binding energy of the cluster. We
will henceforth use the fractional binding energy \begin{equation} f =
\frac{M_p(R)-M(R)}{M_p(R)} \end{equation} as a measure for the stability of a
cluster since analytical and numerical work has shown that radial
instability sets in in clusters around the first maximum of $f$
\citep{fa69,st85}.
From $p_\mu p^\mu = (mc)^2$, $J=r^2 p^3=mru^\phi$, and $p^0c = E e^{-\phi}$, it
follows that for a circular orbit in the $\theta=\pi/2$ plane the
angular momentum is given by \begin{equation} J = \frac{r}{c} E \sqrt{ e^{-\phi}
- \left( \frac{mc^2}{E} \right)^2 }. \end{equation} For a given radius $r$,
the energy of the circular orbit with that radius can be found by
setting $dJ/dr=0$. This leads to \begin{equation} \frac{mc^2}{E} = \sqrt{
\left(1-\frac{r}{2} \frac{d\phi}{dr} \right) e^{-\phi} }. \end{equation}
Plugging this into the expression for the angular momentum yields
\begin{equation} \frac{u^\phi}{c} = \frac{E}{mc^2} \sqrt{ \frac{r}{2}
\frac{d\phi}{dr} e^{-\phi}}. \end{equation} From the viewpoint of a distant observer,
the velocity of a star on a circular orbit with radius $r$ is given by
\begin{equation} v_{\sf circ}(r) = \frac{d\tau_P}{dt} u^\phi = \frac{mc^2}{E} e^\phi
u^\phi = c\sqrt{ \frac{r}{2} \frac{d\phi}{dr} e^\phi} \label{vcgr} \end{equation} since the
derivative of the star's proper time $\tau_P$ with respect to
coordinate time is \begin{equation} \frac{d\tau_P}{dt} = \frac{mc}{p^0} =
\frac{mc^2}{E} e^\phi. \end{equation}
The radiation of a light source at rest at radius $r$, is observed at
infinity to have undergone a gravitational redshift \begin{equation} z(r) =
e^{-\phi(r)/2}-1. \end{equation} This ``redshift-from-rest'' is a measure for
how ``relativistic'' a given cluster model is. Where it was first
thought that no stable models with a central redshift-from-rest
$z(0) \gtrsim 0.5$ can exist \citep{zp65,ip69,os74}, more recent
work has shown that arbitrarily large values for the central
``redshift-from-rest'' are possible in stable models. The first hint
that large redshifts are possible came from numerical integrations
of the relativistic Boltzmann equation \citep{rst89} that was later
on backed up by detailed analytical calculations
\citep{mr95}. It was subsequently shown in \citet{bk98,bkm06} that arbitrarily
large central redshifts are possible in stable models with a
distribution function of the form $F(E) \propto \exp(-E/T)$, with
$T$ the uniform kinetic temperature as observed from infinity, only
if $T/mc^2 \lesssim 0.06$. ``Hotter'' models are stable only for
red\-shifts below $\approx 0.5$.
\subsection{The Newtonian limit}
In the Newtonian limit, we can employ the approximation
\begin{align}
\frac{d\tau_P}{dt} &= \frac{mc^2}{cp^0} = \frac{mc^2}{E}e^\phi
\nonumber \\ &= \frac{1}{c}\frac{ds}{dt}\approx \sqrt{ e^\phi - \left(
\frac{v}{c} \right)^2},
\end{align}
or, in other words, \begin{align} E &\approx \frac{mc^2 e^\phi}{\sqrt{ e^\phi -
\left( \frac{v}{c} \right)^2}} \approx mc^2\left( 1 +
\frac{1}{2}\phi + \frac{1}{2} \frac{v^2}{c^2} \right) \nonumber \\ &\approx mc^2 +
m\left( \psi + \frac{1}{2}v^2 \right) = mc^2 + m \epsilon.
\end{align}
Here, $\epsilon$ is the Newtonian energy per unit mass. Moreover,
$\hat{p}_0c \approx mc^2+\frac{1}{2}mv^2$.
These results can be used to calculate the Newtonian approximation for
the isotropic momentum moments of the DF, given by expression (\ref{mu2q}):
\begin{align} \mu_{k,2q} &\approx 4 \pi (mc)^{k-1} \int F(E) p^{2q+2} dp \nonumber \\
&\approx (mc)^{k-1} \mu^N_{2q}(\psi). \label{mckf}
\end{align} Except for the inconsequential forefactor $(mc)^{k-1}$,
this is the correct expression for the Newtonian isotropic momentum
moment $\mu^N_{2q}(\psi)$.
Taking together $E \approx E_0 \approx mc^2$, $D_{E^2} \approx
\frac{1}{2m^2c^2} D_\epsilon$, $dE_0^2 \approx 2 m^2 c^2 d\psi$, and
(\ref{mckf}), the inversion formula for the DF can be written in the
form \begin{equation} F(\epsilon) \approx
\frac{1}{2^{q+2}\pi^{\frac{3}{2}}\Gamma\left(q+\frac{3}{2}\right)
m^{2q+3}} D_\epsilon^{q+2} \int
\frac{\mu^N_{2q}(\psi)}{\sqrt{2(\psi-\epsilon)}} d\psi, \end{equation} the
correct Newtonian expression for the DF in terms of a Newtonian
momentum moment. For $q=0$, one obtains the important special case
\begin{equation} F(\epsilon) \approx \frac{1}{2\pi^2m^3} D_\epsilon^2 \int
\frac{\rho(\psi)}{\sqrt{2(\psi-\epsilon)}} d\psi, \end{equation} with $\rho$
the mass density.
In the Newtonian limit, the distribution function (\ref{propdf}) becomes $F(E)
\approx f_0 \left[ 2(\Psi-\epsilon) \right]^\alpha$, with
$\Psi=\psi(R)$, the value of the Newtonian gravitational potential at
the outer edge of the cluster. For an infinitely extended system with
$\alpha=7/2$ and $\Psi=\psi(\infty)=0$ this is $f_{\text{Plum}}(E)=f_0
(-2\epsilon)^{7/2}$, the distribution function of the Newtonian
Plummer model. The Newtonian limit of the distribution function does
not depend on the parameter $\beta$:~it only serves to change the
slope of the DF for the most strongly relativistic models. In the
Newtonian limit, the density reduces to \begin{equation} \rho(\psi) \approx
\frac{3}{4\pi} \left( -\frac{a}{GM} ( \Psi-\psi )
\right)^{\alpha+\frac{3}{2}}\frac{M}{a^3}. \end{equation} For a Plummer model,
with $\alpha=7/2$, we retrieve the relation \begin{equation} \rho_{\text{Plum}}
\approx \frac{3}{4\pi} \left( -\frac{a}{GM} \psi
\right)^5\frac{M}{a^3}. \end{equation} The proper density $nm$ reduces to the
same expression as the gravitating mass density $\rho$, as it
should. The Newtonian expression for the pressure is found to be \begin{equation}
P \approx \frac{3}{2\pi(2\alpha+5)} \left( -\frac{a}{GM} (\Psi-\psi)
\right)^{\alpha+\frac{5}{2}}\frac{GM^2}{a^4}. \end{equation} For a Plummer
model, we find \begin{equation} P_{\text{Plum}} \approx \frac{1}{8\pi} \left(
-\frac{a}{GM} \psi \right)^6\frac{GM^2}{a^4} \propto
\rho_{\text{Plum}}^{\frac{6}{5}}. \end{equation}
Clearly, these Newtonian models have equations of state of the form
\begin{equation} P = K \rho^{\frac{2\alpha+5}{2\alpha+3}} = K
\rho^{1+\frac{1}{n}} \end{equation} for some constant $K$. They are polytropes
with polytropic index \begin{equation} n = \alpha+\frac{3}{2} \ge 0. \label{nalpha} \end{equation} The
general relativistic cluster models, due to the presence of the
hypergeometric functions in the expressions for the density and
pressure, are not polytropes and have more complicated equations of
state. Newtonian polytropes have finite mass for $n \in [0,5]$ and
finite radius for $n \in [0,5[$. The Plummer model, with $n=5$ is the
first polytropic model with infinite radius but still with finite
mass. It is generally assumed that the condition $df/dE<0$ is a
prerequisite for the radial stability of a cluster
\citep{ip69,fa71}. We therefore limit ourselves to models with
$\alpha \ge 0$, and hence $n \ge \frac{3}{2}$, for which this
condition is definitely fulfilled.
As is well known, the structure of a polytrope with index $n$ and
equation of state $P = K \rho^{1+\frac{1}{n}}$ for some constant
forefactor $K$ is given by the Lane-Emden equation \begin{equation}
\frac{1}{\xi^2} \frac{d}{d\xi} \left( \xi^2 \frac{d\theta}{d\xi}
\right) = - \theta^n. \end{equation} Here, $\xi$ is a dimensionless radius
related to the radius $r$ via \begin{equation} \xi = \sqrt{ \frac{2 \pi c^2
a^3}{(n+1){\cal A}M}\frac{\rho_c^2}{P_c} } \frac{r}{a} \label{xix}, \end{equation}
with $\rho_c$ and $P_c$ the central density and pressure,
respectively. This equation must be integrated numerically for the
function $\theta(r)$ out to its first zero, which then defines the
outer radius $R$ of the cluster. Then the density is given by $\rho(r)
= \rho_c \theta^n(r)$, and the gravitational potential by $\psi(r) =
-(n+1)K \rho_c^{\frac{1}{n}} \theta(r) + \psi(R)$. The circular
velocity profile, $v_{\sf circ}(r)$, then follows from the relation
\begin{equation} v_{\sf circ}(r) = \sqrt{ r \frac{d\psi}{dr} }. \end{equation}
\begin{figure}
\includegraphics[width=0.49333\textwidth]{fig1_a.pdf}
\includegraphics[width=0.49333\textwidth]{fig1_b.pdf}
\includegraphics[width=0.49333\textwidth]{fig1_c.pdf}
\caption{Central redshift-from-rest versus fractional binding energy
$f=(M_p(R)-M(R))/M_p(R)$ for all models with $\alpha=0.5$ (top),
$\alpha=2.75$ (middle), and $\alpha=3.4$ (bottom). For $\alpha=0.5$,
the effect of different $\beta$-values, between 0.5 and 5.0, is
explored. For all other $\alpha$-values, only $\beta=0.5$ was used.
The color scale of the data points indicates the value of the
potential at the outer boundary of the model, $e^\Phi$. The model
with the smallest value for $e^\Phi$ for each $\alpha$-value is
indicated with a white dot in each panel. Models to the left of this
white dot have shallower potentials; those to the right of it have
deeper potentials.
\label{fig:bifurc_1.67.eps}}
\end{figure}
With which Newtonian model should a given relativistic cluster be
compared? A natural choice for the polytropic index is given by
(\ref{nalpha}). From (\ref{xix}), it is obvious that the dimensionless
radius $\xi$ can be rescaled to the dimensionless radius $x$ with the
scale depending on the central pressure and density. We rescale the
density profile such that the total mass equals unity, something we
will also do with the relativistic models, giving us a value for
$\rho_c$. We then adopt a value for the constant $K$ such that $P_c =
K \rho_c^{1+\frac{1}{n}}$. In this case, $2\psi(R)/c^2 = -1/{\cal A}X$
with $X=R/a$ the dimensionless outer boundary of the Newtonian cluster
(which, obviously, does not need to coincide with the outer boundary
of the relativistic cluster).
One further remark concerns the fact that in the case of Newtonian
stellar clusters, one can choose the mass-scale $M$ and the
length-scale $a$ independently from each other whereas in the general
relativistic models presented here these two parameters are linked by
the parameter $\cal A$, defined as (\ref{Qparam}), and they cannot be
chosen freely. However, in the Newtonian limit, which can be defined
formally as the limit $c \rightarrow \infty$, the parameter $1/{\cal A}$
goes to zero, \begin{equation} \lim_{c \rightarrow \infty} \frac{1}{\cal A} = \lim_{c
\rightarrow \infty} \frac{2GM}{c^2a} = 0, \end{equation} for a finite
mass-scale $M$ and non-zero length-scale $a$. In that limit, $M$ and
$a$ are effectively decoupled since $1/\cal A$ is always zero, irrespective of
which mass and length-scale one chooses.
\section{Solving the field equations} \label{solve}
The two relevant field equations, as shown in e.g. \cite{os74}, are
\begin{align}
\frac{dM}{dr}(r) &= 4 \pi r^2
\rho \label{cofi00a}\\ \frac{d\phi}{dr}(r) &= \frac{2G}{c^2r^2}
\left[M(r)+ 4\pi r^3
\frac{P}{c^2}\right]
\left[1-\frac{2GM(r)}{rc^2}\right]^{-1} . \label{cofi11a}
\end{align}
If we denote the dimensionless radius by $x=r/a$, the scaled mass by
${\cal M}(r)=M(r)/M$, the scaled density by $\tilde{\rho} = \rho
a^3/M$, and the scaled pressure by $\tilde{P} = P a^3/Mc^2$, we can
rewrite these equations in a fully dimensionless form as
\begin{align}
\frac{d{\cal M}}{dx} &= 4 \pi x^2 \tilde{\rho} \\
\frac{d\phi}{dx} &= \frac{1}{{\cal A}x^2} \left[ {\cal M} +
4\pi x^3 \tilde{P} \right]\left[ 1 - \frac{\cal M}{{\cal A}x} \right]^{-1}.
\end{align}
These equations must be integrated numerically starting from the
initial conditions
\begin{align}
{\cal M}(0) &= 0, \nonumber \\
\phi(0) &= \phi_0
\end{align}
where $\phi_0$ must be chosen such that \begin{equation} \exp(\phi(X)) =
\exp(\Phi) = 1-\frac{{\cal M}(X)}{{\cal A}X}, \label{ephiqm} \end{equation} with $X=R/a$ the scaled
radius at the cluster's outer boundary. This ensures that the
``internal'' solution smoothly goes over into the ``external''
Schwarzschild solution. This precludes the retrieval of infinitely
extended models, especially if they have a diverging mass.
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{fig2_a.pdf}
\includegraphics[width=0.85\textwidth]{fig2_b.pdf}
\includegraphics[width=0.85\textwidth]{fig2_c.pdf}
\caption{The potential, $\phi$, gravitating and proper mass profiles,
${\cal M}$ and ${\cal M}_p$, circular-velocity profile, $v_{\sf
circ}$, density $\rho$ and pressure $P$ profiles, and effective
polytropic index $n$ for models with $\alpha=0.5$, 2.75, and
3.4. Where visible, the vertical line indicates the outer boundary
of the cluster model. The dashed curves indicate the
circular-velocity profile, density profile, and polytropic index of
the corresponding Newtonian cluster with the same total gravitating
mass and the same central pressure. For each model, its values for
$\alpha$, the boundary potential $e^\Phi$, the central potential
$\phi(0)$, and the fractional binding energy $f$ are indicated. All
models have $\beta=1/2$.
\label{fig:models.eps}}
\end{figure*}
By explicitly pulling out the $\cal A$-dependence of the density and
pressure, it becomes clear that by rescaling the mass and radius
according to
\begin{align}
x' &= {\cal A}^{(1+2\alpha)/4}x \\
{\cal M}' &= {\cal A}^{(2\alpha-3)/4}{\cal M},
\end{align}
the parameter $\cal A$ can be completely removed from the
dimensionless field equations. So one can always set ${\cal A}=1$ in
the field equations, solve them, and then afterwards rescale to that
particular value of ${\cal A}$ for which ${\cal M}(X)=1$. In that
case, the mass scale $M$ equals the total gravitating mass of the
cluster and ${\cal A}$ has the meaning of the ratio of $a$ to $R_S$.
We wrote a small Python program to numerically integrate these
equations and to determine the central value of the potential using a
least squares minimizer.
\section{Discussion} \label{discussion}
\subsection{Existence of solutions} \label{sub:exist}
For each choice of $\alpha$, the only free parameter in the field
equations is the value of the potential at the outer boundary of the
cluster, in the form $e^\Phi$. Our numerical work shows that the field
equations presented in the previous section have a bifurcation at some
$\alpha$-dependent critical value, $e^{\Phi_0(\alpha)}$. For $e^\Phi <
e^{\Phi_0(\alpha)}$, no solutions exist. At $e^\Phi =
e^{\Phi_0(\alpha)}$, a single solution appears. For choices $1 \ge
e^\Phi > e^{\Phi_0(\alpha)}$, two solutions, with different central
potential values $\phi_0$, exist. This can be seen in
Fig. \ref{fig:bifurc_1.67.eps} in which the fractional binding energy
$f$ is plotted versus the central redshift-from-rest $z_c$ for models
with $\alpha=0.5$, 2.75, and 3.4. We always adopt the value
$\beta=1/2$ except in the top panel, where the effects of different
$\beta$-values are explored. In each panel, the model with the
smallest value for $e^\Phi$ is indicated by a white dot. The color of
the other data points corresponds to their $e^\Phi$-value, as
indicated by the colorbar. To the left of the white dot are models
with shallower potential wells with the Newtonian $f=0$, $z_c=0$ model
as limit. To the right of the white dot are models with deeper
potential wells and correspondingly higher central redshifts.
This situation is reminiscent to that of the family of models
discussed by \citet{bk98} which also exhibits both bifurcations
(i.e. more than one solution for a given set of model parameters)
and limiting values for a parameter connected to the energy at the
outer boundary.
For small values for the power $\alpha$, below $\alpha \sim 3$, the
$f-z_c$-curve has a maximum around $z_c \approx 0.5$. As $\alpha$
increases, the right side of the $f-z_c$-curve appears to curl up from
right to left until this maximum disappears and the $f-z_c$-relation
is monotonically rising. In the limit $\alpha \rightarrow 7/2$, only
the Newtonian $f=0$, $z_c=0$ model exists. This means that the Plummer
model is a purely Newtonian construct:~no relativistic models with
$\alpha=7/2$ exist. Also in the Newtonian is the Plummer model a
limiting case. As the polytropic index $n$ is increased from zero, it
is the first solution of the Lane-Emden equation with infinite
extent. It is also the last model with a finite total mass. This
appears also to be true relativistically. By construction we are
searching for models with a finite total mass by trying to match the
solutions of the field equations to an external Schwarzschild
metric. No such solutions exist for $\alpha > 7/2$.
This is true for different values of the power $\beta$. However,
increasing $\beta$ shifts the high-$z_c$ end of the $f-z_c$-relation
in the direction of smaller $z_c$, i.e. towards models with shallower
potentials. An increase of $\beta$ also raises the $e^\Phi$-value of
those most relativistic cluster models which means they become less
compact (see paragraph \ref{sub:radius}).
The general conclusion we can draw from this is that the steeper the
distribution function $F(E)$ varies as a function of energy $E$, the
more the solutions are confined towards the Newtonian limit
$(f=0,z_c=0)$ and that no relativistic models with finite mass exist
with $\alpha>7/2$.
\subsection{Model properties}
\begin{figure*}
\includegraphics[width=\textwidth]{fig3.pdf}
\caption{The ratio of the scale radius to the Schwarzschild radius,
${\cal A}=a/R_S$ versus outer boundary radius $R/R_S$ (left panel)
and the central redshift-from-rest $z_c$ (right panel) for the
models with $\alpha=\beta=0.5$. The color scale of the data points
indicates the value of the potential at the outer boundary of the
model, $e^{\Phi}$.
\label{fig:A_zc.eps}}
\end{figure*}
In Fig. \ref{fig:models.eps}, we present the potential function
$\phi$, the gravitating and proper mass profiles, ${\cal M}$ and
${\cal M}_p$, the circular-velocity profile, $v_{\sf circ}$, the
density $\rho$ and pressure $P$ profiles, and the effective polytropic
index $n$ for models with $\alpha=0.5$, 2.75, and 3.4. For all models,
we adopt $\beta=1/2$. The dashed curves indicate the circular-velocity
profile, density profile, and polytropic index of the corresponding
Newtonian cluster with the same total gravitating mass and the same
central pressure. The effective polytropic index $n$ is here defined
as \begin{equation} 1+\frac{1}{n} = \frac{d\ln P}{d\ln \rho} \end{equation} which can be
compared with the index (\ref{nalpha}) derived from the power $\alpha$
in the expression for the distribution function. In the Newtonian
limit, both indices coincide.
The $\alpha=3.4$ model shown in Fig. \ref{fig:models.eps} has a very
shallow potential and is essentially Newtonian. Therefore, it is
indistinguishable from the Newtonian solution of the Lane-Emden
equation. The models with $\alpha=0.5$ and $\alpha=2.75$ have much
deeper gravitational wells and are well in the general relativistic
regime. Clearly, these models do not have polytropic equations of
state and their effective polytropic indices can differ significantly
from the value expected from their $\alpha$-value. For the same total
mass, their density profiles are less steep than those of the
Newtonian models. This, combined with the gravitational time
dilatation effect in eqn. (\ref{vcgr}) for the circular velocity,
causes the relativistic circular-velocity curve to be much flatter
than its Newtonian counterpart.
\begin{figure}
\includegraphics[width=0.53\textwidth]{fig4_bis.pdf}
\caption{The fractional binding energy $f = (M_p(R)-M(R))/M_p(R)$ as a
function of central redshift $z_c$ and the potential at the outer
edge of the mass distribution, quantified by $\exp(\Phi)$. The color
scale measures $f$ in percentages; the open circles indicate the
positions of the models that were actually constructed. The
different model sequences have different values for the power
$\alpha$.
\label{fig:z_f.eps}}
\end{figure}
The ratio of the scale-length to the Schwarzschild radius, quantified
by ${\cal A}=a/R_S$, is plotted as a function of the ratio of the
outer boundary radius to the Schwarzschild radius, $R/R_S$, and of the
central redshift-from-rest, $z_c$, in Fig. \ref{fig:A_zc.eps} for the
$\alpha=\beta=0.5$ models. $a/R_S$ shows a non-trivial behavior in the
sense that the model with the smallest scale-length is neither the
most tightly bound model (the one with the largest fractional binding
energy $f$) nor the most compact one (the one with the smallest
$e^\Phi$ or $R/R_S$ value). $a/R_S$ diverges for $z_c \rightarrow 0$
since the Schwarzschild radius tends to zero in the Newtonian
limit. In the limit of extremely compact models, $a/R_S$ increases
again. Apparently, only models with very flat-topped density profiles,
with $a \gtrsim R$, can exist in this regime.
\subsection{Binding energy}
We plot the fractional binding energy $f = (M_p(R)-M(R))/M_p(R)$ as a
function of central redshift $z_c$ and the potential at the outer edge
of the mass distribution, quantified by $\exp(\Phi)$, in
Fig. \ref{fig:z_f.eps}. The open circles in this figure indicate the
loci of the models that were actually constructed. The different model
sequences have different values for the power $\alpha$, the leftmost
corresponding to $\alpha=0.05$. The 2D map of the binding energy was
constructed by applying a bicubic spline interpolator to the model
points. The models nicely cover the first maximum of $f$, where
dynamical instability is expected to set in \citep{bk98,bkm06}.
The grey line connects the models which, for a given $\alpha$, attain
the maximum fractional binding energy. The models with $\alpha$ in the
range $0.05$ to $\approx 3.0$ have central redshift-from-rest values
between $\approx 0.5$ and $\approx 0.55$. For higher $\alpha$-values,
the maximum central redshift rapidly drops to zero. As the power
$\alpha$ approaches the value of 7/2, the Plummer model value, both
the central redshift-from-rest and the fractional binding energy go to
zero, the Newtonian limit. The overall maximum central
redshift-from-rest is achieved by the model with $\alpha=2.75$.
This behavior is caused by the $\alpha$-dependence of the shape of the
$f-z_c$-relation which was discussed in paragraph \ref{sub:exist}. At
first, steepening the distribution function by increasing $\alpha$
above zero leads to a deepening of the potential well and therefore to
a slight increase of $z_c$. Above $\alpha \approx 2.75$, a further
steepening of the distribution function and of the density profile
limits the models more and more to the Newtonian limit, thus reducing
$z_c$.
\subsection{The radius} \label{sub:radius}
Each model is labelled by a unique $\cal A$-value for which the mass
scale $M$ coincides with the model's total mass. If we select this
particular value for $\cal A$ or, equivalently, $M$ , the quantity
\begin{equation} R_S = \frac{2GM}{c^2} \end{equation} has the physical meaning of being
the model's Schwarzschild radius. Numerically integrating the field
equations yields the dimensionless outer radius $X$. Multiplying this
radius with the scale-length $a$ gives the physical value for the
radius $R=aX$. It then follows that \begin{equation} \frac{1}{\cal A} = \frac{2GM}{c^2a}
= \frac{2GM}{c^2R}X \end{equation} and consequently \begin{equation} \frac{R}{R_S} =
{\cal A}X = \frac{1}{1-\exp(\Phi)}, \end{equation} where we made
use of eqn. (\ref{ephiqm}).
For each value of $\alpha$, there exists a minimum value for $\Phi$
below which no solutions to the field equations can be found. Using
the above, this corresponds to a minimum value for $R/R_S$. As
$\alpha$ tends to zero, the minimum radius shrinks to $R \approx 3.6
R_S$, as can be seen in Fig. \ref{fig:alpha_phimin_Rmin.eps}. Hence,
models with very ``flat'' distribution functions and density and
pressure profiles can be very small, with radii only a few times
larger than their Schwarzschild radius. As the distribution function
and the corresponding density and pressure profiles are steepened by
increasing the value of the power $\alpha$, this minimum radius
steadily increases. In the limit $\alpha \rightarrow 7/2$ the only
possible solution is the Newtonian Plummer model and the minimum
radius grows to infinity.
\begin{figure*}
\includegraphics[width=\textwidth]{fig5.pdf}
\caption{The minimum value for $e^\Phi$ versus $\alpha$ (left) and the
smallest possible radius, divided by the model's Schwarzschild
radius versus $\alpha$ (right).
\label{fig:alpha_phimin_Rmin.eps}}
\end{figure*}
\section{Conclusions} \label{conc}
We show that the equations underlying the general relativistic theory
of spherically symmetric isotropic stellar clusters can be cast in a
form analogous to that of the Newtonian theory. Using the mathematical
formalism developed for the latter, we prove that the distribution
function can be derived from any isotropic momentum moment
$\tilde{\mu}_{k,2q}$. This is a direct generalization of the inversion
relations derived by \citet{fa68} and \citet{pk96}. Moreover, every
higher-order moment $\tilde{\mu}_{k,2q}$, with $q>0$, can be written
as an integral over the corresponding zeroth-order moment
$\tilde{\mu}_{k,0}$.
We propose a mathematically simple expression for the distribution
function of a family of isotropic cluster models which is guaranteed
to be positive everywhere in phase space. The distribution function of
each model is basically defined by two parameters:~the slope $\alpha$
and the value of the potential at the boundary, $\Phi$. In the
Newtonian limit, these models reduce to the family of polytropic
models. In the relativistic regime, however, these models do not have
a polytropic equation of state. We derive the Newtonian limits of the
general equations underlying the cluster dynamics and the density and
pressure profiles of the polytropic cluster models.
For a given $\alpha$, the field equations for these general
relativistic cluster models only allow solutions if $\Phi >
\Phi_0(\alpha)$, with $\Phi_0(\alpha)$ an $\alpha$-dependent minimum
value for the potential at the outer boundary. In other words:~for a
given slope of the distribution function, a model cannot be made
arbitrarily compact. The ratio of the minimum outer radius to the
model's Schwarzschild radius is a rising function of $\alpha$,
increasing from $R/R_S \approx 3.6$ for $\alpha=0$ to $R/R_s=\infty$
for $\alpha=3.5$. For less compact models, always two solutions to the
field equations exist:~one with a higher central redshift than the
most compact model and one with a lower central redshift.
The models we constructed, for $\alpha$-values between $0.05$ and
$3.5$, fully cover the first maximum of the fractional binding, where
dynamical instability is expected to set in. This first maximum is
achieved by models which all have a central redshift below $z_c
\approx 0.55$. The most strongly bound model is characterized by
$\alpha=2.75$ and a central redshift $z_c \approx 0.55$. Models with
steeper distribution functions have lower fractional binding energies
than the $\alpha=2.75$ model whereas models with flatter distribution
functions have higher fractional binding energies. In the limit
$\alpha \rightarrow 3.5$, the binding energy and the central redshift
both tend to zero. This indicates that in this limit the distribution
function has become too steep to allow for anything but the Newtonian
solution:~no models with a finite mass exist for $\alpha>3.5$. Hence,
we can conclude that, at least within the context of this family of
models, the Plummer model by necessity is a purely Newtonian
construct.
\section*{Acknowledgements}
The authors wish to thank H. Dejonghe for his insightful suggestions.
This research has been funded by the Interuniversity Attraction Poles
Programme initiated by the Belgian Science Policy Office (IAP P7/08
CHARM).
|
2,877,628,089,499 | arxiv | \section{Introduction}
The directed movement of cells in response to a chemical stimulus is known in biology as chemotaxis. More specifically, if the cells move towards regions of high chemical concentration, the motion is called chemoattraction, while if the cells move towards regions of lower chemical concentration, the motion is called chemorepulsion. Models for chemotaxis motion has been studied in literature (see \cite{Cristian,HP,Horst,KS,Lan1,Lan2,Win1,Win2} and references therein). One of the most important characteristics of chemoattractant models is that the finite blow up of solutions can happen in space dimension greater or equal to $2$; while in chemorepulsion models this phenomenon is not expected. Many works have been devoted to study in what cases and how blow up takes place (see for instance \cite{Win1b,Lin,Via,Win1c,Win1a,XIXI,Zhao}). \\
In those cases in which blow-up phenomenon does not happen, it is interesting to study the asymptotic behaviour of the solutions of the model. In fact, in \cite{Osaki}, Osaki and Yagi studied the convergence of the solution of the Keller-Segel model to a stationary solution in the one-dimensional case. In \cite{HJ}, the convergence of the solution of the Keller-Segel model with an additional cross-diffusion term to a steady state was shown. In \cite{Cristian} the authors proved the convergence to constant state for a chemorepulsion model with linear production. Therefore, taking into account the results above, the first aim of this paper is to study the asymptotic behaviour of the following parabolic-parabolic repulsive-productive chemotaxis model (with quadratic signal production):
\begin{equation} \label{modelf00}
\left\{
\begin{array}
[c]{lll}%
\partial_t u - \Delta u = \nabla\cdot (u\nabla v)\ \ \mbox{in}\ \Omega,\ t>0,\\
\partial_t v - \Delta v + v = u^{2} \ \mbox{in}\ \ \Omega,\ t>0,\\
\displaystyle \frac{\partial u}{\partial \mathbf{n}}=\frac{\partial v}{\partial \mathbf{n}}=0\ \ \mbox{on}\ \partial\Omega,\ t>0,\\
u({\boldsymbol x},0)=u_0({\boldsymbol x})\geq 0,\ v({\boldsymbol x},0)=v_0({\boldsymbol x})\geq 0\ \ \mbox{in}\ \Omega,
\end{array}
\right. \end{equation}
where $\Omega$ is a $n-$dimensional open bounded domain, $n=1,2,3$, with boundary $\partial \Omega$; and the unknowns are $u(\boldsymbol x, t) \geq 0$, the cell density, and $v(\boldsymbol x, t) \geq 0$, the chemical concentration. This model has been studied in \cite{FMD}. There, the authors shown that model (\ref{modelf00}) is well-posed: there exists global in time weak-strong solution (in the sense of Definition \ref{ws00} below) and, for $1D$ or $2D$ domains, there exists a unique global in time regular solution.\\
On the other hand, another interesting topic is the study of fully discrete FE schemes approximating model (\ref{modelf00}), conserving properties of the continuous problem such as: mass-conservation, energy-stability, positivity and long time behaviour. In fact, in \cite{FMD2} it was studied a fully discrete FE scheme for model (\ref{modelf00}), which is mass-conservative and energy-stable with respect to a modified energy given in terms of the auxiliary variable ${\boldsymbol{\sigma}}=\nabla v$. However, neither energy-stability with respect to the primitive variables $(u,v)$ (see (\ref{eneruva}) below) nor positivity (or approximated positivity) were proved. Moreover, as far as we know, there are not another works studying FE aproximations for problem (\ref{modelf00}).
For this reason, the second aim of this paper is to present two new fully discrete FE schemes, that have better properties than the scheme proposed in \cite{FMD2}, in terms of energy-stability, positivity and asymptotic behaviour of the scheme.\\
The asymptotic behaviour of fully discrete numerical schemes has been studied in different contexts. In fact, in \cite{GS} Guill\'en-Gonz\'alez and Samsidy proved asymptotic convergence for a fully discrete FE scheme for a Ginzburg-Landau model for nematic liquid crystal flow. In \cite{MP} Merlet and Pierre studied the asymptotic behaviour of the backward Euler scheme applied to gradient flows. It is important to notice that, in chemotaxis models, there are few works studying large-time behaviour for fully discrete schemes. We refer to \cite{BJ}, where the authors shown conditional stability and convergence at infinite time of a finite volume scheme for a Keller-Segel model with an additional cross-diffusion term. Meanwhile, the behavior at infinite time of a fully discrete scheme for model (\ref{modelf00}) seems to be still an open problem. \\
Likewise, the energy-stability property has been studied for fully discrete numerical schemes in the chemotaxis framework. In \cite{GRR1}, the authors studied unconditionally energy stables FE schemes for a chemo-repulsion model with linear production. A finite volume scheme for a Keller-Segel model with an additional cross-diffusion term satisfying the energy-stablity property (conditionally) has been studied in \cite{BJ}. In \cite{FMD2}, it was studied an unconditionally energy-stable FE scheme for model (\ref{modelf00}) with respect to a modified energy written in terms of the auxiliary variable ${\boldsymbol{\sigma}}=\nabla v$. However, up our knowledge, the energy-stability in FE schemes, with respect to the $(u,v)$-energy given in (\ref{eneruva}) below, is so far an open problem. \\
In terms of positive or approximately positive numerical schemes on chemotaxis context we refer to \cite{Saad1,Zuhr,SG1,FMD,GRR1}. In \cite{Zuhr}, the nonnegativity of numerical methods, using FE techniques, to a generalized Keller-Segel model was analyzed. A discrete maximum principle for a fully discrete numerical scheme (combining the finite volume method and the nonconforming finite element
method) approaching a chemotaxis-swimming bacteria model was obtained in \cite{Saad1}. In \cite{GRR1}, aproximated positivity of FE schemes for a chemo-repulsion model with linear production was proved. The positivity of a finite volume scheme for a parabolic-elliptic chemotaxis system was studied in \cite{SG1}. In \cite{FMD}, positivity of only time-discrete schemes associated to model (\ref{modelf00}) was proved.
However, there are not works studying positive (or approximately positive) FE schemes for model (\ref{modelf00}).
\\
Consequently, the main novelties in this paper are the following:
\begin{itemize}
\item The introduction of a FE scheme (see scheme \textbf{UV} in Section \ref{NEs} below) which is energy-stable with respect to the $(u,v)$-energy of the continuous problem (\ref{modelf00}) given in (\ref{eneruva}), under a ``compatibility'' condition on the FE spaces, namely taking $(\mathbb{P}_m,\mathbb{P}_{2m})$-continuous FE (with $m\geq 1$) for $(u,v)$.
\item The introduction of another FE scheme (see scheme \textbf{US}$_\varepsilon$ in Section \ref{schemeUSe} below) which has the ``approximated'' positivity property, and it is energy-stable (with respect to a modified $(u,{\boldsymbol{\sigma}})$-energy).
\item The proof of the long time behaviour for the schemes previously mentioned, obtaining exponential convergence to constant states as time goes to infinity.
\end{itemize}
The outline of this paper is as follows: In Section \ref{CMo}, we study (formally) the asymptotic behavior of the global solutions for the model (\ref{modelf00}), and we prove the exponential convergence as time goes to infinity to constant states. In Section \ref{NEs}, we study a fully discrete scheme associated to model (\ref{modelf00}), corresponding to the nonlinear backward Euler in time and FE in space in the variables $(u,v)$. The analysis includes the well-posedness of the scheme and some properties such as $u$-conservation, energy stability, convergence and long time behaviour. In Section \ref{schemeUSe}, we propose another fully discrete FE approximation of model (\ref{modelf00}), which is obtained combining the scheme \textbf{US} proposed in \cite{FMD2} with a regularization technique. For this scheme, we can prove, in addition to the properties proved for the previous scheme, the approximated positivity. In Section \ref{NSi}, we compare the numerical schemes throughout several numerical simulations, giving the main conclusions in Section \ref{Con}.
\subsection{Notation}
We recall some functional spaces which will be used throughout this paper. We will consider the usual Sobolev spaces $H^m(\Omega)$ and Lebesgue spaces $L^p(\Omega),$
$1\leq p\leq \infty,$ with norms $\Vert\cdot\Vert_{m}$ and $\Vert\cdot \Vert_{L^p}$, respectively. In particular, the $L^2(\Omega)$-norm will be denoted by $\Vert
\cdot\Vert_0$. We denote by ${\boldsymbol H}^{1}_{\sigma}(\Omega):=\{\mathbf{u}\in {\boldsymbol H}^{1}(\Omega): \mathbf{u}\cdot \mathbf{n}=0 \mbox{ on } \partial\Omega\}$ and we will use the following equivalent norms in $H^1(\Omega)$ and ${\bf H}_{\sigma}^1(\Omega)$, respectively (see \cite{necas} and \cite[Corollary 3.5]{Nour}, respectively):
\begin{equation}
\Vert u \Vert_{1}^2=\Vert \nabla u\Vert_{0}^2 + \left( \int_\Omega u\right)^2, \ \ \forall u\in H^1(\Omega),
\end{equation}
\begin{equation}\label{H1div}
\Vert {\boldsymbol\sigma} \Vert_{1}^2=\Vert {\boldsymbol\sigma}\Vert_{0}^2 + \Vert \mbox{rot }{\boldsymbol\sigma}\Vert_0^2 + \Vert \nabla \cdot {\boldsymbol\sigma}\Vert_0^2, \ \ \forall {\boldsymbol\sigma}\in {\boldsymbol H}^{1}_{\sigma}(\Omega).
\end{equation}
In particular, (\ref{H1div}) implies that
\begin{equation*}
\Vert\nabla v\Vert_{1}^2=\Vert \nabla v\Vert_{0}^2 + \Vert \Delta v\Vert_0^2, \ \ \forall v:\nabla v\in {\boldsymbol H}^{1}_{\sigma}(\Omega).
\end{equation*}
If $Z$ is a
general Banach space, its topological dual will be denoted by $Z'$.
Moreover,the letters $C,C_i,K_i$ will denote different positive constants depending on the data $(\Omega,u_0, v_0)$,
but independent of the discrete parameters $(k, h)$ and time step $n$, which may change from line to line (or even within the same line).
\section{Continuous problem}\label{CMo}
In this section some fundamental concepts associated to problem (\ref{modelf00}) are presented, including the definition of weak-strong solutions and some qualitative properties such as $u$-conservation, positivity and large time behaviour. In particular, exponential convergence to constant states as time goes to infinity is obtained.
\subsection{Some properties}
Problem (\ref{modelf00}) conserves in time the total mass $\int_\Omega u$. In fact, defining
\begin{equation}\label{m0}
m_0=\frac1{|\Omega|} \int_{\Omega} u_0,
\end{equation}
and integrating (\ref{modelf00})$_1$ in $\Omega$,
\begin{equation*}
\frac{d}{dt}\left(\int_\Omega u\right)=0, \ \ \mbox{ i.e. } \
\int_\Omega u(t)=\int_\Omega u_0:= m_0\vert \Omega\vert, \ \ \forall t>0.
\end{equation*}
Now, the definition of weak-strong solutions for problem (\ref{modelf00}) is presented.
\begin{defi} \label{ws00}{\bf (Weak-strong solutions of (\ref{modelf00}))}
Given $(u_0, v_0)\in L^2(\Omega)\times H^1(\Omega)$ with $u_0\geq 0$, $v_0\geq 0$ a.e.~$\boldsymbol x\in \Omega$. A pair $(u,v)$ is called weak-strong solution of problem (\ref{modelf00}) in $(0,+\infty)$, if $u\geq 0$, $v\geq 0$ a.e.~$(t,\boldsymbol x)\in (0,+\infty)\times \Omega$,
\begin{equation}\label{wsa}
(u-m_0,v-m_0^2) \in L^{\infty}(0,+\infty;L^2(\Omega)\times H^1(\Omega))
\cap L^{2}(0,+\infty;H^1(\Omega)\times H^2(\Omega)),
\end{equation}
\begin{equation}\label{wsa-bis}
(\partial_t u, \partial_t v) \in L^{q'}(0,T;H^1(\Omega)' \times L^2(\Omega)), \ \ \forall T>0,
\end{equation}
where $q'=2$ in the $2$-dimensional case $(2D)$ and $q'=4/3$ in the $3$-dimensional case $(3D)$ ($q'$ is the conjugate exponent of $q=2$ in $2D$ and $q=4$ in $3D$); the following variational formulation holds
\begin{equation}\label{wf01}
\int_0^T \langle \partial_t u,\overline{u}\rangle + \int_0^T (\nabla u, \nabla \overline{u}) +\int_0^T (u\nabla v,\nabla \overline{u})=0, \ \ \forall \overline{u}\in L^q(0,T;H^{1}(\Omega)), \ \ \forall T>0,
\end{equation}
the following equation holds pointwisely
\begin{equation}\label{wf02}
\partial_t v +A v=u^2 \ \ \mbox{ a.e. } (t,\boldsymbol x)\in (0,+\infty)\times\Omega,
\end{equation}
the initial conditions $(\ref{modelf00})_4$ are satisfied and the following energy inequality (in integral version) holds a.e.~$t_0,t_1$ with $t_1\geq t_0\geq 0$:
\begin{equation}\label{wsd}
\mathcal{E}(u(t_1),v(t_1)) - \mathcal{E}(u(t_0),v(t_0))
+ \int_{t_0}^{t_1} \left(\Vert \nabla u(s) \Vert_{0}^2
+\frac{1}{2} \Vert \nabla v(s) \Vert_{1}^2
\right)\ ds \leq0,
\end{equation}
where
\begin{equation}\label{eneruva}
\mathcal{E}(u,v)=\displaystyle
\frac{1}{2}\Vert u\Vert_{0}^2 + \frac{1}{4}\Vert \nabla v\Vert_{0}^{2}.
\end{equation}
\end{defi}
\begin{obs}
In particular, the energy inequality (\ref{wsd}) is valid for $t_0=0$. Moreover, (\ref{wsd}) shows the dissipative character of the model with respect to the total energy $\mathcal{E}(u(t), v(t))$.
\end{obs}
\begin{obs} {\bf (Positivity)}\label{OBSP}
$u\geq 0$ in $1D$ and $2D$ domains and $v\geq 0$ in any ($1D$, $2D$ or $3D$) dimension are a consequence of (\ref{wsa})-(\ref{wf02}). Indeed, this follows from the fact that in these cases we can test (\ref{wf01}) by $u_{-}:= \min\{u,0\} \in L^{2}(0,T;H^1(\Omega))$ and (\ref{wf02}) by $v_{-}:= \min\{v,0\}\in L^{2}(0,T;H^2(\Omega))\hookrightarrow L^{2}(0,T;L^2(\Omega))$. Notice that in 3D domains, $u_{-}$ has no the sufficient regularity in order to take it as test function. Hence the positivity of $u$ cannot be deduced from (\ref{wsa})-(\ref{wf01}), which must be explicitly imposed.
\end{obs}
In \cite{FMD}, it was proved the existence of weak-strong solutions of problem (\ref{modelf00}) (satisfying in particular the energy inequality (\ref{wsd})), through convergence of a time-discrete numerical scheme associated to model (\ref{modelf00}). Hereafter, in order to abbreviate, we will use the following notation:
$$\hat{u}:= u - m_0, \ \ \hat{v}=v- m_0^2$$
for $m_0$ defined in (\ref{m0}).
\subsection{Convergence at infinite time}\label{subcc}
In this subsection, the asymptotic analysis of problem (\ref{modelf00}) is going to be analyzed in a formal manner, without justifying the computations and assuming sufficient regularity for the exact solution $(u,v)$. Our main interest is to reproduce the long time behaviour in fully discrete numerical schemes.
\
First, we define:
$$E(t):= \Vert \hat{u}(t)\Vert_0^2 + \frac{1}{2}\Vert \nabla v(t)\Vert_0^2 \ \mbox{ and } \ F(t):= \Vert \nabla \hat{u}(t)\Vert_0^2 + \frac{1}{2}\Vert \nabla v(t)\Vert_1^2.$$ Then, taking $\bar{u}=\hat{u}$ in (\ref{wf01}) and testing (\ref{wf02}) by $\bar{v}=-\frac{1}{2}\Delta v$, one arrives at
\begin{equation}\label{weakcont2}
\frac{1}{2} E'(t)+ F(t) =0.
\end{equation}
Therefore, using the Poincar\'e inequality $\Vert \nabla \hat{u}\Vert_0^2\geq C_p \Vert \hat{u}\Vert_0^2$ one has that $ 2F(t)\geq 2(C_p \Vert \hat{u}(t)\Vert_0^2 + \frac{1}{2}\Vert \nabla v(t)\Vert_1^2)\geq 2K_p E(t)$ (with $K_p=\min\{C_p,1\}$), and from (\ref{weakcont2}) one can deduce
\begin{equation}\label{c21}
E(t)\leq \Vert (\hat{u}_0,\nabla v_0)\Vert_0^2e^{-2K_p t}, \ \ \forall t\geq 0.
\end{equation}
Moreover, testing (\ref{wf02}) by $\bar{v}=\hat{v}$ and using (\ref{wsa}) and (\ref{c21}), one has
$$
\frac{d}{dt}\Vert \hat{v}\Vert_0^2 + \Vert \hat{v}\Vert_1^2 \leq C\Vert \hat{u} \Vert_{0}^2\Vert \hat{u} + 2m_0\Vert_{L^3}^2\leq C e^{-2K_pt}(1+\Vert \hat{u}\Vert_1^2),
$$
from which one arrives at
\begin{equation}\label{newaa}
\Vert \hat{v}(t)\Vert_0^2 \leq \Vert \hat{v}_0\Vert_0^2 e^{-t} + C e^{-t} \int_{0}^{t} e^{(1-2K_p)s} \, ds + C e^{-t} \int_{0}^{t} e^{(1-2K_p)s} \Vert \hat{u}(s)\Vert_1^2 \, ds.
\end{equation}
The last two terms on the right hand side of (\ref{newaa}) are bounded by
\begin{equation}\label{B001}
C e^{-t} \int_{0}^{t} e^{(1-2K_p)s} \, ds\leq
\left\{\begin{array}{l}
C e^{-t} \ \mbox{ if } 2K_p>1,\\
Cte^{-t} \ \mbox{ if } 2K_p=1,\\
C e^{-2K_pt} \ \mbox{ if } 2K_p<1,
\end{array}\right.
\end{equation}
and
\begin{equation}\label{B001-a}
C e^{-t} \int_{0}^{t} e^{(1-2K_p)s} \Vert \hat{u}(s)\Vert_1^2 \, ds\leq
\left\{\begin{array}{l}
C e^{-t} \ \mbox{ if } 2K_p>1,\\
Ce^{-t} \ \mbox{ if } 2K_p=1,\\
C e^{-2K_pt} \ \mbox{ if } 2K_p<1,
\end{array}\right.
\end{equation}
where (\ref{wsa}) was used in (\ref{B001-a}). Thus, from (\ref{newaa})-(\ref{B001-a}) one can deduce that, for any t>1,
\begin{equation*}
\Vert \hat{v}(t)\Vert_0^2 \leq \Vert \hat{v}_0\Vert_0^2 e^{-t} +
\left\{\begin{array}{l}
C e^{-t} \ \mbox{ if } 2K_p>1,\\
Cte^{-t} \ \mbox{ if } 2K_p=1,\\
C e^{-2K_pt} \ \mbox{ if } 2K_p<1,
\end{array}\right.
\leq
C\left\{\begin{array}{l}
e^{-t} \ \mbox{ if } 2K_p>1,\\
te^{-t} \ \mbox{ if } 2K_p=1,\\
e^{-2K_pt} \ \mbox{ if } 2K_p<1.
\end{array}\right.
\end{equation*}
\
\section{Scheme \textbf{UV}}\label{NEs}
The first scheme that will be studied in this paper is obtained by using FE in space and backward Euler in time for the system (\ref{modelf00})
(considered for simplicity on a uniform partition of $[0,+\infty)$ given by $t_n=nk$, where $k>0$ denotes the time step). Concerning the space discretization, we consider a family of shape-regular and quasi-uniform triangulations $\{\mathcal{T}_h\}_{h>0}$ of $\overline{\Omega}$ made up of simplexes
(intervals in one dimension,
triangles in two dimensions and tetrahedra in three dimensions), so that $\overline{\Omega}= \cup_{K\in \mathcal{T}_h} K$, where $h = \max_{K\in \mathcal{T}_h} h_K$, with $h_K$ being the diameter of $K$.
We choose FE spaces for $u$ and $v$, which we denote by
\begin{equation*}
(U_h, V_h) \subset H^1 \times W^{1,6} \mbox{ generated by } (\mathbb{P}_m,\mathbb{P}_{2m})\mbox{-continuous FE, with } m\geq 1.
\end{equation*}
With this choice, $(u_h^n)^2 \in V_h$ is guaranteed, which will be the key point to prove the energy stability of this scheme (see Lemma \ref{estinc1uv} below). Then, the following first order in time, nonlinear and coupled scheme is considered (hereafter, we denote $\delta_t a^n= (a^n - a^{n-1})/k$):
\begin{itemize}
\item{\underline{\emph{Scheme \textbf{UV}:}}\\
{\bf Initialization}: Let $(u^{0}_h,v^0_h)\in U_h\times V_h$ be a suitable approximation of $(u_0,v_0)\in L^2(\Omega) \times H^1(\Omega)$, as $h\rightarrow 0$, with $\displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u^0_h = \displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u_0 = m_0$. \\
{\bf Time step} n: Given $(u^{n-1}_h,v^{n-1}_h)\in U_h\times V_h$, compute $(u^{n}_h,v^{n}_h)\in U_h\times V_h$ solving
\begin{equation}
\left\{
\begin{array}
[c]{lll}%
(\delta_t u^n_h,\bar{u}_h) + (\nabla u^n_h, \nabla \bar{u}_h) +(u^n_h\nabla v^n_h,\nabla \bar{u}_h)=0, \ \ \forall \bar{u}_h\in U_h,\\
(\delta_t v^n_h,\bar{v}_h) +(\nabla v^n_h, \nabla \bar{v}_h) + (v^n_h,\bar{v}_h) -((u^n_h)^2,\bar{v}_h) = 0, \ \ \forall
\bar{v}_h\in V_h.
\end{array}
\right. \label{modelf02uv}
\end{equation}
}
\end{itemize}
\subsection{Mass-conservation, well-posedness, energy-stability and convergence}\label{EESuv}
In this subsection, we follow the arguments presented in \cite{FMD2}. Consequently, the results will be presented omiting technical details. Assuming that $1\in U_h$ and $1\in V_h$, the scheme \textbf{UV} satisfies
\begin{equation}\label{consuuv}
\int_\Omega u^n_h=\int_\Omega u^{n-1}_h=\cdot\cdot\cdot=\int_\Omega
u^{0}_h=m_0\vert \Omega\vert,
\end{equation}
and
\begin{equation}\label{compv1uv}
\delta_t \left(\int_\Omega v^n_h \right)= \int_\Omega (u^n_h)^2 - \int_\Omega v^n_h.
\end{equation}
\begin{tma} {\bf(Unconditional solvability and conditional uniqueness)}
There exists \linebreak{$(u^n_h,v^n_h) \in U_h\times V_h$} solution of the scheme \textbf{UV}. Moreover, if
\begin{equation*}
k\Vert (u^n_h,\nabla v^n_h)
\Vert_{1}^4 \quad \hbox{is small enough,}
\end{equation*}
then the solution is unique.
\end{tma}
\begin{proof}
The proof follows the arguments of Theorem 4.4 of \cite{FMD}.
\end{proof}
Let $A_h: H^1(\Omega) \rightarrow V_h$ be the linear operator defined as follows
\begin{equation}\label{discVh}
(A_h v_h, \bar{v}_h)=(\nabla v_h,\nabla \bar{v}_h)+( v_h, \bar{v}_h) , \ \ \forall \bar{v}_h\in V_h.
\end{equation}
Then, the discrete chemical equation (\ref{modelf02uv})$_2$ can be rewritten as
\begin{equation}\label{refvh}
(\delta_t v^n_h,\bar{v}_h) +(A_h v^n_h, \bar{v}_h) -((u^n_h)^2,\bar{v}_h) = 0, \ \ \forall
\bar{v}_h\in V_h,
\end{equation}
and the following estimate holds (see for instance, Lemma 3.1 in \cite{FMD2}):
\begin{equation}\label{Ah1}
\Vert v_h \Vert_{W^{1,6}}\leq C \Vert A_h v_h\Vert_0, \ \ \forall v_h\in V_h.
\end{equation}
\begin{defi}\label{enesf00}
A numerical scheme with solution $(u^n_h,v^n_h)$ is called energy-stable if the energy defined in (\ref{eneruva}) is time decreasing, that is,
\begin{equation*}
\mathcal{E}(u^n_h,v^n_h)\leq \mathcal{E}(u^{n-1}_h,v^{n-1}_h), \ \ \forall n\geq 1.
\end{equation*}
\end{defi}
\begin{lem} {\bf (Unconditional stability)} \label{estinc1uv}
If $(u_h^n,v_h^n)$ is generated by $(\mathbb{P}_m,\mathbb{P}_{2m})$-continuous FE, then the scheme \textbf{UV} is unconditionally energy-stable. In fact, if $(u^n_h,v^n_h)$ is any solution of the scheme \textbf{UV}, then the following discrete energy law holds
\begin{eqnarray}\label{lawenerfydisceuv}
&\delta_t \mathcal{E}(\hat{u}^n_h,v^n_h)&\!\!\!\!\!+
\frac{k}{2}
\Vert \delta_t \hat{u}^n_h\Vert_{0}^2 +
\frac{k}{4} \Vert \delta_t \nabla v^n_h\Vert_{0}^2
+ \Vert \hat{u}^n_h\Vert_{1}^{2} +
\displaystyle\frac{1}{2}\Vert (A_h-I) v^n_h\Vert_{0}^{2} +
\displaystyle\frac{1}{2}\Vert \nabla v^n_h\Vert_{0}^{2}=0.
\end{eqnarray}
\end{lem}
\begin{proof}
Taking $\bar{u}_h=\hat{u}^n_h$ in (\ref{modelf02uv})$_1$, $\bar{v}_h= \displaystyle\frac{1}{2}(A_h -I) v^n_h$
in (\ref{refvh}) and using (\ref{discVh}), (\ref{lawenerfydisceuv}) is deduced.
\end{proof}
From the (local in time) discrete energy law (\ref{lawenerfydisceuv}), we deduce the following global in time estimates.
\begin{lem} {\bf(Uniform weak-strong estimates)}
Let $(u^n_h,v^n_h)$ be any solution of the scheme \textbf{UV}. Then, the following estimate holds
\begin{equation}\label{weak01uv}
\Vert (\hat{u}^n_h, v^n_h)\Vert_{0\times 1}^{2}
+ k \underset{m=1}{\overset{n}{\sum}}\left(\Vert \hat{u}^m_h \Vert_{1}^2 + \Vert \hat{v}^m_h \Vert_{W^{1,6}}^2\right) \leq C_0, \ \ \ \forall n\geq 1.
\end{equation}
\end{lem}
\begin{proof}
Multiplying (\ref{lawenerfydisceuv}) by $k$ and summing, one obtains
\begin{equation}\label{weak01uv-a}
\Vert (\hat{u}^n_h, \nabla v^n_h)\Vert_{0}^{2}
+ k \underset{m=1}{\overset{n}{\sum}}\left(\Vert \hat{u}^m_h \Vert_{1}^2 + \Vert \nabla v^m_h \Vert_{0}^2 + \Vert (A_h-I) v^m_h \Vert_{0}^2\right) \leq C_0, \ \ \ \forall n\geq 1.
\end{equation}
On the other hand, rewriting (\ref{modelf02uv}) as
\begin{equation}\label{vm0}
(\delta_t \hat{v}^n_h,\bar{v}_h) +( {A}_h \hat {v}^n_h,\bar{v}_h) =((\hat{u}^n_h+2 m_0)\hat{u}^n_h,\bar{v}_h), \ \ \forall
\bar{v}_h\in V_h,
\end{equation}
and taking $\bar{v}=\hat{v}^n_h$ one has
\begin{eqnarray*}
&\displaystyle\delta_t \Vert \hat{v}^n_h \Vert_0^2 &\!\!\!\!
+ \Vert \hat{v}^n_h\Vert_1^2\leq C\Vert \hat{u}^n_h + 2 m_0\Vert_{L^{3/2}}^2 \Vert \hat{u}^n_h\Vert_{L^6}^2\leq C \Vert \hat{u}^n_h\Vert_{H^1}^2,
\end{eqnarray*}
from which, multiplying by $k$, adding and using (\ref{weak01uv-a}), one can deduce
\begin{equation}\label{weak02UVlinvL2-a}
\Vert v^n_h \Vert_0^2 + k \underset{m=1}{\overset{n}{\sum}}\Vert \hat{v}^m_h \Vert_1^2 \leq K_0, \ \ \ \forall n\geq 1.
\end{equation}
Then, adding (\ref{weak01uv-a}) and (\ref{weak02UVlinvL2-a}) and using (\ref{Ah1}), (\ref{weak01uv}) is obtained.
\end{proof}
Starting from the previous stability estimates, the convergence towards weak solutions of (\ref{modelf00}) can be proved. Concretely, by introducing the functions:
\begin{itemize}
\item $(\widetilde{u}_{h,k},\widetilde{v}_{h,k})$ are continuous functions on $[0,+\infty)$, linear on each interval $(t_n,t_{n+1})$ and equal to $(u^n_h,{v}^n_h)$ at $t=t_n$, $n\geq 0$;
\item $({u}_{h,k},{v}_{h,k})$ are the piecewise constant functions taking values $(u^{n}_h,{v}^n_h)$ on $(t_{n-1},t_n]$, $n\geq 1$,
\end{itemize}
the following result holds:
\begin{tma} {\bf (Convergence)}
There exist a subsequence $(k',h')$ of $(k,h)$, with $k',h'\downarrow 0$, and a weak-strong solution $(u,v)$ of (\ref{modelf00}) in $(0,+\infty)$, such that $(\widetilde{ u}_{h',k'}-m_0,\widetilde{v}_{h',k'}-m_0^2)$ and $(u_{h',k'}-m_0,v_{h',k'}-m_0^2)$ converge to $(u-m_0,v-m_0^2)$ weakly-$\star$ in $L^\infty(0,+\infty;L^2(\Omega)\times H^1(\Omega))$, weakly in $L^2(0,+\infty;H^1(\Omega)\times W^{1,6}(\Omega))$ and strongly in $L^2(0,T;L^2(\Omega)\times L^p(\Omega)) \cap C([0,T];H^1(\Omega)' \times L^q(\Omega))$, for any $T>0$, $1\leq p<+\infty$ and $1\leq q<6$.
\end{tma}
\begin{obs}
Note that, since the positivity of $u^n_h$ cannot be assured, then the positivity of the limit function $u$ cannot be proven in the 3D case (see Remark \ref{OBSP}).
\end{obs}
\begin{proof}
Proceeding as in Theorem 4.11 of \cite{FMD} (whose proof follows the arguments of \cite{Tem}), one can prove that there exist a subsequence $(k',h')$ of $(k,h)$, with $k',h'\downarrow 0$, and $(u,v)$ satisfying (\ref{wf01}), (\ref{wf02}) and the initial conditions (\ref{modelf00})$_4$, such that $(\widetilde u_{h',k'}-m_0,\widetilde{v}_{h',k'}-m_0^2)$ and $(u_{h',k'}-m_0,v_{h',k'}-m_0^2)$ converge to $(u-m_0,v-m_0)$ weakly-* in $L^\infty(0,+\infty;L^2(\Omega)\times H^1(\Omega))$, weakly in $L^2(0,+\infty;H^1(\Omega)\times W^{1,6}(\Omega))$ and strongly in $L^2(0,T;L^2(\Omega)\times L^p(\Omega)) \cap C([0,T];H^1(\Omega)' \times L^q(\Omega))$, for any $T>0$, $1\leq p<+\infty$ and $1\leq q<6$. Moreover, it holds
\begin{eqnarray*}
&\displaystyle\frac{d}{dt}\left( \frac{1}{2}\Vert \widetilde{u}_{k',h'}(t) \Vert_0^2 + \frac{1}{4} \Vert\nabla \widetilde{v}_{k',h'}(t)\Vert_0^2\right)&\!\!\! + \frac{(t_n - t)}{2}\Vert (\delta_t u_n ,\delta_t \nabla {v}_n) \Vert_0^2 \nonumber\\
&&\hspace{-2 cm} + \Vert \nabla u_{k',h'}(t)\Vert_{0}^{2} +
\displaystyle\frac{1}{2}\Vert (A_h - I) {v}_{k',h'}(t)\Vert_{0}^{2}+
\displaystyle\frac{1}{2}\Vert \nabla {v}_{k',h'}(t)\Vert_{0}^{2}= 0.
\end{eqnarray*}
In order to obtain that $(u,v)$ satisfies the energy inequality (\ref{wsd}), it is necessary to prove that
\begin{equation}\label{uny}
\underset{(k',h')\rightarrow (0,0)}{\lim \mbox{inf} } \int_{t_0}^{t_1} \Vert(A_h - I) v_{k',h'}(t)\Vert_{0}^{2} \geq \int_{t_0}^{t_1} \Vert\Delta v(t)\Vert_{0}^{2}.
\end{equation}
Taking into account that $\{(A_h - I) v_{k',h'}\}$ is bounded in $L^2(0,T;L^2(\Omega))$, one has that there exists $w\in L^2(0,T;L^2(\Omega)$ such that for some subsequence of $(k',h')$, still denoted by $(k',h')$,
\begin{equation}\label{ca3}
(A_h - I) v_{k',h'} \rightarrow w \ \ \mbox { weakly in } \ L^2(0,T;L^2(\Omega).
\end{equation}
Since $u^2\in L^2(0,T;L^{3/2}(\Omega))\hookrightarrow L^2(0,T;H^{1}(\Omega)')$, one has
\begin{equation}\label{ca4}
\partial_t v - \Delta v +v=u^2 \ \ \mbox{ in } L^2(H^1)',
\end{equation}
and, on the other hand, using (\ref{ca3}), one can deduce
\begin{equation}\label{ca5}
\partial_t v +w +v=u^2 \ \ \mbox{ in } L^2(H^1)'.
\end{equation}
Thus, from (\ref{ca4})-(\ref{ca5}), one can deduce that $w=-\Delta v$ in $\mathcal{D}'(\Omega)$, which implies $-\Delta v \in L^2(0,T;L^2(\Omega))$ because of $w\in L^2(0,T;L^2(\Omega)$. Therefore, $(u,v)$ satisfies the regularity (\ref{wsa}) and taking into account (\ref{ca3}), (\ref{uny}) is concluded. Finally, using (\ref{uny}) and arguing as in the last part of the proof of Theorem 4.11 of \cite{FMD}, it can be obtained that $(u,v)$ satisfies the energy inequality (\ref{wsd}), and therefore, $(u,v)$ is a weak-strong solution of (\ref{modelf00}).
\end{proof}
\subsection{Large-time behavior of the scheme UV}
In this subsection, exponential bounds for any solution $(u_h^n,v_h^n)$ of the scheme \textbf{UV} in weak-strong norms are proved.
\begin{tma}\label{CoUV}
Let $(u_h^n,v^n_h)$ be a solution of the scheme \textbf{UV} associated to an initial data $(u^{0}_h,v^0_h)$, with $\displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u^0_h = \displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u_0 = m_0$. Then,
\begin{equation}\label{LTuv1}
\displaystyle\Vert (\hat{u}^n_h,\nabla v^n_h)\Vert_{0}^{2}\leq C_0(1+2K_p k)^{-n}, \ \ \forall n\geq 0,
\end{equation}
\begin{equation}\label{LTuv2}
\displaystyle\Vert \hat{v}^n_h \Vert_{0}^{2}\leq
\left\{\begin{array}{l}
C (1+ k)^{-n} \ \mbox{ if } 2K_p>1,\\
C(kn) (1+ k)^{-n} \ \mbox{ if } 2K_p=1,\\
C (1+ 2K_p k)^{-n} \ \mbox{ if } 2K_p<1,
\end{array}\right.
\end{equation}
where the constant $K_p>0$ was defined in Subsection \ref{subcc}.
\end{tma}
\begin{proof}
Taking $\bar{u}_h=\hat{u}^n_h$ in (\ref{modelf02uv})$_1$, $\bar{v}_h= \displaystyle\frac{1}{2}(A_h -I) v^n_h$
in (\ref{refvh}) and using (\ref{consuuv}) and (\ref{discVh}), one obtains
\begin{eqnarray}\label{Com1}
&\delta_t \Big( \displaystyle
\frac{1}{2}\Vert \hat{u}^n_h\Vert_{0}^2 + \frac{1}{4}\Vert \nabla v^n_h\Vert_{0}^{2} \Big)&\!\!\!\!\!+
\frac{k}{2}
\Vert \delta_t \hat{u}^n_h\Vert_{0}^2 +
\frac{k}{4} \Vert \delta_t \nabla v^n_h\Vert_{0}^2 \nonumber\\
&&\!\!\!
+ \Vert \hat{u}^n_h\Vert_{1}^{2} +
\displaystyle\frac{1}{2}\Vert (A_h-I) v^n_h\Vert_{0}^{2} +
\displaystyle\frac{1}{2}\Vert \nabla v^n_h\Vert_{0}^{2}=0.
\end{eqnarray}
To get (\ref{Com1}), the fact that $(u^n_h)^2 \in V_h$ is essential (which comes from the choice $(\mathbb{P}_m,\mathbb{P}_{2m})$ approximation for $(U_h,V_h)$) in order to cancel the terms $(u^n_h\nabla v^n_h,\nabla \hat{u}^n_h)$ and $-\frac{1}{2}((u^n_h)^2,(A_h -I) v^n_h)$. Then, from (\ref{Com1}) one arrives at
\begin{equation*}
(1+2K_p k) \Big( \displaystyle\Vert \hat{u}^n_h\Vert_{0}^2 + \frac{1}{2} \Vert \nabla v^n_h\Vert_{0}^{2} \Big) - \Big( \Vert \hat{u}^{n-1}_h\Vert_{0}^2 + \frac{1}{2} \Vert \nabla v^{n-1}_h\Vert_{0}^{2} \Big) \leq 0,
\end{equation*}
from which, multiplying by $(1+2K_p k)^{n-1}$ and summing, one has for all $n\geq 0$,
\begin{equation}\label{e33}
\displaystyle\Vert \hat{u}^n_h\Vert_{0}^2 + \frac{1}{2} \Vert \nabla v^n_h\Vert_{0}^{2}\leq (1+2K_p k)^{-n} \Big( \Vert \hat{u}^0_h\Vert_{0}^2 + \frac{1}{2} \Vert \nabla v^0_h\Vert_{0}^{2} \Big)
\end{equation}
and (\ref{LTuv1}) is obtained. Moreover, taking $\bar{v}_h= \hat{v}^n_h$ in (\ref{vm0}), one has
$$
\frac{1}{2} \delta_t \Vert \hat{v}^n_h\Vert_0^2 + \Vert \hat{v}^n_h\Vert_1^2 =\int_\Omega (\hat{u}^n_h + 2m_0)\hat{u}^n_h \hat{v}^n_h,
$$
which, using the H\"older and Young inequalities, implies that
\begin{equation}\label{Com4}
(1+ k) \Vert \hat{v}^n_h\Vert_0^2 - \Vert \hat{v}^{n-1}_h\Vert_0^2\leq kC \Vert \hat{u}^n_h +2m_0\Vert_{L^3}^2\Vert \hat{u}^n_h \Vert_0^2.
\end{equation}
Then, multiplying (\ref{Com4}) by $(1+ k)^{n-1}$, summing and using (\ref{LTuv1}), one obtains
\begin{equation}\label{Newv1}
(1+ k)^{n}\Vert \hat{v}^n_h\Vert_0^2\leq \Vert \hat{v}^0_h\Vert_0^2 + \frac{C}{1+2K_pk} k \underset{m=1}{\overset{n}{\sum}} \left(\frac{1 +k}{1+2K_pk}\right)^{m-1}(1 +\Vert \hat{u}^m_h\Vert^2_1).
\end{equation}
Then, in order to obtain (\ref{LTuv2}) we split the argument in three cases:
\begin{enumerate}
\item Case 1: If $2K_p= 1$, using (\ref{weak01uv}) in (\ref{Newv1}) one has that for any $t_n=nk>1$,
\begin{equation}\label{Newv2}
\Vert \hat{v}^n_h\Vert_0^2\leq (1+ k)^{-n} (C+ C(kn))\leq C(kn) (1+ k)^{-n}.
\end{equation}
\item Case 2: If $2K_p>1$, using (\ref{weak01uv}) in (\ref{Newv1}) one obtains
\begin{equation}\label{Newv3}
\Vert \hat{v}^n_h\Vert_0^2\leq (1+ k)^{-n}\!\left(C_0 + \frac{C}{2K_p-1}\!\left[1 \!-\!\left(\frac{1+k}{1+2K_pk}\right)^n\right] +\frac{C}{1+2K_pk} \right)\leq C(1+ k)^{-n}.
\end{equation}
\item Case 3: If $2K_p<1$, one rewrites (\ref{Newv1}) as
\begin{equation*}
(1+ 2K_p k)^{n}\Vert \hat{v}^n_h\Vert_0^2\leq \left(\frac{1+ 2K_p k}{1+k}\right)^{n}\Vert \hat{v}^0_h\Vert_0^2 + \frac{C}{1+2K_pk} k \underset{m=1}{\overset{n}{\sum}} \left(\frac{1 +2K_p k}{1+k}\right)^{n-m+1}\!\!(1 +\Vert \hat{u}^m_h\Vert^2_1),
\end{equation*}
and proceeding as in (\ref{Newv3}), taking into account that $\frac{1+ 2K_p k}{1+k}<1$, one arrives at
\begin{equation}\label{Newv4}
\Vert \hat{v}^n_h\Vert_0^2\leq C(1+2K_p k)^{-n}.
\end{equation}
\end{enumerate}
Therefore, from (\ref{Newv2})-(\ref{Newv4}), (\ref{LTuv2}) is deduced.
\end{proof}
\begin{cor}
Under conditions of Theorem \ref{CoUV}, the following estimates hold
\begin{equation*}
\displaystyle\Vert (\hat{u}^n_h,\nabla v^n_h)\Vert_{0}^{2}\leq C_0e^{- \frac{2K_p}{1+2K_pk}kn}, \ \ \forall n\geq 0,
\end{equation*}
\begin{equation*}\label{LTuv2-cor}
\displaystyle\Vert \hat{v}^n_h \Vert_{0}^{2}\leq
\left\{\begin{array}{l}
C e^{- \frac{1}{1+k}kn} \ \ \mbox{ if } 2K_p>1,\\
C(kn) e^{- \frac{1}{1+k}kn} \ \mbox{ if } 2K_p=1,\\
C e^{- \frac{2K_p}{1+2K_p k}kn} \ \mbox{ if } 2K_p<1.
\end{array}\right.
\end{equation*}
\end{cor}
\begin{proof}
Using the inequality $1-x\leq e^{-x}$ for all $x\geq 0$, from (\ref{LTuv1}) one has
\begin{equation}\label{cnewu}
\displaystyle\Vert (\hat{u}^n_h,\nabla v^n_h)\Vert_{0}^{2}\leq C_0(1+2K_p k)^{-n}\leq C_0\Big(1 - \frac{2K_p}{1+2K_p k}k\Big)^{n}\leq C_0e^{- \frac{2K_p}{1+2K_p k}kn}.
\end{equation}
Analogously, (\ref{LTuv2}) can be deduced.
\end{proof}
\section{Scheme US$_\varepsilon$}\label{schemeUSe}
Up to our knowledge, there is not previous works studying FE schemes for model (\ref{modelf00}), with positive or approximately positive discrete solutions. In fact, for the scheme \textbf{UV} analyzed in this paper or the scheme \textbf{US} studied in \cite{FMD2}, it is not clear how to prove any of these properties. For this reason, in this section we propose an unconditionally energy-stable scheme with the property of ``approximated positivity''; this scheme is constructed as a modification of the scheme \textbf{US} (\cite{FMD2}), by introducing the auxiliary variable ${\boldsymbol{\sigma}}=\nabla v$ and applying a regularization procedure.
\
We consider a fully discrete approximation using FE in space and backward Euler in time for a reformulated problem in $(u,{\boldsymbol{\sigma}})$-variables. Moreover, in this case we will assume the following hypothesis on the space discretization:
\begin{enumerate}
\item[({\bf H})]{The triangulation is structured in the sense that all simplices have a right angle.}
\end{enumerate}
We choose the following continuous FE spaces for $u$, ${\boldsymbol{\sigma}}$ and $v$:
$$ (U_h, {\boldsymbol\Sigma}_h, V_h) \subset H^1 \times \mathbf{H}^1_{\sigma}\times W^{1,6} \quad \hbox{generated by $\mathbb{P}_1$-continuous FE.}
$$
\begin{obs}
The right angled requirement and the choice of $\mathbb{P}_1$-continuous FE for $U_h$ are nece\-ssa\-ry in order to obtain the relation (\ref{PL1}) below, which is essential in order to obtain the approximated positivity (see Theorem \ref{AAPP} below).
\end{obs}
We consider the Lagrange interpolation operator $\Pi^h: C(\overline{\Omega})\rightarrow U_h$, and we introduce the discrete semi-inner product on $C(\overline{\Omega})$ (which is an inner product in $U_h$) and its induced discrete seminorm (norm in $U_h$):
\begin{equation*}
(u_1,u_2)^h:=\int_\Omega \Pi^h (u_1 u_2), \ \vert u \vert_h=\sqrt{(u,u)^h}.
\end{equation*}
\begin{obs}\label{eqh2}
In $U_h$, the norms $\vert \cdot\vert_h$ and $\Vert \cdot\Vert_0$ are equivalents uniformly with respect to $h$ (see \cite{PB}).
\end{obs}
We consider also the $L^2$-projection $Q^h:L^2(\Omega)\rightarrow U_h$ given by
\begin{equation*}
(Q^h u,\bar{u})^h=(u,\bar{u}), \ \ \forall \bar{u}\in U_h,
\end{equation*}
the standard $L^2$-projection $\widetilde{Q}^h:L^2(\Omega)\rightarrow {\boldsymbol\Sigma}_h$. Moreover, following the ideas of Barrett and Blowey \cite{BB}, we consider the truncated function $\lambda_\varepsilon:\mathbb{R}\rightarrow [\varepsilon,\varepsilon^{-1}]$ (with $\varepsilon\in (0,1)$) given by
\begin{equation*}
\lambda_\varepsilon(s)\ := \
\left\{\begin{array}{lcl}
\varepsilon & \mbox{ if } & s\leq \varepsilon,\\
s & \mbox{ if } & \varepsilon\leq s\leq \varepsilon^{-1},\\
\varepsilon^{-1} & \mbox{ if } & s\geq \varepsilon^{-1}.
\end{array}\right.
\end{equation*}
If we define
\begin{equation} \label{F2pE}
F''_\varepsilon(s):= \frac{1}{\lambda_\varepsilon(s)},
\end{equation}
then, we can integrate twice in (\ref{F2pE}), imposing the conditions $F'_\varepsilon(1)=F_\varepsilon(1)=0$, and we obtain a convex function $F_\varepsilon: \mathbb{R}\rightarrow [0,+\infty)$, such that $F_\varepsilon \in C^{2}(\mathbb{R})$. Even more, for $\varepsilon\in (0,e^{-2})$, it holds (see \cite{BB})
\begin{equation}\label{PNa}
F_\varepsilon(s)\geq \frac{\varepsilon}{2}s^2 - 2\ \ \forall s\geq 0 \ \ \mbox{ and } \ \ F_\varepsilon(s)\geq \frac{s^2}{2\varepsilon} \ \ \forall s\leq 0.
\end{equation}
Then, for each $\varepsilon\in (0,1)$ we consider the construction of the operator $\Lambda_\varepsilon: U_h\rightarrow L^\infty(\Omega)^{d\times d}$ given in \cite{BB}, satisfying that $\Lambda_\varepsilon u^h$ is a piecewise constant matrix for all $u^h\in U_h$, such that the following relation holds
\begin{equation}\label{PL1}
(\Lambda_\varepsilon u^h) \nabla \Pi^h (F'_\varepsilon(u^h))=\nabla u^h \ \ \mbox{ in } \Omega.
\end{equation}
Basically, $\Lambda_\varepsilon u^h$ is a constant by elements symmetric and positive definite matrix such that (\ref{PL1}) holds by elements. We highlight that (\ref{PL1}) is satisfied due to the right angled constraint requirement ({\bf H}) and the choice of $\mathbb{P}_1$-continuous FE for $U_h$. We recall the result below concerning to $\Lambda_\varepsilon(\cdot)$ (see \cite[Lemma 2.1]{BB}).
\begin{lem}\label{lemconv}
Let $\Vert \cdot \Vert$ denote the spectral norm on $\mathbb{R}^{d\times d}$. Then for any given $\varepsilon\in (0,1)$ the function $\Lambda_\varepsilon:U_h\rightarrow [L^\infty(\Omega)]^{d\times d}$ is continuous and satisfies
\begin{equation}\label{D}
\varepsilon \xi^T \xi \leq \xi^T \Lambda_\varepsilon(u^h) \xi \leq \varepsilon^{-1} \xi^T \xi, \ \ \forall \xi \in \mathbb{R}^d, \ \forall u^h\in U_h.
\end{equation}
\end{lem}
Then, the following first order in time, nonlinear and coupled scheme is considered:
\begin{itemize}
\item{\underline{\emph{Scheme \textbf{US}$_\varepsilon$}:}\\
{\bf Initialization}: Let $(u^{0}_h,{\boldsymbol \sigma}_h^{0},{v}_h^{0})=(Q^h u_0,\widetilde{Q}^h (\nabla v_0),Q^h {v}_{0})\in U_h\times {\boldsymbol\Sigma}_h\times V_h$. Then, $\displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u^0_h = \displaystyle\frac{1}{\vert\Omega\vert}\int_\Omega u_0 = m_0$, $u^0_h\geq 0$ and $v^0_h\geq 0$.\\
{\bf Time step} n: Given $(u^{n-1}_\varepsilon,{\boldsymbol \sigma}^{n-1}_\varepsilon)\in U_h\times {\boldsymbol\Sigma}_h$, compute $(u^{n}_\varepsilon,{\boldsymbol \sigma}^{n}_\varepsilon)\in U_h \times {\boldsymbol\Sigma}_h$ solving
\begin{equation}
\left\{
\begin{array}
[c]{lll}%
(\delta_t u^n_\varepsilon,\bar{u})^h + (\nabla u^n_\varepsilon,\nabla \bar{u}) + (\Lambda_\varepsilon (u^{n}_\varepsilon){\boldsymbol\sigma}^n_\varepsilon,\nabla \bar{u})= 0, \ \ \forall \bar{u}\in U_h,\\
(\delta_t {\boldsymbol \sigma}^n_\varepsilon,\bar{\boldsymbol \sigma}) + ( B_h {\boldsymbol \sigma}^n_\varepsilon,\bar{\boldsymbol \sigma}) =
2(\Lambda_\varepsilon (u^{n}_\varepsilon) \nabla u^n_\varepsilon,\bar{\boldsymbol \sigma}),\ \ \forall
\bar{\boldsymbol \sigma}\in \Sigma_h,
\end{array}
\right. \label{modelf02a}
\end{equation}}
where the linear operator $B_h:{\boldsymbol\Sigma}_h \rightarrow {\boldsymbol\Sigma}_h$ is defined as
$$( B_h {\boldsymbol \sigma}^n_\varepsilon,\bar{\boldsymbol \sigma}) =(\nabla \cdot {\boldsymbol \sigma}^n_\varepsilon,\nabla \cdot \bar{\boldsymbol\sigma}) + (\mbox{rot }{\boldsymbol \sigma}^n_\varepsilon,\mbox{rot }\bar{\boldsymbol\sigma}) + ({\boldsymbol \sigma}^n_\varepsilon,\bar{\boldsymbol\sigma}), \ \ \forall \bar{\boldsymbol\sigma} \in {\boldsymbol\Sigma}_h.$$
\end{itemize}
Once the scheme \textbf{US} is solved, given $v^{n-1}_\varepsilon\in V_h$, we can recover $v^n_\varepsilon=v^n_\varepsilon((u^n_\varepsilon)^2) \in V_h$ solving:
\begin{equation}\label{edovfUSe}
(\delta_t v^n_\varepsilon,\bar{v})^h +(\nabla v^n_\varepsilon,\nabla \bar{v}) + (v^n_\varepsilon,\bar{v})^h =((u^n_\varepsilon)^2,\bar{v}), \ \ \forall
\bar{v}\in V_h.
\end{equation}
\subsection{Mass-conservation, well-posedness and energy-stability}\label{ELus}
In this subsection, we are going to present some properties of the scheme \textbf{US}$_\varepsilon$, whose proofs follow the ideas of the scheme \textbf{US} studied in \cite{FMD2}. We highlight that these properties can be obtained independently of the choice of $\Lambda_\varepsilon(u^n_\varepsilon)$ approximating $u^n_\varepsilon$.
\
Since $\bar{u}=1\in U_h$ and $\bar{v}=1 \in V_h$, then the scheme \textbf{US}$_\varepsilon$ is conservative in $u^n_\varepsilon$, that is,
\begin{equation}\label{conu1N}
(u_\varepsilon^n,1)=(u^n_\varepsilon,1)^h= (u^{n-1}_\varepsilon,1)^h=\cdot\cdot\cdot= (u^{0}_h,1)^h=(u_h^0,1)=(Q^hu_0,1)=(u_0,1)=m_0\vert \Omega\vert,
\end{equation}
and also has the behavior for $\int_\Omega v^n_\varepsilon$ given in \eqref{compv1uv} (with $u^n_\varepsilon$ and $v^n_\varepsilon$ instead of $u^n_h$ and $v^n_h$ respectively).
\
In the following results we stablish the well-posedness of problems (\ref{modelf02a}) and (\ref{edovfUSe}).
\begin{tma} {\bf (Unconditional solvability and conditional uniqueness of (\ref{modelf02a}))}
There exists at least one solution $(u^n_\varepsilon,{\boldsymbol\sigma}^n_\varepsilon)$ of scheme \textbf{US}$_\varepsilon$. Moreover, if $k\, f(h,\varepsilon)<1$ (where $f(h,\varepsilon)\uparrow +\infty$ when $h\downarrow 0$ or $\varepsilon\downarrow 0$), then the solution $(u^n_\varepsilon,{\boldsymbol \sigma}_\varepsilon^n)$ of the scheme \textbf{US}$_\varepsilon$ is unique.
\end{tma}
\begin{proof}
The proof of solvability follows as in Theorem 3.13 (see also Theorem 4.6) of \cite{GRR1}, and the uniqueness
follows as in Lemma 3.14 (see also Lemma 4.7) of \cite{GRR1}.
\end{proof}
\begin{lem}{\bf (Well-posedness of (\ref{edovfUSe}))}
Given $u^n_\varepsilon\in U_h$ and $v^{n-1}_\varepsilon\in V_h$, there exists a unique $v^n_\varepsilon \in V_h$ solution of (\ref{edovfUSe}).
\end{lem}
\begin{proof}
The proof follows from Lax-Milgram theorem.
\end{proof}
\begin{defi}\label{enesf00}
A numerical scheme with solution $(u^n_\varepsilon,{\boldsymbol{\sigma}}^n_\varepsilon)$ is called energy-stable if the energy
\begin{equation}\label{nueva-2}
\widetilde{\mathcal{E}}(u,{\boldsymbol \sigma})=\frac{1}{2}\Vert u\Vert_{0}^2 + \frac{1}{4}\Vert {\boldsymbol
\sigma}\Vert_{0}^{2}
\end{equation}
is time decreasing, that is,
\begin{equation}\label{stabf0200}
\widetilde{\mathcal{E}}(u^n_\varepsilon,{\boldsymbol \sigma}^n_\varepsilon)\leq \widetilde{\mathcal{E}}(u^{n-1}_\varepsilon,{\boldsymbol \sigma}^{n-1}_\varepsilon), \ \ \forall n\geq 1.
\end{equation}
\end{defi}
\begin{tma} {\bf (Unconditional stability)} \label{estinc1us}
The scheme \textbf{US}$_\varepsilon$ is unconditionally energy stable with respect to the modified energy $\widetilde{\mathcal{E}}(u,{\boldsymbol\sigma})$ given in (\ref{nueva-2}). In fact, if $(u^n_\varepsilon,{\boldsymbol\sigma}^n_\varepsilon)$ is a solution of \textbf{US}$_\varepsilon$, then the following discrete energy law holds
\begin{equation}\label{delus}
\delta_t \widetilde{\mathcal{E}}(\hat{u}^n_\varepsilon,{\boldsymbol\sigma}^n_\varepsilon) +\frac{k}{2}\Vert \delta_t \hat{u}^n_\varepsilon\Vert_0^2 + \frac{k}{4} \Vert \delta_t {\boldsymbol\sigma}^n_\varepsilon\Vert_0^2 + \Vert \hat{u}^n_\varepsilon \Vert_1^2 +\frac{1}{2}\Vert {\boldsymbol\sigma}^n_\varepsilon\Vert_1^2 \leq 0.
\end{equation}
\end{tma}
\begin{proof}
Testing (\ref{modelf02a})$_1$ by $\bar{u}= \hat{u}^n_\varepsilon$, (\ref{modelf02a})$_2$ by $\bar{\boldsymbol\sigma}=\frac{1}{2}{\boldsymbol\sigma}^n_\varepsilon$ and adding, the terms
$(\Lambda_\varepsilon (u^{n}_\varepsilon) \nabla \hat{u}_\varepsilon^n,{\boldsymbol \sigma}^n_\varepsilon)$ cancel, and taking into account Remark \ref{eqh2}, (\ref{delus}) is obtained.
\end{proof}
From (\ref{delus}), multiplying by $k$ and summing, one can deduce the following global energy law:
\begin{cor} \label{welemUS} {\bf(Global energy law) }
Assume that $(u_0,v_0)\in L^2(\Omega)\times H^1(\Omega)$. Let $(u^n_\varepsilon,{\boldsymbol\sigma}^n_\varepsilon)$ be any solution of scheme \textbf{US}$_\varepsilon$. Then, the following estimate holds
\begin{equation*}\label{gest}
\Vert (\hat{u}^n_\varepsilon, {\boldsymbol
\sigma}^n_\varepsilon)\Vert_{0}^{2}
+ k \underset{m=1}{\overset{n}{\sum}}\Vert (\hat{u}^m_\varepsilon, {\boldsymbol \sigma}^m_\varepsilon)\Vert_{1}^2\leq C_0, \ \ \ \forall n\geq 1.
\end{equation*}
\end{cor}
\subsection{Large-time behavior of scheme \textbf{US}$_\varepsilon$}
\begin{tma}\label{LTBusEPS}
Let $(u^n_\varepsilon,{\boldsymbol \sigma}^n_\varepsilon)$ be any solution of the scheme \textbf{US}$_\varepsilon$. Then, the following estimate holds
\begin{equation}\label{LTusEPS}
\displaystyle\Vert (\hat{u}^n_\varepsilon,{ \boldsymbol \sigma}^n_\varepsilon)\Vert_{0}^{2}\leq C_0e^{- \frac{2K_p}{1+2K_p k}kn}, \ \ \forall n\geq 0,
\end{equation}
where the constant $K_p>0$ was defined in Subsection \ref{subcc}.
\end{tma}
\begin{proof}
Taking $\bar{u}_h=\hat{u}^n_\varepsilon$ in (\ref{modelf02a})$_1$, $\bar{\boldsymbol \sigma}_h= \displaystyle\frac{1}{2}{\boldsymbol \sigma}^n_\varepsilon$
in (\ref{modelf02a})$_2$ and using (\ref{conu1N}) as well as Remark \ref{eqh2}, one obtains
\begin{equation*}
\delta_t \Big( \displaystyle
\frac{1}{2}\Vert \hat{u}^n_\varepsilon\Vert_{0}^2 + \frac{1}{4}\Vert{\boldsymbol \sigma}^n_\varepsilon\Vert_{0}^{2} \Big)+
\frac{k}{2}
\Vert \delta_t \hat{u}^n_\varepsilon\Vert_{0}^2 +
\frac{k}{4} \Vert \delta_t {\boldsymbol \sigma}^n_\varepsilon\Vert_{0}^2
+ \Vert \hat{u}^n_\varepsilon\Vert_{1}^{2} +
\displaystyle\frac{1}{2}\Vert{ \boldsymbol \sigma}^n_\varepsilon\Vert_{1}^{2}=0.
\end{equation*}
Then, proceeding as in (\ref{e33}) and (\ref{cnewu}), one arrives at (\ref{LTusEPS}).
\end{proof}
\begin{cor}
Let $v^n_\varepsilon=v^n_\varepsilon((u^n_\varepsilon)^2)$ be a solution of (\ref{edovfUSe}). Then, it holds
\begin{equation*}
\displaystyle\Vert \hat{v}^n_\varepsilon \Vert_{0}^{2}\leq
\left\{\begin{array}{l}
C e^{- \frac{1}{1+k}kn} \ \ \mbox{ if } 2K_p>1,\\
C(kn) e^{- \frac{1}{1+k}kn} \ \mbox{ if } 2K_p=1,\\
C e^{- \frac{2K_p}{1+2K_p k}kn} \ \mbox{ if } 2K_p<1,
\end{array}\right.
\end{equation*}
where the constant $K_p>0$ was defined in Subsection \ref{subcc}.
\end{cor}
\begin{proof}
The proof follows as in Theorem \ref{CoUV}; using, in this case, Remark \ref{eqh2}.
\end{proof}
\subsection{Positivity of $v^n_\varepsilon$ and approximated positivity of $u^n_\varepsilon$}
First, the positivity of the discrete chemical signal will be proved. For this, it will be essential that the interior angles of the triangles or tetrahedra be less than or equal to $\pi/2$. Since we impose the right angled constraint (\textbf{H}), then this property holds.
\begin{lem}{\bf (Positivity of $v^n_\varepsilon$)}
Given $u^n_\varepsilon\in U_h$ and $v^{n-1}_\varepsilon\in V_h$, the unique $v^n_\varepsilon \in V_h$ solution of (\ref{edovfUSe}) satisfies $v^n_\varepsilon\geq 0$.
\end{lem}
\begin{proof}
We define $v^n_{\varepsilon-}:=\min\{v^n_{\varepsilon},0\}$ and $v^n_{\varepsilon+}:=\max\{v^n_{\varepsilon},0\}$. Then, testing (\ref{edovfUSe}) by $\bar{v}=\Pi^h (v^n_{\varepsilon-}) \in V_h$, and taking into account that $(\nabla \Pi^h (v^n_{\varepsilon+}), \nabla \Pi^h (v^n_{\varepsilon-}))\geq 0$ (owing to the interior angles of the triangles or tetrahedra are less than or equal to $\pi/2$), and using that $(\Pi^h (v))^2\leq \Pi^h(v^2)$ for all $v\in C(\overline{\Omega})$, one has
\begin{eqnarray*}
&\displaystyle\Big(\frac{1}{k} + 1 \Big) \Vert \Pi^h (v^n_{\varepsilon-})\Vert_0^2&\!\!\! + \Vert \nabla \Pi^h (v^n_{\varepsilon-})\Vert_0^2 \leq 0,
\end{eqnarray*}
and the proof is concluded.
\end{proof}
Notice that the above properties were proved independently of the choice of $\Lambda_\varepsilon(u^n_\varepsilon)$ approximating $u^n_\varepsilon$. Now, in order to obtain aproximated positivity for the discrete cell density $u^n_\varepsilon$, we need to consider $\Lambda_\varepsilon(u^n_\varepsilon)$ satisfying (\ref{PL1}) and (\ref{D}). The main idea in the proof is to get the following bound
\begin{equation}\label{pppu}
(F_\varepsilon(u^n_\varepsilon),1)^h\leq C
\end{equation}
which, following the ideas of Corallary 3.9 and Remark 3.12 of \cite{GRR1}, implies the estimate (\ref{UPosi}) below, from which one can deduce that $u^n_{\varepsilon -}\rightarrow 0$ as $\varepsilon\rightarrow 0$ in the $L^2(\Omega)$-norm.
\begin{tma}{\bf(Approximated positivity of $u^n_\varepsilon$) }\label{AAPP}
Let $(u^n_\varepsilon,{\boldsymbol\sigma}^n_\varepsilon)$ any solution of the scheme \textbf{US}$_\varepsilon$. If $\varepsilon\in (0,e^{-2})$, the following estimate holds
\begin{equation}\label{UPosi}
\max_{n\geq 0} \Vert \Pi^h (u^n_{\varepsilon-})\Vert_0^2 \leq C_0\varepsilon,
\end{equation}
where the constant $C_0$ depends on the data $(\Omega,u_0, v_0)$, but is independent of $k,h,n$ and $\varepsilon$.
\end{tma}
\begin{proof}
Testing (\ref{modelf02a})$_1$ by $\bar{u}= \Pi^h (F'_\varepsilon (u^n_\varepsilon))$ and taking into account that $\Lambda_\varepsilon(u^n_\varepsilon)$ is symmetric as well as (\ref{PL1}) (which implies that $\nabla \Pi^h (F'_\varepsilon (u^n_\varepsilon))=\Lambda_\varepsilon^{-1} (u^{n}_\varepsilon) \nabla u^n_\varepsilon$), one obtains
\begin{eqnarray}\label{I01a}
(\delta_t u^n_\varepsilon,\Pi^h (F'_\varepsilon (u^n_\varepsilon)))^h+ \int_\Omega (\nabla u^n_\varepsilon)^T\!\cdot\!\Lambda_\varepsilon^{-1} (u^{n}_\varepsilon)\!\cdot\! \nabla u^n_\varepsilon d\boldsymbol x = -\int_\Omega {\boldsymbol{\sigma}^n_\varepsilon}\cdot \nabla u^n_\varepsilon d\boldsymbol x.
\end{eqnarray}
By using the Taylor formula and taking into account that $\Pi^h$ is linear and $F''_\varepsilon(s)\geq \varepsilon$ for all $s\in \mathbb{R}$, one has (following \cite[Theorem 3.8]{GRR1})
\begin{equation*}
(\delta_t u^n_\varepsilon,\Pi^h (F'_\varepsilon (u^n_\varepsilon)))^h \geq \delta_t (F_\varepsilon(u^n_\varepsilon),1)^h + \varepsilon\frac{k}{2}\vert \delta_t u^n_\varepsilon \vert_h^2,
\end{equation*}
which, together with (\ref{D}), (\ref{I01a}) and Remark \ref{eqh2}, imply that
\begin{equation}\label{deluvNN}
\delta_t (F_\varepsilon(u^n_\varepsilon),1)^h+ \varepsilon\frac{k}{2}\Vert \delta_t u^n_\varepsilon \Vert_0^2 + \varepsilon\Vert \nabla u^n_\varepsilon\Vert_0^2 \leq \frac{1}{2} \Vert \nabla u^n_\varepsilon\Vert_0^2 + \frac{1}{2} \Vert{\boldsymbol{\sigma}}^n_\varepsilon\Vert_0^2.
\end{equation}
Then, multiplying (\ref{deluvNN}) by $k$, adding for $n=1,\cdot\cdot\cdot,m$ and using Corollary \ref{welemUS}, one arrives at
\begin{equation*}
(F_\varepsilon(u^m_\varepsilon),1)^h \leq (F_\varepsilon(u^0_h),1)^h+k \underset{n=1}{\overset{m}{\sum}}\left( \frac{1}{2} \Vert \nabla u^n_\varepsilon\Vert_0^2 + \frac{1}{2} \Vert{\boldsymbol{\sigma}}^n_\varepsilon\Vert_0^2\right) \leq C_0,
\end{equation*}
where $C_0>0$ is a constant depending on the data $(\Omega, u_0, v_0)$, but independent of $k,h,n$ and $\varepsilon$. Thus, (\ref{pppu}) is obtained. Therefore, if $\varepsilon\in (0,e^{-2})$, from (\ref{PNa})$_2$ and following the proof of Corollary 3.9 and Remark 3.12 of \cite{GRR1}, (\ref{UPosi}) is deduced.
\end{proof}
\section{Numerical Simulations}\label{NSi}
In this section we will compare the results of several numerical si\-mu\-lations that we have carried out using the schemes studied in the paper.
We are considering
$(\mathbb{P}_1,\mathbb{P}_2)$-continuous approximation for $(u^n_h, v_h^n)$. Moreover, we have chosen the domain $\Omega=[0,2]^2$ using a structured mesh, and all the simulations are carried out using $\textbf{FreeFem++}$ software. We will also compare with the scheme \textbf{US} studied in \cite{FMD2}. We use Newton's method to approach the nonlinear schemes \textbf{US} and \textbf{UV}; while for the scheme \textbf{US}$_\varepsilon$, we use the following Picard method:\\
\begin{itemize}
\item \underline{Picard method to approach a solution $(u^{n}_\varepsilon,{\boldsymbol{\sigma}}^{n}_\varepsilon)$ of the scheme \textbf{US}$_\varepsilon$}:\\
{\bf Initialization ($l=0$):} Set $(u^{0}_\varepsilon,{\boldsymbol{\sigma}}^{0}_\varepsilon)=(u^{n-1}_\varepsilon,{\boldsymbol{\sigma}}^{n-1}_\varepsilon)\in U_h\times {\boldsymbol{\Sigma}}_h$.\\
{\bf Algorithm:} Given $(u^{l}_\varepsilon,{\boldsymbol{\sigma}}^{l}_\varepsilon)\in U_h\times {\boldsymbol{\Sigma}}_h$, compute $(u^{l+1}_\varepsilon,{\boldsymbol{\sigma}}^{l+1}_\varepsilon)\in U_h\times {\boldsymbol{\Sigma}}_h$ such that
$$
\left\{
\begin{array}
[c]{lll}%
\frac{1}{k}(u^{l+1}_\varepsilon,\bar{u})^h + (\nabla u^{l+1}_\varepsilon,\nabla \bar{u}) = \frac{1}{k}(u^{n-1}_\varepsilon,\bar{u})^h -(\Lambda_\varepsilon (u^{l}_\varepsilon){\boldsymbol{\sigma}}^{l}_\varepsilon,\nabla \bar{u}), \ \ \forall \bar{u}\in U_h,\\
\frac{1}{k}({\boldsymbol \sigma}^{l+1}_\varepsilon,\bar{\boldsymbol \sigma}) + (B{\boldsymbol \sigma}^{l+1}_\varepsilon,\bar{\boldsymbol \sigma}) =\frac{1}{k}({\boldsymbol \sigma}^{n-1}_\varepsilon,\bar{\boldsymbol \sigma}) +
(\Lambda_\varepsilon (u^{l+1}_\varepsilon) \nabla u^{l+1}_\varepsilon,\bar{\boldsymbol \sigma}),\ \ \forall
\bar{\boldsymbol \sigma}\in \Sigma_h,
\end{array}
\right.
$$
until the stopping criterion $\max\left\{\displaystyle\frac{\Vert
u^{l+1} - u^{l}\Vert_{0}}{\Vert u^{l}\Vert_{0}},\displaystyle\frac{\Vert
{\boldsymbol{\sigma}}^{l+1} - {\boldsymbol{\sigma}}^{l}\Vert_{0}}{\Vert
{\boldsymbol{\sigma}}^{l}\Vert_{0}}\right\}\leq tol$.
\end{itemize}
In all the cases, we consider $tol=10^{-4}$.
\subsection{Positivity}\label{SimPos}
The aim of this subsection is to compare the fully discrete schemes $\textbf{UV}$, $\textbf{US}$ and \textbf{US}$_\varepsilon$ in terms of positivity.
Theoretically, for all schemes, is not clear the positivity of the variable $u^n_h$. However, for the scheme \textbf{US}$_\varepsilon$, it was proved that $\Pi^h(u^n_{\varepsilon-})\rightarrow 0$ in $L^2(\Omega)$ as $\varepsilon\rightarrow 0$ (see Theorem \ref{AAPP}). For this reason, in Figure \ref{fig:PosiU1} we compare the positivity of the variable $u^n$ in the schemes, taking the spatial parameter $h=1/20$, a small time step $k = 10^{-5}$ (in order to see the differences in the spatial approximations), and the initial conditions (see Figure \ref{fig:initcond1}):
$$u_0\!\!=\!\!-10xy(2-x)(2-y)exp(-10(y-1)^2-10(x-1)^2)+10.0001$$
and
$$v_0\!\!=\!\!100xy(2-x)(2-y)exp(-30(y-1)^2-30(x-1)^2)+0.0001.$$
\begin{figure}[h]
\begin{center}
{\includegraphics[height=0.4\linewidth]{IniCond1}}
\caption{Cross section at $y=1$ of the initial cell density $u_0$ and chemical concentration $v_0$.
\label{fig:initcond1}}
\end{center}
\end{figure}
Note that $u_0,v_0>0$ in $\Omega$, $\min (u_0)=u_0(1,1)=0.0001$ and $\max (v_0)=v_0(1,1)=100.0001$. We obtain that (see Figure \ref{fig:PosiU1}):
\begin{enumerate}
\item In all schemes, the discrete cell density $u_h^n$ takes negative values for some $\boldsymbol x \in \Omega$ in some times $t_n>0$.
\item In the scheme \textbf{US}$_\varepsilon$, the negative values of $u^n_{\varepsilon}$ are closer to $0$ as $\varepsilon \rightarrow 0$.
\item The scheme \textbf{US}$_\varepsilon$ evidence ``better positivity'' than the schemes \textbf{UV} and \textbf{US}, because the ``greater'' negative values for \textbf{US}$_\varepsilon$ are of order $10^{-2}$, while the another schemes reach values greater than $-1$.
\end{enumerate}
\begin{figure}[htbp]
\centering
\subfigure[Scheme \textbf{US}$_\varepsilon$]{\includegraphics[width=78mm]{MinuUSE}} \hspace{0.1 cm}
\subfigure[Schemes \textbf{UV} and \textbf{US}]{\includegraphics[width=78mm]{MinuUVS}}
\caption{Minimum values of $u^n_h$} \label{fig:PosiU1}
\end{figure}
\subsection{$\varepsilon$-aproximated positivity vs spurious oscillations}
In this subsection, we present some numerical experiments relating the results concerning to the negativity of the discrete cell density observed in Subsection \ref{SimPos} with the spurious oscillations that could appear. With this aim, we consider $k=10^{-5}$, $h=\frac{1}{25}$, $\varepsilon=10^{-6}$ (for the scheme \textbf{US}$_\varepsilon$) and the following initial conditions:
$$u_0=5cos(2\pi x)cos(2\pi y)+5.0001 \ \ \mbox{and} \ \
v_0=-170cos(2 \pi x)cos(2 \pi y))+170.0001$$
in which, the places with the highest chemical concentration have lower cell density, in order to force to the cell density to be very close to zero. Note that $u_0,v_0>0$ in $\Omega$, $\min (u_0)=u_0(1,1)=0.0001$ and $\max (v_0)=v_0(1,1)=170.0001$.
\
We observe that, in the case of the schemes \textbf{UV} and \textbf{US}, some spuriuos oscillations appear when the discrete cell density takes negative values (which makes simulations unreliable in this ``extreme'' case); while, in the case of the scheme \textbf{US}$_\varepsilon$, the $\varepsilon$-aproximated positivity favors the non-appearance of spurious oscillations (see Figure \ref{fig:NC1}).
\begin{minipage}{\textwidth}
\begin{tabular}{p{1.15cm} p{4.3cm}p{4.3cm} p{4.3cm}}
Time & \hspace{0.9 cm} Scheme \textbf{UV}&\hspace{0.9 cm} Scheme \textbf{US} & \hspace{0.9 cm} Scheme \textbf{US}$_\varepsilon$\\[1mm]
\scriptsize{t=0}& \includegraphics[width=44mm]{UV1} & \hspace{-2mm} \includegraphics[width=44mm]{UV1} & \hspace{-2mm} \includegraphics[width=44mm]{UV1} \\[2mm]
\scriptsize{t=7e-5}& \includegraphics[width=44mm]{UV2} & \includegraphics[width=43mm]{US1} & \includegraphics[width=44mm]{USe1}
\\[2mm]
\scriptsize{t=2.4e-4}& \includegraphics[width=44mm]{UV3} & \includegraphics[width=44mm]{US2} & \includegraphics[width=44mm]{USe2}
\\[2mm]
\scriptsize{t=4.4e-4}& \includegraphics[width=44mm]{UV4} & \includegraphics[width=44mm]{US3} & \includegraphics[width=44mm]{USe3}
\\[2mm]
\scriptsize{t=6.9e-4} & \includegraphics[width=44mm]{UV5} & \includegraphics[width=44mm]{US4} & \includegraphics[width=44mm]{USe4}
\\[-2mm]
\end{tabular}
\figcaption{Positivity vs spurious oscillations of the discrete cell density at different times.} \label{fig:NC1}
\end{minipage}
\subsection{Energy-Stability}
Previously, it was proved that the scheme \textbf{UV} is unconditionally energy-stable with respect to
the energy $\mathcal{E}(u,v)$ given in (\ref{eneruva}) (in the primitive variables $(u,v)$), while the schemes
\textbf{US} and \textbf{US}$_\varepsilon$ are unconditionally energy-stables with respect to the modified energy $\widetilde{\mathcal{E}}(u,{\boldsymbol \sigma})$ given in (\ref{nueva-2}). In this section, we compare numerically the energy stability of the schemes with respect to the ``exact'' energy $\mathcal{E}(u,v)$ which comes from the continuous problem, and to study the behaviour of the corresponding discrete residual of the energy law (\ref{wsd}):
\begin{equation*}
RE^n:=\delta_t \mathcal{E}(u^n_h,{v}^n_h)+ \Vert \nabla u^n_h\Vert_{0}^{2}
+
\displaystyle\frac{1}{2}\Vert (A_h-I) v^n_h\Vert_{0}^{2} +
\displaystyle\frac{1}{2}\Vert \nabla v^n_h\Vert_{0}^{2}.
\end{equation*}
With this aim, we consider the parameters $k=10^{-4}$, $h=\frac{1}{30}$ and the initial conditions
$$u_0\!\!=\!\!-10xy(2-x)(2-y)exp(-10(y-1)^2-10(x-1)^2)+10.0001$$
and
$$v_0\!\!=\!\!20xy(2-x)(2-y)exp(-30(y-1)^2-30(x-1)^2)+0.0001,$$
obtaining that:
\begin{enumerate}
\item[(a)] {
All schemes satisfy the energy decreasing in time property for the energy $\mathcal{E}(u,v)$, that is, $\mathcal{E}(u^n_h,v^n_h)\le \mathcal{E}(u^{n-1}_h,v^{n-1}_h)$ for all $n$, see Figure \ref{fig:EnE}(a).}
\item[(b)]{The schemes \textbf{UV} and \textbf{US} satisfy the discrete energy law $RE^n \leq 0$ for all $n\geq 1$; while the scheme \textbf{US}$_\varepsilon$ evidence positive values for $RE^n$ for some $n\geq 1$, but these values are very close to $0$ (see Figure \ref{fig:EnE})(b).}
\end{enumerate}
\begin{figure}[htbp]
\centering
\subfigure[Energy $\mathcal{E}(u^n_h,{v}^n_h)$]{\includegraphics[width=78mm]{Energy}} \hspace{0,1 cm}
\subfigure[Discrete residual $RE^n$]{\includegraphics[width=78mm]{RE}}
\caption{Energy-stability of the schemes \textbf{UV}, \textbf{US} and \textbf{US}$_\varepsilon$.} \label{fig:EnE}
\end{figure}
\subsection{Asymptotic behaviour}
In this subsection, we present some numerical experiments in order to illustrate the large-time behavior of approximated solutions computed by using the schemes \textbf{UV}, \textbf{US} and \textbf{US}$_\varepsilon$ in two different situations. In the first test, we consider the initial conditions such that the places with the highest chemical concentration have lower cell density; while in the second test, the places with the highest initial chemical concentration have the highest initial cell density. In both situations, we consider $k=10^{-3}$ and $h=\frac{1}{25}$. Moreover, for the scheme \textbf{US}$_\varepsilon$, we consider $\varepsilon=10^{-5}$.
\begin{itemize}
\item Test 1: We choose the initial conditions (see Figure \ref{fig:initcond3}):
$$u^1_0=5cos(2\pi x)cos(2\pi y)+5.0001 \ \ \mbox{and} \ \
v^1_0=-15cos(2 \pi x)cos(2 \pi y))+24.$$
\item Test 2: We choose the initial conditions:
$$u^2_0=u_0^1 \ \ \mbox{and} \ \
v^2_0=15cos(2 \pi x)cos(2 \pi y))+24.$$
\end{itemize}
\begin{figure}[h]
\begin{center}
{\includegraphics[height=0.35\linewidth]{InCAB}}
\caption{Cross section at $y=1$ of the initial cell densities $u_0^1=u_0^2$ and chemical concentrations $v_0^1$, $v_0^2$.
\label{fig:initcond3}}
\end{center}
\end{figure}
In both cases, we observe that $\Vert (u^n_h - m_0,\nabla v^n_h) \Vert_0^2$
decreases to $0$ faster than $\Vert v^n_h - (m_0)^2 \Vert_0^2$.
In Figures \ref{fig:T1u}-\ref{fig:T2u} we observe an exponential decay (at least) of $(u^n_h,v^n_h)$ to $(m_0,(m_0)^2)$. These facts are in agreement with the theoretical results proved in this paper.
\begin{figure}[htbp]
\centering
{\includegraphics[width=78mm]{DebUgVT1}} \hspace{0,1 cm}
{\includegraphics[width=78mm]{DebVT1}} \hspace{0,1 cm}
\caption{Evolution of $\Vert (u^n_h - m_0,\nabla v^n_h) \Vert_0^2$
and $\Vert v^n_h - (m_0)^2 \Vert_0^2$
in test 1.} \label{fig:T1u}
\end{figure}
\begin{figure}[htbp]
\centering
{\includegraphics[width=78mm]{DebUgVT2}} \hspace{0,1 cm}
{\includegraphics[width=78mm]{DebVT2}} \hspace{0,1 cm}
\caption{Evolution of $\Vert (u^n_h - m_0,\nabla v^n_h) \Vert_0^2$
and $\Vert v^n_h - (m_0)^2 \Vert_0^2$
in test 2.} \label{fig:T2u}
\end{figure}
\section{Conclusions}\label{Con}
In this paper, we study two fully discrete FE schemes for a repulsive chemotaxis model with quadratic signal production, called \textbf{UV} (the FE backward Euler in variables $(u,v)$) and \textbf{US}$_\varepsilon$ (obtained by mixing the scheme \textbf{US} proposed in\cite{FMD2} with a regularization technique). For these numerical schemes we obtain better properties than proved for the scheme \textbf{US} in \cite{FMD2}. Specifically, the comparison between the numerical schemes \textbf{UV} and \textbf{US}$_\varepsilon$, and the scheme \textbf{US}, allows us to conclude that, from the theoretical point of view:
\begin{enumerate}
\item\label{one} By imposing the ``compatibility'' condition $(\mathbb{P}_m,\mathbb{P}_{2m})$-continuous FE (with $m\geq 1$) for $(u,v)$, the scheme \textbf{UV} is energy-stable (in the primitive variables $(u,v)$). In the case of the schemes \textbf{US} and \textbf{US}$_\varepsilon$, it can be obtained energy-stability but with respect to a modified energy written in terms of $(u,{\boldsymbol{\sigma}})$.
\item As a consequence of item \ref{one}, the exponential convergence of the scheme \textbf{UV} to the constant states $m_0$ and $(m_0)^2$ (when the time goes to infinity) can be proved in weak norms for $u$ and strong norms for $v$ (equal than the continuous case); while in the schemes \textbf{US} and \textbf{US}$_\varepsilon$, can be proved also exponential convergence towards $m_0$ and $(m_0)^2$, but only in weak norms for $u$ and $v$.
\item Aproximate positivity for the discrete solutions is proved for the scheme \textbf{US}$_\varepsilon$, but it is not clear how to prove neither positivity nor approximated positivity for the schemes \textbf{US} and \textbf{UV}.
\end{enumerate}
From the numerical point of view, we have obtained that:
\begin{enumerate}
\item The scheme \textbf{US}$_\varepsilon$ evidence ``better positivity'' than the schemes \textbf{UV} and \textbf{US}. Moreover, for the scheme \textbf{US$_\varepsilon$} it was observed numerically that $\underset{\overline{\Omega}\times[0,T]}{\min}\ u^n_{\varepsilon} \rightarrow 0$ as $\varepsilon\rightarrow 0$.
\item In some cases, for example when negative values are obtained for $u_h$, some spurious oscillations are observed in the schemes \textbf{UV} and \textbf{US}; while in the scheme \textbf{US}$_\varepsilon$, the approximated positivity of $u_h$ favors the non-appearance of spurious oscillations.
\item The three schemes have decreasing in time energy $\mathcal{E}(u,v)$.
\item It is observed, for the three schemes, an exponential decay (at least) of $(u^n_h,v^n_h)$ in weak-strong norm to $(m_0,(m_0)^2)$.
\end{enumerate}
\section*{Acknowledgements}
The authors have been partially supported by MINECO grant MTM2015-69875-P
(Ministerio de Econom\'{\i}a y Competitividad, Spain) with the participation of FEDER. The first and second authors have also been supported by PGC2018-098308-B-I00 (MCI/AEI/FEDER, UE); and
the third author has also been supported by Vicerrector\'ia de Investigaci\'on y Extensi\'on of Universidad Industrial de Santander.
|
2,877,628,089,500 | arxiv | \section{\label{sec:intro}INTRODUCTION}
Machine learning (ML) means, essentially, computer programs that improve their performance automatically with increasing exposition to data. The algorithmic improvements over the years combined with faster and more powerful hardware \cite{aaron,highbias,breiman,altman,breiman_dt,geurts,rosenblatt,tpot,adam,ert,ensemble,rumelhart} allows now the possibility of extracting useful information out of the monumental and ever-expanding amount of data. It is the fastest-growing and most active field in a variety of research areas, ranging from computer science and statistics, to physics, chemistry, biology, medicine and social sciences \cite{Jordan}. In physics, the applications are abundant, including gravitational waves and cosmology \cite{kamdar2015,kamdar2016,kelleher,lochner,charnock,biswas,carrillo,gauci,ball,banerji, petrillo}, quantum information \cite{torlai2018neural,canabarro,iten2018discovering} and in condensed matter physics \cite{carleo2017solving,torlai2016learning,ghiringhelli2015big} most prominently in the characterization of different phases of matter and their transitions \cite{Carrasquilla,broecker2017machine,PhysRevX.7.031038,deng2017machine,huembeli2018identifying}.
Classifying phase transitions is a central topic in many-body physics and yet a very much open problem, specially in the quantum case due the curse of dimensionality of exponentially growing size of the Hilbert space of quantum systems. In some cases, phase transitions are clearly visible if the relevant local order parameters are known and one looks for non-analyticities (discontinuities or singularities) in the order parameters or in their derivatives. More generally, however, unconventional {transitions such as infinite order (e.g., Kosterlitz–Thouless) transitions}, are much harder to be identified. Typically, they appear at considerably large lattice sizes, a demanding computational task which machine learning has been proven to provide a novel approach \cite{Carrasquilla,broecker2017machine,PhysRevX.7.031038,deng2017machine,huembeli2018identifying}. For instance, neural networks can detect local and global order parameters directly from the raw state configurations \cite{Carrasquilla}. They can also be used to perform transfer learning, for example, to detect transition temperatures of a Hubbard model away from half-filling even though the machine is only trained at half-filling (average density of the lattice sites occupation) \cite{PhysRevX.7.031038}.
In this paper our aim is to unveil the phase transitions through machine learning, for the first time, in the axial next-nearest neighbor Ising (ANNNI) model \cite{Selke,Suzuki}. Its relevance stems from the fact that it is the simplest model that combines the effects of quantum fluctuations (induced by the transverse field) and competing, frustrated exchange interactions (the interaction is ferromagnetic for nearest neighbors, but antiferromagnetic for next-nearest neighbors). This combination leads to a rich ground state phase diagram which has been investigated by various analytical and numerical approaches \cite{Villain,Allen,Rieger, Guimaraes,Beccaria,Nagy}. The ANNNI model finds application, for instance, in explaining the magnetic order in some quasi-one-dimensional spin ladder materials \cite{Wen}. Moreover, it has recently been used to study dynamical phase transitions \cite{Karrasch} as well as the effects of interactions between Majorana edge modes in arrays of Kitaev chains \cite{Hassler,Milsted}.
Using the ANNNI model as a benchmark, we propose a machine learning framework employing unsupervised, supervised and transfer learning approaches. In all cases, the input data to the machine is considerably small {and simple}, the {raw} pairwise correlation functions between the spins for lattices up to $12$ sites. First, we show how unsupervised learning can detect, with great accuracy, the three main phases of the ANNNI model: ferromagnetic, paramagnetic and the clustered antiphase/floating phase. As we show, the unsupervised approach also identifies, at least qualitatively, two regions within the paramagnetic phase associated with commensurate and in-commensurate areas separated by the Peschel-Emery line \cite{Peschel}, a subtle change in the correlation functions which is hard to discern by conventional methods using only data from small chains (as we do here). Finally, we show how transfer learning becomes possible: by training the machine with nearest-neighbour interactions only, we can also accurately predict the phase transitions happening at regions including next-nearest-neighbour interactions.
The paper is organized as follows. In Sec. \ref{sec:model} we describe in details the ANNNI model. In Sec. \ref{sec:ML} we provide a succinct but comprehensive overview of the main machine learning concepts and tools we employ in this work (with more technical details presented in the Appendix). In Sec. \ref{sec:results} we present our results: in Sec. \ref{sec:dataset} we explain the data set given as input to the algorithms; in Sec. \ref{sec:unsup} we discuss the unsupervised approach followed in Sec. \ref{sec:dbscan} by an automatized manner to find the best number of clusters; in Sec. \ref{sec:sup} the supervised/transfer learning part is presented. Finally, in Sec. \ref{sec:disc} we summarize our findings and discuss their relevance.
\section{\label{sec:model}The ANNNI Model}
The axial next-nearest-neighbor Ising (ANNNI) model is defined by the Hamiltonian \cite{Selke,Suzuki}
\begin{equation}
H=-J\sum_{j=1}^N\left(\sigma_j^z\sigma_{j+1}^z-\kappa \sigma_j^z\sigma_{j+2}^z+g \sigma_j^x\right).\label{ANNNI}
\end{equation}
Here $\sigma^{a}_j$, with $a=x,y,z$, are Pauli matrices that act on the spin-$1/2$ degree of freedom located at site $j$ of a one-dimensional lattice with $N$ sites and periodic boundary conditions. The coupling constant $J>0$ of the nearest-neighbor ferromagnetic exchange interaction sets the energy scale (we use $J=1$), while $\kappa$ and
$g$ are the dimensionless coupling constants associated with the next-nearest-neighbor interaction and the transverse magnetic field, respectively.
The {groundstate} phase diagram of the ANNNI model exhibits four phases: ferromagnetic, antiphase, paramagnetic, and floating phase. In both the ferromagnetic phase and the antiphase, the $\mathbb Z_2$ spin inversion symmetry $\sigma_j^z\mapsto -\sigma_j^z$ of the model is spontaneously broken in the thermodynamic limit $N\to \infty$. However, these two ordered phases have different order parameters. While the ferromagnetic phase is characterized by a uniform spontaneous magnetization, with one of the ground states represented schematically by $\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow$, the antiphase breaks the lattice translational symmetry and has long-range order with a four-site periodicity in the form $\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow$. On the other hand, the paramagnetic phase is disordered and has a unique ground state that can be pictured as spins pointing predominantly along the direction of the field. Inside the three phases described so far, the energy gap is finite and all correlation functions decay exponentially. By contrast, the floating phase is a critical (gapless) phase with quasi-long-range order, i.e., power-law decay of correlation functions at large distances. This phase is described by a conformal field theory with central charge $c=1$ (a Luttinger liquid \cite{Giamarchi} with an emergent U(1) symmetry).
The quantum phase transitions in the ANNNI model are well understood. For $\kappa=0$, the model is integrable since it reduces to the transverse field Ising model \cite{Sachdev}. The latter is exactly solvable by mapping to noninteracting spinless fermions. Along the $\kappa=0$ line of the phase diagram, a second-order phase transition in the Ising universality class occurs at $g=1$. It separates the ferromagnetic phase at $g<1$ from the paramagnetic phase at $g>1$. Right at the critical point, the energy gap vanishes and the low-energy properties of a long chain are described by a conformal field theory with central charge $c=1/2$.
Another simplification is obtained by setting $g=0$. In this case, the model becomes classical in the sense that it only contains $\sigma_j^z$ operators that commute with one another. For $g=0$, there is a transition between the ferromagnetic phase at small $\kappa$ and the antiphase at large $\kappa$ that occurs exactly at $\kappa=1/2$. At this classical transition point, the ground state degeneracy grows exponentially with the system size: any configuration that does not have three consecutive spins pointing in the same direction is a ground state.
For $g\neq 0$ and $\kappa\neq0$, the model is not integrable and the critical lines have to be determined numerically. In the region $0\leq\kappa\leq1/2$, the Ising transition between paramagnetic and ferromagnetic phases extends from the exactly solvable point $g=1$, $\kappa=0$ down to the macroscopically degenerate point $g=0$, $\kappa=1/2$. The latter actually becomes a multicritical point at which several critical lines meet. For fixed $\kappa>1/2$ and increasing $g>0$, one finds a second-order commensurate-incommensurate (CIC) transition \cite{Schulz} (with dynamical exponent $z=2$) from the antiphase to the floating phase, followed by a Berezinsky-Kosterlitz-Thouless (BKT) transition from the floating to the paramagnetic phase.
In summary, the ANNNI model has four phases se\-parated by three quantum phase transitions. Approximate expressions for the critical lines in the phase diagram have been obtained by applying perturbation theory in the regime $\kappa<1/2$ \cite{Suzuki} or by fitting numerical results for large chains (obtained by density matrix renormalization group methods \cite{Beccaria}) in the regime $\kappa>1/2$. The critical value of $g$ for the Ising transition for $0\leq \kappa\leq 1/2$ is given approximately by \cite{Suzuki}
\begin{equation}
\label{trans1}
g_{\textrm{I}}(\kappa)\approx \frac{1-\kappa}{\kappa}\left(1-\sqrt{\frac{1-3\kappa+4\kappa^2}{1-\kappa}}\right).
\end{equation}
This approximation agrees well with the numerical estimates based on exact diagonalization for small chains \cite{Guimaraes}. The critical values of $g$ for the CIC and BKT transitions for $1/2<\kappa\lesssim 3/2$ are approximated respectively by \cite{Beccaria}
\begin{eqnarray}
\label{trans2}
g_{\textrm{CIC}}(\kappa)&\approx& 1.05\left(\kappa-0.5\right),\\
\label{transBKT}
g_{\textrm{BKT}}(\kappa)&\approx& 1.05\sqrt{ (\kappa-0.5 ) (\kappa-0.1)}.
\end{eqnarray}
In addition, the paramagnetic phase is sometimes divided into two regions, distinguished by the presence of commensurate versus incommensurate oscillations in the exponentially decaying correlations. These two regions are separated by the exactly known Peschel-Emery line \cite{Peschel}, which does not correspond to a true phase transition because the energy gap remains finite and there is no symmetry breaking across this line. The exact expression for the Peschel-Emery line is
\begin{eqnarray}
\label{trans4}
g_{\textrm{PE}}(\kappa)&=& \frac{1}{4\kappa} - \kappa.
\end{eqnarray}
While the Ising transition is captured correctly if one only has access to numerical results for short chains, cf. \cite{Guimaraes}, detecting the CIC and BKT transitions using standard approaches requires computing observables for significantly longer chains \cite{Beccaria}.
\section{\label{sec:ML}Machine Learning Review}
\begin{figure}[h!]
\includegraphics*[scale=0.8]{fig0_af.jpg}
\includegraphics*[scale=0.8]{fig0_bf.jpg}
\caption{Schematic representation of machine learning techniques. (a) Supervised learning. (b) Unsupervised learning (clustering).}
\label{fig:fig0}
\end{figure}
Machine learning is defined as algorithms that identify patterns/relations from data without being specifically programmed to. By using a variety of statistical/analytical methods, learners improve their performance $p$ in solving a task $T$ by just being exposed to experiences $E$. Heuristically speaking, machine learning occurs whenever $p(T) \propto E$, i. e. the performance in solving task $T$ enhances with increasing training data.
The state-of-the-art of a typical ML project has four somewhat independent components: i) the data set $\mathbf{X}$, ii) a model $m(\mathbf{w})$, iii) a cost function $J(\mathbf{X};m(\mathbf{w}))$ and iv) an optimization procedure. Our aim is to find the best model parameters, $\mathbf{w}$, which minimizes the cost function for the given data set. This optimization procedure is, generally, numerical and uses variations of the well known gradient descent algorithm. In this manner, by combining distinct ingredients for each component in this recipe, we end up with a myriad of possible machine learning pipelines \cite{aaron,highbias}.
In machine learning research, there are two main approaches named as supervised and unsupervised learning, related to the experience passed to the learner. The central difference between them is that supervised learning is performed with prior knowledge of the correct output values for a subset of given inputs. In this case, the objective is to find a function that best approximates the relationship between input and output for the data set. In turn, unsupervised learning, does not have labeled outputs. Its goal is to infer the underlying structure in the data set, in other words, discover hidden patterns in the data, see Fig. \ref{fig:fig0} for a pictorial distinction. In this work, in order to propose a general framework, we used both approaches in a complementary manner.
Using the supervised learning as a prototype, one can depict the general lines of a ML project. We first split the data set into two non-intersecting sets: $\mathbf{X}_{\text{train}}$ and $\mathbf{X}_{\text{test}}$, named training and test sets, respectively. Typically, the test set corresponds to $10 \%-30 \%$ of the data set. Then, we minimize the cost function with the training set, producing the model, $m(\mathbf{w}^*)$, where $\mathbf{w}^* = \text{argmin}_w \{J(\mathbf{X}_{\text{train}};m(\mathbf{w}))\}$. We evaluate the cost function of this model in the test set in order to measure its performance with out-of-sample data. In the end, we seek a model that performs adequately in both sets, i. e., a model that generalizes well. In other words, the model works fine with the data we already have as well as with any future data. This quality, called generalization, is at the core of the difference between the machine learning approach and a simple optimization solution using the whole data set.
The performance of the model is made by evaluating the cost function in the test set. This is generally done by computing the mean square error (MSE) between the prediction made by the model and the known answer (target), $\epsilon_{\text{test}} = \langle J(\mathbf{X}_{\text{test}};m(\mathbf{w}^*))\rangle$. In a ML project, we are dealing, generally, with complex systems for which we have \textit{a priori} no plausible assumption about the underlying mathematical model. Therefore, it is common to test various types of models ($m_1,m_2, ...$) and compare their performance on the test set to decide which is the most suitable one. In fact, it is even possible to combine them (manually or automatically) in order to achieve better results by reducing bias and variance \cite{aaron,highbias,canabarro}.
One should be careful with what is generally called overfitting, that is, some models may present small values for $\epsilon_{\text{train}}$, but $\epsilon_{\text{test}} \gg \epsilon_{\text{train}}$. It happens because some mo\-dels (often very complex) can deal well with data we already have, however produce large error with unobserved data. Overfitting is a key issue in machine learning and various methods have been developed to reduce the test error, often causing an increase in the training error, but reducing the generalization error as a whole. Making ensemble of multiple models is one of such techniques and as has been already successfully demonstrated \cite{aaron, canabarro}. On the opposite trend, we might also have underfitting, often happening with very simple models where $\epsilon_{\text{test}} \sim \epsilon_{\text{train}}$ are both large. Although extremely important, the discussion about the bias-variance trade-off is left to the good review in Refs. \cite{aaron,highbias}.
Overall, the success of a ML project depends on the quality/quantity of available data and also our prior knowledge about the underlying mechanisms of the system. In the Appendix, we provide a brief but intuitive description of all machine learning steps involved in our work. These include the tasks (classification and clustering), the experiences (supervised and unsupervised learning), the machine learning algorithms (multi-layer perceptron, random forest, and so on) and also the performance measures. For additional reading and more profound and/or picturesque discussions we refer to \cite{aaron,highbias} and as well as to the Appendix.
\section{\label{sec:results}Machine Learning phase transitions in the ANNNI model}
\subsection{\label{sec:dataset}Our Data Set}
We use the pairwise correlations among all spins in the lattice as the data set to design our models. Thus, the set of observables used is given by $\left\{ \langle \sigma^{x}_{i}\sigma^{x}_{j} \rangle,\langle \sigma^{y}_{i}\sigma^{y}_{j } \rangle, \langle \sigma^{z}_{i}\sigma^{z}_{j} \rangle \right\}$ with, $j>i$ and $i=[1,N-1]$ where $N$ is the number of spins/qubits in the lattice and $\langle \sigma^{x}_{i}\sigma^{x}_{j} \rangle =\langle\lambda_{0}|\sigma_{i}^{x}\sigma_{j}^{x}|\lambda_{0}\rangle$ is the expectation value of the spin correlation for the Hamiltonian ground state $\vert \lambda_{0}\rangle$ (analogously to {$\langle \sigma^{y}_{i}\sigma^{y}_{j} \rangle$ and $\langle \sigma^{z}_{i}\sigma^{z}_{j} \rangle$}).
It is easy to see that the number of features is given by $3\sum_{k=1}^{N-1}k$ since that for $8$, $10$, and $12$ sites we have $84$, $135$, and $198$ features respectively.
\subsection{\label{sec:unsup}Unsupervised Approach}
Unsupervised learning is the branch of machine learning dealing with data that has not been labeled, classified or categorized. Simply from the features (the components) of the input data, represented by a vector $\mathbf{X}$, one can extract useful properties about it. Quite generally, we seek for the entire probability distribution $p(\mathbf{X})$ that generated and generalizes the data set. Clustering the data into groups of similar or related examples is a common task in an unsupervised project. Self-labeling is another crucial application of unsupervised learning, opening the possibility of combining unsupervised and supervised algorithms to speed up and/or improve the learning process, an approach known as semi-supervised learning (SSL). As we shall demonstrate, using the ANNNI model as a benchmark, unsupervised learning offers a valuable framework to analyze complex phase diagrams even in situations where only few and low dimensional input data is available.
Briefly describing, the algorithm is used to partition $n$ samples into $K$ clusters, fixed \textit{a priori}. In this manner, $K$ centroids are defined, one for each cluster. The next step is to associate each point of the data set to the nearest centroid. Having all points associated to a centroid, we update the centroids to the barycenters of the clusters. So, the $K$ centroids change their position step by step until no more changes are done, i.e. until a convergence is achieved. In other words, centroids do not move more {within} a predetermined threshold. See Appendix for more details.
\begin{figure}[t!]
\includegraphics*[scale=0.6]{fig1.png}
\caption{
Comparison among the approximate solutions \eqref{trans1} and \eqref{transBKT} and the unsupervised learning trained $N=12$ sites in the lattice. The Ising transition \eqref{trans1} is almost perfectly reproduced. The machine results for the BKT transition \eqref{transBKT} shows a smaller accuracy; nonetheless, it is qualitatively accurate.}
\label{fig:fig1}
\end{figure}
For distinct pairs of the coupling parameters ($g;\kappa$) of the Hamiltonian \eqref{ANNNI}, we explicitly compute all the pairwise spin correlations described in section ~\ref{sec:dataset}. Since the computation of correlations is computationally very expensive, the coupling parameters were varied with step size of $10^{-2}$ in the range $\kappa, g \in [0,1]$. So, in total, we are training the learner with a modest number of 10000 examples. Equipped with that, we investigate the capacity of an unsupervised algorithm to retrieve the phase diagram. In Fig.~\ref{fig:fig1}, we show the phase diagram produced by using $k$-means algorithm \cite{kmeans} and focusing on a lattice size $N=12$. Providing $K=3$ to the learner (assuming three phases), the algorithm returns the best clustering based on the similarities it could find in the features. Strikingly, given the few data points and the relatively small size of spin chain used to generated the data, it finds three well distinct clusters in very good accordance with the three main phases of the ANNNI model.
Indeed, since we imposed to the method the gathering in only three groups, the K-means algorithm detects the ferromagnetic phase, the paramagnetic phase, and a third one which clustered the floating phase with the antiphase. It is quite surprising because the boundary between the paramagnetic and the floating phases is the BKT transition, therefore it detects a transition which is notoriously hard to pinpoint, as the correlation length diverges exponentially at the critical point \cite{koster1}. Moreover, the unsupervised approach almost perfectly recovers the curve corresponding to the ferromagnetic-paramagnetic transition and its analytical critical value of $g_{\mathrm{crit}}=1$ (with $\kappa = 0$). As well, it gives very accurate quantitative predictions for the analytical tricritical value $\kappa_{\mathrm{crit}}=1/2$, at which the transition between the ferromagnetic, paramagnetic and antiphase regions happens.
\begin{figure}[t!]
\includegraphics[scale=0.6]{fig2.png}
\caption{Comparison among the approximate solutions \eqref{trans1}, \eqref{transBKT} and \eqref{trans4} with the unsupervised results for $K=4$. As one can see, at least qualitatively, the machine is finding a new region associated with the Emery-Peschel line \eqref{trans4}.}
\label{fig:fig1b}
\end{figure}
We have also tested the unsupervised prediction by setting $K=4$, that is, assuming four phases for the ANNNI model as described in Section \ref{sec:model}. The result is shown in Fig. \eqref{fig:fig1b}.
As we can see, the algorithm does not separate the floating phase and the antiphase but, instead, the new critical line that appears for $K=4$ divides the paramagnetic phase into two regions which, at least qualitatively, can be identified with the commensurate and incommensurate regions separated by the Peschel-Emery line \eqref{trans4}. This result is remarkable because it tells us that machine learning approach manages to detect a subtle change in the correlation functions which is hard to discern by conventional methods using only data for small chains, up to $N=12$. On the other hand, recall that such change in the correlation function does not correspond to a phase transition in the strict sense.
The results presented in this section indicate that unsupervised learning approach is a good candidate when one
knows in advance a good estimate for the number of phases $K$, as it is requested upfront for various unsupervised algorithms. In the next sections, we show how supervised learning can be used as a validation step for the unsupervised results, also providing surprisingly accurate results. However, we first address the task of how we can use a complementary unsupervised approach to cope with the limitation mentioned above, that is, when one has no \textit{a priori} knowledge of a reasonable number of existing phases.
\subsection{\label{sec:dbscan}Density-based clustering}
To make our framework applicable to cases in which we have no guess of how many phases we {can} possibly expect, we propose to use a density-based (DB) clustering technique \cite{dbscan2,dbscan} to estimate an initial number of clusters. Density clustering makes the intuitive assumption that clusters are defined by regions of space with higher density of data points, meaning that outliers are expected to form regions of low density. The main input in such kind of algorithms is the critical distance $\epsilon$ above which a point is taken as an outlier of a given cluster. In fact, it corresponds to the maximum distance between two samples for them to be labelled as in the same neighborhood. One of its most relevant output is the estimated number of distinct labels, that is, the number of clusters/phases. Therefore, it can be taken as a complementary technique for the use of the $k$-means or any other unsupervised approach which requires one to specify the number of clusters expected in the data.
DBSCAN is one of the most common DB clustering algorithms and is known to deal well with data which contains clusters of similar density \cite{dbscan}, making it suitable for our case. We use DBSCAN to retrieve the number of clusters we should input in the unsupervised KNN algorithm, thus assuming no prior knowledge of how many phases one expects. For that, we feed the DBSCAN with a critical distance $\epsilon$ of the order of the step size used to span the training data, i. e., $\epsilon = 10^{-2}$. As a result, the algorithm returned 3 clusters as the optimal solution, thus coinciding with the three main phases present in the ANNNI model, precisely those that one can expect to recognize at the small lattice size we have employed. It also coincides with the Elbow curve for estimating the optimal number of clusters, see Appendix for details.
\subsection{\label{sec:sup}Supervised Approach}
In spite of the clear success of unsupervised ML in identifying the phases in the ANNNI model, a natural question arises. How can we trust the machine predictions in the absence of a more {explicit} knowledge about the Hamiltonian under scrutiny? {Could} partial knowledge help in validating the ML results?
Typically, limiting cases of a Hamiltonian of interest
are simpler and thus more likely to have a known solution. That is precisely the case of the ANNNI model, which for $\kappa=0$ is fully understood, in particular the fact that at $g=1$ there is a phase transition between the ferromagnetic and paramagnetic phases. Can a machine trained with such limited knowledge ($\kappa=0$) make any meaningful predictions to the more general model ($\kappa \geq 0$)?
The best one can hope for in this situation is that the unsupervised and supervised approaches point out similar solutions, a cross validation enhancing our confidence. In the following we show that this is indeed possible by investigating supervised learning algorithms as a complementary approach to the unsupervised framework introduced above.
In supervised machine learning, the learner experiences a data set of features $\mathbf{X}$ and also the target or label vector $\mathbf{y}$, provided by a "teacher", hence the term "supervised". In other words, the learner is presented with example inputs and their known outputs and the aim is to create a general rule that maps inputs to outputs, by generally estimating the conditional probability $p(\mathbf{y} | \mathbf{X})$. One of the main differences between a ML algorithm and a canonical algorithm is that in the second we provide inputs and rules and receive answers and in the first we insert inputs and answers and retrieve rules (see Appendix for more details).
Our aim is to understand whether transfer learning is possible (training with $\kappa=0$ to predict at regions where $\kappa \geq 0$). Both the unsupervised approach as well as the analytical solution to $\kappa=0$, point out that a transition occurs at $g \approx 1$. With this information, we train the supervised algorithms with $g$ ranging in the interval $[0.5,1.5]$. Given that we don't have to vary over $\kappa$, we reduce the step size (in comparison with the unsupervised approach) to $10^{-3}$, generating an evenly distributed training data with equal number of samples, 500 in each phase (ferromagnetic for $g < 1$ and paramagnetic for $g> 1$ ). The main drawback of this supervised approach is that it always performs binary classification, known as one-vs-all classification. For instance, for handing writings digits it is similar to the case in which a learner can simply identify whether or not the number 5 has been written.
\begin{figure}[t!]
\begin{center}
\includegraphics*[scale=0.6]{fig3_vf.png}
\end{center}
\caption{Detecting the critical transverse magnetic field coupling parameter $g$ at which a phase transition occurs. The machine was trained at $\kappa=0$ and asked to predict where the transition happens at $\kappa=0.1$, by considering where the machine is most uncertain, that is, when the probabilities $p_1=p_2=1/2$. Here the ferromagnetic (paramagnetic) phase is labeled as $0$ $(1)$.}
\label{fig:fig3}
\end{figure}
Motivated by the sound results in its unsupervised version, we first tried the KNN algorithm (vaguely related to the $k$-means method) as well as different methods such as the multilayer perceptron (MLP, a deep learning algorithm), random forest (RF) and extreme gradient boosting (XGB). Once the model is trained, we use the same data set used in section ~\ref{sec:unsup} to predict the corresponding phases. Actually, for a given instance $X^{'}$, the trained model, $m$, returns $m(X^{'})= (p_1,p_2)$, where $(p_1,p_2)$ is a normalized probability vector and the assigned phase corresponds to the component with largest value. To determine when we are facing a transition, we plot both the probability components and check when they cross, as shown in Fig.~\ref{fig:fig3}. As can be seen in Fig.~\ref{fig:fig2}, the different ML methods successfully recover the left part ($\kappa < 0.5$) of the phase diagram, exactly corresponding to the ferromagnetic/paramagnetic transition over which the machine has be trained for the case $\kappa=0$. However, what happens as we approach the tricritical point at $\kappa=0.5$ at which new phases (the antiphase and the floating phase) appears? As can be seen in Fig.~\ref{fig:fig2}, near this point the predictions of the different methods start to differ.
\begin{figure}[t!]
\begin{center}
\includegraphics*[scale=0.6]{fig4.png}
\end{center}
\caption{
Phase diagrams produced with diverse ML algorithms when trained only with $\kappa =0$: KNN (black circles), Random Forest (cyan down stars), Multilayer Perceptron (yellow squares) and Extreme Gradient Boosting (blue diagonal stars) and two different analytical solutions (Ising (solid blue) and BKT (solid orange triangles). All different methods recover the ferro/paramagnetic transition very well while the transition between the paragmanetic and the BKT are only recovered by the KNN and RF methods (see main text for more details).}
\label{fig:fig2}
\end{figure}
To understand what is going on, we highlight that since the machine can only give one out of two answers (the ones it has been trained for), the best it can do is to identify {the clustered antiphase/floating phase (here labeled as phase '2')} with either the ferromagnetic (phase '0') or the paramagnetic (phase '1') cases. Since the models were never trained for the antiphase, it is a good test to check the learner's ability to classify a new phase as an outlier.
For $\kappa \geq 0.5$, we are in a region where only phases '0' and '2' are present. So, the best the machine model can do is to output '0' fs the phase is indeed '0' and '1' otherwise. As we already remarked, it is a drawback of the supervised approach in comparison to the unsupervised one, but it is still useful to validate the emergence of a new phase, as suggested by the unsupervised technique. In this sense, the KNN and RF methods perform quite well.
As seen in Fig.~\ref{fig:fig2} the transition between the paramagnetic and the clustered antiphase/floating is qualitatively recovered even though the machine has never been exposed to these phases before.
In Table 1 we present the average {$\ell_1$-norm} for the distinct algorithms taking as benchmark the approximate analytical solutions for the three main phases given by Eqs. \eqref{trans1} and \eqref{transBKT}. One can observe that the two best approaches are the supervised RF and the unsupervised KNN trained with 12 sites, which reinforces our framework of using the unsupervised and supervised methodologies complementarily. It is worth mentioning that the better performance of the supervised approach is related to the fact that we provided more training data, accounting for a more precise transition, as the step size over $g$ is reduced.
\begin{table}[!t]
\caption{Performance (average $\ell_1$-norm with relation to the analytical approximations given by Eqs. 2 and 3) computed for the three main phases and different ML approaches. See Appendix for details. Two best ones in boldface.}
\label{general_analysis}
\setlength\tabcolsep{0pt}
\footnotesize\centering
\smallskip
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lcc}
\hline
\diagbox[width=8em]{Technique}& average $\ell_1$-norm \\
\hline
\hline
RF (supervised) & \textbf{0.03375(9)} \\
KNN (supervised) & 0.07461(4) \\
MLP (supervised) & 0.18571(4) \\
XGB (supervised) & 0.19507(7) \\
KNN (unsupervised - 12 sites) & \textbf{0.03474(4)} \\
KNN (unsupervised - 8 sites) & 0.07904(1) \\
KNN (unsupervised - 10 sites) & 0.16350(2) \\
\end{tabular*}
\end{table}
\section{Discussion}
\label{sec:disc}
In this paper we have proposed a machine learning framework to analyze the phase diagram and phase transitions of quantum Hamiltonians. Taking the ANNNI model as a benchmark, we have shown how the combination of a variety of ML methods can characterize, both qualitatively and quantitatively, the complex phases appearing in this Hamiltonian. First, we demonstrated how the unsupervised approach can recover accurate predictions about the 3 main phases {(ferromagnetic, paramagnetic and the clustered antiphase/floating)} of the ANNNI model. It can also recover qualitatively different regions within the paramagnetic phase as described by the Emery-Peschel line, even though there is no true phase transition in the thermodynamic limit. This is remarkable, given that typically this transition needs large lattice sizes to be evidenced. Here, however, we achieve that using comparatively very small lattice sizes (up to 12 spins). Finally, we have also considered supervised/transfer learning and showed how transfer learning becomes possible: by training the machine at $\kappa=0$ ($\kappa$ representing the next-neighbour interaction strength) we can also accurately {determine} the phase transitions happening at $\kappa \geq 0$.
{Surprisingly, the machine trained to distinguish the ferromagnetic and paramagnetic phase is also able to identify the BKT transition, which is
notoriously hard to pinpoint, since the correlation length
diverges exponentially at the critical point. Moreover, we note that the machine is not simply testing the order parameter because it is able to distinguish between the ferromagnetic phase and the antiphase which are both ordered but with a different order parameter. Indeed, the machine is identifying a new pattern in the data.}
{Taken together, these results suggest that the ML investigation should start with the unsupervised techniques (DBSCAN + Clustering), as it is a great initial exploratory entry point in situations where it is either impossible or impractical for a human or an ordinary algorithm to propose trends and/or have insights with the raw data. After some hypotheses are collected with the unsupervised approach, one should now proceed to get more data and apply supervised techniques as to consolidate the unsupervised outputs. See Appendix for more discussions.}
Overall, we see that both the unsupervised and the various supervised machine learning predictions are in very good agreement. Not only do they recover similar critical lines but also discover the multicritical point of the phase diagram. Clearly, we can only say that because the precise results are known for the ANNNI model. The general problem of mapping out phase diagrams of quantum many-body systems is still very much open, even more so with the discovery of topological phases. The methods investigated here may contribute to advancing this field and hope to motivate the application of this framework to Hamiltonians where the different phases have not yet been completely sorted out. If as it happens for the ANNNI model, if all these multitude of evidence obtained by considerably different ML methods point out similar predictions, this arguably give us good confidence about the correctness of the results. The predictions given by the ML approach we propose here can be seen as guide, an initial educated guess of regions in the space of physical parameters where phase transitions could be happening.
\section*{Acknowledgements}
The authors acknowledge the Brazilian ministries MEC and MCTIC, funding agency CNPq (AC's Universal grant No. $423713/2016-7$, RC's grants No. $307172/2017-1$ and $No 406574/2018-9$ and INCT-IQ) and UFAL (AC's paid license for scientific cooperation at UFRN). This work was supported by the Serrapilheira Institute (grant number Serra-1708-15763) and the John Templeton Foundation via the grant Q-CAUSAl No 61084.
|
2,877,628,089,501 | arxiv | \section{Introduction}
Planetary nebulae (PNe) arise from intermediate- and low-mass stars,
which makes them an excellent probe of the dynamics of these stars in
nearby galaxies. Their bright emission lines can provide accurate
line-of-sight velocities with a minimum of telescope time. Therefore
spectroscopy of PNe can be used as a powerful tool for the study of
the kinematics of nearby galaxies \citep[e.g.,][ hereafter HK04,
PFF04, and M06]{HK04,PFF04,Mer06} and for the mapping of stellar
streams around massive galaxies such as M31\ \citep{Mer03,Mor03}. The
spectra of individual PNe in nearby galaxies also provide chemical
abundances of certain elements in their progenitor stars
\citep[e.g.,][]{W97,Richer98,Fornax,SALT_Sgr,Sa09,Magr09a,Kwitter12,Sanders12}.
They complement photometric or spectroscopic metallicity information
traditionally derived from old red giants or from H\,{\sc ii} regions
by providing a probe of the abundances of intermediate-age
populations.
Searches for PNe are usually based on narrow-band imaging in the
H$\alpha$ and [O\,{\sc iii}] $\lambda$5007 bandpasses in which PNe can
emit 15--20\% of their total luminosity
\citep[e.g.,][]{Magr03,Magr05a}. Occasionally special instruments like
the Planetary Nebula Spectrograph \citep[PM.S;][]{Douglas02} are
employed for the simultaneous identification of PNe and the
measurement of their radial velocities \citep[e.g.,][]{Mer06}, or
integral field spectrographs are used in small fields of view with
particularly high crowding in order to identify and measure [O\,{\sc
iii}] $\lambda$5007 emission against a pronounced stellar background
\citep{Pastorello13}. Other searches, especially photometric ones,
aim at covering as large an area as possible.
The Sloan Digital Sky Survey (SDSS) \citep{york00,stou02} was an
imaging and spectroscopic survey that covered about one quarter of the
celestial sphere. The imaging data were collected in drift-scan mode
in the five bandpasses $u, \ g, \ r, \ i$, and $z$
\citep{fuku96,gunn98,hogg01}. The images were subsequently processed
with special data reduction pipelines to measure the photometric,
astrometric, and structural properties of all detectable sources
\citep{lupt02, stou02, smit02, pier03} and to identify targets for
spectroscopy. The SDSS passbands were carefully chosen to provide a
wide color baseline, to avoid night sky lines and atmospheric OH
bands, to match passbands of photographic surveys, and to guarantee
good transformability to existing extragalactic studies.
The SDSS has been used extensively for the detection and
characterization of objects with special characteristics, including
different types of emission-line objects
\citep[e.g.,][]{Rich02,Kni03,Kni04,PT11,Tanaka12,Zhao13}.
Since the detected flux from PNe comes almost entirely from nebular
emission lines in the optical, the range of colors characteristic of
PNe is defined by the ratios of these emission lines and their
corresponding contributions in different SDSS passbands. Some of these
colors may be expected to be similar to the colors of emission-line
galaxies \citep[ELGs; see, e.g.,][]{Kni04} and should be usable for
the detection of PNe based on SDSS photometry.
In this paper, we present a method designed to detect PN candidates
based on their SDSS colors. Using a sample of known PNe, we isolate
a region in the SDSS $ugri$ two-color diagram in which the probability
of an object to be a PN is very high. In \S\ref{txt:method} we
describe the detection method. We apply this method to the M31\ region
scanned by the SDSS in 2002 \citep[DR6,][]{DR6}. In
\S\ref{txt:obs_red} the follow-up observations and data reduction are
described. In \S\ref{txt:compar} we compare our data with other
surveys for PNe in M31. In \S\ref{txt:efficiency} the resulting
detection efficiency of the method is discussed. In
\S\ref{txt:M31PNe} we present our results for the new PNe in M31\ and
discuss them. A summary is presented in \S\ref{txt:summary}.
For the remainder of this paper, we assume a distance to M31\ of
760\,kpc \citep{vand99}.
\section{The Method}
\label{txt:method}
\subsection{Primary Selection Criteria}
\label{txt:Sel_determ}
In order to develop a method for the selection of PN candidates from
Sloan Digital Sky Survey (SDSS) imaging data based on photometric
criteria, we used an SDSS scan of M31\ reduced with the standard SDSS
photometric pipeline \citep[see][]{Zucker04a}. Using 37 PNe in M31\
from \citet[][NF87 hereafter]{NF87} and one PN from \citet{JF86} we
constructed a test sample of previously known PNe in this region. We
re-identified these PNe in the SDSS data and developed selection
criteria on the basis of their SDSS colors and magnitudes. Because the
standard SDSS pipeline (PHOTO) does not work properly in very crowded
fields, there are {\em no}\ SDSS data in the central area of M31\
where many of the previously identified PNe in M31\ are located. In
addition, eight PNe from the list of 37 from \citet{NF87} could not be
recovered since they are either located close to diffraction spikes of
bright stars on SDSS images or lie in regions where the SDSS source
detections are incomplete due to crowding. Our final test sample
contained 30 PNe.
The location of the known PNe from our test sample and of other
stellar sources from the SDSS M31\ data in different color-magnitude
diagrams (CMDs) is shown in Figure~\ref{fig:Sel_crit}. Throughout
this paper, we use magnitudes resulting from point spread function
(PSF) fitting in the SDSS photometry pipeline, as these magnitudes
give the best results for point sources in the SDSS. To minimize any
reddening effects, the measured magnitudes for each object and each
filter were corrected using the extinction values from the
\citet{Schl98} maps prior to further analysis.
One of the basic characteristics for PNe is the magnitude m$_{5007}$,
which is an equivalent of the V-magnitude calculated from the flux of
the emission line [O\,{\sc iii}] $\lambda$5007 \citep{J89}. The SDSS
filter $g$ (central wavelength 4686 \AA) covers this emission line in
a region very close to the maximum of its response. Since PNe at the
distance of M31\ are objects without detected continuum we assume
that the $g$ magnitudes for these objects have to be very close to the
m$_{5007}$ magnitudes (see Section~\ref{txt:compar} for the final
comparison). Our final criteria to recover all objects of the test
sample from the SDSS photometric database are:
\begin{eqnarray}
\label{eqn:final_select}
{\rm Object~type} & = & {\rm star} \nonumber \\
{\rm Magnitude~type} & = & {\rm PSF} \nonumber \\
19\fm9 \le g_0 & \le & 21\fm6 \nonumber \\
(g - r)_0 & \le & -0\fm4 \\
(r - i)_0 & \le & -0\fm2 \nonumber \\
(u - g)_0 & \ge & 1\fm0 \nonumber
\end{eqnarray}
As Figure~\ref{fig:Sel_crit} demonstrates the most important CMD is
$(u - g)_0$ versus $g_0$, in which the locus of all PNe from our test
sample is clearly separated from the location of most other stars in
this diagram. The very red $(u - g)_0$ colors of PNe are defined by
the existence of strong [O\,{\sc iii}] $\lambda$5007 emission in the
$g$ filter and the absence of any strong emission lines in the $u$
filter. The limiting magnitude for the bright end of the PN
distribution, $g_0 = 19\fm9$, was selected on the basis of the
distance modulus $\rm (m - M)_0 = 24.4$ to M31\ \citep[][]{vand99} and
the absolute magnitude cut-off of the PN [O\,{\sc iii}] $\lambda$5007
luminosity function (PNLF) in massive galaxies with a large population
of PNe \citep[$M_{5007} = -4\fm47$;][]{Ciar02}. The limiting
magnitude for the faint end and the bluest limiting $(u - g)_0$ color
of the PNe was selected such as to recover all objects of the test
sample.
The locus of the PNe from the test sample defines the selection
parameter range for our ``first priority'' candidates. Since these are
objects whose photometric properties match those of the known PNe from
the test sample, they are considered to be very likely PNe
as well. However, it is conceivable that ``true'' PNe are located not
only in this region of parameters, but have some spread around it.
PNe could have brighter $g_0$ magnitudes because $g$ is not exactly
identical with a narrow-band m$_{5007}$ filter. We also expect PNe
with fainter $g_0$ magnitudes, because the PNLF for M31\ has been traced down
to 6 mag fainter from its bright cut-off \citep[][]{Mer06}. PNe could
also have redder $(u - g)_0$ colors in the case that the [O\,{\sc
iii}] $\lambda$5007 line is not as strong as for objects from the test
sample.
For these reasons, based only on the distribution plotted in the $(u -
g)_0$ vs.\ $g_0$ CMD, we additionally defined more relaxed selection
criteria of $19\fm4 \le g_0 \le 22\fm2$ and $(u - g)_0 \ge 0\fm6$ to
identify ``second priority'' candidates. We realized that we can
assess how good or bad these relaxed criteria are only after getting
additional information or confirmation observations for the selected
candidates. These softer criteria for the selection of the second
priority candidates are indicated with dotted lines in the
bottom-right $(u-g)_0$ vs.\ $g_0$ diagram in
Figure~\ref{fig:Sel_crit}. As can be seen from this figure, for
magnitudes fainter than $g_0 = 22\fm2$, the CMD data become very
uncertain because of the incompleteness of the SDSS M31\ data
themselves. The completeness of the SDSS M31\ data varies from field
to field because of, for instance, variable seeing during the
observations and photometry pipeline problems caused by crowding.
Because of the high stellar density in the M31\ region, the
incompleteness is higher than in standard SDSS imaging data,
which have 95\% completeness for point sources at the level
of $r \sim 22\fm2$ \citep{DR2}.
We then applied our criteria to the whole area of M31\ observed by the
SDSS in 2002 \citep[][]{DR6} in order to select PN candidates. Since
the star-galaxy separation in the SDSS is better than 90\% at $r =
21\fm6$ \citep{DR1}, but worsens for fainter magnitudes, fainter PNe
may have been wrongly identified as (slightly) extended sources in the
SDSS database. In our case $r = 21\fm6$ can be transformed to $g_0
\sim 20\fm6$ using $(g-r)_0 \sim -1$ from Figure~\ref{fig:Sel_crit},
and could be even brighter. In order not to lose such potential PNe,
we applied the same color and magnitude selection criteria as listed
above to extended objects and selected additional PN candidates of
first and second priority. Subsequently all candidates were visually
inspected to eliminate recognizably false detections, such as
diffraction spikes of bright stars and clearly extended objects.
We also visually checked our selected candidates using the images of
the ``Survey of Local Group Galaxies Currently Forming Stars''
\citep[SLGG hereafter;][]{Mas06,Mas07}. In this survey, imaging was
obtained in broadband {\em UBVRI} filters and in narrow-band filters
centered on the H$\alpha$ and [O\,{\sc iii}]$\lambda$5007 lines. Using
the SLGG we marked those of our candidates with obvious emission in
the [O\,{\sc iii}] $\lambda$5007 and H$\alpha$ narrow-band filters and
removed all candidates without such emission. This work was done only
for that part of our sample covered by the SLGG images.
In the end, we detected a total of 167 PN candidates, 100 first
priority and 67 second priority. By design, all PNe from the
test sample were detected as candidates of the first priority. We
list all selected M31\ PN candidates in the SDSS data in
Table~\ref{tbl-1}. This table contains the designations (column 1),
coordinates (columns 2--3), PSF magnitudes corrected for the
extinction and magnitude uncertainties as given by the SDSS (columns
4--8), and the priority type (column 9). The table also shows the
result of our spectroscopic follow-up observations and visual checks
with data from the SLGG (column 10; see Section \ref{txt:obs} for
detailed explanations), cross-identifications with other catalogs of
PNe in M31\ (column 12), and the possible association of objects from
our sample with some structures of the outer region of M31\ (column
11). In Figure~\ref{fig:Sel_pos} we show the positions of all
selected candidates of both priorities in our survey using a standard
coordinate projection centered on M31.
\subsection{Possible Contaminants}
\label{txt:contam}
In order to understand possible contamination by point sources such as
QSOs and stars, we analyzed the distribution of our candidates as compared to the
known loci of quasars of different redshifts as shown and discussed in
\citet{Rich02} and \citet{QSO10}. Our analysis shows that our
criteria select star-like sources that are far from both the QSO loci
and the loci of the Galactic stars fainter than $g_0 \sim 15.3$ mag.
Only a few data points from stars are located in our areas of
interest, which would yield perhaps 1--10 objects per 1000 deg$^2$.
Since the SDSS M31\ data studied in this paper cover only
about 26 deg$^2$ in total \citep{DR6},
including the central part of M31, such contamination should thus be
extremely small.
For stars brighter than $g_0 \sim 15.3$ mag foreground stars from the
Galaxy begin to contribute as shown in Figure~\ref{fig:All_stars},
where the CMD for all available stellar sources from the M31\ SDSS
data is plotted after applying the criteria $(g - r)_0 \le
-0\fm4~~and~~(r - i)_0 \le -0\fm2$. Figure~\ref{fig:All_stars} shows
that $g_0 \sim 15.3$ mag is a natural limit of our color-selection
method and is far away from the limiting magnitude of 19\fm4 we
used for the M31\ data.
We also estimated the amount of possible contamination by background
ELGs using data from \citet{Kni04} for galaxies with strong emission
lines from SDSS Data Release 1 \citep{DR1}. With our color criteria,
only two of the ELGs from the SDSS catalog of \citet{Kni04} would be
selected. These two have very high ([O\,{\sc iii}]
$\lambda$5007/H$\beta$) line ratios of 6.5 and 7.2, respectively,
which are very close to the characteristics of the PNe from the test
sample (see Section~\ref{txt:efficiency}).
Considering that the above ELG catalog is based on the SDSS Data
Release 1, which covers an area of 1360 deg$^2$, the number of
possible contaminants in our M31\ region is obviously very small.
Furthermore, the two recovered ELGs are clearly very extended objects
and would therefore certainly have been rejected during the visual
inspection. These two ELGs are both relatively bright sources with
total magnitudes $r \le 17\fm77$ (this was the spectroscopic target
magnitude limit for galaxies in the SDSS-I). In contrast, it is
conceivable that very distant, very faint ELGs might appear as
star-like sources instead. As can be seen in
Figure~\ref{fig:ELGs_red}, the $(g-r)_0$ color for ELGs increases with
redshift. Beyond a redshift of 0.1 all ELGs will lie outside of our
color selection area. Hence we may assume that contamination of our
PN sample by faint ELGs is negligible.
We conclude that our color-selection method for PN candidates using
SDSS $ugri$ filters appears to work very well for point-like sources
at the distance of M31\ with a very low probability for the selected
sources to be contaminated with other types of objects. Potentially,
this method can work up to the bright limit of $g_0 \sim 15.3$, which
corresponds to a distance of $\sim 90$~kpc (for the absolute magnitude
cutoff of the PNLF of $M_{5007} = -4\fm47$).
For extended sources at the distance of M31, the probability that the
selected candidates could be contaminated by nearby (redshift $\le$
0.1) ELGs with strong emission lines is also very low. But this
contamination will surely grow when the $(u-g)_0$ color criterion is
relaxed (see Section~\ref{txt:efficiency} for more details).
\section{Spectroscopic Follow-up Observations}
\label{txt:obs_red}
\subsection{Observations}
\label{txt:obs}
Spectroscopic follow-up observations of a subset of our PN candidates in
M31\ were carried out in 2004 October 7 to 14 at Calar Alto
Observatory with the Calar Alto Faint Object Spectrograph (CAFOS) at
the 2.2\,m telescope. During the eight nights of observations in
service mode a total of 80 PN candidates of both first and second
priorities were observed under variable weather conditions. The seeing
varied from 0\farcs8 to 2\farcs5. A long slit whose width was
adjusted depending on the seeing (1\arcsec\ -- 2.5\arcsec) was used in
combination with a G-100 grism (87\,\AA\,mm$^{-1}$, first order). The
spatial scale along the slit was $0\farcs53$~pixel$^{-1}$. The
detector was a SITE 2K$\times$2K CCD, which we used without binning.
The resulting wavelength coverage was $\lambda$\,4200 --
$\lambda$\,6800\,\AA\ with maximum sensitivity at $\sim$~6000\,\AA.
The obtained dispersion was $\sim$1.9--2\,\AA/pixel, leading to a
spectral resolution of $\sim$~4 -- 6\,\AA\ (FWHM). The exposure times
were adjusted according to the target brightness and weather
conditions and ranged from 15 to 30 minutes per target. In addition,
the flux standard star Hiltner~102 was observed at least once per
night, and Hg--Cd reference spectra for wavelength calibration were
obtained, complemented by the usual dome flatfields, bias, and dark
exposures.
We have marked all spectroscopically observed PNe in Table~\ref{tbl-1}.
Column 10 specifies which of the PNe were observed at Calar Alto
Observatory during our spectroscopic follow-up campaign. Confirmed
PNe are marked with the flag value ``1''. Three candidates that were
not detected as PNe in our follow-up but that were confirmed later
via cross-identifications with other catalogues as real PNe are marked
with the flag value ``2''. PN candidates that were not found in our
follow-up observations are marked with the flag value ``3''. We did
not reject these unconfirmed candidates because all of them are
fainter than g=21\fm5 and possibly were not identified correctly with
the 2.2\,m telescope under poor weather conditions.
\subsection{Data Reduction}
\label{txt:red}
The two-dimensional spectra were bias-subtracted and flat-field
corrected using IRAF\footnote{IRAF: the Image Reduction and Analysis
Facility is distributed by the National Optical Astronomy Observatory,
which is operated by the Association of Universities for Research in
Astronomy, Inc. (AURA) under cooperative agreement with the National
Science Foundation (NSF).}. Cosmic ray removal was done with the
FILTER/COSMIC task in MIDAS\footnote{MIDAS is an acronym for the
European Southern Observatory data reduction package -- Munich Image
Data Analysis System.}. We used the IRAF software routines IDENTIFY,
REIDENTIFY, FITCOORD, and TRANSFORM to perform the wavelength
calibration and to correct each frame for distortion and tilt. The
accuracy of the velocity determination depends on careful wavelength
calibration of the spectra. The rms error in fitting the dispersion
curve was always less than 0.3~\AA, or 18~km~s$^{-1}$ at a wavelength
of 5000~\AA. After flux calibration, one-dimensional (1D) spectra
were extracted from the reduced frames using the IRAF APALL routine to
allow us to measure the total flux. The resulting reduced and
extracted spectra of typical observed PNe of different $g$ magnitudes
are shown in Figure~\ref{fig:SDSS_PN_spec}.
After 1D spectra were extracted, we used our standard method for
measuring emission-line intensities \citep{Kni04,Kni05}. Briefly, our
programs determine the location of the continuum, perform a robust
noise estimation, and fit separate emission lines by a single Gaussian
superimposed on the continuum-subtracted spectrum. The emission lines
H$\alpha$ $\lambda$6563 and [N\,{\sc ii}] $\lambda\lambda$6548,6584
were fitted simultaneously as a blend of three Gaussian features. The
quoted uncertainties of the individual line intensities $\sigma_{\rm
tot}$ include two components: $\sigma_{\rm p}$ caused by the Poisson
statistics of line photon flux, and $\sigma_{\rm c}$, the uncertainty
resulting from the creation of the underlying continuum and calculated
using the Absolute Median Deviation (AMD) estimator.
Since our data are not of good quality, only the strongest emission
lines are detected in our spectra. The emission lines [O\,{\sc iii}]
$\lambda$4959,5007 and H$\alpha$ are seen in all spectra. The
emission lines H$\beta$ and [N\,{\sc ii}] $\lambda\lambda$6548,6584
are detected in most (but not all) spectra. The average
signal-to-noise ratios for the detected lines are 33.3 for [O\,{\sc
iii}] $\lambda$5007, 13.2 for H$\alpha$ and 3.8 for H$\beta$. Lines
with a signal-to-noise ratio less than one were rejected. The strong
emission line He\,{\sc ii} $\lambda$4686 was also found in the
spectrum of SDSS~J005123+435321, which is located in the area of
Andromeda~NE \citep{Zucker04a} shown in Fig.~\ref{fig:SDSS_PN_spec}.
The observed flux of this line is about 65\% of the flux of the
H$\beta$ emission line.
The quoted velocities were derived as mean values weighted by the
velocities determined from the individual lines. The weights are
inversely proportional to the velocity accuracy for each line. The
observed velocities were further corrected for the motion of the Earth
and transformed to heliocentric velocities. The resulting radial
heliocentric velocities and their errors (column 2), the observed line
fluxes (columns 3 -- 6), and the derived extinction coefficient
$C$(H$\beta$) (column 7) based on the H$\alpha$/H$\beta$ ratio are
listed in Table~\ref{tbl-2}. The distributions of the observed
[O\,{\sc iii}] $\lambda$5007~/~H$\alpha$ and [N\,{\sc ii}]
$\lambda\lambda$6548,6584~/~H$\alpha$ line ratios are shown in
Figure~\ref{fig:lines_ratio}. The observed and the
extinction-corrected [O\,{\sc iii}] $\lambda$5007~/~H$\alpha$ line
ratios are plotted versus the $g_0$ magnitude in
Figure~\ref{fig:mag_ratio}.
It is worth noting here that in all cases of non-detection we see no
other emission lines in the spectra and do not see any continuum. For
this reason we are not able to conclude anything about nature of the
candidates that are not identified as genuine PNe: no obvious ELGs,
QSOs, or identifiable special types of stars are detected.
\section{Comparison with other data sets}
\label{txt:compar}
After we constructed our sample and obtained follow-up observations,
data from four additional surveys for PNe in M31\ were published.
These new data allow us to compare and to check for different
systematic effects or to test the external accuracy. The fifth
new survey of the center of M31\ \citep{Pastorello13} is not included
here since the central regions are not resolved in our data.
\citet[][ hereafter HK04]{HK04} present positions and radial
velocities of a sample of 135 PNe, which were selected using
narrow-band imaging and follow-up spectroscopy in the area located to
the South and East of the nucleus of M31. \citet[][ hereafter
H06]{Hall06} published positions and velocities for 723 PNe located in
the disk and bulge of M31. H06 used the conventional approach of
narrow-band imaging and fibre-fed spectroscopy. \citet[][ hereafter
M06]{Mer06} present a catalogue of positions, magnitudes, and
velocities for 3300 emission-line objects (of which 2730 are probably
PNe) found by the Planetary Nebula Spectrograph in the area of M31.
In our cross-identification work
we used a $2.5''$ search box, which is larger than the cited
astrometric accuracy of of both HK04 and M06.
We did not compare our data with the data of H06, because the M06
sample includes 99\% of the H06 sample and the H06 sample is located
more in the central region of M31. We also do not compare with the
outer disk sample of \citet{Kwitter12}, since their 16 PNe were
selected from the M06 data.
Finally, we do not compare our data with the survey for PNe in globular
clusters in M31 \citep{Jacoby13} since due to crowding none of those
PNe and PN candidates are in our sample.
Figure~\ref{fig:Sel_pos1} shows the spatial distribution of all PNe
from our sample, and of the samples of HK04, H06 and M06 relative to
the center and orientation of M31.
The PN candidates from our
sample are shown with red (first priority) and green (second priority)
squares. All observed PNe from our sample that are real PNe are
marked by blue squares that are larger in size than the other symbols.
All PNe from M06 and H06 are shown with plus signs (+). All PNe from
HK are indicated by crosses (x). In this figure, it is easy to see
the PNe that belong to both samples (as indicated by square symbols of
any color with a cross or plus inside) or that are new ones that were
discovered during this work (empty squares).
\subsection{Velocities}
As was described in Section~\ref{txt:Sel_determ} we used PNe from NF87
to define our selection criteria. Therefore the PNe from the NF87
sample are also in our sample. We have nine PNe in common that were
re-observed at Calar Alto Observatory. The weighted mean velocity
difference $\Delta v$(Our$-$NF87) is $14.3\pm6.6$ km~s$^{-1}$. This
is very close to the systematic difference of 10.4 km~s$^{-1}$ that
HK04 found comparing their velocities with those of NF87.
In our final sample we have 17 PNe in common with HK04 (six of them
are from NF87), but only eight PNe that were re-observed at Calar
Alto. The weighted mean velocity difference $\Delta v$(Our$-$HK04) is
$3.7\pm3.7$~km~s$^{-1}$, which means that we do not have any
substantial systematic offset in the velocities of our and of the HK04
data. We have 12 PNe in common with H06, seven of which were
re-observed at Calar Alto. Similarly, the weighted mean difference
$\Delta v$(Our$-$H06) is $1.2\pm2.4$~km~s$^{-1}$. All these PNe from
H06 in our sample are also found by M06.
We have 66 PNe in common with M06 (many of them are in common with
NF87 and/or HK04 as well) of which 43 were re-observed at Calar Alto.
The weighted mean difference $\Delta v$(Our$-$M06) is $1.6\pm3.0$
km~s$^{-1}$ with a combined dispersion of 19.5~km~s$^{-1}$, implying
that we do not have any systematic offset between our and the M06
data. M06 found that the PN.S data have an uncertainty of
14~km~s$^{-1}$, thus our data have about the same 14~km~s$^{-1}$
uncertainty.
\subsection{Astrometry and $g$ versus m$_{5007}$}
The PN.S astrometry can also be compared with SDSS astrometry. Both
the differences in right ascension (RA) and in declination (Dec) for
our 66 PNe in common with M06 are extremely small ($\Delta {\rm
RA}$(Our$-$M06) = $0.004\pm0.9$ arcsec and $\Delta {\rm
Dec}$(Our$-$M06) = $0.08\pm0.8$ arcsec) and do not show any systematic
effects.
Data from the SDSS together with data from M06 can be used to check
our basic hypothesis that the SDSS $g$ magnitudes for PNe are very
close to $m_{5007}$ magnitudes. Figure~\ref{fig:g_v_5007} shows a
comparison between $g$ magnitudes from the SDSS without extinction
correction and $m_{5007}$ magnitudes from M06 for 66 PNe that are
common to both samples. As can be seen in the top panel of the
figure, most of the points are located around the line of slope unity
where both values would equal each other. To check more accurately
for possible systematic differences, the value $\Delta
$($g-$m$_{5007}$) is drawn in the bottom panel. As this panel shows,
the difference does not reveal any systematic trends up to
$g\sim22.5$~mag, which is around our detection limit.
Only five data points show an underestimation of their flux in the
PN.S data starting from m$_{5007}\sim22.0$~mag, or an overestimation
of the $g$ magnitude by the SDSS. Since the difference amounts to up to 2.3
mag, but all these PNe were observed with the 2.2m telescope, it seems
more likely that the flux was underestimated by M06. Without these
five points the $g$ and $m_{5007}$ magnitudes agree within a
weighted standard deviation of 0.09 mag, which supports our assumption
of the approximate equivalence of the $g$ and $m_{5007}$
magnitudes.
\section{The Efficiency of the Method}
\label{txt:efficiency}
\subsection{Spectral observations}
Our spectroscopic follow-up observations of part of our sample allow
us to estimate the efficiency of our color-selection method.
Altogether, out of the 80 observed PN candidates in the SDSS M31 data,
70 objects turned out to be genuine PNe, resulting in an estimated
detection efficiency of $\sim88\%$. The efficiency is different for
the first ($\sim95\%$) and the second priority ($\sim63\%$) candidates
and depends obviously on the magnitude and color of the selected
candidates. The histogram distribution of the PN candidates as a
function of magnitude in the $g$ band is shown in
Figure~\ref{fig:Obs_hist}. Cross-hatched bins indicate observed PN
candidates that turned out not to be PNe in our follow-up spectroscopy
(no obvious emission lines). As can be seen from this figure the
detection efficiency is essentially 100\% for magnitudes brighter than
$g_0 = 21\fm6$, but shows a pronounced decrease for fainter
magnitudes.
There are two possibilities inherent to our method and data set that
help to explain this trend with luminosity. Firstly, the decreasing
number of detected true PNe with decreasing luminosity may reflect the
increasing photometric errors for fainter magnitudes and the thus
increasing number of false detections. This affects in particular the
SDSS $u$ band, since this band has the lowest sensitivity of all the
passbands used in our detection method. At the same time, the
incompleteness of true detections is likely to increase towards
fainter magnitudes, since especially PNe with weak emission lines may
remain unrecognized in the photometric data.
Secondly, the small telescope employed for our follow-up observations
contributes to the difficulty of confirming fainter candidates. There
is at least one PN candidate with $g = 22\fm0$, which we re-observed
with an exposure time of 1800~s after a 900~s exposure did not reveal
emission lines. In the longer exposure emission lines were detected.
Furthermore, three additional, known PNe from the lists of HK04 and
M06 do not show any emission in our spectra (even though two of them
lie within the selection box of our ``first priority'' objects -- see
Figure~\ref{fig:Obs_hist}). These three PNe are shown in
Table~\ref{tbl-1} with a flag value ``2'', and our total efficiency is
$\sim91\%$ ($\sim98\%$ for the first priority candidates and
$\sim68\%$ for the second) after taking them into account.
Hence we cannot be certain that our faint PN candidates without
detected emission lines in their spectra are indeed false
identifications -- deeper observations or observations with larger
telescopes may uncover weak emission lines after all and may thus
improve our detection statistics. In this sense, our listed numbers
may, in fact, only be lower limits. Therefore we did not remove
these candidates from Table~\ref{tbl-1}.
As explained in Section~\ref{txt:Sel_determ}, very red $(u-g)_0$
colors of PNe are caused by the strong emission line [O\,{\sc iii}]
$\lambda$5007 in the wavelength range covered by the $g$ filter and by
the absence of any strong emission lines in the $u$ band. The
H$\beta$ emission line is also located in the $g$ filter, close to the
position of [O\,{\sc iii}] $\lambda$5007, but the ([O\,{\sc iii}]
$\lambda$5007/H$\beta$) line ratio is usually much stronger for
spectra of PNe as compared to spectra of ELGs. This contributes to
the efficiency of our color-selection method to select PNe as opposed
to ELGs.
Using our observational data, we can try to evaluate our $(u-g)_0$
color criterion in terms of this ratio. The distribution of the
observed ([O\,{\sc iii}] $\lambda$5007/H$\beta$) line ratio versus the
$(u - g)_0$ color is shown in Figure~\ref{fig:5007_ug}. Taking into
account that ELGs with strong emission lines have a mean line ratio of
4.06$\pm$1.11 \citep{Kni04}, we can conclude that $(u - g)_0 =
0\fm6-1\fm0$ is approximately the limit where ELGs with strong lines
and PNe start to be comparable in SDSS colors.
\subsection{Cross-identification with other data}
Additional cross identifications with data from HK04 and M06 (73
identifications in total) and visual checks using images of the SLGG
(22 identifications in total) provide further possibilities to
evaluate our method.
A CMD of the SDSS M31\ data with all our PN
candidates and those currently known as genuine PNe is shown in
Figure~\ref{fig:obs1}. All PN candidates are shown as crosses in
the selected color-magnitude area. All PNe from the test sample and true
PNe confirmed with spectroscopic follow-up observations are shown with
red filled circles. All PNe that were identified in the HK04 and/or
M06 samples are marked with empty blue circles. All PNe candidates
that were not observed or not identified by HK04 and/or M06, but that
showed obvious emission in the [O\,{\sc iii}] $\lambda$5007 and
H$\alpha$ images in the SLGG data are shown with green filled circles.
Observed PN candidates without obvious emission lines that are still
in the sample are shown as empty red circles. All PNe candidates that
were deleted from the sample after the visual inspection of the SLGG
data are depicted by empty black circles.
In total, 103 candidates were selected as first priority candidates,
out of which three were rejected after checking the SLGG data. Only
five first priority candidates still remain to be confirmed.
Altogether, this results in an efficiency of our method in the area of
the first priority candidates of at least 92\%. The efficiency for
the area of the second priority candidates drops from $\sim100\%$ for
magnitudes $g<20\fm0$ to $\sim30\%$ for magnitudes $g>21\fm6$.
Finally, we made one more cross-identification search using the HK04
and M06 sample, but this time for {\em all} available M31\ SDSS data
(both stellar and extended sources). We identified only one
additional PN as compared to those previously identified in our
sample. We thus conclude that with our selection criteria we are able
to select 99\% of the known PNe in the SDSS M31\ data.
We show a comparison of the PNLF from our work with the PNLF from M06
in Figure~\ref{fig:PN_5007}. All data were binned in 0.25 mag
intervals. The data were not corrected for reddening.
The PNLF from M06 is plotted as a blue line. The black line shows the
PNLF for our sample assuming that all selected PN candidates are real
PNe. The red line shows the PNLF for confirmed PNe from our sample.
As expected for a presumably universal curve regardless of the region
sampled, at the bright end our PNLF and the one of M06 show excellent
agreement. Our completeness limit is about $g = 21\fm0-21\fm2$.
\section{A Few Characteristics of the Planetary Nebulae in M31}
\label{txt:M31PNe}
\subsection{Spatial Distribution}
\label{txt:space}
The spatial distribution of all newly discovered PNe is shown in
Figure~\ref{fig:Obs_final}. They are overplotted on top of all stars
detected by the SDSS. It is obvious that the discovered PNe really
trace the distribution of stars in the outer regions of M31. In part,
many of them are seen superimposed on various of the recently
uncovered well-known morphological features like the Northern Spur,
the NE Shelf, the NGC\,205 Loop, the G1 Clump, etc. For a more
detailed description of these features and of their stellar
populations, see \citet{Fer02,Fer05,Ibata05,Ibata07,Rich08}.
Certain structures stand out in the spatial distribution of the PNe: A
large number of PNe is seen in the area of the Northern Spur and three
PN candidates are identified at the location of Andromeda NE
\citep{Zucker04a}. Those newly discovered PNe, which could be
associated with known structures in the outer regions of M31, are
marked in Table~\ref{tbl-1}. However, whether individual PNe are
associated with M\,31's disk itself or alternatively with tidal
streams (or other M31\ components) cannot be decided solely based on
their location or spatial coincidence. Such examples will be
discussed in the next sections.
There is a certain asymmetry in the distribution of newly detected PNe
around M31: we find more in the upper and central portion of
Figure~\ref{fig:Obs_final} (corresponding to the northwestern part of
M31) than in the lower portion. Hardly any PNe appear to be
associated with the giant stellar stream, while there are several in
the area of the NE shelf. \citet{Fer05} point out that the stellar
populations of the NE shelf and of the Giant Stream are very similar,
but that the Giant Stream is approximately 60 kpc more distant from us
than the shelf. \citet{Tan10} found the distance to the Giant Stream
to be even twice as large: 883$\pm$45 kpc in total. Deeper data might
reveal more PNe in these two metal-rich features whose stellar
population properties appear to be so similar. Alternatively, the
lower number of detected PNe could also be an effect of the PNLF
affected by the inclination of M31's disk relative to the observer
\citep{Mer06}, and a higher extinction in the more distant part as
seen from our perspective.
\subsection{A minor axis density profile for M31}
\label{txt:minor_axis}
The very extended, diffuse, and disturbed outer regions of M31, which
are visible in, e.g., SDSS star counts as shown in
Figure~\ref{fig:Obs_final}, is not a unique occurence. For instance, deep optical
surveys of nearby face-on and edge-on dIrrs and disk galaxies have also
found that their stellar distributions are much more extended than
previously thought, e.g., NGC\,6822 \citep{deBlok06}, Leo\,A
\citep{Vans04}, IC\,1613 \citep{Batt07}, Pegasus \citep{Kniaz09}, the
Magellanic Clouds \citep[][and references therein]{Casetti-Dinescu12}
and the works of \citet{Malin99,Ti05,Ti06}. Moreover, warps and
flaring have been shown to be common features of galactic disks
\citep[e.g.,][and references therein]{Guijarro10,vanderKruit11}. In
addition, accretion features are now commonly detected
\citep[e.g.,][]{Delgado08,Delgado09,Delgado10,Mouhcine10,Miskolczi11,Ludwig12}.
In the case of M31\ we have the opportunity to use PNe as tracers of
intermediate-age stellar components in these various structures.
In the case of M31\ we appear to see an extension of the disk, which
has been shown to possess a complex structure with considerable
warping, both in the optical and in H\,{\sc i}
\citep[e.g.,][]{Brinks84,Walterbos88,Braun91,Morris94,Carb10},
probably caused and modified by interactions
\citep[e.g.,][]{McConnachie09,Richardson11,Qu11}.
In recent years the surroundings of M31\
have been mapped using deep ground-based photometric
surveys of the Andromeda galaxy
\citep{Fer02,Irwin05,Ibata07,McConnachie09} with the wide-field
cameras of the Isaac Newton Telescope (INT) and the Canada France
Hawaii Telescope (CFHT). \citet{Ibata05,Ibata07} presented a surface
brightness profile for M31\ and concluded that along the minor axis,
in the region $0.2^\circ<R<0.4^\circ$, the classical inner (thin) disk
of M31\ contributes to the profile, but at $0.5^\circ<R<1.3^\circ$ the
extended disk component becomes dominant.
Since PNe trace the distribution of the underlying intermediate-age
stellar populations, we construct a PN density profile for M31\ using
the catalogue of PNe we compiled from our sample and from the samples
of HK04, H06, and M06 (see Section~\ref{txt:compar}). We used the
same method as in \citet{Kniaz09}, where the density profile for the
outer parts of the Pegasus dIrr was constructed using star counts: PN
densities were calculated within elliptical apertures with a fixed
aspect ratio. The following assumptions and rules were used during
this procedure: (1) We assume that the PNLF of M31\ does not vary
throughout the entire M31\ area. (2) For the central part of M31\ ,
i.e., the region inside of the classical disk (within an ellipse with
a semimajor axis of two degrees in Figure~\ref{fig:Sel_pos1}), where
the standard SDSS software does not work properly due to crowding, the
sample from M06 provides most of the PNe and our data added only very
few objects. The aspect ratio for this region was chosen to be the
same as for the optical disk. All PNe from the compact elliptical
galaxy M\,32 were excluded. The PN densities were calculated in
elliptical apertures with a stepwise axis increase of 0.01 degrees.
(3) For all PNe outside of this inner region we first limited the
sample to PNe brighter than $g = m_{5007} = 22.5$~mag (see
Figure~\ref{fig:PN_5007}). In addition, all PNe from the dwarf elliptical
galaxy NGC\,205 were excluded. An aspect ratio of 3:5 was used for
this region \citep{Fer02,Irwin05,Ibata07}. PN densities were
calculated within elliptical apertures with a step of 0.2 degrees.
The PN data cover only part of the studied region and this geometrical
incompleteness increases with radius. To correct for this effect, we
used an ``incompleteness factor'' -- the ratio between the total area
for the elliptical annulus and the actually covered area. This factor
is about 1.0 at $0.5^\circ<R<1.0^\circ$, but increases drastically
after that. We included this factor in the error propagation. All
subsequent fitting was done with weights of $w_k = \sigma_k^{-1}$,
where $\sigma_k$ is the uncertainty calculated for each level
\citep{LSBs}.
The final density distribution was normalized such that the surface
brightness level at a distance of 0.5$^\circ$ from the center of M31\
is $\sim25$ mag arcsec$^{-2}$. This normalization was chosen in order
to be comparable with the V-band minor-axis surface brightness profile
shown in Figure~51 of \citet{Ibata07}. Our calculated density
distribution along the minor axis is shown in
Figure~\ref{fig:exp_disk}. Data calculated for the central part of
M31\ are marked by small open circles, and data for the outer part are
shown as blue circles. Error bars indicating the uncertainties for
each point are also plotted.
Our profile looks very similar to the $V$-band minor-axis surface
brightness profile of \citet[][their Figure 51]{Ibata07}: in the very
center at minor axis radii $R<0.1^\circ$ we clearly see the bulge.
Farther out in the region of $0.1^\circ<R<0.5^\circ$ the classical
inner disk (with bumps caused by spiral arms) contributes to the
profile. Then the extended disk becomes dominant, and our profile
shows an exponential decline out to the $R=20$~kpc, where our data, in
principle, still trace the extended disk. Fitting the data for the
region $8<R<20$~kpc, we measure an exponential scale length of
3.21$\pm$0.14~kpc, which is quite similar to what was found for the
same region by \citet{Irwin05} and \citet{Ibata07}, who found a scale
length of 3.22$\pm$0.02~kpc along the minor axis profile using
photometric data from the INT Wide Field Camera survey of M31. This
structure has a very low central surface brightness at a level of
$\mu_0\sim23$~mag arcsec$^{-2}$.
Various scenarios for the formation of the extended disk were
discussed by \citet{Ibata05}. These authors concluded that the most
probable scenario is the formation via accretion of many small
subgalactic structures. \citet*{Pen06} interpreted the extended disk
as a possible result of a single dwarf satellite merger (with a mass of
$10^9-10^{10} M_{\sun}$) and suggested that the inner disk would not
be strongly affected by such an accretion event \citep[see
also][]{Hammer10}. \citet{Magr05b} correlate the number of PNe within
four magnitudes of the absolute magnitude of the PNLF bright-end
cut-off with the total stellar mass for different galaxies (their
Fig.\ 5). Taking the number of PNe detected in the outer region of M31\
into account ($\sim$200), the fact that most of them belong to the extended disk,
and considering the relation derived by \citet{Magr05b}, we can
conclude that the PNe in this area trace a total stellar mass
comparable to that of M\,33 ($\sim10^{10} M_{\sun}$) or to the
stellar mass estimate inferred for M31's thick disk component as
analyzed by \citet{Collins11}. If we then assume that the
abundance distribution for PNe in the extended disk was also similar
to the distribution for M\,33, we would expect a mean oxygen abundance
value close to 12+log(O/H)=8.2--8.4 dex following \citet{Magr09b}.
Finally, we note that a similar, very extended (up to 40--50 kpc) and
rotationally supported disk-like structure was found with PN data
in the halo of the nearby peculiar giant elliptical galaxy
Centaurus\,A \citep[][]{PFF04}.
\subsection{Extended disk or rotating spheroid?}
\label{txt:M31velocities}
A number of studies of the M31\ surface brightness profile and of the
stellar populations in M31\ have suggested that fields as far out as 20~kpc
were still dominated by the bulge \citep[e.g.][]{PB94, Durrell01,
Irwin05}. Deep HST imaging studies of selected minor-axis fields out
to 35 kpc suggest that both the spheroid and the giant stellar stream
contain, in part, similar (though not identical) intermediate-age
populations younger than 10 Gyr and that the spheroid populations are
polluted with stars from the progenitor of the stream
\citep[e.g.,][]{Brown06,Brown07,Brown08}. These photometric findings
are supported by the spectroscopic studies of red giants along the
minor axis by \citet{Gilbert07} and \citet{Fardal12}.
Altogether, out to 20 kpc from the center of M31\ we see the
superimposed contributions of stars belonging to M31's spheroid (bulge
and halo components), disk components, and accreted components. In
their kinematic study, \citet{Collins11} identify a dynamically hotter
thick disk component in M31\ (in addition to the colder classical thin
disk and extended disk). \citet{Dorman12}, in another kinematic study
based on red giants, find that although the disk components dominate
in the inner 20~kpc of M31, the inner spheroid can be traced
throughout this region as a rotating, hot component.
HK04 had a sample of 135 PNe that only cover a fraction of M31, but
their data include a similar range of minor and major axis distances
from the center of M31\ as the above studies that are based on red
giants. HK04 carried out dynamical modeling and conclude that a
standard model for the thin and thick disks and bulge can not
reproduce the observed PNe kinematics. They suggest that the majority
of the PNe in their sample are probably members of a very extended
bulge, which rotates rapidly at large distances and dominates over the
halo out to at least 20 kpc. (In the more recent literature this hot
rotating component is commonly referred to as ``inner spheroid''
\citep[see, e.g.,][]{Dorman12,Fardal12}.
\citet{Ibata05} discovered an additional extended disk population and
reassessed HK04's data in the light of this discovery. They concluded
HK04's data may favor the disk interpretation, but could not confirm
or exclude whether a rotating spheroid is still needed. \citet{Mer06}
obtained velocities for a vast number of PNe and found that M31's
extended bulge (or spheroid) component can be traced out to ten
effective bulge radii or approximately 15~kpc. \citet{Far07} conclude
that \citet{Mer06}'s global PNe kinematics can be best represented as
a combination of M31\ components plus debris from accreted satellites,
including the counterrotating shelf structures.
In their kinematic PN study, \citet{Hall06} argue that the
substantial drop in the PN velocity dispersion from the center of
M31\ ($\sim 130$~km~s$^{-1}$) out to about 11~kpc ($\sim
50$~km~s$^{-1}$) along the major axis does not support the existence
of a dynamically hot PN halo, but they, too, find evidence for an
extended bulge component akin to HK04. For the PNe belonging to the
disk component \citet{Hall06} find a rotation velocity of $\sim
140$~km~s$^{-1}$. As pointed out by \citet{Kwitter12}, the PN
velocity dispersions at large radii and those of the thick disk red
giants measured by \citet{Collins11} are roughly in agreement.
A full dynamical model fit to all PN data is beyond the scope of this
paper. But considering the data shown in Figures~\ref{fig:Sel_pos1},
\ref{fig:Obs_final}, \ref{fig:exp_disk}, and \ref{fig:gen_vels} we
suggest that the domination by an immense rotating spheroid traced by
PNe out to $R\sim20$~kpc along the minor axis looks less probable than
the extended disk scenario: (1) The spatial PN distribution follows
that of the stars with an aspect ratio close to 3:5 for the outer part
of M31. (2) The density profile along the minor axis for the PNe is
very similar to the surface brightness profile of M31\ for the region
$0^\circ<R<0.5^\circ$ and shows a similar exponential-like profile in
the region $0.5^\circ<R<1.5^\circ$. (3) As also found in the studies
mentioned earlier, clearly the bulk of the PNe does not show the
signature of a kinematically hot halo where all objects would have
random velocities except for the very central part dominated by the
bulge population; (4) most PNe (except for those in the very central
part) exhibit a distribution indicating that they belong to a
component that is rotationally supported.
In Figure~\ref{fig:gen_vels} we show the velocity distribution of all
PNe from HK04, H06, M06, and our sample along the major axis of M31.
All shown velocities were corrected for the systemic velocity of M31\
of V$_{sys} = -306$ km s$^{-1}$ \citep{Carb10}. This plot shows that
the majority of the PNe belongs to the rotationally supported system,
where most of our new PNe (blue circles) have velocities
systematically shifted to lower values. This may be the expected
situation in the case of, for example, a simple model for the velocity
of stars on circular orbits around M31\ \citep{Ibata05}.
Alternatively, we may be seeing the difference between the kinematics
of the inner part of M31\ and the extended disk as predicted by
\citet{Pen06}. Which of these scenarios is the more likely one cannot
yet be answered with the existing data.
There are also indications for the presence of an intermediate-age
population in the halo of M31\ that may amount to up to 30\% of the
stars \citep{Brown03}. The few PNe that deviate from the rotational
signature might be members of the halo of M31\ or of its ``extended
spheroid'' \citep[see discussion of the vast extent of M31\
in][]{W05,Tan10}. Considering the apparent presence of an
intermediate-age population in M31's halo, PNe should exist there as
well, but since the number of PNe strongly depends on the luminosity
of the underlying population \citep[e.g.,][]{Ciardullo89} only a
handful of PNe can be expected. However, whether the intermediate-age
population is indeed part of the halo is still being debated,
especially when considering that M31's disk may be even more extended
than commonly thought, that the existing pencil-beam pointings may be
contaminated by stream stars, and that there are apparent large-scale
variations in the stellar populations in M31's outer regions
\citep{Durrell04,Fer05,Chapman05}. We find very few PNe with
distances of more than 30 kpc from M31's center.
\subsection{PNe in Andromeda NE}
\label{txt:AndNE_ab}
We identified three PN candidates at the location of Andromeda NE
\citep{Zucker04a}. The nature of this diffuse low-surface brightness
structure in the outer regions of M31\ is still unclear. It may be a
very low-mass and low surface-brightness galaxy, a portion of an
extended stellar tidal stream, or possibly just turn-off material from
the disk of M31. Andromeda NE has an absolute luminosity in the
$g$-band of $\sim -11.6$~mag and a central surface brightness of only
$\sim 29$ mag arcsec$^{-2}$ \citep{Zucker04a}.
According to \citet{RB86} the number of stars n$_P$ in any
post-main-sequence phase $P$ can be calculated with the equation
$ n_P = \eta\,L_T\,t_P$,
where $\eta$ is the number of stars per unit luminosity that leave the
main sequence per year (between 5$\times10^{-12}$ to 2$\times10^{-11}$
yr$^{-1}$ $L_{\odot}^{-1}$), $L_T$ is the total luminosity of the
galaxy (5$\times10^6 L_{\odot}$ for Andromeda NE), and $t_P$ is the
duration of the evolutionary phase ($\le$~20,000 years for PNe).
Applying this formula shows that the number of PNe that might be
expected in Andromeda NE is 0.5--2. These numbers are in good
agreement with the number of PN candidates that we found in this area.
Both our 2.2m spectroscopy in this paper and later 3.5m (Calar Alto,
Spain) and 6m Russian telescope spectroscopy (Kniazev et al., in
preparation) confirm that all these candidates are real PNe, which
could be used for kinematical and chemical studies of Andromeda NE.
Two of the PNe presented in this work are located at projected
distances of $\sim$48~kpc and $\sim$41~kpc from the center of M31\ and
are the most distant PNe in M31\ found up to now.
\citet{Ibata05} obtained stellar spectra of presumed M31\ disk stars
with the Keck DEIMOS multi-object spectrograph. Only one of their
DEIMOS fields ($16.7\times5$ arcmin) was located in the area of
Andromeda NE. The measured heliocentric velocities for 92 stars in
this field show a narrow distribution in the region $V_h=-100$ to
$-200$~km~s$^{-1}$. This velocity distribution peaks at about
$V_h=-150$~km~s$^{-1}$ \citep[Figure~24 in][]{Ibata05} though there
may be some contamination from Galactic foreground stars. All our
newly found PNe in the area of Andromeda NE have heliocentric
velocities in a very narrow velocity range close to
$V_h=-150$~km~s$^{-1}$, which suggests that Andromeda NE has an
average velocity close to that value.
\subsection{Could Andromeda NE be the core or a remnant of the Giant Stream?}
\label{txt:AndNE_core}
In Figure~\ref{fig:gen_vels} we show all PNe identified as a possible
continuation of the Giant Stream by \cite{Mer03}. The Giant Stream
was first detected by \citet{Ibata01} at the southeastern outer part
of M31, close to the minor axis, and was photometrically and
spectroscopically studied at various locations along the stream by,
e.g., \citet{McC03,Ibata04,Guha06}. The continuation of this stream
in the internal part of M31\ is rather uncertain. It is also unknown
whether the progenitor of the stream has survived, and if so where it
is. For these reasons various possible scenarios have been suggested and
studied and different models for the Giant Stream have been
calculated \citep[e.g.,][]{Mer03,Fer02,Ibata04,Far06,Far07,Far08}.
To constrain the stream's orbit \citet{Far06} carried out N-body
simulations. They find that the PN distribution of \citet{Mer03} can
be fit well when assuming that it is part of the expected extension of
the stream. They also note that it is easy to make the orbit of this
stream pass through Andromeda NE but warn that there is no strong
evidence that Andromeda NE is indeed part of the stream since it is
not visibly connected with it.
In Figure~\ref{fig:gen_vels}, we plot all our PNe as green circles
that are located within the same region of ($X$, $V_{\rm los}$)
parameters as the previously identified PNe that may represent a
continuation of the Giant Stream \citep[see][]{Far06}. Three of our
PNe were identified before, and our velocities for them are very close
to the values of \citet{Mer06}. Two of our PNe are new. It is easy to
see that positions of the two new PNe in Andromeda NE are located
along the extension of the line that goes through the ``continuation
of the Giant Stream'' PN sample.
In Figure~\ref{fig:gen_stream} we also plot positions of all these PNe
relative to the center and orientation of M31. All PNe from the
``continuation of the Giant Stream'' sample show a cone-like
structure, beginning close to the southeastern minor axis area, then
expanding to northeastern direction and covering the Northern Spur and
Andromeda NE area.
Taking into account the results of \citet{Far06} and our own plots, we
suggest that the PN data support the notion that Andromeda NE could be
a remnant of the Giant Stream. If this suggestion is correct, then
there are altogether 24 PNe in this putative stream sample.
Using PN surveys of (other) LG galaxies, \citet{Magr05b} infer a
relation between the number of PNe and the stellar mass of the
intermediate-age population of a galaxy. Applying their relation, we
estimate that the Giant Stream progenitor had a mass of $\approx10^9
M_{\sun}$ \citep{Magr05b}, close to the mass of the NGC\,205 dwarf
elliptical galaxy. Similarly, using the relation between the number
of PNe in a given galaxy and its $V$-band luminosity derived by
\citet{Magr03}, we estimate a total luminosity $10^8-10^{8.5}
L_{\odot}$ for the progenitor of the Giant Stream. Comparing this
value to its currently measured luminosity, we suggest that about 90\%
of its stars have been lost during the interaction with M31. This
estimate is very close to the calculated dynamical mass of the Giant
Stream progenitor, $M_s \approx 10^9 M_{\sun}$, from \citet{Far06}.
Our estimate of the possible luminosity range of the progenitor is of
the order of the luminosities of the dwarf elliptical companions of
M31\ as well as the Local Group dwarf irregular galaxy IC\,10. The
estimated lower limits of the metallicity of the PNe in NGC\,185 and
NGC\,205 are $12+\log$\,(O/H) = 8.2 and 8.6, respectively
\citep{Richer95}. For the PNe in NGC\,147 \citet{Goncalves07} find a
mean metallicity of $12+\log$\,(O/H) $=8.06^{+0.09}_{-0.12}$.
In IC\,10, only one PN metallicity based on a direct
measurement of $T_{\rm eff}$ has been published so far, yielding
$12+\log$\,(O/H) = 7.96 \citep{Magr09a}. The lower limits estimated
for the remaining PNe without direct $T_{\rm eff}$ measurements are
higher. We recall that for a given luminosity early-type dwarfs tend
to have higher metallicities than late-type dwarfs by up to 0.5 dex
\citep[e.g.,][]{Richer99, Grebel03}. In any case, regardless of the
morphological type of the progenitor, these sparse data
on known or estimated PN abundances in dwarf galaxies in the infered
luminosity range of the Giant Stream progenitor suggest that the
intermediate-age stellar populations in the progenitor may have had
metallicities $12+\log$\,(O/H) of the order of 8.0 to 8.6 dex.
Finally, we would like to emphasize two findings resulting from our PN
analysis: (1) The estimated difference in mass between the Giant Stream
progenitor ($\approx20$ PNe; $\sim10^9$~M$_{\sun}$) and the extended
disk of M31\ ($\approx200$ PNe; $\sim10^{10}$~M$_{\sun}$) is an
order of magnitude. For that reason the extended disk cannot be the
result of a merger of the Giant Stream progenitor and M31\ as a number
of models have suggested \citep[e.g.,][]{Far06,Far08}. (2) Some part
of the Giant Stream progenitor may be spread across the extended disk,
but its contribution has to be small compared to the material of the
extended disk itself.
\subsection{The Giant Stream and the Northern Spur connection}
\label{txt:GS_NE_connect}
The Northern Spur was first noticed by \citet{Walterbos88}. This
structure is located near the northeastern major axis of M31. The
direction of the gaseous warp \citep[e.g.,][]{Carb10} provides strong
support for the association of the Northern Spur with a warp in the
outer stellar disk. \citet{Fer02} were the first to show that this
feature is an excess of stars at the same distance as M31\ and that it
is about a factor of 1.5--2 times more overdense than the G1 clump.
They also suggested that the Northern Spur possibly consists of
intermediate-age stars of moderate metallicity, [Fe/H]$\ge-0.7$ dex.
Analyzing a DEIMOS field ($16.7\times5$ arcmin) in this area
\citet{Ibata05} found that the stellar velocities are similar to the
velocities of other fields located in the outer part of M31.
\citet{Fer02, Zucker04a} and \citet{Ibata07} suggested that the
metallicity of the Northern Spur agrees well with those of many other
parts of the extended disk of M31\ such as the G1 clump and Andromeda
NE. \citet{Rich08} analyzed HST data of a number of fields sampling
different areas of the outer part of M31\ and distinguished two types
of fields based on the morphology seen in the color-magnitude
diagrams. Their ``stream-like'' fields resemble the populations found
in the Giant Stream, while their ``disk-like'' fields reveal prominent
internmediate-age and recent star formation. The Northern Spur
belongs to their ``disk-like'' fields, just as the G1 Clump and
Andromeda NE.
Our data provide further support for the presence of intermediate-age
populations, since we found 3--4 PNe in the area of the G1 Clump and
7--10 PNe in the area of the Northern Spur. Our
Figures~\ref{fig:Sel_pos1} and \ref{fig:Obs_final} show the high
density of detected PNe in the area of the Northern Spur as compared
to the neighboring regions, and Figures~\ref{fig:gen_vels} and
\ref{fig:gen_stream} show that three of these PNe are kinematically
different and belong to the ``continuation of the Giant Stream''
sample. In this scenario the progenitor of the Giant Stream
passed over, near, or through the Northern Spur area and lost there a
sufficiently large part of its mass to account for the PNe.
Considering the detected PNe, the total mass of the Northern Spur may
be estimated to be $6\times10^8 M_{\sun}$, of which 20--30\% could
have been contributed by the Giant Stream. Since the resulting mixture
of populations spread over a sufficiently large area and the resulting
range of the metallicities is unknown the detection of such a
proposed accreted component would be difficult observationally.
However, it would be interesting to compare abundances of different PNe
from the Northern Spur area with those from an in-situ population in
the same area and with PNe from Anromeda NE.
\section{Summary and Conclusions}
\label{txt:summary}
In this paper we present a method to identify extragalactic PNe based
on $ugri$ SDSS photometry and results from follow-up studies. Our
results and conclusions can be summarized as follows:
1. We have developed a method to identify PN candidates in imaging
data of the SDSS using their unique characteristics in $ugri$
photometric data. We apply and test this technique using M31\ and its
large number of PNe. Altogether, we identify 167 PN candidates in the
M31\ area.
2. We demonstrate that our color-selection method for PN candidates
using SDSS $ugri$ filters can work very well for point-like sources at
distances of 90--800~kpc. The probability for the selected sources
to be contaminated with other types of objects is very low. For
extended sources the probability that the selected candidates could be
contaminated by ELGs with strong emission lines and with redshift
$\le$ 0.1 is also very low. But this contamination will surely grow
when the $(u-g)_0$ color criterion is relaxed: $(u - g)_0 =
0\fm6-1\fm0$ is approximately the limit at which ELGs with strong
lines and PNe start to be comparable in SDSS colors.
3. We obtained spectroscopic follow-up observations of 80 PN
candidates in the M31\ area. These observations, additional
cross-identification work with other samples, and visual checks with
narrow-band images show that the efficiency of our method is at least
92\% for the area of our ``first priority'' candidates. The
efficiency for the area of our ``second priority'' candidates drops
from $\sim100\%$ for magnitudes of $g<20\fm0$ to $\sim30\%$ for
magnitudes of $g>21\fm6$.
4. In general, the distribution of PNe in the outer region of M31,
i.e., $8<R<20$~kpc along the minor axis follows the rotationally
supported low surface-brightness structure with an exponential scale
length of 3.21$\pm$0.14~kpc, the so-called extended disk suggested by
\citet{W05}. This disk-like component is also visible in photometric
data from the Isaac Newton Telescope Wide Field Camera survey of M31\
\citep[][]{Ibata05,Ibata07}. We estimate the total stellar mass of
this structure to be $\sim10^{10} M_{\sun}$, which is equivalent of
the mass of M\,33.
5. Our spectroscopy confirms that we have found two new PNe in the
area of Andromeda NE \citep{Zucker04a}, a number consistent with the
number of PNe that can be expected in a stellar structure of this low
a luminosity ($\sim 5\times10^6 L_\odot$). These two PNe are located
at projected distances of $\sim$46 Kpc and $\sim$40 Kpc along the
major axis from the center of M31.
6. With the new PN data at hand we see a possible kinematic connection
between the Giant Stream and PNe in Andromeda\,NE suggesting that
Andromeda\,NE could be the core or remnant of the Giant Stream. Using
the PN data we estimate the total mass of the Giant Stream progenitor
to be $\approx10^9 M_{\sun}$, which would imply that about 90\% of
stars were lost during the interaction with M31.
6. Our data show an obvious kinematic connection between the
continuation of the Giant Stream and the Northern Spur. We suggest
that 20 -- 30\% of stars in the Northern Spur area could belong to the
Giant Stream.
\acknowledgments
The Sloan Digital Sky Survey (SDSS) is a joint project of The
University of Chicago, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, The Johns Hopkins University, the
Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute
for Astrophysics (MPA), New Mexico State University, Princeton
University, the United States Naval Observatory, and the University of
Washington. Apache Point Observatory, site of the SDSS telescopes, is
operated by the Astrophysical Research Consortium (ARC).
Funding for the project has been provided by the Alfred P.\ Sloan
Foundation, the SDSS member institutions, the National Aeronautics and
Space Administration, the National Science Foundation, the U.S.\
Department of Energy, the Japanese Monbukagakusho, and the Max Planck
Society. The SDSS Web site is http://www.sdss.org/.
A.Y.K acknowledges the support
from the National Research Foundation (NRF) of South Africa
and support from the Collaborative Research Area
``The Milky Way System'' (SFB 881) of the German Research Foundation
(DFG) during his visits to Heidelberg.
EKG acknowledges
support from the Swiss National Science Foundation through the grants
200021-101924/1 and 200020-105260/1.
|
2,877,628,089,502 | arxiv | \section{Introduction}
\label{sec:intro}
The linear polarization of optical light is an important element in the study of quasars and other active galatic nuclei (AGN). Polarization is a key feature in AGN unification models since it provides a periscopic view of the AGN core \citep{1985Antonucci,1993Antonucci,2005Zakamska}. Several types of quasars, such as blazars, broad absorption line (BAL) quasars, and red quasars \citep{1981Moore, 1984Stockman, 1998bHutsemekers, 2018Alexandroff}, are characterized by a polarization higher than normal, which reveals specific physical processes (jets, outflows, etc). When attributed to scattering, polarization is related to the object symmetry axis, even if the object is spatially unresolved. Correlated to the object morphology, polarization allows us to study alignments of distant quasars with the large-scale structures in which they are embedded \citep{2005Hutsemekers,2014Hutsemekers}. Polarization is also sensitive to tiny physical effects that can be revealed by the huge travel distance of photons that are emitted by quasars, and it is therefore a useful tool for testing cosmic birefringence that is caused by departures from the Einstein equivalence principle or by hypothetical axion-like particles, among others \citep{2010DiSerego,2015DiSerego,2012Payez}.
We here report new optical linear polarization measurements of quasars obtained with the FOcal Reducer and low-dispersion Spectrographs FORS1 and FORS2 attached to the Very Large Telescope (VLT) installed at the Paranal European Southern Observatory (ESO). The observations were designed for different scientific goals, but the quality of the data is homogeneous.
In Sect.~\ref{sec:obs} we outline the observing procedure. Data reduction and measurements are summarized in Sect.~\ref{sec:reduc}. The online table with the final measurements is described in Sect.~\ref{sec:data}.
\section{Observations}
\label{sec:obs}
The polarimetric observations were carried out at the ESO VLT using FORS1 in May 2008 (visitor mode) and October 2008 (service mode), and with FORS2 from April to July 2015 (service mode). Both FORS1 and FORS2 were mounted at the Cassegrain focus of the Unit Telescopes (UTs). Linear polarimetry was performed by inserting a Wollaston prism into the parallel beam. This prism splits the incoming light rays into two orthogonally polarized beams. Each object in the field therefore has two orthogonally polarized images on the CCD detector, separated by 22$\arcsec$. To avoid image overlapping, the Multi-Object Spectroscopy (MOS) slits were used to create a mask of alternating transparent and opaque parallel strips whose widths correspond to the splitting. The final CCD image consists of alternate orthogonally polarized strips of the sky, two of them containing the polarized images of the object itself. Because the two orthogonally polarized images of the object are recorded simultaneously, the polarization measurements do not depend on variable atmospheric transparency or seeing. In order to derive the normalized Stokes parameters $q$ and $u$, four frames were obtained with the half-wave plate (HWP) at four different position angles (0$\degr$, 22.5$\degr$, 45$\degr$, and 67.5$\degr$). Two different orientations of the HWP are sufficient to measure the linear polarization, but the two additional orientations allowed us to remove most of the instrumental polarization. The targets were positioned at the center of the field to avoid the significant off-axis instrumental polarization generated by the FORS optics \citep{2006Patat}. In May 2008, polarized and unpolarized standard stars were observed to check the zero-point of the polarization position angle, to estimate the residual instrumental polarization, and to verify the whole observing and reduction process. Standard stars were also occasionally observed in service mode (Table~{\ref{tab:std}}).
Most targets are quasars with redshifts between one and three and V magnitudes between 17.5 and 19.5. All but two (3C138 and SDSSJ204727.54-052118.8) are at high galactic latitudes $|b_{\rm gal}| > 30 \degr$. They were mostly selected among broad absorption line, radio-loud, or red quasars that are more likely to be significantly polarized, or among quasars belonging to large quasar groups \citep{2014Einasto,2014Hutsemekers}. All observations were obtained through the V-high$+114$ filter with typical exposure times per frame ranging between 30 seconds and 10 minutes. The FORS1 CCD was a 4k$\times$4k E2V mosaic, used with binning 1$\times$1 in May 2008 and 2$\times$2 in October 2008. With the collimator standard resolution, the pixel size was 0$\farcs$125 and 0$\farcs$25 on the sky, respectively. The FORS2 CCD was a 4k$\times$4k MIT mosaic, used with binning 2$\times$2 corresponding to a pixel size of 0$\farcs$25 on the sky.
\section{Data reduction and measurements}
\label{sec:reduc}
\begin{table*}[t]
\caption[ ]{Observed standard stars}
\label{tab:std}
\begin{tabular}{lll}
\hline\hline
Date & Polarized & Unpolarized\\
yyyy-mm-dd & &\\
\hline\\
2008-05-08 & Ve6$-$23, BD$-$12$\degr$5133 & WD0752$-$676, WD2149$+$021 \\
2008-05-09 & Ve6$-$23, & WD0752$-$676 \\
2008-05-10 & Ve6$-$23, BD$-$12$\degr$5133, BD$-$14$\degr$4922 & WD0752$-$676 \\
2008-10-29 & NGC2024$-$1 & \\
2008-10-30 & NGC2024$-$1 & \\
2015-05-15 & & WD2039$-$202 \\
2015-06-10 & & WD1344$+$106 \\
2015-06-18 & BD$-$12$\degr$5133 & \\
2015-07-10 & BD$-$12$\degr$5133 & WD2149$+$021 \\
\hline
\end{tabular}
\tablebib{\citet{2007Fossati}}
\end{table*}
The $q$ and $u$ Stokes parameters were computed from the ratios of the integrated intensities of the orthogonally polarized images of the object, measured for the four different orientations of the HWP. They were calculated with respect to the instrumental reference frame according to
\begin{eqnarray}
q & = & \frac{R_q - 1}{R_q + 1} \hspace{0.5cm} \mbox{where} \hspace{0.5cm}
R_q^2 = \frac{I_{\scriptscriptstyle\rm 0}^{\rm u}/I_{\scriptscriptstyle\rm 0}^{\rm l}}
{I_{\scriptscriptstyle\rm 45}^{\rm u}/I_{\scriptscriptstyle\rm 45}^{\rm l}},\;\;\; \rm{and} \nonumber\\
& &\\
u & = & \frac{R_u - 1}{R_u + 1} \hspace{0.5cm} \mbox{where} \hspace{0.5cm}
R_u^2 = \frac{I_{\scriptscriptstyle\rm 22.5}^{\rm u}/I_{\scriptscriptstyle\rm 22.5}^{\rm l}}
{I_{\scriptscriptstyle\rm 67.5}^{\rm u}/I_{\scriptscriptstyle\rm 67.5}^{\rm l}},\nonumber
\end{eqnarray}
where $I^{\rm u}$ and $I^{\rm l}$ refer to the intensities (electron counts) integrated over the upper and lower images of the object, respectively. This combination of measurements from the four frames secured with different HWP orientations removes most of the instrumental polarization, and corrects for the effects of image distortions that can be generated by the HWP \citep{1989diSerego,1999Lamy}. The intensity measurements were performed using the procedures described in \citet{1999Lamy} and \citet{2005Sluse}. Specifically, the positions of the upper and lower images were measured at subpixel precision by fitting two-dimensional Gaussian profiles. The intensities were then integrated in circles centered on the upper and lower images, and the Stokes parameters computed for increasing values of the aperture radius. Since the Stokes parameters are most often found to be stable for increasing aperture radii, we adopted a fixed aperture radius of $3.0 \times [(2 \ln 2)^{-1/2}\, \rm{HWHM}],$ where HWHM is the mean half-width at half-maximum of the two-dimensional Gaussian profile. In a few cases, the Stokes parameters strongly fluctuated when the aperture radius was changed, which made these measurements unreliable. The uncertainties $\sigma_q$ and $\sigma_u$ were estimated by computing the errors on the intensities $I^{\rm u}$ and $I^{\rm l}$ from the read-out noise and the photon noise in the object and the sky background, and then by propagating these errors. Typical uncertainties are around 0.1 \% for either $q$ or $u$.
\begin{table*}[t]
\caption[ ]{Polarization of distant stars}
\label{tab:halo}
\begin{tabular}{lccccccccccr}
\hline\hline
Reference-number & RA & DEC & Distance & Obs. Date & $q$ & $u$ & $p$ & $\sigma_p$ & $p_0$ & $\theta$ & $\sigma_{\theta}$ \\
& h m s & $\degr$ $\arcmin$ $\arcsec$ & kpc & & \% & \% & \% & \% & \% & \degr & \degr \\
\hline \\
Beers-745 & 11 15 47.10 & $-$17 55 56.7 & 19.6 & 2008-05-08 & $+$0.01 & $-$0.04 & 0.04 & 0.05 & 0.00 & - & - \\
Beers-747 & 11 16 46.59 & $-$16 59 52.4 & 13.8 & 2008-05-08 & $-$0.06 & $+$0.02 & 0.06 & 0.07 & 0.00 & - & - \\
Beers-748 & 11 16 59.95 & $-$19 33 57.7 & 21.7 & 2008-05-10 & $-$0.04 & $-$0.07 & 0.08 & 0.08 & 0.00 & - & - \\
Beers-752 & 11 19 32.13 & $-$17 05 11.5 & 19.4 & 2008-05-10 & $+$0.02 & $+$0.34 & 0.34 & 0.09 & 0.33 & 43 & 8 \\
Clewley-CF789-045 & 12 43 23.35 & $-$04 11 29.8 & 15.1 & 2008-05-09 & $-$0.08 & $-$0.06 & 0.10 & 0.07 & 0.08 & 107 & 25 \\
Clewley-CF789-041 & 12 44 20.10 & $-$03 21 38.1 & 18.2 & 2008-05-10 & $-$0.06 & $+$0.12 & 0.14 & 0.08 & 0.12 & 58 & 19 \\
\hline
\end{tabular}
\tablebib{\citet{2000Beers,2004Clewley}}
\end{table*}
A zero-point angle offset was then applied to the normalized Stokes parameters $q$ and $u$ in order to convert the polarization angle measured in the instrumental reference frame into the equatorial reference frame. For the V filter, the offset was 1.8$\degr$, according to the FORS user manual. This angle offset was checked using polarized standard stars (Table~{\ref{tab:std}}). For all standard stars, the measured values of the polarization angles corrected with that offset are within 1\degr\ of their nominal values. The polarization of unpolarized standard stars (Table~{\ref{tab:std}}) is around 0.10 $\pm$ 0.05 \% for all runs, indicating that the residual instrumental polarization is small at the center of the field.
Then, the polarization degree was computed using $p = (q^2+u^2)^{1/2}$ and the associated error $\sigma_p \simeq \sigma_q \simeq \sigma_u$. The debiased value $p_{0}$ of the polarization degree was obtained using the \citet{1974Wardle} estimator, which is a reasonably good estimator of the true polarization degree \citep{1985Simmons}. The polarization position angle $\theta$ was obtained by solving the equations $q = p\cos 2\theta$ and $u = p \sin 2\theta$. The uncertainty of the polarization position angle $\theta$ was estimated from the standard \citet{1962Serkowski} formula, where the debiased value $p_{0}$ was conservatively used instead of $p$, that is, $\sigma_{\theta} = 28.65\degr \, \sigma_p / p_{0}$ \citep[see also][]{1974Wardle}.
As in \citet{2017Hutsemekers}, we secured the V-band polarization of a few distant stars ($d > $ 10 kpc) to check the magnitude of the interstellar polarization in the direction of our targets. These measurements are reported in Table~\ref{tab:halo}. All these stars have low polarization. Although the sample is small, this confirms that on average, contamination by interstellar polarization is essentially negligible for quasars at high galactic latitudes ($|b_{\rm gal}| > 30 \degr$) and with polarization degrees higher than 0.6\% \citep{1990Berriman,2000Lamy,2005Sluse,2017Pelgrims}.
\section{Polarization data}
\label{sec:data}
The full Table~3, available at the Strasbourg astronomical Data Center (CDS), contains 87 polarization measurements obtained for 86 quasars (63 measurements in May 2008, 4 in October 2008, and 20 in 2015). Unreliable measurements were discarded (measurements for which no stable value of the Stokes parameters could be secured; see Sect.~\ref{sec:reduc}). Thirty-seven quasars have $p \geq 0.6\%$, 9 have $p \geq 2\%$, and 1 has $p \geq 10\%$. Column~(1) gives the quasar name from the NASA/IPAC Extragalactic Database (NED), Cols.~(2) and~(3) the equatorial coordinates (J2000), Col.~(4) the redshift $z$, Col.~(5) the filter, and Col.~(6) the date of observation (year-month-day). Columns~(7) and~(8) give the normalized Stokes parameters $q$ and $u$ in percent. The normalized Stokes parameters are expressed in the equatorial reference frame. Columns~(9) and~(10) give the polarization degree $p$ and its error $\sigma_p$ in percent. Column~(11) gives the debiased polarization degree $p_0$ in percent. Columns~(12) and~(13) give the polarization position angle $\theta$ east-of-north and its error $\sigma_{\theta}$, in degree. When $p < \sigma_p$, the polarization angle is undefined and its value set to 999.
The five objects with $p \geq 3\%$ are reported in the excerpt of Table~\ref{tab:qsos}. For all of them, polarization measurements are secured for the first time. SDSS J112738.76+013537.9 is a BAL quasar \citep{2006Trump}. PKS 2054-377, WISE J121043.78-275858.9, PKS 1336-237, and WISE J115217.19-084103.1 are Parkes radio sources. The last two objects also belong to the Fermi Gamma-ray Space Telescope source catalogue \citep{2015Acero}.
\begin{sidewaystable}
\caption[ ]{Polarization of quasars}
\label{tab:qsos}
\centering
\begin{tabular}{lcccccrrrrrrr}
\hline\hline
Name & RA & DEC & $z$ & Filter & Obs. Date & $q$ & $u$ & $p$ & $\sigma_p$ & $p_0$ & $\theta$ & $\sigma_{\theta}$ \\
& h m s & $\degr$ $\arcmin$ $\arcsec$ & & & & \% & \% & \% & \% & \% & \degr & \degr \\
\hline \\
EIS J033252.61-273846.5 & 03 32 52.60 & $-$27 38 46.2 & 1.023300 & V & 2008-10-30 & 0.78 & $-$0.14 & 0.79 & 0.85 & 0.00 & 999 & 999 \\
SDSS J112738.76+013537.9 & 11 27 38.76 & $+$01 35 38.0 & 2.014061 & V & 2008-05-09 & $-$6.75 & $-$2.32 & 7.14 & 0.17 & 7.14 & 100 & 1 \\
WISE J115217.19-084103.1 & 11 52 17.21 & $-$08 41 03.3 & 2.370000 & V & 2008-05-08 & 6.05 & $-$11.78 & 13.24 & 0.10 & 13.24 & 149 & 1 \\
WISE J121043.78-275858.9 & 12 10 43.61 & $-$27 58 54.6 & 0.828000 & V & 2008-05-09 & $-$7.80 & 4.36 & 8.94 & 0.12 & 8.94 & 75 & 1 \\
PKS 1336-237 & 13 39 01.75 & $-$24 01 14.0 & 0.657000 & V & 2008-05-09 & $-$9.26 & 0.16 & 9.26 & 0.10 & 9.26 & 90 & 1 \\
PKS 2054-377 & 20 57 41.60 & $-$37 34 03.0 & 1.071000 & V & 2008-05-09 & 2.42 & $-$3.48 & 4.24 & 0.33 & 4.23 & 152 & 2 \\
\hline
\end{tabular}
\tablefoot{This table gives the polarization measurements for six quasars. It contains the five objects with $p \geq 3\%$. The complete table is available electronically at the CDS. }
\end{sidewaystable}
\begin{acknowledgements}
This research has made use of the 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.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,877,628,089,503 | arxiv | \section{Introduction}
In recent years, Gaussian graphical model has been an important tool to capture the conditional dependency structure among variables. The edges of the Gaussian graphical network are characterized by the inverse covariances for each pair of nodes. To be more specific, for Gaussian graphical model, the joint distribution of $p$ random variables $(X_1,\ldots,X_p)^\top$ is assumed to be multivariate Gaussian $N(\zero,\bOmega^{-1})$, where $\bOmega$ is the inverse of the covariance matrix and is called precision matrix. It is known that for Gaussian graphical model, the conditional dependency structure is completely encoded in the precision matrix, i.e., for each pair of nodes $X_a$ and $X_b$, they are conditionally independent given all other variables if and only if the $(a,b)$-th entry of $\bOmega$ is equal to zero. A growing number of literature has focused on the support recovery and link strength estimation of Gaussian graphical model in high-dimensional setting, see, for example, \cite{meinshausen2006high,yuan2007model,friedman2008sparse,yuan2010high,
cai2011constrained,cai2012estimating,Liu2013Gaussian}, among many others. For more detailed discussions and comparisons of these methods, we refer to \cite{Ren2015Asymptotic} and \cite{Fan2016Innovated}. The works mentioned above focus on analyzing one particular Gaussian graph. However, in some cases, it is of greater interest to investigate how the network of connected node pairs change from one state to another. For example, in genomic studies, it is more meaningful to investigate how the network of connected gene pairs change from different experimental condition,
which provides deeper insights on an underlying biological process, e.g., identification of pathways that correspond to the condition change. Indeed, differential networking modeling has drawn much attention as an important tool to analyze a set of changes in graph structure. The differential network is typically modeled as the difference of two precision matrices and this type of model has been used by \cite{Li2007Finding,Ideker2012Differential, Danaher2014The,zhao2014direct,Xia2015Testing,Tian2016Identifying}. To investigate the differential network, in the first step, we need to identify whether there exists any network change, which is equivalent to test the equality of two precision matrices:,
\begin{equation}\label{hyp:omega2test0}
\Hb_0:\bOmega_1=\bOmega_2.
\end{equation}
{Although the equality of two precision matrices is equivalent to the equality of two covariance matrices from mathematical view, the test problem could be very different due to the fundamental difference between conditional
and unconditional dependencies.} Literatures on testing equality of two covariance matrices in high-dimensional setting mainly falls into two categories, sum-of-square type testing and maximum type testing . The sum-of-square type testing are particular powerful under dense alternative where the two covariance matrices differ in a large
number of entries \citep{Schott2007A,Srivastava2010Testing,chen2012two} while the maximum type testing are particular powerful under sparse alternative where the two covariance matrices differ only in a small
number of entries \citep{Tony2013Two}. Literature \cite{Zhou2018Unified} proposed a unified framework for developing tests based on U-statistics, which includes testing the equality of two covariance matrices as a special case. The tests are powerful against a large
variety of alternative scenarios. This research area is very active, and as a result, this list of references is illustrative rather than comprehensive. In contrast, literatures on testing equality of two precision matrices rarely exists. Literature \cite{Xia2015Testing} proposed a maximum-type testing which is powerful against alternative where $\bDelta=\bOmega_1-\bOmega_2$ is sparse.
As far as we know, this is the unique existing work on testing the equality of two precision matrices in the high-dimensional setting. In other word, a powerful testing for hypothesis (\ref{hyp:omega2test0}) under dense alternative still don't exist, which urges us to consider such a problem.
In this paper, we propose a testing procedure for hypothesis (\ref{hyp:omega2test0}) which is powerful against a large variety of
alternative scenarios in high dimensions. Both theoretical results and numerical simulation show the advantage of proposed test against existing methods. The rest of the paper is organized as follows. In section \ref{sec:background}, we introduce some notations and briefly review the test statistic proposed by \cite{Xia2015Testing}.
In Section \ref{sec:methodology} we present our test statistic and the multiplier bootstrap procedure to obtain the critical value or $p$-value of the test. Section \ref{sec::theoretical} gives the theoretical analysis of the test. In Section \ref{sec:experiments}, we conduct thorough numerical simulation to investigate the empirical performance of the test. A real gene expression data set is analyzed to illustrate the usefulness of the test. At last we discuss possible future directions in the last section.
\section{Background}\label{sec:background}
\subsection{Notation}
For a vector $\vb = (v_1,\dots,v_d)^\top\in {\mathbb R}^d$, let $\|{\vb}\|_p = \big(\sum_{j=1}^d |v_j|^p\big )^{1/p}$ as the $L_p$-norm. As $p=\infty$, we set $\|\vb\|_{\infty}=\max_{1\le j\le d}|v_j|$. As $p=0$, we set $\|\vb\|_0=\sum_{j=1}^d I\{v_j\neq0\}$. We use $v^{(1)}, v^{(2)},\ldots, v^{(d)}$ to denote the order statistics of the absolute value of $\vb$'s entries with $v^{(1)}\le v^{(2)}\le \ldots\le v^{(d)}$. Apparently, we have $v^{(j)}\ge 0$ for $j=1,\ldots,d$. We define the $(s_0,p)$-norm of $\vb$ as $\|{\vb}\|_{(s_0,p)}=\big(\sum_{j=d-s_0+1}^d(v^{(j)})^p\big )^{1/p}$. As $p=\infty$, we set $\|{\vb}\|_{(s_0,p)}=\|\vb\|_{\infty}=v^{(d)}$ for any $s_0$. We denote
$\mathbb{S}^{d-1}:=\{\vb\in{\mathbb R}^d:\|\vb\|_2=1\}$ as the spherical surface in ${\mathbb R}^d$. For any vector $\bmu_m \in \mathbb{R}^d$, let $\bmu_{m,-i}$ denote the $(d-1)\times 1$ vector by removing the $i$-th entry from $\bmu_m$. For a data matrix $\Ub=(\bU_1,\ldots,\bU_n)^\top\in \mathbb{R}^{n\times d}$, let $\Ub_{\cdot,-i}=(\bU_{1,-i},\ldots,\bU_{n,-i})^\top$ with dimension $(n\times(d-1))$, $\bar{\bU}_{\cdot,-i}=n^{-1}\sum_{k=1}^n\bU_{k,-i}$ with dimension $(d-1)\times 1$, $\bU_{(i)}=(U_{1,i},\ldots,U_{n,i})^\top$ with dimension $n\times 1$, $\bar{\bU}_{(i)}=(\bar{U}_i,\ldots,\bar{U}_i)^\top$ with dimension $n\times 1$ where $\bar{U}_i=n^{-1}\sum_{k=1}^nU_{k,i}$ and $\bar{\bU}_{(\cdot,-i)}=(\bar{\bU}_{\cdot,-i},\ldots,\bar{\bU}_{\cdot,-i})^\top$ with dimension $n\times (d-1)$. For tuning parameter $\lambda$, let $\lambda_{n_m,i,m}$ represent the $i$-th tuning parameter for binary trait $m$, which depends on the sample size $n_m$.
For a matrix $\bA=[a_{i,j}]\in \mathbb{R}^{d\times d}$, we denote the matrix $\ell_1$ norm, the matrix element-wise infinity norm and the matrix element-wise $\ell_1$-norm by $\|\bA\|=\mathrm{max}_{1\leq j\leq d}\sum_{i=1}^d|a_{i,j}|$, $|\bA|_\infty=\max_{i,j}|a_{i,j}|$ and $|\bA|_1=\sum_{i=1}^d\sum_{j=1}^d|a_{i,j}|$ respectively. $\bA_{i,-j}$ denote the $i$-th row of $\bA$ with its $j$-th entry removed and $\bA_{-i,j}$ denote the $j$-th column of $\bA$ with its $i$-th entry removed. $\bA_{-i,-j}$ denotes a $(d-1)\times(d-1)$ matrix obtained by removing the $i$-th row and $j$-th column of $\bA$. We say $\bA$ is $k$-sparse if each row/column has at most $k$ nonzero entries.
For a symmetric matrix $\bA\in \mathbb{R}^{d\times d}$, we use $\lambda_{\mathrm{min}}(\bA)$ and $\lambda_{\max}(\bA)$ to denote the smallest and largest eigenvalues of $\bA$ respectively. Besides, we define a $d(d-1)/2$-dimension vector
\[
\begin{array}{ll}
{\rm trivec}(\bA)&=(a_{21}, \ldots,a_{d1},a_{32}, \ldots, a_{3d},\ldots, a_{(d-1)d})^\top
\end{array}
\]
which is obtained by concatenating the lower triangular part of $\bA$ column by column. We use $a_{i_sj_s}$ to denote the $s$-th entry of ${\rm trivec}(\bA)$.
For two sequences of real numbers $\{a_n\}$ and $\{b_n\}$, we write $a_n=O(b_n)$ if there exists a constant $C$ such that $|a_n|\leq C|b_n|$ holds for all $n$, write $a_n=\bo(b_n)$ if $\lim_{n\rightarrow \infty} a_n/b_n=0$, and write $a_n \asymp b_n$ if there exist constants $c$ and $C$ such that $c\leq a_n/b_n \leq C$ for all $n$.
For a sequence of random variables $\{\xi_1,\xi_2,\ldots\}$, we use $\lim_{n\rightarrow\infty}\xi_n=\xi$ to denote that the sequence $\{\xi_n\}$ converges in probability towards $\xi$ as $n\rightarrow\infty$. For simplicity, we also use $\xi_n=o_p(1)$ to denote $\lim_{n\rightarrow\infty}\xi_n=0$. For random variables $\xi$ and $\eta$, we use ${\rm Cov}(\xi,\eta)$ and ${\rm Corr}(\xi,\eta)$ to denote the covariance and correlation coefficients between $\xi$ and $\eta$. Let ${\rm Var}(\xi)$ be the variance of random variable $\xi$.
For a set $\mathcal{H}$, denote by $\#\{\mathcal{H}\}$ the cardinality of $\mathcal{H}$.
\subsection{General Setup}
Let $\bX=(X_1,\ldots,X_d)^\top$ and $\bY=(Y_1,\ldots,Y_d)^\top$ be two $d$-dimensional random vectors
independent of each other. $\bX_1,\ldots,\bX_{n_1}$ are independent and identically distributed (i.i.d.)
random samples from $\bX\thicksim N(\bu_1,\bSigma_1)$ with $\bX_k=(X_{k,1},X_{k,2},\ldots,X_{k,d})^\top$. Similarly, $\bY_1,\ldots, \bY_{n_2}$ are i.i.d. random
samples from $\bY\thicksim (\bu_2,\bSigma_2)$ with $\bY_k=(Y_{k,1},Y_{k,2},\ldots,Y_{k,d})^\top$. Let $\Xb=(\bX_1,\ldots,\bX_{n_1})^\top$ and $\Yb=(\bY_1,\ldots,\bY_{n_1})^\top$ denote the data matrices. Let $\bSigma_m=(\sigma_{i,j,m})$ and $\bOmega_m=(\omega_{i,j,m})=\bSigma_m^{-1}$ for $m=1,2$. Let $\bbeta_{i,1}=(\beta_{1,i,1},\ldots,\beta_{d-1,i,1})^\top$ denote the regression coefficients of $X_{k,i}$ regressed on the rest of the entries of $\bX_k$ and let $\bbeta_{i,2}=(\beta_{1,i,2},\ldots,\beta_{d-1,i,2})^\top$ denote the regression coefficients of $Y_{k,i}$ regressed on the rest of the entries of $\bY_k$.
In the Gaussian setting, the precision matrix can be described in terms of regression models. Specifically:
\begin{equation}\label{PrecisonExpression1}
\begin{aligned}
X_{k,i}&=\alpha_{i,1}+\bX_{k,-i}^\top\bbeta_{i,1}+\epsilon_{k,i,1}, \\%& (i=1,\ldots,d;\ \ k=1,\ldots n_1),\\
Y_{k,i}&=\alpha_{i,2}+\bY_{k,-i}^\top\bbeta_{i,2}+\epsilon_{k,i,2},
\end{aligned}
\end{equation}
where the error terms $\epsilon_{k,i,m}$ follow normal distribution with mean zero and variance $$\{\sigma_{i,i,m}-\bSigma_{i,-i,m}(\bSigma_{-i,-i,m})^{-1}\bSigma_{-i,i,m}\}.$$ and $\epsilon_{k,i,1}$, $\epsilon_{k,i,2}$ are independent of $\bX_{k,-i}$ and $\bY_{k,-i}$ respectively.
Besides, we have $\alpha_{i,m}=\mu_{i,m}-\bSigma_{i,-i,m}\bSigma_{-i,-i,m}^{-1}\bmu_{-i,m}$. The regression coefficient vectors $\bbeta_{i,m}$ and error terms $\epsilon_{k,i,m}$ satisfy
\begin{equation*}
\begin{aligned}
\bbeta_{i,m}=-\omega_{i,i,m}^{-1}\bOmega_{-i,i,m}, \qquad \qquad
r_{i,j,m}=\Cov\big(\epsilon_{k,i,m},\epsilon_{k,j,m}\big)=\frac{\omega_{i,j,m}}{\omega_{i,i,m}\omega_{j,j,m}}.
\end{aligned}
\end{equation*}
We aim to test the null hypothesis:
\[
\Hb_0:\bOmega_1=\bOmega_2 \ \ \mathrm{or} \ \ \mathrm{equivalentlly } \ \ \bDelta=\bOmega_1-\bOmega_2=0.
\]
Let $\widehat \bbeta_{i,m}=(\widehat{\beta}_{1,i,m},\ldots,\widehat{\beta}_{d-1,i,m})^\top$ be estimators of $\bbeta_{i,m}$ by Lasso or Dantzig selector satisfying
\begin{equation}\label{betaestimatebound}
\begin{aligned}
\mathop{\mathrm{max}}_{1\leq i\leq d}\big\|\widehat \bbeta_{i,m}-\bbeta_{i,m}\big\|_1=\bo_{p}\big\{(\log d)^{-1}\big\},
\qquad \qquad
\mathop{\mathrm{\max}}_{1\leq i\leq d}\big\|\widehat \bbeta_{i,m}-\bbeta_{i,m}\big\|_2=\bo_{p}\big\{(n_m\log d)^{-1/4}\big\}.
\end{aligned}
\end{equation}
Under the sparsity conditions $\max_{1\le i\le d}|\beta_{i}|_{0}=\bo\big(n^{1/2}/(\log d)^{3/2}\big)$, together with with Assumption \textbf{(B)} in Section \ref{subsec::assumption}, both the Lasso and Dantzig selector estimators satisfy the condition in \eqref{betaestimatebound} according to the Proposition 4.1 in \cite{Liu2013Gaussian}.
With the $\widehat \bbeta_{i,m}$, define the residuals by
\begin{equation}\label{equ:residual}
\begin{aligned}
\widehat \epsilon_{k,i,1}&=X_{k,i}-\bar{X}_i-(\bX_{k,-i}-\bar{\bX}_{\cdot,-i})^\top\widehat \bbeta_{i,1}, \\ \widehat \epsilon_{k,i,2}&=Y_{k,i}-\bar{Y}_i-(\bY_{k,-i}-\bar{\bY}_{\cdot,-i})^\top\widehat \bbeta_{i,2}.
\end{aligned}
\end{equation}
Let $\widetilde{r}_{i,j,m}=(1/{n_m})\sum_{k=1}^{n_m}\widehat \epsilon_{k,i,m}\widehat \epsilon_{k,j,m}$ be the empirical covariance between $\{\widehat \epsilon_{k,i,m}:k=1,\ldots,n_m\}$ and $\{\widehat \epsilon_{k,j,m}:k=1,\ldots,n_m\}$. Similarly, let $\widetilde{R}_{i,j,m}=(1/{n_m})\sum_{k=1}^{n_m}(\epsilon_{k,i,m}-\bar{\epsilon}_{i,m})(\epsilon_{k,j,m}-\bar{\epsilon}_{j,m})$ be the empirical covariance between $\{\epsilon_{k,i,m}:k=1,\ldots,n_m\}$ and $\{\epsilon_{k,j,m}:k=1,\ldots,n_m\}$.
Lemma 2 in \cite{Xia2015Testing} shows that
\begin{equation}\label{rhatRt}
\begin{aligned}
\widetilde{r}_{i,j,m}=\widetilde{R}_{i,j,m}-\widetilde{r}_{i,i,m}
(\widehat{\beta}_{i,j,m}-\beta_{i,j,m})-\widetilde{r}_{j,j,m}(\widehat{\beta}_{j-1,i,m}-\beta_{j-1,i,m})+\bo_{p}\big\{
(n_m\log d)^{-1/2}\big\}.
\end{aligned}
\end{equation}
For $1\leq i<j\leq d$, it can be shown that
$$\beta_{i,j,m}=-\omega_{i,j,m}/\omega_{j,j,m}, \ \ \beta_{j-1,i,m}=-\omega_{i,j,m}/\omega_{i,i,m}.$$
A bias-corrected estimator of $r_{i,j,m}(1\leq i<j\leq d)$ is initially proposed by \cite{Liu2013Gaussian}:
\begin{equation}\label{rijmestimatebyXia}
\widehat{r}_{i,j,m}\!=\!\!-\!\big(\widetilde{r}_{i,j,m}\!+\!\widetilde{r}_{i,i,m}\widehat{\beta}_{i,j,m}\!+\!\widetilde{r}_{j,j,m}\widehat{\beta}_{j\!-\!1,i,m}\big),
\end{equation}
For $i=j$, the Lemma 2 of \cite{Xia2015Testing} showed that
\begin{equation}\label{tilderriibound}
\mathop{\mathrm{\max}}_{1\leq i\leq d}\big|\widetilde{r}_{i,i,m}-r_{i,i,m}\big|=\bO_{p}\big\{(\log d/n_m)^{1/2}\big\},
\end{equation}
which implies that $\widehat{r}_{i,i,m}=\widetilde{r}_{i,i,m}$ is a nearly unbiased estimator of $r_{i,i,m}$.
Thus one can naturally estimate $\omega_{i,j,m}$ by
\begin{equation}\label{def:Tijm}
T_{i,j,m}=\frac{\widehat{r}_{i,j,m}}{\widehat{r}_{i,i,m}\widehat{r}_{j,j,m}}
, 1\leq i\leq j\leq d,
\end{equation}
and test $\Hb_0:\bDelta=0$ based on the estimators $\mathcal{T}=\{T_{i,j,1}-T_{i,j,2},1\leq i\leq j\leq d\}$.
Considering the heteroscedasticity of estimators in $\mathcal{T}$, Literature \cite{Xia2015Testing} proposed the following test statistic for the null hypothesis $\Hb_0$:
\[
M_n=\mathop{\mathrm{\max}}_{1\leq i \le j \leq d}W_{i,j}^2=\mathop{\mathrm{\max}}_{1\leq i\le j\leq d}\frac{(T_{i,j,1}-T_{i,j,2})^2}{\widehat{\theta}_{i,j,1}+\widehat{\theta}_{i,j,2}},
\]
where
\begin{equation}\label{Wij}
\begin{aligned}
W_{i,j}=\frac{T_{i,j,1}-T_{i,j,2}}{(\widehat{\theta}_{i,j,1}+\widehat{\theta}_{i,j,2})^{1/2}}, \qquad \qquad \widehat{\theta}_{i,j,m}=\text{Var}(T_{i,j,m})=
\frac{1+\widehat{\beta}_{i,j,m}^2\widehat{r}_{i,i,m}/\widehat{r}_{j,j,m}}{n_m\widehat{r}_{i,i,m}\widehat{r}_{j,j,m}}.
\end{aligned}
\end{equation}
Literature \cite{Xia2015Testing} obtained the asymptotic null distribution of $M_n$ under suitable conditions, which is type I extreme value distribution. However, this limiting distribution of maximum-type statistic based approach has
two fatal limitations. Firstly, the convergence rate of extreme-value statistics is notoriously slow
and the process of getting the limiting distribution ignores the correlation between coordinates. Secondly, the maximum-type statistic is particularly powerful against large and sparse signal alternatives, however, it is powerless against small and dense signal alternatives.
In this paper we develop new tests for hypothesis (\ref{hyp:omega2test0}), which
are adaptive to a large variety of alternative scenarios in high dimensions. We utilize the multiplier bootstrap method to approximate the asymptotic distribution of the proposed test statistics and thus overcomes the limitation of the extreme-value-type statistic $M_n$.
\section{Methodology}\label{sec:methodology}
As the extrem-value-type statistic is only powerful against the sparse large alternatives, we aim to provide a data-driven adaptive test for the hypothesis (\ref{hyp:omega2test0}) in this section. In Section \ref{sec:individual test}, a family of tests based on $(s_0,p)$-norm are proposed. The $(s_0,p)$-norm was first introduced in \cite{Zhou2018Unified}. The tests based on different $p$ have different powers under different alternative scenarios. For example, $(s_0,\infty)$-norm based test are sensitive to large perturbations on a small number of entries of $\bOmega_1-\bOmega_2$. Moreover, $(s_0,2)$-norm are sensitive to small perturbations on a large number of entries of $\bOmega_1-\bOmega_2$. By combining a family of $(s_0,p)$-norm based tests with various $p$, we present our adaptive test in Section \ref{sec:adaptive test}
\subsection{The $(s_0,p)$-norm based test statistics}\label{sec:individual test}
In this section, we provide some ($s_0, p$)-norm based tests. Recall that we have defined the $W_{i,j}$ in (\ref{Wij}). Based on the statistics in $\mathcal{W}=\{W_{i,j},1\leq i\leq j\leq d\}$, define $\Wb=(W_{i,j})_{d\times d}$. we then propose our test statistic based on $(s_0,p)$-norm of the vector $\mathrm{trivec}(\Wb)$. Specifically, we propose the $(s_0,p)$-norm based test statistic is
\begin{equation}\label{teststatistic}
N_{(s_0,p)}=\big\|\mathrm{trivec}(\Wb)\big\|_{(s_0,p)}.
\end{equation}
With the proposed test statistic, we still need to obtain the critical value or $P$-value to test (\ref{hyp:omega2test0}). To this end, we develop a multiplier bootstrap method to approximate the limiting distribution of the test statistic $N_{(s_0,p)}$.
In the high dimensional setting, \cite{chernozukov2014central} introduced the multiplier bootstrap method for the sum of independent random vectors. In detail, let $\bZ_1, \ldots,\bZ_{n}$ be independent zero mean random vectors in ${\mathbb R}^d$ with
$\bZ_k=(Z_{k1},\ldots,Z_{kd})^\top$. The bootstrap sample for the sample mean
$n^{-1}\sum_{k=1}^n\bZ_k$ then becomes $n^{-1}\sum_{k=1}^n\varepsilon_k\bZ_k$, where $\varepsilon_1, \varepsilon_2,\ldots,\varepsilon_n$ are independent standard normal random variables. Inspired by the multiplier bootstrap method in \citep{chernozukov2014central}, we propose a specific multiplier bootstrap procedure for the problem here. In detail, we generate independent samples $\eta_{1,1}^b,\ldots,\eta_{1,n_1}^b$ and
$\eta_{2,1}^b,\ldots,\eta_{2,n_2}^b$ from $\eta\sim N(0,1)$ for $b=1,\ldots,B$. Similarly, we set the $b$-th multiplier bootstrap sample for $\widetilde{r}_{i,j,m}, 1\leq i\leq j\leq d$ as
\begin{equation}\label{def:originalbootsr}
\widetilde{r}_{i,j,m}^b=\frac{1}{n_m}\sum_{k=1}^{n_m}\eta_{m,k}^b(\widehat \epsilon_{k,i,m}\widehat \epsilon_{k,j,m}-\widetilde{r}_{i,j,m}).
\end{equation}
Considering the definitions of $\widehat{r}_{i,j,m}$ in \eqref{rijmestimatebyXia}, we set its $b$-th bootstrap sample as
\[
\widehat{r}_{i,j,m}^b=-\big(\widetilde{r}_{i,j,m}^b+\widetilde{r}_{i,i,m}^b\widehat{\beta}_{i,j,m}+
\widetilde{r}_{j,j,m}^b\widehat{\beta}_{j-1,i,m}\big)
\]
for $1\leq i<j\leq d$ and $\widehat{r}_{i,i,m}^b=\widetilde{r}_{i,j,m}^b$.
Further, by the definitions of $T_{i,j,m}$ and $W_{i,j}$ in \eqref{def:Tijm} and \eqref{Wij} respectively, we then get the $b$-th bootstrap sample of $T_{i,j,m}$ and $W_{i,j}$ as
\begin{equation}\label{Wbootsineqj}
\begin{aligned}
T_{i,j,m}^b=\frac{\widehat{r}_{i,j,m}^b}{\widehat{r}_{i,i,m}\widehat{r}_{j,j,m}}, 1\leq i\leq j\leq d, \qquad \qquad
W_{i,j}^b=\frac{T_{i,j,1}^b-T_{i,j,2}^b}{(\widehat{\theta}_{i,j,1}+\widehat{\theta}_{i,j,2})^{1/2}}, 1\leq i\leq j\leq d.
\end{aligned}
\end{equation}
With $W_{i,j}^b$, we set $\Wb^b=(W_{i,j}^b)_{d\times d}$ and finally obtain the bootstrap samples of $N_{(s_0,p)}$ as
\begin{equation}\label{test-statistic-bootstrap-sample}
N_{(s_0,p)}^b=\|\mathrm{trivec}(\Wb^b)\|_{(s_0,p)}, \hspace{2em} b=1\ldots B.
\end{equation}
Given the significance level $\alpha$,
we use $t^{N}_{\alpha,(s_0,p)}$ to denote the oracle critical values of $N_{(s_0,p)}$ . Given the bootstrap samples, we then estimate $t^{N}_{\alpha,(s_0,p)}$ by
\begin{equation}
\hat{t}^{N}_{\alpha,(s_0,p)}=\inf\Big\{t\in {\mathbb R}: \frac{1}{B}\sum_{b=1}^B\ind \{N^b_{(s_0,p)}\le t\}> 1-\alpha\Big\}.
\end{equation}
Therefore, we obtain the $(s_0,p)$-norm based tests for (\ref{hyp:omega2test0}) as
\begin{equation}
T^{N}_{(s_0,p)}=\ind\big\{N_{(s_0,p)}\ge \hat{t}^N_{\alpha,(s_0,p)}\big\}.
\end{equation}
We reject $\Hb_0$ of (\ref{hyp:omega2test0}) if and only if $T^N_{(s_0,p)}=1$.
Accordingly, we estimate $N_{(s_0,p)}$'s oracle $P$-values $P^N_{(s_0,p)}$ by
\begin{equation}\label{def:PhatN}
\hat{P}^N_{(s_0,p)}=\frac{\sum_{b=1}^B\ind\{N^b_{(s_0,p)}> N_{(s_0,p)}\}}{B+1}.
\end{equation}
Therefore, given a significance level $\alpha$, we reject $\Hb_{0}$ of (\ref{hyp:omega2test0}) if and only if $\hat{P}^N_{(s_0,p)}\le
\alpha$.
\begin{algorithm}[ht]
\caption{A bootstrap procedure to obtain $N_{\rm ad}$}\label{alg:first}
\raggedright
{\bf Input:} $\mathcal{X}$.\\
{\bf Output:} $N^1_{(s_0,p)},\ldots,N^B_{(s_0,p)}$ with $p\in \mathcal{P}$, and $N_{\rm ad}$.\\
\begin{algorithmic}[1]
\Procedure{}{}
\State $N_{(s_0,p)} \!=\! \|{\mathrm{trivec}(\Wb)}\|_{(s_0,p)} \text{ with } {\Wb} \!=\! (W_{i,j})_{d\times d}^\top$ and $W_{i,j} \!=\! {(T_{i,j,1} \!-\! T_{i,j,2})}\big/{(\widehat{\theta}_{i,j,1} \!+\! \widehat{\theta}_{i,j,2})^{1/2}}$.
\For{ $b\leftarrow 1$ {\bf to} $B$}
\State Sample independent standard normal random variables $\{\eta^b_{1,1},\ldots,\eta^b_{1,n_m}\}$ for $m=1,2$.
\State For $1\leq i\leq j\leq d$, set $\widetilde{r}_{i,j,m}^b=({1}/{n_m})\sum_{k=1}^{n_m}\eta_{m,k}^b\big(\widehat \epsilon_{k,i,m}\widehat \epsilon_{k,j,m}-\widetilde{r}_{i,j,m}\big)$.
\State Set $\widehat{r}_{i,i,m}^b=\widetilde{r}_{i,i,m}^b$, and set $\widehat{r}_{i,j,m}^b\!=\!\!-\!\big(\widetilde{r}_{i,j,m}^b\!+\!\widetilde{r}_{i,i,m}^b\widehat{\beta}_{i,j,m}\!+\!\widetilde{r}_{j,j,m}^b\widehat{\beta}_{j-1,i,m}\big)$ for $1\leq i<j\leq d$.
\State Set $T_{i,j,m}^b\!=\!{\widehat{r}_{i,j,m}^b}\big/({\widehat{r}_{i,i,m}\widehat{r}_{j,j,m}})$, $1\leq i\leq j\leq d$.
\State Set $W_{i,j}^b=({T_{i,j,1}^b-T_{i,j,2}^b})\big/{(\widehat{\theta}_{i,j,1}+\widehat{\theta}_{i,j,2})^{1/2}}$, $1\leq i\le j\leq d$ and $\Wb^b=(W_{i,j}^b)_{d\times d}$.
\For{$p$ {\bf in} $\mathcal{P}$}
\State $N^b_{(s_0,p)}=\|{\mathrm{trivec}(\Wb}^b)\|_{(s_0,p)}$ with ${\Wb}^b=(W_{i,j}^b)_{d\times d}$.
\EndFor
\EndFor
\State $\hat{P}^N_{(s_0,p)}={\sum_{b=1}^B\ind\{N^b_{(s_0,p)}> N_{(s_0,p)}\}}/{(B+1)}$ for $p\in \mathcal{P}$.
\State $N_{\rm ad}=\min_{p\in\mathcal{P}}\hat{P}^N_{(s_0,p)}$.
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Data adaptive combined test}\label{sec:adaptive test}
After providing the $(s_0,p)$-norm based tests for each individual $p$, we propose a data-driven adaptive test by combining a group of the $(s_0,p)$-norm based tests in this section.
Set $\mathcal{P}=\{p_1,p_2,\cdots\}$ as a finite set of positive numbers, and set the size of $\cP$ as a finite fixed constant. Then we combine the $(s_0,p)$-norm based test with $p \in \mathcal{P}$ by taking the minimum $P$-value of these tests. Specifically, we set the data-adaptive test statistic $N_{\rm ad}$ as
\begin{equation}\label{def:Nad}
N_{\rm ad}=\min_{p\in\mathcal{P}}\hat{P}^N_{(s_0,p)}.
\end{equation}
The detail process of getting $N_{\rm ad}$ is in Algorithm \ref{alg:first}.
The set $\{\mathcal{P}\}$ can be chosen by users with prior information about the alternative patterns. If one knows the alternative pattern, then he/she can choose the set $\mathcal{P}$ accordingly to improve the power performance of the data adaptive test. For example, let $\mathcal{P}$ consists of large values of $p$ with prior information that the alternative pattern is sparse. If one knows nothing about the alternative pattern, then a balanced set $\mathcal{P}$ containing both large and small $p$ is recommended. For example, one may choose the set $\mathcal{P}$ to be $\{1,2,3,4,5,\infty\}$.
For the data adaptive test, we need to get the $P$-value. It's difficult to get the limiting distribution for the $(s_0,p)$-norm based statistics, not to mention for the data adaptive test statistic. Hence, the intuitive way is to do a double loop bootstrap procedure to get the empirical distribution for our data adaptive test. But this way is too costly for computation. As is shown by Algorithm \ref{alg:first}, in addition to the data adaptive statistic $N_{\rm ad}$, we also obtain the bootstrap samples for $N_{(s_0,p)}$, i.e, $\big\{N^1_{(s_0,p)},\ldots,N^B_{(s_0,p)}\big\}$. Therefore, we can recycle the bootstrap samples to accelerate our computation speed. Specifically, for $b=1,\ldots,B$ and $p\in\mathcal{P}$, we set
\[
\hat{P}^{b,N}_{(s_0,p)}=\frac{\sum_{b_1\ne b} \ind\{N^{b_1}_{(s_0,p)}> N^{b}_{(s_0,p)}\}}{B}.
\]
We use $N^b_{\rm ad}=\min_{p\in\mathcal{P}
}\hat{P}^{b,N}_{(s_0,p)}$ as the bootstrap sample for $N_{\rm ad}$. We then
estimate the oracle $P$-value of $N_{\rm ad}$ by
\begin{equation}\label{def:hatPWad}
\hat{P}_{\rm ad}^N = \frac{\big(\sum_{b=1}^{B}\ind\{N^b_{\rm ad}\le N_{\rm ad}\}\big)+1}{B+1}.
\end{equation}
For more details, see Algorithm \ref{alg:adjust}. The samples $N^1_{\rm ad},\ldots,N^B_{\rm ad}$ are nonindependent. But as $n,B\rightarrow\infty$, they are asymptotically independent. Hence, it dosen't affect the consistency of $\hat{P}^N_{\rm ad}$.
After getting the estimated $P$-values of the data-adaptive tests $N_{\rm ad}$, given the significance level $\alpha$, we set
\begin{equation}\label{def:TWNAd}
T^N_{\rm ad}=\ind\{\hat{P}^N_{\rm ad}\le\alpha\}.
\end{equation}
Therefore, we reject $\Hb_0$ of (\ref{hyp:omega2test0}) if and only if $T^N_{\rm ad}=1$.
\begin{algorithm}[ht]
\caption{A low-cost bootstrap procedure}\label{alg:adjust}
\raggedright
{\bf Input:} $\mathcal{X}$ and $N^1_{(s_0,p)},\ldots,N^B_{(s_0,p)}$ for $p\in \mathcal{P}$.\\
\raggedright
{\bf Output:} $N_{\rm ad}^1,\ldots, N_{\rm ad}^B$.
\begin{algorithmic}[1]
\Procedure{}{}
\For{$b \leftarrow 1$ {\bf to} $B$}
\For{ $p$ {\bf in} $\mathcal{P}$}
\State $\hat{P}^{b,N}_{(s_0,p)}={\sum_{b_1\ne b} \ind\{N^{b_1}_{(s_0,p)}> N^{b}_{(s_0,p)}\}}/{B}$.
\EndFor
\State $N^b_{\rm ad}=\min_{p\in\mathcal{P}}\hat{P}^{b,N}_{(s_0,p)}$.
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Theoretical properties}\label{sec::theoretical}
In this section, we investigate the theoretical properties of our proposed test. Firstly, some assumptions are introduced in Section \ref{subsec::assumption}. In Section \ref{subsec:theoretical}, we verify the validity of multiplier bootstrap which is used in Section \ref{sec:methodology} and then analyze the theoretical properties of the proposed test.
\subsection{Assumptions}\label{subsec::assumption}
In this section, we introduce some assumptions that are commonly used in high-dimensional analysis.
\vspace{0.8em}
\textbf{(A)} Set $n =\mathrm{max}(n_1,n_2)$, there exists some $0<\delta<1/7$ such that $s_{0}^{2}\log(d)=\bo(n^{\delta})$ hold, where $n_1\asymp n_2\asymp n$.
\vspace{0.8em}
Assumption \textbf{A} allows $s_0$ and $d$ go to infinity as long as $s_{0}^{2}\log(d)=\bo(n^{\delta})$ hold. By using the multiplier bootstrap to get the critical values for our tests, we need some more assumptions compared with \cite{Xia2015Testing}. Other than the Assumption \textbf{(A)}, we also introduce a more strong Assumption \textbf{(A)$'$} to state the scaling of $s_0$, $d$ and $n$. Before stating the next assumption, we need some additional notations. Let $U_{i,j,m}=\frac{1}{n_{m}} \sum_{k=1}^{n_m} \Big\{ \epsilon_{k,i,m} \epsilon_{k,j,m} - \mathbb{E}\big(\epsilon_{k,i,m}\epsilon_{k,j,m}\big) \Big\}$ and define $\tilde{U}_{i,j,m}=(r_{i,j,m}-U_{i,j,m})/r_{i,i,m}r_{j,j,m}$ with $1\le i,j\le d$, $m=1,2$.
Define $\tilde{\Ub}_{m}=\big( \tilde{U}_{i,j,m}\big)$ as a square matrix of order $d$ and denote
the covariance matrix of $\text{trivec}(\tilde{\Ub}_{m})$ as $\bSigma_{m}^{\tilde{U}}=(\sigma^{\tilde{U}}_{s,t,m})_{1\le s,t\le d(d-1)/2}$, where
\[
\sigma^{\tilde{U}}_{s,t,m} = \left\{\
\begin{aligned}
\theta_{i,j,m}=
\frac{1+ \beta_{i,j,m}^{2} r_{i,i,m}/r_{j,j,m}}{n_m r_{i,i,m}r_{j,j,m}}, s=t,\\
\frac{r_{i_{1},i_{2},m} r_{j_{1},j_{2},m}+r_{i_{1 }, j_{2},m}r_{i_{2},j_{1},m}}{n_m r_{i_1,i_1,m} r_{j_1,j_1,m} r_{i_2,i_2,m} r_{j_2,j_2,m}}, s\neq t,
\end{aligned}
\right.
\]
with $1\le i<j\le d$, $1\le i_1<j_1\le d$, $1\le i_2<j_2\le d$, $i_1\neq i_2$, $j_1\neq j_2$, $m=1,2$.
Let $\bG$ be a Gaussian random vector in ${\mathbb R}^{d(d-1)/2}$ with mean zero and covariance matrix $\mathbf{R}^{\tilde{U}}_{12}$, where $\mathbf{R}^{\tilde{U}}_{12}=(\mathbf{D}^{\tilde{U}}_{12})^{-1/2}\bSigma^{\tilde{U}}_{12}(\mathbf{D}^{\tilde{U}}_{12})^{-1/2}$ with $\bSigma^{\tilde{U}}_{12}=\bSigma^{\tilde{U}}_1/n_1+\bSigma^{\tilde{U}}_2/n_2$ and $\mathbf{D}^{\tilde{U}}_{12}=\mathrm{Diag}(\bSigma^{\tilde{U}}_{12})$. Set the probability density function (PDF) and the $\alpha$-quantile of $\|\bG\|_{(s_0,p)}$ as $f_{\bG,(s_0,p)}$ and $c_{ (s_0,p)}(\alpha)$ respectively. We then define $h_{T}(z)$ as
\begin{equation*}
\begin{aligned}
h_{T}(z)=\max_{p\in \mathcal{P}}\max_{x\in C_{(s_0,p)}(z)}f^{-1}_{\bG,(s_0,p)}(x) \\
\text{with}\quad C_{(s_0,p)}(z)=[c_{(s_0,p)}(z),c_{(s_0,p)}(1-z)].
\end{aligned}
\end{equation*}
With these new notations, we introduce the following assumption.
\vspace{0.8em}
\textbf{(A)$'$} Define $n =\mathrm{max}(n_1,n_2)$. We assume that $h_{T}^{0.6}(z)s_{0}^{2}\log d=\bo(n^{1/10})$ holds for any $0<z<1$ as $n, d\rightarrow\infty$ and $n_1\asymp n_2\asymp n$.
\vspace{0.8em}
{Assumption {\bf (A)$'$} is more stringent. It is critical to guarantee the uniform convergence of the distribution functions and the corresponding quantile functions of the test statistics $N_{(s_0,p)}$ for any $p \in \mathcal{P}$. }
The next two mild assumptions are often used in high dimensional setting, especially when the inference for covariance matrix and precision matrix are involved.
\vspace{0.8em}
\textbf{(B)} {
There exist some positive constants $C_0<C_1$,
such that $\lambda_{\min}(\bOmega_m)\ge C_0$ and $\lambda_{\max}(\bOmega_m) \le C_1$, with $m=1,2$.}
There exists some $\tau>0$ such that $|A_\tau| = \bo(d^{1/16})$ where $A_{\tau}= \{(i,j):|w_{i,j,m}|\ge(\log d)^{-2-\tau},1\le i<j\le d, \text{ for } m=1 \text{ or } 2\}$.
\vspace{0.8em}
\vspace{0.8em}
\textbf{(C)} Let $D_m$ be the diagonal of $\bOmega_{m}$ and let $(\eta_{i,j,m})=D^{-1/2}_{m}\bOmega_{m}D^{-1/2}_{m}$, for $m=1,2$. Assume that $\max_{1\le i\le j\le d}|\eta_{i,j,m}|\le \eta_{m}\le c$, where $0<c<1$ is a constant.
\vspace{0.8em}
{\textbf{(D)} Suppose $\max_{1\le i\le d}s_{i,m}=\big(n^{1/2}/(\log d)^{3/2}\big)$, where $s_{i,m}$ is sparsity for the $i$-th row or column of $\bOmega_{m}$ for $m=1,2$.
}
{Note that $\bbeta_{i,m}=-\omega_{i,i,m}^{-1}\bOmega_{-i,i,m}$, then the sparsity conditions of the Proposition 4.1 in \cite{Liu2013Gaussian} are automatically satisfied under Assumption \textbf{(D)}.}
\subsection{Theoretical analysis}\label{subsec:theoretical}
After introducing some needing assumptions, we analyze the theoretical properties of our test. Due to the complicated structure of our test statistics, we use the multiplier bootstrap to get the critical values for our test in Section \ref{sec:methodology}. But this procedure is different from \cite{Zhou2018Unified}. Specifically, other than the testing statistics cannot be rewritten as a sum of independent random variables, there are also some bias correction terms. Hence, we need to justify the validity of this multiplier bootstrap.
\begin{theorem}\label{theorem:bootstrap}
Suppose Assumptions {\bf (A)}-{\bf(D)} hold. Under the null hypothesis $\Hb_0$ of (\ref{hyp:omega2test0}), we have as $n, d\rightarrow\infty$,
\begin{equation}\label{equ:bootstrap}
\mathop{\mathrm{sup}} \limits_{z\in(0,\infty)}\Big|{\mathbb P}(N_{(s_0,p)}\leq z)\!-\!{\mathbb P}(N^{b}_{(s_0,p)} \!\leq\! z|\mathcal{X},\mathcal{Y})\Big|\!=\!\bo(1).
\end{equation}
\end{theorem}
Under the Gaussian distribution setting, it is easily to check that the sub-exponential distribution assumption and the moment condition in \cite{chernozukov2014central} are satisfied. Hence, under milder conditions, Theorem \ref{theorem:bootstrap} verifies the validity of the multiplier bootstrap method. The proof of Theorem \ref{theorem:bootstrap} is in the Appendix.
{By Theorem \eqref{theorem:bootstrap} hold, it's easy to prove that the size of $T^{N}_{(s_0,p)}$ asymptotically coverges to pre-specified significance level $\alpha$.}
\begin{corollary}
Suppose Assumptions {\bf (A)}-{\bf(D)} hold. Under the null hypothesis $\Hb_0$ of (\ref{hyp:omega2test0}), we have
\begin{equation*}
\mathbb{P}_{\Hb_0}(T^{N}_{(d_0,p)}=1)\rightarrow \alpha,
\end{equation*}
as $n, B\rightarrow\infty$.
\end{corollary}
With Theorem \ref{theorem:bootstrap}, we then show the theoretical properties of our data-adaptive test $T^{N}_{\rm ad}$. By the definition of $\hat{P}_{\rm ad}^N$ and $T_{\rm{ad}}^{N}$ in \eqref{def:hatPWad} and \eqref{def:TWNAd}, it can be seen that $T^{N}_{\rm ad}$ relies on the estimated $P$-values of $N_{(s_0,p)}$.
Therefore, we suppose the more stringent Assumption {\bf (A)$'$} holds, which guarantees the uniform convergence of the distribution
functions and the corresponding quantile functions of the test statistics $N_{(s_0,p)}$ for any $p \in \mathcal{P}$. Under this condition, we show that the empirical size of the data-adaptive test $T^{N}_{\rm ad}$ approximates to the pre-specified level $\alpha$.
\begin{theorem}\label{theorem:sizead}
Suppose Assumptions \textbf{(A)$'$}, \textbf{(B)}-\textbf{(D)} hold. Under the null hypothesis $\Hb_0$ of (\ref{hyp:omega2test0}), we have
\begin{equation*}
\mathbb{P}_{\Hb_0}(T^{N}_{\rm ad}=1)\rightarrow \alpha,
\end{equation*}
as $n, B\rightarrow\infty$.
\end{theorem}
After analyzing the asymptotic sizes of {the tests in Section \ref{sec:methodology},}
we summarize the asymptotic power properties in the following theorem. To analyze the power performance of $T^{N}_{\rm ad}$, we need to introduce some other notations. Define $\Wb^{*}=(W^{*}_{i,j})_{d\times d}$ with
\begin{equation*}\label{def:powerAssumption}
W^{*}_{i,j}=\big|\tilde{U}_{i,j,1}-\tilde{U}_{i,j,2}\big|\Big/\sqrt{\theta_{i,j,1}/n_1+\theta_{i,j,2}/n_2},
\end{equation*}
where $\theta_{i,j,m}$ are the diagonal elements of $\bSigma_{m}^{\tilde{U}}$, $m=1,2$. We then introduce the following theorem to characterize the asymptotic power properties of {$T^{N}_{(s_0,p)}$ and} $T^{N}_{\rm ad}$.
\begin{theorem}\label{thm:powerad}
Suppose Assumptions {\bf (B)}-{\bf(D)} hold and assume $\varepsilon_n=\bo(1)$, $\varepsilon_n\sqrt{\log d^2}\rightarrow\infty$ as $n,d\rightarrow\infty$.
\textbf{(a)} As $n,d \rightarrow \infty$, there exists some $\delta_{1}>0$ such that $\log(d)=\bo(n^{1/3})$ and $n=\bO(d^{2\delta_{1}})$. Assume $s_0=\bO\big((\log d)^{\delta_{2}}\big)$ for some postive constant $\delta_{2}$. Under the alternative hypothesis $\Hb_1$ of (\ref{hyp:omega2test0}) and with
\begin{equation*}
\begin{aligned}
\big\|{\rm trivec}(\Wb^{*})\big\|_{(s_0,p)}\ge s_0 (1+\varepsilon_n)
\big(\sqrt{2\log(d(d-1)/2)}+\sqrt{2\log (1/\alpha)}\big),
\end{aligned}
\end{equation*}
hold, we have ${\mathbb P}_{\Hb_1}\big(T^{N}_{(s_0,p)}=1\big)\rightarrow 1$ as $n, d, B\rightarrow \infty$.
\textbf{(b)} With Assumption {\bf (A)$'$} hold, and under the alternative hypothesis $\Hb_1$ of (\ref{hyp:omega2test0}) and suppose
\begin{equation*}
\begin{aligned}
\big\|{\rm trivec}(\Wb^{*})\big\|_{(s_0,p)}\!\ge\! s_0 (1\!+\!\varepsilon_n)\big(\sqrt{2\log(d(d\!-\!1)/2)}+\sqrt{2\log (\#\{\mathcal{P}\}/\alpha)}\big),
\end{aligned}
\end{equation*}
hold, we have ${\mathbb P}_{\Hb_1}\big(T^{N}_{\rm ad}=1\big)\rightarrow 1$ as $n, d, B\rightarrow \infty$.
\end{theorem}
By Theorem \ref{thm:powerad}, we show that the asymptotic
{ powers of $T^{N}_{(s_0,p)}$ and $T^{N}_{\rm ad}$ converge to 1 under the minimum signal condition on $\|{\rm trivec}(\Wb^{*})\|_{(s_0,p)}$.}
\section{Experiments}\label{sec:experiments}
\subsection{Simulation study} \label{subsec:simulation}
In this section, we conduct simulation study to investigate the empirical size and power of the proposed test. To show the adaptivity of our method, we compare it with recently developed method proposed by \cite{Xia2015Testing} under various model settings.
We denote the test proposed by \cite{Xia2015Testing} as $T_{\rm CX}$ for simplifing notations. To distinguish the adaptive test with different $s_0$, we denote the adaptive test with any fixed $s_0$ as $TD^{N}_{s_0,{\rm ad}}$.
\begin{figure*}[hbpt]
\includegraphics[width=16cm, height=18cm]{model1.pdf}
\caption{Empirical powers of various tests for {\bf Model 1}. The orange line with circles represents the adaptive test $T^{N}_{10,{\rm ad}}$, the blue line with triangles represents the adaptive test $T^{N}_{100,{\rm ad}}$, the red line with crosses represents the adaptive test $T^{N}_{500,{\rm ad}}$, the green line with diamonds represents the adaptive test $T^{N}_{1000,{\rm ad}}$, the black line with stars represents the $T_{\rm CX}$ test.$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$}\label{fig:model1}
\end{figure*}
In the simulation study, the sample sizes are set to be $n_1=n_2=200$, while the dimension $d=100$. Although the dimension $d$ seems to be small compared with the sample size, the parameters in the precision matrix which we are interested in are already much larger than the sample size ($d(d-1)/2$). In all simulations, the bootstrap sample sizes $B$ are set to be $1000$ and all the simulation results are based on $1000$ replications. Under the null hypothesis $\rm{\mathbf{H}_{0}}$, we set $\bOmega_2=\bOmega_1=\bOmega$. Under the alternative hypothesis $\rm{\mathbf{H}_{1}}$, we set $\bOmega_1=\bOmega+\delta \bI$ and $\bOmega_2=\bOmega+\bGamma+\delta \bI$, where $\bGamma=(\gamma_{i,j})_{d\times d}$ is a nonzero matrix and $\delta=|\lambda_{\min}(\bOmega+\bGamma)|+0.05$. Suppose there are $m_t$ nonzero entries of $\bGamma$. Specially, we random sample $m_t/2$ locations in the upper triangle of $\bGamma$ and set each with a magnitude $r$. By the symmetric requirement of $\bOmega_2$, the location and magnitude of the other $m_{t}/2$ nonzero entries of $\bGamma$ can be determined by its upper triangle. To show that our test is adaptive to various alternative patterns, we set the nonzero entries of the $\bGamma$ as $m_t=20,200,1000,2000$. The $m_t=20,200$ are to illustrate the sparse alternatives and $m_t=1000,2000$ are to represent the dense alternatives. For all the simulations, simulation data are generated from multivariate Gaussian distributions with mean $\zero$ and covariance matrices $\bSigma_1=(\bOmega_1)^{-1}$ and $\bSigma_2=(\bOmega_2)^{-1}$. The nominal significance level for all the tests are set to be $\alpha=0.05$. To study the empirical performance of the test, the following three models of $\bOmega$ are considered.
\vspace{1em}
\textbf{Model 1:} $\bOmega^*=(\omega_{i,j}^{*})$ where $\omega_{i,i}^{*}=1$, $\omega_{i,j}^{*}=0.5\times \text{Bernoulli}(1,0.5)$ for $i<j$ and $\omega_{j,i}^{*}=\omega_{i,j}^{*}$. $\bOmega=(\bOmega^*+\delta \bI)/(1+\delta)$ with $\delta=|\lambda_{\min}(\bOmega^*)|+0.05$.
\vspace{1em}
\textbf{Model 2:} $\bSigma^*=(\sigma^{*(1)}_{i,j})$ where $\sigma^{*(1)}_{i,i}=1$, $\sigma^{*(1)}_{i,j}=0.5$ for $2(k-1)+1\leq i\neq j \leq 2k$, where $k=1,\ldots,[p/2]$ and $\sigma^{*(1)}_{i,j}=0$ otherwise. $\bOmega=\{(\bSigma^*+\delta \bI)/(1+\delta)\}^{-1}$ with $\delta=|\lambda_{\min}(\bSigma_1^*)|+0.05$.
\vspace{1em}
\textbf{Model 3:} $\bOmega^*=(\omega_{i,j}^{*(1)})$ where $\omega_{i,i}^{*(1)}=1$, $\omega_{i,j}^{*(1)}=0.5\times \text{Bernoulli}(1,0.3)$ for $i<j$ and $\omega_{j,i}^{*(1)}=\omega_{i,j}^{1}$. Other than that, we set $\omega_{i,j}^{*(1)}=\omega_{j,i}^{*(1)}=0.5$ for $i=20(k-1)+1$ and $20(k-1)+2\leq j \leq20(k-1)+20$, $1\leq k\leq p/20$. $\omega_{i,j}^{*(1)}=0$ otherwise. $\bOmega=(\bOmega^*+\delta \bI)/(1+\delta)$ with $\delta=|\lambda_{\min}(\bOmega^*)|+0.05$.
\vspace{1em}
The performances of the test methods under various alternative patterns for Model 1 are shown in Figure \ref{fig:model1}.
In Figure \ref{fig:model1}, the orange line with circles represents the adaptive test $T^{N}_{10,{\rm ad}}$, the blue line with triangles represents the adaptive test $T^{N}_{100,{\rm ad}}$, the red line with crosses represents the adaptive test $T^{N}_{500,{\rm ad}}$, the green line with diamonds represents the adaptive test $T^{N}_{1000,{\rm ad}}$, the black line with stars represents the $T_{\rm CX}$ test proposed by \cite{Xia2015Testing}. The horizontal axis represents magnitude $r$ in the upper triangle of $\bGamma$, a larger value of $r$ indicates a stronger signal. The vertical axis represents the empirical powers of different tests, while $r=0$ corresponds to the empirical sizes.
From Figure \ref{fig:model1}, we can see that all the empirical sizes of different tests are under control. Under sparse alternative setting with $m_t=20$ (corresponds to the upper left panel in Figure \ref{fig:model1} with 20 non-equal signals), the empirical powers of maximum-norm based test $T_{\rm CX}$ are the highest and the empirical powers of the adaptive test with $s_0=10$ are still comparable though a little bit lower than CX test. Besides, with $s_0$ decreasing, the adaptive test tends to more powerful under sparse alternative setting. As the non-equal number $m_t$ becomes bigger, the empirical powers of the adaptive test with larger $s_0$ are getting better and better. Under the dense alternative (corresponds to the upper right panel and lower panels in Figure \ref{fig:model1} with more than 200 non-equal signals), the empirical powers of the adaptive tests are greater than those of the maximum-norm based CX tests with the magnitude $r$ larger than certain threshold. Although the empirical powers of adaptive test with different $s_0$ have some difference, the empirical powers of the adaptive test show some robustness for the small changes of $s_0$. From Figure \ref{fig:model1}, we can also see that the empirical power of $T^{N}_{500,{\rm ad}}$ and $T^{N}_{1000,{\rm ad}}$ are almost equal to each other.
At last, we point out that the influence of the parameter $s_0$ on the power performance is more complicated. However, by choosing $s_0$ close to half of the true number of nonzero signals $m_t/2$, the tests enjoy good performance. In practice, we can determine $s_0$ by the prior information or some empirical information.
The empirical results for Model $2$ and Model $3$ are similar as for Model 1 and thus are presented in the supplementary materials for saving space here.
\begin{figure}[!ht]
\centering
\includegraphics[width=12cm, height=8cm]{zmj.png}
\caption{The differential networks estimated for the Wnt signaling pathway. Orange edges show an increase in conditional dependency from control group to lung cancer patient group; grey edges show a decrease.}\label{fig1}
\end{figure}
\subsection{Real data analysis}\label{subsec:realdata}
In this section, we apply our adaptive test method to a gene expression data set which is associated with lung cancer. The data set is publicly available from the Gene Expression Omnibus (\url{https://www.ncbi.nlm.nih.gov/geo/}) at accession number GDS2771. The data set is made up of 22,283 microarray-derived gene expression measurements from large airway epithelial cells sampled from 97 patients with lung cancer, and 90 control patients. \cite{mazieres2005wnt} showed that the Wnt pathway associated with lung cancer and many other lung diseases such as interstitial lung disease (ILD) and asthma. The Wnt pathway is also implicated in the development of several types of cancers, such as gastric cancer \citep{Clements2002Beta}, breast cancer \citep{Howe2004Wnt}.
Hence, in this paper, we focus our analysis on the $188$ genes in the Wnt signaling pathway, with $97$ patients with lung cancer and $90$ control patients. Gene expression levels were analyzed on a logarithmic scale and each gene feature was standardized within each group. Although the true conditional dependence relationships are unknown, we believe that there exists some specific links among genes in the Wnt signaling pathway of the patients with lung cancer. Hence, we use our method to test whether the underlying precision matrices of the patients with cancer or not are equal to each other. {In the real example, we know nothing about the underlying alternative patterns. Hence, as to the choice of $s_0$, we propose to tune $s_0$ in a finite set $\mathcal{S}$. Specifically, we set the doubly tuned data-adaptive test statistic as}
\begin{equation}\label{equ:adaptiveall}
TN_{\rm ad}= \min_{p \in \mathcal{P}, s_0 \in \mathcal{S}} \widehat{P}^{N}_{(s_0,p)}.
\end{equation}
{As long as the cardinality of the set $\mathcal{S}$ is fixed, all the theoretical properties for the adaptive test $N_{\rm ad}$ with fixed $s_0$ still hold for $TN_{\rm ad}$. Furthermore, the simulation study in Section \ref{subsec:simulation} showed that the empirical powers of the adaptive tests show robustness to different $s_0$. Hence, assuming the cardinality of the set $\mathcal{S}$ to be finite is reasonable. Specifically, we choose $\mathcal{S}=\{10,50,100,500,1000,2000\}$ here. By the adaptive test $TN_{\rm ad}$, }
we reject the null hypothesis and think that there are difference for the underlying conditional dependence relationships. Hence, we use the differential network estimation approach in \cite{Zhao2015Direct} to estimate the differential network between the control group and the patient group. In detail, we choose the tuning parameter by the Bayesian information criterion (BIC) using the element-wise $L_{1}$ loss function. The differential network structure is given in Figure \ref{fig1}, in which the black edges represents the conditional correlations in lung cancer group are stronger compared with those in the control group, and the gray edges the other way around. From Figure \ref{fig1}, many potentially important genes for lung cancer are detected, such as WNT1, WNT2, WNT5A etc, see \cite{mazieres2005wnt}. By Figure \ref{fig1}, we see that RHOA and FZD6 are two hubs in this graph. Hence, we may conclude that the connections of these two genes to other genes are important for identifying the lung cancer. Actually the importance of FDZ and RHOA can be referred to \cite{Corda2017Non}, \cite{Rapp2017WNT}.
\section{Discussion}\label{sec:discussion}
In this paper we propose an adaptive approach for testing the equality of two precision matrices, i.e. to investigate whether the network of connected node pairs change from one state to another. In the Gaussian setting, the precision matrix can be described in terms of regression models and the elements of the precision matrix have a direct correspondence connection with the correlations of the error term. By Lasso or Dantzig selector, the regression coefficient estimator and the corresponding estimated regression errors are obtained. Based on the bias corrected estimator of the correlations of the error terms, we propose to construct a family of $(s_0,p)$-norm based test statistics with different $p$. By taking the minimum $P$-value of these tests, we construct an adaptive test statistics. We utilize multiplier bootstrap method to approximate the limiting distribution of the test statistic. Theoretical guarantees are provided for the proposed procedure and numerical study illustrates its good empirical performance under various alternatives.
The current work relies heavily on the Gaussian graph assumption which is sometimes restrictive in real application. In the future, we will consider the adaptive test of more general graphical models.
\section*{Acknowledgements}
Yong He's research is partially supported by the grant of the National Science Foundation of China (NSFC 11801316) and National Statistical Scientific Research Project (2018LY63). Xinsheng Zhang's research is partially supported by the grant of the National Science Foundation of China (NSFC 11571080).
\bibliographystyle{plain}
|
2,877,628,089,504 | arxiv | \section{Introduction}
Throughout all groups considered are finite. $G$ always denotes a group and its derived subgroup is denoted by $G'$; $p$ is a prime number, and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$ is a cyclic group of order $p$ as well as a finite field with $p$-elements.
To find all the groups of a given order is one of the oldest and core problems in the theory of groups which dates back to late nineteenth century. Cayley (1878) call it the \emph{general problem} of groups. Amounts of work has been done on this general problem since, and many problems in the theory of (finite) groups are closely related to it, among which the classification of finite simple groups
and the Burnside problem and its variants, if not the only, are the most influential ones.
The Burnside problem, originally posed by William Burnside in 1902, asks given positive integer $m, n$, whether and when the free Burnside group $B(m,n)$ is finite, where $B(m,n)$ is the group generated by $m$ generators $x_1, \cdots, x_m$ subject to relations $x^n$ for each word $x$ in $x_1, \cdots, x_m$. The full answer to this problem is not known yet, although P. Novikov, S. Adian, S. Ivanov and I. Lys\"{e}nko have shown the infiniteness of $B(m,n)$ for $m>1$ and sufficiently large $n$. Another invariant of the Burnside problem, namely the restricted Burnside problem was posed in 1930s. It asks whether there exists an upper bound for the order of finite groups with $m$ generators and exponent $n$. Recall that a group $G$ is said to have exponent $n$ if $n$ is the least positive number such that $g^n=1$ for all $g\in G$. The answer to the restricted Burnside problem is affirmative. In 1958 A. Kostrikin gave a solution in case $n$ is a prime number, and later in 1989, E. Zelmanov for arbitrary $n$.
The general problem can be deduced to the one to find all groups of given order and given exponent. Although it is known that there are only finitely many isoclasses of groups with given number of generators and exponent, it is still far from having a full answer to the general problem. The simplest case of the general problem is to determine all groups of prime-power order, or more restricted, the ones with prime exponent. A group of order $p^k$ for some prime $p$ and $k\le 1$ is usually called a $p$-group. A finite group of exponent $p$ is clearly a $p$-group. We are interested in the following basic problem in the theory of $p$-groups.
\noindent {\bf Problem.}
Determine all non-isomorphic groups of exponent $p$.
The classification of $p$-groups has been studied by many authors, see \cite{new,vish,wil,lee1,lee2,lee3,lee4} and the references therein. Newman gave a good survey to early development of this problem, and he also showed several methods to produce $p$-groups from the ones of lower order \cite{new}. Vishnevetskii obtained a classification of class 2 groups of exponent $p$ with derived groups of order $p^2$ \cite{vish}. In fact, he completely classified the groups of this form which cannot be expressed as a central product of two proper subgroups. Wilkinson gave a list of the groups of exponent $p$ and order $p^7$ for all $p$ \cite{wil}. Recently, Vaughan-Lee \cite{lee2} studied the groups of order $p^8$ and exponent $p$ based on the $p$-group generation algorithm as introduced in \cite{ob} and give the formula to calculate the number of the groups of order $p^8$ and exponent $p$ $(p > 7)$.
All known methods to construct $p$-groups rely heavily on the fact that a $p$-group is nilpotent, so that it can be written as iterated extension by elementary $p$-groups or more specifically by $\mathbb{Z}_p$'s, the cyclic group with $p$-elements.
Let $H$ and $K$ be groups. Recall that an extension of $H$ by $K$ is by definition an exact sequence $1\to K\to G\to H\to 1$, and $K$ is called the kernel of the extension. To find all extensions of $H$ by $K$ for given $H$ and $K$ is called the extension problem. With the classification of finite simple groups, a full solution to the extension problem will give a full solution to the general problem of groups for every finite group has a composite series.
The extension problem is very hard in general. However, if the extension has abelian kernel, say $K$ is an abelian group, then the conjugate action of $G$ on $K$ induces an action of $H$ on $K$, and
the famous Schreier theorem says that the equivalence classes of extensions of $H$ by $K$ is in one-to-one correspondence with $H^2(H, K)$, the second cohomology group of $H$ with coefficients in $K$.
Thus to find all extensions of $H$ by $K$, it is equivalent to find all possible actions $\rho\colon H\to \operatorname{Aut}(K)$ and to calculate $H^2(H,K)$ for each $\rho$.
Now let $0\to A\to G\to H\to 1$ be an extension with abelian kernel, where $A$ is viewed as an additive group with the identity element denoted by 0. Let $\rho\colon H\to \operatorname{Aut}(A)$ be the homomorphism induced by the $H$-action on $A$, and $\varphi\in H^2(H, A)$ the corresponding cohomology class. Such a quadruple $(H, A, \rho, \varphi)$ is called a \emph{group extension datum}, and the extension $0\to A\to G\to H\to 1$ is called a \emph{realization} of the datum. Any two realizations of a given datum are isomorphic. Moreover, given a datum $(H, A, \rho, \varphi)$, there exists a canonical realization $0\to A\to A\rtimes_{\rho,\varphi}H\to H\to 1$, where the middle group $G(H, A, \rho, \varphi)=A\rtimes_{\rho,\varphi}H$ is a generalization of semiproduct and called the \emph{realizing group} of the datum.
Recall that by definition, two extensions with abelian kernel are equivalent if and only if the corresponding extension data are isomorphic. We refer to Section 3.2 for more detail.
Recall that $G$ is solvable if and only if it has a subnormal series with abelian factor groups, i.e., $$G\cong A_r\rtimes_{\rho_r,\varphi_r}(A_{r-1}\rtimes_{\rho_{r-1},\varphi_{r-1}}(\cdots (A_1\rtimes_{\rho_1,\varphi_1} A_0)\cdots))$$ for some abelian (or even cyclic) groups $A_0, \cdots, A_r$, and some actions $\rho_1, \cdots, \rho_r$ and 2-cocycles $\varphi_1, \cdots, \varphi_r$.
It is highly nontrivial to classify all modules for a given group in general, even for abelian groups. Fortunately, all groups concerned in this work are nilpotent and we need only to consider trivial actions. In fact, $G$ is nilpotent if and only if it is obtained by iterated central extensions, or equivalently, $$G\cong Z_s\rtimes_{\varphi_s}(Z_{s-1}\rtimes_{\varphi_{s-1}}(\cdots (Z_1\rtimes_{\varphi_1} Z_0)\cdots))$$ for some abelian (or even cyclic) groups $Z_0, \cdots, Z_{s}$ with trivial actions and some 2-cocycles $\varphi_1, \cdots\varphi_s$. Here we simply write $\rtimes_{1,\varphi}$ as $\rtimes_{\varphi}$. The price is that the length $s$ may be much bigger than the one $r$ if we allow nontrivial actions as above.
The calculation of the cohomology group $H^2(H, A)$ is another big problem. The cohomology group of finite abelian groups can be calculated by using the K\"unneth formula or the well known Lyndon-Hochschild-Serre spectral sequence \cite{lyn, hs}, while for nonabelian groups quite a few is known yet. Moreover, for our purpose we need to understand the structure (the group table, group presentation ect.) of the realizing group. Then only knowing the group structure of $H^2(H, A)$ is not enough, one needs an explicit formula for the 2-cocylces. To find the cocycle formula is far more difficult than the calculation of the cohomology group. In fact, for a finite abelian group $H$, the cohomology group $H^*(H, A)$ has been long known, while the cocycle formula just appeared very recently, see for instance \cite{hlyy,hwy}.
Another problem should be taken into account is that nonequivalent extensions can give isomorphic middle group, c.f. \cite[Exercise 7.40]{rot}. Thus we also need to answer the question when the middle groups of two nonequivalent extensions are isomorphic.
In this paper, we mainly deal with extensions of the form $0\to A\to G\to H\to 1$ such that $A$ and $H$ are both abelian groups and $G$ has prime exponent $p$. Note that $H$ is abelian if and only if $A\ge G'$, the derived subgroup of $G$. Clearly $G$ has exponent $p$ implies that both $A$ and $H$ have exponent $p$, and hence $A\cong \mathbb{Z}_p^n$ and $H\cong \mathbb{Z}_p^m$ for some $m, n\ge 0$. In this case, $A$ can be viewed as an $n$-dimensional representation of $H$ over $\mathbb{Z}_p$, or equivalently an $n$-dimensional module of the algebra $\mathbb{Z}_p[T_1, \cdots, T_m]/\langle T_1^p, \cdots, T_m^p\rangle$, and hence the representation theory of finite dimensional algebra may apply. Moreover, since $H$ is a finite abelian group, the cocycle formula obtained in \cite{hlyy} enables the calculation of the structure of the middle group $G$.
First we consider the problem when two group extension datum have isomorphic realizing group. By introducing an equivalence relation on the set of group extension data, we can give an answer to this problem: under certain mild condition, two group extension data are equivalent if and only if their realizing groups are isomorphic.
\begin{proposition}[Proposition \ref{prop-equidata-isogp} and Corollary \ref{cor-equidata-isogp}] Let $\mathcal{D} = (H, A, \rho, \varphi)$ and $\tilde{D}=(\tilde H,\tilde A, \tilde\rho, \tilde\varphi)$ be two data, and let $G=A\rtimes_{\rho, \varphi}H$ and $\tilde G=\tilde A\rtimes_{\tilde \rho, \tilde\varphi} \tilde H$. Assume that one of the following holds:
\begin{enumerate} \item $A=G'$ and $\tilde A=\tilde G'$;
\item $A= Z(G)$ and $\tilde A= Z(\tilde G)$.
\end{enumerate}
Then $G$ and $\tilde{G}$ are isomorphic if and only if $\mathcal D$ and $\tilde {\mathcal D}$ are equivalent.
\end{proposition}
We describe the derived subgroup $(A\rtimes_{\rho,\varphi}H)'$, and consequently we obtain an equivalent condition for a group extension datum satisfying $A=(A\rtimes_{\rho,\varphi}H)'$, cf. Proposition \ref{prop-der-abext} and Corollary \ref{cor-der-abext}. Moreover, by using the cocycle formula for abelian groups, we give criteria for a realizing group having exponent $p$, see Theorem \ref{thm-expp} and Corollary \ref{cor-expp2}. Thus combined with the above proposition, the isoclasses of groups of exponent $p$ whose and with abelian derived subgroup are in one-to-one correspondence with the equivalence classes of certain group extension data.
It is well known that the nilpotency classs of a group of exponent $p$ is no greater than $p$ in case $p=3$, while the statement is not true for general $p$, cf. Section 2.2. However, the statement still holds true for groups which can be expressed as extension of two elementary $p$-groups, cf. Proposition \ref{prop-cn-mgp}.
One of the main purposes of this paper is to introduce the notion of matrix presentations for group extension data of the form $(\mathbb{Z}_p^m, \mathbb{Z}_p^n, \rho, \varphi)$, so that we can restate the classification problem of groups of exponent $p$ as a problem in linear algebra over the field $\mathbb{Z}_p$. Note that one can read a presentation of the realizing group of a data from its matrix presentation, cf. Proposition \ref{prop-pres-from-mrep}. In particular, we establish a connection between isoclasses of class 2 groups of exponent $p$
and the orbits of certain Grassmannian of anti-symmetric matrices under the congruence action.
\begin{theorem}[cf. Theorem \ref{thm-data-1to1-grorbits}] Let $m, n\ge 1$ and $0\le d\le n$ be integers. Then there exist one-to-one correspondences between:
\begin{enumerate}
\item $\mathcal E_p(d;m,n)$ and $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$;
\item $\mathcal E_p(m,n)$ and $\operatorname{Gr}(\le n, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$;
\item $\mathcal G_2(p;m,n)$ and $\operatorname{Gr}(n, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$.
\end{enumerate}
\end{theorem}
Note that $\mathcal G_2(p;m,n)$ denotes the set of isoclasses of class 2 groups of order $p^{m+n}$ and exponent $p$ with derived subgroup isomorphic to $\mathbb{Z}_p^n$; $\mathcal E_p(m, n)$ denotes the equivalence classes of data of the form $\mathcal{D}=(\mathbb{Z}_p^m, \mathbb{Z}_p^n, 1, \varphi)$ with $G(\mathcal{D})$ of exponent $p$; and $\mathcal E_p(d; m, n)$ denotes the subclasses with $(G(\mathcal{D}))' = \mathbb{Z}_p^d$ for $0\le d\le m$.
Let $M_m(\mathbb{Z}_p)$ be the space of square matrices of order $m$ over $\mathbb{Z}_p$, and $\operatorname{AS}_m(\mathbb{Z}_p)$ be the subspace consisting of anti-symmetric ones. Let $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))$ and $\operatorname{Gr}(\le d, \operatorname{AS}_m(\mathbb{Z}_p))$ denote respectively the set of $d$-dimensional subspaces and the set of subspaces of dimension less than or equal to $d$. Then
the congruence action of $\operatorname{GL}_m(\mathbb{Z}_p)$ on $\operatorname{AS}_m(\mathbb{Z}_p)$ induces an action of $\operatorname{GL}_m(\mathbb{Z}_p)$ on $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))$ as well as on $\operatorname{Gr}(\le d, \operatorname{AS}_m(\mathbb{Z}_p))$. We use $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ and $\operatorname{Gr}(\le d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ to denote the orbit spaces respectively.
By the above theorem, to find class 2 groups of exponent $p$ is equivalent to the problem to find a complete set of representatives of the orbits of $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))$ under the congruence action of $\operatorname{GL}_m(\mathbb{Z}_p)$. The latter may have its own interest from different perspectives.
In fact, if $d=1$, then it is equivalent to find the normal form of a nonzero $m\times m$ anti-symmetric matrix under the congruence action. The answer has been long known: any such a matrix is congruent to a unique matrix of the form
\[\operatorname{diag}\left(\begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix},
\cdots,
\begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix},
0, 0, \cdots, 0
\right). \]
If $d=2$, then the problem is closely related but not equivalent to the problem to find the canonical form of pencils of anti-symmetric matrices. The canonical form of pencils of anti-symmetric matrices is well-known, see for instance \cite[Chap XII]{gan}. However, in our case we need further to consider the $\operatorname{GL}_2(\mathbb{Z}_p)$ action on the set of canonical forms of pencils.
In the case $d\ge 3$, no results are known to our knowledge.
We can give a description of $\operatorname{Gr}(n, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ for $n=1, 2$, and hence obtain a classification of $\mathcal G_2(p;m,1)$ and $\mathcal G_2(p;m,2)$.
\begin{theorem}(1)[cf. Theorem \ref{thm-1dim-dergp}]
The set $\{W_{m,k}\mid 1\le k\le \frac m 2\}$ gives a complete set of representatives of $\operatorname{Gr}(1, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$, and $\{G_{m,k}\mid 1\le k\le \frac m 2\}$ gives a complete set of representatives of $\mathcal G_2(p;m,1)$.
(2) [cf. Theorem \ref{thm-2dim-dergp}] There exists a bijection between $\operatorname{Gr}(2, \operatorname{AS}_m(\mathbbm{k}))/\operatorname{GL}_m(\mathbbm{k})$ and the set of equivalence classes of canonical forms of pencils of anti-symmetric matrices. Consequently, $\mathcal G_2(p; m, 2)$ is in one-to-one correspondence with the equivalence classes of canonical forms.
\end{theorem}
We refer to Section 5.3 and 6.1 for unexplained notions.
We mention that in the above theorem, (1) should be well known to experts; and (2) generalizes to some extend the result obtained by Vishnevetskii in \cite{vish}, which provides a classification of groups in $\mathcal G_2(2; m, n)$ that can not be expressed as a central product of proper subgroups.
\begin{theorem}[cf. Theorem \ref{thm-3of4}, Theorem \ref{thm-4of4}, and Theorem \ref{thm-3of5}] We obtain a complete set of representatives of the orbits of $\operatorname{Gr}(3, \operatorname{AS}_4(\mathbb{Z}_3))/\operatorname{GL}_4(\mathbb{Z}_3)$, $\operatorname{Gr}(4, \operatorname{AS}_4(\mathbb{Z}_3))/\operatorname{GL}_4(\mathbb{Z}_3)$ and $\operatorname{Gr}(3, \operatorname{AS}_5(\mathbb{Z}_3))/\operatorname{GL}_5(\mathbb{Z}_3)$.
\end{theorem}
The proof is given by traditional ``hand" calculation. Consequently, we obtain a classification of groups of exponent $3$ and of order up to $3^8$, which recovers the results obtained in \cite{wil} and \cite{lee2} in case $p=3$, cf. Theorem \ref{thm-ord1to7} and Theorem \ref{thm-ord8}.
Note that we use different methods here. It is not known to us whether our method can be used to improve the existing $p$-groups generation algorithms.
The paper is organized as follows.
In Section 2, we recall some basics on groups and modules. In Section 3, we introduce the notion of group extension datum, which is a reformulation of extensions of group with abelian kernel. We show that under some mild assumption, two group extension data are equivalent if and only if the realizing group are isomorphic. We also discuss the derived subgroup and center of the realizing group of an abelin datum.
In Section 4, by using the cocycle formula for finite abelian groups, we show a criterion for the realizing group of an abelian datum having exponent $p$. We also show when the derived subgroup of the realizing group of an abelian extension is equal to the kernel. We give a bound to the nilpotency class of the realizing group of a $p$-elementary data.
In Section 5, we introduce matrix presentations for a $p$-elementary data, and show that when two matrix presentations give equivalent data. As a consequence, we show that class 2 groups of exponent $p$ are in one-to-one correspondence with congruence classes of subspaces of anti-symmetric matrices over $\mathbb{Z}_p$.
In Section 6, we give a description of the congruences classes of 2-dimensional subspaces of $\operatorname{AS}_m(\mathbb{Z}_p)$. Consequently we can find all groups of exponent $p$ and with derived subgroup of rank 2. As an easy consequence, we give a complete set of groups of order $3^{m}$ and exponent $3$ whose derived subgroup is isomorphic to $\mathbb{Z}_3^2$ for $m\le 8$. Such groups are shown to have nilpotency class 2.
In Section 7 and Section 8, we give a complete set of representatives of isoclasses of groups of exponent $3$ and order $\le 3^8$.
In the last two sections we calculate the orbits of $\operatorname{Gr}(n, \operatorname{AS}_m(\mathbb{Z}_3))/\operatorname{GL}_m(\mathbb{Z}_3)$ for small $m, n$. More precisely, we give representatives of all congruence classes of 3 and 4-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$ in Section 9, and the ones of 3-dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$ in Section 10.
\section{Basics on groups and modules}
We recall some basic facts on finite groups and modules in this section.
\subsection{Nilpotent group and nilpotency class} Let $G$ be a group.
For elements $g, h\in G$, by the \emph{commutator} of $g$ and $h$ we mean the element $[g,h]=ghg^{-1}h^{-1}$. Clearly $[g,h]=1$ if and only if $gh=hg$. For subsets $X, Y\subset G$, we use $[X,Y]$ to denote the subgroup of $G$ generated by elements of the form $[g,h]$ with $g\in X$ and $h\in Y$. The subgroup $[G,G]$ is called the \emph{derived subgroup} (or the commutator subgroup) of $G$, and denoted by $G'$ or $G^{(1)}$. We may also define the $n$-th derived subgroup $G^{(n)}=(G^{(n-1)})'$ of $G$ inductively. The following well-known lemma explains the importance of the derived subgroup.
\begin{lemma} Let $G$ be a group. Then $G'$ is a normal subgroup; and for any normal subgroup $N\trianglelefteq G$, the quotient group $G/N$ is abelian if and only if $G'\le N$.
\end{lemma}
Recall that the \emph{lower central series} of $G$ is the descending series of subgroups
\[ G=G_1\trianglerighteq G_2\trianglerighteq \cdots G_n\trianglerighteq\cdots,
\]
where each $G_{n+1}=[G_n,G]$. By definition, $G$ is nilpotent if and only if the lower central series terminates, i.e., $G_n=1$ for some $n$. The following notion of nilpotency class measures the nilpotency of a group, see for instance Definition 3.10 in \cite{khu}.
\begin{definition}
If a group $G$ satisfies $G_{c+1}=1$, then we say that $G$ is \emph{nilpotent} of class $\le c$; the least such number $c$ is called the \emph{nilpotency class} of $G$.
\end{definition}
\begin{remark}
(1) There are other equivalent definitions for nilpotent groups by using upper cental series or any other central series.
(2) If a group $G$ has nilpotency class $\le c$, then it is sometimes called a \emph{nil-$c$ group}.
\end{remark}
We collect some well-known facts on groups of lower nilpotency class without proof.
\begin{lemma}\label{lem-facts-nilclass} Let $G$ be a finite nilpotent group. Then
\begin{enumerate}
\item[(0)] $G$ has nilpotency class $0$ if and only if $G$ is the trivial group.
\item[(1)] $G$ has nilpotency class $\le 1$ if and only if $G$ is an abelian group.
\item[(2)] $G$ has nilpotency class $\le 2$ if and only if the derived subgroup $G'$ is contained in the center of $G$.
\item[(3)] If $G$ has nilpotency class $3$, then $G'$ is commutative.
\end{enumerate}
\end{lemma}
\subsection{Exponent of a group} The \emph{exponent} of a group $G$ is defined to be the least common multiple of the orders of all elements. If there is no least common multiple, the exponent is defined to be infinity. For finite groups, the exponent is a devisor of the order of the group and hence finite. Clearly, abelian groups of prime exponent $p$ are exactly the elementary $p$-groups.
We recall the following well-known result, which can be directly deduced from Theorem 6.5 and Theorem 6.6 in \cite[III]{HU}.
\begin{lemma} Any finite group of exponent $3$ is nilpotent of nilpotency class at most $3$.
\end{lemma}
We draw the following consequence.
\begin{corollary}\label{cor-exp3-dersubgp}
Let $G$ be a group of exponent 3. Then $G'$ is an elementary abelian 3-group.
\end{corollary}
\begin{proof}
Since $G$ is a group of exponent 3, the above lemma implies that $G$ is nilpotent of class at most 3. Then $[G',G']\le G_4=1$ by \cite[III, Satz 2.8]{HU}, which means that $G'$ is an abelian group, and hence an elementary abelian 3-group for $G$ has exponent 3.
\end{proof}
\subsection{Modules of a group and $p$-subgroups of $\operatorname{GL}_n(\mathbb{Z}_p)$} A \emph{module} of a given group $G$ is by definition an abelian group $M$ together with a map $G\times M\to M$, $(g,m)\mapsto g\cdot m$, such that $1_G\cdot m=m$ and $(gh)\cdot m=g\cdot(h\cdot m)$ for any $g,h\in G$ and $m\in M$. We also say that $G$ acts linearly on the abelian group $M$. We simply write $g\cdot m$ as $gm$ when there is no risk of confusion.
Clearly a $G$-module structure on an abelian group $M$ is equivalent to a group homomorphism $\rho\colon G\to \operatorname{Aut}(M)$ with $\rho$ given by $\rho(g)(m) = g m$ for any $g\in G$ and $m\in M$. In case we need to specify $\rho$, we also say that $A$ is a $G$-module via the homomorphism $\rho$, or simply say that $(A,\rho)$ is a $G$-module.
We remark that any group homomorphism $\rho\colon G\to \operatorname{Aut}(M)$ extends to a ring homomorphism
from $\mathbb{Z} [G]$ to $\mathrm{End} (M)$, where $\mathbb{Z} [G]$ is the group ring of $G$ over $\mathbb{Z}$ and $\mathrm{End} (M)$ is the ring of endomorphisms from $M$ to itself. Thus a $G$-module is equivalent to a $\mathbb{Z} [G]$-module.
Any finite abelian group can be written as a direct sum of cyclic subgroups. For any positive integer $m$, we use $\mathbb{Z}_m=\mathbb{Z}/m\mathbb{Z}$ to denote the cyclic group of order $m$. In fact, $\mathbb{Z}_m$ inherits a ring structure from $\mathbb{Z}$, in which the underlying additive group is exactly a cyclic group of order $m$.
Now let $p$ be a prime number. Then $\mathbb{Z}_p$ is the unique prime field of characteristic $p$, and any elementary $p$-group is equivalent to a vector space over $\mathbb{Z}_p$. In particular, the cyclic group $\mathbb{Z}_p$ is viewed as a one dimensional vector space over $\mathbb{Z}_p$.
Let $A\cong \mathbb{Z}_p^n$ be an $n$-dimensional $\mathbb{Z}_p$-vector space. Clearly $\operatorname{Aut}(A)=\operatorname{GL}(A)$, the general linear group of $A$. Any basis of $A$ gives an isomorphism $\operatorname{GL}(A)\cong \operatorname{GL}_n(\mathbb{Z}_p)$, mapping a linear transformation to its matrix under this basis, where $\operatorname{GL}_n(\mathbb{Z}_p)$ denotes the general linear group of degree $n$ over $\mathbb{Z}_p$.
By calculating the orders, it is easy to show that $UT(n,\mathbb{Z}_p)$, the subgroup consisting of $n\times n$ unipotent upper triangular matrices, is a Sylow $p$-subgroup of $\operatorname{GL}_n(\mathbb{Z}_p)$. Unipotent means that all entries in the main diagonal are 1. The following useful result is easy.
\begin{lemma} Let $G$ be a $p$-group, $A\cong \mathbb{Z}^n_p$, and $\rho\colon G\to \operatorname{Aut} A$ a group homomorphism. Then there exists a basis of $A$, under which $G$ maps into $UT(n,\mathbb{Z}_p)$. In particular,
if $A\cong \mathbb{Z}_p$, then $G$ acts trivially on $A$.
\end{lemma}
For a proof we use the fact that the matrices of a linear transformation under two bases are conjugate, and the fact that any two Sylow-$p$ subgroups of a finite group are conjugate to each other. We need also the following result for later use.
\begin{proposition}\label{prop-rad0}
Let $G$ be an abelian subgroup of $\operatorname{GL}_n(\mathbb{Z}_p)$. Then the following are equivalent:
\begin{enumerate}
\item $1+ g + \cdots + g^{p-1} =0$ for any $g\in G$;
\item $(g-1)^{p-1}=0$ for any $g\in G$;
\item $(g_1-1)\cdots (g_{p-1}-1) =0$ for any $g_i\in G$;
\item $(g_1-1)\cdots (g_{p-1}-1) =0$ for any $g_i\in X$, where $X$ is a set of generators of $G$.
\end{enumerate}
\end{proposition}
\begin{proof}
First by the binomial expansion we have
\[(1-g)^{p-1}=\sum_{i=0}^{p-1} (-1)^i \frac{(p-1)(p-2)\cdots (p-i)}{1\cdot 2\cdots i} g^i= \sum_{i=0}^{p-1} g^i,\]
where we use the fact $p-i=-i$ in $\mathbb{Z}_p$. Then the equivalence $(1)\Longleftrightarrow(2)$ is obvious.
$(2)\Longrightarrow(3)$ follows from Lemma \ref{lemma-strongnil} below and $(3)\Longrightarrow (2), (4)$ is obvious.
We are left to show that $(4)\Longrightarrow (3)$. Assume (4) holds, and let $g_1,\cdots, g_{p-1}$ be in $G$. Since $X$
generates $G$, each $g_i= g_{i1}g_{i2}\cdots g_{i, t_i}$ for some $g_{i1},g_{i2},\cdots ,g_{i, t_i}\in X$.
Then \[g_i-1= g_{i1}g_{i2}\cdots g_{i, t_i} -1= \sum_{k=1}^{t_i} g_{i1}g_{i2}\cdots g_{i,k-1}(g_{ik}-1),\]
and it follows from (4) that $(1-g_1)\cdots (1-g_{p-1}) =0$.
\end{proof}
\begin{remark} (1) A group satisfies the equivalent conditions in the proposition has exponent $p$, and hence
isomorphic to a finite dimensional $\mathbb{Z}_p$-vector space.
(2) In the proposition the assumption that $G$ is commutative cannot be dropped. For instance, for any $a, b ,c\in \mathbb{Z}_3$, we set
\[g_{a,b,c}=\begin{pmatrix}
1 & a & b & c \\
& 1 & & -b\\
& & 1 & a \\
& & & 1 \\
\end{pmatrix}\in \operatorname{GL}_4(\mathbb{Z}_3).\]
Consider the subgroup $G=\{g_{a,b,c}\mid a,b,c\in \mathbb{Z}_3\}\le \operatorname{GL}_4(\mathbb{Z}_3)$.
Then for any $g\in G$, $(g-1)^2=0$, while $(g_{1,0,0}-1)(g_{0,1,0}-1)\ne 0$.
\end{remark}
\begin{lemma}\label{lemma-strongnil} Let $G\le \operatorname{GL}_n(\mathbb{Z}_p)$ be an abelian subgroup, $r\le p-1$ a positive integer, and $M$ an $n\times n$ matrix
over $\mathbb{Z}_p$. Assume that $(g-1)^rM=0$ for all $g\in G$. Then $$(g-1)^{r-1}(h-1)M=0$$ for all $g, h\in G$. Consequently,
$(g_1-1)(g_2-1)\cdots (g_r-1)M=0$ for all $g_1, g_2\cdots, g_r\in G$.
\end{lemma}
\begin{proof} Given $g, h\in G$, we write $x=g-1$, and $y= h-1$. Then $g=1+x, h=1+y$. We set $w_k=1 + h + \cdots+ h^{k-1}$ for $1\le \mathbbm{k}\le r$. Clearly $w_k$ is invertible, and $w_k-w_l$ is invertible for any $1\le k\neq l\le r$.
By assumption, $(g-h^k)^{r}M=g^r(1-g^{-1}h^k)^rM=0$ for any $k$. It is direct to check that $g-h^k=x - yw_k$ and hence we get
\[
\begin{pmatrix} w_1& w_1^2 & \cdots& w_1^{r-1}\\
w_2& w_2^2 & \cdots &w_2^{r-1}\\
\cdots& \cdots& \cdots& \cdots\\
w_{r-1}&w_{r-1}^2 & \cdots & w_{r-1}^{r-1}
\end{pmatrix}
\begin{pmatrix} -C_r^{r-1}x^{r-1}yM\\C_r^{r-2}x^{r-2}y^2M\\ \cdots \\(-1)^{r-1} C_r^1xy^{r-1}M
\end{pmatrix}
= 0,
\]
where $C_r^i = \frac {r!}{i!(r-i)!}$ is the binomial coefficient. Using the Van der Monde determinant, we know that each
$C_r^i x^i y ^{r-i}M=0$, hence $x^i y ^{r-i}M=0$. In particular, we have the desired $(g-1)^{r-1}(h-1) M=0$. The last conclusion follows easily by using induction on $r$.
\end{proof}
\subsection{Radical and socle} We collect some well-known facts on finite length modules, cf. \cite{ars}. Let $R$ be a finite dimensional algebra over an arbitrary field $\mathbbm{k}$, and $M$ an $R$-module of finite length.
Recall that the \emph{socle} of $M$, denoted by $\operatorname{soc}(_RM)$ or simply $\operatorname{soc}(M)$ when there is no confusion, is by definition the sum of all simple submodules of $M$; and the radical $\operatorname{rad}(_RM)$ (or $\operatorname{rad}(M)$) is the intersection of all its maximal submodules. Then $\operatorname{rad}(_RR)=\operatorname{rad}(R_R)$ is equal to the intersection of all two-sided maximal ideal of $R$, and we denote it by $\operatorname{rad}(R)$. Moreover, $\operatorname{rad}(_RM)=\operatorname{rad}(R)M$ for any left $R$-module $M$ and $\operatorname{rad}(N_R)=N\operatorname{rad}(R)$ for any right $R$-module $N$.
The filtration $0= \operatorname{rad}^{l}(M)\subseteq \operatorname{rad}^{l-1}(M)\subseteq\cdots\subseteq \operatorname{rad}(M)\subseteq M$ is called the radical filtration of an $R$-moduled $M$, where $\operatorname{rad}^{i}(M)=\operatorname{rad}(\operatorname{rad}^{i-1}M)=(\operatorname{rad}(R))^iM$ for each $i$, and the least
$l$ with $\operatorname{rad}^{l}(M)=0$ is called the Loewy length of $M$. We use $\ell\ell(M)$ to denote the Loewy length of $M$.
The coradical filtration is the filtration $0=\operatorname{soc}^0(M)\subseteq \operatorname{soc}^1(M)\subseteq\cdots \subseteq \operatorname{soc}^l(M)= M$ of submodules such that $\operatorname{soc}^1(M)=\operatorname{soc}(M)$, and $\operatorname{soc}^i(M)/\operatorname{soc}^{i-1}(M)= \operatorname{soc}(M/\operatorname{soc}^{i-1}(M))$ for each $i$. The least $l$ with $\operatorname{soc}^l(M)= M$ is called the length of the coradical filtration. It is well known that the radical filtration and the coradical filtration have equal length, cf. \cite[Chapter II, Proposition 4.7]{ars}.
\begin{proposition}\label{prop-loewylength} Let $G$ be a finite abelian group, $\mathbb{Z}_p[G]$ be the group algebra over $\mathbb{Z}_p$, and $(M,\rho)$ be a $\mathbb{Z}_p[G]$-module. Let $1\le r\le p-1$ be a given positive integer.
\begin{enumerate}
\item[(1)] For any $g\in G$, if $(\rho(g)-1)^p= 0$, then $\rho(g)^{p}=1$;
\item[(2)] If $(\rho(g)-1)^r= 0$ for all $g\in G$, then $\ell\ell(M)\le r$.
\end{enumerate}
\end{proposition}
\begin{proof} (1) follows from the fact that the Frobenius map respects the addition, say $\rho(g)^p-1 = (\rho(g)-1)^p =0$.
(2) Let $T_g=\rho(g)-1$. Extend $\rho$ to a ring homomorphism from $\mathbb{Z}_p[G]$ to $\operatorname{End}(M)$, and set $R=\rho(\mathbb{Z}_p[G])$. Then $\operatorname{rad}(R)=\sum_{g\in G}RT_g$, and $\operatorname{rad}^{k}(R)=\sum_{g_1, g_2,\cdots, g_k\in G} RT_{g_1}T_{g_2}\cdots T_{g_k}$ for any $k\ge 1$. Therefore by Lemma \ref{lemma-strongnil} we have $\operatorname{rad}^{r}(M)=\operatorname{rad}^r(R)M=0$ and hence $\ell\ell(M)\le r$.
\end{proof}
\begin{remark} (1) We mention that if we drop the assumption $r\le p-1$ on $r$ in Proposition \ref{prop-loewylength} (2), then the conclusion does not hold in general. For instance, we may take $r= p$ and some $m> 1$. Consider the action of the group $\mathbb{Z}_p^m$ on the group ring $\mathbb{Z}_p[\mathbb{Z}_p^m]$ given by multiplication. Then $(\rho(g)-1)^p=0$ for any $g\in \mathbb{Z}_p$, while $\ell\ell(\mathbb{Z}_p[\mathbb{Z}_p^m])=(p-1)^m+1 > p$.
(2) Let $f\colon R\to R'$ be a homomorphism of rings, and $M$ be an $R'$-module of finite length. Then $M$ can be viewed as an $R$-module. Clearly $_RM$ is simple if and only if $_{R'}M$ is simple. It follows that $\operatorname{rad}(_RM)=\operatorname{rad}(_{R'}M)$ and $\operatorname{soc}(_RM)=\operatorname{soc}(_{R'}M)$ as subgroups, and hence $\ell\ell(_RM)=\ell\ell(_{R'}M)$.
(3) Let $M$ be a nonzero $R$-module with finite length and $X$ a semisimple module. Then $\ell\ell(M)=\ell\ell(M+X)$.
\end{remark}
\section{Group cohomology and group extension}
In this section, we recall some basics on group cohomology and group extension. One can consult Chapter 6 in \cite{wei} and Chapter 7 in \cite{rot}. We reformulate the group extension and introduce the notion of group extension datum, which plays a crucial role in this work.
\subsection{The bar resolution}
Let $G$ be a group. Consider the following bar resolution $B_*$ of the trivial $G$-module $\mathbb{Z}$.
\[ \cdots\to B_3\xrightarrow{d} B_2 \xrightarrow{d} B_1 \xrightarrow{d} B_0 \xrightarrow{\varepsilon} \mathbb{Z}.
\]
For each $n\ge1$, $B_n$ is the free $\mathbb{Z} [G]$-module on the set of all symbols $[g_1,\cdots ,g_n]$ with $g_i\in G$; and $B_0$ is the free $\mathbb{Z} [G]$-module of rank one and with basis $[\ ]$. The differentials are given by $\varepsilon([\ ])= 1$, $d([g])= g[\ ]-[\ ]$, and \[d([g_1,\cdots, g_n])= g_1[g_2,\cdots, g_n] +\sum_{1\le i\le n-1}(-1)^i[g_1,\cdots, g_ig_{i+1}, \cdots, g_n] +(-1)^n[g_1,\cdots, g_{n-1}].\]
Let $A$ be a $G$-module. An \emph{$n$-cochain} is a set map $\varphi$ from $G^n=G\times\cdots \times G$ to $A$; the set of $n$-cochains is identified with $\hom_G(B_n, A)$. A cochain $\varphi$ is normalized if $\varphi(g_1, \cdots)$ vanishes whenever some $g_i=1$. The differential $d\varphi$ of $\varphi$ is an $n+1$-cochian given by
\[ d\varphi(g_0, \cdots, g_n)=g_0\varphi(g_1,\cdots, g_n)+ \sum(-1)^{i+1}\varphi(\cdots, g_ig_{i+1},\cdots) + (-1)^n\varphi(g_0,\cdots, g_{n-1}).
\]
By definition, \emph{$n$-cocycles} are those $n$-cochains such that $d\varphi=0$, and \emph{$n$-coboundaries} are the ones of the form $d\varphi$. We denote by $B^n(G; A)$ and $Z^n(G; A)$ the set of $n$-coboundaries and the set of $n$-cocycles of $G$ (with coefficients in $A$) respectively. Clearly, $B^n(G; A)\subseteq Z^n(G; A)$, and the quotient $H^n(G; A)=Z^n(G; A)/B^n(G; A)$ is called the $n$-th cohomology group of $G$ with coefficients in $A$. By definition $H^n(G; A)=\mathrm{Ext}_{\mathbb{Z} [G]}^n(\mathbb{Z}, A)$. We mention that any $n$-cocycle is cohomologous to a normalized one.
\begin{example} Let $A$ be a $G$-module, and $\varphi\colon G\times G\to A$ a map. Then
$\varphi$ is a 2-cocycle if and only if $g\varphi(h,k)-\varphi(gh,k)+\varphi(g,hk)-\varphi(g,h)=0$ for all $g,h,k\in A$; and $\varphi$ is a 2-coboundary if and only if there exists some $f\colon G\to A$ such that $\varphi(g,h)=g f(h)-f(gh)+ f(g)$ for all $g$, $h\in G$.
\end{example}
\subsection{Group extension datum} Let $K$ and $H$ be groups. An extension of $H$ by $K$ is a short exact sequence $1\to K\to G\to H\to 1$ of groups, and $K$ is usually called the kernel of the extension.
The question to find all group extensions of given groups $H$ and $K$ is called the extension problem, which has been studied heavily since the late nineteenth century.
By Jordan-H\"older theorem, any finite group is obtained iteratively by group extension from simple groups. The solution to the extension problem would give us a complete classification of all finite groups. The extension problem is a very hard problem, and no general theory exists which treats all possible extensions at one time. However, group extensions with abelian kernel can be studied by using the group cohomology. We mainly deal with the case that both $K$ and $H$ are elementary $p$-groups here.
Now let $A$ be an abelian group. As usual, we use + to denote the multiplication and 0 the identity element in $A$. Let $0\to A \xrightarrow{\iota} G\xrightarrow{\pi} H\to 1$ be a group extension. We may identify $A$ with a normal subgroup of $G$. Then $G$ acts on $A$ by conjugate in $G$. To avoid confusion, we write $^g\!a$ for the conjugate $gag^{-1}$ in $G$. This induces an $H$-module structure on $A$, and the induced $H$-action is written as $h\triangleright a$ for $h\in H$ and $a\in A$.
\begin{remark} By definition, the conjugate action is trivial if and only if $A$ is contained in the center of $G$. A group extension of this type is called a \emph{central extension}.
\end{remark}
Let $s\colon H\to G$ be a (set-theoretic) section, i.e., a set map such that $\pi\circ s= \operatorname{id}_H$. Then we obtain a well-defined map $[\ ]_s\colon H\times H\to A$ by setting $[h_1,h_2]_s=s(h_1)s(h_2)s(h_1h_2)^{-1}$. We call $[\ ]_s$ the \emph{factor set} determined by the group extension and $s$. It is direct to check that $[\ ]_s\in Z^2(H; A)$ is a 2-cocycle, and $[\ ]_s$ is normalized if $s(1_H)=1_G$. Moreover, $[\ ]_s-[\ ]_{s'}$ is a 2-coboundary for any other section $s'$.
We say that two extensions of $H$ by $A$ are equivalent if we have a commutative diagram
\[\begin{CD}
0 @>>> A@>>>G@>>>H@>>>1\\
@. @V\operatorname{id}_AVV @VV\sigma V @VV\operatorname{id}_HV @.\\
0 @>>> A@>>>\tilde{G}@>>>H@>>>1
\end{CD}
\]
of group homomorphisms. Clearly $\sigma$ is an isomorphism in this case. Then we have the following well-known classification result.
\begin{proposition} Let $H$ be a group and $A$ an $H$-module. Then the equivalence classes of
group extensions such that the induced $H$-action on $A$ agrees with the $H$-module structure
are in 1-1 correspondence with the cohomology group $H^2(G, A)$.
\end{proposition}
We aim to apply group extensions to the classification of $p$-groups, and for this we need a generalized
version of the above proposition. We use the following notion of extension datum, which is essentially a combination of the notions of ``data'' and ``factor set'' as introduced in Page 179-180 in \cite{rot}.
\begin{definition} A \emph{group extension datum} (\emph{datum} for short) is by definition a quadruple $\mathcal D= (H, A, \rho, \varphi)$, where $H$ is a group, $A$ is an abelian group with an $H$-action given by $\rho\colon H\to \operatorname{Aut}(A)$, and $\varphi\colon H\times H\to A$ is a normalized 2-cocycle. $A$ is called the \emph{kernel} of the datum and $H$ is called the \emph{cokernel} of the datum.
If $H$ is an abelian group, then we call $\mathcal D$ an \emph{abelian datum}; If $H\cong \mathbb{Z}_p^m$ and $A\cong \mathbb{Z}_p^n$ are both elementary $p$-groups, then we call $\mathcal D$ a \emph{$p$-elementary datum of type $(m,n)$}.
\end{definition}
Let $0\to A\to G\to H\to 1$ be a group extension with abelian kernel. As we have shown above, any set-theoretic section $s\colon H\to G$ with $s(1_H)=1_G$ gives to a datum $(H, A, \rho, \varphi)$. We say that the group extension $0\to A\to G\to H\to 1$ is a realization of $(H, A, \rho, \varphi)$. Clearly $\rho$ is uniquely determined by the extension, while $\varphi$ depends on the choice of $s$.
\begin{lemma-definition} Given a datum $\mathcal D= (H, A, \rho, \varphi)$. We can define a product on the set $A\times H$ by $(a,h)(b,k)= (a+ h\triangleright b +\varphi(h,k), hk)$. Then
\begin{enumerate}
\item The product makes $A\times H$ a group with the identity element $(0, 1)$, and the inverse given by $(a, h)^{-1}=(-h^{-1}\triangleright(a+\varphi(h,h^{-1})), h^{-1})$. The resulting group is called the \emph{realizing group} of the datum $\mathcal{D}$ ( or the group realizing $\mathcal{D}$), and is denoted by $G({\mathcal D})$ or $A\rtimes_{\rho,\varphi} H$.
\item $A\rtimes_{\rho, 0} H$ gives the semidirect product of $A$ and $H$.
\item $(A,1)\subseteq Z(G(\mathcal D))$, the center of $G(\mathcal D)$, if and only if $H$ acts on $A$ trivially.
\end{enumerate}
\end{lemma-definition}
\begin{proof} (1) and (2) are trivial and we omit the proof here. We only prove (3). First assume that $A$ is a trivial module, say $h\triangleright a=a$ for any $a\in A$ and $h\in H$. Then
\[(a,1)(b,h)= (a+b+\varphi(1,h), h)=(a+b, h)=(h\triangleright a+b+ \varphi(h,1), h)= (b,h)(a,1)\]
holds for any $(b,h)\in G(\mathcal D)$, hence $(a,1)\in Z(G(\mathcal D))$. Note that $\varphi$ is assumed to be normalized here.
Conversely, for any $a\in A$, if $(a,1)\in Z(G(\mathcal D))$, then we have $$(a+b+\varphi(1,h), h)=(h\triangleright a+b+ \varphi(h,1), h)$$ for any $b\in A, h\in H$. In particular, $h\triangleright a = a$ for any $x\in H$, which means that the $H$-action is trivial.
\end{proof}
\begin{remark} \label{rem-data}(1) We simply write $\rtimes_{\varphi}=\rtimes_{1,\varphi}$ and $\rtimes_{\rho}=\rtimes_{\rho,0}$ when there is no confusion. Sometimes we also use $a\rtimes h$ to denote the element $(a,h)\in A\rtimes_{\rho,\varphi} H$.
(2) It is not hard to show that $A\rtimes_{\rho,\varphi} H \cong A\rtimes_{\rho,\varphi'} H$ if $\varphi$ and $\varphi'$ are cohomologous. We freely identify $A$ as the subgroup $(A, 1)$ of $A\rtimes_{\rho,\varphi} H$
and $H$ a subset via the maps $a\mapsto (a, 1_H)$ and $h\mapsto (0,h)$ for any $a\in A$ and $h\in H$.
We mention that $(0, H)$ is not a subgroup in general, it is if and only if $\varphi=0$.
(3) The above construction works well if $\varphi$ is not normalized. In this case, $(A,1)$ will not be a subgroup. In fact, $\varphi$ is normalized if and only if $1_{A\rtimes_{\rho,\varphi} H} =(0,1_H)$.
(4) Extensions with non-abelian kernel relate also closely to certain cohomology theory, cf. \cite[Section IV.6]{br}.
\end{remark}
By definition, any solvable group is realized as an iterated extension of abelian groups, and hence of cyclic groups. This fact seems to motivate (at least partly) the early development of group cohomology theory.
\begin{proposition} Let $G$ be a finite group. Then
(1) $G$ is solvable if and only if
\[G\cong A_n\rtimes_{\rho_n,\varphi_n}(A_{n-1}\rtimes_{\rho_{n-1},\varphi_{n-1}}(\cdots (A_2\rtimes_{\rho_1,\varphi_1}(A_1\rtimes_{\rho_1,\varphi_1}A_0))\cdots))\]
for some cyclic groups $A_0, A_1,\cdots, A_n$, group actions $\rho_1, \cdots, \rho_n$ and 2-cocycles $\varphi_1, \cdots, \varphi_n$.
(2) $G$ is nilpotent if and only if all $\rho_i$'s above can be chosen trivial, say
\[G\cong A_n\rtimes_{\varphi_n}(A_{n-1}\rtimes_{\varphi_{n-1}}(\cdots (A_2\rtimes_{\varphi_1}(A_1\rtimes_{\varphi_1}A_0))\cdots))\]
for some cyclic groups $A_0, A_1,\cdots, A_n$ and 2-cocycles $\varphi_1, \cdots, \varphi_n$.
\end{proposition}
We end this subsection with the construction of product of group extension data. Let $\mathcal{D}_1= (H_1, A_1, \rho_1, \varphi_1)$ and $\mathcal{D}_2= (H_2, A_2, \rho_2, \varphi_2)$ be data. We set $H= H_1\times H_2$ and $p_1\colon H\to H_1$ and $p_2\colon H\to H_2$ to be the projection maps. Then $A_1$ and $A_2$ are $H$-modules with the obvious action given by $\rho_1\circ p_1$ and $\rho_2\circ p_2$
respectively. Set $A= A_1\oplus A_2$. It is direct to show that the map $\varphi\colon H\times H\to A, \varphi((h_1, h_2), (h_1', h_2')) = (\varphi_1(h_1, h_1'), \varphi_2(h_2, h_2'))$ gives a 2-cocycle of $H$. Then we have a datum $(H, A, \rho, \varphi)$, which is called the product datum of $\mathcal{D}_1$ and $\mathcal{D}_2$ and denoted by $\mathcal{D}_1\times \mathcal{D}_2$.
\begin{proposition}\label{prop-prod-data} Keep the above notations. Then
(1) $G(\mathcal{D}_1\times \mathcal{D}_2)\cong G(\mathcal{D}_1)\times G(\mathcal{D}_2)$.
(2) Given a datum $\mathcal D= (H, A, \rho, \varphi)$. Let $A_1$ and $A_2$ be $H$-submodules of $A$, such that $A= A_1\oplus A_2$, $H$ acts trivially on $A_2$, and $\mathrm{Im}(\varphi)\subseteq A_1$. Then $G(\mathcal{D})\cong G(\mathcal{D}_1)\times A_2$, where $\mathcal{D}_1= (H, A_1, \rho, \varphi)$.
\end{proposition}
The proof is easy. In fact, the map $(a_1, a_2)\rtimes(h_1,h_2))\mapsto (a_1\rtimes h_1, a_2\rtimes h_2)$ is a desired isomorphism in $(1)$; and (2) is the special case of $(1)$ with $H_1=H$ and $H_2=\{1\}$.
\subsection{Equivalence of data}
Let $\sigma_A\colon A\to \tilde A$ and $\sigma_H\colon H\to \tilde H$ be isomorphisms of groups. Clearly $\sigma_A$ induces an isomorphism $\operatorname{Aut}(A)\to \operatorname{Aut}(\tilde A)$, mapping $\tau\in \operatorname{Aut}(A)$ to the composition map $\sigma_A\circ\tau\circ\sigma_A^{-1}$. The composition with $\rho\circ \sigma_H^{-1}$ gives a group homomorphism $\tilde \rho\colon \tilde H\to \operatorname{Aut}(\tilde A)$, making $\tilde A$ an $\tilde H$-module. More precisely, the $\tilde H$-action is given by $\tilde h\triangleright\tilde a=\sigma_A(\sigma_H^{-1}(\tilde h)\triangleright \sigma_A^{-1}(a))$ for any $\tilde h\in \tilde H$ and $\tilde a\in \tilde A$. Consider the map $\tilde\varphi\colon \tilde H\times \tilde H\to \tilde A$, $\tilde \varphi(\tilde h, \tilde k)= \sigma_A(\varphi(\sigma_H^{-1}(\tilde{h}), \sigma_H^{-1}(\tilde{k})))$. It is direct to check that $\tilde\varphi\in Z^2(\tilde H, \tilde A)$, and $A\rtimes_{\rho,\varphi} H\cong \tilde A\rtimes_{\tilde \rho,\tilde \varphi} \tilde H$. This suggests the following definition.
\begin{definition}\label{def-equi-data}
Two data $\mathcal{D}=(H, A, \rho, \varphi)$ and $\tilde{\mathcal{D}}=(\tilde H, \tilde A, \tilde\rho, \tilde\varphi)$ are said to be \emph{equivalent} if
there exist group isomorphisms $\sigma_H\colon H\to \tilde H$ and $\sigma_A\colon A\to \tilde A$ and a map $f\colon H\to\tilde A$ , such that
$\sigma_A(h\triangleright a)=\sigma_H(h)\triangleright \sigma_A(a)$ and \[\tilde \varphi(\sigma_H(h), \sigma_H(k)) -\sigma_A(\varphi(h, k))=\sigma_H(h)\triangleright f(k)-f(hk)+f(h)\] for all $a\in A$ and $h, k\in H$; if furthermore, $f$ can be chosen to be 0, then $\mathcal D$ and $\tilde {\mathcal D}$ are said to be \emph{isomorphic}.
\end{definition}
\begin{remark}\label{rem-equi-data} We denoted two equivalent data by $\mathcal{D}\sim\tilde D$, or $\mathcal{D} \stackrel{(\sigma_H, \sigma_A, f)}{\sim} \tilde D$ in case one needs to specify $\sigma_H, \sigma_A$ and $f$. It is easy to show that $``\sim "$ is an equivalence relation, and hence provides a partition of the set of data into equivalence classes. As usual, we use $[\mathcal{D}]$ to denote the equivalence class containing $\mathcal{D}$, that is $[\mathcal{D}]=\{\tilde\mathcal{D} \mid D\sim \tilde D\}$.
\end{remark}
We need the following result. The proof is straightforward and we omit it here.
\begin{proposition}\label{prop-equidata-isogp} Let $\mathcal{D}=(H, A, \rho, \varphi)$ and $\tilde{\mathcal{D}}=(\tilde H, \tilde A, \tilde\rho, \tilde\varphi)$ be data. Then $\mathcal D$ and $\tilde {\mathcal D}$ are equivalent if and only if there exists a group isomorphism $\sigma\colon G(\mathcal D) \to G(\tilde{\mathcal D})$ that restricts to an isomorphism $\sigma_A\colon A\to\tilde A$.
\end{proposition}
\begin{remark} We mention that not all isomorphisms $\sigma\colon G(\mathcal D)\to G(\tilde{\mathcal D})$ will restrict to an isomorphism between $A$ and $\tilde A$. It will be the case if $A=G(\mathcal D)'$ and $\tilde A= G(\tilde{\mathcal D})'$, or if
$A= Z(G(\mathcal D))$ and $\tilde A= Z(G(\tilde{\mathcal D}))$.
An obvious counterexample is as follows. Let $A_1$ and $A_2$ be nonisomorphic abelian groups. Consider the trivial datum $\mathcal D= (A_1, A_2, \rho, \varphi)$ and $\tilde{\mathcal D}=(A_2, A_1, \tilde\rho, \tilde\varphi)$, where trivial means that the group actions and the 2-cocyles are all trivial. Then the natural isomorphism $A_2\times A_1\cong A_1\times A_2$ gives an isomorphism $G(\mathcal D)\cong G(\tilde{\mathcal D})$, which does not restrict to an isomorphism from $A_2$ to $A_1$.
\end{remark}
We may draw an easy consequence, which says that in many cases we are interested, isomorphisms of realizing groups is equivalent to equivalences of data.
\begin{corollary}\label{cor-equidata-isogp} Let $\mathcal{D}=(H, A, \rho, \varphi)$ and $\tilde{\mathcal{D}}=(\tilde H, \tilde A, \tilde\rho, \tilde\varphi)$ be data. Assume that one of the following holds:
\begin{enumerate} \item $A=G(\mathcal D)'$ and $\tilde A= G(\tilde{\mathcal D})'$;
\item $A= Z(G(\mathcal D))$ and $\tilde A= Z(G(\tilde{\mathcal D}))$.
\end{enumerate}
Then $G(\mathcal D)$ and $G(\tilde{\mathcal D})$ are isomorphic if and only if $\mathcal D$ and $\tilde {\mathcal D}$ are equivalent.
\end{corollary}
\subsection{Abelian data} We are interested in groups that are extensions of abelian groups, or equivalently, realizing groups of abelian data. For instance, by Lemma \ref{lem-facts-nilclass} and Corollary \ref{cor-exp3-dersubgp} any finite group of exponent 3 or any nilpotent group of class 3 is such a group.
\begin{lemma}\label{lem-abel-extension} Let $\mathcal{D}=(H, A, \rho, \varphi)$ be an abelian datum. Then
\begin{enumerate}
\item For any $(a,k), (b,h)\in G(\mathcal D)$, we have \[[(a,k),(b,h)]=((1-h)\triangleright a+(k-1)\triangleright b+\varphi(k,h)-\varphi(h,k),1);\]
\item The derived subgroup of $G(\mathcal D)$ is abelian;
\item The center of $G(\mathcal{D})$ is given by \[\{(a,k)\in G(\mathcal{D})\mid k\triangleright b= b\ \forall b\in A,\ \varphi(k,h)-\varphi(h,k)= (h-1)\triangleright a\ \forall h\in G\}.\]
Moreover, there exist some $\tilde\varphi$ cohomologous to $\varphi$ and a subgroup $K\le H$, such that
(a) $K$ acts on $A$ trivially;
(b) $\tilde\varphi(k,h)\in \operatorname{soc}(A)$ and $\tilde\varphi(k, h)=\tilde\varphi(h, k)$ for any $k\in K$ and $h\in H$;
(c) $Z(A\rtimes_{\rho,\tilde\varphi}H)=\{(a,k)\mid a\in\operatorname{soc}(A), k\in K\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) By definition $(a,k)(b,h)= [(a,k),(b,h)](b,h)(a,k)$. Write $[(a,k),(b,h)]=(c,g)$. Then we have
\begin{align*}(a+ k\triangleright b+\varphi(k,h), kh) = &(c,g)(b+ h\triangleright a+\varphi(h,k), hk)
\\ = &(c + g\triangleright(b+ h\triangleright a+\varphi(h,k)) + \varphi(g, hk), ghk).
\end{align*}
It follows that $g=1$, and $a+ k\triangleright b+\varphi(k,h)=c + b+ h\triangleright a+\varphi(h,k)$, and we obtain the desired equality. Note that we use the assumptions $hk=kh$ and $\varphi$ is normalized here.
(2) By (1) $G(\mathcal D)'$ is a subgroup of $A$ and hence abelian.
(3) Note that $(a, k)\in Z(G)$ if and only if $[(a,k), (0, h)]=(0,1)= [(a,k), (b, 1)]$ for any $b\in A$ and $k\in G$. The first conclusion follows easily from in (1).
It is direct to show that $(a, 0)\in Z(G(\mathcal{D}))$ if and only if $(h-1)\triangleright a= 0$ for any $h\in H$, which is equivalent to $a\in\operatorname{soc}(A)$. For $a, a'\in A$, if $(a,k), (a',k)\in Z(G(\mathcal{D}))$ for some $k\in K$, then $a-a'\in \operatorname{soc}(A)$.
Set $K=\{k\in H\mid (a, k)\in Z(G(\mathcal{D})) \text{ for some } a\in A\}$. Then $K\le H$ is a subgroup. In fact,
consider the projection from $G(\mathcal{D})$ to $H$, then $K$ is the image of $Z(G(\mathcal{D}))$. Clearly (a) is true by the first assertion.
For any $k\in K$, we fix some $a(k)\in A$ such that $(a(k),k)\in Z(G(\mathcal{D}))$ and we choose $a(1_H)=0$.
Then $(a(k), k)(a(k'), k')=(a(k)+ a(k') +\varphi(k, k'), kk')\in Z(G(\mathcal{D}))$ and hence $a(kk')- a(k)- a(k') - \varphi(k, k')\in \operatorname{soc}(A)$.
Let $S\subseteq H$ be a complete set of representatives of left cosets of $K$ in $H$.
Then each $h\in H$ is uniquely written as $ks$ for some $k\in K$ and $s\in S$.
We then have a map $f\colon H\to K$ given by $f(ks)= a(k) +\varphi(k,s)$.
We claim that $\varphi+d(f)$ gives the desired $\tilde\varphi$.
In fact, for any $k, k'\in K$ and $s\in S$,
\begin{align*}
\tilde\varphi(k, k's)= &\varphi(k,k's)+ k\triangleright(a(k')+ \varphi(k', s))- (a(kk')+ \varphi(kk', s))+ a(k)\\
=& \varphi(k', s)- \varphi(kk', s)+ \varphi(k,k's)-\varphi(k, k') - a(kk') + a(k)+ a(k')+\varphi(k, k')\\
=& a(k)+ a(k')+\varphi(k, k') - a(kk')\in \operatorname{soc}(A),
\end{align*}
and
\begin{align*}
\tilde\varphi(k's, k)= \varphi(k's, k)+ k's\triangleright a(k) - (a(kk')+ \varphi(kk', s))+ (a(k')+ \varphi(k', s)),
\end{align*}
therefore we have
\[\tilde\varphi(k's, k)-\tilde\varphi(k, k's)= \varphi(k's, k)- \varphi(k,k's) + (k's-1)\triangleright a(k)=0,\]
which proves (b).
Now (c) follows from (b) and the first assertion of (3).
\end{proof}
Let $\tilde\mathcal{D}=(\tilde H, \tilde A, \tilde \rho, \tilde\varphi)$ be another abelian data. Then combined with Proposition \ref{prop-prod-data}, we have the following result, generalizing Corollary \ref{cor-equidata-isogp}(1).
\begin{proposition}\label{prop-equiabeldata-isogp} Let $\mathcal{D}=(H, A, \rho, \varphi)$ and $\tilde\mathcal{D}=(\tilde H, \tilde A, \tilde \rho, \tilde\varphi)$ be abelian data with $A\cong \tilde A$ as abelian groups. Assume that $ A= G(\mathcal{D})'\oplus A_1$ and $\tilde A= G(\tilde\mathcal{D})'\oplus \tilde A_1$ for some trivial modules $_H\!A_1$ and $_{\tilde H}\!\tilde A_1$, and $\mathrm{Im}(\varphi)\subseteq G(\mathcal{D})'$ and $\mathrm{Im}(\tilde\varphi)\subseteq G(\tilde\mathcal{D})'$. Then $G(\mathcal{D})\cong G(\tilde D)$ if and only if $\mathcal{D}\sim\tilde\mathcal{D}$.
\end{proposition}
\begin{proof} The sufficiency follows from Proposition \ref{prop-equidata-isogp}. We need only to prove the necessity. Assume $G(\mathcal{D})\cong G(\tilde{\mathcal{D}})$, we will show that $\mathcal{D}\sim \tilde \mathcal{D}$.
We set $A_0=G(D)'$. Let $\rho_0\colon H\to \operatorname{Aut}(A_0)$ and $\varphi_0\colon H\times H\to A_0$ be the obvious maps induced by $\rho$ and $\varphi$. Then $\mathcal{D}_0=(H, A_0, \rho_0, \varphi_0)$ is a datum. It is easy to check that $G(\mathcal{D})\cong G(\mathcal{D}_0)\times A_1$ and $G(\mathcal{D}_0)'=A_0$. Moreover, we have a decomposition $\mathcal{D}=\mathcal{D}_0\times \mathcal{D}_1$, where $\mathcal{D}_1=(\{1\}, A_1, 1, 0)$ is a trivial datum.
Similarly we have $\tilde\mathcal{D}_0=(\tilde H, \tilde A_0, \tilde \rho_0, \tilde\varphi_0)$, and $G(\tilde\mathcal{D})\cong G(\tilde\mathcal{D}_0)\times \tilde A_1$ and $G(\tilde\mathcal{D}_0)'=\tilde A_0$. Now $G(\mathcal{D})\cong G(\tilde{\mathcal{D}})$ implies that $A_0\cong \tilde A_0$, and hence $A_1\cong \tilde A_1$ as abelian groups. Applying Jordan-H\"{o}lder theorem, we obtain an group isomorphism $G(\mathcal{D}_0) \cong G(\tilde \mathcal{D}_0)$. By Corollary \ref{cor-equidata-isogp}, we know that $\mathcal{D}_0\sim\tilde \mathcal{D}_0$, and the conclusion $\mathcal{D}\sim \tilde\mathcal{D}$ follows easily.
\end{proof}
\subsection{Data with realizing group having exponent $p$}
Since we are interested in groups of exponent $p$, we discuss in this subsection when a realizing group has exponent $p$.
\begin{proposition}\label{prop-ext-p} Let $\mathcal D = (H, A, \rho, \varphi)$ be a datum with $A$ and $H$ nontrivial. Then $G(\mathcal D)$ has exponent $p$ if and only if the following conditions hold:
\begin{enumerate}
\item Both $H$ and $A$ have exponent $p$;
\item $1+ \rho(h) + \rho(h^2) +\cdots +\rho(h^{p-1})=0$ for all $h\in H$;
\item $\varphi(h,h)+\varphi(h^2,h)+\cdots +\varphi(h^{p-1},h) =0$ for all $h\in H$.
\end{enumerate}
\end{proposition}
\begin{proof} First we prove the necessity. We assume that $G(\mathcal D)$ has exponent $p$. Then (1) is obvious since any nontrivial subgroup and quotient group of $G(\mathcal D)$ will have exponent $p$. Now for any $(a,h)\in G(\mathcal D)$, it is direct to check that
\[(a, h)^p= (a+h\triangleright a + \cdots+ h^{p-1}\triangleright a + \varphi(h,h)+\varphi(h^2,h)+\cdots +\varphi(h^{p-1},h), h^p),\]
which equals $(0,1)$ by assumption.
By taking $a=0$ we will obtain (3), and (2) follows by considering arbitrary $a$. The above calculation also proves the sufficiency.
\end{proof}
\section{A formula for 2-cocycles and application to abelian data}
\subsection{2-cocycles of a finite abelian group} Let $H$ be a finite abelian group. Then $H$ is a direct product of nontrivial cyclic subgroups $H=H_1\times H_2\times \cdots\times H_m$. Assume that for any $1\le r\le m$, $H_r=\langle h_r\rangle \cong \mathbb{Z}_{n_r}$ for some $h_r\in H_r$ and some positive integer $n_r$. Clearly $h_r$ has order $n_r$.
Given any $h\in H$, $h=h_1^{i_1}h_2^{i_2}\cdots h_m^{i_m}$ for some array $(i_1,i_2,\cdots, i_m)\in \mathbb{Z}^m$, which is uniquely determined if we require that $0\le i_r\le n_r-1$ for any $1\le r\le m$.
We mention that the decomposition $H=H_1\times H_2\times \cdots\times H_m$ is not unique in general.
Given a ring $R$ and an element $x\in R$, we set $(x)_n= 1 + x +\cdots +x^{n-1}\in R$ for any positive integer $n$. For consistency of notations, we set $(x)_0=0$. Then $1-x^n=(1-x)(x)_n$ for any $x\in R$ and $n\ge 0$.
Let $(A,\rho)$ be an $H$-module. Let $(\varphi_{rs})_{1\le r\le s\le m}$ be a collection of elements in $A$.
Then $(\varphi_{st})$ gives a normalized 2-cochain $\varphi\colon H\times H\to A$ by setting
\begin{align}\label{formula-cochain}
\notag &\varphi(h_1^{i_1}h_2^{i_2}\cdots h_m^{i_m}, h_1^{j_1}h_2^{j_2}\cdots h_m^{j_m})\\
=&\sum_{r=1}^m [\frac{i_r+j_r}{n_r}](h_1^{i_1+j_1}\cdots h_{r-1}^{i_{r-1}+j_{r-1}})\triangleright \varphi_{rr} \\
\notag
-&\sum_{1\le r<s\le m} (h_1^{i_1}\cdots h_{s-1}^{i_{s-1}}h_1^{j_1}\cdots h_{r-1}^{j_{r-1}}(h_r)_{j_r}(h_s)_{i_s})\triangleright\varphi_{rs}
\end{align}
for any $0\le i_r, j_r\le n_r-1$, $r=1, 2, \cdots, m$, where $[\frac{i_r+j_r}{n_r}]$ denotes the greatest integer which is less than or equal to $\frac{i_r+j_r}{n_r}$.
We write $T_{r}=h_{r}-1$ and $N_{r}=(h_r)_{n_r} =1+ h_r + h_r^2 +\cdots + h_{r}^{n_r-1}$. The following characterization is essentially given in \cite{hlyy, hwy}.
\begin{proposition}\label{prop-2cocycle} Let $H$, $A$ be as above. Then
\begin{enumerate}
\item The 2-cochain $\varphi$ is a $2$-cocycle if and only if
\begin{enumerate}
\item[(i)] $T_r\triangleright\varphi_{rr}=0$ for $1\le r\le m$;
\item[(ii)] $N_r\triangleright\varphi_{rs}+T_s\triangleright \varphi_{rr}=0$, $T_r\triangleright \varphi_{ss}-N_s\triangleright \varphi_{rs}=0$ for $1\le r<s\le m$; and
\item[(iii)] $T_r\triangleright \varphi_{st}-T_s\triangleright \varphi_{rt}+T_t\triangleright \varphi_{rs}=0$ for $1\le r<s<t\le m$.
\end{enumerate}
\item $\varphi$ is a $2$-coboundary if and only if
there exist $a_1, a_2,\cdots, a_m\in A$, such that
\begin{enumerate}
\item[(i)] $\varphi_{rr}=N_r\triangleright a_r$ for $1\le r\le m$, and
\item[(ii)] $\varphi_{rs}=T_r\triangleright a_s-T_s\triangleright a_r$ for $1\le r<s\le m$.
\end{enumerate}
\item Let $\phi\colon H\times H\to A$ be a 2-cocycle. Then $\phi$ is cohomologous to the $2$-cochain corresponding to $(\phi_{rs})_{1\le r\le s\le m}$, where $\phi_{rr} = \phi(1,h_r) + \phi(h_r, h_r) + \cdots + \phi(h_r^{n_r-1}, h_r)$ for $1\le r\le m$, and $\phi_{rs}= \phi(h_r, h_s)-\phi(h_s, h_r)$ for $1\le r<s\le m$.
\item If moreover $H$ acts on $A$ trivially. Then $H^2(H; A)$ is in one-to-one correspondence with the set of collections $(\varphi_{rs})_{1\le r\le t\le m}$.
\end{enumerate}
\end{proposition}
Combined with Lemma \ref{lem-abel-extension}, we obtain a characterization of the derived subgroup of an extension of abelian groups.
\begin{proposition}\label{prop-der-abext} Let $H$, $A$, $(\varphi_{rs})_{1\le r\le s\le m}$ be as above. Assume that the corresponding 2-cochain $\varphi$ is a 2-cocycle. Then the derived subgroup $(A\rtimes_{\rho,\varphi} H)'$ is exactly the $H$-submodule of $A$ generated by $\{\varphi_{rs}\mid 1\le r< s\le m\}$ and $\{T_r\triangleright a\mid 1\le r\le m, a\in A\}$.
\end{proposition}
\begin{proof} Set $G=A\rtimes_{\rho,\varphi} H$, and let $M$ be the $H$-submodule of $A$ which is generated by $\{\varphi_{rs}\mid 1\le r< s\le m\}$ and $\{T_r\triangleright a\mid 1\le r\le m, a\in A\}$.
By Lemma \ref{lem-abel-extension}, for any $a, b\in A$ and $h, k\in H$, we have \[[(a,h),(b,k)]=((1-k)\triangleright a+(h-1)\triangleright b+\varphi(h,k)-\varphi(k,h),1).\]
Then $(T_r\triangleright a,1)= [(0,h_r),(a,1)]\in G'$ and $(\varphi_{rs},1)=[(0,h_r),(0, h_s)]\in G'$ for $1\le r<s\le m$, which implies $M\subseteq G'$.
Conversely, for any $h=h_1^{i_1}\cdots h_m^{i_m}$, $k=h_1^{j_1}\cdots h_m^{j_m}$ and any $a, b\in A$,
\[(h-1)\triangleright b= \sum_{u=1}^m h_1^{i_1}\cdots h_{u-1}^{i_{u-1}}(h_u^{i_{u}}-1)\triangleright b= \sum_{u=1}^m h_1^{i_1}\cdots h_{u-1}^{i_{u-1}}(h_u)_{i_u}(h_u-1)\triangleright b\in M,\]
and similarly $(1-k)\triangleright a\in M$. Moreover, we have
\begin{align*}\varphi(h,k)&=\varphi(h_1^{i_1}h_2^{i_2}\cdots h_m^{i_m}, h_1^{j_1}h_2^{j_2}\cdots h_m^{j_m})\\
&= \sum_{r=1}^m [\frac{i_r+j_r}{n_r}](h_1^{i_1+j_1}\cdots g_{r-1}^{i_{r-1}+j_{r-1}})\triangleright \varphi_{rr} \\
&
-\sum_{1\le r<s\le m} (h_1^{i_1}\cdots h_{s-1}^{i_{s-1}}h_1^{j_1}\cdots h_{r-1}^{j_{r-1}}(h_r)_{j_r}(h_s)_{i_s})\triangleright\varphi_{rs},
\end{align*}
and
\begin{align*}\varphi(k,h)
&= \sum_{r=1}^m [\frac{j_r+i_r}{n_r}](h_1^{j_1+i_1}\cdots g_{r-1}^{j_{r-1}+i_{r-1}})\triangleright \varphi_{rr} \\
&
-\sum_{1\le r<s\le m} (h_1^{j_1}\cdots h_{s-1}^{j_{s-1}}h_1^{i_1}\cdots h_{r-1}^{i_{r-1}}(h_r)_{i_r}(h_s)_{j_s})\triangleright\varphi_{rs},
\end{align*}
and therefore $\varphi(h,k)-\varphi(k,h)\in M$. It follows that
$G'\subseteq M$ and hence $G'=M$.
\end{proof}
\begin{remark} We mention that in the above proposition, the $H$-submodule of $A$ generated by $\{T_r\triangleright a|1\le r\le m, a\in A\}$ is exactly the radical $\operatorname{rad}(A)$ of $A$, i.e., the intersection of all maximal submodule of $A$.
\end{remark}
\subsection{A technical lemma}
The following observation is handful. It is equivalent to certain combinatorial identities in characteristic $p$, which are probably well-known to experts. For the convenience of the readers we also include a proof here.
\begin{lemma}\label{keylem} Let $R$ be a commutative ring over $\mathbb{Z}_p$, and $x, y\in R$ be two nonzero elements such that $x^iy^{p-1-i}=0$ for any $0\le i\le p-1$. Then the following statements are equivalent:
(1) $\sum_{r=1}^{p-1}(1+x)^{ri}(1+y)_{rj}=0$ for any $1\le i, j\le p-1$;
(2) $\sum_{r=1}^{p-1}(1+x)^{ri}(1+y)_{r}=0$ for any $1\le i\le p-1$;
(3) $\sum_{r=1}^{p-1}(1+x)^{r}(1+y)_{rj}=0$ for any $1\le j\le p-1$;
(4) $x^iy^{p-2-i}=0$ for any $0\le i\le p-2$.
\end{lemma}
\begin{proof} Let $z= f(x,y)$ be in $R$ such that $f$ is a polynomial in $x$ and $y$ with zero constant term. It follows from the assumption on $x$ and $y$ that $z^{p-1}=0$, hence $(1+z)$ is invertible. If $z\ne 0$, then $1+z$ has order $p$, and $(1+z)^i=(1+z)^{i'}$ if and only if $\bar i=\bar {i'}$ in $\mathbb{Z}_p$.
There is no difficulty to show that $(1+z)_i$ is invertible for any $1\le i\le p-1$. In fact, $(1+z)_i = i + \tilde f(x,y)$ for some polynomial $\tilde f$ in $x$ and $y$ with zero constant term, hence $(1+z)_i = i(1 +i^{-1}\tilde f(x,y))$ is invertible. Moreover,
\[(1+z)_p = 1 + (1+z) + \cdots +(1+z)^{p-1}= ((1+z)-1)^{p-1} = f(x,y)^{p-1}= 0.\]
In particular, $(1+y)_{i+p}- (1+y)_{i} = (1+y)^i(1+y)_p =0$, and therefore $(1+y)_i= (1+y)_{i'}$ whenever $\bar i=\bar {i'}$ in $\mathbb{Z}_p$. Thus $(1)\Longleftrightarrow (2)\Longleftrightarrow(3)$ follows.
Next we show the equivalence $(2)\Longleftrightarrow (4)$. For each $1\le i\le p-1$, we have
\[\sum_{r=1}^{p-1}(1+x)^{ri}(1+y)_{r} =\sum_{u,v\ge 0} \lambda^i_{u,v} x^uy^v,
\]
where $\lambda^i_{u,v}$'s are integer coefficients which are independent on $x, y$ and the ring $R$.
This inspires us to consider the ring $S=\mathbb{Z}_p[X,Y]/\langle X^{p-1}, X^{p-2}Y, \cdots, Y^{p-1} \rangle$. It is easy to show that $S$ has a $\mathbb{Z}_p$-basis $X^iY^j, 0\le i, j\le p-2, 0\le i+j\le p-2$. Clearly, the subring $\mathbb{Z}_p[x,y]$ of $R$ can be viewed as a quotient ring of $S$ under the map $X\mapsto x$ and $Y\mapsto y$.
By direct calculation, we have
\begin{align*}
&((1+X)^i-1)\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}\\
=& \sum_{r=2}^{p}(1+X)^{ri}(1+Y)_{r-1} - \sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}\\
=& (1+X)^{pi}(1+Y)_{p-1} - \sum_{r=2}^{p-1}(1+X)^{ri}((1+Y)_r- (1+Y)_{r-1}) -(1+X)^i(1+Y)_1\\
=& (1+Y)_{p-1} - \sum_{r=2}^{p-1}(1+X)^{ri}(1+Y)^{r-1} - (1+X)^i \\
=& -(1+Y)^{p-1}- \sum_{r=2}^{p-1}(1+X)^{ri}(1+Y)^{r-1} - (1+X)^i \\
=& -(1+Y)^{-1}(1+ \sum_{r=2}^{p-1}(1+X)^{ri}(1+Y)^{r} + (1+X)^i(1+Y))\\
=& -(1+Y)((1+X)^{i}(1+Y))_{p}\\
=& 0
\end{align*}
for each $1\le i\le p-1$. Using the facts $(1+X)^i-1 = X(1+X)_i$ and $(1+X)_i$ is invertible, we know that \[X\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}=0.\]
Note that the converse is also true, say $X\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}=0$ if and only if $((1+X)^i-1)\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}$. Similarly,
\begin{align*}
Y\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}
=& ((1+Y) -1)\sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)_{r}\\
=& \sum_{r=1}^{p-1}(1+X)^{ri}((1+Y)^{r}-1)\\
=& \sum_{r=1}^{p-1}(1+X)^{ri}(1+Y)^{r}- \sum_{r=1}^{p-1}(1+X)^{ri}\\
=& ((1+X)^i(1+Y))_p - ((1+X)^i)_p\\
=& 0.
\end{align*}
It forces that $\bar \lambda^i_{u, v} = 0$ in $\mathbb{Z}_p$ for any $u+v< p-2$.
We are left to determine $\lambda^i_{u, v}$'s with $u+v=p-2$. Note that the higher terms vanish automatically in $S$ as well in the subring $\mathbb{Z}_p[x,y]$.
For simplicity we set $g= 1+X$ and $h=1+Y$. We write the elements $\sum_{r=1}^{p-1}g^{ri}(h)_r$, $i=1, \cdots, p-1$ into a column vector $(F_1,F_2,\cdots, F_{p-1})^T$ with entries in $S$, say
\[
\begin{pmatrix}
F_1 \\
F_2\\
\cdots\\
F_{p-1}\\
\end{pmatrix}
=\begin{pmatrix}
g & g^2 & \cdots & g^{p-1} \\
g^2 & g^4 & \cdots & g^{2(p-1)} \\
\cdots & \cdots & \cdots & \cdots \\
g^{p-1} & g^{(p-1)2} & \cdots & g^{(p-1)(p-1)} \\
\end{pmatrix}
\begin{pmatrix}
(h)_1 \\
(h)_2\\
\cdots\\
(h)_{p-1}\\
\end{pmatrix}.
\]
Consider the following column vector obtained by elementary row operations
\begin{align*}
\begin{pmatrix}
E_1\\E_2\\ \cdots\\ E_{p-1}
\end{pmatrix}
=&\begin{pmatrix}
1 & & & & & \\
1 & -1 & & & & \\
\vdots & \vdots & \ddots & & & \\
1 & -C_i^1 & \cdots & (-1)^iC_i^i & & \\
\vdots & \vdots & \vdots & \vdots & \ddots & \\
1 & -C_{p-2}^1 & \cdots & (-1)^iC_{p-2}^i & \cdots & -C_{p-2}^{p-2} \\
\end{pmatrix}
\begin{pmatrix}
F_1 \\
F_2\\
\cdots\\
F_{p-1}\\
\end{pmatrix} \\
=& \begin{pmatrix}
g & g^2 & \cdots & g^{p-1} \\
g(1-g) & g^2(1-g^2) & \cdots & g^{p-1}(1-g^{p-1}) \\
g(1-g)^2& g^2(1-g^2)^2 & \cdots & g^{p-1}(1-g^{p-1})^2 \\
\cdots & \cdots & \cdots & \cdots \\
g(1-g)^{p-2}& g^2(1-g^2)^{p-2} & \cdots & g^{p-1}(1-g^{p-1})^{p-2} \\
\end{pmatrix}
\begin{pmatrix}
(h)_1 \\
(h)_2\\
\cdots\\
(h)_{p-1}\\
\end{pmatrix}.
\end{align*}
Again we may expand each $E_i$ into a polynomial in variables $X, Y$ with integer coefficients, which is independent on the ring $S$. Assume that
\[E_i= \sum_{u\ge i-1} \mu^i_{u,v} X^uY^v= X^{i-1}f_i(Y)+ X^{i}f'_i(X,Y),\] where $f_i(Y)$ and $f'_i(X,Y)$ are polynomials.
Clearly $XE_i=YE_i=0$ for each $i$, and hence $\mu^i_{u,v}=0$ in $\mathbb{Z}_p$ for any $u+v< p-2$.
It is direct to check that
\[\begin{pmatrix}
(h)_1 \\
(h)_2\\
\cdots\\
(h)_{p-1}\\
\end{pmatrix}
=\begin{pmatrix}
C_1^1 & & & \\
C_2^1 & C_2^2 & & \\
\vdots &\vdots & \ddots & \\
C_{p-1}^1 &C_{p-1}^2 &\cdots & C_{p-1}^{p-1} \\
\end{pmatrix}
\begin{pmatrix}
1 \\
Y \\
\cdots \\
Y^{p-2} \\
\end{pmatrix},
\]
then we have
\begin{align*}\begin{pmatrix}
f_1(Y)\\
f_2(Y)\\
\cdots\\
f_{p-1}(Y)
\end{pmatrix}
=&\begin{pmatrix}
1 & 1 & \cdots & 1 \\
1 & 2 & \cdots & p-1 \\
\cdots &\cdots & \cdots & \cdots \\
1 & 2^{p-2} & \cdots & (p-1)^{p-2} \\
\end{pmatrix}
\begin{pmatrix}
C_1^1 & & & \\
C_2^1 & C_2^2 & & \\
\vdots &\vdots & \ddots & \\
C_{p-1}^1 &C_{p-1}^2 &\cdots & C_{p-1}^{p-1} \\
\end{pmatrix}
\begin{pmatrix}
1 \\
Y \\
\cdots \\
Y^{p-2} \\
\end{pmatrix} \\
=& \begin{pmatrix}
& & & \mu^1_{0,p-2} \\
& & \mu^2_{1,p-3} & * \\
& \cdots &\cdots &\cdots \\
\mu^{p-1}_{p-2,0} & \cdots & * & * \\
\end{pmatrix}
\begin{pmatrix}
1 \\
Y \\
\cdots \\
Y^{p-2} \\
\end{pmatrix}.
\end{align*}
It follows that
\[\begin{pmatrix}
E_1\\E_2\\ \cdots\\ E_{p-1}
\end{pmatrix}
=\begin{pmatrix}
* & \cdots & * & \mu^1_{0,p-2} \\
* & \cdots & \mu^2_{1,p-3} & \\
\cdots &\cdots & & \\
\mu^{p-1}_{p-2,0} & & & \\
\end{pmatrix}
\begin{pmatrix}
X^{p-2} \\
X^{p-3}Y \\
\cdots \\
Y^{p-2} \\
\end{pmatrix}.
\]
Applying the Van der Monde determinant, we know that $\mu^1_{0,p-2} \mu^2_{1,p-3}\cdots \mu^{p-1}_{p-2,0}\ne 0$ in $\mathbb{Z}_p$, and hence the following matrix
\[\Lambda =
\begin{pmatrix}
\lambda^1_{p-2,0} & \lambda^2_{p-3,1} & \cdots & \lambda^1_{0,p-2} \\
\lambda^2_{p-2,0} & \lambda^2_{p-3,1} & \cdots & \lambda^2_{0,p-2} \\
\cdots & \cdots & \cdots & \cdots \\
\lambda^{p-1}_{p-2,0} & \lambda^{p-1}_{p-3,1} & \cdots & \lambda^{p-1}_{0,p-2}\\
\end{pmatrix}
\]
is invertible in $\mathbb{Z}_p$. Now the equivalence $(2)\Longleftrightarrow (4)$ follows from the equality
\[
\begin{pmatrix}
\sum_{r=1}^{p-1}(1+x)^{r}(1+y)_{r}\\
\sum_{r=1}^{p-1}(1+x)^{2r}(1+y)_{r}\\
\cdots\\
\sum_{r=1}^{p-1}(1+x)^{(p-1)r}(1+y)_{r}
\end{pmatrix}
=\Lambda
\begin{pmatrix}
x^{p-2}\\
x^{p-3}y\\
\cdots\\
y^{p-2}.
\end{pmatrix}.
\]
\end{proof}
\subsection{A criterion for a realizing group having exponent $p$} Now we consider the special case that $H$ and $A$ are both elementary $p$-groups.
Let $H=\langle h_1\rangle\times \langle h_2\rangle\times\cdots \times\langle h_m\rangle $ be an elementary $p$-group with a basis $h_1, h_2, \cdots, h_m$. Then it is well known that there exists a ring isomomorphism \[\mathbb{Z}_p[H]\xrightarrow{\cong} \mathbb{Z}_p[x_1, \cdots, x_m]/\langle x_1^{p},\cdots, x_m^p\rangle,\ \ h_i\mapsto x_i+1, 1\le r\le m, \]
where $\mathbb{Z}_p[H]$ is the group ring of $H$ over $\mathbb{Z}_p$. Clearly, the Jacobson radical $\mathrm{rad}(\mathbb{Z}_p[H])$ of $\mathbb{Z}_p[H]$ is the ideal generated by $h_1-1, \cdots, h_m-1$. Recall that the Jacobson radical of a ring is by definition the intersection of all its maximal ideals.
Combing Proposition \ref{prop-ext-p} and Proposition \ref{prop-2cocycle}, we obtain the following criterion to determine when a realizing group has exponent $p$.
\begin{theorem} \label{thm-expp} Let $H=\langle h_1\rangle\times \langle h_2\rangle\times\cdots \times\langle h_m\rangle $ be an elementary $p$-group with a basis $h_1, h_2,\cdots, h_m$. Set $T_r=h_r-1$ for $r=1, 2, \cdots, m$. Let $(A,\rho)$ be an $H$-module, and let $(\varphi_{rs})_{1\le r\le s\le m}$ be a family of elements in $A$ and $\varphi$ the corresponding 2-cochain.
Then $\varphi$ is a 2-cocyle such that $A\rtimes_{\rho, \varphi} H$ has exponent $p$ if and only if
\begin{enumerate}
\item $A$ is an elementary $p$-group;
\item $T_{r_1}T_{r_2}\cdots T_{r_{p-1}} \triangleright a=0$ for all $1\le r_1\le r_2\le \cdots\le r_{p-1}\le m$ and $a\in A$;
\item $\varphi_{rs}$'s satisfy the following conditions:
\begin{enumerate}
\item $\varphi_{rr}=0$ for any $1\le s\le m$;
\item For any $1\le l\le p-1$, $1\le r_1<r_2<\cdots<r_{l+1}\le m$, and $i_1,\cdots, i_{l+1}\ge0$ with $i_1+i_2+\cdots+ i_{l+1}= p-l-1$,
\[T_{r_1}^{i_1}T_{r_2}^{i_2}\cdots T_{r_{l+1}}^{i_{l+1}}\triangleright (\sum_{u=1}^l(i_u+1)T_{r_1}\cdots T_{r_{u-1}}T_{r_{u+1}}\cdots T_{r_l}\triangleright \varphi_{r_u,r_{l+1}})=0;\]
\item $T_r\triangleright\varphi_{st} - T_s\triangleright\varphi_{rt}+T_t\triangleright\varphi_{rs}=0$ for any $1\le r<s<t\le m$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof} By Proposition \ref{prop-2cocycle}, $\varphi$ is a 2-cocyle if and only if
\begin{enumerate}
\item[(I.1)] $T_r\triangleright\varphi_{rr}=0$ for $1\le r\le m$;
\item[(I.2)] $N_r\triangleright\varphi_{rs}+T_s\triangleright \varphi_{rr}=0$, $T_r\triangleright \varphi_{ss}-N_s\triangleright \varphi_{rs}=0$ for $1\le r<s\le m$; and
\item[(I.3)] $T_r\triangleright \varphi_{st}-T_s\triangleright \varphi_{rt}+T_t\triangleright \varphi_{rs}=0$ for $1\le r<s<t\le m$.
\end{enumerate}
By Proposition \ref{prop-ext-p}, the realizing group $A\rtimes_{\rho, \varphi} H$ has exponent $p$ if and only if
\begin{enumerate}
\item[(II.1)] $A$ has exponent $p$;
\item[(II.2)] $1+ \rho(h) + \rho(h^2) +\cdots +\rho(h^{p-1})=0$ for all $h\in H$;
\item[(II.3)] $\varphi(h,h)+\varphi(h^2,h)+\cdots +\varphi(h^{p-1},h) =0$ for all $h\in H$.
\end{enumerate}
First we claim that assuming (II.1) and (II.2), then $\varphi$ satisfies (II.3) if and only if
$\varphi_{rr} = 0$ for any $1\le r\le m$, and
\begin{equation}\label{eqM1}(\sum_{k=1}^{p-1}
(h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{r}^{\lambda_{r}})^k (h_{r+1})_{k\lambda_{r+1}})\triangleright (\sum_{u=1}^{r}
h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{u-1}^{\lambda_{u-1}} (h_{u})_{\lambda_u}\triangleright \varphi_{u,r+1})=0
\end{equation}
for any $1\le r\le m-1$, $0\le \lambda_1, \cdots, \lambda_{r+1}\le p-1$.
We mention that $(\ref{eqM1})$ holds true automatically if $\lambda_{r+1}=0$, or if $\lambda_u=0$ for all $1\le u\le r$. The proof of the claim is straightforward. In fact, by applying (II.3) to each $h_r$ we have
\[0 = \sum_{k=1}^{p-1}\varphi(h_r^k, h_r) = \sum_{k=1}^{p-1}[\frac{k+1}{p}]\varphi_{rr}=\varphi_{rr}.
\]
For any $1\le r\le m-1$, and any $0\le \lambda_1, \cdots, \lambda_{r+1}\le p-1$, by applying (II.3) to $h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{r}^{\lambda_r}$ and $h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{r}^{\lambda_r}h_{r+1}^{\lambda_{r+1}}$, we have
\[\sum_{k=1}^{p-1} \sum_{1\le u<v\le r}
(h_{1}^{k\lambda_1}h_{2}^{k\lambda_2}\cdots h_{v-1}^{k\lambda_{v-1}} (h_{v})_{k\lambda_v}
h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{u-1}^{\lambda_{u-1}} (h_{u})_{\lambda_u})\triangleright \varphi_{u,v}=0,
\]
and
\[\sum_{k=1}^{p-1} \sum_{1\le u<v\le r+1}
(h_{1}^{k\lambda_1}h_{2}^{k\lambda_2}\cdots h_{v-1}^{k\lambda_{v-1}} (h_{v})_{k\lambda_v}
h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{u-1}^{\lambda_{u-1}} (h_{u})_{\lambda_u})\triangleright \varphi_{u,v}=0,
\]
and $(\ref{eqM1})$ follows by taking the difference of the above equalities. Thus we have proved the necessity, and the argument works well for the sufficiency.
Now we can prove the theorem. Clearly (II.1) is the same as the condition (1) in the theorem, and by Proposition \ref{prop-rad0}, (II.2) is equivalent to (2). We will prove that under the assumptions (1) and (2), the condition (3) is equivalent to (I.1)-(I.3) and (II.3).
Assume (I.1)-(I.3) and (II.1)-(II.3). We need to prove (3). Clearly (3)(a) follows from the above statement and (3)(c) is the same as (I.3), and we are left to prove (3)(b).
By the above claim, $(\ref{eqM1})$ holds for any $1\le r\le m-1$, $0\le \lambda_1, \cdots, \lambda_{r+1}\le p-1$. By Lemma \ref{keylem}, this is equivalent to
\begin{equation}\label{eqM2}(h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{r}^{\lambda_{r}}-1)^i(h_{r+1}-1)^j\triangleright
(\sum_{u=1}^{l}
h_{1}^{\lambda_1}h_{2}^{\lambda_2}\cdots h_{u-1}^{\lambda_{u-1}} (h_{u})_{\lambda_u}\triangleright \varphi_{u,r+1})=0
\end{equation}
holds for any $0\le \lambda_1, \cdots, \lambda_r\le p-1$ and for any $i+j=p-2$.
Under the assumption $(2)$, the above equality is equivalent to
\begin{equation}\label{eqM3}(\lambda_1T_{1} + \cdots +\lambda_rT_{r})^iT_{r+1}^j\triangleright
(\sum_{u=1}^{l}
\lambda_u\varphi_{u,r+1})=0,
\end{equation}
and by expanding the left hand side, it turns into
\begin{equation}\label{eqM4}\sum_{j_1,j_2,\cdots, j_r\ge 0\atop j_1+ j_2 +\cdots +j_r=i+1}
\lambda_1^{j_1}\cdots \lambda_r^{j_r}
T_{r+1}^j \sum_{1\le u\le r\atop j_u>0} C_{i; j_1,\cdots, j_u-1,\cdots, j_r} T_{r_1}^{j_1}\cdots T_{u}^{j_u-1}\cdots T_{r}^{j_r}\triangleright\varphi_{u, r+1}=0,
\end{equation}
where $C_{i; u_1, u_2, \cdots, u_r} =\frac{i!}{u_1! u_2!\cdots u_r!}$ is the multinomial coefficient for any $u_1+\cdots+u_l=i$.
Since the above equality holds for arbitrary $\lambda_1, \cdots, \lambda_r$, by using the Van der Monde determinant, $(\ref{eqM4})$ is equivalent to
\begin{equation}\label{eqM5}T_{r+1}^j \sum_{1\le u\le r\atop j_u>0} C_{i; j_1,j_2,\cdots, j_u-1,\cdots, j_r} T_{1}^{j_1}T_{2}^{j_2}\cdots T_{u}^{j_u-1}\cdots T_{r}^{j_r}\triangleright\varphi_{u, r+1}=0
\end{equation}
for any $j_1, j_2, \cdots, j_r, j\ge 0$ such that $j_1+\cdots + j_r + j = p-2$.
Given $j_1, j_2, \cdots, j_l, j\ge 0$ such that $j_1+\cdots + j_l + j = p-2$. Consider the subset $X=\{1\le u\le r\mid j_u>0\}= \{r_1, r_2,\cdots, r_l\}$ with $r_1<r_2<\cdots <r_l$. We set $r_{l+1}=r+1$, $i_u = j_{r_u}-1$ for $1\le u\le l$ and $i_{l+1} = j$, then $i_1+i_2 +\cdots + i_{l+1} = p-l-1$, and the above equality reads as
\begin{equation}\label{eqM6}T_{r_1}^{i_1}\cdots T_{r_{l+1}}^{i_{l+1}}\triangleright (\sum_{u=1}^l\frac{(i_1+\cdots+i_l+l-1)(i_u+1)}{(i_1+1)!(i_2+1)!\cdots(i_l+1)!}
T_{r_1}\cdots T_{r_{u-1}}T_{r_{u+1}}\cdots T_{r_l}\triangleright \varphi_{r_u,r_{l+1}})=0,
\end{equation}
which is obviously equivalent to $(3)(b)$.
Conversely, assume that the assumptions (1)-(3) hold, we will prove (I.1)-(I.3) and (II.1)-(II.3). As we have shown above, it suffices to prove (I.1)-(I.3) and (II.3).
Obviously (I.1) holds for each $\varphi_{rr}=0$ by the assumption (3)(a); (I.2) follows from the assumption (2) and (3)(a); and (I.3) is the same as (3)(c). We are left to prove (II.3).
By the claim in the begin of the proof, it suffices to show that (\ref{eqM1}) holds for any $1\le r\le m-1$, $0\le \lambda_1, \cdots, \lambda_{r+1}\le p-1$, which has been shown to be equivalent to (3)(b). This completes the proof.
\end{proof}
Then combined with Proposition \ref{prop-der-abext}, we have a description of the derived subgroup of the resulting group of an abelian datum.
\begin{corollary}\label{cor-der-abext} Let $(H, A, \rho, \varphi)$ be an abelian datum such that $G=A\rtimes_{\rho,\varphi} H$ has exponent $p$.
Then $G'$ is exactly the $H$-submodule of $A$ generated by $\mathrm{Im}(\varphi)$ and $\operatorname{rad}(A)$. In particular, $A=G'$ if and only if $A$ is generated by $\mathrm{Im}(\varphi)$ as an $H$-module.
If moreover, $\rho$ is a trivial action, then $A=G'$ if and only if $A=\mathrm{Im}(\varphi)$.
\end{corollary}
If the group action $\rho$ is trivial, then $\rho(T_r)=0$ for all $r$, and hence all conditions concerning $T_r$'s in the theorem automatically hold.
\begin{corollary}\label{cor-expp2} Let $H$ and $A$ be elementary $p$-groups such that $H$ acts on $A$ trivially. Let $(\varphi_{rs})_{1\le r\le s\le m}$ be a collection of elements in $A$, and $\varphi$ the
corresponding 2-cochain defined by (\ref{formula-cochain}). Then
\begin{enumerate}
\item $\varphi$ is a 2-cocycle; and $\varphi$ is a 2-coboundary if and only if $\varphi=0$, if and only if $\varphi_{rs}=0$ for all $r, s$;
\item $A\rtimes_{\varphi} H$ has exponent $p$ if and only if $\varphi_{rr}=0$ for $r=1, 2, \cdots, m$.
\end{enumerate}
\end{corollary}
In case $p=3$, the conditions in Theorem \ref{thm-expp} are much more simplified.
\begin{corollary}\label{cor-expp1} Let $H=\langle h_1\rangle\times \langle h_2\rangle\times\cdots \times\langle h_m\rangle $ be an elementary $3$-group, where $h_1, h_2,\cdots, h_m$ is a basis. Let $(A,\rho)$ be an $H$-module, and let $(\varphi_{rs})_{1\le r\le s\le m}$ be a family of elements in $A$ and $\varphi$ the corresponding 2-cochain.
Then $\varphi$ is a 2-cocyle such that $A\rtimes_{\rho, \varphi} H$ has exponent $3$ if and only if
\begin{enumerate}
\item $A$ is an elementary $3$-group;
\item $(h_r-1)(h_s-1) \triangleright a=0$ for any $1\le r\le s\le m$ and $a\in A$;
\item $\varphi_{st}$'s satisfy the following conditions:
\begin{enumerate}
\item $\varphi_{rr}=0$ for any $1\le r\le m$;
\item $h_r\triangleright \varphi_{rs} = h_s\triangleright \varphi_{rs}= \varphi_{rs}$ for any $1\le r< s\le m$;
\item $(h_r-1)\triangleright \varphi_{st}=-(h_s-1)\varphi_{rt}=(h_t-1)\varphi_{rs}$ for any $1\le r<s<t\le m$.
\end{enumerate}
\end{enumerate}
\end{corollary}
\begin{proof} Consider the condition $(3)(b)$ in Theorem \ref{thm-expp}. The case $l=1$ reads as
$T_r\triangleright\varphi_{rs} = 0$ and $T_s\triangleright\varphi_{rs}=0$ for any $1\le r< s\le m$, which is equivalent to $h_r\triangleright \varphi_{rs} = h_s\triangleright \varphi_{rs}= \varphi_{rs}$. The case $l=2$ reads as
$T_r\triangleright\varphi_{st} + T_s\triangleright\varphi_{rt}=0$ for all $1\le r<s<t\le m$, which gives $T_r\triangleright\varphi_{st}=- T_s\triangleright\varphi_{rt}$. Now $T_r\triangleright\varphi_{st} - T_s\triangleright\varphi_{rt}+T_t\triangleright\varphi_{rs}=0$ is equivalent to $T_t\triangleright\varphi_{rs}=2T_s\triangleright\varphi_{rt}=-T_s\triangleright\varphi_{rt}$.
\end{proof}
We also draw the following consequence, which says that for a group of exponent 3 and order $\le 3^6$, the derived subgroup will be contained in its center.
\begin{corollary}\label{cor-expp3}
Let $\mathcal{D}= (H, A,\rho, \varphi)$ be a datum such that $G=H\rtimes_{\varphi} A$ has exponent $p$ and $G'=A$. Assume $A=\mathbb{Z}_3^n$ and $H=\mathbb{Z}_3^m$. Then $\rho$ is trivial unless $m\ge3$ and $n\ge4$.
\end{corollary}
\begin{proof} Keep the above notations $h_1, \cdots, h_m$ and $a_1, \cdots, a_n$, and $\varphi_{rs}$, $1<r<s\le m$. Set $T_r= h_r-1$. Then $H$ acts trivially on $A$ if and only if $T_r\triangleright a_u=0$ for all $1\le r\le m$ and $1\le u\le n$.
Assume that $\rho$ is nontrivial. We will show that $m\ge3$ and $n\ge4$.
First we prove $m\ge 3$. By Proposition \ref{prop-der-abext}, $G(\mathcal{D})'$ is the submodule generated by $T_r\triangleright a_u$'s and $\varphi_{rs}$'s. If $m=1$, then $G(\mathcal{D})'=T_1\triangleright A=\mathrm{rad}(A)$, which never equals $A$ by Nakayama Lemma. If $m=2$, then $T_1(\varphi_{12})= T_2\triangleright\varphi_{12}=0$, which means that $\varphi_{12}\in \mathrm{soc}(A)$, the socle of the $H$-module $A$, say the sum of simple submodules of $A$. Thus $G(\mathcal{D})'\subseteq \mathrm{soc}(A) + \mathrm{rad}(A)$, which equals $A$ only if $A$ is a simi-simple $H$-module, or equivalently, $H$ acts on $A$ trivially. Thus we must have $m\ge 3$.
Now we assume $m\ge 3$. We claim that there exist some $1\le r< s< t\le m$ such that $\varphi_{rs}, \varphi_{st}, \varphi_{st}$ are linearly independent in the quotient space $A/\mathrm{soc}(A)$. Otherwise, for any $1\le r< s <t \le m$, we have $ \varphi_{rs} \in \mathbb{Z}_p \varphi_{rt} + \mathbb{Z}_p\varphi_{st} +\mathrm{soc}(A)$. Recall that $a\in \mathrm{soc}(A)$ if and only if $T_r\triangleright a=0$ for all $1\le r\le m$. Then $T_t\triangleright \varphi_{rs} = 0$, for $T_t\triangleright \varphi_{rt}= T_t\triangleright\varphi_{st} = 0$ and $T_t\triangleright(\mathrm{soc}(A)) = 0$. It follows that $T_s\triangleright \varphi_{rt} = T_r\triangleright\varphi_{st}=0$, and hence $\varphi_{rs}\in \mathrm{soc}(A)$. Therefore $A = G(\mathcal{D})'\subseteq \mathrm{soc}(A)$, which forces that $A= \mathrm{soc}(A)$ which happens only when $A$ is a semi-simple module.
Let $1\le r< s< t\le m$ be such that $\varphi_{rs}, \varphi_{st}, \varphi_{st}$ are linearly independent in the quotient space $A/\mathrm{soc}(A)$. Using the fact that the socle of a finite dimensional module is always nonzero, we know that $n= \dim_{\mathbb{Z}_p} A\ge 4$.
\end{proof}
\begin{remark} We mention that the above corollary does not hold true in case $p>3$. For instance, let $G$ be the group
\[\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, [a,b]=c, [a,c]=d, [a,d]=[b,c]=[b,d]=[c,d]=1\rangle.
\]
It is easy to show that $G'=\langle c,d\rangle$, and $Z(G)=\langle d \rangle$. Now $|G'|=p^2$, and the $G$-action on $G'$ is nontrivial.
\end{remark}
\subsection{Nilpotency class of an extension of elementary $p$-groups} In this subsection, we will show that the nilpotency class of the realizing group $A\rtimes_{\rho, \varphi}H$ relates closely to the Loewy length of $A$ as an $H$-module.
\begin{proposition}\label{prop-cn-mgp} Let $(A, H, \rho, \varphi)$ be an abelian datum with $A=\mathbb{Z}_p^m$ for some $m>0$. Let $c$ be the nilpotency class number of $A\rtimes_{\rho,\varphi}H$, and $l=\ell\ell(_HA)$ the Loewy length. Then
\begin{enumerate}
\item $l\le c\le l+1$, and $c=l$ if and only if there exists some $\psi$ cohomologous to $\varphi$ such that $\psi(h_1,h_2)\in \operatorname{soc}^l(A)$ and $\psi(h_1, h_2)- \psi(h_2, h_1)\in \operatorname{soc}^{l-1}(A)$ for any $h_1, h_2\in H$;
\item $c\le p$ if in addition $A\rtimes_{\rho,\varphi}H$ has exponent $p$.
\end{enumerate}
\end{proposition}
\begin{proof} (1) We set $G= A\rtimes_{\rho,\varphi}H$. Then $l=\ell\ell(_HA)=\ell\ell(_GA)$. By definition $\operatorname{soc}^{l-1}(A)\ne \operatorname{soc}^l(A)=A$. We use induction on $l$.
If $l=1$, then $A$ is semisimple and $H$ acts trivially on $A$. Then $A\in Z(G)$ and $H\cong G/A$ is ableian, hence $c=1$ or $2$. Clearly $c=1$ if and only if $G$ is commutative, if and only if $\varphi(h_1,h_2)=\varphi(h_2, h_1)$ for all $h_1, h_2\in H$.
Now we assume (1) holds for data with $\ell\ell(A)\le l-1$. Suppose $\ell\ell(A)=l\ge 2$. By Lemma \ref{lem-abel-extension},
there exist some $\tilde\varphi$ cohomolgous to $\varphi$ and a subgroup $K\le H$, such that
\begin{enumerate}
\item[(a)] $K$ acts on $A$ trivially;
\item[(b)] $\tilde\varphi(k,h)\in \operatorname{soc}(A)$ and $\tilde\varphi(k, h)=\tilde\varphi(h, k)$ for any $k\in K$ and $h\in H$;
\item[(c)] $Z(A\rtimes_{\rho,\tilde\varphi}H)=\{(a,k)\mid a\in\operatorname{soc}(A), k\in K\}$.
\end{enumerate}
Note that $A\rtimes_{\rho,\tilde\varphi}H\cong A\rtimes_{\rho,\varphi}H$. Then
$$G/Z(G)\cong \frac{A\rtimes_{\rho,\tilde\varphi}H}{Z(A\rtimes_{\rho,\tilde\varphi}H)}
\cong\frac {A}{\operatorname{soc}(A)}\rtimes_{\rho', \varphi'}\frac H K,$$
where $\rho'$ is the induced action of $H/K$ on $A/\operatorname{soc}(A)$, and $\varphi'\colon H/K \times H/K\to A/\operatorname{soc}(A)$ the 2-cocycle induced from $\tilde\varphi$.
By definition $\ell\ell(A/\operatorname{soc}(A))=l-1$. By the induction hypothesis, the class number of the quotient group $G/Z(G)$ is equal to $l-1$ or $l$, and hence the class number of $G$ is $l$ or $l+1$.
Now assume $\varphi$ is cohomologous to some $\psi$ with $\psi(h_1,h_2)\in \operatorname{soc}^l(A)$ and $\psi(h_1, h_2)- \psi(h_2, h_1)\in \operatorname{soc}^{l-1}(A)$ for any $h_1, h_2\in H$. Then by easy calculation, $\operatorname{soc}^{l-1}(A)\vartriangleleft A\rtimes_{\rho, \psi} H$, and $\tilde H=A\rtimes_{\rho, \psi} H/\operatorname{soc}^{l-1}(A)$ is abelian, and hence $G\cong \operatorname{soc}^{l-1}(A)\rtimes_{\tilde \rho, \tilde\psi} \tilde H$ for some induced action $\tilde\rho$ and 2-cocycle $\tilde\psi$. Since $\ell\ell(\operatorname{soc}^{l-1}(A))=l-1$, by the induction hypothesis, we have $l-1\le c\le l$ and it forces that $c=l$.
Conversely, we assume $c=l$. Then by definition $G/Z(G)$ has nilpotency class number $c-1=l-1$ and $\ell\ell(A/\operatorname{soc}(A))=l-1$. Now by the induction hypotheses, there exists some $\varphi_1\colon H/K\times H/K\to A/\operatorname{soc}(A)$, which is cohomologous to $\varphi'$ and
\[\varphi_1(\bar h_1, \bar h_2)\in \operatorname{soc}^{l-1}(A/\operatorname{soc}(A))=\operatorname{soc}^l(A)/\operatorname{soc}(A),\]
and \[ \varphi_1(\bar h_1, \bar h_2)-\varphi_1(\bar h_2, \bar h_1)\in \operatorname{soc}^{l-2}(A/\operatorname{soc}(A))=\operatorname{soc}^{l-1}(A)/\operatorname{soc}(A)\]
for any $h_1, h_2\in H$, where $\bar h_1, \bar h_2$ are the image of $h_1$ and $h_2$ under the canonical projection map $\pi_H\colon H\to H/K$.
Since $\varphi'$ is cohomolgous to $\varphi_1$, we have $\varphi_1-\varphi' = d(\bar f)$ for some $\bar f\colon H/K\to A/\operatorname{soc}(A)$. Let $\pi_A\colon A\to A/\operatorname{soc}(A)$ be the canonical projection map. Take a set theoretic section map $s\colon A/\operatorname{soc}(A)\to A$, say $\pi_A\circ s= \operatorname{id}_{A/\operatorname{soc}(A)}$.
Clearly $f=s\circ \pi_A\circ \pi_H\colon H\to A$ lifts the map $\bar f$. Then $\psi= \tilde\varphi+ d(f)$ is the desired 2-cocycle, that is, $\psi(h_1,h_2)\in \operatorname{soc}^l(A)$ and $\psi(h_1, h_2)- \psi(h_2, h_1)\in \operatorname{soc}^{l-1}(A)$ for any $h_1, h_2\in H$.
(2) If in addition $G$ has exponent $p$, then $H\cong \mathbb{Z}_p^n$ for some $n$, and applying Theorem \ref{thm-expp} (2) and Proposition \ref{prop-loewylength} we have $\ell\ell(A)\le p-1$, and hence $c\le p$.
\end{proof}
\begin{remark} If we consider an arbitrary abelian datum, then the inequality $c\le \ell\ell(A)+1$ in the above proposition still holds, while the one $c\ge \ell\ell(A)$ may not hold true in general. For instance, we take $A=\mathbb{Z}/p^2\mathbb{Z}$ and $H$ to be the trivial group. Then $\ell\ell(A)=2$ while the realizing group is abelian and hence has nilpotency class 1.
\end{remark}
\section{Matrix presentation and extensions of elementary $p$-groups}
In this section, we discuss the classification problem of extensions of elementary $p$-groups, say groups of the form $\mathbb{Z}_p^n\rtimes_{\rho,\varphi}\mathbb{Z}_p^m$. From now on, $H$ and $A$ are both assumed to be elementary $p$-groups without otherwise stated, where $p\ge 3$ is a prime number.
\subsection{Matrix presentation of a $p$-elementary datum}
Let $\mathcal{D}=(H, A, \rho, \varphi)$ be a $p$-elementary datum of type $(m,n)$ for some positive integer $m$ and $n$, say $H\cong\mathbb{Z}_p^m$ and $A\cong\mathbb{Z}_p^n$. Let $\mathcal{B}_H=\{h_1, h_2,\cdots, h_m\}$ be a basis of $H$ and $\mathcal{B}_A=\{a_1, a_2,\cdots, a_n\}$ be a basis of $A$. We mention that all bases discussed here will be ordered.
Then under the basis $\mathcal{B}_A$, any linear operator $\rho(h)\colon A\to A$, $h\in H$, is represented by some $n\times n$ matrix $\Gamma_{\mathcal{B}_A}(h)$. Set $\Gamma^{(r)}=\Gamma_{\mathcal{B}_A}(h_r)$. Thus for fixed $\mathcal{B}_H$ and $\mathcal{B}_A$, $\rho$ is uniquely determined by an (ordered) collection of matrices $\Gamma^{(1)},\Gamma^{(2)}, \cdots, \Gamma^{(m)}$.
Set $\varphi_{rr} = \varphi(1,h_r) + \varphi(h_r, h_r) + \cdots + \varphi(h_r^{p-1}, h_r)$ for $1\le r\le m$, and $\varphi_{rs}= \varphi(h_r, h_s)-\varphi(h_s, h_r)$ for $1\le r\neq s\le m$. Clearly $\varphi_{rs}=-\varphi_{sr}$ for $r\ne s$. By Proposition \ref{prop-2cocycle}, $\varphi$ is cohomologous to the cocycle corresponding to $(\varphi_{rs})_{1\le r< s\le m}$. Since $\mathcal{B}_A$ is a basis of $A$, each $\varphi_{rs}=\varphi^1_{rs}a_1 + \cdots +\varphi^n_{rs}a_n$ for some uniquely determined $\varphi^i_{rs}\in \mathbb{Z}_p$, $i=1,2,\cdots, n$. Set
$\Phi^{(i)}=(\varphi^i_{rs})_{m\times m}\in M_m(\mathbb{Z}_p)$ to be the $m\times m$ matrix with the $(r,s)$-entry given by $\varphi^i_{rs}$.
\begin{definition} The collection of matrices $(\Gamma^{(1)}, \cdots, \Gamma^{(m)}; \Phi^{{(1)}}, \cdots, \Phi^{(n)})$, denoted by $\mathcal{M}(\mathcal{D};\mathcal{B}_H,\mathcal{B}_A)$, is called the \emph{matrix presentation} of the datum $\mathcal{D}=(H, A, \rho, \varphi)$ with respect to the bases $\mathcal{B}_H$ and $\mathcal{B}_A$, where $\Gamma^{(r)}$'s and $\Phi^{(i)}$'s are defined as above.
\end{definition}
Conversely, let $\mathcal{M}=(\Gamma^{(1)}, \cdots, \Gamma^{(m)}; \Phi^{{(1)}}, \cdots, \Phi^{(n)})$ be a collection with $\Gamma^{(1)}, \cdots, \Gamma^{(m)}\in \operatorname{GL}_n(\mathbb{Z}_p)$, and $\Phi^{{(1)}}, \cdots, \Phi^{(n)}\in M_m(\mathbb{Z}_p)$ such that $\Phi^{(i)}_{rs}=-\Phi^{(i)}_{sr}$ for any $i$ and $r\ne s$. Then $\mathcal{M}$ is a matrix presentation of some datum with respect to suitable bases.
One can easily read a presentation of the realizing group of $\mathcal{D}$ from its matrix presentation. Note that we may identify $A$ and $H$ with subsets of $G(\mathcal{D})$, cf. Remark \ref{rem-data}.
\begin{proposition}\label{prop-pres-from-mrep} Keep the above notations. Then the group $G(\mathcal D)$ is generated by elements $a_1, a_2, \cdots, a_n$ and $h_1, h_2, \cdots, h_m$, subject to the relations:
\[a_i^p=h_r^p= [a_i, a_j]=1, [h_r, h_s] = \prod_{i=1}^na_i^{\varphi^i_{rs}}, [h_r,a_i]= a_i^{-1}\prod_{j=1}^{n} a_j^{(\Gamma^{(r)})_{ij}},
\]
$1\le i,j\le n, 1\le r,s\le m$, where $(\Gamma^{(r)})_{ij}$ is the $(i, j)$-entry of $\Gamma^{(r)}$.
\end{proposition}
\begin{proof}
Let $G$ be the group generated by $a_1, a_2, \cdots, a_n$ and $h_1, h_2, \cdots, h_m$ and subject to the relations given in the proposition. Clearly, by Lemma \ref{lem-abel-extension} (2) there exists a natural epimorphism from $G$ to $G(\mathcal D)$. Moreover,
each element in $g$ can be written in the form $a_1^{\alpha_1}\cdots a_n^{\alpha_n} h_1^{\beta_1}\cdots h_m^{\beta_m}$, it follows that $|G|\le p^{m+n} = |G(\mathcal D)|$, and hence $G\cong G(\mathcal D)$.
\end{proof}
Let $\tilde{\mathcal D}=(\tilde H, \tilde A, \tilde\rho, \tilde\varphi)$ be another datum which is equivalent to $\mathcal{D}$ via isomorphisms $\sigma_H\colon H\to \tilde H$ and $\sigma_A\colon A\to \tilde A$. Let $\mathcal{B}_{\tilde H}=\sigma_H(\mathcal{B}_H)$ and $\mathcal{B}_{\tilde A}= \sigma_A(\mathcal{B}_A)$ be the corresponding bases in $H$ and $A$. Let $\mathcal{M}(\tilde\mathcal{D};\mathcal{B}_{\tilde H}, \mathcal{B}_{\tilde A})=(\tilde\Gamma^{(1)},\cdots, \tilde\Gamma^{(m)}; \tilde\Phi^{(1)}, \cdots, \tilde\Phi^{(n)})$ be the matrix presentation of $\tilde\mathcal{D}$ with respect to the bases $\mathcal{B}_{\tilde H}$ and $\mathcal{B}_{\tilde A}$. We have the following observation.
\begin{proposition}\label{prop-mr-eq} Keep the above notations. Then
\begin{enumerate}
\item $\tilde \Gamma^{(r)}=\Gamma^{(r)}$ for any $1\le r\le m$;
\item If $(1-\rho(h_r))^{p-1}=0$ for any $1\le r\le m$, then there exists some $n\times m$ matrix $X$ over $\mathbb{Z}_p$, such that $\Phi^{(i)}-\tilde\Phi^{(i)}= W_iX-(W_iX)^T-X+X^T$ for each $i$, where $X^T$ is the transpose of $X$ and $W_i$ is an $m\times n$ matrix with $(W_i)_{rj}=(\Gamma^{(r)})_{ij}$.
\item If $H$ acts trivially on $A$, then $\Gamma^{(r)}= \tilde\Gamma^{(r)}=I_n$ for any $1\le r\le m$, and $\Phi^{(i)}=\tilde\Phi^{(i)}$ for any $1\le i\le n$.
\end{enumerate}
\end{proposition}
The proposition is an easy consequence of Proposition \ref{prop-2cocycle} and Definition \ref{def-equi-data}. The proof is given by routine check and we omit it here.
We can rewrite the above equalities by using the notion of \emph{block transpose} of a matrix. Let $M$ be a $m\times n$ matrix. Assume $m=rm'$ and $n=sn'$ for some positive integers $r, s, m', n'$. We may view $M$ as a $r\times s$ block matrix with all blocks of size $m'\times n'$, say
\[M=\begin{pmatrix}
M_{11} & M_{12} & \cdots & M_{1s} \\
M_{21} & M_{22} & \cdots & M_{2s} \\
\cdots & \cdots & \cdots & \cdots \\
M_{r1} & M_{r2} & \cdots & M_{rs} \\
\end{pmatrix},
\]
where each $M_{ij}$ is a $m'\times n'$ matrix. We set
\[ T_{m',n'}(M)= \begin{pmatrix}
M_{11} & M_{21} & \cdots & M_{r1} \\
M_{12} & M_{22} & \cdots & M_{r2} \\
\cdots & \cdots & \cdots & \cdots \\
M_{1s} & M_{2s} & \cdots & M_{rs} \\
\end{pmatrix},
\]
the $sm'\times rn'$ matrix obtained by transposing the blocks of $M$, called the block transpose of $M$ of size $m'\times n'$. In particular, $T_{1,1}(M) = M^T$, the transpose of $M$, and $T_{m,n}(M) = M$.
\begin{remark}\label{rem-mr-eq} We may put $\Phi^{(i)}$'s into an $m\times m$ block matrix $\Phi$ with the $(r,s)$-entry given by the $n$-dimensional column vector $(\varphi_{rs}^1, \varphi_{rs}^2, \cdots, \varphi_{rs}^n)^T$, or equivalently, an $mn\times m$ matrix with the $((r-1)m+i, s)$-entry $\varphi_{rs}^i$. Similarly, we put $\tilde\Phi^{(i)}$'s into one matrix $\tilde\Phi$.
Conversely, given an $mn\times n$ matrix $\Phi$, we may obtain a sequence of $m\times m$ matrix $\Phi^{(1)}, \cdots, \Phi^{(n)}$ such that $(\Phi^{(i)})_{rs} = (\Phi)_{(r-1)n +i, s}$ for any $1\le i\le n$ and $1\le r, s\le m$, where $(\Phi^{(i)})_{rs}$ is the $(r,s)$-entry of $\Phi^{(i)}$ and $(\Phi)_{(r-1)n +i, s}$ is the $((r-1)n +i, s)$-entry of $\Phi$.
Then the equalities in Proposition \ref{prop-mr-eq} (2) can be rewritten as $$\Phi-\tilde\Phi= \Gamma X- T_{n,1}(\Gamma X),$$ where $\Gamma$ is the block matrix $\begin{pmatrix}
\Gamma^{(1)}-I\\
\Gamma^{(2)}-I\\
\cdots\\
\Gamma^{(m)} -I\\
\end{pmatrix}.$
\end{remark}
\subsection{Changing of basis} We will use the matrix presentations to study group extensions. Then a naive question arises: given two matrix presentations, how to detect whether they come from equivalent data. Due to Proposition \ref{prop-mr-eq}, we need only to consider the matrix presentations of a given $p$-elementary datum $\mathcal{D}=(H, A, \rho, \varphi)$ under different choices of bases of $A$ and $H$.
Let $\mathcal{B}_H'=(h_1', \cdots, h_m')$ and $\mathcal{B}_A'=(a_1', \cdots, a_n')$ be another bases of $H$ and $A$ respectively. Let $C_H$ and $C_A$ be the transition matrices, say $\mathcal{B}_H'=\mathcal{B}_HC_H$ and $\mathcal{B}_A'=\mathcal{B}_AC_A$. Assume $C_H=(\beta_{uv})_{m\times m}$ and $C_A=(\alpha_{ij})_{n\times n}$. Then $h_r'=\prod_{u=1}^mh_u^{\beta_{ur}}$ for any $1\le r\le m$, and $a_i'=\sum_{l=1}^n\alpha_{li}a_l$. Notice that we write $H$ as a multiplicative group and $A$ an additive group.
Assume $\mathcal{M}(\mathcal{D};\mathcal{B}_H', \mathcal{B}_A')=(\Gamma'^{(1)}, \cdots, \Gamma'^{(m)}; \Phi'^{{(1)}}, \cdots, \Phi'^{(n)})$, and let $\Phi$ and $\Phi'$ be the $mn\times m$ matrices obtained from $(\Phi^{(1)}, \cdots, \Phi^{(n)})$ and $(\Phi'^{(1)}, \cdots, \Phi'^{(n)})$ as in Remark \ref{rem-mr-eq}.
For any $1\le r, u\le m$, we set $h_{r;u}=h_1^{\beta_{1r}}h_2^{\beta_{2r}}\cdots h_{u-1}^{\beta_{u-1, r}} (h_u)_{\beta_{ur}}$, and use $\Sigma_{ru}$ to denote the matrix of $\rho(h_{r;u})$ under the basis $\mathcal{B}_A$. Set $\Sigma= (\Sigma_{ru})_{m\times m}$ to be the $m\times m$ blocks matrix.
\begin{proposition}\label{prop-mr-bc} Keep the above notations. Assume that $1+\rho(h)+\cdots + \rho(h)^{p-1}=0$ and $\varphi(1,h) + \varphi(h, h) + \cdots + \varphi(h^{p-1}, h)=0$ for all $h\in H$, and $\varphi$ is given by (\ref{formula-cochain}). Then
\begin{enumerate}\item $\Gamma'^{(r)}= C_A^{-1}(\Gamma^{(1)})^{\beta_{1r}}(\Gamma^{(2)})^{\beta_{2r}}\cdots (\Gamma^{(m)})^{\beta_{mr}}C_A$ for any $1\le r\le m$;
\item $ \Phi' = YT_{m,1}(\Sigma T_{m, 1}(\Sigma \Phi))$, where $Y= \operatorname{diag}(C_A^{-1}, C_A^{-1}, \cdots, C_A^{-1})$ is the $m\times m$ block diagonal matrix with equal main diagonal block $C_A^{-1}$.
\end{enumerate}
\end{proposition}
\begin{proof} The proof of (1) follows from the fact that the matrices of a linear transformation under two bases are conjugate to each other. We sketch the proof of (2). By assumption,
$$\varphi_{rs} = \varphi(h_r, h_s) -\varphi(h_s, h_r)= -\varphi_{sr}$$ for any $1\le r\ne s\le m$, and $\varphi_{rr}=0$ for any $r$. Then for $r\ne s$ we have
\begin{align*}\varphi'_{rs} &= \varphi(h_r', h_s') -\varphi(h_s', h_r')\\
&=- \sum_{1\le u<v\le m}(h_{r;v}h_{s;u}\triangleright \varphi_{uv})+ \sum_{1\le u<v\le m}(h_{r;u}h_{s;v}\triangleright \varphi_{uv})\\
&=\sum_{u=1}^{m}\sum_{v=1}^{m}h_{r;u}h_{s;v}\triangleright \varphi_{uv},\end{align*}
and
\[\varphi'_{rr} = 0 = \sum_{u=1}^{m}\sum_{v=1}^{m}h_{r;u}h_{r;v}\triangleright \varphi_{uv}\]
for any $r$, where we use the assumptions $\varphi_{vu}=-\varphi_{uv}$ and $\varphi_{rr}=0$. Now we write the above equalities into the matrix form to obtain $(2)$.
\end{proof}
\begin{remark} \label{rem-mr-bc} By Proposition \ref{prop-2cocycle}(3), $\varphi$ is cohomologous to the cocycle obtained from $(\varphi_{rs})_{1\le r\le s\le m}$, thus without loss of generality we may assume that $\varphi$ is given by (\ref{formula-cochain}). Moreover, if the realizing group $G(\mathcal{D})$ has exponent $p$, then $1+\rho(h)+\cdots + \rho(h)^{p-1}=0$ and $\varphi(h,h)+\varphi(h^2,h)+\cdots +\varphi(h^{p-1},h) =0$ for all $h\in H$, see Proposition \ref{prop-ext-p}. In this case, all $\Phi^{(i)}$'s are $m\times m$ anti-symmetric matrices.
\end{remark}
The formula in the proposition is a bit complicated, while if the group action $\rho$ is trivial, then the it becomes much easier to understand.
\begin{corollary}\label{cor-mr-ta-bc} Keep assumptions as in Proposition \ref{prop-mr-bc}. Assume that $H$ acts trivially on $A$. Then
\begin{enumerate}\item $\Gamma^{(r)}=\Gamma'^{(r)}= I_n$ for any $1\le r\le m$;
\item $\Phi'^{(i)}=\sum_{l=1}^n \hat \alpha_{i,l}C_H\Phi^{(l)}C_H^T$ for any $1\le i\le n$, where the matrix $(\hat \alpha_{ij})_{n\times n}$ is the inverse of $C_A$.
\end{enumerate}
\end{corollary}
In fact, since $H$ acts trivially on $A$, then $\Sigma_{ru}=\beta_{ur}I_n$ for any $1\le r, u\le m$, and the conclusion follows.
\subsection{The classification theorem: trivial action case} In this subsection, we focus on the case such that $H$ acts trivially on $A$. In this situation, the realizing group has nilpotency class 2, and any class 2 groups of exponent $p$ is obtained from such a datum.
Let $\mathcal{D}=(H, A, 1, \varphi)$ be a $p$-elementary datum of type $(m,n)$ with realizing group having exponent $p$. Consider its matrix presentation. Obviously $\Gamma^{(r)}=I_n$ for $r=1, 2,\cdots, m$ for any choice of bases $\mathcal{B}_H$ and $\mathcal{B}_A$. Thus $\mathcal{M}(\mathcal{D}; \mathcal{B}_H, \mathcal{B}_A)= (I_n, \cdots, I_n; \Phi^{(1)}, \Phi^{(2)}, \cdots, \Phi^{(n)})$. As explained in Remark \ref{rem-mr-eq}, we may assume $\varphi$ is given by (\ref{formula-cochain}), and every $\Phi^{(i)}$ is anti-symmetric.
As it was shown in Proposition \ref{prop-der-abext}, $(G(\mathcal{D}))'= \langle \varphi_{rs}\mid 1\le r,s\le m \rangle$, the linear span of $\varphi_{rs}$'s, which is equal to the $H$-submodule generated by $\{\varphi_{rs}\mid 1\le r, s\le m\}$. Note that the $H$-action on $A$ is assumed to be trivial. We have an easy observation.
\begin{lemma} Keep the above notations. Then $\dim((G(\mathcal{D}))')=\operatorname{rank}(\Phi^{(1)}, \Phi^{(2)}, \cdots, \Phi^{(n)})$,
the dimension of the subspace of $\operatorname{AS}_m$ spanned by $\Phi^{(1)}, \Phi^{(2)}, \cdots, \Phi^{(n)}$. In particular,
$G(\mathcal{D})'=A$ if and only if $\Phi^{(1)}, \Phi^{(2)}, \cdots, \Phi^{(n)}$ are linearly independent.
\end{lemma}
\begin{proof} The proof is given by elementary linear algebra. By applying $T_{1,m}$, we turn $\Phi^{(i)}$ into a row vector, say
$$R_i= T_{1,m}(\Phi^{(i)}) = (\varphi_{11}^{i},\cdots, \varphi_{1m}^{i},\varphi_{21}^{i}, \cdots, \varphi_{2m}^{i},\cdots, \varphi_{m1}^{i},\cdots, \varphi_{mm}^{i}).$$
Then $\operatorname{rank}(\Phi^{(1)}, \Phi^{(2)}, \cdots, \Phi^{(n)})=\operatorname{rank}(R_1, \cdots, R_n)$. We consider the block matrix
$$X=\begin{pmatrix}
R_1 \\
R_2 \\
\cdots \\
R_m \\
\end{pmatrix}.$$
Clearly $\dim(\langle \varphi_{rs}\mid 1\le r,s\le m \rangle)$ is equal to the column rank of the matrix $X$, which is equal to the row rank of $X$, and the conclusion follows.
\end{proof}
Let $A_1= \langle \varphi_{rs}\mid 1\le r,s\le m \rangle$ and $A_2$ be a complement of $A_1$ in $A$, say $A= A_1\oplus A_2$. Let $\varphi_1\colon H\times H\to A_1$ be the 2-cocycle whose composite with the embedding $A_1\subseteq A$ gives rise to $\varphi$. Then $G(\mathcal{D})= (A_1, H) \times (A_2, 1)\cong G(\tilde \mathcal{D}) \times A_2$, where $\tilde\mathcal{D}=(H, A_1, 1, \varphi_1)$. Thus we have proved the following result.
\begin{lemma} Let $\mathcal{D}=(H, A, 1, \varphi)$ be a $p$-elementary data of type $(m, n)$. Then $G(\mathcal{D})'\cong \mathbb{Z}_p^d$ for some $d\le n$, and $G(\mathcal{D})\cong G(\tilde \mathcal{D})\times \mathbb{Z}_p^{n-d}$ for some $p$-elementary datum $\tilde\mathcal{D}$ of type $(m,d)$.
\end{lemma}
We denote by $\mathcal E_p(m, n)$ the equivalence classes of those $p$-elementary data $\mathcal{D}=(H, A, 1, \phi)$ of type $(m, n)$ such that $G(\mathcal{D})$ has exponent $p$, or equivalently, $\varphi(h,h)+\varphi(h^2,h)+\cdots +\varphi(h^{p-1},h) =0$ for all $h\in H$; and denote by $\mathcal E_p(d; m, n)$ the subclasses such that $(G(\mathcal{D}))' = \mathbb{Z}_p^d$ for any $0\le d\le m$. By definition, data in $\mathcal E_p(m, n)$
exactly give all central extensions of $\mathbb{Z}_p^m$ by $\mathbb{Z}_p^n$.
\begin{lemma}\label{lem-class2todata} (1) For any $[\mathcal{D}]\in \mathcal E_p(m, n)$, the realizing group $G(\mathcal{D})$ has nilpotency class 2 and exponent $p$.
(2) Assume $[\mathcal{D}_1], [\mathcal{D}_2]\in \mathcal E_p(m, n)$. Then $G(\mathcal{D}_1)\cong G(\mathcal{D}_2)$ if and only if $[\mathcal{D}_1]=[\mathcal{D}_2]$.
(3) Let $G$ be a group with exponent $p$ and nilpotencey class 2 such that $|G|=p^{m+n}$ and $|G'|= p^n$. Then $G\cong G(\mathcal{D})$ for some $[\mathcal{D}]\in \mathcal E_p(n; m, n)$.
\end{lemma}
\begin{remark} We use $\mathcal G_2(p;m,n)$ to denote the isoclasses of class 2 groups which have order $p^{m+n}$ and exponent $p$ and whose derived subgroup is isomorphic to $\mathbb{Z}_p^n$. Clearly the disjoint union $\bigsqcup_{0\le d< n} \mathcal G_2(p;m-d,d)$ gives isocalsses of all class 2 groups of order $p^n$ and exponent $p$.
\end{remark}
Recall the notion of Grassmannian. Let $V$ a vector space over a field $\mathbbm k$. We use $\operatorname{Gr}(d, V)$ to denote the set of all $d$-dimensional subspaces of $V$ for any integer $d\ge 0$, and $\operatorname{Gr}(\le d, V)$ the set of all subspaces with dimension no greater than $d$.
We use $\operatorname{AS}_m(\mathbb{Z}_p)$, or simply $\operatorname{AS}_m$ when no confusion arises, to denote the subspace of $M_m(\mathbb{Z}_p)$ consisting of $m\times m$ anti-symmetric matrices. Clearly $\operatorname{GL}_m(\mathbb{Z}_p)$ acts on $\operatorname{AS}_m(\mathbb{Z}_p)$ congruently, that is, $P\cdot M = PMP^T$ for any $P\in \operatorname{GL}_m(\mathbb{Z}_p)$ and any $M\in \operatorname{AS}_m(\mathbb{Z}_p)$.
The congruence action induces an action of $\operatorname{GL}_m(\mathbb{Z}_p)$ on $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))$ as well as on $\operatorname{Gr}(\le d, \operatorname{AS}_m(\mathbb{Z}_p))$. For any subspace $V\subset\operatorname{AS}_m(\mathbb{Z}_p)$, we denote by $[V]$ the orbit of $V$ under the congruence action, i.e., $[V]=\{PVP^T\mid P\in \operatorname{GL}_m(\mathbb{Z}_p)\}$. We use $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ and $\operatorname{Gr}(\le d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ to denote the sets of orbits respectively.
Combing Proposition \ref{prop-mr-eq} and Corollary \ref{cor-mr-ta-bc}, we have the following observation, which plays a crucial role in our classification on groups of exponent $p$.
\begin{theorem}\label{thm-data-1to1-grorbits} Let $m, n\ge 1$ and $0\le d\le n$ be integers. Then there exist one-to-one correspondences between:
\begin{enumerate}
\item $\mathcal E_p(d;m,n)$ and $\operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$;
\item $\mathcal E_p(m,n)$ and $\operatorname{Gr}(\le n, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$;
\item $\operatorname{Gr}(n, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$ and $\mathcal G_2(p;m,n)$.
\end{enumerate}
\end{theorem}
\begin{proof} We need only to prove (1), and the rest follows from (1) and Lemma \ref{lem-class2todata} (3).
Given $[\mathcal{D}]\in \mathcal E_p(d;m,n)$, where $\mathcal{D}=(H, A, 1, \varphi)$ is a $p$-elementary datum of type $(m,n)$ with $H$ acts on $A$ trivially. By assumption, $H\cong \mathbb{Z}_p^m$, $A\cong \mathbb{Z}_p^n$, and $\varphi$ is defined by (\ref{formula-cochain}) for some collection $(\varphi_{rs})_{1\le r\le s\le m}$ of elements in $A$ with $\varphi_{rr}=0$ for $1\le r\le m$.
Choose a basis $\mathcal{B}_H$ of $H$ and a basis $\mathcal{B}_A$ of $A$. Then the matrix presentation of $\mathcal{D}$ is $$\mathcal{M}(\mathcal{D}; \mathcal{B}_H, \mathcal{B}_A) = (I_n,\cdots, I_n; \Phi^{(1)}, \cdots, \Phi^{(n)}),$$ where $\Phi^{(i)}$, $i=1, 2, \cdots, n,$ are anti-symmetric $m\times m$ matrices. Set $V$ to be the linear subspace of $\operatorname{AS}_m(\mathbb{Z}_p)$ spanned by $\Phi^{(1)}, \cdots, \Phi^{(n)}$. Then we can define a map
\[\Theta\colon \mathcal E_p(d;m,n)\to \operatorname{Gr}(d, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)\]
by sending $[\mathcal{D}]$ to $[V]$.
First we claim that $\Theta$ is well-defined, that is, $[V]$ is independent on the choice of $\mathcal{B}_H$ and $\mathcal{B}_H$. In fact, by Corollary \ref{cor-mr-ta-bc}(2), $V$ is independent on the choice of $B_A$; and if we take another basis $\mathcal{B}'_H$ of $H$ and let $V'$ be the corresponding subspace of $\operatorname{AS}_m(\mathbb{Z}_p)$, again by Corollary \ref{cor-mr-ta-bc}, $V'$ is congruent to $V$, i.e., $[V]=[V']$.
Next we show that $\Theta$ is surjective. Let $V$ be a subspace of $\operatorname{AS}_m(\mathbb{Z}_p)$ of dimension $d$. Take $n$ matrices $\Phi^{(1)}, \cdots, \Phi^{(n)}\in V$ which span $V$. Let $H=\mathbb{Z}_p^m$ and $A=\mathbb{Z}_p^n$ and let $\mathcal{B}_H$ and $\mathcal{B}_A$ be the standard bases. Consider the trivial action of $H$ on $A$. Clearly $\Phi^{(1)}, \cdots, \Phi^{(n)}\in V$ will define a 2-cocycle $\varphi\colon H\times H\to A$. Thus we obtain a datum $\mathcal{D}$, which is a preimage of $[V]$.
We are left to show that $\Theta$ is injective.
For this it suffices to show that if $\mathcal{M}(\mathcal{D}; \mathcal{B}_H, \mathcal{B}_A) = \mathcal{M}(\tilde \mathcal{D}; \mathcal{B}_{\tilde H}, \mathcal{B}_{\tilde A)}$, then $\mathcal{D}\sim \tilde \mathcal{D}$. Assume $\mathcal{B}_H=(h_1, \cdots, h_r)$, $\mathcal{B}_A=(a_1,\cdots, a_r)$, $\mathcal{B}_{\tilde H}=(\tilde h_1, \cdots, \tilde h_r)$ and $\mathcal{B}_{\tilde A} =(\tilde a_1,\cdots, \tilde a_r)$. Then we may define $\sigma_H\colon H\to \tilde H$ by setting $\sigma_H(h_r)= \tilde h_r$ for $1\le r\le m$, and define $\sigma_A\colon A\to \tilde A$ by setting $\sigma_A(a_i) =\tilde a_i$ for any $1\le i\le n$. Now it is direct to show that $(\sigma_H, \sigma_A, 0)$ defines an equivalence from $\mathcal{D}$ to $\tilde\mathcal{D}$, which completes the proof.
\end{proof}
For a $p$-elementary data of type $(m, 1)$, the group action is automatically trivial, and hence the above theorem applies.
\begin{theorem}\label{thm-1dim-dergp}(1) Set $W_{m,k}$ to be the one dimensional subspace of $\operatorname{AS}_m(\mathbb{Z}_p)$ consisting of anti-symmetric block diagonal matrices of the form \[\operatorname{diag}\left(\underbrace{\begin{pmatrix}
0 & a \\
-a & 0 \\
\end{pmatrix},
\cdots,
\begin{pmatrix}
0 & a \\
-a & 0 \\
\end{pmatrix}}_k,
\underbrace{0, 0, \cdots, 0}_{m-2k}
\right). \]
Then $\{W_{m,k}\mid 0\le k\le \dfrac m 2\}$ is a complete set of representatives of $\operatorname{Gr}(\le 1, \operatorname{AS}_m(\mathbb{Z}_p))/\operatorname{GL}_m(\mathbb{Z}_p)$.
(2) Let $G$ be a group of order $p^{m+1}$ and exponent $p$. Assume that $G'\cong Z_p$. Then there exists a unique $1\le k\le \dfrac m 2$, such that $G$ is isomorphic to the group
\[G_{m,k}=\left\langle {a, b_1, \cdots, b_{2k},\atop c_1, \cdots, c_{m-2k}}\left| {{a^p=b_i^p=c_j^p=[a, b_i]=[a, c_j]=[b_i, c_j]=[c_i,c_j]=1, \forall i, j }\atop {[b_{2i-1}, b_{2i}]= a\ \forall i, [b_{2i-1}, b_j]=[b_{2i}, b_j]=1, \forall j > 2i}}\right. \right\rangle.
\]
\end{theorem}
\begin{proof} (1) Let $V$ be a one dimensional subspace of $\operatorname{AS}_m(\mathbb{Z}_p)$, and $X\in V$ a nonzero anti-symmetric matrix of rank $2k$. It is easy to show that $X$ is congruent to the matrix \[\operatorname{diag}\left(\underbrace{\begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix},
\cdots,
\begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix}}_k,
\underbrace{0, 0, \cdots, 0}_{m-2k}
\right), \] and hence $V$ is equivalent to $W_{m,k}$. On the other hand side, congruent matrices have equal rank, hence $W_{m,k}$'s are not equivalent to each other and the assertion follows.
(2) is an easy consequence of Proposition \ref{prop-pres-from-mrep} and Theorem \ref{thm-data-1to1-grorbits}.
\end{proof}
\begin{remark}\label{rmk-1dim-dergp} $G_{m,k}\cong \mathbb{Z}_3^{m-2k}\times G_{2k, k}$ and $G_{2k, k}\cong\underbrace{M_p(1,1,1)*\cdots*M_p(1,1,1)}_k$, where $M_p(1,1,1)$ is the unique
non-abelian group of order $p^3$ and exponent $p$, and $*$ denotes the (iterated) central product.
\end{remark}
\section{Two dimensional subspaces of anti-symmetric matrices}
In this section, $\mathbbm{k}$ is a fixed base field. We will give a description of congruence classes of 2 dimensional subspaces of anti-symmetric matrices of order $m$ over $\mathbbm{k}$. Consequently, we obtain a description of $\mathcal G_2(p;m,2)$, the isoclasses of class 2 groups of exponent $p$ and order $p^{m+2}$ and with derived subgroup isomorphic to $\mathbb{Z}_p^{2}$. We also calculate detailed examples in the case $m=3$ for arbitrary base field $\mathbbm{k}$, and in the case $m=4,5,6$ for $\mathbbm{k}=\mathbb{Z}_3$.
Recall that in \cite{vish}, Vishnevetskii gave a classification of ``indecomposable" class 2 groups of prime exponent with derived subgroup (commutator subgroup) of rank 2, here a group is indecomposable means that it is not a central product of proper subgroups. Moreover, he showed that any class 2 group of prime exponent with derived subgroup of rank two is a central product of ``indecomposable" ones in a unique way. However, it was not shown there when two central products are isomorphic. Our description on $\mathcal G_2(p;m,2)$ recovers and strengthens to some extend the results by Vishnevetskii. We stress that the notion ``indecomposable group" used by Vishnevetskii is different from the one we used in the present paper, which refers to those groups that are not direct product of proper subgroups, cf. Section 7.
We heavily depends on the theory of so-called ``pencil", which gives a standard form for a pair of anti-symmetric matrices under the congruence. For more details we refer to \cite{gan}, \cite{schar} and \cite{vish}.
\subsection{Canonical form of a pair of anti-symmetric matrices.}\
We first introduce some notations. For any $k\le 1$, let $L_k = (I_k, 0)^T, R_k=(0, I_k)^T$ be $(k+1)\times k$ matrices over $\mathbbm{k}$, $N_k= \begin{pmatrix}
0 & & & \\
1 & 0 & & \\
& \ddots & \ddots & \\
& & 1& 0 \\
\end{pmatrix}
$ be the Jordan block of size $k$, where $I_k$ is the identity matrix of order $k$. For any monic polynomial $f= x^k+ a_{k-1} x^{k-1} + \cdots + a_0\in \mathbbm{k}[x]$, the companion matrix of characteristic polynomial $f(x)$ is defined to be \[C(f) = \begin{pmatrix}
0 & 1 & 0 &\cdots & 0\\
0 & 0 & 1 &\cdots & 0 \\
0 & 0 & 0 &\ddots & 0 \\
\vdots &\vdots &\vdots &\ddots & 1 \\
-a_0 & -a_1 & -a_2 &\cdots & -a_{k-1} \\
\end{pmatrix}.
\]
Let $f(x,y)= a_kx^k+ a_{k-1}x^{k-1}y+\cdots + a_0 y^{k}\in \mathbbm{k}[x,y]$ be a homogeneous polynomial of degree $k\ge 1$ which is a power of an irreducible one. Then $f=a_0y^k, a_0\ne 0$ or $a_k\ne 0$, and in the latter case $\frac 1 {a_k} f(x,-1)\in \mathbbm{k}[x]$ is a monic polynomial of degree $k$. We set $L(f)= N_k$ and $R(f)= I_k$ if $f= a_0 y^k$; and $L(f)= I_k$ and $R(f)= C(\frac{1} {a_k} f(x, -1))$ else. For two polynomials $f_1, f_2\in \mathbbm{k}[x,y]$, we say $f_1\sim f_2$ if $f_1=\lambda f_2$ for some $0\neq\lambda\in\mathbbm{k}$.
Clearly, $L(f_1) = L(f_1)$ and $R(f_1)= R(f_2)$ if and only if $f_1\sim f_2$.
We use $\operatorname{Irr}(\mathbbm{k}[x,y])$ to denote the set of irreducible homogenous polynomials in two variables $x, y$, and $\operatorname{Irr}(\mathbbm{k}[x,y]_k)$ the subset of the ones of degree $k$ for each $k\ge1$.
\begin{lemma} The set of irreducible homogenous polynomials in $\mathbb{Z}_3$ of degree $\le3$ are listed as follows:
\noindent(1) $\operatorname{Irr}(\mathbb{Z}_3[x,y]_1) = \{\pm x, \pm y\}$;
\noindent(2) $\operatorname{Irr}(\mathbb{Z}_3[x,y]_2) = \{\pm(x^2+y^2), \pm(x^2+xy-y^2), \pm(x^2-xy-y^2)\}$;
\noindent(3) $\operatorname{Irr}(\mathbb{Z}_3[x,y]_3) = \left\{\pm(x^3+x^2y+xy^2-y^3),\ \pm(x^3+x^2y-xy^2+y^3),\ \pm(x^3+x^2y-y^3),\ \pm(x^3-x^2y+y^3),\atop \pm(x^3-x^2y-xy^2-y^3),\ \pm(x^3-x^2y+xy^2+y^3),\ \pm(x^3-xy^2+y^3),\ \pm(x^3-xy^2-y^3)\right\}$
\end{lemma}
Let $A, B\in \operatorname{AS}_m(\mathbbm{k})$ be a pair of anti-symmetric matrices. We say two pairs $(A, B)$ and $(\tilde A,\tilde B)$ are congruently equivalent, or congruent for short, if
there exists an invertible matrix $P\in M_m(\mathbbm{k})$ such that $\tilde A = PAP^T$ and $\tilde B = PBP^T$.
\begin{definition} Let $k_1\ge \cdots\ge k_r\ge 1, d_1, \cdots, d_s \ge 1$ be integers, and let $f_1, f_2, \cdots, f_s\in\operatorname{Irr}(\mathbbm{k}[x,y])$ be irreducible homogeneous polynomials. The pair of $m\times m$ anti-symmetric matrices
$$\left(\begin{pmatrix}0 &C_1 &0\\ -C_1^T &0 &0 \\ 0&0&0\end{pmatrix}, \begin{pmatrix}0 &C_2 &0\\ -C_2^T &0 &0 \\ 0&0&0\end{pmatrix}\right)$$
is called a \emph{canonical form} of type $(m; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$, where $$C_1= \operatorname{diag}(L_{k_1}, \cdots, L_{k_r}, L(f_1^{d_1}), \cdots, L(f_s^{d_s})),$$
$$C_2= \operatorname{diag}(R_{k_1}, \cdots, R_{k_r}, R(f_1^{d_1}), \cdots, R(f_s^{d_s}))$$ are block diagonal matrices.
\end{definition}
Clearly, $m\ge 2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j$.
The following result says that any pair of anti-symmetric matrices is congruent to a unique canonical form.
\begin{theorem} \cite[9.1]{schar} Let $(A, B)$ be a pair of $m\times m$ anti-symmetric matrices. Then there exist integers $k_1\ge \cdots\ge k_r\ge 1, d_1, \cdots, d_s \ge 1$, and $f_1, f_2, \cdots, f_s\in\operatorname{Irr}(\mathbbm{k}[x,y])$, such that $(A,B)$ is congruent to the canonical form of type $(m; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$.
If $(A, B) $ is congruent to another canonical form of type $(m; k'_1, \cdots k'_{r'}; {f'_1}^{d'_1}, \cdots, {f'_{s'}}^{d'_{s'}})$, then $r=r'$, $s= s'$, and $k'_i=k_i$ for $1\le i\le r$, and there exists some permutation $\sigma\in S_{\{1, 2,\cdots, s\}}$, such that $d_j= d'_{\sigma(j)}$ and $f_j\sim f'_{\sigma(j)}$ for $1\le j\le s$.
\end{theorem}
Consider the natural action of $\operatorname{GL}_2(\mathbbm{k})$ on $\mathbbm{k}[x, y]$. Precisely, for any $P=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
$ and any $f(x,y)\in \mathbbm{k}[x,y]$, $(P\cdot f)(x, y) = f(ax+by, cx +dy)$. Clearly $f(x,y)$ is irreducible if and only if $P\cdot f$ is. Thus $\operatorname{GL}_2(k)$ acts on $\operatorname{Irr}(\mathbbm{k}[x,y])$ and on each $\operatorname{Irr}(\mathbbm{k}[x,y]_k)$ as well.
\begin{remark} One can check that $\operatorname{GL}_2(\mathbb{Z}_3)$ acts transitively on the set $\operatorname{Irr}(\mathbb{Z}_3[x,y]_k)/\{\pm 1\}$ for $k=1, 2, 3$.
\end{remark}
\begin{definition}Two types $(m; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$ and $ (m; k'_1, \cdots k'_{r'}; {f'_1}^{d'_1}, \cdots, {f'_{s'}}^{d'_{s'}})$ are said to be \emph{equivalent} if $r=r'$, $s= s'$, $k'_i=k_i$ for $1\le i\le r$, and there exists some permutation $\sigma\in S_{\{1, 2,\cdots, s\}}$ and some $P\in \operatorname{GL}_2(k)$, such that $d_j= d'_{\sigma(j)}$ and $f_j\sim P\cdot f'_{\sigma(j)}$ for $1\le j\le s$.
\end{definition}
Applying the above theorem, we obtain a description of congruence classes of two dimensional subspaces of anti-symmetric matrices.
\begin{theorem}\label{thm-2dim-dergp} There exists a bijection between $\operatorname{Gr}(2, \operatorname{AS}_m(\mathbbm{k}))/\operatorname{GL}_m(\mathbbm{k})$ and the set of equivalence classes of canonical forms. Consequently, $\mathcal G_2(p; m, 2)$ is in one-to-one correspondence with the equivalence classes of canonical forms over $\mathbb{Z}_p$.
\end{theorem}
\begin{proof} Let $V\in \operatorname{Gr}(2, \operatorname{AS}_m(\mathbbm{k}))$, and $A, B\in V$ be a basis of $V$. Then $(A,B)$ is congruent to some canonical form $(m; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$, and $V$ is congruent to some space spanned by the canonical forms determined by $(A, B)$. Let $(\tilde A, \tilde B)$ be another basis of $V$, then there exists some $P=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
\in \operatorname{GL}_2(\mathbbm{k})$, such that $\tilde A= aA+ bB$ and $\tilde B= cA +dB$.
We need to determine the canonical form of $(\tilde A, \tilde B)$, and it suffices to consider the case
\[A(k) = \begin{pmatrix}
0 & L_{k} \\
-L_{k}^T & 0 \\
\end{pmatrix},
B(k) = \begin{pmatrix}
0 & R_{k} \\
-R_{k}^T & 0 \\
\end{pmatrix}\]
for some positive integer $k\ge 0$, and the case
\[A(f^d)= \begin{pmatrix}
0 & L(f^{d}) \\
-L(f^{d})^T & 0 \\
\end{pmatrix},
B(f^d)= \begin{pmatrix}
0 & R(f^{d}) \\
-R(f^{d})^T & 0 \\
\end{pmatrix}\]
for some irreducible homogeneous polynomial $f\in\mathbbm{k}[x,y]$ and some positive integer $d$.
It is direct to check that the canonical form of $(aA(k)+bB(k), cA(k)+dB(k))$ is $(A(k), B(k))$,
and the one of $(aA(f^d) + bB(f^d), cA(f^d) + dB(f^d))$ is $(A((P\cdot f)^d), B((P\cdot f)^d))$.
The conclusion follows.
\end{proof}
\subsection{m=3.} \label{sec-2of3}\ In this case, $\mathbbm{k}$ can be an arbitrary field.
Consider the type $(3; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$. Since $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j\le3$,
we have $r=1, k_1=1$ and $s=0$. Thus there exists only one congruence class of subspaces of $\operatorname{AS}_3(\mathbbm{k})$, say
\[\begin{pmatrix}&&a\\&&b\\-a&-b&\end{pmatrix}=\left\{\left.a\begin{pmatrix}
& &1 \\
& 0& \\
-1& & \\
\end{pmatrix}
+ b\begin{pmatrix}
0 & & \\
& &1 \\
& -1& \\
\end{pmatrix} \right| a, b\in \mathbbm{k}\right\}.
\]
In the rest of this section, we assume $\mathbbm{k}=\mathbb{Z}_p$. We will give a full list of representatives for
congruence classes of 2 dimensional subspaces of $\operatorname{AS}_m(\mathbb{Z}_3)$ for $m=4, 5$, and $6$.
\subsection{m=4.}\label{sec-2of4}
Consider the type $(4; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$.
Since $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j\le4$,
there are 4 possible cases:
\begin{enumerate}
\item[(1)] $r=1$, $k_1=1$, $s=0$;
\item[(2)] $r=0$, $s=2$, $d_1= d_2 =1$, and $\deg(f_1)=\deg(f_2)=1$;
\item[(3)] $r=0$, $s=1$, $d_1=2$, and $\deg(f_1)=1$;
\item[(4)] $r=0$, $s=1$, $d_1=1$, and $\deg(f_1)=2$.
\end{enumerate} The corresponding 2-dimensional subspace of $\operatorname{AS}_4(\mathbb{Z}_3)$ are listed as follows.
$$\left(\begin{array}{cccc}&&a&\\&&b&\\-a&-b&&\\&&&\end{array}\right),
\left(\begin{array}{cccc}&&a&\\&&&b\\-a&&&\\&-b&&\end{array}\right),
\left(\begin{array}{cccc}&&a&b\\&&&a\\-a&&&\\-b&-a&&\end{array}\right),
\left(\begin{array}{cccc}&&a&b\\&&-b&a\\-a&b&&\\-b&-a&&\end{array}\right).$$
We mention that in Case (2), $f_1\ne \pm f_2$, otherwise it gives a one dimensional subspace. In Case (3), we need only to take $f_1(x, y)= x$, since $\operatorname{GL}_2(\mathbb{Z}_3)$ acts transitively on the set $\operatorname{Irr}(\mathbb{Z}_3[x,y]_1)$. In Case (4), we need only to take $f(x, y)= x^2+y^2$, since $\operatorname{GL}_2(\mathbb{Z}_3)$ acts transitively on the set $\operatorname{Irr}(\mathbb{Z}_3[x,y]_2)/\{\pm1\}$.
\subsection{m=5.}\label{sec-2of5}
We claim that there are six congruence classes of 2-dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$ which are listed as follows.
$$\begin{pmatrix}&&a&&\\&&b&&\\-a&-b&&&\\&&&0&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&a&&\\&&&b&\\-a&&&&\\&-b&&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&a&b&\\&&&a&\\-a&&&&\\-b&-a&&&\\&&&&0\end{pmatrix},$$
$$\begin{pmatrix}&&a&b&\\&&-b&a&\\-a&b&&&\\-b&-a&&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a\\&&&& b\\-a&-b&&&\\&-a&-b&&\end{pmatrix}.$$
In fact, for any type $(5; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$, $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j\le5$. The cases that $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j \le 4$ exactly correspond to the first four subspaces in the above list.
Now we are left to consider the cases that the $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j = 5$. In this situation we have two subcases:
\begin{enumerate}
\item[(1)] $r=s=1$, $k_1=1$, $d_1=1$, $\deg(f_1)=1$, and
\item[(2)] $r=1, k_1=2, s=0$,
\end{enumerate}
which correspond to the last two subspaces in the above list respectively. Note that in Case (1), we may take $f_1= x$, for $\operatorname{GL}_2(\mathbb{Z}_3)$ acts transitively on $\operatorname{Irr}(\mathbb{Z}_3[x,y]_1)$.
\subsection{m=6.}\label{sec-2of6} \
Let $(6; k_1,\cdots, k_r; f_1^{d_1}, \cdots, f_s^{d_s})$ be a type. Then $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j\le 6$. As above, the case $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j\le 5$ are essentially obtained in last section, and the corresponding subspaces are listed as follows.
$$\begin{pmatrix}&&&a&&\\&&&b&&\\&&&&&\\-a&-b&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&a&&&\\&&&b&&\\-a&&&&&\\&-b&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&a&b&&\\&&&a&&\\-a&&&&&\\-b&-a&&&&\\&&&&0&\\&&&&&0\end{pmatrix},$$
$$\begin{pmatrix} &&a&b&&\\&&-b&a&&\\-a&b&&&&\\-b&-a&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&&a&&\\&&&b&&\\&&&&a&\\-a&-b&&&&\\&&-a&&&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&&a&&\\&&&b&a&\\&&&& b&\\-a&-b&&&&\\&-a&-b&&&\\&&&&&0\end{pmatrix}.
$$
We are left to consider the case $2\sum_{i=1}^r k_i + r + 2\sum_{j=1}^s \deg(f_j)d_j= 6$.
Easy calculation shows that there are the following cases:
(a) $r=0$, $s=3$, $\deg(f_1)=\deg(f_2)=\deg(f_3)=d_1=d_2=d_3=1$. There are two subcases: (a.1) $ f_1\neq\pm f_2\neq\pm f_3$, in this case, we may choose $f_1= x$, $f_2=y$, and $f_3=x+y$, and all possible such choices are equivalent; and (a.2) $f_1 = \pm f_2\neq\pm f_3$, in this case, we may choose $f_1=f_2= x$, and $f_3=y$.
(b) $r=0$, $s=2$, $\deg(f_1)=1, d_1=2, \deg(f_2)= d_2=1$. There are two subcases: (b.1) $f_1=\pm f_2$, in this case we may choose $f_1=f_2=x$; and (b.2) $f_1\neq \pm f_2$, in this case we may choose $f_1=x$, $f_2=y$.
(c) $r=0$, $s=2$, $\deg(f_1)=2, d_1=1, \deg(f_2)= d_2=1$. In this case, we may choose $f_1=x^2+y^2$ and $f_2= x$.
(d) $r=0$, $s=1$, $\deg(f_1)=3, d_1=1$. We may choose $f_1= x^3-xy^2 +y^3$ in this case.
(e) $r=0$, $s=1$, $\deg(f_1)=1, d_1=3$. We may choose $f_1= x$ in this case.
(f) $r=2$, $s=0$, $k_1= k_2 =1$. This case is uniquely determined.
The corresponding subspaces are listed as follows.
\[(a.1): \begin{pmatrix}
& & & a & & \\
& & & & b & \\
& & & & & a+b \\
-a& & & & & \\
&-b& & & & \\
& &-a-b& & & \\
\end{pmatrix},\quad
(a.2): \begin{pmatrix}
& & & a & & \\
& & & & a & \\
& & & & & b \\
-a& & & & & \\
&-a& & & & \\
& & -b& & & \\
\end{pmatrix},
\]
\[(b.1): \begin{pmatrix}
& & & a & b & \\
& & & & a & \\
& & & & & a \\
-a& & & & & \\
-b&-a& & & & \\
& & -a & & & \\
\end{pmatrix},\quad
(b.2): \begin{pmatrix}
& & & a & b & \\
& & & & a & \\
& & & & & b \\
-a& & & & & \\
-b&-a& & & & \\
& & -b& & & \\
\end{pmatrix}
\]
\[(c): \begin{pmatrix}
& & & a & b & \\
& & &-b & a & \\
& & & & & a \\
-a& b& & & & \\
-b&-a& & & & \\
& & -a & & & \\
\end{pmatrix},\quad
(d): \begin{pmatrix}
& & & a & b & \\
& & & & a & b \\
& & & b & b & a \\
-a& & -b& & & \\
-b&-a& -b& & & \\
&-b& -a& & & \\
\end{pmatrix}
\]
\[(e): \begin{pmatrix}
& & & a & b & \\
& & & & a & b \\
& & & & & a \\
-a& & & & & \\
-b&-a& & & & \\
&-b& -a & & & \\
\end{pmatrix},\quad
(f): \begin{pmatrix}
& & & & a & \\
& & & & b & \\
& & & & & a \\
& & & & & b \\
-a&-b& & & & \\
& & -a& -b& & \\
\end{pmatrix}
\]
In summary, $\operatorname{AS}_6(\mathbb{Z}_3)$ has 14 congruence classes of two dimensional subspaces which are listed as above.
\section{Groups of order $3^3$, $3^4$, $3^5$, $3^6$, $3^7$ and exponent 3}
In this section we will give a full list of groups of exponent $3$ and order $\le 3^7$. From now on, $p=3$, and all groups considered are of exponent 3.
Let $G$ be a group of exponent 3. Set $A=G'$ and $H=G/G'$.
Assume $A=\mathbb{Z}_3^{n}$ and $H=\mathbb{Z}_3^m$. Then $G=\mathbb{Z}_3^{n}\rtimes_{\rho,f}\mathbb{Z}_3^m$, and $|G|=3^{m+n}$. We will deal with the cases $m+n\le 7$ in this section. By Corollary \ref{cor-expp3}, in case $m+n\le 6$, it suffices to consider the trivial action, where Theorem \ref{thm-data-1to1-grorbits} applies.
Since $\dim\operatorname{AS}_m(\mathbb{Z}_3)=\frac{m(m-1)}{2}$, we only need to consider the case $n\le \frac{m(m-1)}{2}$.
We say that a group $G$ is \emph{indecomposable} if $G$ is not a product of proper subgroups, otherwise we say that $G$ is \emph{decomposable}. Clearly, for a nilpotent group, if $G$ is decomposable, then so is its center $Z(G)$.
\begin{example} \label{eg-dec-1dim-dergp}
It is easy to show that the group $G_{m,k}$ defined in Theorem \ref{thm-1dim-dergp} is indecomposable if and only if $m=2k$. In fact, by Remark \ref{rmk-1dim-dergp}, we need only to show that $G_{2k,k}$ is indecomposable, which is true for $Z(G_{2k,k})\cong \mathbb{Z}_p$.
\end{example}
Clearly to classify all groups of exponent $3$, it suffices to
classify the indecomposable ones. In fact, by Krull-Schmidt Theorem, any finite group is a unique product of indecomposable ones. We will make a full list of indecomposable groups of exponent $3$.
\subsection{$\mathbf{|G|=3}$, or $\mathbf{3^2}$} \ The following result is obvious.
\begin{proposition} $\mathbb{Z}_3$ is the unique indecomposable group of order 3 up to isomorphism, and there exist no indecomposable groups of order $3^2$ and exponent 3.
\end{proposition}
For consistency of notations, we set $I_1= \mathbb{Z}_3$.
\subsection{$\mathbf{|G|=3^3}$}\quad In this case $n=0$, or $1$.
If $n=0$, then $G\cong \mathbb{Z}_3^3$ and hence is decomposable.
If $n=1$, then $m=2$ and
$\operatorname{AS}_2(\mathbb{Z}_3)$ has a unique subspace of dimension $1$. We denote by $I_3$ the group given by the only element in $\mathcal E_3(1;2,1)$. It is well known that $I_3$ is the unique nonabelian group of order 27 and exponent 3. By Proposition \ref{prop-pres-from-mrep}, \[I_3 = \langle a, x, y\mid a^3=x^3=y^3=[x,a]=[y,a]=1, [x,y]=a\rangle.\] Thus we have shown the following result.
\begin{proposition} $I_3$ is the unique indecomposable group of order $3^3$ and exponent $3$ up to isomorphism.
\end{proposition}
\subsection{$\mathbf{|G|=3^4}$} \quad In this case $n=0$ or $1$, for $n\le \frac{m(m-1)}{2}$.
If $n=0$, then $G$ is abelian and hence decomposable. If $n=1$, then by Theorem \ref{thm-1dim-dergp}, we know that $G$ is isomorphic to $G_{3,1}$ which is decomposable by Example \ref{eg-dec-1dim-dergp}.
\begin{proposition} There exist no indecomposable groups of exponent 3 and order $3^4$.
\end{proposition}
\subsection{$\mathbf{|G|=3^5}$}\quad In this case, $n=0, 1$ or $2$.
If $n=0$, then $G\cong \mathbb{Z}_3^5$ and hence is decomposable.
If $n=1$, then $G$ is isomorphic to one of the groups $G_{4,1}$ or $G_{4,2}$ in Theorem \ref{thm-1dim-dergp} (2). By Example \ref{eg-dec-1dim-dergp}, only
$G_{4,2}$ is indecomposable. We denote $G_{4,2}$ by $I_{5.1}$.
Now assume $n=2$. Then there is only one congruence class of 2 dimensional subspaces of $\operatorname{AS}_3(\mathbb{Z}_3)$, say
$\left(\begin{array}{ccc}&a&b\\-a&&\\-b&&\end{array}\right)$. The corresponding group, denoted by $I_{5.2}$, is
\[I_{5.2} = \langle a_1,a_2, h_1,h_2,h_3\mid a_i^3, h_j^3, [a_i, h_j], [a_1,a_2], [h_2, h_3], [h_1,h_2]= a_1, [h_1, h_3]=a_2
\rangle.\]
\begin{proposition} $I_{5.1}$ and $I_{5.2}$ give a complete set of isoclasses of irreducible groups of exponent 3 and order $3^5$.
\end{proposition}
\subsection{$\mathbf{|G|=3^6}$}\quad In this case, $n=0, 1, 2, 3$.
If $n=0$, then $G=\mathbb{Z}_3^6$ and hence is decomposable.
If $n=1$, then $m=5$ and $G = G_{m,1}$ or $G_{m,2}$, and $G$ is decomposable in either case.
If $n=2$, then $m=4$. By Section \ref{sec-2of4} there are 4 congruence classes of 2 dimensional subspaces of the space of $\operatorname{AS}_4(\mathbb{Z}_3)$, say
$$\left(\begin{array}{cccc}&&a&\\&&b&\\-a&-b&&\\&&&0\end{array}\right),
\left(\begin{array}{cccc}&&a&\\&&&b\\-a&&&\\&-b&&\end{array}\right),
\left(\begin{array}{cccc}&&a&b\\&&&a\\-a&&&\\-b&-a&&\end{array}\right),
\left(\begin{array}{cccc}&&a&b\\&&-b&a\\-a&b&&\\-b&-a&&\end{array}\right).$$
Let $G_1, G_2, G_3, G_4$ be the corresponding groups. It is easy to show that $G_1\cong I_{5.2}\times \mathbb{Z}_3$ and $G_2\cong I_3\times I_3$, and $G_3$ and $G_4$ are both indecomposable. We set
\[I_{6.1}=G_3=\left\langle a_1, a_2, h_1, h_2, h_3, h_4\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_3, h_4], [h_2, h_3], \atop [h_1, h_3]=[h_2, h_4] = a_1, [h_1, h_4]=a_2}\right.
\right\rangle,\]
\[I_{6.2}= G_4\left\langle a_1, a_2, h_1, h_2, h_3, h_4\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_3, h_4], \atop [h_1, h_3]=[h_2, h_4] = a_1, [h_1, h_4]=[h_3, h_2]=a_2}\right.
\right\rangle.\]
Now we assume $n=3$, then $m=3$. There is only one congruence class of 3-dimensional subspaces of $\operatorname{AS}_3(\mathbb{Z}_3)$, say
$\left(\begin{array}{ccc}&a&b\\-a&&c\\-b&-c&\end{array}\right)$. Denote the corresponding group by $I_{6.3}$. By Proposition \ref{prop-pres-from-mrep},
\[I_{6.3}=\left\langle\left.{ a_1, a_2, a_3,\atop h_1,h_2,h_3}\right | {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1,h_2]= a_1,\atop [h_1, h_3]=a_2, [h_2, h_3]=a_3}
\right\rangle.\]
\begin{proposition} $I_{6.1}$, $I_{6.2}$ and $I_{6.3}$ give a complete set of isoclasses of irreducible groups of exponent 3 and order $3^6$.
\end{proposition}
\subsection{$\mathbf{|G|=3^7}$}{\ } Then $n =0, 1, 2, 3, 4$.
In fact, if $n\ge 5$, then $m\le 2$, and
by Corollary \ref{cor-expp3}, $H$ acts on $A$ trivially, this case will not
happen since $\dim(\operatorname{AS}_m(\mathbb{Z}_p))< n$.
If $n=0$, then $G=\mathbb{Z}_3^7$, and hence is decomposable.
If $n=1$, then $m=6$, and $G= G_{6,1}, G_{6,2}$ or $G_{6,3}$ as in Theorem \ref{thm-1dim-dergp} (2). By Example \ref{eg-dec-1dim-dergp},
only $G_{6,3}$ is indecomposable, and we denote it by $I_{7.1}$.
If $n=2$, then $m=5$. By Section \ref{sec-2of5}, there are 6 congruence classes of 2 dimensional subspaces of the space $\operatorname{AS}_5(\mathbb{Z}_3)$:
$$\begin{pmatrix}&&a&&\\&&b&&\\-a&-b&&&\\&&&0&\\&&&&0\end{pmatrix},\quad
\begin{pmatrix}&&a&&\\&&&b&\\-a&&&&\\&-b&&&\\&&&&0\end{pmatrix},\quad \begin{pmatrix}&&a&b&\\&&&a&\\-a&&&&\\-b&-a&&&\\&&&&0\end{pmatrix},$$
$$\begin{pmatrix}&&a&b&\\&&-b&a&\\-a&b&&&\\-b&-a&&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a\\&&&& b\\-a&-b&&&\\&-a&-b&&\end{pmatrix}.$$
Let $G_i, 1\le i\le 6$ be the corresponding groups. Then one checks that $G_1\cong I_{5.2}\times \mathbb{Z}_3^2$, $G_2\cong I_3\times I_3\times Z_3$, $G_3\cong I_{6.1}\times \mathbb{Z}_3$, $G_4\cong I_{6.2}\times \mathbb{Z}_3$, and $G_5$ and $G_6$ are indecomposable. Denote $G_5$ and $G_6$ by $I_{7.2}$ and $I_{7.3}$ respectively. Then
\[I_{7.2}=\left\langle {a_1, a_2, h_1, h_2,\atop h_3, h_4, h_5}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_1, h_5], [h_2, h_3], [h_2, h_4], [h_3, h_4],\atop [h_3, h_5], [h_4, h_5], [h_1, h_3]=[h_2, h_5] = a_1, [h_1, h_4]=a_2}\right.
\right\rangle,\]
\[I_{7.3}=\left\langle {a_1, a_2, h_1, h_2,\atop h_3, h_4, h_5}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_1, h_5], [h_2, h_3], [h_3, h_4], [h_3, h_5],\atop [h_4, h_5], [h_1, h_3]=[h_2, h_4] = a_1,[h_1, h_4]=[h_2, h_5]=a_2}\right.
\right\rangle.\]
If $n=3$, then $m=4$. By Theorem \ref{thm-3of4}, there are 6 congruence classes of 3-dimensional subspaces of the space of $\operatorname{AS}_4(\mathbb{Z}_3)$:
$$\begin{pmatrix}&c&a&\\-c&&b&\\-a&-b&&\\&&&\end{pmatrix},
\begin{pmatrix}&&a&\\&&b&\\-a&-b&&c\\&&-c&\end{pmatrix},
\begin{pmatrix}&c&a&\\-c&&b&\\-a&-b&&c\\&&-c&\end{pmatrix},$$
$$\begin{pmatrix}&&a&c\\&&b&\\-a&-b&&\\-c&&&\end{pmatrix},
\begin{pmatrix}&c&a&\\-c&&&b\\-a&&&c\\&-b&-c&\end{pmatrix},
\begin{pmatrix}&c&a&b\\-c&&&a\\-a&&&-c\\-b&-a&c&\end{pmatrix}.$$
The first one corresponds to the group $I_{6.3}\times \mathbb{Z}_3$, and the others are indecomposable and correspond to the following groups respectively:
\[I_{7.4}=\left\langle {a_1, a_2, a_3,\atop h_1, h_2, h_3, h_4}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_1, h_4], [h_2, h_4], \atop [h_1, h_3]=a_1, [h_2, h_3] = a_2, [h_3, h_4]=a_3}\right.
\right\rangle,\]
\[I_{7.5}=\left\langle {a_1, a_2, a_3,\atop h_1, h_2, h_3, h_4}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_4], [h_2, h_4], \atop [h_1, h_2]= [h_3, h_4]=a_1, [h_1, h_3]=a_2, [h_2, h_3] = a_3}\right.
\right\rangle,\]
\[I_{7.6}=\left\langle {a_1, a_2, a_3,\atop h_1, h_2, h_3, h_4}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_2], [h_2, h_4], [h_3, h_4], \atop [h_1, h_3]=a_1, [h_1, h_4] = a_3, [h_2, h_3]=a_2}\right.
\right\rangle,\]
\[I_{7.7}=\left\langle {a_1, a_2, a_3,\atop h_1, h_2, h_3, h_4}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_1, h_4], [h_2, h_3], \atop [h_1, h_2]=[h_3, h_4]=a_1, [h_1, h_3] = a_2, [h_2, h_4]=a_3}\right.
\right\rangle,\]
\[I_{7.8}=\left\langle {a_1, a_2, a_3,\atop h_1, h_2, h_3, h_4}\left| {a_i^3, h_j^3, [a_i, h_j], [a_i,a_j], [h_2, h_3], [h_1, h_2]=[h_4, h_3]=a_1\atop, [h_1, h_3] = [h_2, h_4]= a_2, [h_1, h_4]=a_3}\right.
\right\rangle.\]
Now assume $n=4$ and $m=3$.
Then the action $\rho$ of $H$ on $A$ must be nontrivial. Otherwise, $A$ can not be the derived subgroup since $\dim(\operatorname{AS}_3(\mathbb{Z}_3)) = 3 < 4$. Let $h_1$, $h_2$ and $h_3$ be a basis of $H$. Then by Corollary \ref{cor-expp3}, $\varphi_{12}$, $\varphi_{13}$ and $\varphi_{23}$ are linearly independent over $A/\mathrm{soc}(A)$. It forces $\dim(\mathrm{soc}(A))=1$.
We set $a_1=\varphi_{23}$, $a_2 = -\varphi_{13}$, $a_3 = \varphi_{12}$,
and
$$a_4= h_1\triangleright\varphi_{23} -\varphi_{23} =\varphi_{13} - h_2\triangleright\varphi_{13}= h_3\triangleright\varphi_{12} -\varphi_{12}.$$ Now under the basis $a_1, a_2, a_3, a_4$, the action $\rho$ of $H$ on $A$ is given by
$$\rho(h_1)=\left(\begin{array}{cccc}1&&&\\&1&&\\&&1&\\1&&&1\end{array}\right),\
\rho(h_2)=\left(\begin{array}{cccc}1&&&\\&1&&\\&&1&\\&1&&1\end{array}\right),\
\rho(h_3)=\left(\begin{array}{cccc}1&&&\\&1&&\\&&1&\\&&1&1\end{array}\right).$$
The corresponding group, denoted by $I_{7.9}$, is indecomposable and has presentation
\[I_{7.9}=\left\langle {a_1, a_2, a_3, a_4\atop h_1, h_2, h_3}\left| {a_i^3, h_j^3, [h_i, a_j] \ (i\ne j) , [h_i, a_i]=a_4\ (i=1,2,3),\atop [h_1, h_2]=a_3, [h_1, h_3]=-a_2, [h_2,h_3]=a_1}\right.
\right\rangle.\]
Thus we have shown the following propostion.
\begin{proposition} $I_{7.1}, I_{7.2}, \cdots, I_{7.9}$ give a complete set of isoclasses of irreducible groups of exponent 3 and order $3^7$.
\end{proposition}
\subsection{ } We summarize the classification results in this section as follows.
\begin{theorem}\label{thm-ord1to7} \cite{wil} The isoclasses of groups of order $3^{k}$($k\le 7$) and of exponent 3 are listed as follows, where $n$ is the dimension of the derived subgroup.
\begin{center}
{
\begin{tabular}{|c|c|c|c|}
\hline
$k$ & $n$& Decomposable & Indecomposable \\ \hline
$1$ & $0$ & & $I_1=\mathbb{Z}_3$\\ \hline
$2$ & $0$ &$I_1^2$ &\\ \hline
$3$ & $0$ &$I_1^3$ &\\ \hline
$3$ & $1$ & & $I_3$\\ \hline
$4$ & $0$ &$I_1^4$ & \\ \hline
& 1 &$I_1\times I_3$ & \\ \hline
$5$ & $0$ &$I_1^5$ & \\ \hline
& $1$ &$I_1^2\times I_3$ & $I_{5.1}$\\ \hline
& $2$ & & $I_{5.2}$\\ \hline
$6$ & $0$ &$I_1^6$ & \\ \hline
& $1$ &$I_1^3\times I_3, I_1\times I_{5.1}$ & \\ \hline
& $2$ &$I_1\times I_{5.2}, I_3\times I_3$ & $I_{6.1}, I_{6.2}$\\ \hline
& $3$ & & $I_{6.3}$\\ \hline
$7$ & $0$ &$I_1^7$ & \\ \hline
& $1$ &$I_1^4\times I_3, I_1^2\times I_{5.1}$ & $I_{7.1}$\\ \hline
& $2$ &$I_1^2\times I_{5.2}, I_1\times I_3\times I_3, I_1\times I_{6.1}, I_1\times I_{6.2}$ & $I_{7.2}, I_{7.3}$ \\ \hline
& $3$ &$I_1\times I_{6.3}$ & $I_{7.4}, I_{7.5}, I_{7.6}, I_{7.7}, I_{7.8}$\\ \hline
& $4$ & & $I_{7.9}$\\ \hline
\end{tabular}
\footnotesize Table 1. Isoclasses of groups of order $3^k$ ($k\le 7$) and exponent 3}
\end{center}
\end{theorem}
\section{Groups of order $3^8$ and exponent 3}
In this section, we aim to classify groups of order $3^8$ and exponent 3.
Let $G$ be such a group, and $A=G'$, $H=G/G'$.
Assume $A=\mathbb{Z}_3^n$, $H=\mathbb{Z}_3^m$, and $G=\mathbb{Z}_3^n\rtimes_{\rho,f}\mathbb{Z}_3^m$.
Then $n\le 5$, otherwise it is easy to show that $A$ can not be the derived subgroup by Corollary \ref{cor-expp3} and by comparing the dimension.
We work on $n$ case by case. Note that if $n\le 3$, then $\rho$ is trivial, and in this case, the classification
of isoclasses of $G$ is equivalent to the classification of certain congruence classes of subspaces.
\subsection{n=0, or 1}\
This case follows from Theorem \ref{thm-1dim-dergp} easily.
\begin{lemma} Keep the above notations.
\begin{enumerate}
\item[(1)] If $n=0$, then $G = \mathbb{Z}_3^8$ and is decomposable.
\item[(2)] If $n=1$, then $G = G_{7, 1}, G_{7, 2}$, or $G_{7, 3}$, and $G$ is decomposable.
\end{enumerate}
\end{lemma}
\subsection{n=2}\
Then $m=6$, and $G=\mathbb{Z}_3^2\rtimes_{\rho,f}\mathbb{Z}_3^6$.
By Section \ref{sec-2of6}, there are 14 congruence classes of 2 dimensional subspaces of $\operatorname{AS}_6(\mathbb{Z}_3)$:
\[\begin{pmatrix}&&&a&&\\&&&b&&\\&&&&&\\-a&-b&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&a&&&\\&&&b&&\\-a&&&&&\\&-b&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&a&b&&\\&&&a&&\\-a&&&&&\\-b&-a&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\]
\[
\begin{pmatrix} &&a&b&&\\&&-b&a&&\\-a&b&&&&\\-b&-a&&&&\\&&&&0&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&&a&&\\&&&b&&\\&&&&a&\\-a&-b&&&&\\&&-a&&&\\&&&&&0\end{pmatrix},
\begin{pmatrix} &&&a&&\\&&&b&a&\\&&&& b&\\-a&-b&&&&\\&-a&-b&&&\\&&&&&0\end{pmatrix},
\]
\[
\begin{pmatrix}
& & & a & & \\
& & & & a & \\
& & & & & b \\
-a& & & & & \\
&-a& & & & \\
& & -b& & & \\
\end{pmatrix},\quad
\begin{pmatrix}
& & & a & & \\
& & & & b & \\
& & & & & a+b \\
-a& & & & & \\
&-b& & & & \\
& &-a-b& & & \\
\end{pmatrix},
\]
\[
\begin{pmatrix}
& & & a & b & \\
& & & & a & \\
& & & & & a \\
-a& & & & & \\
-b&-a& & & & \\
& & -a & & & \\
\end{pmatrix},\quad
\begin{pmatrix}
& & & a & b & \\
& & & & a & \\
& & & & & b \\
-a& & & & & \\
-b&-a& & & & \\
& & -b& & & \\
\end{pmatrix},\quad
\begin{pmatrix}
& & & a & b & \\
& & &-b & a & \\
& & & & & a \\
-a& b& & & & \\
-b&-a& & & & \\
& & -a & & & \\
\end{pmatrix},
\]
\[
\begin{pmatrix}
& & & a & b & \\
& & & & a & b \\
& & & b & b & a \\
-a& & -b& & & \\
-b&-a& -b& & & \\
&-b& -a& & & \\
\end{pmatrix},\quad
\begin{pmatrix}
& & & a & b & \\
& & & & a & b \\
& & & & & a \\
-a& & & & & \\
-b&-a& & & & \\
&-b& -a & & & \\
\end{pmatrix},\quad
\begin{pmatrix}
& & & & a & \\
& & & & b & \\
& & & & & a \\
& & & & & b \\
-a&-b& & & & \\
& & -a& -b& & \\
\end{pmatrix}.
\]
Let $G_1, G_2, \cdots, G_{14}$ be the corresponding groups respectively. Then
$G_1= I_{5.2}\times \mathbb{Z}_3^3$, $G_2 = I_3\times I_3\times \mathbb{Z}_3^2$, $G_3= I_{6.1}\times \mathbb{Z}_3^2$, $G_4=I_{6.2}\times \mathbb{Z}_3^2$,
$G_5= I_{7.2}\times \mathbb{Z}_3$, $G_6= I_{7.3}\times \mathbb{Z}_3$ , and $G_7 = I_3\times I_{5.1}$ are decomposable, and the others are all indecomposable. We denote $G_8,\cdots, G_{14}$ by $I_{8.1},\cdots, I_{8.7}$ respectively. The presentations of these groups can be easily read by Proposition \ref{prop-pres-from-mrep} and we left it to the readers.
\subsection{n=3}
In this case, $m=5$.
By Theorem \ref{thm-3of5}, there are 22 congruence classes of 3-dimensional subspaces of the space $\operatorname{AS}_5(\mathbb{Z}_3)$, say
\[
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&-d\\&-a&-b&d&\end{pmatrix},
\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&&&a&\\&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&&&a&d\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\-d&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&d\\&&&&b\\-a&-b&&&\\&-d&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&&&a&\\&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&-d&b\\-a&-b&d&&\\&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\&-a-d&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix} &d&&a&\\-d&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\begin{pmatrix} &&&a&\\&&d&b&\\&-d&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\]
\[
\begin{pmatrix} &&&a&\\&&&b&\\&&&&a\\-a&-b&&&d\\&&-a&-d&\end{pmatrix},
\begin{pmatrix} &&&a&\\&&&b&\\&&&&d\\-a&-b&&&\\&&-d&&\end{pmatrix},
\]
\[
\begin{pmatrix}&c&a&&\\-c&&b&&\\-a&-b&&&\\&&&0&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&a&&\\&&b&&\\-a&-b&&c&\\&&-c&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&&\\-c&&b&&\\-a&-b&&c&\\&&-c&&\\&&&&0\end{pmatrix},
\]
\[
\begin{pmatrix}&&a&c&\\&&b&&\\-a&-b&&&\\-c&&&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&&\\-c&&&b&\\-a&&&c&\\&-b&-c&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&b&\\-c&&&a&\\-a&&&-c&\\-b&-a&c&&\\&&&&0\end{pmatrix}.
\]
Let $G_1, G_2, \cdots, G_{22}$ be the corresponding groups. Then $G_{16}= I_3\times I_{5.2}$, $G_{17}= I_{6.3}\times \mathbb{Z}_3^2$, $G_{18}= I_{7.4}\times \mathbb{Z}_3$, $G_{19}= I_{7.5}\times \mathbb{Z}_3$, $G_{20}= I_{7.6}\times \mathbb{Z}_3$, $G_{21}= I_{7.7}\times \mathbb{Z}_3$, and $G_{22}= I_{7.8}\times \mathbb{Z}_3$ are decomposable, and the others are indecomposable. We denote the groups $G_1,G_2,\cdots,G_{15}$ by $I_{8.8}, \cdots, I_{8.22}$ respectively. Moreover, the presentations of these groups can be read easily from the matrix presentations by Proposition \ref{prop-pres-from-mrep}.
\subsection{n=4}
Then $m=4$.
In this case, there are two cases, say the action $\rho$ of $H$ on $A$ is trivial, or nontrivial.
If $\rho$ is trivial, then the isoclasses of such groups are in one-to-one correspondence with the following four congruence class of 4-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$ as listed in Theorem \ref{thm-4of4}:
\[\begin{pmatrix}&&a&c\\&&b&d\\-a&-b&&\\-c&-d&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&\\-c&&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&d\\-c&&-d&\end{pmatrix},
\begin{pmatrix}&c&a&d\\-c&&d&b\\-a&-d&&c\\-d&-b&-c&\end{pmatrix}.
\]
The groups are all indecomposable, and denoted by $I_{8.23}, I_{8.24}, I_{8.25}$ and $I_{8.26}$ respectively.
Now assume $\rho$ is nontrivial.
Let $h_1$, $h_2$ and $h_3$ and $h_4$ be a basis of $H$. Then by Corollary \ref{cor-expp3}, there exist some $r\le s\le t$ such that $\varphi_{rs}$, $\varphi_{rt}$ and $\varphi_{st}$ are linearly independent over $A/\mathrm{soc}(A)$. It follows that $\dim(\mathrm{soc}(A))=1$. Without loss of generality, we assume $(r,s,t)= (1,2,3)$.
Let $a_1=\varphi_{23}$, $a_2 = -\varphi_{13}$, $a_3 = \varphi_{12}$,
and $a_4= h_1\triangleright\varphi_{23} -\varphi_{23} =\varphi_{13} - h_2\triangleright\varphi_{13}= h_3\triangleright\varphi_{12} -\varphi_{12}$. Under the basis $a_1, a_2, a_3, a_4$, the action $\rho$ of $H$ on $A$ is given by
$$\rho(h_1)=\begin{pmatrix}1&&&\\&1&&\\&&1&\\1&&&1\end{pmatrix},\
\rho(h_2)=\begin{pmatrix}1&&&\\&1&&\\&&1&\\&1&&1\end{pmatrix},\
\rho(h_3)=\begin{pmatrix}1&&&\\&1&&\\&&1&\\&&1&1\end{pmatrix}.$$
We may assume $\rho(h_4)= \operatorname{id}_A$ after changing basis. In fact,
$\rho(h_4)$ commutes with $\rho(h_1)$, $\rho(h_2)$ and $\rho(h_3)$ for $H$ is abelian. Then $\rho(h_4) =\left(\begin{array}{cccc}\lambda &&&\\\lambda_1&\lambda&&\\ \lambda_2&&\lambda&\\ \lambda_3&&&\lambda\end{array}\right)$ for some $\lambda\ne 0$ and $\lambda_1, \lambda_2$ and $\lambda_3$. Moreover, $h_4^3=1$ implies $\lambda=1$. Clearly $\tilde h_4 = h_4h_1^{-\lambda_1}h_2^{-\lambda_2}h_3^{-\lambda_3}$ acts trivially on $A$, and $h_1, h_2, h_3, \tilde h_4$ also form a basis of $H$.
By Corollary \ref{cor-expp1}, we have $(h_1-1)\triangleright \varphi_{14} = (h_4-1)\triangleright \varphi_{14} =0$ and $$(h_2-1)\triangleright \varphi_{14}= -(h_4-1)\triangleright \varphi_{12}=0, $$ Similarly we have $(h_3-1)\triangleright \varphi_{14}=0$.
This means that $\varphi_{14}\in \mathrm{soc}(A)$ and hence $\varphi_{14}= \mu_1 a_4$, $\varphi_{24}= \mu_2 a_4$ and $\varphi_{34}=\mu_3 a_4$ for some
$\mu_1, \mu_2, \mu_3\in \mathbb{Z}_3$.
Set $\tilde \varphi_{ij} = \varphi_{ij}$ for $1\le i, j\le 3$, and $\tilde \varphi_{ij} = 0$ for $i=4$ or $j=4$. We claim that $\varphi$ and $\tilde\varphi$ differ by a cocycle.
In fact, by the formula \ref{formula-cochain}, we have
\[(\varphi-\tilde\varphi)(h_1^{i_1}h_2^{i_2}h_3^{i_3} h_4^{i_4}, h_1^{j_1}h_2^{j_2}h_3^{j_3} h_4^{j_4}) = -i_4(j_1 \mu_1 + j_2 \mu_2 + j_3 \mu_3) \mu_3 a_4.
\]
Let $f\colon H\to A$ be given by $f(h_1^{i_1} h_2^{i_2} h_3^{i_3} h_4^{i_4}) = i_4 (\mu_1a_1+ \mu_2a_2+\mu_3a_3)
+ i_4(\mu_1i_1 + \mu_2i_2 + \mu_3i_3)a_4$.
It is direct to show that
\begin{align*}
&d(f)(h_1^{i_1}h_2^{i_2}h_3^{i_3} h_4^{i_4}, h_1^{j_1}h_2^{j_2}h_3^{j_3} h_4^{j_4})\\
=& (h_1^{i_1}h_2^{i_2}h_3^{i_3} h_4^{i_4})\triangleright f(h_1^{j_1}h_2^{j_2}h_3^{j_3} h_4^{j_4}) - f(h_1^{i_1+j_1}h_2^{i_2+j_2}h_3^{i_3+j_3} h_4^{i_4+j_4})
+ f(h_1^{i_1} h_2^{i_2} h_3^{i_3} h_4^{i_4})\\
=& (h_1^{i_1}h_2^{i_2}h_3^{i_3} h_4^{i_4})\triangleright [j_4(\mu_1a_1+ \mu_2a_2+\mu_3a_3)
+ j_4(\mu_1j_1 + \mu_2j_2 + \mu_3j_3)a_4] \\
& - [(i_4+j_4)(\mu_1a_1+ \mu_2a_2+\mu_3a_3) \\
& + (i_4+j_4)(\mu_1(i_1+j_1) + \mu_2(i_2+j_2) + \mu_3(i_3+j_3))a_4] \\
&+ [i_4(\mu_1a_1+ \mu_2a_2+\mu_3a_3)
+ i_4(\mu_1i_1 + \mu_2i_2 + \mu_3i_3)a_4]\\
=& -i_4(j_1 \mu_1 + j_2 \mu_2 + j_3 \mu_3) \mu_3 a_4,
\end{align*}
and the claim follows.
Thus $G \cong A\rtimes_{\rho, \tilde\varphi} H \cong I_{7.9}\times \mathbb{Z}_3$, and hence $G$ is decomposable.
\subsection{n=5}
In this case, $m=3$ and the action $\rho$ must be nontrivial.
Let $h_1$, $h_2$ and $h_3$ be a basis of $H$. Then by Corollary \ref{cor-expp3}, $\varphi_{12}$, $\varphi_{13}$ and $\varphi_{23}$ are linearly independent over $A/\mathrm{soc}(A)$, and $h_1\triangleright\varphi_{23} -\varphi_{23} =\varphi_{13} - h_2\triangleright\varphi_{13}= h_3\triangleright\varphi_{12} -\varphi_{12}\ne 0$.
Let $a_1=\varphi_{23}$, $a_2 = -\varphi_{13}$, $a_3 = \varphi_{12}$,
and $a_4= h_1\triangleright\varphi_{23} -\varphi_{23}$. Let $\tilde A$ denote the subspace spanned by $a_1, a_2, a_3$ and $a_4$. We pick some $a_5\in A$ such that $a_1, a_2, a_3, a_4, a_5$ form a basis of $A$. Then $(h_1-1)\triangleright a_5 = \sum_{i=1}^5 \lambda_i a_i$. Clearly we have $\lambda_5=0$ for $0= (h_1-1)^3\triangleright a_5 = \lambda_5^3 a_5$, i.e., $(h_1-1)\triangleright a_5 \in \tilde A$. Similary $(h_2-1)\triangleright a_5, (h_3-1)\triangleright a_5\in \tilde A$. Then by Proposition \ref{prop-der-abext}, the derived subgroup of $G'= \tilde A$, which leads to a contradiction. Thus we have proved the following result.
\begin{proposition} There exists no group of exponent $3$ and order $3^8$ whose derived subgroup has order $3^5$.
\end{proposition}
\subsection{} In summary, we have the following classification.
\begin{theorem}\label{thm-ord8} The isoclasses of groups of exponent 3 and of order $3^8$ are listed as follows, where $n$ is the dimension of the derived subgroup.
\begin{center}
{
\begin{tabular}{|c|c|c|}
\hline
$n$ & Decomposable & Indecomposable \\ \hline
$0$ & $I_1^8$ & \\ \hline
$1$ & $I_1\times I_{7.1}, I_1^3\times I_{5.1}, I_1^5\times I_3$ & \\ \hline
$2$ & $I_1^3\times I_{5.2}, I_1^2\times I_3^2, I_1^2 \times I_{6.1}, I_1^2 \times I_{6.2}, I_1 \times I_{7.2}, I_1 \times I_{7.3}, I_3\times I_{5.1}$ & $I_{8.1}, \cdots, I_{8.7}$ \\ \hline
$3$ & $I_1^2 \times I_{6.3}, I_1\times I_{7.4}, I_1\times I_{7.5}, I_1\times I_{7.6}, I_1\times I_{7.7}, I_1\times I_{7.8}, I_3\times I_{5.2}$ & $I_{8.8}, \cdots, I_{8.22}$\\ \hline
$4$ &$I_1\times I_{7.9}$ & $I_{8.23},\cdots, I_{8.26}$\\ \hline
\end{tabular}
\footnotesize Table 2. Isoclasses of groups of order $3^8$ and exponent 3}
\end{center}
\end{theorem}
\section{Congruence classes of 3 and 4-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$}
We will give a complete set of representatives of congruence classes of $3$ and $4$ dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3))$ in this section. We need some basic notions.
\subsection{Pfaffian of anti-symmetric matrices of order 4} In this subsection and the next, $\mathbbm{k}$ can be an arbitrary base field.
Let $X= (x_{ij})\in \operatorname{AS}_4(\mathbbm{k})$ be an anti-symmetric matrix over $\mathbbm{k}$. Recall that the \emph{Pfaffian} of $X$ is defined to be $\mathrm{pf}(X) = x_{12}x_{34}- x_{13}x_{24}+ x_{14}x_{23}$, and $\det(X)=\mathrm{pf}(X)^2$. We remark that the notion of Pfaffian is defined for anti-symmetric matrices of arbitrary even order.
\begin{lemma} Let $V\subseteq \operatorname{AS}_4(\mathbbm{k})$ be an $r$-dimensional subspace with a basis $X_1, \cdots, X_r$. Then $\mathrm{pf}(x_1X_1+ \cdots+ x_rX_r)$ defines a quadratic form in variables $x_1, \cdots, x_r$.
Moreover, let $\tilde V$ be a subspace of $\operatorname{AS}_4(\mathbbm{k})$ congruent to $V$, and let $\tilde X_1, \cdots, \tilde X_r$ be a basis of $\tilde V$. Then the quadratic forms $\mathrm{pf}(x_1\tilde X_1+ \cdots+ x_r\tilde X_r)$ and $\mathrm{pf}(x_1X_1+ \cdots+ x_rX_r)$ are equivalent up to a scalar.
\end{lemma}
We use $E_{ij}\in M_m(\mathbbm{k})$ to denote the matrix with $(i,j)$-entry being 1 and other entries being 0. Then the $m\times m$ identity matrix $I_m= E_{11} + E_{22} + E_{33} + \cdots +E_{mm}$. We set
\begin{enumerate}
\item $P_{ij}= I_m -E_{ii}- E_{jj}+E_{ij}+E_{ji}$,
\item $D_i(\lambda)= I_m + (\lambda-1) E_{ii}$ for $0\ne\lambda\in\mathbbm{k}$,
\item $T_{ij}(\lambda) = I_m + \lambda E_{ij}$ for $\lambda\in \mathbbm{k}$.
\end{enumerate}
Matrices of the above form are known as elementary matrices. Any congruence transformation is a composite of series congruence transformations by elementary matrices.
It is direct to show that $\mathrm{pf}(P_{ij}XP_{ij}^T) = -\mathrm{pf}(X)$, $\mathrm{pf}(D_{i}(\lambda)XD_i{\lambda}^T) = \lambda\mathrm{pf}(X)$ and $\mathrm{pf}(T_{ij}(\lambda)XT_{ij}{\lambda}^T) = \mathrm{pf}(X)$ for any $X\in \operatorname{AS}_4(\mathbbm{k})$ and $\lambda\in\mathbbm{k}$. It follows that $\mathrm{pf}(C^TXC)=\det(C)\mathrm{pf}(X)$ for any $C\in \operatorname{GL}_4(\mathbbm{k})$.
\subsection{Orthogonal complement and radical.} $\mathbbm{k}$ is an arbitrary field in this subsection.
Let $m$ be a positive integer, and $X\in \operatorname{AS}_m(\mathbbm{k})$ be an anti-symmetric matrix. For any $u\in \mathbbm{k}^m$, the subspace $u^{\perp_X}=\{v\in \mathbbm{k}^n\mid uXv^T=0\}$ is called the \emph{orthogonal complement} of $u$ with respect to $X$, or $X$-complement of $u$ for short. Let $\mathbf{U}\subseteq \mathbbm{k}^m$ and $V\subseteq \operatorname{AS}_m(\mathbbm{k})$ be subspaces. Then the $V$-complement of $u$, the $X$-complement and the $V$-complement of $\mathbf U$ are defined by
\[u^{\perp_V}=\bigcap_{X\in V}u^{\perp_X}, \quad
{\mathbf U}^{\perp_X}= \bigcap_{u\in \mathbf U}u^{\perp_X}, \quad
{\mathbf U}^{\perp_V}= \bigcap_{u\in \mathbf U, X\in V}u^{\perp_X}.\]
\begin{definition}
Let $\mathbbm{k}$ be an arbitrary field. Let $X\in \operatorname{AS}_m(\mathbbm{k})$ be an anti-symmetric matrix, where $m$ is a positive integer.
The subspace \[\mathrm{rad}(X) = \{v\in \mathbbm{k}^m\mid vXu^T=0, \forall u\in \mathbbm{k}^m\} \]
is called the \emph{radical} of $X$. Similarly, for a subspace $V\subseteq \operatorname{AS}_m(\mathbbm{k})$, the space $\mathrm{rad}(V)=\bigcap\limits_{X\in V}\mathrm{rad}(X)$
is called the radical of $V$.
\end{definition}
We mention that the above definitions coincide with the ones of quadratic forms.
\begin{remark}\label{rem-complement-inv}
(1) For any $X\in \operatorname{AS}_m(\mathbbm{k})$, $P\in \operatorname{GL}_m(\mathbbm{k})$ and $u\in \mathbbm{k}^{m}$, $(uP)^{\perp_X} = (u^{\perp_{PXP^T}})P$.
(2) By definition, $\mathrm{rad}(X)= (\mathbbm{k}^m)^{\perp_X}$, and by (1) we have $\mathrm{rad}(PXP^T)P = \mathrm{rad}(X)$. Therefore the radical of an anti-symmetric matrix or a subspace of $\operatorname{AS}_m(\mathbbm{k})$ is invariant under congruence transformation.
(3) If $m$ is odd, then $\mathrm{rad}(X)\ne 0$ for any $X\in \operatorname{AS}_m(\mathbbm{k})$ for any anti-symmetric matrix of odd degree is always degenerate.
\end{remark}
We have the following obvious observation.
\begin{lemma}\label{lem-dimincreasing} Let $V\subseteq \operatorname{AS}_m(\mathbbm{k})$ be an $r$-dimensional subspace. Then $$V^\natural= \{\operatorname{diag}(X,0)\mid X\in \operatorname{AS}_m(\mathbbm{k})\}$$ is an $r$-dimensional subspace of $\operatorname{AS}_{m+1}(\mathbbm{k})$, and $V^\natural_1$ is congruent to $V^\natural_2$ if and only if $V_1$ is congruent to $V_2$. Moreover, let $W\subseteq\operatorname{AS}_{m+1}(\mathbbm{k})$ be an $r$ dimensional subspace with $\mathrm{rad}(W)\ne 0$, then $W= V^\natural$ for some $V\subseteq\operatorname{AS}_m(\mathbbm{k})$.
\end{lemma}
We return to the classification of $n$-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$ for $n\le 4$.
\subsection{n=3}\label{sec-3of4}
By Section \ref{sec-2of4}, there are 4 congruence classes of 2-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$, say
$$
\begin{pmatrix}&&a&\\&&b&\\-a&-b&&\\&&&0\end{pmatrix},
\begin{pmatrix}&&a&\\&&&b\\-a&&&\\&-b&&\end{pmatrix},
\begin{pmatrix}&&a&b\\&&&a\\-a&&&\\-b&-a&&\end{pmatrix},
\begin{pmatrix}&&a&b\\&&-b&a\\-a&b&&\\-b&-a&&\end{pmatrix}.
$$
Now we consider the congruence classes of 3-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$. The key point is that any such a 3-dimensional subspace is obtained from one of the above 2-dimensional subspaces by adding a one dimensional subspace.
Let $V\subseteq \operatorname{AS}_4(\mathbb{Z}_3)$ be a 3-dimensional subspace. Without loss of generality, we may assume that $V$ contains a 2-dimensional subspace $W$ as listed above. Thus we have the following four cases.
\textbf{Case 1.}\quad Assume $W = \left(\begin{array}{cccc}&&a&\\&&b&\\-a&-b&&\\&&&0\end{array}\right)$.
Then there exists $(\alpha,\beta,\gamma,\delta)\ne (0,0,0,0)$, such that $V= W + \mathbb{Z}_3\begin{pmatrix}&\alpha && \beta\\-\alpha&&&\gamma\\&&&\delta\\-\beta&-\gamma&-\delta&\end{pmatrix}$, and we denote $V$ of this form by $\langle\alpha,\beta,\gamma,\delta\rangle$.
It is easy to show that $\langle-\alpha,-\beta,-\gamma,-\delta\rangle=\langle\alpha,\beta,\gamma,\delta\rangle\overset{D_4(-1)}\sim\langle\alpha,-\beta,-\gamma,-\delta\rangle$, where the notation $V\overset{P}\sim W$ means that $W=PVP^T$.
If $\beta\ne0$, then
$$\left(\begin{array}{cccc}&c\alpha&a&c\beta\\-c\alpha&&b&c\gamma\\-a&-b&&c\delta\\-c\beta&-c\gamma&-c\delta&\end{array}\right)
\overset{T_{21}(-\beta\gamma)}\sim\left(\begin{array}{cccc}&c\alpha&a&c\beta\\-c\alpha&&b-a\beta\gamma&\\-a&-b+a\beta\gamma&&c\delta\\-c\beta&&-c\delta&\end{array}\right)$$
$$\overset{T_{24}(-\beta\alpha)}\sim\left(\begin{array}{cccc}&&a&c\beta\\&&b-a\beta\gamma+c\alpha\beta\delta&\\-a&-b+a\beta\gamma-c\alpha\beta\delta&&\\-c\beta&&&\end{array}\right)$$
Hence $\langle\alpha,\beta,\gamma,\delta\rangle\sim\langle0,\beta,0,0\rangle=\langle0,1,0,0\rangle$.
Similarly, if $\gamma\ne0$, then $\langle\alpha,\beta,\gamma,\delta\rangle\sim\langle0,0,\gamma,0\rangle=\langle0,0,1,0\rangle$.
If $\beta=\gamma=0$, then clearly $\langle\alpha,\beta,\gamma,\delta\rangle\sim \langle1,0,0,0\rangle$, $\langle0,0,0,1\rangle$ or $\langle1,0,0,1\rangle$.
Combined with the fact $\langle0,1,0,0\rangle\overset{P_{12}}\sim\langle0,0,1,0\rangle$, we know that $\langle\alpha,\beta,\gamma,\delta\rangle$ is congruent to one of $\langle1,0,0,0\rangle$, $\langle0,1,0,0\rangle$, $\langle0,0,0,1\rangle$ and $\langle1,0,0,1\rangle$. We claim that these four subspaces are not congruent to each other.
The subspaces $\langle1,0,0,0\rangle$, $\langle0,1,0,0\rangle$, $\langle0,0,0,1\rangle$ and $\langle1,0,0,1\rangle$ are given by
\[\begin{pmatrix}
& c& a& \\
c& & b& \\
-a&-b& & \\
& & &0\\
\end{pmatrix},
\begin{pmatrix}
& & a&c \\
& & b& \\
-a&-b& & \\
-c& & & \\
\end{pmatrix},
\begin{pmatrix}
& & a& \\
& & b& \\
-a&-b& &c\\
& &-c& \\
\end{pmatrix},
\begin{pmatrix}
& c& a& \\
c& & b& \\
-a&-b& &c\\
& &-c& \\
\end{pmatrix},
\]
and the Pfaffians are $0$, $bc$, $0$ and $c^2$ respectively. By comparing the Pfaffians, to prove the claim it suffices to show that $\langle1,0,0,0\rangle\not\sim\langle0,0,0,1\rangle$. Easy calculation shows that $$\mathrm{rad}(\langle1,0,0,0\rangle)=(0,0,0,\mathbb{Z}_3):=\{(0,0,0,a)\mid a\in\mathbb{Z}_3\},$$
while $\mathrm{rad}(\langle0,0,0,1\rangle)=0$, and it follows that $\langle1,0,0,0\rangle\not\sim\langle0,0,0,1\rangle$.
\textbf{Case 2.}\quad
Assume $W=\left(\begin{array}{cccc}&&a&\\&&&b\\-a&&&\\&-b&&\end{array}\right)$.
Then $V= W+ \mathbb{Z}_3\begin{pmatrix}&\alpha&&\beta\\-\alpha&&\gamma&\\&-\gamma&&\delta\\-\beta&&-\delta&\end{pmatrix}$, and we denote this space by $(\alpha,\beta,\gamma,\delta)$.
First we have $(\alpha,\beta,\gamma,\delta)=(-\alpha,-\beta,-\gamma,-\delta)$, and moreover, \[(\alpha,\beta,\gamma,\delta)\overset{P_{24}}\sim(\beta,\alpha,-\delta,-\gamma)
\overset{P_{13}}\sim (\delta,-\gamma,-\beta,\alpha)\overset{P_{24}}\sim
(-\gamma,\delta,-\alpha,\beta)
,\]
\[(-\alpha,-\beta,\gamma,\delta)\overset {D_1(-1)}\sim(\alpha,\beta,\gamma,\delta)\overset {D_4(-1)}\sim(\alpha,-\beta,\gamma,-\delta).\]
Thus we may assume that $\alpha\ne0$. Without loss of generality, we can take $\alpha=1$. Now we have $(1,\beta,\gamma,\delta)\overset{T_{42}(-\beta)}\sim
(1, 0, \gamma, \delta+ \beta\gamma)\overset{T_{31}(-\gamma)}\sim(1,0,0,\delta+ \beta\gamma)$. Hence there are at most two cases, say $(1,0,0,0)$ and $(1,0,0,1)$. Clearly, $(1,0,0,0)\overset{P_{24}}\sim(0,1,0,0)\overset{P_{34}}\sim\langle0,1,0,0\rangle$.
The subspace $(1,0,0,1)$ is given by
$\begin{pmatrix}
& c& a& \\
-c& & &b \\
-a& & &c\\
&-b&-c& \\
\end{pmatrix},$
and the Pfaffian of $(1,0,0,1)$ is $c^2-ab$, by comparing the Pfaffians, we know that $(1,0,0,0)$ is congruent to none of $\langle1,0,0,0\rangle$, $\langle0,1,0,0\rangle$, $\langle0,0,0,1\rangle$ and $\langle1,0,0,1\rangle$.
\textbf{Case 3.}\quad
Assume $W=\left(\begin{array}{cccc}&&a&b\\&&&a\\-a&&&\\-b&-a&&\end{array}\right)$, then $V=W + \mathbb{Z}_3\begin{pmatrix} &\alpha& \beta& \\- \alpha & & \gamma &\\- \beta&- \gamma&& \delta\\&&- \delta&\end{pmatrix}$ for some $\alpha, \beta, \delta, \gamma$. We denote this 3-dimensional subspace by $[\alpha,\beta,\gamma,\delta]$.
By direct calculation, we have \[[\alpha,\beta,\gamma,\delta]\overset{D_{1,2}(-1)}\sim[\alpha,-\beta,-\gamma,\delta]
\overset{D_{1,3}(-1)}\sim[-\alpha,-\beta,\gamma,-\delta]\overset{D_{1,2}(-1)}\sim[-\alpha,\beta,-\gamma,-\delta],\]
where $D_{1,2}(-1)=\operatorname{diag}(-1,-1,1,1)$ and $D_{1,3}(-1)=\operatorname{diag}(-1,1,-1,1)$. Consider
$$\mathrm{pf}\begin{pmatrix}&c\alpha&a+c\beta&\\-c\alpha&&c\gamma&a\\
-a-c\beta&-c\gamma&&c\delta\\&-a&-c\delta&\end{pmatrix}
=c^2\alpha\delta-a^2-ac\beta=-(a-c\beta)^2+c^2(\beta^2+\alpha\delta).$$
If $\beta^2+\alpha\delta=0$, then we take $a=\beta,c=1$ to get a rank two element in the 2-dimensional subspace $\begin{pmatrix}&c\alpha&a+c\beta&\\-c\alpha&&c\gamma&a\\-a-c\beta&-c\gamma&&c\delta\\&-a&-c\delta&\end{pmatrix}$,
and together with $\left(\begin{array}{cccc}&&&1\\&&0&\\&0&&\\-1&&&\end{array}\right)$ it spans a 2-dimensional subspace, which is equivalent to the first two classes of 2-dimensional subspaces. Hence this case reduces to the first two cases.
If $\beta^2+\alpha\delta=1$, then by taking $a=\beta+1,c=1$ we get a rank two element in the 2-dimensional subspace $\left(\begin{array}{cccc}&c\alpha&a+c\beta&\\-c\alpha&&c\gamma&a\\-a-c\beta&-c\gamma&&c\delta\\&-a&-c\delta&\end{array}\right)$.
Together with $\left(\begin{array}{cccc}&&&1\\&&0&\\&0&&\\-1&&&\end{array}\right)$, it spans a 2-dimensional subspace, which is equivalent to the first two classes of 2-dimensional subspaces. Hence this case reduces to the first two cases.
It remains to consider the case $\beta^2+\alpha\delta=-1$. Then $\beta=0$, $\alpha\delta=-1$ or $\beta\ne0$, $\alpha\delta=1$. In either case, $\alpha\ne0$ and $\delta\ne0$.
We may assume $\gamma=0$. Otherwise if $\gamma\ne0$, then $[\alpha,\beta,\gamma,\delta]\overset{T_{42}(\delta \gamma)}\sim[\alpha,\beta,\gamma,0]$, and we will go back to the cases $\beta^2+\alpha\delta=0$ or $1$.
Now we have $[\alpha,\beta,\gamma,\delta]\sim [1,0,0,-1]$ or $[1,1,0,1]$.
We claim that $[1,1,0,1]=[1,0,0,-1]$. In fact,
\begin{align*}
[1,1,0,1]=&\begin{pmatrix}&c&a+c&b\\-c&&&a\\-a-c&&&c\\-b&-a&-c&\end{pmatrix}
\overset{T_{32}(1)}\sim
\begin{pmatrix}&c&a-c&b\\-c&&&a+c\\-a+c&&&c\\-b&-a-c&-c& \end{pmatrix}\\
\overset{T_{41}(\delta \gamma)}\sim &
\begin{pmatrix}&c&a-c&b\\-c&&&a-c\\c-a&&&-c\\-b&c-a&c&\end{pmatrix}
=\begin{pmatrix}&c&a&b\\-c&&&a\\-a&&&-c\\-b&-a&c&\end{pmatrix}=[1,0,0,-1].
\end{align*}
\textbf{Case 4.} \quad Assume $W=\begin{pmatrix}&&a&b\\&&-b&a\\-a&b&&\\-b&-a&&\end{pmatrix}$, then
$V= W+ \mathbb{Z}_3\begin{pmatrix}&c\alpha&c\beta&c\gamma\\-c\alpha&&&\\-c\beta&&&c\delta\\-c\gamma&&-c\delta&\end{pmatrix}$.
Consider the determinant $$\det\begin{pmatrix}&c\alpha&a+c\beta&b+c\gamma\\
-c\alpha&&-b&a\\-a-c\beta&b&&c\delta\\-b-c\gamma&-a&-c\delta&\end{pmatrix}
=(-(a-c\beta)^2-(b-c\gamma)^2+c^2(\alpha\delta+\beta^2+\gamma^2))^2.$$
Then there always exists some element of rank two element in the 3-dimensional subspace. Hence this case reduces to the first three cases.
In summary, we have the following classification.
\begin{theorem}\label{thm-3of4}
There are 6 congruence classes of 3-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$:
$$\begin{pmatrix}&c&a&\\-c&&b&\\-a&-b&&\\&&&0\end{pmatrix},
\begin{pmatrix}&&a&\\&&b&\\-a&-b&&c\\&&-c&\end{pmatrix},
\begin{pmatrix}&c&a&\\-c&&b&\\-a&-b&&c\\&&-c&\end{pmatrix},$$
$$\begin{pmatrix}&&a&c\\&&b&\\-a&-b&&\\-c&&&\end{pmatrix},
\begin{pmatrix}&c&a&\\-c&&&b\\-a&&&c\\&-b&-c&\end{pmatrix},
\begin{pmatrix}&c&a&b\\-c&&&a\\-a&&&-c\\-b&-a&c&\end{pmatrix}.$$
Moreover, the Pfaffians are $0,0,c^2,bc,c^2-ab,-c^2-a^2$ respectively, which are pairwise nonequivalent except the first two.
\end{theorem}
\subsection{n=4.}\label{sec-4of4} In this subsection, we will give a full list of congruence classes of 4-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$.
Let $V\subset\operatorname{AS}_4(\mathbb{Z}_3)$ be of dimension 4, and $W$ be a subspace of $V$ dimension 3. We may assume that $W$ is of one of the six subspaces as listed in Theorem \ref{thm-3of4}.
\textbf{Case 1.}\quad Assume $W= \begin{pmatrix}&&a&c\\&&b&\\-a&-b&&\\-c&&&\end{pmatrix}$. Then $V= W + \mathbb{Z}_3 \begin{pmatrix}&\alpha&&\\-\alpha&&&\beta\\&&&\gamma\\&-\beta&-\gamma&\end{pmatrix}$
for some $\alpha, \beta, \gamma\in\mathbb{Z}_3$. We denote $V$ of this form by $\langle\alpha,\beta,\gamma\rangle$.
Applying the congruence transformation by $P_{13}P_{24}$ we obtain that $\langle\alpha,\beta,\gamma\rangle\sim\langle\gamma,-\beta,\alpha\rangle$, and applying $D_1(-1), D_2(-1)$ and $D_3(-1)$, we obtain that
$\langle\alpha,\beta,\gamma\rangle\sim\langle-\alpha,\beta,\gamma\rangle\sim
\langle\alpha,-\beta,\gamma\rangle\sim\langle\alpha,\beta,-\gamma\rangle.
$
If $\beta\ne0$, then
$\langle\alpha,\beta,\gamma\rangle\overset{T_{32}
(-\gamma\beta)}\sim\langle\alpha,\beta,0\rangle\overset{T_{14}(\alpha\beta)}\sim
\langle0,\beta,0\rangle\sim\langle0,1,0\rangle$; else if $\beta=0$, then
$\langle\alpha,\beta,\gamma\rangle\sim\langle1,0,0\rangle$ or $\langle1, 0, 1\rangle$.
The subspaces $\langle0,1,0\rangle,\langle1,0,0\rangle$ and $\langle1,0,1\rangle$ are
\[\begin{pmatrix}&&a&c\\&&b&d\\-a&-b&&\\-c&-d&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&\\-c&&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&d\\-c&&-d&\end{pmatrix}.
\]
\textbf{Case 2.}\quad
Assume $W=\left(\begin{array}{cccc}&c&a&\\-c&&&b\\-a&&&c\\&-b&-c&\end{array}\right)$, then
$V= W +\mathbb{Z}_3
\left(\begin{array}{cccc}&&&\alpha\\&&\beta&\\&-\beta&&\gamma\\-\alpha&&-\gamma&\end{array}\right)$, for some $\alpha, \beta$ and $\gamma$. We denote the subspace of this form by $(\alpha,\beta,\gamma)$.
It is easy to check that $(\alpha,\beta,\gamma)\overset{P_{12}P_{34}}\sim(\beta,\alpha,-\gamma)$, and \[(\alpha,\beta,\gamma)\overset{D_1(-1)D_4(-1)}\sim(\alpha,\beta,-\gamma)\overset{D_1(-1)D_2(-1)}\sim(-\alpha,-\beta,-\gamma).\]
If $\gamma=0$, then $(\alpha,\beta,\gamma)\sim (1,1,0)$, $(1,-1,0)$ or $(1,0,0)$.
If $\gamma\ne0$, then $(\alpha,\beta,\gamma)\sim (1,1,1)$, $(1,-1,1)$, $(1,0,1)$ or $(0,0,1)$.
Now we compare $(\alpha, \beta, \gamma)$'s with $\langle\alpha, \beta, \gamma\rangle$'s.
(i) $(0,0,1)\overset{P_{13}}\sim \langle0,1,0\rangle$.
(ii) $(1,1,1)\sim\langle1,0,1\rangle$ for
\begin{align*}
(1,1,1)=& \begin{pmatrix}&c&a&d\\-c&&d&b\\-a&-d&&c+d\\-d&-b&-c-d&\end{pmatrix}\overset{T_{31}(1)T_{42}(1)}\sim
\begin{pmatrix}&c&a&d+c\\-c&&d-c&b\\-a&c-d&&d-c\\-d-c&-b&c-d& \end{pmatrix}\\
\overset{T_{42}(1)} \sim&
\begin{pmatrix}&c&a&d-c\\-c&&d-c&b\\-a&c-d&&\\c-d&-b&&\end{pmatrix}
\overset{P_{34}P_{24}} \sim \begin{pmatrix}&d-c&c&a\\c-d&&-b&\\-c&b&&d-c\\-a&&c-d&\end{pmatrix}
=\langle1,0,1\rangle
\end{align*}
(iii) $(1,1,0)\sim (1,-1,1)$ for
\begin{align*}(1,1,0)=&\begin{pmatrix}&c&a&d\\-c&&d&b\\-a&-d&&c\\-d&-b&-c&\end{pmatrix}
\overset{T_{24}(-1)}\sim
\begin{pmatrix}&c-d&a&d\\-c+d&&d+c&b\\-a&-d-c&&c\\-d&-b&-c&\end{pmatrix}\\
\overset{T_{31}(1)}\sim
&\begin{pmatrix}&c-d&a&d\\-c+d&&-d&b\\-a&-d&&(c-d)-d\\-d&-b&d-(c-d)&\end{pmatrix} = (1,-1,-1)\sim (1,-1,1).
\end{align*}
(iv) $(1,0,1)\sim(0,0,1)\sim\langle0,1,0\rangle$ for
\begin{align*}(1,0,1)=&\begin{pmatrix}&c&a&d\\-c&&&b\\-a&&&c+d\\-d&-b&-c-d&\end{pmatrix}
\overset{T_{41}(1)}\sim\begin{pmatrix}&c&a&c+d\\-c&&&b\\-a&&&c+d\\-c-d&-b&-c-d&\end{pmatrix}\\
\overset{T_{13}(-1)}\sim&
\begin{pmatrix}&c&a&\\-c&&&b\\-a&&&c+d\\&-b&-c-d&\end{pmatrix}=(0,0,1)\sim \langle0,1,0\rangle.
\end{align*}
(v) $(1,-1,0)\sim\langle0,1,0\rangle$ for
\begin{align*} (1,-1,0)=&\begin{pmatrix}&c&a&d\\-c&&-d&b\\-a&d&&c\\-d&-b&-c&\end{pmatrix}
\overset{T_{42}(1)} \sim \begin{pmatrix}&c&a&d+c\\-c&&-d&b\\-a&d&&c+d\\-c-d&-b&-c-d&\end{pmatrix}\\
\overset{T_{13}(-1)} \sim
&\begin{pmatrix}&(c-d)&a&\\d-c&&-d&b\\-a&d&&(c-d)-d\\&-b&d-(c-d)&\end{pmatrix}=(0,1,1)\sim(1,0,1)
\sim\langle0,1,0\rangle.
\end{align*}
(vi) $(1,0,0) \sim\langle1,0,1\rangle$ for
$$(1,0,0)= \begin{pmatrix}&c&a&d\\-c&&&b\\-a&&&c\\-d&-b&-c&\end{pmatrix}
\overset{P_{34}}\sim
\begin{pmatrix}&c&d&a\\-c&&b&\\-d&-b&&-c\\-a&&c&\end{pmatrix}= \langle1,0,-1\rangle\sim \langle1,0,1\rangle.$$
\textbf{Case 3.}\quad
Assume $W= \left(\begin{array}{cccc}&c&a&b\\-c&&&a\\-a&&&-c\\-b&-a&c&\end{array}\right)$, and
$V= W + \mathbb{Z}_3 \begin{pmatrix}&&&\\&&\alpha&\beta\\&-\alpha&&\gamma\\&-\beta&-\gamma&\end{pmatrix}$, we denote the 4-dimensional subspace of this form by $[\alpha,\beta,\gamma]$.
We note that $$[\alpha,\beta,\gamma]\overset{D_1(-1)D_2(-1)}\sim [-\alpha,-\beta,\gamma]\overset{D_{1}(-1)D_3(-1)}\sim[\alpha,-\beta,-\gamma]\overset{D_1(-1)D_2(-1)}\sim [-\alpha,\beta,-\gamma].$$
If $\alpha\ne1$, then we may assume $\alpha=1$ for $[\alpha,\beta,\gamma]\sim[-\alpha,-\beta,\gamma]$, and $[1,\beta,\alpha]\sim(1,1,0)$ for
\begin{align*} [1,\beta,\gamma]=&\begin{pmatrix}&c&a&b\\-c&&d&a+d\beta\\-a&-d&&-c+d\gamma\\-b&-a-d\beta&c-d\gamma&\end{pmatrix}
\overset{T_{43}(\beta)}\sim \begin{pmatrix}&c&a&b-a\beta\\-c&&d&a\\-a&-d&&-c+d\gamma\\a\beta-b&-a&c-d\gamma&\end{pmatrix}\\
\overset{T_{42}(\gamma)}\sim & \begin{pmatrix}&c&a&b-a\beta+c\gamma\\-c&&d&a\\-a&-d&&-c\\-b+a\beta-c\gamma&-a&c&\end{pmatrix}
= \begin{pmatrix}&c&a&b\\-c&&d&a\\-a&-d&&-c\\-b&-a&c&\end{pmatrix}\\
\overset{P_{34}}\sim &
\begin{pmatrix}&c&b&a\\-c&&a&d\\-b&-a&&c\\-a&-d&-c&\end{pmatrix} = (1,1,0).
\end{align*}
If $\alpha=0$, then $[0,\beta,\gamma]\sim[0,0,1], [0,1,0]$ or $[0,1,1]$.
We can show that
(i) $[0,0,1]\overset{P_{34}P_{23}}\sim \langle1,0,1\rangle$;
(ii) $[0,1,0]\overset{P_{34}}\sim \langle1,0,1\rangle$;
(iii) $[0,1,1]\sim \langle1,0,1\rangle$ for
\begin{align*}
[0,1,1]=&\begin{pmatrix} &c&a&b\\-c&&&a+d\\-a&&&-c+d\\-b&-a-d&c-d&\end{pmatrix}
\overset{T_{32}(-1)}\sim \begin{pmatrix} &c&a-c&b\\-c&&&a+d\\c-a&&&-c-a\\-b&-a-d&a+c&\end{pmatrix}\\
\overset{T_{23}(-1)}\sim &\begin{pmatrix} &-a-c&a-c&b\\a+c&&&-a+c+d\\c-a&&&-a-c\\-b&a-c-d&a+c&\end{pmatrix}
=\begin{pmatrix} &c&a&b\\-c&&&d\\-a&&&-c\\-b&-d&c&\end{pmatrix} \overset{P_{34}}\sim \langle1,0,1\rangle.
\end{align*}
\textbf{Case 4.}
Assume $W=\left(\begin{array}{cccc}&c&a&\\-c&&b&\\-a&-b&&c\\&&-c&\end{array}\right)$, and $V= \begin{pmatrix}&c&a&d\alpha\\-c&&b&d\beta\\-a&-b&&c+d\gamma\\-d\alpha&-d\beta&-c-d\gamma&\end{pmatrix}$ for some $\alpha, \beta, \gamma$. Then
the Pfaffian is $c^2+cd\gamma-ad\beta+bd\alpha$.
If $\alpha\ne0$, then the 3-dimensional subspace $\{a=0\}$ is equivalent to the 3-dimensional subspaces considered in the first three cases, since its Pfaffian is neither equivalent to $c^2$ nor to 0. Hence this case reduces to the first three cases.
If $\beta\ne0$, the 3-dimensional subspace $\{b=0\}$ is equivalent to the 3-dimensional subspaces considered in the first three cases, since its Pfaffian is neither equivalent to $c^2$ nor to 0. Hence this case reduces to the first three cases.
If $\alpha=\beta=0$, then $\gamma\ne0$, and hence the 3-dimensional subspace $\{a=0\}$ is equivalent to the 3-dimensional subspaces considered in the first three cases, since its Pfaffian is neither equivalent to $c^2$ nor 0. Hence this case reduces to the first three cases.
\textbf{Case 5.}
Assume $W=\begin{pmatrix}&c&a&\\-c&&b&\\-a&-b&&\\&&&0\end{pmatrix}$, and $V= \begin{pmatrix}&c&a&d\alpha\\-c&&b&d\beta\\-a&-b&&d\gamma\\-d\alpha&-d\beta&-d\gamma&\end{pmatrix}$ for some $\alpha, \beta$ and $\gamma$. The Pfaffian is $cd\gamma-ad\beta+bd\alpha$.
We claim that there always exists a full rank element in $V$ and hence this case reduces to the first four cases discussed above.
In fact, if $\alpha\ne0$, then we can take $a=c=0$, $b=d=1$; if $\beta\ne0$, then we can take $b=c=0$, $a=d=1$; and if $\gamma\ne0$, then we can take $a=b=0$, $c=d=1$. It is direct to check that in each case, the corresponding matrix is of full rank.
\textbf{Case 6.} Assume $W= \begin{pmatrix}&&a&\\&&b&\\-a&-b&&c\\&&-c&\end{pmatrix}$, then
$V=\begin{pmatrix}&d\alpha&a&d\beta\\-d\alpha&&b&d\gamma\\-a&-b&&c\\-d\beta&-d\gamma&-c&\end{pmatrix}$ for some
$\alpha, \beta$ and $\gamma$, and the Pfaffian is
$cd\alpha-ad\gamma+bd\beta$. Similar arguments as in Case 5 shows that this case reduces to the first four cases.
Now we have obtained all possible equivalence subspaces, say $\langle0,1,0\rangle$, $\langle1,0,0\rangle$, $\langle1,0,1\rangle$ and $(1,1,0)$. Moreover, they are pairwise noncongruent.
\begin{theorem}\label{thm-4of4}
There are 4 congruence classes of 4-dimensional subspaces of the space of $\operatorname{AS}_4(\mathbb{Z}_3)$:
\[\begin{pmatrix}&&a&c\\&&b&d\\-a&-b&&\\-c&-d&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&\\-c&&&\end{pmatrix},
\begin{pmatrix}&d&a&c\\-d&&b&\\-a&-b&&d\\-c&&-d&\end{pmatrix},
\begin{pmatrix}&c&a&d\\-c&&d&b\\-a&-d&&c\\-d&-b&-c&\end{pmatrix}.
\]
Moreover, the Pfaffians are $bc-ad$, $bc$, $bc+d^2$ and $c^2-ab+d^2$ respectively which are pairwise nonequivalent.
\end{theorem}
\begin{remark}
There is a non-degenerate symmetric bilinear form $(\, ,\,)$ on $\operatorname{AS}_4(\mathbb{Z}_3)$ given by $(X,Y) =\mathrm{tr}(XY)$. For any $V\subseteq\operatorname{AS}_4(\mathbb{Z}_3)$, set $V^\perp= \{Y\in \operatorname{AS}_4(\mathbb{Z}_3)\mid \mathrm{tr}(XY)=0, \forall X\in V\}$ to be its orthogonal complement. Clearly $\dim V + \dim V^\perp = 6$. Since $\mathrm{tr}(PXP^{-1})=\mathrm{tr}(X)$ for any $P\in \operatorname{GL}_4(\mathbb{Z}_3)$, we have $(PVP^T)^\perp=(P^{-1})^T V^\perp P^{-1}$, which implies a one-to-one correspondence between congruence classes of $k$-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$ and the $6-k$ dimensional ones under taking $()^\perp$. We may compare Theorem \ref{thm-4of4} and Section 6.3.
\end{remark}
\section{Congruence classes of 3-dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$}
In this section, we will give a complete set of representatives of congruence classes of 3-dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$.
\subsection{A technical lemma} First we have the following lemma.
\begin{lemma}\label{lem-trivialradical}
Let $V\subseteq\operatorname{AS}_5(\mathbb{Z}_3)$ be a subspace with $\mathrm{rad}(V)=0$.
\begin{enumerate}
\item[(1)] If $V$ has a basis $X, Y, Z$ such that $\operatorname{rank}(aX+bY)\le 2$ for all $a, b\in \mathbb{Z}_3$ and $\operatorname{rank}(Z)=2$, then
$V$ is congruent to $\operatorname{diag}(\begin{pmatrix}&&a\\&&b\\-a&-b&\end{pmatrix},\begin{pmatrix}&c\\-c&\end{pmatrix})$.
\item[(2)] There exist some $X, Y\in V$ such that $\operatorname{rank}(X)=4$ and $\mathrm{rad}(X)\cap\mathrm{rad}(Y)=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $W$ be the two dimensional subspace spanned by $X$ and $Y$. By Section \ref{sec-2of5}, we may assume $W = \operatorname{diag}(\begin{pmatrix}&&a\\&&b\\-a&-b&\end{pmatrix},0)$. Since $\operatorname{rank}(Z)=2$, there exists some $P\in \operatorname{GL}_5(\mathbb{Z}_3)$ such that $PZ P^T=\operatorname{diag}(0,\begin{pmatrix}&1\\-1&\end{pmatrix})$. We may write $P =\begin{pmatrix}P_1&P_2\\P_3&P_4\end{pmatrix}$, where $P_1\in M_{3\times 3}(\mathbb{Z}_3)$, $P_2\in M_{3\times 2}(\mathbb{Z}_3)$, $P_3\in M_{2\times 3}(\mathbb{Z}_3)$ and $P_4\in M_{2\times 2}(\mathbb{Z}_3)$ are matrices.
Since $\dim\mathrm{rad}(W)=2$ and $\mathrm{rad}(W)\cap \mathrm{rad}(Z)=0$, we can show that $P_1$ is invertible. Set $Q=\begin{pmatrix}P_1^{-1}&\\-P_3P_1^{-1}& I\end{pmatrix}P$. Then $Q$ has the form $\begin{pmatrix}I_3 & Q_2 \\ &Q_4\end{pmatrix}$, and $QZQ^{T}= \operatorname{diag}(0,\begin{pmatrix}&1\\-1&\end{pmatrix})$. Now the conclusion follows from the fact that $W$ is invariant under the congruence transformation by $Q$.
(2) From the proof of (1) we show that there exists some $X\in V$ with $\operatorname{rank}(X)=4$.
If $\mathrm{rad}(X)\cap\mathrm{rad}(Y)\ne0$ for any $Y\in V$, then $\mathrm{rad}(X)\subseteq\mathrm{rad}(Y)$ for any $Y$, which means that $\mathrm{rad}(V)=\mathrm{rad}(X_1)\ne 0$, contradicts to the assumption $\mathrm{rad}(V)=0$.
Thus there must exist some $Y\in V$ such that $\mathrm{rad}(X)\cap\mathrm{rad}(Y)=0$.
\end{proof}
In Section 9, we have obtained all congruence classes of 3 and 4-dimensional subspaces of $\operatorname{AS}_4(\mathbb{Z}_3)$. By Lemma \ref{lem-dimincreasing}, to determine the congruence classes of subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$, we need only to consider those ones which have trivial radicals.
Let $V\subseteq \operatorname{AS}_5(\mathbb{Z}_3)$ be a 3-dimensional subspace with $\mathrm{rad}(V)=0$. Then by Lemma \ref{lem-trivialradical} and Section \ref{sec-2of5}, $V$ has a 2-dimensional subspace $W$ which is congruent to one of the following two subspaces:
\[W_1= \begin{pmatrix}&&&a&\\&&&b&a\\&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix}, \ W_2 = \begin{pmatrix}&&&a&\\&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix}.\]
In the following two subsections, we will deal with these two cases separately.
\subsection{Case 1: $V$ has a two dimensional subspace congruent to $W_1$.}\
In this subsection, we assume that $V$ contains a 2-dimensional subspace $W$ such that any nonzero element has rank 4. Then we may assume $W= W_1$.
We recall a group homomorphism $\operatorname{GL}_2(\mathbb{Z}_3)\to \operatorname{GL}_3(\mathbb{Z}_3)$ induced by the natural action of $\operatorname{GL}_2(\mathbb{Z}_3)$ on the homogeneous polynomials of degree 2 in two variables. Precisely, any $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{GL}_2(\mathbb{Z}_3)$ maps to
$\hat X= \begin{pmatrix}a^2&ab&b^2\\2ac&ad+bc&2bd\\c^2&cd&d^2\end{pmatrix}\in \operatorname{GL}_3(\mathbb{Z}_3)$. For instance,
\[\widehat{\begin{pmatrix}1&\\&-1\end{pmatrix}} = \begin{pmatrix}1&&\\&-1&\\&&1\end{pmatrix},
\qquad \widehat{\begin{pmatrix}&1\\1&\end{pmatrix}} = \begin{pmatrix}&&1\\&1&\\1&&\end{pmatrix},\]
\[\widehat{\begin{pmatrix}1&1\\&1\end{pmatrix}} = \begin{pmatrix}1&1&1\\&1&2\\&&1\end{pmatrix},
\qquad\quad\ \widehat{\begin{pmatrix}1&\\1&1\end{pmatrix}} = \begin{pmatrix}1&&\\2&1&\\1&1&1\end{pmatrix}.
\]
Then $\operatorname{GL}_2(\mathbb{Z}_3)$ acts on $M_{3\times2}(\mathbb{Z}_3)$ by $X\cdot M = \hat X M X^{-1}$ for any $X\in \operatorname{GL}_2(\mathbb{Z}_3)$ and $M\in M_{3\times2}(\mathbb{Z}_3)$.
Let $E=\left\{\left.\begin{pmatrix}a&\\b&a\\&b\end{pmatrix}\right|a,b\in\mathbb{Z}_3\right\}$ be a subspace of $M_{3\times2}(\mathbb{Z}_3)$. Then $\hat{X}E=EX$ for any $X\in \operatorname{GL}_2(\mathbb{Z}_3)$, or in other words, $E$ is a subrepresentation.
Consequently, $\operatorname{diag}(\hat X, (X^{-1})^T) W \operatorname{diag}({\hat X}^T, (X^{-1})^T) = W$.
\begin{remark} \label{rem-E-stabilizer} Clearly, let $Z$ be a square matrix of order 3. If $ZE\subseteq E$, then $Z=\lambda I_3$ for some $\lambda\in\mathbb{Z}_3$. Consequently, if $ZEX^{-1}\subseteq E$ for some $X\in \operatorname{GL}_2(\mathbb{Z}_3)$, then $Z=\lambda\hat X$ for some $\lambda\in \mathbb{Z}_3$.
\end{remark}
Define an equivalence relation ``$\sim$" on $\mathbb{Z}_3^3$ by $(\alpha,\beta,\gamma)\sim(\alpha',\beta',\gamma')$ if
$$\left(\begin{array}{ccc}0&\alpha'&\beta'\\-\alpha'&0&\gamma'\\-\beta'&-\gamma'&0\end{array}\right)=\hat{X}\left(\begin{array}{ccc}0&\alpha&\beta\\-\alpha&0&\gamma\\-\beta&-\gamma&0\end{array}\right)\hat{X}^T$$
for some $X\in\text{GL}_2(\mathbb{Z}_3)$.
By applying the action by $\begin{pmatrix}1&\\&-1\end{pmatrix}$, $\begin{pmatrix}&1\\1&\end{pmatrix}$, $\begin{pmatrix}1&1\\&1\end{pmatrix}$ and $\begin{pmatrix}1&\\1&1\end{pmatrix}$ iteratively, one shows easily that \begin{align*}(\alpha,\beta,\gamma)\sim&(-\alpha,\beta,-\gamma)\sim(-\gamma,-\beta,-\alpha)\sim(\gamma,-\beta,\alpha)\\
\sim&(\alpha-\beta+\gamma,\beta+\gamma,\gamma)\sim(\alpha+\beta+\gamma,\beta-\gamma,\gamma)\\
\sim&(\alpha,\alpha+\beta,\alpha-\beta+\gamma)\sim(\alpha,\alpha-\beta,\alpha+\beta+\gamma).
\end{align*}
If $\alpha\ne0$ or $\gamma\ne0$, then $(\alpha,\beta,\gamma)\sim(1,0,*)$, otherwise $(\alpha,\beta,\gamma)\sim(0,1,0)$ or $(0,0,0)$.
We note that $(1,0,-1)\sim(-1,0,1)\sim(0,1,1)\sim(0,-1,1)\sim(0,1,0)$.
Hence any $(\alpha, \beta, \gamma)$ is equivalent to one of $(1,0,0)$,$(1,0,1)$, $(0,1,0)$ and $(0,0,0)$.
We claim that these four classes are not congruent to each other.
Firstly, by comparing the rank, we know that $(0,0,0)$ is not equivalent to other 3 classes. By direct calculation,
\begin{align*}&\begin{pmatrix}a^2&ab&b^2\\2ac&ad+bc&2bd\\c^2&cd&d^2\end{pmatrix}
\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}
\begin{pmatrix}a^2&2ac&c^2\\ab&ad+bc&cd\\b^2&2bd&d^2\end{pmatrix}\\
=&\begin{pmatrix}0&a^2(ad-bc)&ac(ad-bc)\\ *&0&c^2(ad-bc)\\ *&*&0\end{pmatrix}
\end{align*}
cannot be equal to $\left(\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right)$ or $\left(\begin{array}{ccc}0&1&0\\-1&0&1\\0&-1&0\end{array}\right)$. Similarly,
\begin{align*}
&\begin{pmatrix}a^2&ab&b^2\\2ac&ad+bc&2bd\\c^2&cd&d^2\end{pmatrix}
\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix}
\begin{pmatrix}a^2&2ac&c^2\\ab&ad+bc&cd\\b^2&2bd&d^2\end{pmatrix}\\
=&\begin{pmatrix}0&2ab(ad-bc)&(ad+bc)(ad-bc)\\ *&0&2cd(ad-bc)\\ *&*&0\end{pmatrix}
\end{align*}
cannot be equal to $\begin{pmatrix}0&1&0\\-1&0&1\\0&-1&0\end{pmatrix}$, and the claim follows.
Therefore we may assume that $V$ is congruent to one of the following:
\begin{description}
\item[Case 1.1]
$\begin{pmatrix}&d&&a&d\alpha\\-d&&d&b&a+d\beta\\&-d&&d\gamma&b+d\delta
\\-a&-b&-d\gamma&&d\epsilon\\-d\alpha&-a-d\beta&-b-d\delta&-d\epsilon&\end{pmatrix},$
\item[Case 1.2]
$\begin{pmatrix}&d&&a&d\alpha\\-d&&&b&a+d\beta\\&&&d\gamma&b+d\delta
\\-a&-b&-d\gamma&&d\epsilon\\-d\alpha&-a-d\beta&-b-d\delta&-d\epsilon&\end{pmatrix},$
\item[Case 1.3]
$\begin{pmatrix}&&d&a&d\alpha\\&&&b&a+d\beta\\-d&&&\gamma&b+d\delta
\\-a&-b&-d\gamma&&d\epsilon\\-d\alpha&-a-d\beta&-b-d\delta&-d\epsilon&\end{pmatrix},$
\item[Case 1.4]
$\begin{pmatrix}&&&a&d\alpha\\&&&b&a+d\beta\\&&&d\gamma&b+d\delta
\\-a&-b&-d\gamma&&d\epsilon\\-d\alpha&-a-d\beta&-b-d\delta&-d\epsilon&\end{pmatrix}.$
\end{description}
We will find out all possible congruence classes case by case.
\noindent\textbf{Case 1.1}\quad
In this case, we can assume $\alpha=\beta=\gamma=\delta=0$. In fact, after taking the congruence transformation by $T_{52}(-\alpha)T_{43}(-\alpha)$ and $T_{42}(\gamma)T_{51}(\gamma)$, we can eliminate
$\alpha$ and $\gamma$. Now we may assume $\alpha=\gamma=0$, then we can take the congruence transformation by $T_{41}(-\delta)$ to eliminate $\gamma$ and by $T_{53}(-\beta)$ to eliminate $\beta$. Thus $V$ is congruent to
$$\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&d\epsilon\\&-a&-b&-d\epsilon&\end{pmatrix}$$
for some $\epsilon\in\{0,\pm1\}$. We denote by $V_{1.1.1}, V_{1.1.2}$ and $V_{1.1.3}$ the subspaces corresponding to
the cases $\epsilon=0$, $\epsilon=1$ and $\epsilon=-1$ respectively.
\noindent\textbf{Case 1.2}\quad
In this case, we can assume $\alpha=\beta=\delta=0$. By taking the congruence transformation by $T_{52}(-\alpha)T_{43}(-\alpha)$, $T_{51}(\beta)T_{42}(\beta)$ and $T_{41}(-\delta)$ we may eliminate $\alpha$, $\beta$ and $\gamma$ respectively.
If $\gamma\ne 0$, we may assume $\gamma=1$, otherwise we may take the congruence transformation by $D_3(-1)D_5(-1)$. Then \[V\sim\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&d&b\\-a&-b&-d&&d\epsilon\\&-a&-b&-d\epsilon&\end{pmatrix}
\overset{T_{53}(\epsilon)}\sim
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&-d&b\\-a&-b&d&&\\&-a&-b&&\end{pmatrix}
.\]
We denote this subspace by $V_{1.2.0}$.
If $\gamma=0$, then $V$ is equivalent to
$$V\sim\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&d\epsilon\\&-a&-b&-d\epsilon&\end{pmatrix}$$
for some $\epsilon\in\{0,\pm1\}$. The subspaces corresponding to $\epsilon=0$, $\epsilon= 1$ and $\epsilon=-1$ are denoted by $V_{1.2.1}$, $V_{1.2.2}$ and $V_{1.2.3}$ respectively.
\noindent\textbf{Case 1.3}\quad
In this case, we can assume $\alpha=\beta=\gamma=\delta$. In fact, we can take congruence transformation by $T_{53}(-\alpha)$, $T_{43}(\beta)T_{52}(\beta)$, $T_{41}(\gamma)$ and $T_{51}(\delta)T_{42}(\delta)$ to
eliminate $\alpha$, $\beta$, $\gamma$ and $\delta$ respectively. Thus in this case, $V$ is equivalent to \[\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&d\epsilon\\&-a&-b&-d\epsilon&\end{pmatrix}\]
for some $\epsilon\in\{0,1\}$. Note that the case $\epsilon=1$ is equivalent to the one $\epsilon=-1$ by taking congruence transformation by $D_2(-1)D_4(-1)$. We denote the subspaces corresponding to $\epsilon=0$ and $\epsilon=1$ by $V_{1.3.1}$ and $V_{1.3.2}$ respectively.
\begin{remark}\label{rem-eqeg-5case1} We observe that $V_{1.1.2}\sim V_{1.1.3}\sim V_{1.2.0}\sim V_{1.3.2}$. In fact, we have
\begin{align*}
&\begin{pmatrix}0&-1&0&1&1\\-1&1&-1&0&0\\0&-1&0&-1&-1\\0&0&1&1&-1\\1&0&1&-1&1
\end{pmatrix}
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&d\\&-a&-b&-d&
\end{pmatrix}
\begin{pmatrix}0&-1&0&0&1\\-1&1&-1&0&0\\0&-1&0&1&1\\1&0&-1&1&-1\\1&0&-1&-1&1
\end{pmatrix}\\
=&\begin{pmatrix}&&-a-b&b-a-d&\\&&&d-a+b&b-a-d\\a+b&&&&d-a+b\\a+d-b&a-b-d&&&-a-b\\&a+d-b&a-b-d&a+b&
\end{pmatrix}
=\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&d\\&-a&-b&-d&
\end{pmatrix};
\end{align*}
\begin{align*}
&\begin{pmatrix}1&0&-1&0&1\\1&0&0&0&0\\1&0&-1&0&-1\\0&1&0&-1&0\\0&1&0&1&0
\end{pmatrix}
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&-d\\&-a&-b&d&
\end{pmatrix}
\begin{pmatrix}1&1&1&0&0\\0&0&0&1&1\\-1&0&-1&0&0\\0&0&0&-1&1\\1&0&-1&0&0
\end{pmatrix}\\
=&\begin{pmatrix}&&-b&a+d&\\&&&d-a&a+d\\b&&&&d-a\\-a-d&a-d&&&-b\\&-a-d&a-d&b&
\end{pmatrix}
=\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&d\\&-a&-b&-d&
\end{pmatrix};
\end{align*}
\begin{align*}
&\begin{pmatrix}0&1&0&0&0\\0&0&0&0&1\\1&0&0&0&0\\0&0&0&1&0\\0&0&-1&0&0
\end{pmatrix}
\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&d\\&-a&-b&-d&
\end{pmatrix}
\begin{pmatrix}0&0&1&0&0\\1&0&0&0&0\\0&0&0&0&-1\\0&0&0&1&0\\0&1&0&0&0
\end{pmatrix}\\
=&\begin{pmatrix}&a&&b&\\-a&&&-d&b\\&&&a&-d\\-b&d&-a&&\\&-b&d&&
\end{pmatrix}
=\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&
\end{pmatrix}.
\end{align*}
\end{remark}
\noindent\textbf{Case 1.4}\quad
Note that if one of $\alpha$, $\beta$, $\gamma$, and $\delta$ is nonzero, then we can assume $\epsilon=0$.
In fact, when $\alpha$ (\emph{resp.} $\beta$, $\gamma$, $\delta$) is nonzero, we can take the congruence transformation by $T_{41}(-\alpha\epsilon)$ (\emph{resp.} $T_{42}(\beta\epsilon)T_{51}(\beta\epsilon)$, $T_{53}(\gamma\epsilon)$, $T_{43}(-\delta\epsilon)T_{52}(-\delta\epsilon)$) to eliminate $\epsilon$. Otherwise $\alpha=\beta=\gamma=\delta=0$, then we can make $\epsilon=1$ (taking the congruence transformation by $D_2(-1)D_4(-1)$ if necessary), and $V$ is equivalent to
$$\begin{pmatrix}&&&a&\\&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},$$ which is denoted by $V_{1.4.1}$.
Next we consider the case $\epsilon=0$. We define an equivalence relation on $\mathbb{Z}_3^4$ by $(\alpha,\beta,\gamma,\delta)\sim(\alpha',\beta',\gamma',\delta')$ if
$$\hat{X}\begin{pmatrix}a&d\alpha\\b&a+d\beta\\d\gamma&b+d\delta\end{pmatrix}X^{-1}=
\begin{pmatrix}a'&d'\alpha'\\b'&a'+d'\beta'\\d'\gamma'&b'+d'\delta'\end{pmatrix}$$
for some $X\in\text{GL}_2(\mathbb{Z}_3)$.
By applying the matrices $\begin{pmatrix}1&\\&-1\end{pmatrix}$, $\begin{pmatrix}&1\\1&\end{pmatrix}$, $\begin{pmatrix}1&1\\&1\end{pmatrix}$ and $\begin{pmatrix}1&\\1&1\end{pmatrix}$ iteratively,
we obtain that
\begin{align*}
(\alpha,\beta,\gamma,\delta)\sim&(-\alpha,-\beta,-\gamma,-\delta)
\sim(-\alpha,\beta,\gamma,-\delta)\sim(\gamma,-\delta,\alpha,-\beta)
\sim(-\gamma,-\delta,\alpha,\beta)\\
\sim&(\alpha+\beta-\gamma+\delta,\beta-\delta,\gamma,\delta)
\sim(\alpha-\beta+\gamma+\delta,\beta+\delta,\gamma,\delta)\\
\sim&(\alpha,\beta,\gamma-\alpha-\beta-\delta,\delta-\beta)
\sim(\alpha,\beta,\gamma+\alpha-\beta+\delta,\delta+\beta).
\end{align*}
If $\beta=\delta=0$, $(\alpha,0,\gamma,0)\sim(\alpha-\gamma,0,\gamma,0)\sim(\alpha,0,\gamma-\alpha,0)$. Since $(\alpha,0,\gamma,0)\sim(\gamma,0,\alpha,0)$, we can assume $\gamma\ne0$, then $(\alpha,0,\gamma,0)\sim(0,0,\gamma,0)\sim(0,0,1,0)$.
Otherwise, one of $\beta$ and $\delta$ is nonzero. Since $(\alpha,\beta,\gamma,\delta)\sim(\gamma,-\delta,\alpha,-\beta)$, we can assume $\beta\ne0$, then $(\alpha,\beta,\gamma,\delta)\sim(\alpha,\beta,\gamma',0)$.
Since $(\alpha,\beta,\gamma,0)\sim(\alpha+\beta-\gamma,\beta,\gamma,0)$, if $\beta\ne\gamma$, then $(\alpha,\beta,\gamma,0)\sim(0,\beta,\gamma,0)\sim(0,1,0,0)$ or $(0,1,-1,0)$.
If $\beta=\gamma$, since $(\alpha,\beta,\beta,0)\sim(-\alpha,\beta,\beta,0)$, we have $(\alpha,\beta,\beta,0)\sim(0,1,1,0)$ or $(1,1,1,0)$.
Then $(\alpha,\beta,\delta,\gamma)$ is equivalent to one of $(1,1,1,0)$, $(0,0,1,0)$, $(0,1,-1,0)$, $(0,1,0,0)$, $(0,1,1,0)$, and hence $V$ is
equivalent to one of the corresponding subspaces, say
\[\begin{pmatrix}&&&a&d\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\-d&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&d\\&&&&b\\-a&-b&&&\\&-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&\end{pmatrix}
\]
\[\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&-d&b\\-a&-b&d&&\\&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\&-a-d&-b&&\end{pmatrix}.
\]
We denote these subspaces by $V_{1.4.2}$, $V_{1.4.3}$, $V_{1.4.4}$, $V_{1.4.5}$ and $V_{1.4.6}$ respectively.
\subsection{Case 2: any two dimensional subspace of $V$ contains rank 2 matrices}\
In this subsection, we assume $V$ does not occur in Case 1, say any two dimensional subspace of $V$ has a element of rank 2. Then by Lemma \ref{lem-trivialradical}, we may assume $V$ contains $W_2$ as a subspace, and $V=\begin{pmatrix}&d\phi&d\theta&a&d\alpha\\-d\phi&&d\psi&b&d\beta\\
-d\theta&-d\psi&&d\gamma&a+d\delta\\-a&-b&-d\gamma&&d\epsilon\\
-d\alpha&-d\beta&-a-d\delta&-d\epsilon&\end{pmatrix}$ for some $\alpha, \beta, \gamma, \delta, \epsilon, \theta, \phi $ and $\psi\in \mathbb{Z}_3$. We write $V=[\alpha, \beta, \gamma, \delta, \epsilon, \theta, \phi, \psi]$ for short.
We recall a well-known result which follows easily from the Cauchy-Binet formula.
\begin{lemma}\label{lem-maxprincipalminor}
Let $X\in\operatorname{AS}_5(\mathbb{Z}_3)$ be an anti-symmetric matrix. Then $\operatorname{rank}(X)=4$ if and only if $X$ has a nonzero principal minor of order 4.
\end{lemma}
We may draw the following consequence.
\begin{lemma} Assume that any 2-dimensional subspace of $V = [\alpha, \beta, \gamma, \delta, \epsilon, \theta, \phi, \psi]$ contains a matrix of rank 2. Then $\alpha=\theta=0$.
\end{lemma}
\begin{proof} If $\theta\neq0$, then $V\cong [0, \beta', 0, 0, \epsilon', \theta, \phi', 0]$ for some $\beta', \epsilon'$ and $\phi'$. In fact, we can take the congruence transformation by $T_{53}(-\alpha\theta)T_{41}(\gamma\theta)T_{21}(-\psi\theta)$) to eliminate $\alpha, \gamma$ and $\theta$, and then
take the congruence transformation by $T_{43}(-\delta\theta)T_{51}(-\delta\theta)$ to eliminate $\delta$. Thus without loss of generality, we may assume $V=[0, \beta, 0, 0, \epsilon, 1, \phi, 0]$, which reads as
\[\begin{pmatrix}
&d\phi&d&a&\\-d\phi&&&b&d\beta\\-d&&&&a\\-a&-b&&&d\epsilon\\&-d\beta&-a&-d\epsilon
\end{pmatrix}.
\]
The Pfaffians of the principal minors of order $4$ are $ba,d^2\epsilon-a^2$, $d^2\phi\epsilon-ad\beta$, $d\phi a-d^2\beta$ and $bd$. Clearly, if $\epsilon\ne0$ or $\beta\ne0$, then 2-dimensional subspace $\{a=b\}$ contains no matrices of rank 2, and if $\epsilon=\beta=0$, then the subspace $\{b=d\}$ contains no matrices of rank 2.
If $\alpha\ne0$, then by taking the congruence transformation by $T_{35}(1)$ or $T_{35}(-1)$, we show that $V$ is congruent to some $[\alpha, \beta, \gamma', \delta', \epsilon, \theta', \phi, \psi']$ with $\theta'\ne 0$, hence $V$ contains some 2-dimensional subspace which contains no matrices of rank 2.
\end{proof}
Now we may assume $\alpha=\theta=0$ thanks to the lemma. By more detailed analysis, we have the following result.
\begin{proposition}\label{prop-3-asym5.1} Assume any two dimensional subspace of $V$ contains a rank two element. Then $V$ is equivalent to one of
$V_{2.1}=[0,0,0,0,0,0,1,0]$, $V_{2.2}=[0,0,0,0,0,0,0,1]$, $V_{2.3}=[0,0,0,1,0,0,0,0]$ and $V_{2.4}=[0,0,0,0,1,0,0,0]$, say
\[V_{2.1}=\begin{pmatrix} &d&&a&\\-d&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&
\end{pmatrix},\quad
V_{2.2}=\begin{pmatrix} &&&a&\\&&d&b&\\&-d&&&a\\-a&-b&&&\\&&-a&&
\end{pmatrix},
\]
\[V_{2.3}=\begin{pmatrix} &&&a&\\&&&b&\\&&&&d\\-a&-b&&&\\&&-d&&
\end{pmatrix}, \
V_{2.4}=\begin{pmatrix} &&&a&\\&&&b&\\&&&&a\\-a&-b&&&d\\&&-a&-d&
\end{pmatrix}.
\]
\end{proposition}
\begin{proof}
By the above lemma, we may assume $\alpha=\theta=0$. We have the following cases.
\noindent\textbf{Case 2.1: $\phi\ne0$.}\quad
By taking congruence actions by $T_{51}(\phi\beta)T_{43}(\phi\beta)$, $T_{45}(-\phi\psi)T_{31}(\phi\psi)$ and $T_{42}(\phi\delta)$ one after another, we show that
$V$ is equivalent to some $[0, 0, \gamma', 0, \epsilon', 0, \phi, 0]$. Thus without loss of
generality, we may assume $\phi=1$, $\beta=\delta=\psi=0$, and $V$ reads as
\[\begin{pmatrix}&d&&a&\\-d&&&b&\\&&&d\gamma&a\\-a&-b&-d\gamma&&d\epsilon\\&&-a&-d\epsilon&\end{pmatrix}.\]
The Pfaffians of the principal blocks of order 4 are $ab$, $a^2$, $d^2\epsilon$, $ad$, and $d^2\gamma$.
If $\epsilon\ne 0$ or $\gamma\ne0$, then the two dimensional subspace $\{b=0\}$ has no elements of rank 2, which
contradicts the assumption on $V$. Therefore, $V$ is congruent to $V_{2.1}=[0,0,0,0,0,0,1,0]$.
Note that the Pfaffians of the principal minors of $V_{2.1}$ of order 4 are $ab$, $a^2$, 0, $ad$, and $0$. Then clearly any nonzero element in the two dimensional subspace $\{a=0\}$ has rank $2$. Thus any two dimensional subspace of $V_{2.1}$ has an element of rank 2, since it will have nonzero intersection with the space $\{a=0\}$.
\noindent\textbf{Case 2.2: $\phi=0$, $\psi\ne0$.}\quad
In this case, we can take the congruence transformation by $T_{52}(-\psi\beta)T_{42}(\psi\gamma)$ to show that $V$ is congruent to
some $[0, 0, 0, \delta', \epsilon', 0, 0, \psi]$. Thus we may assume $\psi=1$ and $\phi=\beta=\gamma=0$, and $V$ reads as
\[\begin{pmatrix} &&&a&\\&&d&b&\\&-d&&&a+d\delta\\-a&-b&&&d\epsilon\\&&-a-d\delta&-d\epsilon&
\end{pmatrix}.
\]
The Pfaffians of the principal minors of order 4 are $d^2\epsilon-b(a+d\delta)$, $a(a+d\delta)$, $0$, $0$ and $ad$.
If $\epsilon\ne0$, then any matrix in the 2-dimensional subspace $\{b=0\}$ has rank 4;
if $\epsilon=0$ and $\delta\ne0$, then any matrix in the 2-dimensional subspace $\{b=d\}$ has rank 4.
It forces that $\epsilon=\delta=0$, and hence $V$ is congruent to $V_{2.2}=[0,0,0,0,0,0,0,1]$.
Now the Pfaffians of the principal minors of $V_{2.2}$ of order 4 are $-ba$, $a^2$, $0$, $0$ and $ad$. Then any nonzero elements of the subspace $\{a=0\}$ of $V_{2.2}$ has rank 2, and hence any two dimensional subspace of $V_{2.2}$ contains an element of rank 2.
\noindent\textbf{Case 2.3: $\phi=\psi=0$.} We may assume $\beta=0$. In fact, if $\beta\ne0$, then we take the congruence transformation by $T_{25}(1)$ to make $\psi$ nonzero, and it reduces to the Case 2.2. Now $V=[0, 0, \gamma, \delta, \epsilon, 0, 0, 0]$ reads as
\[V=\begin{pmatrix}&&&a&\\&&&b&\\
&&&d\gamma&a+d\delta\\-a&-b&-d\gamma&&d\epsilon\\
&&-a-d\delta&-d\epsilon&\end{pmatrix}.\]
Now we have two subcases.
(i) $\delta\ne0$. After taking the congruence transformations by $T_{31}(-\gamma\delta)T_{45}(-\gamma\delta)$ and $T_{43}(-\epsilon\delta)T_{51}(-\epsilon\delta)$, we show that $V$ is equivalent to
$[0, 0, 0, \delta, 0, 0, 0, 0]$, and hence to $V_{2.3}=[0, 0, 0, 1, 0, 0, 0, 0]$.
The Pfaffians of the principal minors of $[0, 0, 0, 1, 0, 0, 0, 0]$ of order 4 are $b(a+d)$, $a(a+d), 0, 0, 0$. Thus any nonzero element in the subspace $\{a+d=0\}$ of $[0, 0, 0, 1, 0, 0, 0, 0]$ has rank 2, and hence any two dimensional subspace contains an element of rank 2.
(ii) $\delta=0$. Write $(\gamma, \epsilon)= [0, 0, \gamma, 0, \epsilon, 0, 0, 0]$ for short. Then it is easy to show that $(\gamma, \epsilon)\overset{T_{35}(1)}\sim (\gamma-\epsilon, \epsilon)$ and $(\gamma, \epsilon)\overset{T_{53}(1)}\sim(\gamma, \epsilon-\gamma)$. Applying these two equivalences iteratively, we show that $V$ is equivalent to $V_{2.4}=(0, 1)= [0, 0, 0, 0, 1, 0, 0, 0]$.
The Pfaffians of the principal minors of $V_{2.4}$ of order 4 are $ba$, $a^2, 0, 0, 0$. Then any nonzero element in the subspace $\{a=0\}$ of $V_{2.4}$ has rank 2, and hence any two dimensional subspace contains an element of rank 2.
\end{proof}
We denote the subspaces in the above proposition by $V_{2.1}, V_{2.2}, V_{2,3}$ and $V_{2.4}$ respectively.
\begin{proposition}\label{prop-3-asym5.2} $V_{2.1}, V_{2.2}, V_{2,3}$ and $V_{2.4}$ are not congruent to each other.
\end{proposition}
\begin{proof} For any subspace $V\subseteq \operatorname{AS}_5(\mathbb{Z}_3)$, we set $N(V)=\#\{X\in V\mid \operatorname{rank}(X)\ne 4\}$. Then $N(V)$ is invariant under congruence equivalence. By Lemma \ref{lem-maxprincipalminor}, $X$ has rank 2 if and only if all its principal minor of order 4 is 0. Easy calculation shows that
$N(V_{2.1})=N(V_{2.2})=9$, $N(V_{2.3})=11$, $N(V_{2.4})=9$. Thus $V_{2.3}$ is not congruent to other three.
For short we write $\perp_1=\perp_{V_{2.1}}$, $\perp_2=\perp_{V_{2.2}}$ and $\perp_4=\perp_{V_{2.4}}$ temporarily.
Set \[\mathbf U= (\mathbb{Z}_3, \mathbb{Z}_3, 0, 0, 0)= \{(u_1, u_2, 0, 0, 0)\in\mathbb{Z}_3^5\mid u_1, u_2\in \mathbb{Z}_3\}, \mathbf V= (\mathbb{Z}_3, \mathbb{Z}_3, 0, 0, \mathbb{Z}_3).\] It is easy to show that $\mathbf U^{\perp_4}=(\mathbb{Z}_3,\mathbb{Z}_3,0,0,\mathbb{Z}_3)$ has dimension $4$, and
$\mathbf V^{\perp_2}=(\mathbb{Z}_3,\mathbb{Z}_3,0,0,\mathbb{Z}_3)$ has dimension 3. Moreover, for any $\mathbf W\subseteq \mathbb{Z}_3^5$, if $\dim(W)=2$, then $\dim(W^{\perp_1})\le 3$ and $\dim(W^{\perp_2})\le 3$; if $\dim(W)=3$, then $\dim(W^{\perp_1})<3$.
Then the assertion follows from Remark \ref{rem-complement-inv}.
\end{proof}
\subsection{The classification result.}
We have shown that any three dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$ is equivalent to one as listed in Section 10.2 and 10.3. Moreover, the subspaces $V_{2.1}$, $V_{2.2}$, $V_{2.3}$ and $V_{2.4}$ as in Proposition \ref{prop-3-asym5.1} are pairwise noncongruent, and they are not congruent to the ones given in Section 10.2. Now to complete our classification, it suffices to show that the ones in Case 1.1 through 1.4 are pairwise noncongruent, except the ones as listed in Remark \ref{rem-eqeg-5case1}.
\begin{proposition}\label{prop-noneq-5case1} The 12 subspaces $V_{1.1.1}$, $V_{1.1.2}$, $V_{1.2.1}$, $V_{1.2.2}$, $V_{1.2.3}$, $V_{1.3.1}$, $V_{1.4.1}$, $V_{1.4.2}$, $V_{1.4.3}$, $V_{1.4.4}$, $V_{1.4.5}$, $V_{1.4.6}$ are not congruent to each other.
\end{proposition}
\begin{proof} First the Pfaffians of the principal minors of order 4 and the number of elements with rank strictly less than 4 are listed as follows.
\begin{align*}
&V_{1.1.1}: -b^2, ab, -a^2, bd, ad; &N= 3.\\
&V_{1.1.2}: d^2-b^2, ab, d^2-a^2, bd, ad; &N= 1.\\
&V_{1.2.1}: -b^2, -ab, - a^2, bd, 0; &N= 3.\\
&V_{1.2.2}: -b^2, -ab, d^2- a^2, bd, 0; &N= 5.\\
&V_{1.2.3}: -b^2, -ab, -d^2- a^2, bd, 0; &N= 1.\\
&V_{1.3.1}: -b^2, -ab, -a^2, -da, -bd; &N= 3.\\
&V_{1.4.1}: -b^2, -ab, -a^2, 0, 0; &N=3.\\
&V_{1.4.2}: d(a+d)-b^2, d^2-ad, bd-a(a+d), 0, 0; &N= 1.\\
&V_{1.4.3}: -b^2, -ab, -ad, 0, 0 ; &N=5.\\
&V_{1.4.4}: ad-b^2, -ab, -a^2, 0, 0; &N= 3.\\
&V_{1.4.5}: -d(a+d)-b^2, -ab, -a(a+d), 0, 0; &N= 3.\\
&V_{1.4.6}: d(a+d)-b^2, -ab, -a(a+d), 0, 0; &N= 7.
\end{align*}
We listed the spaces according to $N(V)$ as follows:
$N=1, V= V_{1.1.2}\sim V_{1.2.0}, V_{1.2.3}, V_{1.4.2}:$
\begin{align*}
&V_{1.1.2}=\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix}
\sim \begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&
\end{pmatrix}= V_{1.2.0}, \\
& V_{1.2.3}= \begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&-d\\&-a&-b&d&\end{pmatrix},
V_{1.4.2} \begin{pmatrix}&&&a&d\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\-d&-a-d&-b&&\end{pmatrix}.
\end{align*}
$N=3, V= V_{1.1.1}, V_{1.2.1}, V_{1.3.1}, V_{1.4.1}, V_{1.4.4}, V_{1.4.5}:$
\begin{align*}
&V_{1.1.1}= \begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
&V_{1.2.1}=\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},\\
&V_{1.3.1}=\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
&V_{1.4.1}=\begin{pmatrix}&&&a&\\&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},\\
&V_{1.4.4}=\begin{pmatrix}&&&a&\\&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&\end{pmatrix},
&V_{1.4.5}=\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&-d&b\\-a&-b&d&&\\&-a-d&-b&&\end{pmatrix}.
\end{align*}
$N=5, V= V_{1.2.2}, V_{1.4.3}:$
\[V_{1.2.2}=\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},\qquad \qquad
V_{1.4.3}=\begin{pmatrix}&&&a&\\&&&b&d\\&&&&b\\-a&-b&&&\\&-d&-b&&\end{pmatrix}.
\]
$N=7, V= V_{1.4.6}:$
\[V_{1.4.6}=\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\&-a-d&-b&&\end{pmatrix}.\]
Clearly subspaces with different number of rank 4 elements will not be congruent. Thus the subspace $V_{1.4.6}$ is not congruent to others. Now we discuss on $N$ case by case.
\noindent{\bf Case N=1:}
$V_{1.4.2}$ is not congruent to $V_{1.1.2}$ and $V_{1.2.3}$, since $\mathbb{Z}_3^5 = (\mathbb{Z}_3,\mathbb{Z}_3,\mathbb{Z}_3, 0, 0)\oplus (0,0,0,\mathbb{Z}_3,\mathbb{Z}_3)$ is a direct sum of a 3-dimensional isotropy subspace and a 2-dimensional isotropy subspace with respect to $V_{1.4.2}$, while for $V_{1.1.2}$ and $V_{1.2.3}$ there is no such a decomposition.
Moreover, $\mathbb{Z}_3^5$ has a pair of 2-dimensional isotropy subspace with respect to $V_{1.2.0}$, say $(0,\mathbb{Z}_3,\mathbb{Z}_3, 0, 0)$ and $(0,0,0,\mathbb{Z}_3,\mathbb{Z}_3)$, whose intersection is zero, while for $V_{1.2.3}$, there are no such isotopy subspaces. Hence $V_{1.1.2}$ is not congruent to $V_{1.2.3}$.
\noindent{\bf Case N=5:}
$\mathbb{Z}_3^5 = (\mathbb{Z}_3,\mathbb{Z}_3,\mathbb{Z}_3, 0, 0)\oplus (0,0,0,\mathbb{Z}_3,\mathbb{Z}_3)$ is a direct sum of isotropy subspaces with respect to $V_{1.4.3}$, while for $V_{1.2.2}$ there is no such a decomposition. Thus $V_{1.2.2}$ and
$V_{1.4.3}$ are not congruent.
\noindent{\bf Case N=3:}
Similar argument as above shows that $V_{1.4.4}$ and $V_{1.4.5}$ are not congruent to $V_{1.1.1}$, $V_{1.2.1}$, $V_{1.3.1}$ and $V_{1.4.1}$. Moreover,
\[V_{1.4.5}\overset{T_{31}(-1)}\sim
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&-(a+d)&b\\-a&-b&a+d&&\\&-a-d&-b&&\end{pmatrix}
=\begin{pmatrix}&&&a&\\&&&b&d\\&&&-d&b\\-a&-b&d&&\\&-d&-b&&\end{pmatrix},\]
and the latter contains a subspace $W=\begin{pmatrix}&&&&\\&&&b&d\\&&&-d&b\\&-b&d&&\\&-d&-b&&\end{pmatrix}$, such that $\mathrm{rad}(W)\ne 0$ and every nonzero element of $W$ has rank 4. There is no difficulty to show that $V_{1.4.4}$ does not have such a two dimensional subspace, therefore $V_{1.4.4}$ is not congruent to $V_{1.4.5}$.
Note that $V_{1.4.1}$ has a three dimensional isotropy subspace $(\mathbb{Z}_3, \mathbb{Z}_3, \mathbb{Z}_3, 0, 0)$, while $V_{1.1.1}$, $V_{1.2.1}$ and $V_{1.3.1}$ do not have, thus $V_{1.4.1}$ is not congruent to the others.
$V_{1.2.1}$ has a two dimensional subspace $\{b=0\}$ whose radical is $(0,0,\mathbb{Z}_3,0,0)\ne 0$, while any two dimensional subspace of $V_{1.1.1}$ or $V_{1.3.1}$ has trivial radical, thus $V_{1.2.1}$ is neither congruent to $V_{1.1.1}$ nor to $V_{1.3.1}$.
Now we are left to prove that $V_{1.1.1}$ and $V_{1.3.1}$ are not congruent. Otherwise, if $V_{1.3.1}$ is congruent to $V_{1.1.1}$, then there exists some $P\in \operatorname{GL}_5(\mathbb{Z}_3)$, such that $PXP^T\in V_{1.1.1}$ for any $X\in V_{1.3.1}$.
We write
\[D=\begin{pmatrix}&&1\\&&\\-1&&\end{pmatrix}, D'=\begin{pmatrix}&1&\\-1&&1\\&-1&\end{pmatrix},
E(a,b)=\begin{pmatrix}a&\\b&a\\&b\end{pmatrix}\]
for $a, b\in \mathbb{Z}_3$. Then
$P\begin{pmatrix}dD &E(a,b)\\ -E(a,b)^T&\end{pmatrix}P^T =\begin{pmatrix}d'D'& E(a',b')\\ -E(a',b')^T&\end{pmatrix}$, and $(a,b,d)=(a',b',d')Q$ for some $Q\in\operatorname{GL}_3(\mathbb{Z}_3)$. Write $P$ as a block matrix, say $P=\begin{pmatrix}P_1& P_2\\ P_3&P_4 \end{pmatrix}$, where $P_1, P_2, P_3$ and $P_4$ are $3\times 3$, $3\times 2$, $2\times 3$ and $2\times 2$ matrices respectively.
By comparing the rank, $P\begin{pmatrix}D & \\ &\end{pmatrix}P^T =\begin{pmatrix}d'D'& \\ &\end{pmatrix}$ for some $d'$. Then $\begin{pmatrix}P_1\\P_3\end{pmatrix}DP_3^T=0$, hence $DP_3^T=0$, $P_3=\left(\begin{array}{ccc}0&r_1&0\\0&r_2&0\end{array}\right)$.
Let $P_4=\left(\begin{array}{cc}s_{11}&s_{12}\\s_{21}&s_{22}\end{array}\right)$, then from
$$\begin{pmatrix}P_1&P_2\\P_3&P_4\end{pmatrix}\begin{pmatrix}dD&E(a,b)\\-E(a,b)^T&\end{pmatrix}
\begin{pmatrix}P_1^T&P_3^T\\P_2^T&P_4^T\end{pmatrix}=\begin{pmatrix}d'D'&E(a',b')\\-E(a',b')^T&\end{pmatrix},$$
we get $P_3E(a,b)P_4^T=P_4E(a,b)^TP_3^T$ for any $a, b$.
Thus $r_1s_{21}=r_2s_{11}$, $r_1s_{22}=r_2s_{12}$ and
$(P_3,P_4)=\left(\begin{array}{ccccc}0&r_1&0&s_{11}&s_{12}\\0&r_2&0&s_{21}&s_{22}\end{array}\right)$.
If $(r_1,r_2)\ne(0,0)$, then $\operatorname{rank}(P_3,P_4)=1$, which leads to a contradiction.
Therefore $r_1=r_2=0$, $P_3=0$. Hence $P_1(dD)P_1^T=d'D'$ and $P_1E(a,b)P_4^T=E'(a',b')$. By Remark \ref{rem-E-stabilizer}, $D'$ is a multiple of some $\hat X$, and $P_1DP_1^T= \hat X D \hat X^T$, contradicts to the fact shown in Section 10.2 that $D$ and $D'$ are not equivalent under the congruence action by $\hat X$'s.
\end{proof}
In summary, we have the following classification.
\begin{theorem}\label{thm-3of5}
There are 16 congruence classes of 3-dimensional subspaces of $\operatorname{AS}_5(\mathbb{Z}_3)$ with trivial radical:
\[
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&d&b&a\\&-d&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&d&&a&\\-d&&&b&a\\&&&&b\\-a&-b&&&-d\\&-a&-b&d&\end{pmatrix},
\begin{pmatrix}&&d&a&\\&&&b&a\\-d&&&&b\\-a&-b&&&\\&-a&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&&&a&\\&&&b&a\\&&&&b\\-a&-b&&&d\\&-a&-b&-d&\end{pmatrix},
\begin{pmatrix}&&&a&d\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\-d&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&d\\&&&&b\\-a&-b&&&\\&-d&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix}&&&a&\\&&&b&a\\&&&d&b\\-a&-b&-d&&\\&-a&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&-d&b\\-a&-b&d&&\\&-a-d&-b&&\end{pmatrix},
\begin{pmatrix}&&&a&\\&&&b&a+d\\&&&d&b\\-a&-b&-d&&\\&-a-d&-b&&\end{pmatrix},
\]
\[
\begin{pmatrix} &d&&a&\\-d&&&b&\\&&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\begin{pmatrix} &&&a&\\&&d&b&\\&-d&&&a\\-a&-b&&&\\&&-a&&\end{pmatrix},
\]
\[
\begin{pmatrix} &&&a&\\&&&b&\\&&&&d\\-a&-b&&&\\&&-d&&\end{pmatrix},
\begin{pmatrix} &&&a&\\&&&b&\\&&&&a\\-a&-b&&&d\\&&-a&-d&\end{pmatrix},
\]
and 6 congruence classes of 3-dimensional subspaces with nontrivial radical:
$$\begin{pmatrix}&c&a&&\\-c&&b&&\\-a&-b&&&\\&&&0&\\&&&&0\end{pmatrix},
\begin{pmatrix}&&a&&\\&&b&&\\-a&-b&&c&\\&&-c&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&&\\-c&&b&&\\-a&-b&&c&\\&&-c&&\\&&&&0\end{pmatrix},$$
$$\begin{pmatrix}&&a&c&\\&&b&&\\-a&-b&&&\\-c&&&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&&\\-c&&&b&\\-a&&&c&\\&-b&-c&&\\&&&&0\end{pmatrix},
\begin{pmatrix}&c&a&b&\\-c&&&a&\\-a&&&-c&\\-b&-a&c&&\\&&&&0\end{pmatrix}.$$
\end{theorem}
\vskip20pt
\subsection*{Acknowledgments}
Z. Wan is supported by the Shuimu Tsinghua Scholar Program. Y. Ye is partially supported by the Natural
Science Foundation of China (Grant No.11971449).
C. Zhang is supported by the Fundamental Research Funds for the Central Universities (No. 2020QN20).
|
2,877,628,089,505 | arxiv | \section{Introduction}\label{intro}
Alongside with the realization that the quark gluon plasma (QGP) created at RHIC should be described as a strongly coupled liquid rather than a gas of weakly interacting quasiparticles \cite{Tannenbaum}, holographic methods have become a standard tool in making qualitative --- and in a few cases even quantitative --- predictions for heavy ion physics \cite{CasalderreySolana:2011us}. Perhaps the best known example of this is the famous conjecture of a lower limit for the shear viscosity to entropy ratio, $\eta/s\geq 1/(4\pi)$ \cite{Kovtun:2004de}, which the QGP appears to almost saturate (see also \cite{Romatschke:2009im,Kovtun:2011np,Rebhan:2011vd}). In addition, holography has been used to address many complicated dynamical problems out of the reach of conventional field theory methods, such as strong coupling thermalization and particle production (see e.g.~\cite{Chesler:2010bi,Steineder:2012si} and references therein). Within thermal equilibrium, it has finally been observed that the behavior of many bulk
thermodynamic quantities near the critical temperature of the deconfinement transition, measurable with lattice methods, can be reproduced to a very good accuracy using holographic models with broken supersymmetry and conformal invariance \cite{kiri2,kiri3,Alanen:2009xs}.
On the field theory side, a major obstacle in the quantitative description of the thermalizing plasma is the inapplicability of lattice methods to real time physics --- or even to the determination of transport coefficients. In the latter case, discussed extensively e.g.~in \cite{Meyer:2011gj}, some progress has recently been made in combining lattice measurements of Euclidean correlators with perturbative results for the corresponding spectral functions \cite{Burnier:2011jq,Burnier:2012ts}. Despite this, important hydrodynamic parameters such as the shear and bulk viscosities are still outside the realm of accurate first principles calculations. Recalling that these quantities are readily available in the strongly coupled limit of a class of large-$N_c$ field theories via the AdS/CFT conjecture, it is not surprising that quite some attention has lately turned towards a quantitative comparison of lattice, perturbative and gauge/gravity predictions for various (mostly Euclidean) correlation functions. Recent
lattice studies of energy momentum tensor correlators include at least \cite{Huebner:2008as,Iqbal:2009xz,Meyer:2010ii,meyershear}, while related perturbative work has been performed in
\cite{Laine:2010fe,Laine:2010tc,Laine:2011xm,Schroder:2011ht,Zhu:2012be} and holographic calculations in
\cite{teaney,Gubser:2008sz,Gursoy:2009kk,Kajantie:2010nx,Kajantie:2011nx,Springer:2010mf,Springer:2010mw}. Finally, closely related studies of sum rules that the associated spectral functions must obey can be found e.g.~from \cite{Romatschke:2009ng,Meyer:2010gu}, while an analytic study of the UV limit of different correlators was performed in \cite{CaronHuot:2009ns}.
A particularly interesting comparison of lattice, weak coupling and holographic correlators was reported in \cite{Iqbal:2009xz}. There, it was found that lattice data for various Euclidean Green's functions just above the deconfinement temperature of SU(3) Yang-Mills plasma is better described by infinitely strongly coupled ${\mathcal N}=4$ Super Yang-Mills (SYM) theory than by a leading order perturbative calculation in the original theory. Even though there are indications that the inclusion of further perturbative orders acts in the direction of closing the gap between the weak coupling and lattice results \cite{Laine:2010tc,Laine:2011xm}, it is equally worthwhile to attempt to improve the description of the system on the strong coupling side. To this end, in \cite{Kajantie:2011nx} we addressed the determination of the shear channel spectral function and the corresponding imaginary time and coordinate space correlators in the so-called Improved Holographic QCD (IHQCD) model, which exhibits a dynamical
dilaton field that has the effect of breaking conformal invariance and supersymmetry \cite{kiri2,kiri3}. The study revealed important quantitative effects originating from the loss of conformal invariance near $T_c$, and in addition highlighted the importance of performing similar computations in the more complicated bulk channel, where conformal theories (such as ${\mathcal N}=4$ SYM) lead to vanishing correlation functions.
In the paper at hand, our aim is to continue and extend the treatment of \cite{Kajantie:2011nx} by performing a detailed analysis of the bulk channel of the IHQCD model, concentrating in particular on the spectral function at vanishing external three-momentum and the associated imaginary time correlator. The latter quantity is of special interest to us due to the recent emergence of the corresponding perturbative and lattice results \cite{Meyer:2010ii,Laine:2011xm}, to which we can compare our holographic predictions. In addition to this, we will briefly revisit the shear channel, where a Next-to-Leading Order (NLO) result for the spectral function has been determined since the appearance of \cite{Kajantie:2011nx}, motivating a reanalysis of the IHQCD calculation.
Our paper is organized as follows. In section 2, we review the setup both on the field theory and gravity sides, recalling the most important aspects of the IHQCD model. In section 3, we next write down the fluctuation equations, from which both the shear and bulk spectral functions are determined, and in addition explain the most important steps of the holographic calculation. Section 4 then reviews the existing perturbative results for the quantities of our interest, while sections 5-6 contain our holographic results in the two channels. A more comprehensive discussion of the results is finally left to section 7, where we also draw our conclusions.
Our notation follows closely that explained in section 1 of \cite{Kajantie:2011nx}. In particular, with the exception of the introductory section 2, we will in the following set the AdS radius ${\mathcal L}=1$.
\section{Setup}\label{setup}
\subsection{Field theory}
We work within pure SU($N_c$) Yang-Mills theory at a nonzero temperature $T$, defined by the Euclidean Lagrangian
\ba
S_\mathrm{E} &=& \int_{0}^{\beta} \! \mathrm{d} \tau \int \! {\rm d}^{3}\vec{x}
\, \frac{1}{4} F^a_{\mu\nu} F^a_{\mu\nu} \,,\quad F^a_{\mu\nu} \,\equiv\, \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g_\rmii{B} f^{abc} A^b_\mu A^c_\nu\, ,
\ea
with $\beta\equiv 1/T$. The energy momentum tensor of the theory takes the form
\begin{equation}\la{eq:T}
T_{\mu\nu}(x) = \frac{1}{4} \delta_{\mu\nu} F^a_{\alpha\beta} F^a_{\alpha\beta} -F^a_{\mu\alpha} F^a_{\nu\alpha} =\theta_{\mu\nu}(x)+\frac{1}{4}\delta_{\mu\nu}\theta(x)\, ,
\end{equation}
where we have in the latter stage separated the traceless part $\theta_{\mu\nu}$ and the anomalous trace
\begin{equation}
\theta(x)\equiv T_{\mu\mu}=\frac{\beta(g)}{2g}F_{\mu\nu}^aF_{\mu\nu}^a\, ,
\end{equation}
in which $\beta(g)$ denotes the beta function of the theory.
In this paper, we are interested in correlation functions of the shear and bulk operators $T_{12}$ and $\theta$ of the above theory, of which the latter we can furthermore replace by the simpler quantity $T_{ii}$, as correlators of $T_{00}$ are known to reduce to contact terms (see e.g.~\cite{Meyer:2010ii}). This implies that the retarded correlators we study obtain the forms
\ba
G_s^R(\omega,\mathbf{k}=0)&=&-i\int\! {\rm d}^4x\, e^{i \omega t} \theta (t) \langle[
T_{12}(t,\vec{x}),T_{12}(0,0) ]\rangle \label{Gsdef}\, \\
G_b^R(\omega,\mathbf{k}=0)&=&-i\int\! {\rm d}^4x\, e^{i \omega t} \theta (t) \langle[\frac{1}{3}
T_{ii}(t,\vec{x}),\frac{1}{3}T_{jj}(0,0) ]\rangle \label{Gbdef} \, ,
\ea
while the corresponding (zero three-momentum) spectral functions read
\ba
\rho_{s,b} (\omega,T)&=&\mathop{\mbox{Im}} G_{s,b}^R(\omega,\mathbf{k}=0)\, .
\ea
The relation between these functions and the associated transport coefficients (the shear and bulk viscosities) is finally given by
\ba
\eta&=&\lim_{\omega\to 0}\frac{\rho_s(\omega,T)}{\omega}\, ,\label{eta} \\
\zeta&=&\lim_{\omega\to 0}\frac{\rho_b(\omega,T)}{\omega}\, .\label{zeta}
\ea
It is also good to recall that in conformal theories, in which the beta function is zero, we have $T_{\mu\mu}=0$, and thus vanishing bulk correlators and viscosity.
\subsection{Dual gravity system} \label{dual}
In the IHQCD model of \cite{kiri2,kiri3}, the gravitational system involves a gravity+dilaton type action
\begin{equation}
S=\frac{1}{16 \pi G_5}\int {\rm d}^5x\sqrt{-g} \left[R -\frac{4}{3}(\partial\phi)^2 +V(\phi) \right]\, , \label{action}
\end{equation}
while the background metric takes the generic form
\begin{equation}
ds^2=b^2(z)\left[-f(z)dt^2+d{\bf x}^2 +{dz^2\over f(z)}\right]\, , \label{metric}
\end{equation}
where the radial coordinate $z$ is chosen so that the boundary is located at $z=0$. The functions $\phi(z)$, $f(z)$ and $b(z)$ appearing here are determined from the Einstein equations
\begin{eqnarray}
\dot{W}&=& 4 b W^2 -\frac{1}{f}(W\dot{f} +\frac{1}{3}b V ), \label{Einstein1}\\
\dot{b} &=&-b^2 W\,, \label{Einstein2}\\
\dot{\lambda} &=&\frac{3}{2}\lambda \sqrt{b \dot{W}}\,,\label{Einstein3}\\
\ddot{f} &=& 3\dot{f}b W\,,\label{Einstein4}
\end{eqnarray}
in which the dot denotes a derivative with respect to $z$, and we have defined $\lambda (z)= e^{\phi(z)}$. As the notation suggests, this function is found to be dual to the 't Hooft coupling on the field theory side, $\lambda_c\equiv g^2 N_c$, while
\begin{equation}
\beta(\lambda)\equiv\frac{\dot{\lambda}}{\dot{b}/b}
\la{betafn}
\end{equation}
is related to the field theory beta function.
In the vicinity of the boundary, the metric function $b(z)$ is required to satisfy
\begin{equation}
b(z)\stackrel[z\to 0]{}{\rightarrow} \frac{\mathcal{L}}{z}\, ,
\end{equation}
where $\mathcal{L}$ is the curvature radius of AdS space. For the other functions, the UV limits are obtained from the running of the field theory coupling, while the IR behaviors are determined by requiring that the model satisfy the confinement criterion of a linear glueball spectrum $m^2\sim \mathrm{integer}$ \cite{kiri2} (see also appendices A and B of \cite{Kajantie:2011nx}). The dilaton potential we use follows the choice of \cite{Jarvinen:2011qe}, reading
\begin{equation}
V(\lambda)=\frac{12}{\mathcal{L}^2}\left[
1+\frac{88}{27}\lambda + \frac{4619}{729}\lambda^2\frac{ \sqrt{1+\ln(1+\lambda)}}{(1+\lambda)^{2/3}}
\right]\, . \label{V}
\end{equation}
It is constructed to reproduce the small $\lambda$ expansion
\begin{equation}
V(\lambda)=\frac{12}{\mathcal{L}^2}\left[
1+\frac{88}{27}\lambda + \frac{4619}{729}\lambda^2+{\mathcal O}(\lambda^3)
\right]\, ,
\end{equation}
where the coefficients in front of $\lambda$ and $\lambda^2$ are determined by matching the holographic beta function of eq.~\nr{betafn} to the perturbative 2-loop result with the identification $\lambda=\lambda_c/(8\pi^2)$. The $\lambda^2$ term in eq.~(\ref{V}) is finally multiplied by the factor
\begin{equation}
\frac{\sqrt{1+\ln(1+\lambda)}}{(1+\lambda)^{2/3}}
\end{equation}
to ensure that the model satisfies the confinement criterion.
With the gravity system specified, we can now use eqs.~(\ref{Einstein1})-(\ref{Einstein4}) to solve for those bulk field configurations that exhibit a horizon at $z=z_h$ (i.e.~satisfy $f(z_h)=0$) and thus correspond to an equilibrium state of the field theory. This enables us to determine a host of equilibrium thermodynamical quantities such as the pressure of the system \cite{kiri3,Alanen:2011hh}, which we match to its leading order counterpart in perturbative large-$N_c$ Yang-Mills theory. This leads to one further matching relation
\begin{equation}
\frac{\mathcal{L}^3}{4\pi G_5}=\frac{4 N_c^2}{45 \pi^2}\, ,
\la{ellcube}
\end{equation}
which fixes the last unknown parameter of the model.
\section{The calculations}\label{calculations}
To determine the correlation functions of the field theory operators $T_{12}$ and $T_{ii}$ using holography, we follow the steps laid out in \cite{Gubser:2008sz}. First, we introduce perturbations around the background metric of eq.~(\ref{metric}),
\ba
g_{00} &=& b^2 f \left(1+\epsilon H_{00}\right)\,, \;\;
g_{11} = b^2 \left(1+\epsilon H_{11}\right)\, , \;\;
g_{12}=\epsilon b^2 H_{12}\, , \;\;
g_{55} =\frac{b^2}{f} \left(1+\epsilon H_{55}\right)\,,\;\;\; \label{bulkpert}
\ea
where $\epsilon$ is a power counting parameter. With these definitions, the perturbation $H_{11}$ becomes dual to the operator $\frac{1}{3}T_{ii}$, while $H_{12}$ corresponds to the shear operator $T_{12}$. Expanding the Einstein equations to first order in $\epsilon$, we then find that the metric fluctuations must satisfy the equations
\ba
&&\ddot H_{12}+ {d\over dz}\log(b^3f)\dot H_{12}+
{\omega^2\over f^2} H_{12}=0 \,, \label{fluctshear} \\
&&\ddot H_{11}+ {d\over dz}\log(b^3fX^2)\dot H_{11}+
\biggl({\omega^2\over f^2} -{\dot f\,\dot X\over fX}\biggr)H_{11}=0\,,\label{fluctbulk}
\ea
where we have set the corresponding three-momentum $\mathbf{k}$ to zero and defined
\ba
X(\lambda) &\equiv& \frac{\beta (\lambda)}{3 \lambda}. \label{Xdef}
\ea
These equations are to be solved using purely infalling boundary conditions at the horizon, most conveniently implemented via an analytic expansion around $z= z_h$,
\begin{equation}
H_{12/11}(z\rightarrow z_h)=(z-z_h)^{i\omega/\dot{f}_h}[1 +d_1(z-z_h)+ d_2(z-z_h)^2 +\dots ]\, . \label{Hexp}
\end{equation}
The equations of motion (\ref{fluctshear})-(\ref{fluctbulk}) are then straightforwardly solvable using Mathematica.
With the metric fluctuations at hand, a standard recipe provides us with rather simple forms for the shear and bulk spectral functions,
\ba
\rho_s(\omega,T)&=&\frac{f(z) b(z)^3}{16 \pi G_5}\, \frac{\mathop{\mbox{Im}} \dot{H}_{12}(z)H^*_{12}(z)}{|H_{12}(z\rightarrow 0)|^2}\label{rhos}\, ,\\
\rho_b(\omega,T)&=&\frac{6X(z)^2\,f(z) b(z)^3}{16 \pi G_5} \, \frac{\mathop{\mbox{Im}} \dot{H}_{11}(z)H^*_{11}(z)}{|H_{11}(z\rightarrow 0)|^2}\, ,\label{rhob}
\ea
in which the factors $|H_{12/11}(z\rightarrow 0)|^{-2}$ account for the fact that the functions in eq.~(\ref{Hexp}) have not been normalized to unity at the boundary. It can furthermore be shown that the expressions (\ref{rhos}) and (\ref{rhob}) are in fact independent of $z$, i.e.~can be evaluated at any value of the radial coordinate (cf.~\cite{Kajantie:2011nx} for a more detailed discussion of this issue). For practical reasons, we choose to do so infinitesimally close to the horizon, where a use of eq.~(\ref{Hexp}) as well as the identity $f(z\to z_h) = \dot{f}(z_h)(z-z_h)+{\mathcal O}((z-z_h)^2)$ leads us to the expressions
\ba
\rho_s(\omega,T)&=&\frac{s(T)}{4 \pi}\frac{\omega}{|H_{12}(z\rightarrow 0)|^2}\,, \label{rhos1} \\
\rho_b(\omega,T)&=&6X_h^2 \frac{s(T)}{4 \pi}\frac{\omega}{|H_{11}(z\rightarrow 0)|^2}\, ,\label{rhob1}
\ea
in which $X_h\equiv X(z_h)$, and $s(T)=b_h^3/(4G_5)$ denotes the entropy. One should note that these results mix the IR and UV scales of the system in a way that will be seen to result in very interesting large-$\omega$ behavior of the spectral functions in the following sections.
\section{Perturbative limit}\label{results}
Before proceeding to the results of our holographic calculations, let us briefly review what is known about the behavior of the shear and bulk spectral functions in weakly coupled SU($N_c$) Yang-Mills theory. This is helpful in particular for the analysis of the UV (large-$\omega$) behavior of our results, as due to asymptotic freedom all physical correlators are expected to reduce to their perturbative limits as $\omega\to\infty$.
At the moment, the $\mathbf{k}=0$ spectral functions of both channels are known up to and including their respective NLO terms in perturbation theory \cite{Laine:2011xm,Zhu:2012be}. In the shear case, we can read off the result from eq.~(4.1) of \cite{Zhu:2012be}, obtaining (note an additional factor of -1/16 due to differing definitions of the shear operator)
\ba
\frac{\rho_s(\omega,T)}{d_A}&=&\frac{\omega^4}{160\pi}\bigl( 1 + 2 n_{\frac{\omega}{2}} \bigr)\Bigg\{1-\frac{10\lambda_c}{16\pi^2}\bigg(\frac{2}{9}+\phi_T^\eta({\omega\over T})\bigg)\Bigg\}+
{\mathcal O}(\lambda_c^2)\label{rhosper0} \\
&\stackrel[\omega\to \infty]{}{\rightarrow}& \frac{1}{160\pi}\,\omega^4 \, , \label{rhosper1}
\ea
where $d_A\equiv N_c^2-1$, $n_x\equiv 1/(e^{x/T}-1)$, and $\phi_T^\eta(\omega/T)$ is a numerically evaluatable dimensionless function that behaves like $T^6/\omega^6$ in the $\omega\to\infty$ limit. It should be noted that this result misses a number of terms proportional to $\omega\,\delta(\omega)$, which give important contributions to the shear sum rule but are only known to leading order, cf.~e.g.~eq.~(4) of \cite{meyershear}.
In the bulk channel, the perturbative spectral function consistent with our earlier definitions is obtainable from eq.~(4.1) of \cite{Laine:2011xm}. Multiplying this result by $1/9$ and choosing the constant $c_\theta$ as $g^2 c_\theta = \frac{\beta(\lambda_c)}{4\lambda_c}$, where $\beta(\lambda_c)$ is the beta function of Yang-Mills theory, we obtain
\ba
\frac{\rho_b(\omega,T)}{d_A}&=&\frac{\omega^4}{576\pi}\frac{\beta(\lambda_c)^2}{\lambda_c^2}\bigl( 1 + 2 n_{\frac{\omega}{2}} \bigr)\Bigg\{1+\frac{\lambda_c}{8\pi^2}\bigg(\frac{44}{3}\ln\frac{\bar{\mu}}{\omega}+\frac{73}{3}+8\phi_T^\theta({\omega\over T})\bigg)\Bigg\}\nonumber \\
&+&{\mathcal O}(\lambda_c^4) \label{rhobper0} \\
&\stackrel[\omega\to \infty]{}{\rightarrow}&
\frac{121\omega^4}{324(4\pi)^5}\lambda_c^2 , \label{rhobper1}
\ea
where $\phi_T^\theta(\omega/T)$ is again a numerical function, whose behavior was analyzed in quite some detail in \cite{Laine:2011xm}. An important difference to the shear channel result is clearly the appearance of the 't Hooft coupling in the leading large-$\omega$ behavior of eq.~(\ref{rhobper1}). Together with the realization that the renormalization scale, with which the coupling runs, is in the limit $\omega\gg T$ necessarily proportional to $\omega$, this implies that the leading UV behavior of the bulk spectral function takes the form of a $T$-independent constant times $\omega^4/(\ln\,\omega/\Lambda_{\overline{\mbox{\tiny\rm{MS}}}})^2$. In addition, one should note that in the bulk channel, no terms of the type $\omega\delta(\omega)$ appear at least at the orders considered above.
When analyzing the perturbative results for both the shear and bulk spectral functions, an important thing to note is that even at high temperatures --- and thus weak coupling --- the above expressions are not valid in the limit of very small $\omega$. This is due to the multitude of soft scales that enter the calculation at small momentum exchange and require complicated resummations to be performed when entering the regions of $\omega$ of order $gT$, $g^2T$ and ultimately $g^4T$ (see e.g.~\cite{Arnold:2003zc}). While with the Hard Thermal Loop resummation performed in \cite{Laine:2011xm} the above bulk result should be correct down to momenta of order $\omega\sim gT$, it is clear from the plots of \cite{Laine:2011xm,Zhu:2012be} that in both the shear and bulk cases, the perturbative results begin to lose accuracy when $\omega\lesssim T$. In particular, this implies that even the leading order transport coefficients are not available from the above expressions, and that when comparing our holographic
results to them, one should only expect quantitative agreement at $\omega\gg T$.
\section{Holographic results in the shear channel}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Fig1a.eps}$\;\;\;$ \includegraphics[width=0.48\textwidth]{Fig1b.eps}
\caption{\small Left: the IHQCD shear channel spectral function displayed in the region of small frequencies for three different temperatures. The dashed curve represents the large-$\omega$ limit of the SYM result, cf.~\cite{Kajantie:2010nx}. Right: the behavior of the spectral
function at large frequencies for the case of $T=3T_c$. The black curve stands for our IHQCD result, while the two dashed red lines denote the NLO perturbative result evaluated with two different renormalization scales \cite{Zhu:2012be}.}
\label{figshear}
\end{figure}
The shear spectral function was first determined within IHQCD in \cite{Kajantie:2011nx}, and is reproduced for vanishing external three-momentum in fig.~\ref{figshear}. Comparing to the corresponding result in the conformal ${\cal N}=4$ SYM theory \cite{teaney,Kajantie:2010nx}, whose asymptotic behavior is represented by the dashed blue curve on the left, we see that the effects of conformal invariance breaking are largest near the IHQCD deconfinement temperature $T_c$, while already at $T=10T_c$ the IHQCD result is rather close to the SYM one. In the $\omega\to 0$ limit, each of the curves furthermore reproduces the well known result of $\eta/s = 1/(4\pi)$, equally valid in IHQCD as in the SYM theory.
Proceeding to larger values of $\omega$, we display the behavior of the shear spectral function on a log-log scale in fig.~\ref{figshear} (right), where it is further compared with the perturbative result of eq.~(\ref{rhosper0}). This reveals a clear discrepancy between the two results in the UV region, which one can understand using the analytic WKB calculation of \cite{Kajantie:2011nx}. With the help of eq.~(\ref{ellcube}), one namely easily obtains as the limiting behavior of the IHQCD shear spectral function
\ba
\rho_s(\omega,T)&\stackrel[\omega\to \infty]{}{\rightarrow}& \frac{N_c^2}{360\pi}\,\omega^4 ,\label{rhosas}
\ea
which deviates from the perturbative limit of eq.~(\ref{rhosper1}) by a factor of $4/9$. This should not come as a surprise considering that IHQCD is only a two-derivative model, but nevertheless highlights its limitations in describing the UV dynamics of the physical theory.
\section{Holographic results in the bulk channel}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\textwidth]{Fig2.eps}
\end{center}
\caption{The bulk spectral function evaluated for a set of different temperatures in the region of small $\omega$. The temperature dependence of the bulk viscosity can be read off from intercepts of the curves at $\omega=0$.}
\label{figrhobsmallomega}
\end{figure}
The bulk channel has been studied in various conformality breaking holographic models already in several works \cite{Gubser:2008sz,Gursoy:2009kk,buchelgursoy}, but typically concentrating only on the behavior of the bulk viscosity to entropy ratio as a function of $T$. In this section, our aim is to extend this treatment to the evaluation of the full IHQCD bulk spectral function.
Beginning from the limit of small frequencies, we first display in fig.~\ref{figrhobsmallomega} the behavior of $\rho_b(\omega,T)$ for temperatures ranging from $T_c$ to $10T_c$ (cf.~also fig.~\ref{figshear} (left)). In accordance with our expectations, we observe a decrease in the values of the function with increasing $T$, signifying the approach of the system towards the conformal limit. From the intercepts of the curves at $\omega=0$, one can furthermore read off the values of the bulk viscosity $\zeta$ at different temperatures, leading to a behavior consistent with that shown in fig.~8 of \cite{Gursoy:2009kk}.
Next, we study the large-$\omega$ behavior of the bulk spectral function in fig.~\ref{figrhoblargeomega}. On the left, we demonstrate, how the combination $\rho_b(\omega,T)/\omega^4$ undergoes a sharp transition from a $T$-dependent $1/\omega^3$ behavior at small frequencies towards a $T$-independent
$1/(\ln\omega/T_c)^2$ limit at $\omega\gg T$. On the right, we on the other hand specialize to the case of $T=3T_c$, displaying the holographic result together with the perturbative one, eq.~(\ref{rhobper0}). This figure demonstrates a remarkable fact: not only is the form of the asymptotic $1/(\ln\omega/\Lambda_{\overline{\mbox{\tiny\rm{MS}}}})^2$ behavior (note that $T_c\sim\Lambda_{\overline{\mbox{\tiny\rm{MS}}}}$) of the perturbative result reproduced by our IHQCD calculation, but even the overall coefficient in eq.~(\ref{rhobper1}) appears to agree with our numerics. We find this possibly coincidential fact very surprising, considering the missing $4/9$ factor encountered in the shear spectral function. We see no a priori reason, why corrections from higher derivative terms in the holographic action should be present in the shear channel but absent from the bulk one.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig3a.eps}$\;\;\;$ \includegraphics[width=0.48\textwidth]{Fig3b.eps}
\end{center}
\caption{Left: the behavior of the IHQCD bulk spectral function shown over a wide range of frequencies at three different temperatures. Note that unlike in most of our other plots, $\omega$ has here been scaled dimensionless by $T_c$ and not $T$. Right: a comparison of the IHQCD (solid black curve) and perturbative (red dashed lines) results for the bulk spectral function. The perturbative result is taken from \cite{Laine:2011xm}.}
\label{figrhoblargeomega}
\end{figure}
Ideally, it would of course be pleasing to be able to derive the logarithmic UV behavior of the bulk spectral function analytically, following a WKB expansion similar to that performed in the shear channel in \cite{Kajantie:2011nx}. In the bulk case, this, however, turns out to be a rather demanding task due to the appearance of logarithmic terms in the fluctuation equation, originating from the $z\to0$ limit of the quantity
\be
3X= {\beta\over\lambda}={d\log\lambda\over d\log b}\stackrel[z\to 0]{}{\rightarrow}{1\over\log z}\, .
\ee
Scaling the radial variable according to $z\to z'=\omega z$, the large-$\omega$ (i.e.~small-$z$) limit of eq.~(\ref{fluctbulk}) namely becomes (with $\Lambda$ denoting an arbitrary scale parameter)
\ba
\ddot H_{11}+\Bigg\{-{3\over z}\biggl(1+{4\over 9(\log \Lambda z/\omega )^2}\biggr)
+{2\over z|\log\Lambda z/\omega |}\Bigg\}\dot H_{11}+H_{11}&=&0 \, ,
\ea
which without the logarithmic terms would lead to the usual $\omega^4$ behavior of the spectral function. In the presence of the conformality breaking $\omega$-dependence, solving the equation however becomes much harder, and in particular leads to the numerically verified appearance of logarithmic suppression in the spectral function. It is in any case worth noting that the resulting $1/(\ln\omega/T_c)^2$ behavior of $\rho_b(\omega,T)$ enters eq.~(\ref{rhob1}) solely through the $z\to0$ limit of $H_{11}(z)$, and not via the factor $X_h^2\sim \beta(\lambda(z_h))^2/\lambda(z_h)^2$. This is in clear contrast with the perturbative limit in eq.~(\ref{rhobper1}), in which the logarithmic behavior is due to the running of the gauge coupling.
Having the bulk spectral function now at hand, a natural application is clearly the determination of the corresponding imaginary time correlator, for which both perturbative and lattice results exist. To this end, we plug our function $\rho_b(\omega,T)$ to the relation
\ba
G(\tau,T)&=&\int_0^\infty\frac{d\omega}{\pi}\rho_b(\omega,T)\frac{\cosh\left[\left(
\frac{\beta}{2} -\tau\right) \pi \omega\right]}{\sinh\left( \frac{\beta}{2}\omega\right)}\, ,\quad \beta\equiv 1/T\, ,
\label{Gtau}
\ea
obtaining the result displayed in fig.~\ref{figbim}. Our holographic prediction
is seen to agree with the lattice data better than the weak coupling result over a wide range of temperatures, the difference being (not surprisingly) most pronounced close to $T_c$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48 \textwidth]{Fig4a.eps}
$\;\;\;$
\includegraphics[width=0.48 \textwidth]{Fig4b.eps}
\end{center}
\caption{The bulk channel imaginary time correlator of eq.~(\ref{Gtau}) evaluated for two different temperatures in IHQCD (solid black curves) and perturbation theory (red dashed curves), and compared with the lattice data points of \cite{Meyer:2010ii}. The perturbative result is again taken from \cite{Laine:2011xm}.}
\label{figbim}
\end{figure}
Finally, a different way of inspecting the imaginary time correlator is to look at its value at the symmetry point $\tau=1/(2T)$ as a function of temperature. This we do in fig.~\ref{figbrhosymmpoint}, where $G(\tau=1/(2T),T)$ is displayed, normalized dimensionless by $T^5$. The plot indicates a rapid decrease in the quantity as the temperature is raised above $T_c$, to be contrasted with the slow increase of the corresponding quantity in the shear channel, shown in fig.~6 of \cite{Kajantie:2011nx}. This fact can clearly be attributed to the system approaching conformality in the limit of high temperatures.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\textwidth]{Fig5.eps}
\end{center}
\caption{The imaginary time correlator of eq.~(\ref{Gtau}), normalized by $T^5$ and plotted as a function of temperature at the symmetry point $\tau=1/(2T)$.}
\label{figbrhosymmpoint}
\end{figure}
\section{Conclusions}\label{conclusions}
In the paper at hand, we have studied finite temperature correlation functions of the energy momentum tensor of large-$N_c$ Yang-Mills theory, concentrating on the shear $\langle T_{12}T_{12}\rangle$ and bulk $\langle T_{ii}T_{jj}\rangle$ channels at vanishing external three-momentum. In particular, after determining the spectral functions and associated imaginary time correlators in the Improved Holographic QCD (IHQCD) model \cite{kiri2,kiri3}, we performed a detailed comparison of our results with state-of-the-art perturbative and lattice works. Clearly, the domains of validity of the three methods do not always overlap. Perturbation theory requires the gauge coupling to be small, which is formally only realized at asymptotically large $T$ or $\omega$, while holographic methods work best in the strongly coupled, yet conformal limit, to which systematic corrections are accounted for in the IHQCD model. Finally, while being a fundamentally nonperturbative first principles method, lattice QCD is unfortunately
restricted to the Euclidean formulation of the theory, and thus only provides results for a limited set of observables.
Comparing the IHQCD shear and bulk spectral functions with their perturbative counterparts, cf.~figs.~\ref{figshear}-\ref{figrhoblargeomega}, we witnessed an expected pattern, in which conformal invariance breaking effects were seen to be largest near $T_c$, but rapidly decrease with increasing temperature. Furthermore, we saw that in the large-$\omega$ limit of both the shear and bulk channels, the parametric dependence of the perturbative spectral functions on $\omega$ ($\omega^4$ and $\omega^4/(\ln\,\omega/\Lambda_{\overline{\mbox{\tiny\rm{MS}}}})^2$, respectively) was correctly reproduced by IHQCD. A closer inspection further revealed that while in the shear channel the $\omega\to\infty$ limit of the perturbative result was larger than the IHQCD one by a factor 9/4, surprisinly in the bulk channel the asymptotic limits perfectly coincide. We find this quite remarkable, considering that the nonzero value of the bulk correlator is entirely due to the conformal invariance breaking built into IHQCD.
For a set of Euclidean quantities --- the imaginary time correlation functions --- we were able to perform comparisons between IHQCD, perturbation theory and lattice Monte Carlo results. A direct comparison of the bulk channel correlator $G_b(\tau,T)$ was performed at two temperatures, $1.65T_c$ and $3.2T_c$. The results showed the lattice data consistently prefering the holographic prediction, though at higher temperatures the difference was seen to somewhat diminish.
Finally, we note that in a recent paper \cite{Gursoy:2012bt}, an IHQCD calculation closely related to ours was performed for the correlators of the pseudoscalar operator $\tr F_{\mu\nu}\tilde{F}_{\mu\nu}$. It is interesting to compare the results reported in section 4 of this paper to ours, concerning in particular the asymptotic large-$\omega$ behavior of the $\mathbf{k}=0$ spectral function. While perturbative arguments suggest that the pseudoscalar spectral function should behave similarly to our bulk result (cf.~ref.~\cite{Laine:2011xm}), the authors of \cite{Gursoy:2012bt} argue that their numerical data at large frequencies is consistent with a pure $\omega^4$ behavior. This being the case, it would clearly be crucial to understand the physical origin of the differing behavior, perhaps by performing a WKB type expansion in the asymptotic region of both channels. This calculation, as well as an analysis of the correlation functions of the $\tr F_{\mu\nu}F_{\mu\nu}$ operator, we however leave for the
future.
\section*{Acknowledgments}
We thank U.~G\"ursoy, E.~Kiritsis, and Yan Zhu for useful discussions, as well as H.~Meyer for providing us his lattice results in a tabulated form. This work was supported by the Sofja Kovalevskaja programme
of the Alexander von Humboldt foundation, the DFG graduate school \textit{Quantum Fields and Strongly Interacting Matter} as well as the ESF network \textit{Holographic methods for strongly coupled systems (HoloGrav)}.
|
2,877,628,089,506 | arxiv | \section{Introduction}
\label{sec:introduction}
The large-scale structure (LSS) of the Universe, as traced for example
by the distribution of galaxies, is the focus of several ongoing and
upcoming observational campaigns. In addition to furthering our
understanding of the cosmic web these projects seek to investigate
fundamental physics, including the properties of the initial
conditions, the imprint of (massive) neutrinos becoming
non-relativistic, and the behavior of the mysterious dark energy.
Improving our theoretical understanding of the LSS will enhance the
scientific return of these projects. In particular, a more detailed
understanding of the anisotropy in the observed clustering is of great
interest, as the imprint of peculiar velocities in redshift survey
maps (known as redshift-space distortions: RSD) allows a consistency
test in general relativity between the expansion history and growth of
perturbations; such tests could provide support for modified gravity
theories as an explanation for the observed cosmic expansion.
Moreover, a precise understanding of the peculiar velocity induced
anisotropy in galaxy clustering would improve our ability to measure
the geometrically induced anisotropy known as the Alcock-Paczynski
effect \citep{1979Natur.281..358A}, and thus constrain the expansion
rate $H(z)$ directly \citep[for further details, see
e.g.][]{2011MNRAS.410.1993S}.
In this work we shall investigate an analytic model to predict the
two-point function of biased tracers of large-scale structure based on
perturbation theory. There is a large literature using perturbative
techniques to study RSD \citep[see for example reviews
in][]{1998ASSL..231..185H, 2002PhR...367....1B,2009PhRvD..80d3531C}.
Standard perturbation theory (SPT) adopts an Eulerian description of
fluids, focusing on the velocity field and density contrast
(e.g.~\citealt{1980lssu.book.....P} for linear theory and
\citealt{1981MNRAS.197..931J,1983MNRAS.203..345V,1986ApJ...311....6G,
1992PhRvD..46..585M,1994ApJ...431..495J} for higher orders). On the
other hand, Lagrangian perturbation theory performs an expansion in
the Lagrangian displacement field \citep{1992MNRAS.254..729B,
1994MNRAS.267..811B, 1995A&A...296..575B}. Lagrangian perturbation
theory (LPT) and SPT give identical results for the matter power
spectrum in real space when expanded to the same order
\citep{2008PhRvD..77f3530M}. However, it is easier to include
redshift space distortions in LPT: a time derivative of the original
displacement field is simply added in the line-of-sight direction.
Furthermore, current theories of galaxy formation rely on the cooling
of gas within dark matter potential wells to form galaxies.
Therefore, like dark matter halos, galaxies are biased tracers of the
underlying matter distribution. A local Lagrangian bias model seems
to provide a better description of dark matter halo clustering than a
local Eulerian bias
\citep[e.g.][]{2011MNRAS.415..829R,2012arXiv1201.4827B,
2012PhRvD..85h3509C, 2012PhRvD..86d3508W}, although additional terms
involving the tidal tensor may also become important for high mass
halos \citep{2012arXiv1207.7117S}.
Recently Lagrangian perturbation theory was extended by a resummation
scheme known as ``integrated perturbation theory''
\citep[iPT;][]{2008PhRvD..77f3530M,2008PhRvD..78h3519M}. A key
success of iPT is a very accurate description of the redshift-space
two-point correlation function of dark matter halos on scales of
interest for studying baryon acoustic oscillations (BAO).
Unfortunately, the iPT predictions are inaccurate on scales
$20-70\Mpch$, even though deviations from linear theory are still only
$\sim 10$ per cent.
A recent paper by \citet{2012arXiv1209.0780C} introduced convolution
Lagrangian perturbation theory (CLPT) which improves the iPT method by
resumming more terms in the perturbative expansion. CLPT gives
dramatically better results on small scales when compared to N-body
simulations, particularly for the description of the redshift-space
clustering of dark matter. The methodology is easily extendable to
compute properties of the pairwise halo velocity distributions that
generate redshift-space distortions. The primary purpose of this
paper is to examine CLPT's accuracy in predicting these statistics, in
comparison with N-body simulations. We will see that CLPT provides an
accurate description of the velocity distributions. Unfortunately,
the CLPT predictions for the anisotropy in the two-point correlation
function measured by the quadrupole are still inaccurate on the
quasi-linear scales of interest \citep[see figure 5
of][]{2012arXiv1209.0780C}. Therefore, in this paper we combine the
real-space correlation function and the velocity statistics predicted
by CLPT with the non-perturbative approach advocated in \citet[][the
scale-dependent Gaussian streaming model]{2011MNRAS.417.1913R}. This
model convolves the real-space two-point correlation function with an
approximation to the (scale-dependent) velocity distribution functions
to predict redshift space clustering.
This paper is structured as follows. In Sections \ref{sec:review} and
\ref{sec:v-sigma} we provide some analytic prerequisites for
evaluating clustering statistics with CLPT. Section
\ref{sec:evaluate} contains the primary new calculation in this work
-- the prediction of pairwise velocity statistics for biased tracers
in CLPT. We evaluate both auto- and cross-correlation statistics. In
Section \ref{sec:GSM} we review the Gaussian streaming model, the
basis of our final model for the redshift space halo correlation
function. In Section \ref{sec:results-statistics} we show the
numerical evaluation of those perturbative analytic results. Those
CLPT results are input into the Gaussian streaming model in Section
\ref{sec:redshift-space-stat} and compared with values obtained
directly and indirectly by simulations. Section \ref{sec:summary}
gives the summary of this article.
\section{Review}
\label{sec:review} Before we present our calculation of the velocity
moments in CLPT, let us review some background material to set our
notation and conventions.
\subsection{Background}
Throughout this work we will adopt the ``plane-parallel''
approximation, so that the line-of-sight (LOS) is chosen along a
single Cartesian axis ($\hat{\vec{z}}$). While wide-angle effects
could potentially be important in modern surveys
\citep{2008MNRAS.389..292P}, \citet{2012MNRAS.420.2102S} have shown
that in practice these effects are small given current errors. The
redshift-space position $\vec{s}$ of an object differs from its
real-space position $\vec{r}$ due to its peculiar velocity,
\begin{equation}
\label{eq:sx}
\vec{s} = \vec{x} + v_z(\vec{x})\,\hat{\vec{z}},
\end{equation}
where $v_z(\vec{x}) \equiv u_z(\vec{x})/(aH)$ is the
LOS component of object's velocity (assumed non-relativistic) in units
of the Hubble velocity. In linear theory, the peculiar velocity field
is assumed curl-free, and its divergence is sourced by the underlying
matter fluctuations:
\begin{equation}
\nabla \cdot \vec{v} = -f \delta_{m}
\end{equation}
where $f \equiv \d\ln D/\d\ln a$ and $D(a)$ is the growth rate of
fluctuations in linear theory. Measurements of two-point clustering
as a function of angle with respect to the LOS direction can directly
constrain $f$ times the normalization of matter fluctuations
\citep[e.g.][]{2008Natur.451..541G, 2009MNRAS.393..297P,
2009MNRAS.397.1348W}.
In this paper, we will focus on the prediction of the two-point
correlation function:
\begin{equation}
\xi(\vec{r}) = \langle \delta(\vec{x}) \delta(\vec{x}+\vec{r})
\rangle.
\end{equation}
In real-space, $\xi(\vec{r}) \equiv \xi(r)$ is only a function of the
separation length, while in redshift-space $\xi(\vec{s})$ depends on
the cosine of the angle between the pair separation vector and the
LOS, $\mu_s \equiv \hat{\vec{s}} \cdot \hat{\vec{z}}$. It is
convenient and common to condense the information in $\xi(\vec{s})$
into Legendre polynomial moments (we use $L_\ell$ for $\ell$th order
Legendre polynomial to avoid ambiguity):
\begin{equation}
\label{eq:legendre-expansion}
\xi(s,\mu_s) = \sum_{\ell} \xi_{\ell}(s) L_{\ell}(\mu)\ .
\end{equation}
By symmetry, odd $\ell$ moments vanish. In linear theory, only $\ell=
0,2,4$ contribute; we will focus our model predictions on those
moments. In \S\ref{sec:redshift-space-stat} we shall also look at
clustering ``wedges'' \citep[e.g.][]{2012MNRAS.419.3223K}, but these
require no further formalism.
Throughout this paper, we adopt the Einstein summation convention and
the following convention of Fourier transform and its inverse ($n$ is
the number of dimensions, here usually 1 or 3):
\begin{equation}
\label{eq:fourier-def}
\tilde{F}({\bf k}) = \int \d^nx\ F({\bf x})
\e{-i{\bf k} \cdot {\bf x}}\ ,\quad F({\bf x}) = \int
\dfrac{\d^nk}{(2\pi)^n}\ \tilde{F}({\bf k}) \e{i{\bf k} \cdot {\bf
x}}\ .
\end{equation}
\subsection{Integrated Perturbation Theory (iPT): formalism for biased
tracers}
Lagrangian perturbation theories perform a perturbative expansion in
the displacement field $\vec{\Psi} = \vec{\Psi}^{(1)} +
\vec{\Psi}^{(2)} + \vec{\Psi}^{(3)} + \cdots$. Here, $\vec{\Psi}$
relates the Eulerian (final) coordinates $\vec{x}$ and Lagrangian
(initial) coordinates $\vec{q}$ of a mass element or discrete tracer
object:
\begin{equation}
\label{eq:lagriangian-disp-def}
\vec{x}(\vec{q}, t) = \vec{q} + \vec{\Psi}(\vec{q},t)\ .
\end{equation}
The relation between Eulerian and Lagrangian fields of
matter density contrast ($\delta = \rho/\bar{\rho}-1$) is given by
\begin{equation}
\label{eq:density-field-eulerian-lagrangian}
[1+\delta_m(\vec{x},t)]\ \d^3x = [1+\delta_m(\vec{q},0)]\ \d^3q = \d^3q\ .
\end{equation}
\citet{2008PhRvD..78h3519M,2008PhRvD..77f3530M} laid
out the formalism for including redshift-space distortions and
non-linear local Lagrangian biasing within Lagrangian perturbation
theory. The density contrast of our tracer field in Lagrangian space,
$\delta(\vec{q})$, is related to the underlying Lagrangian matter
density fluctuations smoothed on scale $R$:
\begin{equation}
1 + \delta(\vec{q}) = F[\delta_{m,R}(\vec{q})]\ ,
\end{equation}
where $F(\delta)$ is the bias function. Note that the
smoothing scale $R$ naturally drops out in the final predictions for
all statistics of interest in this paper, but is necessary to keep
intermediate quantities well-behaved. Thus
\begin{equation}
\label{eq:density-field-integration}
1 + \delta(\vec{x},t) = \int
\d^3q\ F[\delta_{m,R}(\vec{q})] \delta^D[\vec{x} - \vec{q} -
\vec{\Psi}(\vec{q}, t)]\ .
\end{equation}
After a coordinate transformation $\{\vec{q}_1,\vec{q}_2\} \rightarrow
\{\vec{q} = \vec{q}_2-\vec{q}_1, \mathbf{Q} = (\vec{q}_1+\vec{q}_2) /
2\}$, and expressing both $F$ and $\delta^D$ in
Eq.~\eqref{eq:density-field-integration} by their Fourier
representations, the two-point correlation function in real-space is
given by [see also equations (15) through (20) in
\citet{2012arXiv1209.0780C}]:
\begin{equation}
\label{eq:xi-fourier}
\begin{split}
1 + \xi(\vec{r}) & = \int \d^3q \int \dfrac{\d^3k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} \int \dfrac{\d \lambda_1}{2\pi}
\dfrac{\d \lambda_2}{2\pi}
\\
& \times\tilde{F}(\lambda_1) \tilde{F}(\lambda_2) \left\langle
\e{i\left(\lambda_1\delta_1 + \lambda_2\delta_2 + \vec{k}\cdot
\vec{\Delta} \right)} \right\rangle\ ,
\end{split}
\end{equation}
where $\delta_{1,2} = \delta(\vec{q}_{1,2})$,
$\vec{\Delta}=\vec{\Psi}(\vec{q}_2) - \vec{\Psi}(\vec{q}_1)$, and
$\tilde{F}(\lambda)$ is the Fourier transform of $F(\delta_R)$ with
coordinates pair $\delta_R$ versus $\lambda$.
\section{Clustering and velocity statistics in CLPT}
\label{sec:v-sigma}
In this section we extend the work of \citet{2008PhRvD..78h3519M} and
\citet{2012arXiv1209.0780C} to enable the calculation of moments of
the pairwise velocity distribution for tracers that are biased in a
local Lagrangian sense. Once the real-space two-point correlation
function and pairwise velocity distributions are known, they determine
the observed two-point clustering in redshift-space. These are also
the key ingredients of the Gaussian streaming model, as we shall
discuss further in \S\ref{sec:GSM}.
\subsection{Velocity moments in CLPT: formalism}
The relative peculiar velocity between two tracers at Eulerian
coordinates $\vec{x}_1$ and $\vec{x}_2$ can be simply expressed in
terms of the time derivative of the displacement field $\vec{\Psi}$:
\begin{equation}
\vec{u}(\vec{x}_2) - \vec{u}(\vec{x}_1) = a \left(
\dot{\vec{x}}_2 - \dot{\vec{x}}_1\right) = a\dot{\vec{\Delta}}.
\end{equation}
In the time-independent approximation to the perturbative kernels
where ${\vec{\Psi}}^{(k)} \propto D^k$ for $D$ the linear growth
function \citep[equation (46) of][]{2008PhRvD..77f3530M},
\begin{equation}
\dot{\vec{\Psi}}^{(k)} = kHf\vec{\Psi}^{(k)}\ .
\end{equation}
Thus we have a perturbative expansion for the Cartesian components of
$\vec{v}_n$ (adopting the units of Eq.~\ref{eq:sx}) in terms of the
components of $\vec{\Delta}_n$
\begin{equation}
\vec{v}_n(\vec{x}_2) - \vec{v}_n(\vec{x}_1) = \sum_k
kf \vec{\Delta}_n^{(k)} = \frac{\dot{\vec{\Delta}}_n}{H}\ .
\end{equation}
We follow common practice and define the velocity generating function
$Z(\vec{r}, \vec{J})$ by
\begin{equation}
\label{eq:Z}
\begin{split}
Z(\vec{r}, \vec{J}) & = \int \d^3q \int \dfrac{\d^3k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} \int \dfrac{\d \lambda_1}{2\pi}
\dfrac{\d \lambda_2}{2\pi}
\\
& \times\tilde{F}(\lambda_1) \tilde{F}(\lambda_2) \left\langle
\e{i\left(\lambda_1\delta_1 + \lambda_2\delta_2 + \vec{k}\cdot
\vec{\Delta} + \vec{J}\cdot\dot{\vec{\Delta}}/H\right)}
\right\rangle\ .
\end{split}
\end{equation}
Note that $\xi(\vec{r}) = Z(\vec{r}, 0) - 1$
(Eq.~\eqref{eq:xi-fourier}), and derivatives of $Z$ give the pairwise
velocity moments of interest (i.e., the numerators in our
Eqs.~\ref{eq:v-infall-def} and \ref{eq:vdisp-sigma-mu}):
\begin{equation}
\label{eq:vLPT}
\begin{split}
& \left\langle [1+\delta(\vec{x})] [1+ \delta(\vec{x}+\vec{r})]
\left\{\prod_{k=1}^{p} [\vec{v}_{i_k}(\vec{x}+\vec{r}) - \vec{
v}_{i_k}(\vec{x})]\right\} \right\rangle
\\
= & \int \d^3q \int \dfrac{\d^3k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} \int \dfrac{\d \lambda_1}{2\pi}
\dfrac{\d \lambda_2}{2\pi}
\\
\times & \tilde{F}(\lambda_1) \tilde{F}(\lambda_2) \left\langle
\left(\prod_{k=1}^{p}
\left(\dfrac{\dot{\vec{\Delta}}_{i_k}}{H}\right)\right)
\e{i\left(\lambda_1\delta_1 + \lambda_2\delta_2 + \vec{k}\cdot
\vec{\Delta} \right)} \right\rangle
\\
= & \left.\prod_{k=1}^{p} \left(-i\dfrac{\partial}{\partial
\vec{J}_{i_k}}\right) Z(\vec{r}, \vec{J}) \;
\right|_{\vec{J} = 0}
\end{split}
\end{equation}
Here the set $\{i_k\}$ specifies the Cartesian coordinate direction
for each derivative with respect to $J_{i_k}$. Before proceeding to
evaluate Eq.~\eqref{eq:vLPT}, we generalize the definitions of the
functions $K$, $L$, and $M$ in \citet{2012arXiv1209.0780C} to include
$\vec{J}$. These functions are convenient shorthand for intermediate
results.
\begin{equation}
\label{eq:perturbation-notation-short}
\begin{split}
& X = \lambda_1\delta_1 + \lambda_2\delta_2 + \vec{k}\cdot
\vec{\Delta} + \vec{J} \cdot \dot{\vec{\Delta}} / H\ ,
\\
& K_{p,\{i_1..i_p\}}(\vec{q}, \vec{k}, \lambda_1, \lambda_2) =
\left.\left\langle \left(-i\dfrac{\partial}{\partial
\vec{J}_{i_k}}\right)^p \e{iX} \right\rangle\
\right|_{\vec{J}=0}\ ,
\\
& L_{p,\{i_1..i_p\}}(\vec{q}, \vec{k}) = \int \dfrac{\d
\lambda_1}{2\pi} \dfrac{\d \lambda_2}{2\pi}
K_{p,\{i_1..i_p\}}(\vec{q}, \vec{k}, \lambda_1, \lambda_2)\ ,
\\
& M_{p,\{i_1..i_p\}}(\vec{r}, \vec{q}) = \int
\dfrac{\d^3k}{(2\pi)^3} \e{i\vec{k}\cdot(\vec{q}-\vec{r})}
L_{p,\{i_1..i_p\}}(\vec{q}, \vec{k})\ ,
\\
& \left\langle [1+\delta(\vec{x})] [1+ \delta(\vec{x}+\vec{r})]
\left\{\prod_{k=1}^{p} [\vec{v}_{i_k}(\vec{x}+\vec{r}) -
\vec{v}_{i_k} (\vec{x})]\right\} \right\rangle
\\
& \quad =\int \d^3q M_{p,\{i_1,..,i_p\}}(\vec{r}, \vec{q})\ .
\end{split}
\end{equation}
The first subscript of these functions indicates the
number of derivative terms $p$, and the second is a list containing
the Cartesian indices of the $p$ derivatives.
\section{Evaluating the CLPT predictions}
\label{sec:evaluate}
\subsection{Evaluating $\xi(r)$ in CLPT}
To begin, we review the calculation of the real-space two-point
correlation function $\xi(r)$ in CLPT, first presented in
\citet{2012arXiv1209.0780C}, building upon the work of
\citet{2008PhRvD..78h3519M,2008PhRvD..77f3530M}. The cumulant
expansion theorem,
\begin{equation}
\label{eq:cumulant-expansion}
\left\langle \e{iX} \right\rangle = \exp\left[ \sum_{N=1}^\infty
\dfrac{i^N}{N!} \langle X^N \rangle_c \right]\ ,
\end{equation} makes the evaluation of $K$ tractable; here $\langle
X^N \rangle_c$ is the $N$th cumulant of the random variable $X$.
Taylor expanding the exponential on the right hand side of
Eq.~\eqref{eq:cumulant-expansion} only for terms that vanish in the
limit $|\vec{q}| \rightarrow \infty$ and keeping only terms up to
$O(P_L^2)$, \citet{2012arXiv1209.0780C}, Sec.~4 obtain
\begin{equation}
\label{eq:xi-kernel-k}
\begin{split}
& K_0 = \left. \left\langle \e{iX} \right\rangle
\right|_{\vec{J}=0}
\\
& = \e{-(1/2)A_{ij}k_ik_j} \e{-(1/2)(\lambda_1^2+\lambda_2^2)
\sigma_R^2} \bigg\{ 1 - \lambda_1\lambda_2 \xi_L +
\dfrac{1}{2}\lambda_1^2\lambda_2^2 \xi_L^2
\\
& - (\lambda_1+\lambda_2) U_ik_i + \dfrac{1}{2}
(\lambda_1+\lambda_2)^2 U_iU_jk_ik_j - \dfrac{i}{6}
W_{ijk}k_ik_jk_k
\\
& + \lambda_1\lambda_2(\lambda_1+\lambda_2) \xi_L U_ik_i -
\dfrac{i}{2} (\lambda_1+\lambda_2) A^{10}_{ij}k_ik_j
\\
& + \dfrac{i}{2}(\lambda_1^2 +\lambda_2^2) U^{20}_ik_i -
i\lambda_1\lambda_2U^{11}_ik_i + O(P_L^3) \bigg\}\ .
\end{split}
\end{equation}
In Eq.~\eqref{eq:xi-kernel-k}, we adopt the following short-hand
definitions:
\begin{equation}
\label{eq:xi-short-def}
\begin{split}
& \langle \delta_1^2 \rangle_c = \langle \delta_2^2 \rangle_c =
\sigma_R^2\ ,\quad \langle \delta_1\delta_2 \rangle_c =
\xi_L(\vec{q})\ ,
\\
& U^{mn}_i = \langle \delta_1^m \delta_2^n \Delta_i \rangle_c\ ,
\quad A^{mn}_{ij} = \langle \delta_1^m \delta_2^n \Delta_i\Delta_j
\rangle_c\ ,
\\
& W^{mn}_{ijk} = \langle \delta_1^m \delta_2^n
\Delta_i\Delta_j\Delta_k \rangle_c\ ,
\\
& U_i = U^{10}_i\ ,\quad A_{ij} = A^{00}_{ij}\ ,\quad W_{ijk} =
W^{00}_{ijk}\ .
\end{split}
\end{equation}
Subsequently we apply
\citep[see][]{2008PhRvD..78h3519M,2012arXiv1209.0780C}
\begin{equation}
\label{eq:bias-expect-identity}
\int \dfrac{\d \lambda}{2\pi} \tilde{F}(\lambda)
\e{\lambda^2\sigma_R^2/2} (i\lambda)^n = \langle F^{(n)} \rangle\ ,
\end{equation}
where $\langle F^{(n)} \rangle$ is the expectation value of the $n$th
derivative of $F(\delta_R)$. This relation enables us to conduct such
transformations from integration with respect to $\lambda$ to bias
parameters:
\begin{equation}
\label{eq:bias-transform}
\begin{split}
& (\lambda_1 + \lambda_2) \rightarrow 2\fp{}\ , \qquad
(\lambda_1^2 + \lambda_2^2 ) \rightarrow 2\fpp{}\ ,
\\
& \lambda_1 \lambda_2 \rightarrow \fp{}^2\ , \qquad \lambda_1^2
\lambda_2^2\rightarrow \fpp{}^2\ ,
\\
& \lambda_1 \lambda_2 (\lambda_1 + \lambda_2) \rightarrow
2\fp{}\fpp{}\ .
\end{split}
\end{equation}
We can hence evaluate $L_0$ analytically:
\begin{equation}
\label{eq:xi-kernel-l}
\begin{split}
& L_0 = \e{-(1/2)A_{ij}k_ik_j} \bigg\{ 1 + \fp{}^2 \xi_L +
\dfrac{1}{2}\fpp{}^2 \xi_L^2 + 2i \fp{} U_ik_i
\\
& - [\fpp{} + \fp{}^2] U_iU_jk_ik_j - \dfrac{i}{6}
W_{ijk}k_ik_jk_k - \fp{} A^{10}_{ij}k_ik_j
\\
& + 2i \fp{} \fpp{} \xi_L U_ik_i + i\fpp{} U^{20}_ik_i
\\
& - i\fp{}^2U^{11}_ik_i + O(P_L^3) \bigg\}\ .
\end{split}
\end{equation}
The integration with respect to $\vec{k}$ is then conducted, using
some basic relations for Gaussian integration \citep[see appendix C
of][] {2012arXiv1209.0780C}, which gives
\begin{equation}
\label{eq:xi-kernel-m}
\begin{split}
& M_0 = \int \dfrac{\d^3 k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} L_0(\vec{q},\vec{k})
\\
& = \dfrac{1}{(2\pi)^{3/2}|A|^{1/2}}\e{-(1/2)(A^{-1})_{ij}
(q_i-r_i) (q_j-r_j)}
\\
&\times \bigg\{ 1 + \fp{}^2 \xi_L + \dfrac{1}{2}\fpp{}^2
\xi_L^2 - 2 \fp{} U_ig_i + \dfrac{1}{6} W_{ijk}\Gamma_{ijk}
\\
& - [\fpp{} + \fp{}^2] U_iU_jG_{ij} - \fp{}^2U^{11}_ig_i - \fpp{}
U^{20}_ig_i
\\
& - 2 \fp{} \fpp{} \xi_L U_ig_i - \fp{} A^{10}_{ij}G_{ij} +
O(P_L^3) \bigg\}\ .
\end{split}
\end{equation}
Here we define
\begin{equation}
\label{eq:gaussian-intg-aux}
\begin{split}
& g_i = (A^{-1})_{ij}(q_j-r_j)\ ,\quad G_{ij} = (A^{-1})_{ij} -
g_ig_j\ ,
\\
& \Gamma_{ijk} = (A^{-1})_{ij} g_k + (A^{-1})_{ki} g_j +
(A^{-1})_{jk} g_i - g_ig_jg_k\ .
\end{split}
\end{equation}
Finally, the desired correlation function is given by
the $\vec{q}$ integration of $M_0$:
\begin{equation}
\label{eq:xi-exp} 1 + \xi(\vec{r}) = \int \d^3q\
M_0(\vec{r},\vec{q})\ .
\end{equation}
In order to evaluate $M_0$, we expand $U^{mn}_{i}$, $A^{mn}_{ij}$ and
$W^{mn}_{ijk}$ in \eqref{eq:xi-short-def} with respect to
$\vec{\Delta}^{(n)}$, i.e.
\begin{equation}
\label{eq:xi-correlator-different-order}
\begin{split}
& U^{mn(p)}_i = \langle \delta_1^m \delta_2^n \Delta_i^{(p)}
\rangle_c\ , \quad A^{mn(pq)}_{ij} = \langle \delta_1^m \delta_2^n
\Delta_i^{(p)} \Delta_j^{(q)} \rangle_c\ ,
\\
& W^{mn(pqr)}_{ijk} = \langle \delta_1^m \delta_2^n \Delta_i^{(p)}
\Delta_j^{(q)} \Delta_k^{(r)} \rangle_c\ ,
\end{split}
\end{equation}
and up to desired order, we have
\begin{equation}
\label{eq:xi-correlator-desired-order}
\begin{split}
& U_i = U^{(1)}_i + U^{(3)}_i + \cdots\ ,
\\
& U^{20}_i = U^{20(2)}_i + \cdots\ ,\quad U^{11}_i = U^{11(2)}_i +
\cdots\ ,
\\
& A_{ij} = A^{(11)}_{ij} + A^{(22)}_{ij} + A^{(13)}_{ij} +
A^{(31)}_{ij} + \cdots\ ,
\\
& A^{10}_{ij} = A^{10(12)}_{ij} + A^{10(21)}_{ij} + \cdots\ ,
\\
& W_{ijk} = W^{(112)}_{ijk} + W^{(121)}_{ijk} + W^{(211)}_{ijk} +
\cdots \ .
\end{split}
\end{equation} We refer the readers to appendices B through C in
\cite{2012arXiv1209.0780C} for details of evaluating those
correlators.
\subsection{The mean pairwise velocity in CLPT}
To compute the mean pairwise velocity in CLPT, we first evaluate
$K_{1,n}$, again making use of the cumulant expansion theorem:
\begin{equation}
\label{eq:v-kernel-k-full}
\begin{split}
& K_{1,n}(\lambda_1, \lambda_2, \vec{k}, \vec{q}) =
\\
& = \left. \exp{ \left[ \sum_{N=0}^\infty\dfrac{i^N}{N!}
\left\langle X^N \right\rangle_c \right] } \left[
\sum_{N=0}^\infty\dfrac{i^N}{N!} \left\langle
\dfrac{\dot{\Delta}_n}{H} X^N \right\rangle_c \right]\
\right|_{\vec{J}=0} \ .
\end{split}
\end{equation} Up to the second order of the linear power spectrum
[i.e. $O(P_L^2)$], Eq.~\eqref{eq:v-kernel-k-full} is recast as
\begin{equation}
\label{eq:v-kernel-k-2nd}
\begin{split}
& K_{1,n}(\lambda_1, \lambda_2, \vec{k}, \vec{q})
\\
& = f \e{-(1/2)A_{ij}k_ik_j}
\e{-(1/2)(\lambda_1^2+\lambda_2^2)\sigma_R^2}
\\
& \times \bigg\{ i(\lambda_1+\lambda_2)\dot{U}_n + i k_i
\dot{A}_{in} - \dfrac{1}{2}(\lambda_1^2+\lambda_2^2)
\dot{U}^{20}_n
\\
& - \lambda_1\lambda_2 \dot{U}^{11}_n - \dfrac{1}{2}
k_ik_j\dot{W}_{ijn} -(\lambda_1+\lambda_2) k_i \dot{A}^{10}_{in}
\\
& - i\lambda_1\lambda_2(\lambda_1+\lambda_2) \xi_L \dot{U}_n
-i(\lambda_1+\lambda_2)^2 k_i U_i \dot{U}_n
\\
& - i\lambda_1\lambda_2 \xi_L k_i\dot{A}_{in} -
i(\lambda_1+\lambda_2) k_ik_jU_i\dot{A}_{in} + O(P_L^3) \bigg\}\ .
\end{split}
\end{equation}
where we define (up to desired order)
\begin{equation}
\label{eq:vd-sym-def}
\begin{split}
& \dot{U}_n = \dfrac{\langle \delta_1 \dot{\Delta}_n \rangle}{f} =
U^{(1)}_n + 3 U^{(3)}_n + \cdots\ ,
\\
& \dot{U}^{20}_n = \dfrac{\langle \delta_1^2 \dot{\Delta}_n
\rangle}{f} = U^{20(2)}_n + \cdots\ ,
\\
& \dot{U}^{11}_n = \dfrac{\langle \delta_1 \delta_2 \dot{\Delta}_n
\rangle}{f} = U^{11(2)}_n + \cdots\ ,
\\
& \dot{A}_{in} = \dfrac{\langle \Delta_i \dot{\Delta}_n
\rangle}{f} = A^{(11)}_{in} + 3A^{(13)}_{in} + A^{(31)}_{in} +
2A^{(22)}_{in} + \cdots\ ,
\\
& \dot{A}^{10}_{in} = \dfrac{\langle \delta_1 \Delta_i
\dot{\Delta}_n \rangle}{f} = 2A^{10(12)}_{in} +
A^{10(21)}_{in}+\cdots\ ,
\\
& \dot{W}_{ijn} = \dfrac{\langle \delta_1 \Delta_i \Delta_j
\dot{\Delta}_n \rangle}{f} = 2 W^{(112)}_{ijn} + W^{(121)}_{ijn}
+ W^{(211)}_{ijn}+\cdots\ .
\end{split}
\end{equation}
Integrate with respect to $\lambda_1$ and $\lambda_2$, we have
\citep[see the appendices of][]{2012arXiv1209.0780C}
\begin{equation}
\label{eq:v-kernel-l-2nd}
\begin{split}
& L_{1,n} = \int \dfrac{\d \lambda_1}{2\pi} \dfrac{\d
\lambda_2}{2\pi} \tilde{F}(\lambda_1) \tilde{F}(\lambda_2)
K_{1,n}(\lambda_1, \lambda_2, \vec{k}, \vec{q})
\\
& = f \e{-(1/2)A_{ij}k_ik_j}
\e{-(1/2)(\lambda_1^2+\lambda_2^2)\sigma_R^2}
\\
& \times \bigg\{ 2 \fp{}\dot{U}_n + i k_i \dot{A}_{in} + \fpp{}
\dot{U}^{20}_n + \fp{}^2 \dot{U}^{11}_n
\\
& - \dfrac{1}{2} k_ik_j\dot{W}_{ijn} + 2i \fp{} k_i
\dot{A}^{10}_{in} + 2\fp{} \fpp{} \xi_L \dot{U}_n
\\
& + 2i[\fpp{}+\fp{}^2] k_i U_i \dot{U}_n + i \fp{}^2 \xi_L
k_i\dot{A}_{in}
\\
& - 2\fp{} k_ik_jU_i\dot{A}_{in} + O(P_L^3) \bigg\}\ .
\end{split}
\end{equation}
Then evaluate the integration over $\vec{k}$, we have
\begin{equation}
\label{eq:v-kernel-m-2nd}
\begin{split}
& M_{1,n} = \int \dfrac{\d^3 k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} L_{1,n}(\vec{q},\vec{k})
\\
& = \dfrac{f^2}{(2\pi)^{3/2}|A|^{1/2}}
\e{-(1/2)(A^{-1})_{ij}(q_i-r_i)(q_j-r_j)}
\\
& \times \bigg\{ 2 \fp{}\dot{U}_n - g_i\dot{A}_{in} + \fpp{}
\dot{U}^{20}_n + \fp{}^2 \dot{U}^{11_n}
\\
& - \dfrac{1}{2} G_{ij}\dot{W}_{ijn} - 2 \fp{} g_i
\dot{A}^{10}_{in} + 2\fp{} \fpp{} \xi_L \dot{U}_n
\\
& - 2 [\fpp{}+\fp{}^2] g_i U_i \dot{U}_n - \fp{}^2 \xi_L
g_i\dot{A}_{in}
\\
& - 2\fp{} G_{ij}U_i\dot{A}_{in} + O(P_L^3) \bigg\}\ ,
\end{split}
\end{equation}
and finally,
\begin{equation}
\label{eq:v-exp} v_{12,n}(\vec{r}) = [1+\xi(r)]^{-1} \int \d^3q\
M_{1,n}(\vec{r}, \vec{q})\ .
\end{equation}
Typically $v_{12,n}$ is projected along the direction
of pair separation vector, i.e. $v_{12} = v_{12,n}\hat{r}_n$.
\subsection{The pairwise velocity dispersion in CLPT}
The integration kernel for the velocity dispersion tensor is
\begin{equation}
\label{eq:vd-kernel-k-full}
\begin{split}
& K_{2,nm}(\lambda_1, \lambda_2, \vec{k}, \vec{q}) = \left. \left(
-i\dfrac{\partial}{\partial J_m}\right) \left(
-i\dfrac{\partial}{\partial J_n}\right) \left\langle
\e{iX(\vec{J})}\right\rangle \right|_{\vec{J}\rightarrow 0}
\\
& = \exp{ \left[ \sum_{N=0}^\infty\dfrac{i^N}{N!} \left\langle
X^N \right\rangle_c \right] } \Bigg\{ \left[
\sum_{N=0}^\infty\dfrac{i^N}{N!} \left\langle \dot{\Delta}_n
\dot{\Delta}_m X^N \right\rangle_c \right]
\\
& \quad + \left[ \sum_{N=0}^\infty\dfrac{i^N}{N!} \left\langle
\dot{\Delta}_n X^N \right\rangle_c \right] \left[
\sum_{M=0}^\infty\dfrac{i^M}{M!} \left\langle \dot{\Delta}_m
X^M \right\rangle_c \right] \Bigg\}\ ,
\end{split}
\end{equation}
Expanding Eq.~\eqref{eq:vd-kernel-k-full} to second order in the
linear power spectrum (i.e.~to the order of $O(P_L^2)$):
\begin{equation}
\label{eq:vd-kerner-k-2nd}
\begin{split} &
K_{2,nm}(\lambda_1, \lambda_2, \vec{k}, \vec{q})
\\
& = f^2 \e{-(1/2)A_{ij}k_ik_j}
\e{-(1/2)(\lambda_1^2+\lambda_2^2)\sigma_R^2}
\\
& \times \big\{ (\lambda_1+\lambda_2)^2 \dot{U}_n \dot{U}_m -
(\lambda_1+\lambda_2) ( \dot{A}_{in}k_i\dot{U}_m +
\dot{A}_{im}k_i\dot{U}_n)
\\
& - \dot{A}_{im}k_i\dot{A}_{jn}k_j + [1 -\lambda_1\lambda_2 \xi_L -
(\lambda_1+\lambda_2) U_ik_i ] \ddot{A}_{nm}
\\
& + i (\lambda_1+\lambda_2) \ddot{A}^{10}_{nm} + i
\ddot{W}_{inm}k_i + O(P_L^3) \big\}\ ,
\end{split}
\end{equation}
where we define (up to the desired order)
\begin{equation}
\label{eq:vd-sym-def}
\begin{split}
& \ddot{A}_{nm} = \dfrac{\langle \dot{\Delta}_n\dot{\Delta}_m
\rangle}{f^2} = A^{(11)}_{nm} + 3 A^{(13)}_{nm} + 3
A^{(31)}_{nm} + 4 A^{(22)}_{nm}\ ,
\\
& \ddot{A}_{10,nm} = \dfrac{\langle \delta_1
\dot{\Delta}_n\dot{\Delta}_m \rangle}{f^2} = 2A^{10(12)}_{nm} +
2A^{10(21)}_{nm}\ ,
\\
& \ddot{W}_{inm} = \dfrac{\langle \delta_1 \Delta_i
\dot{\Delta}_n\dot{\Delta}_m \rangle}{f^2} = 2 W^{(112)}_{inm} +
2 W^{(121)}_{inm} + W^{(211)}_{inm}\ .
\end{split}
\end{equation}
Then evaluate the integration with respect to $\lambda_1$,
$\lambda_2$, we have
\begin{equation}
\label{eq:vd-kernel-l-2nd}
\begin{split}
& L_{2,nm} = \int \dfrac{\d \lambda_1}{(2\pi)} \dfrac{\d
\lambda_2}{(2\pi)} \tilde{F}(\lambda_1) \tilde{F}(\lambda_2)
K_{2,nm}(\lambda_1, \lambda_2, \vec{k}, \vec{q})
\\
& = \big\{ 2 [\fp{}^2 + \fpp{} ] \dot{U}_n \dot{U}_m + 2 i \fp{} (
\dot{A}_{in}k_i\dot{U}_m + \dot{A}_{im}k_i\dot{U}_n)
\\
& - \dot{A}_{im}k_i\dot{A}_{jn}k_j + [1 + \fp{}^2 \xi_L + 2i \fp{}
U_ik_i ] \ddot{A}_{nm}
\\
& + 2\fp{} \ddot{A}^{10}_{nm} + i \ddot{W}_{inm}k_i + O(P_L^3)
\big\} f^2 \e{-(1/2)A_{ij}k_ik_j}\ ,
\end{split}
\end{equation}
and then with respect to $\vec{k}$:
\begin{equation}
\label{eq:vd-kernel-m-2nd}
\begin{split}
& M_{2,nm} = \int \dfrac{\d^3 k}{(2\pi)^3}
\e{i\vec{k}\cdot(\vec{q}-\vec{r})} L_{2,nm}(\vec{q},\vec{k})
\\
& = \dfrac{f^2}{(2\pi)^{3/2}|A|^{1/2}}
\e{-(1/2)(A^{-1})_{ij}(q_i-r_i)(q_j-r_j)}
\\
& \times \big\{ 2 [\fp{}^2 + \fpp{} ] \dot{U}_n \dot{U}_m - 2
\fp{} ( \dot{A}_{in}g_i\dot{U}_m + \dot{A}_{im}g_i\dot{U}_n)
\\
& - \dot{A}_{im}\dot{A}_{jn}G_{ij} + [1 + \fp{}^2 \xi_L - 2\fp{}
U_ig_i ] \ddot{A}_{nm}
\\
& + 2\fp{} \ddot{A}^{10}_{nm} - \ddot{W}_{inm}g_i + O(P_L^3)
\big\}\ .
\end{split}
\end{equation}
Finally, $\vd{12,nm}$ can be obtained by
\begin{equation}
\label{eq:vd-exp}
\vd{12,nm}(\vec{r}) = [1+\xi(r)]^{-1} \int \d^3q\
M_{2,nm}(\vec{r}, \vec{q})\ .
\end{equation}
Desired component of pairwise velocity dispersion can be obtained by
different components or kinds of contractions of the tensor
$\vd{12,nm}$. In order to obtain the velocity dispersion components
parallel to and perpendicular to the pairwise separation unit vector
$\hat{r}$, we project $\vd{12,nm}$ into different directions:
\begin{equation}
\label{eq:vd-components}
\vd{\parallel} = \vd{12,nm} \hat{r}_n
\hat{r}_m\ ,\quad \vd{\bot} = ( \vd{12,nm} \delta^K_{nm} -
\vd{\parallel} ) / 2\ .
\end{equation}
\subsection{Cross-correlation of halos with different bias parameters}
\label{sec:cross-corr-theory}
It is a straightforward generalization of the above to handle
cross-correlations between two tracers with different biases. We note
that the displacement field $\vec{\Psi}$ is identical for all species
-- the difference is only in their bias parameters. Therefore, in
this ``cross-correlation'' scenario we have different $\tilde{F_j}$
($j=1$ or $2$) for $\lambda_1$ and $\lambda_2$. Equation
\eqref{eq:bias-expect-identity} is hence recast as:
\begin{equation}
\label{eq:bias-expect-identity-different-species}
\int \dfrac{\d \lambda_j}{2\pi} \tilde{F_j}(\lambda_j)
\e{\lambda_j^2\sigma_R^2/2} (i\lambda_j)^n = \langle F_j^{(n)}
\rangle\ ,\quad j = 1,\ 2\ .
\end{equation}
Hence we can adopt a list of transformations for bias
parameters to obtain cross-correlation between different species,
which can be straightforwardly deduced from
Eq.~\eqref{eq:bias-expect-identity-different-species}:
\begin{equation}
\label{eq:bias-transform-cross-correlation}
\begin{split}
& (\lambda_1 + \lambda_2) \rightarrow [\fp{1} + \fp{2}]\ ,
\\
& (\lambda_1^2 + \lambda_2^2 ) \rightarrow [\fpp{1} + \fpp{2}]\ ,
\\
& \lambda_1 \lambda_2 \rightarrow \fp{1} \fp{2}\ ,
\\
& \lambda_1^2 \lambda_2^2\rightarrow \fpp{1} \fpp{2}\ ,
\\
& \lambda_1 \lambda_2 (\lambda_1 + \lambda_2) \rightarrow
[ \fp{1}\fpp{2} + \fp{2}\fpp{1} ]\ .
\end{split}
\end{equation}
Those transformations can also be derived and verified by using the
symmetry in $\lambda_1$ and $\lambda_2$ of the relevant
expressions. They can be applied to Eqs.~\eqref{eq:xi-kernel-m},
\eqref{eq:v-kernel-m-2nd}, and \eqref{eq:vd-kernel-m-2nd}, with
$\fp{j}$ and $\fpp{j}$ ($j=1\ ,2$) being bias parameters for two
different species. It is easy to verify that the cross-correlation
expressions reduce to the auto-correlation expressions when
$\fp{1}=\fp{2}$ and $\fpp{1}=\fpp{2}$.
\section{The Gaussian streaming model}
\label{sec:GSM}
Both iPT and CLPT have difficulties reproducing the redshift space
clustering of biased tracers on small scales. An alternative is the
``Gaussian streaming model'' introduced in
\citet{2011MNRAS.417.1913R}, which takes as inputs perturbation theory
expressions for the real-space correlation function and the velocity
statistics.
The clustering of a population of objects in redshift-space can be
related to their underlying real-space clustering and the full
pairwise velocity distribution by \citep{1995ApJ...448..494F,
2004PhRvD..70h3007S}
\begin{equation}
1+\xi(r_p, r_\parallel) = \int_{-\infty}^{\infty}dy\
[1+\xi(r)] \mathcal{P}(v_z = r_\parallel - y, \vec{r}).
\end{equation}
Here $r_p$ is the transverse separation in both real and
redshift-space, $r_\parallel$ is the LOS pair separation in redshift
space, and $y$ is the LOS separation in real-space, so that $r^2 =
r_p^2 + y^2$. \citet{2011MNRAS.417.1913R} showed that even though the
true $\mathcal{P}$ is certainly non-Gaussian, approximating it with a
Gaussian provides an accurate description of the redshift space
correlation function of massive halos:
\begin{equation}
\label{eq:streaming-xi-s}
\begin{split}
1 + \xi^s(r_p, r_\parallel) & = \int \dfrac{\d y} {[2\pi
\vd{12}(r,\mu)]^{1/2}} [1 + \xi(r)] \\ & \times \exp \left\{
-\dfrac{[r_\parallel - y - \mu v_{12}(r)]^2}{2\vd{12}(r,\mu)}
\right\}\ ,
\end{split}
\end{equation}
In the scale-dependent Gaussian streaming model, the Gaussian
probability distribution function is centered at $\mu v_{12}(r)$, the
mean LOS velocity between a pair of tracers as a function of their
real space separation:
\begin{equation}
\label{eq:v-infall-def}
v_{12}(r) \vec{\hat{r}} = \frac{\left\langle
[1+\delta(\vec{x})] [1+ \delta(\vec{x}+\vec{r})]
[\vec{v}(\vec{x}+\vec{r}) - \vec{ v}(\vec{x})] \right\rangle}
{\left\langle [1+\delta(\vec{x})] [1+ \delta(\vec{x}+\vec{r})]
\right\rangle}
\end{equation}
The factor $[1+\delta(\vec{x})] [1+ \delta(\vec{x}+\vec{r})]$ in the
numerator and the denominator specifies that we are computing the
average relative velocity over pairs of tracers, rather than over
randomly chosen points in space. By symmetry, the mean velocity is
directed along the pair separation vector; projecting it onto the LOS
brings a factor of $\mu = y/r$ in Eq.~\eqref{eq:streaming-xi-s}.
Similarly, the width of the velocity PDF is different for components
along and perpendicular to the pair separation vector $\hat{r}$, so
the LOS ($\hat{z}$) velocity dispersion can be decomposed as a sum
with contributions from two one-dimensional velocity dispersions.
\begin{equation}
\label{eq:vdisp-sigma-mu}
\begin{split}
\vd{12}(r, \mu) & = \dfrac{\left\langle [1+\delta(\vec{x})] [1+
\delta(\vec{x}+\vec{r})] [v_z(\vec{x}+\vec{r}) -
v_z(\vec{x})]^2 \right\rangle}{\left\langle
[1+\delta(\vec{x})] [1+\delta(\vec{x}+\vec{r})] \right\rangle}
\\
& = \mu^2 \vd{\parallel}(r) + (1-\mu^2) \vd{\bot}(r)\ .
\end{split}
\end{equation}
Linear theory expressions for $v_{12}(r)$ and $\vd{\bot,\parallel}(r)$
are given in \citet{1995ApJ...448..494F, 1988ApJ...332L...7G,
1989ApJ...344....1G, 2011MNRAS.417.1913R}. One finds that the
pairwise mean infall velocity, $v_{12}(r)$, is proportional to $bf$,
while $\vd{12}(r)$ scales as $f^2$ with no dependence on the
large-scale bias $b$ at linear order.
\citet{2011MNRAS.417.1913R} evaluated Eqs.~\eqref{eq:v-infall-def} and
\eqref{eq:vdisp-sigma-mu} in standard perturbation theory under the
assumption of a linear bias $b$ relating the tracer and matter density
fields, $\delta_t(\vec{x}) = b\,\delta_m(\vec{x})$. There were several
shortcomings of this approach however. First, standard perturbation
theory does an unsatisfactory job of describing the smoothing of the
BAO features in the real-space correlation function. As a result, the
analysis of \citet{2012MNRAS.426.2719R} used iPT to model
$\xi^s(r_p, r_\parallel)$
above separations of $70\,\Mpch$. Second, the inaccuracy of the
streaming model results with standard perturbation theory inputs for
the velocity statistics can be traced to inaccuracies in the
perturbative calculation of $v_{12}$ and its derivative, $\d v_{12}/\d
r$. This inaccuracy was smallest for halos with second-order bias
near zero, which raises the question of whether the source of
inaccuracy was the neglect of second order bias terms in the
\citet{2011MNRAS.417.1913R} calculation. CLPT naturally includes
higher-order bias corrections, and will allow us to quantify the size
of the second order contributions to the velocity statistics of
interest. For these reasons, we shall consider the combination of
CLPT statistics within the Gaussian streaming model ansatz.
\section{Results}
\label{sec:results-statistics}
We implemented the formulae above in a C++ code\footnote{The code is
available at {\tt https://github.com/wll745881210/CLPT\_GSRSD.git}},
which numerically evaluates the integrations in
Eqs.~\eqref{eq:xi-exp}, \eqref{eq:v-exp} and \eqref{eq:vd-exp} for
CLPT statistics, and in Eq.~\eqref{eq:streaming-xi-s} for the Gaussian
streaming model. In this section we present the results, and compare
them with pertinent simulation statistics. The $N$-body simulation
set used in this work is described in more detail in
\citep{2011MNRAS.417.1913R, 2011ApJ...728..126W}, in which the
halo catalogues are constructed by FoF method. Table
\ref{table:bias-parameters} lists the halo mass bins we use to compare
with our analytic predictions.
\subsection{Auto-correlation of halos}
\label{sec:auto-corr}
\subsubsection{Real-space auto-correlation statistics}
\label{sec:real-space-auto}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]{xi_r.eps}
\caption{Real-space correlation function for halos in different mass
bins (refer to Table \ref{table:bias-parameters}). CLPT results
are given by heavy dashed curves; simulation results are presented
by shaded bands showing the error range. They are divided by
linear theory results ($\xi_\mathrm{lin}(r)$) to remove the trend,
and different mass bins are elevated by different constants
(labelled in the figure) to show each more clearly. The bias
parameters, $\fp{}$ and $\fpp{}$, for the CLPT model are given in
Table \ref{table:bias-parameters}.}
\label{fig:real-space-xi}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]{v12_r.eps}
\caption{Pairwise infall velocity of halos in five different mass
bins. Our CLPT results are shown by heavy dashed curves and the
SPT results \citep{2011MNRAS.417.1913R} are shown by heavy dotted curves.
Both are to be compared with the simulation results shown by shaded bands
(indicating the error on the mean of the simulations).
All results are divided by the absolute value of
linear theory results ($|v_{12,\mathrm{lin}}(r)|$) for better
comparison, and are elevated by different constants.}
\label{fig:real-space-v12}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm,keepaspectratio]{sigma12_r.eps}
\caption{Pairwise velocity dispersion for halos in five different
mass bins. For clearer presentation we show the values of
$\vd{\parallel} / \vd{\parallel,\mathrm{lin}}$ (upper panel) and
$\vd{\bot} / \vd{\bot,\mathrm{lin}}$ (lower panel). Shaded bands
(very narrow) show the simulation results, the SPT results
\citep{2011MNRAS.417.1913R} are presented by heavy dotted curves,
and our theory prediction is presented by dashed curves. Different
curves are elevated by different constants.}
\label{fig:real-space-sigma12}
\end{figure}
\begin{table}
\small
\centering
\caption{Bias parameters, as well as the uncertainty (denoted by
$\sigma$) obtained by fitting the real-space correlation functions
for different halo mass bins. Please note that $\fp{}$ and
$\fpp{}$ are both free parameters determined by fitting and $\sigma$
is the formal error on the fit assuming Gaussian, uncorrelated errors
on $\xi$ as determined from the variance in the simulations.}
\label{table:bias-parameters}
\vspace{0.3cm}
\begin{tabular}{ccccc}
\hline
$\lg{[M/(M_\odot h^{-1})]}$ & $\fp{}$ & $\sigma(\fp{})$
& $\fpp{}$ & $\sigma(\fpp{})$ \\
\hline
$12.182-12.483$ & $0.341$ & $0.004$ & $0.06$ & $0.06$ \\
$12.484-12.784$ & $0.435$ & $0.004$ & $0.16$ & $0.06$ \\
$12.785-13.085$ & $0.652$ & $0.005$ & $0.16$ & $0.01$ \\
$13.086-13.386$ & $0.965$ & $0.006$ & $0.14$ & $0.09$ \\
$> 13.387$ & $1.738$ & $0.007$ & $-0.10$ & $0.11$ \\
\hline
\end{tabular}
\end{table}
We present the CLPT predictions of real-space statistics in this
section, which will be used as the ``input'' of the Gaussian streaming
redshift-space distortion model. All calculations are compared with
pertinent results in Section \ref{sec:v-sigma}.
We treat $\fp{}$ and $\fpp{}$ as free parameters in our model and fit
them to the real-sapce correlation function, $\xi(r)$, measured in the
N-body simulations for each halo mass bin. We treat all of the
$\xi(r)$ bins as independent and use the inverse variance obtained from
the simulation. While it is incorrect to neglect the
correlations, one can see by eye that the resulting best-fit (and
the value of $\chi^2$) are entirely reasonable. The resulting
values are listed in Table \ref{table:bias-parameters}. We note here
that it is also possible to obtain $\fpp{}$ as a function of $\fp{}$
using the peak-background split relation \citep[as
in][]{2008PhRvD..78h3519M}. While the relation between our best-fit
$\fp{}$ and halo mass is close to that obtained from the
peak-background split the values of $\fpp{}$ can differ significantly.
Imposing the peak-background split value of $\fpp{}$ has only a modest
effect on the shape of the correlation function on the scales of
interest however, and does not change our conclusions in any
qualitative way. $\fpp{}$ is also not well constrained in our fitting
(see Table \ref{table:bias-parameters} for the uncertainty of $\fpp{}$),
which confirms that $\fpp{}$ does not have a considerable impact on the
correlation function on the scales of interest.
Fig.~\ref{fig:real-space-xi} compares the real-space correlation
function predicted by CLPT with that measured in the simulations.
Note that the consistency between CLPT results and simulations is
almost perfect from $\lsim 10\Mpch$ through the BAO scale ($\sim
110\Mpch$), as also seen in \citet{2012arXiv1209.0780C}. The
redshift-space correlation function predicted directly from CLPT was
presented in \citet{2012arXiv1209.0780C}. Here we want to examine the
velocity statistics themselves.
We assume that the values of $\fp{}$ and $\fpp{}$ obtained by fitting
the real-space correlation functions are the right ones for evaluating
the velocity statistics. Using these values in
Eq.~\eqref{eq:v-kernel-m-2nd} and \eqref{eq:vd-kernel-m-2nd}, we
obtain the scale dependence of the pairwise infall velocity and
velocity dispersion. The CLPT results (divided by the linear theory
as fiducial values) are compared with simulations in
Figs.~\ref{fig:real-space-v12} and \ref{fig:real-space-sigma12}. The
CLPT predictions for $v_{12}$ are better than the SPT predictions with
first order bias presented in \citet{2011MNRAS.417.1913R} for all but
the highest mass bin. The CLPT predictions of the pairwise infall
velocity statistics can be slightly improved by varying $\fpp{}$,
but the prediction of velocity dispersion is quite insensitive to
$\fpp{}$.
When comparing the CLPT result to $\vd{\parallel}(r)$ and
$\vd{\bot}(r)$ we add a constant to the predictions so that they take
the same value as the simulation at $r=130\Mpch$. The constant
offsets for $\vd{\parallel}(r)$ and $\vd{\bot}(r)$ are almost the
same, with only $\sim 1$ per cent relative difference. These two
constants are similar to what the authors referred to in
\citet{2011MNRAS.417.1913R}: the CLPT prediction of absolute value of
$\vd{\parallel}(r)$ and $\vd{\bot}(r)$ is not correct, but a constant
shift over the whole range of scales reveals that the CLPT results
have correct trend. The possible reason for this is that the velocity
dispersion component yielded by gravitational evolution on smallest
scales, which should be separated from the overall scale dependence of
pairwise velocity dispersion, is not able to be predicted by
perturbation theory: this suggests that we should evaluate the
constant shift as a fitting parameter. Our CLPT predictions have a
similar accuracy to the SPT predictions in \citet{2011MNRAS.417.1913R}
for the second highest mass bin. However, it is not clear that CLPT
accurately captures the bias-dependence of the deviations from linear
theory for $\vd{\parallel}(r)$ and $\vd{\bot}(r)$.
\subsubsection{Redshift-space distortion for auto-correlation}
\label{sec:redshift-space-stat}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]{xi_s_0.eps}
\caption{Redshift-space statistics obtained by Gaussian streaming
model specified in \citet{2011MNRAS.417.1913R}, showing
$\xi^s_0/\xi^s_{0,\mathrm{lin}}$. The shaded bands present
simulation values (showing the error range) and our CLPT values
are presented by heavy dashed curve. Each mass bin is elevated by
a different constant.}
\label{fig:redshift-monopole}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]{xi_s_2.eps}
\caption{Redshift-space results of $\xi^s_2/\xi^s_{2,\mathrm{lin}}$.
Labels and curve indications are identical to
Fig.~\ref{fig:redshift-monopole}.}
\label{fig:redshift-quadrupole}
\end{figure}
The redshift-space correlation functions depend on the angle between
separation vectors and LOS. This ``direction dependency'' can be
expanded into series with respect to Legendre polynomials
(Eq.~\ref{eq:legendre-expansion}), or, equivalently:
\begin{equation}
\label{eq:multipole-expansion}
\xi^s_l(s) = \dfrac{2l+1}{2}
\int_{-1}^1\d\mu_s\ L_l(\mu_s) \xi^s_l(s,\mu)\ ,
\end{equation}
where $s = (r_p^2 + r_\parallel^2)^{1/2}$ is the redshift-space
distance, and $\mu_s = r_\parallel / s$ is the cosine of the angle
between separation vector and LOS. Generally we are most interested in
the lowest non-zero moments, i.e. $l = 0,\ 2$. In
Figs.~\ref{fig:redshift-monopole} and \ref{fig:redshift-quadrupole},
we present the lowest two non-zero multipole moments divided by linear
theory results \citep[see] [for the linear theory expessions]
{1995ApJ...448..494F, 2011MNRAS.417.1913R}.
We show the results of the multipole expansion in
Figs.~\ref{fig:redshift-monopole} and \ref{fig:redshift-quadrupole}.
The accuracy of the prediction of CLPT with the Gaussian streaming
model is at the several per cent level on scales larger than $\gsim
20\Mpch$, and no worse than $\sim 10$ per cent even down to $\sim
10\Mpch$. The agreement remains equally good at BAO scales, but we
only show $10\Mpch < s < 70\Mpch$ for clearer presentation at smaller
scales. The theory breaks down at $r\lsim 10\Mpch$ where the
correlation function amplitude is approaching $O(1)$. For the
quadrupole moment ($\ell=2$), we observe that the model has reasonable
overlap with the simulations throughout the scales of general interest
($10\Mpch \lsim s \lsim 120\Mpch$).
To further isolate the regions where the theory and N-body simulations
are in good agreement we additionally examine the ``wedge'' statistics
\citep[e.g.][]{2012MNRAS.419.3223K}, defined by
\begin{equation}
\label{eq:wedge-definition}
\xi^s_\mathrm{wedge}(s,\mu_\mathrm{min},\mu_\mathrm{max}) =
\dfrac{1}{\Delta\mu} \int_{\mu_\mathrm{min}}^{\mu_\mathrm{max}}
\xi^s(s,\mu) \d \mu\ ,
\end{equation}
where $\Delta\mu=\mu_\mathrm{max}-\mu_\mathrm{min}$. In this paper we
use three such ``wedges'', which are denoted by
$\xi_\mathrm{w0}=\xi^s_\mathrm{wedge}(s,0,1/3)$,
$\xi_\mathrm{w1}=\xi^s_\mathrm{wedge}(s,1/3,2/3)$ and
$\xi_\mathrm{w2}=\xi^s_\mathrm{wedge}(s,2/3,1)$. The predictions for
the $\xi_{\mathrm{w}i}$ are compared to N-body simulations in
Fig.~\ref{fig:wedge}. Note that the fractional deviations from linear
theory are largest on small scales and when $\mu\simeq 1$. In
addition the inaccuracy of our theoretical prediction for the
quadrupolar moment on about $\sim 10\Mpch$ can be attributed to the
disagreement near $\mu\simeq 1$ (please note that $\xi^s_2$ is
negative around $r\sim 10\Mpch$ but $\xi_\mathrm{w2}$ is positive
there). On scales above $20\Mpch$ our model works well, the
difference between the model and N-body results is less than 5 per
cent for all three wedges.
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{xi_wedge.eps}
\caption{Wedge statistics showing $\xi_\mathrm{w0}$,
$\xi_\mathrm{w1}$ and $\xi_\mathrm{w2}$ (see
Eq.~\ref{eq:wedge-definition}) in redshift-space. The scheme of
presentation is similar to Fig.~\ref{fig:redshift-monopole}.}
\label{fig:wedge}
\end{figure}
In order to provide another view of the disagreement between the model
and simulations, we show in Fig.~\ref{fig:2d-contour} contours of
$\xi$ predicted by the analytic model (dashed contours) and N-body
simulations (solid contours) for two bins in halo mass. We can
clearly observe that, for the halos in the lower mass bin, $\xi^s$ is
less precisely predicted around $\mu\simeq 1$. On larger scales
($s\gsim 20\Mpch$) simulation results are accurately predicted for
both mass bins.
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]{xi_2d.eps}
\caption{Contour plots that compare theoretical predictions (dashed)
and simulation results (solid). The contour lines for $\xi^s=0.8,\
0.4,\ 0.2,\ 0.08\ ,$ and $0.04$ are presented in the figure. We
note that the dashed and solid contours are nicely overlapped in
most areas, except in the regions that $s \lsim 10\ \Mpch$ and
$\mu \lsim 1$. }
\label{fig:2d-contour}
\end{figure}
\subsection{Cross correlation between halos and dark matter particles}
\label{sec:cross-corr-halo-dm}
As shown in subsection \ref{sec:cross-corr-theory}, CLPT theory is
also capable of making predictions for cross-correlations. Here we
compare statistics predicted by CLPT and the Gaussian streaming model
with those given by simulations, for cross-correlations of halos with
dark matter particles in the simulations.
In this subsection we adopt the same bias parameters as in Table
\ref{table:bias-parameters}.
We can obtain a better match to the cross-correlation infall velocities
by adjusting $\fp{}$ and $\fpp{}$, however those values do not provide a
good match to the real-space cross-correlation function suggesting either
that our bias model is too simple or the improved agreement reflects a
breakdown of perturbation theory.
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{xi_r_cross.eps}
\caption{Real-space cross-correlation function between dark matter
and halos (in five different mass bins; each mass bin is elevated
by a specific constant). The scheme of presentation is similar to
Fig.~\ref{fig:real-space-xi}.}
\label{fig:real-space-xi-cross}
\end{figure}
\begin{figure} \centering
\includegraphics[width=80mm, keepaspectratio]
{v12_r_cross.eps}
\caption{Pairwise infall velocity as cross-correlations. Our CLPT
results are shown by a heavy solid curve, compared with
simulations presented by shaded bands showing the error range.
The scheme of presentation is similar to
Fig.~\ref{fig:real-space-v12}.}
\label{fig:real-space-v12-cross}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{sigma12_r_cross.eps}
\caption{Pairwise velocity dispersion as cross-correlations.
Similar to Fig.~\ref{fig:real-space-sigma12}, we also show the
values of $\vd{\parallel} / \vd{\parallel,\mathrm{lin}}$ (upper
panel) and $\vd{\bot} / \vd{\bot,\mathrm{lin}}$ (lower panel),
with shaded bands (simulation results) and dashed curves
(theoretical predictions). Different mass bins are elevated by
different constants. }
\label{fig:real-space-sigma12-cross}
\end{figure}
Real-space statistics are presented by
Figs.~\ref{fig:real-space-xi-cross} through
\ref{fig:real-space-sigma12-cross}. Predictions of CLPT for pairwise
infall velocity (Fig.~\ref{fig:real-space-v12-cross}) in such a
cross-correlation case is not as good as the prediction for
auto-correlation (Fig.~\ref{fig:real-space-v12}), but still
satisfactory; the discrepancy is $\lsim 5$ per cent throughout the
scales of interest. The real-space correlation function, $\xi(r)$,
and the velocity dispersion, $\vd{12}(r)$, on the other hand, are
still accurately predicted by the theory.
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{xi_s_0_cross.eps}
\caption{Monopole moment of redshift-space cross-correlation
function between halos and dark matter. Results are divided by
fiducial linear theory results (i.e. $\xi^s_0/\xi^s_{0,lin}$ is
presented). The scheme of presentation is similar to
Fig.~\ref{fig:redshift-monopole}.}
\label{fig:redshift-monopole-cross}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{xi_s_2_cross.eps}
\caption{Quadrupole moment of redshift-space cross-correlation
function between halos and dark matter (linear theory results as
fiducial value) $\xi^s_2/\xi^s_{2,lin}$. The scheme of
presentation is similar to
Fig.~\ref{fig:redshift-monopole-cross}.}
\label{fig:redshift-quadrupole-cross}
\end{figure}
Inserting the real-space statistics shown by
Figs.~\ref{fig:real-space-xi-cross} through
\ref{fig:real-space-sigma12-cross} into Eq.~\eqref{eq:streaming-xi-s},
we get the redshift-space correlation function, which is also expanded
with respect to Legendre polynomials as in
Eq.~\eqref{eq:multipole-expansion}. Similar to Section
\eqref{sec:redshift-space-stat}, this section also presents monopole
and quadrupole moments in Figs.~\ref{fig:redshift-monopole-cross} and
\ref{fig:redshift-quadrupole-cross}. While we focus on $10\Mpch < s <
70\Mpch$, the agreement remains good on BAO scales. Although the
predictions for the velocity statistics are not as good as in the
auto-correlation case, the behavior of the multipole moments is still
well sketched by CLPT and the Gaussian streaming model (to $\sim 10$
per cent, even on scales of $\sim 10\Mpch$). The manner in which
$\xi^s_0/\xi^s_\mathrm{0,lin}$ and $\xi^s_2/\xi^s_\mathrm{2,lin}$ vary
with $s$ is still correct to quite small $s$.
Similar procedures also produce predictions for halo-halo cross
correlations. In Fig.~\ref{fig:halo-halo-cross} we compare the
statistics as cross-correlations betweeen halos in two different
mass bins: $12.182 < \lg[M/(M_\odot h^{-1})] < 12.483$ and
$\lg[M/(M_\odot h^{-1})] > 13.387$. There are no adjustable parameters
in this comparison, because the values of $\fp{}$ and $\fpp{}$ are fixed
by the auto-correlations.
Good agreement between our theoretical model and the simulations is
still observed, even down to the scale of $s\gsim 10\Mpch$.
\begin{figure}
\centering
\includegraphics[width=80mm, keepaspectratio]
{stats_cross_hh.eps}
\caption{Statistic functions as cross-correlations betweeen halos in
two different mass bins: $12.182 < \lg[M/(M_\odot h^{-1})] <
12.483$ and $\lg[M/(M_\odot h^{-1})] > 13.387$. The curvs, lines
and shaded bands in the panels have similar indications to
Figs.~\ref{fig:real-space-xi} through
\ref{fig:redshift-quadrupole}, which compare our theoretical
predictions with simulations. From top to bottom: real-space
correlation function (first panel); pairwise infall velocity
(second panel); velocity dispersion parallel (third panel) and
perpandicular (fourth panel) to separation vector; monopole (fifth
panel) and quadrupole (sixth panel) moment of redshift-space
correlation function.}
\label{fig:halo-halo-cross}
\end{figure}
\section{Discussion and summary}
\label{sec:summary}
By introducing an auxiliary term $\vec{J}$ in the generating function,
we generalize the CLPT scheme elaborated in
\citet{2012arXiv1209.0780C} to estimate the pairwise infall velocity
and velocity dispersion as functions of pair separation. This allows
a self-consistent calculation of these statistics for biased tracers,
including scale-dependent or higher-order bias terms. Indeed we find
that CLPT gives better estimates for the magnitude of the pairwise
infall velocity, $v_{12}(r)$, than to quasi-linear theory with a
``linear'' bias \citep{2011MNRAS.417.1913R} for a wide range of halo
masses.
The $\xi(r)$, $v_{12}(r)$ and $\vd{12}(r)$ predicted by CLPT can be
used as inputs to Gaussian streaming model
(Eq.~\ref{eq:streaming-xi-s}) to obtain predictions for the
redshift-space correlation function of halos. For the monopole and
quadrupole moments of the correlation function the agreement between
theory and N-body simulations is at the few per cent level down to
$\sim 15\Mpch$, and $\sim 10$ per cent at $\sim 10\Mpch$. We infer
that the Gaussian streaming model of redshift-space distortion is not
sensitively affected by $\vd{12}(r)$, but the small scale statistics
are enhanced by better estimations of $v_{12}(r)$, compared with
semi-linear results in \citet{2011MNRAS.417.1913R}. We attribute the
enhanced results to our inclusion of higher order (one-loop) terms and
the resummation scheme employed in CLPT.
It is worth noting that the argeement between the CLPT-Gaussian
streaming quadrupolar moment and N-body simulation
(e.g.~Fig.~\ref{fig:redshift-quadrupole}) is consideribaly better than
the ``original'' SPT scheme in \citet{2011MNRAS.417.1913R}. From
Figs.~\ref{fig:wedge} and \ref{fig:2d-contour} we observe that the
theoretical predictions are not sufficiently accurate only around the
region where $s\lsim 20\Mpch$ and $\mu\simeq 1$ (or $r_p\simeq 0$) for
the lower mass bins. It was shown in \citet{2011MNRAS.417.1913R}
(e.g.~their figure 6) that the Gaussian streaming model predicted the
redshift-space correlation function well when accurate ``inputs''
(i.e. $v_{12}(r)$ and $\vd{12}(r)$) were used. Prediction of these
inputs using CLPT seems a reliable way of computing redshift-space
statistics for tracers with a local Lagrangian bias.
We also extended the CLPT-Gaussian streaming model to
cross-correlations between differently biased tracers. As an example,
we modelled the monopole and quadrupole moments of the redshift-space
cross-correlation function between halos and dark matter, and
between halos in different mass bins. The agreement with
N-body simulations for $v_{12}(r)$ was not as good as in the
auto-correlation case, as expected, but the distortions were still
accurately revealed in monopole and quadrupole moments. This is not
unexpected: it was already noted by \citet{2011MNRAS.417.1913R} that
the Gaussian approximation for the velocity PDF worked much better for
halos in simulations than for the dark matter particles themselves.
\bibliographystyle{mn2e}
|
2,877,628,089,507 | arxiv | \section{Introduction}\label{sec1}
The emergence of high-throughput technologies has\break made it
feasible to measure molecular signatures of thousands of
genes/\break proteins simultaneously. This provides scientists an
opportunity to study the global genetic regulatory networks,
shedding light on the functional interconnections among the
regulatory genes, and leading to a better understanding of
underlying biological processes. In this paper, we propose a network
building procedure for learning genetic regulatory networks. Our
work is motivated by an expression study of breast cancer (BC) that
aims to infer the network structure based on 414 BC tumor samples
[\citet{Loietal}]. The proposed method enables us to
detect high-confidence edges and well-connected hub genes that
include both those previously implicated in BC and novel ones that
may warrant further follow-up.
In practice, dependency structures of molecular activities such as
correlation matrix and partial correlation matrix have been used to
infer regulatory networks [\citet{Poletal02}, \citet{Kimetal06},
\citet{Varetal05},
Nie, Wu and Zhang (\citeyear{NieWuZha06})]. Such dependency structures are often
represented by graphical models in which nodes of a graph represent
biological components such as genes or proteins, and the edges
represent their interactions. These interactions may be indirect
(e.g., two genes are co-regulated by a third gene) or direct (e.g.,
one gene is regulated by another gene). For the latter case,
Gaussian Graphical Models (GGMs), which represent dependencies
between pairs of nodes conditioning on the remaining of nodes, are
often used.
For the data obtained from high-throughput technologies,
the number of nodes is typically much larger than the number of
samples, which is where the classical GGM theory [\citet{Whi90}]
generally fails [\citet{Fri89}, Scha\"{a}fer and
Strimmer (\citeyear{SchStr05})]. This large-$p$-small-$n$ scenario is usually
addressed by assuming that the conditional dependency structure is
sparse [\citet{Dobetal04}, \citet{LiGui06}, Meinshausen and Bu\"{u}hlmann
(\citeyear{MeiBuh06}), \citet{YuaLin07}, Friedman, Hastie and Tibshirani (\citeyear{FriHasTib08}),
\citet{Rotetal08},
Peng et al. (\citeyear{PenZhoZhu09})]. However, like many high-dimensional regularization problems,
finding the appropriate level of sparsity remains a challenge. This
is particularly true for network structure learning, since the
problem is unsupervised in nature. Traditional methods, such as
Bayes information criteria [\citet{Sch78}] and cross-validation, aim
to find a model that minimizes prediction error or maximizes a
targeted likelihood function. They tend to include many irrelevant
features [e.g., \citet{Efr04N2},
\citet{Efretal04},
Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh06}) and
\citet{Penetal10}], and thus are not
appropriate for learning the interaction structures.
Choosing the amount of regularization by directly controlling the
false positive level would be ideal for structure learning.
Recently, a few model aggregation methods have been proposed, and
some of them provide certain control of false positives. For
example, \citet{Bach08} proposed
\textit{Bolasso}, which chooses variables that
are selected by all the lasso models [\citet{Tib96}] built on
bootstrapped data sets. In the context of network reconstruction,
\citet{Penetal10} proposed choosing
edges that are consistently selected across at least half of the
cross-validation folds. More recently, Meinshausen and
B\"{u}hlmann (\citeyear{MeiBuh10}) proposed the
\textit{stability selection}
procedure to choose
variables with selection frequencies exceeding a threshold. Under
suitable conditions, they derived an upper bound for the expected
number of false positives. In the same paper they also proposed the
randomized lasso penalty, which aggregates models from perturbing
the regularization parameters. Combined with stability selection,
randomized lasso achieves model selection consistency without
requiring the \textit{irrepresentable condition}
[\citet{ZhaYu06}] that is necessary for lasso to achieve model
selection consistency. In another work,
\citet{Wanetal11} proposed a modified
lasso regression---random lasso---by aggregating models based on
bootstrap samples and random subsets of variables. All these works
have greatly advanced research in model selection in the
high-dimensional regime. However, none of these methods provide
direct estimation and control of the false discovery rate (FDR).
In this paper, we address the problem of finding the right amount of
regularization in the context of high-dimension GGMs learning. In a
spirit similar to the aforementioned methods, we first obtain
selection frequencies from a collection of models built by
perturbing both the data and the regularization parameters. We then
model these selection frequencies by a mixture distribution to yield
an estimate of FDR on the selected edges, which is then used to
determine the cut-off threshold for the selection frequencies. This
framework is rather general, as it only depends on the empirical
distribution of the selection frequencies. Thus, it can be applied
to a wide range of problems beyond GGMs.
The rest of this paper is organized as follows. In Section~\ref{sec2} we
describe in detail the proposed method. In Section~\ref{sec3} an extensive
simulation study is conducted to compare the method with the
\textit{stability selection}
procedure and then evaluate its performance under different settings.
In Section~\ref{sec4} the method is illustrated by building a genetic
interaction network based on microarray expression data from a BC
study. The paper is concluded with some discussion in Section~\ref{sec5}.
\section{Method}\label{sec2}
\subsection{Gaussian graphical models}\label{sec2.1}
In a Gaussian Graphical Model (GGM) network construction is
defined by the conditional dependence relationships among the random
variables. Let $Y=(Y_1,\ldots,Y_p)$ denote a $p$-dimension random
vector following a multivariate normal distribution $N(0,\Sigma)$,
where $\Sigma$ is a $p\times p$ positive definite matrix. The
conditional dependence structure among $Y$ is represented by an
undirected graph $G=(U,E)$ with vertices $U=\{1,2,\ldots,p\}$
representing $Y_1,\ldots,Y_p$ and the edge set $E$ defined as
\[
E=\bigl\{(i,j)\dvtx Y_i \mbox{ and } Y_j \mbox{ are
dependent given }Y_{-\{i,j\}
}, 1\leq i,j\leq p\bigr\},
\]
where $Y_{-\{i,j\}}\equiv\{Y_k\dvtx k\not=i,j, 1\leq k\leq p\}$.
The goal of network construction is to identify the edge set $E$.
Under the normality assumption, the conditional independence between
$Y_i$ and $Y_j$ is equivalent to the partial correlation $\rho_{ij}$
between $Y_i$ and $Y_j$ given $Y_{-\{i,j\}}$ being zero. It is also
equivalent to the $(i,j)$ entry of the
\textit{concentration
matrix}
($\Sigma^{-1}$) being zero, that is,
$\sigma_{ij}\equiv (\Sigma^{-1})_{ij}=0$ [\citet{Dem72}, \citet{CoxWer96}],
since $\rho_{ij}= -\frac{\sigma_{ij}}{\sqrt{\sigma_{ii}\sigma_{jj}}}$.
There are two main types of approaches to fitting a GGM. One is the
maximum-likelihood-based approach, which estimates the concentration
matrix directly. The other is the regression-based approach, which
fits the GGM through identifying nonzero regression coefficients of
the following regression:
\[
Y_i=\sum_{j\not=i}\beta_{ij}Y_j+
\varepsilon_i,\qquad 1\leq i\leq p,
\]
where $\varepsilon_i$ is uncorrelated with $Y_{-i}=\{Y_k,k\not=i,1\leq
k \leq p \} $. The nonzero $\beta_{ij}$'s in the above regression
setting correspond to nonzero entries in the concentration matrix
since it can be shown that $\beta_{ij}=-\sigma_{ij}/\sigma_{ii}=\rho_{ij}\sqrt{\sigma_{jj}/\sigma_{ii}}$. In both
approaches, there are $O(p^2)$ parameters to estimate, which
requires proper regularization on the model if $p$ is larger than
the sample size $n$. This can be achieved by making a
\textit{sparsity assumption}
on the network
structure, that is, assuming that most pairs of variables are
conditionally independent given all other variables. Such an
assumption is reasonable for many real life networks, including
genetic regulatory networks [\citet{Garetal03},
Jeong et al. (\citeyear{Jeoetal11}),
\citet{Tegetal}]. Methods have been developed along these lines by using $L_1$
regularization. For example, \citet{YuaLin07} proposed a sparse
estimator of the concentration matrix via maximizing the $L_1$
penalized log-likelihood. Efficient algorithms were subsequently
developed to fit this model with high-dimensional data [Friedman, Hastie and Tibshirani
(\citeyear{FriHasTib08}),
\citet{Rotetal08}]. For regression-based approaches,
Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh06}) considered the
neighborhood selection estimator by minimizing $p$ individual loss
functions
\begin{equation}\label{eq2.1}
L^{(i)}(\beta,Y)=\frac{1}{2}\biggl\Vert Y_i-\sum
_{j:j\not=i}\beta_{ij}Y_j\biggr\Vert^2+
\lambda\sum_{j:j\not=i}|\beta_{ij}|,\qquad i=1,\ldots,p,
\end{equation}
while Peng et al. (\citeyear{PenZhoZhu09}) proposed the
\textit{space}
algorithm by minimizing the joint loss
\begin{equation}\label{eq2.2}
L(Y,\theta)=\frac{1}{2}\Biggl\{\sum_{i=1}^p
\biggl\Vert Y_i-\sum_{j:j\not
=i}\sqrt{
\frac{\sigma_{jj}}{\sigma_{ii}}}\rho_{ij} Y_j\biggr\Vert^2\Biggr\}
+\lambda\sum_{1\leq i<j\leq p}|\rho_{ij}|.
\end{equation}
From objective functions (\ref{eq2.1}) and (\ref{eq2.2}), it is clear that the
selected edge set depends on the regularization parameter $\lambda$.
Since the goal here is to recover the true edge set, ideally
$\lambda$ should be determined based on considerations such as FDR
and power with respect to edge selection. Moreover, when the sample
size is limited, a model-aggregation-based strategy can improve the
selection result compared to simply tuning the regularization
parameter. Thus, in the following section, we introduce a new
model-aggregation-based procedure that selects edges based on
directly controlling the FDRs.
Throughout the rest of this paper, we refer to the set of all pairs
of variables as the
\textit{candidate edge
set}
(denoted by $\Omega$), the subset of
those edges in the true model as the
\textit{true edge
set}
(denoted by $E$) and the rest as the
\textit{null edge set}
(denoted by~$E^c$). We denote the
size of a set of edges by $|\cdot|$. Note that $\Omega=E\cup E^c$
and the total number of edges in $\Omega$ is $N_{\Omega
}=|\Omega|=p(p-1)/2$.
\subsection{Model aggregation}\label{sec2.2}
Consider a good network construction procedure, where good
is in the sense that the true edges are stochastically more likely
to be selected than the null edges. Then it would be reasonable to
choose edges with high selection probabilities. In practice, these
selection probabilities can be estimated by the selection
frequencies over networks constructed based on perturbed data sets.
In the following, we formalize this idea.
Let $A(\lambda)$ be an edge selection procedure with a
regularization parameter $\lambda$ and $S^{\lambda}(Y)\equiv
S^{\lambda}(A(\lambda),Y)$ be the set of selected edges by
applying $A(\lambda)$ to data $Y$. The
\textit{selection
probability}
of edge $(i,j)$ is defined as
\[
p_{ij}= E \bigl(I\bigl\{(i,j)\in S^{\lambda}(Y)\bigr\} \bigr),
\]
where
$I\{\cdot\}$ is the indicator function. Let $R(Y)$ be the space of
resamples from $Y$ (e.g., through bootstrapping or subsampling). For
a random resample $Y'$ from $R(Y)$, we define
\[
\tilde{p}_{ij}= E \bigl(I\bigl\{(i,j)\in S^{\lambda}
\bigl(Y'\bigr)\bigr\} \bigr)= E \bigl(E \bigl(I\bigl\{(i,j)\in
S^{\lambda}\bigl(Y'\bigr)\bigr\}\mid Y \bigr) \bigr).
\]
In many cases (see Section C in the supplemental article [Li et al. (\citeyear{Lietal})]), $p_{ij}$'s and $\tilde{p}_{ij}$'s are close. For these
cases, we can estimate $p_{ij}$ by the
\textit{selection
frequency}
$X_{ij}$, which is the
proportion of $B$ resamples in which the edge $(i,j)$ is selected:
\begin{eqnarray}\label{eq2.3}
X_{ij}^{\lambda}&\equiv& X_{ij} \bigl(A(
\lambda);Y^1,\ldots,Y^B \in R (Y) \bigr)
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&=&
\frac{1}{B}\sum_{k=1}^BI \bigl\{(i,j)
\in S^{\lambda
}\bigl(Y^k\bigr) \bigr\},\qquad 1\leq i<j\leq p.
\end{eqnarray}
The aggregation-based procedures for choosing edges of large
selection frequencies can be represented as
\[
S_c^{\lambda}= \bigl\{(i,j)\dvtx X_{ij}^{\lambda}
\geq c\bigr\}\qquad \mbox{for } c\in(0,1].
\]
$S_c^{\lambda}$ is reasonable as long as most true
edges have selection frequencies greater than or equal to $c$ and
most null edges have selection frequencies less than~$c$. Ideally, we
want to find a threshold $c$ satisfying
\begin{equation}\label{eq2.4}\qquad
\operatorname{Pr} \biggl( \biggl\{\bigcap_{(i,j)\in E} \bigl
\{X_{ij}^{\lambda}\geq c\bigr\} \biggr\} \cap \biggl\{\bigcap
_{(i,j)\in E^c}\bigl\{X_{ij}^{\lambda}< c\bigr\}
\biggr\} \biggr)\rightarrow1\qquad \mbox{as }n\rightarrow\infty,
\end{equation}
so that the corresponding procedure $S_c^{\lambda}$ is consistent,
that is, $\operatorname{Pr}(S_c^{\lambda} =E)\rightarrow1$. In fact, if $A(\lambda)$
is selection consistent and $p_{ij}-\tilde{p}_{ij}\rightarrow0$, then
\begin{equation}\label{eq2.5}\qquad
\operatorname{Pr} \biggl( \biggl\{ \bigcap_{(i,j)\in E}\bigl
\{X_{ij}^{\lambda}=1\bigr\} \biggr\} \cap \biggl\{ \bigcap
_{(i,j)\in E^c}\bigl\{X_{ij}^{\lambda}=0\bigr\} \biggr\}
\biggr)\rightarrow1\qquad \mbox{as }n\rightarrow\infty,
\end{equation}
and thus any $c \in(0,1]$ satisfies (\ref{eq2.4}). Note that (\ref{eq2.4}) is in
general a much weaker condition than (\ref{eq2.5}), which suggests that we
might find a consistent $S_c^{\lambda}$ even when $A(\lambda)$ is not
consistent.
\begin{figure}
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{589f01a.eps}
& \includegraphics{589f01b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}
\end{tabular}
\caption{The distributions of selection frequencies based on a
simulated data set. \textup{(a)} The distribution of selection frequencies of
all edges. \textup{(b)} Distributions of selection frequencies of null and
true edges, respectively (note that these are not observable in
practice). Simulation is based on a power-law network with $p=500$,
$n=200$, and the number of true edges is 495. The space
algorithm with $\lambda=135$ is
used as the original nonaggregation procedure $A(\lambda)$. For
illustrating the tail behavior of these distributions more
effectively, we only show them on the selection frequency range
[0.06, 1], as there are too many edges with selection frequency less
than 0.06.}\label{fig1}
\end{figure}
For the finite data case, an aggregation-based procedure could also
perform better than the original procedure, as illustrated by the
following simulation example (the simulation setup is provided in
Section~\ref{sec3}). Figure~\ref{fig1}(a) shows the empirical distribution of
selection frequencies based on a simulated data set and Figure~\ref{fig1}(b)
shows the empirical distributions of true edges (green triangles)
and null edges (red crosses). Note that most null edges have low
selection frequencies $<0.4$, while most true edges have large
selection frequencies $>0.6$. This suggests that with a properly
chosen $c$ (say, $c\in[0.4,0.6]$), $S_c^{\lambda}$ will select
mostly true edges and only a small number of null edges. In fact, by
simply choosing the cutoff $c=0.5$, $S_c^{\lambda}$ outperforms
$A(\lambda)$ in both FDR and power (Figure~\ref{fig2}).
\begin{figure}
\includegraphics{589f02.eps}
\caption{Power and FDR of the aggregation-based procedure
$S_c^{\lambda}$ with cutoff $c=0.5$ and the original procedure
$A(\lambda)$ for $\lambda= 96, 114, 135, 160$, with the rest of
settings the same as in Figure~\protect\ref{fig1}.} \label{fig2}
\end{figure}
\subsection{Modeling selection frequency}\label{sec2.3}
Now we introduce a mixture model, similar in spirit to \citet{Efr04N1},
for estimating the FDR of an aggregation-based procedure
$S_c^{\lambda}$. We will use this estimate to choose the optimal
$c$ and $\lambda$ by controlling FDR while maximizing power. Assume
that the selection frequencies $\{X_{ij}^{\lambda},(i,j)\in\Omega
\}$,\vspace*{1pt} generated from $B$ resamples, fall into two categories, ``true''
or ``null,'' depending on whether $(i,j)$ is a true edge or a null
edge. Let $\pi$ be the proportion of the true edges. We also assume
that $X_{ij}^{\lambda}$ has density $f_1^{\lambda}(x)$ or
$f_0^{\lambda}(x)$ if it belongs to the ``true'' or the ``null''
categories, respectively. Note that both $f_1^{\lambda}$ and
$f_0^{\lambda}$ depend on the sample size $n$, but such dependence
is not explicitly expressed in order to keep the notation simple.
The mixture density for $X_{ij}^{\lambda}$ can be written as
\begin{equation}\label{eq2.6}
f^{\lambda}(x)=(1-\pi)f_0^{\lambda}(x)+\pi
f_1^{\lambda}(x), \qquad x\in \{0,1/B,2/B,\ldots,1\}.
\end{equation}
Based on this mixture model, the (positive) FDR [\citet{Sto03}] of the
aggregation-based procedure $S_c^{\lambda}$ is
\begin{equation}\label{eq2.7}
\operatorname{FDR}\bigl(S_c^{\lambda}\bigr)=\operatorname{Pr} \bigl((i,j)\in
E^c|(i,j)\in S_c^{\lambda} \bigr)=\frac{\sum_{x\geq c}(1-\pi)f_0^{\lambda}(x)}{\sum_{x\geq
c}f^{\lambda}(x)}.
\end{equation}
Given an estimate $\widehat{\operatorname{FDR}}(S_c^{\lambda})$ (which will be
discussed below) from (\ref{eq2.7}), the number of true edges in $S_c^{\lambda
}$ can be estimated by
\begin{equation}\label{eq2.8}
\widehat{N}_{E}\bigl(S_c^{\lambda}
\bigr)=\bigl|S_c^{\lambda}\bigr| \bigl( 1-\widehat {\operatorname{FDR}}
\bigl(S_c^{\lambda}\bigr) \bigr),
\end{equation}
which can be used to compare the power of $S_c^{\lambda}$ across
various choices of $c$ and $\lambda$, as the total number of true edges
is a constant. Consequently, for a given targeted FDR level $\alpha$,
we first seek for the optimal threshold $c$ for each $\lambda\in
\Lambda $,
\begin{equation}\label{eq2.9}
c^*(\lambda)=\operatorname{min}\bigl\{c\dvtx \widehat{\operatorname{FDR}}\bigl(S_c^{\lambda}
\bigr)\leq\alpha\bigr\},
\end{equation}
and then we find the optimal regularization parameter
\begin{equation}\label{eq2.10}
\lambda^* = \mathop{\operatorname{argmax}}_{\lambda\in\Lambda} \widehat {N}_{E}
\bigl(S_{c^*(\lambda)}^{\lambda}\bigr),
\end{equation}
such that the corresponding procedure $S_{c^*(\lambda^*)}^{\lambda^*}$
achieves the largest power among all competitors with estimated FDR not
exceeding $\alpha$.
The above procedure depends on a good FDR estimate, which in turn
requires good estimates of the mixture density $f^{\lambda}$ and
its null-edge contribution $(1-\pi)f_0^{\lambda}$. A natural
estimator of $f^{\lambda}$ is simply the empirical selection
frequencies, that is,
\[
\hat{f}^{\lambda} \biggl(\frac{k}{B} \biggr)=\frac{n^{\lambda
}_k}{N_{\Omega}},\qquad
k=0,1,\ldots,B,
\]
where $N_{\Omega}=p(p-1)/2$ is the total number of candidate edges and
$n_k^{\lambda}=|\{(i,j)\dvtx X_{ij}^{\lambda}=k/B\}|$ is the number of
edges with selection frequencies equal to $k/B$.
Before describing an approach to estimating $\pi$ and $f_0^{\lambda
}$, we note two observations from Figure~\ref{fig1}(b). First, the
contribution from the true edges to the mixture density $f^{\lambda}$ is small in the
range where the selection frequencies are small. Second, the empirical
distribution of $f_0^{\lambda}$ is monotonically decreasing. These can
be formally summarized as the following condition.
\begin{pc*}
There exist $V_1$ and $V_2$, $0<V_1<V_2<1$, such that as
$n\rightarrow\infty$:
\begin{longlist}[(C1)]
\item[(C1)] $ f_1^{\lambda}\rightarrow0$ on $(V_1,V_2]$;
\item[(C2)] $f_0^{\lambda}$ is monotonically decreasing on $(V_1,1]$.
\end{longlist}
\end{pc*}
This
\textit{proper condition}
is satisfied by a class of procedures as described in the lemma below
(the proof is provided in the \hyperref[app]{Appendix}).
\begin{lemma}\label{le1}
A selection procedure satisfies the
\textit{proper condition}
if, as the sample size increases, $\tilde{p}_{ij}$ tends to one
uniformly for all true edges and has a limit superior strictly less
than one for all null edges.
\end{lemma}
\begin{remark}\label{rem1} It is easy to verify that all consistent procedures
applied to subsampling resamples satisfy the condition in Lemma~\ref{le1}.
Other examples are procedures that use randomized lasso penalties
[Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh10})]. See Section~\ref{sec2.5} for
more details.
\end{remark}
The
\textit{proper condition}
motivates us to
estimate $\pi$ and $f_0^{\lambda}$ by fitting a parametric model
$g_{\theta}$ for $f^{\lambda}$ in the region $(V_1, V_2]$ and then
extrapolating the fit to the region $(V_2,1]$. This is because if C1
is satisfied, then $(1-\pi)f_0^{\lambda}$ can be well approximated
based on the empirical mixture density from the region $(V_1 ,
V_2]$. If C2 is also satisfied, the extrapolation of $g_{\theta}$
will be a good approximation to $(1-\pi)f_0^{\lambda}$ on
$(V_2,1]$ for a reasonably chosen family of $g_{\theta}$.
We choose the parametric family as follows. Given $\tilde{p}_{ij}$,
it is natural to model the selection frequency by a (rescaled)
binomial distribution, denoted by $b_1(\cdot|\tilde{p}_{ij})$, due
to the independent and identical nature of resampling conditional on
the original data. Moreover, we use a
\textit{powered beta}
distribution [i.e., the
distribution of $Q^\gamma$ where $Q\sim$ beta$(a,b)$, $a,b,r>0$]
as the prior for $\tilde{p}_{ij}$'s, denoted by $b_2(\cdot|\theta
)$ with $\theta=(a,b,r)$. This is motivated by the fact that the
beta family is a commonly used conjugate prior for the binomial
family, and the additional power parameter~$\gamma$ simply provides
more flexibility in fitting. Thus, the distribution of selection
frequencies of null edges is modeled as
\[
h_{\theta}(x) =\int_0^1
b_1(x|\tau)b_2(\tau|\theta)\,d \tau.
\]
The null-edge contribution $(1-\pi)f_0^{\lambda}$ can be estimated by
fitting $h_{\theta}$ to the empirical mixture density $\hat{f}^{\lambda
}$ in the
\textit{fitting range}
$(V_1,V_2]$, which, in practice, is determined based on the shape of
$\hat{f}^{\lambda}$ (details are given in Section~\ref{sec2.4}). Specifically,
we estimate $\pi$ and $f_0^{\lambda}$ by $\hat{\pi}$ and $h_{\hat
{\theta}}$, via
\begin{equation}\label{eq2.11}
(\hat{\pi},\hat{\theta})=\mathop{\operatorname{argmin}}_{\pi, \theta}L \bigl(\hat
{f}^{\lambda}(\cdot),(1-\pi)h_{\theta}(\cdot) \bigr),
\end{equation}
where $L(f,g)\equiv-\sum_{x\in(V_1,V_2]}[f(x)\operatorname{ log }g(x)]$,
which amounts to the Kullback--Leibler distance.
\subsection{Proper regularization range}\label{sec2.4}
Following what we propose in Section~\ref{sec2.3}, we can evaluate
the aggregation-based procedure $S_{c}^{\lambda}$ for different
choices of $(\lambda,c)$ with regard to model--selection--based
criteria: the FDR and the number of selected true edges. For the
range of $\lambda$, we consider those that yield
``\textit{U-shaped}''
empirical distributions of
selection frequencies, that is, $\hat{f}^{\lambda}$ decreases in
the small-selection-frequency range and then increases in the
large-selection-frequency range [see Figure~\ref{fig1}(a) and Figure~\ref{fig3} for
examples of ``U-shaped'' distribution]. The decreasing trend is
needed for the
\textit{proper condition}
to hold,
while the increasing trend helps to control the FDR, since an
$S_{c}^{\lambda}$ with $\operatorname{FDR}\leq\alpha$ implies, by (\ref{eq2.7}), that
\begin{equation}\label{eq2.12}
\sum_{x\geq c}f^{\lambda}(x)\geq\frac{(1-\pi)\sum_{x\geq
c}f_0^{\lambda}(x)}{\alpha}.
\end{equation}
\textit{\hspace*{-12pt}\begin{tabular}{@{}l p{340pt}@{}}
\hline
\multicolumn{2}{c}{\begin{bfseries}\textbf{U-shape detection procedure}\end{bfseries}} \\
\hline
1. & \textbf{INPUT} $\hat{f}^\lambda$, the empirical density of selection frequencies. Set $U=1$ (the U-shape indicator).\\
2. & Check U-shape.\\
& 2.1. Check valley point.\\
& \hspace{1cm} 2.1.1. Calculate $v_2=\operatorname{argmin}_x\tilde {f}^\lambda(x)$, the valley point position,\\
& \hspace{2.1cm} where $\tilde {f}^\lambda $ is a smooth curve fitted based on $\hat{f}^\lambda$. (We use\\
& \hspace{2.1cm} the R-function $\tt{smooth.spline ()}$, where the degree of \\
& \hspace{2.1cm} freedom parameter is determined such that the derivative \\
& \hspace{2.1cm} of $\tilde {f}^\lambda $ has only one sign change.)\\
& \hspace{0.95cm} 2.1.2. \textbf{IF} $v_2>0.8$\\
& \hspace{2.1cm} Set $U=0$, \textbf{GOTO} Step 3.\\
& \hspace{2.1cm}\textbf{END IF}\\
& 2.2. Calculate $v_1=\operatorname{argmax}_{x<v_2}\hat {f}^\lambda(x)$, the peak before $v_2$.\\
& 2.3. Check if $\hat{f}^\lambda$ is ``roughly'' decreasing on $(v_1,v_2]$.\\
& \hspace{1cm} 2.3.1. Calculate $\mu_1=(v_1+v_2)/2$, $s_1=\sum_{x\in [v_1,\mu_1]}\hat{f}^\lambda(x)$ and\\
& \hspace{2.1cm} $s_2=\sum_{x\in [\mu_1,v_2]}\hat{f}^\lambda(x)$. \\
& \hspace{0.95cm} 2.3.2. \textbf{IF} $s_1<s_2$\\
& \hspace{2.1cm} Set $U=0$, \textbf{GOTO} Step 3.\\
& \hspace{2.1cm}\textbf{END IF}\\
& 2.4. Check if $\hat{f}^\lambda$ is ``roughly'' increasing on $(v_2,1]$.\\
& \hspace{1cm} 2.4.1. Calculate $\mu_2=(v_2+1)/2$, $s_3=\sum_{x\in [v_2,\mu_2]}\hat{f}^\lambda(x)$ and\\
& \hspace{2.1cm} $s_4=\sum_{x\in [\mu_2,1]}\hat{f}^\lambda(x)$. \\
& \hspace{0.95cm} 2.4.2. \textbf{IF} $s_3>s_4$\\
& \hspace{2.1cm} Set $U=0$, \textbf{GOTO} Step 3.\\
& \hspace{2.1cm}\textbf{END IF}\\
3. & \textbf{RETURN} $v_1, v_2, U$.\\
\hline
\end{tabular}}\vspace*{6pt}
Therefore, if $\hat{f}^{\lambda}$ is not sufficiently large at the
tail, $\operatorname{FDR}\leq\alpha$ may not be achieved for a small value of $\alpha
$. The increasing trend also helps to obtain decent power since it
guarantees a substantial size of $S_{c}^{\lambda}$. Based on our
experience, the $\lambda$ values chosen based on (\ref{eq2.9}) and (\ref{eq2.10})
indeed always corresponds to a ``U-shaped'' empirical selection
frequency distribution.
Thus, we propose the following simple procedure for identifying
``U-shaped'' $\hat{f}^{\lambda}$'s to determine the proper
regularization range in practice. An illustration for this procedure is
given in Figure~\ref{fig3}.
\begin{figure}
\includegraphics{589f03.eps}
\caption{An illustration for the proposed U-shape identification
procedure. The empirical distribution ($\hat{f}^{\lambda}$) is the
same as the one in Figure~\protect\ref{fig1}. The smooth curve ($\tilde{f}^{\lambda}$)
is fitted by the R-function $\tt{smooth.spline}$ with df${}=4$.
Locations of $v_1$, $v_2$, $\mu_1$ and $\mu_2$ are found by following
steps in the U-shape detection procedure.} \label{fig3}
\end{figure}
\begin{remark}\label{rem2} Step 2.1 is based on our extensive simulation where we
find that a large value of $v_2$ often corresponds
to a too-small $\lambda$, yielding too many null edges with high
selection frequencies, which makes (\ref{eq2.12}) difficult to hold for
reasonably small FDR levels $\alpha$ (see Section D1 in the
supplemental article [Li et al. (\citeyear{Lietal})]). If $\hat{f}^{\lambda}$ is not
recognized as ``U-shaped'' for a large range of $\lambda$'s, we
would consider the data as lack of signals where a powerful
$S_{c}^{\lambda}$ is not attainable. One example is the empty
network (see Section~\ref{sec3.2} and Figure S-1 in the supplemental article [Li
et al. (\citeyear{Lietal})]).
\end{remark}
Sections~\ref{sec2.2}--\ref{sec2.4} provide a procedure for network inference based on
directly estimating FDR. We name the procedure as
\textit{BINCO}---Bootstrap Inference for
Network COnstruction, as we suggest to use bootstrap resamples. The
main steps are summarized below.
\subsection{Randomized lasso}\label{sec2.5}
For an $L_1$ regularized procedure $A(\lambda)$, the
\textit{proper condition}
(Section~\ref{sec2.3}) is
satisfied if $A(\lambda)$ is selection consistent, which usually
requires strong conditions, for instance, the well-known
\textit{irrepresentable condition}
under the lasso
regression setting [\citet{ZhaYu06}, \citet{Zou06}, \citet{YuaLin07}, \citet{Wai09}] or the so-called
\textit{neighborhood stability condition}
under the GGM setting [Meinshausen and
B\"{u}hlmann (\citeyear{MeiBuh06}), Peng et al. (\citeyear{PenZhoZhu09})].
Recently, Meinshausen and
B\"{u}hlmann (\citeyear{MeiBuh10}) proposed the
\textit{randomized lasso}, which is a procedure based on
randomly sampled regularization parameters. For example, the
randomized lasso version of
\textit{space}
would be
\begin{equation}\label{eq2.13}
\qquad L(Y,\theta,W)=\frac{1}{2}\Biggl\{\sum_{i=1}^p
\biggl\Vert Y_i-\sum_{j:j\not
=i}\sqrt{
\frac{\sigma_{jj}}{\sigma_{ii}}}\rho_{ij} Y_j\biggr\Vert^2\Biggr\}
+\lambda\sum_{1\leq i<j\leq p}|\rho_{ij}|/w_{ij},
\end{equation}
\textit{\begin{tabular}{l p{333pt}@{}}
\hline
\multicolumn{2}{c}{\textbf{BINCO procedure}} \\
\hline
1. & \textbf{INPUT} $\Lambda =(\lambda_1,\ldots,\lambda_k)$ the initial range of regularization parameter values; $Y_{n \times p}$ the
dataset; and $\alpha$ the desired FDR level. \\
2. & \textbf{FOR} $i = 1$ \textbf{TO} $k$ \\
& 2.1. $\lambda =\lambda _i$ \\
& 2.2. Generate $\hat{f}^{\lambda }$ the empirical density of selection frequencies.\\
& 2.3. Check whether $\hat{f}^{\lambda }$ is U-shaped based on the output $(v_1, v_2, U)$ from\\
& \hspace{0.6cm} the ``U-Shape Detection Procedure.''\\
& 2.4. \textbf{IF} $\hat{f}^\lambda $ is U-shaped (i.e., $U=1$) \\
& \hspace{1cm} 2.4.1. Obtain the null density estimate $\hat{f}^{\lambda }_0$ by (\ref{eq2.11}). \\
& \hspace{0.95cm} 2.4.2. Find the optimal threshold $c^*(\lambda )$ by (\ref{eq2.9}), where the FDR \\
& \hspace{1.9cm} is estimated based on (\ref{eq2.7}) with $f^{\lambda }$ and $f^{\lambda }_0$ replaced by $\hat{f}^{\lambda }$\\
& \hspace{1.9cm} and $\hat{f}^{\lambda }_0$,respectively.\\
& \hspace{1cm} 2.4.3. Obtain $S_{c^*(\lambda )}^\lambda $ and calculate $\hat{N_E}(S_{c^*(\lambda )}^\lambda)$, the estimated\\
& \hspace{1.9cm} number of true edges being selected, based on (\ref{eq2.8}).\\
& \hspace{1cm}\textbf{END IF}\\
& \hspace{1cm}\textbf{ELSE} $\hat{N_E}(S_{c^*(\lambda )}^\lambda)= 0$, $S_{c^*(\lambda )}^\lambda=\emptyset$.\\
& 2.5. \textbf{OUTPUT} $\hat{N_E}(S_{c^*(\lambda )}^\lambda)$ and $S_{c^*(\lambda )}^\lambda$.\\
& \textbf{NEXT} $i$\\
3. & Determine the optimal regularization $\lambda^*$ through (\ref{eq2.10}). The optimal selection is $S_{c^*(\lambda^* )}^{\lambda^*}$.\\
\hline
\end{tabular}}\vspace*{6pt}
\noindent where $w_{ij}$'s are randomly sampled from a probability
distribution $p(w)$ supported on $(l,1]$ for some $l\in(0,1]$
(note that $l=1$ corresponds to the ordinar
$L_1$ penalty). The
advantage of this randomized lasso procedure is that, by
perturbing
the regularization parameters, the irrelevant features may be
decorrelated from the true features in some configurations of
randomly sampled weights such that the irrepresentable condition is
satisfied. Therefore, it selects all true features with probability
tending to 1 and any irrelevant feature with a limiting probability
strictly less than 1. As a result, a consistent aggregation-based
procedure can be achieved under conditions ``typically much weaker
than the standard assumption of the irrepresentable condition''
[Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh10}), Theorem~2]. For
this case, based on Lemma~\ref{le1}, the
\textit{proper condition}
is also satisfied.
If (\ref{eq2.13}) is used as the original (nonaggregated) procedure, an
additional parameter $l$, which controls the amount of perturbation
of the regularization parameter, needs to be chosen. A small $l$
guards better against false positives but damages power, while a
large $l$ may result in a liberal procedure. Here we provide a
two-step data-driven procedure for choosing an appropriate $l$ in
BINCO. We first fix $l=1$, that is, the ordinary $L_1$ penalty, to find
a proper range $\Lambda^*$ for $\lambda$ that corresponds to the
``U-shaped'' empirical mixtures. Then for each $\lambda\in\Lambda^*$, we
consider a set of pairs $\Lambda_2=\{(\lambda_i,l_i), i=1,\ldots,m\}$ such that
$\int_{l_i}^1\frac{\lambda_i}{w}p(w)\,dw=\lambda$, that is, keeping the
average amount of regularization unchanged. For example, in the
simulation study, we use $l_i=i/10, i=1,\ldots,9$. We then
pick the pair $(\lambda^*,l^*)\in\Lambda_2$ such that $l^*$ is
the smallest among those $l$'s that yield U-shaped empirical mixture
distributions. Our simulation shows that such a choice of $(\lambda^*,l^*)$ ensures good power for BINCO while controlling FDR in a
slightly conservative fashion.
\section{Simulation}\label{sec3}
In this section we first compare the performance of BINCO
with
\textit{stability selection}
[Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh10})], and then
investigate the performance of BINCO with respect to various
factors, including the network structure, dimensionality, signal
strength and sample size.
We use
\textit{space}
[Peng et al. (\citeyear{PenZhoZhu09})] coupled with randomized lasso
(\ref{eq2.13}) as the
original nonaggregate procedure, where the random weights
$\frac{1}{w_{ij}}$'s are generated from the uniform distribution
$U[1,1/l]$ for $l\in(0,1]$. The selection frequencies are obtained
based on $B=100$ resamples. Since subsampling of size $[n/2]$ is
proposed for
\textit{stability selection}, we use subsampling
to generate resamples when comparing BINCO and
\textit{stability selection}. For investigating
BINCO's performance, we use bootstrap resamples because it
yields slightly better performance (see Remark~\ref{rem4}).
The performance of both methods are evaluated by true FDRs and power,
since for simulations we know whether an edge is true or null. In
addition, we define
\textit{ideal
power}, which is the best power one can
achieve for $S_c^{\lambda}$ given the true \mbox{$\operatorname{FDR}\leq \alpha$} (in
simulation we consider $\alpha=0.05$ and $\alpha=0.1$). Based on
\textit{ideal power}, we can evaluate the
efficiency of the methods under different settings. For each
simulation setting, results are based on 20 independent simulation
runs.
\subsection{Comparison between BINCO and stability selection}\label{sec3.1}
\textit{Stability selection}
procedure
selects $S_{\mathrm{stable}}^{\Lambda}(t)\equiv\{(i,j)\dvtx \operatorname{max}_{\lambda
\in\Lambda}(X_{ij}^{\lambda})\geq t\}$, a set of edges with the
maximum selection frequency over a prespecified regularization set
$\Lambda$ exceeding a threshold $t$. Assuming an exchangeability
condition upon the irrelevant variables (here the null edges),
Meinshausen and B\"{u}hlmann [(\citeyear{MeiBuh10}), Theorem~1] derived
an upper bound for the expected number of falsely selected variables
for each choice of $t>0.5$. Specifically, under suitable conditions,
the expected number of null edges selected by the set
$S_{\mathrm{stable}}^{\Lambda}(t)$, denoted by $E(V)$, satisfies
\begin{equation}\label{eq3.1}
E(V)\leq\frac{q_{\Lambda}^2}{(2t-1)N_{\Omega}},
\end{equation}
where $N_{\Omega}=p(p-1)/2$ is the total number of candidate edges and
$q_{\Lambda}$ is the expected number of edges selected under at least
one $\lambda\in\Lambda$. In practice, $q_{\Lambda}$ can be estimated
by $\frac{1}{B}\sum_{i=1}^B|\bigcup_{\lambda\in\Lambda}S^{\lambda
}(Y^{i})|$. Dividing both sides of (\ref{eq3.1})\vadjust{\goodbreak} by $|S_{\mathrm{stable}}^{\Lambda
}(t)|$, we obtain
\begin{equation}\label{eq3.2}
\frac{E(V)}{|S_{\mathrm{stable}}^{\Lambda}(t)|}\leq\frac{q_{\Lambda
}^2}{(2t-1)N_{\Omega}\cdot|S_{\mathrm{stable}}^{\Lambda}(t)|}.
\end{equation}
Although
\textit{stability selection}
is intended to control $E(V)$, for an easier comparison with BINCO, we
use $\frac{E(V)}{|S_{\mathrm{stable}}^{\Lambda}(t)|}$ to approximate FDR and
obtain the optimal $S_{\mathrm{stable}}^{\Lambda}(t)$ by finding the smallest
threshold $t$ such that the upper bound on the right-hand side of (\ref{eq3.2})
is less than or equal to $\alpha$.
For data generation, we first consider a
\textit{power-law
network}
with $p=500$ nodes whose degree
(i.e., the number of connected edges for each node) distribution
follows $P(k)\sim k^{-\gamma}$. The scaling exponent $\gamma$ is
set to be 2.3, which is consistent with the findings in the
literature for biological networks [\citet{New03}]. There are in
total 495 true edges in this network and its topology is illustrated
in Figure~\ref{fig32}(a). The sample size is $n=200$. Two settings with
different signal strengths are considered: (1) strong signal, the
mean and standard deviation (SD) of nonzero $|\rho_{ij}|$'s are
0.34 and 0.13, respectively; (2) weak signal, the mean and SD of
nonzero $|\rho_{ij}|$'s are 0.25 and 0.09, respectively. Note both
positive and negative correlations are allowed in this network.
We compare the performance of BINCO and
\textit{stability selection}
at a targeted FDR level of 0.05. For BINCO,
we consider $\Lambda_0=\{40,50,\ldots,100\}$ as the initial range
for $\lambda$ and then obtain the optimal final selection following
the steps at the end of Section~\ref{sec2.4}. For
\textit{stability
selection}, since no specific guidance was provided for
choosing $\Lambda$ and $l$ (the randomized lasso regularization
perturbation parameter), we consider three different values for
$l\in\{0.5,0.8,1\}$ and a collection of intervals
$\Lambda=(\lambda_{\operatorname{min}},\lambda_{\operatorname{max}})$ with $\lambda_{\operatorname{min}}$
varying from 40 to 100 and $\lambda_{\operatorname{max}}=100$. This choice
of $\Lambda$ is due to the fact that the upper bound in (\ref{eq3.2})
cannot be controlled at 0.05 for any $t$ for $\lambda_{\operatorname{min}}<40$, and the performance of
\textit{stability
selection}
is largely invariant for $\lambda_{\operatorname{max}}$.
\begin{figure}
\centering
\begin{tabular}{@{}c@{\hspace*{3pt}}c@{}}
\includegraphics{589f04a.eps}
& \includegraphics{589f04b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}\\[6pt]
\includegraphics{589f04c.eps}
& \includegraphics{589f04d.eps}\\
\footnotesize{(c)} & \footnotesize{(d)}\vspace*{-2pt}
\end{tabular}
\caption{The FDR (top panels) and power (bottom panels) for BINCO and
stability selection
(Stab. Sel.). \textup{(a)} and \textup{(c)} are for the strong signal setting; \textup{(b)} and
\textup{(d)} are for the weak signal setting.} \label{fig31}\vspace*{-2pt}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{@{}ccc@{}}
\includegraphics{589f05a.eps}
& \includegraphics{589f05b.eps} & \includegraphics{589f05c.eps}\\
\footnotesize{(a)} & \footnotesize{(b)} & \footnotesize{(c)}\vspace*{-2pt}
\end{tabular}
\caption{Different network topologies: \textup{(a)} Power-law network, number
of true edges${} = 495$; \textup{(b)}~Empirical network, number of true edges${}=633$;
\textup{(c)} Hub network, number of true edges${}=587$. All three networks have
$p=500$ nodes.} \label{fig32}\vspace*{-2pt}
\end{figure}
When the signals are strong, BINCO gives a conservative $\operatorname{FDR}=0.026$
but still maintains good power${}=0.801$ [Figures~\ref{fig31}(a) and~\ref{fig31}(c)]. The
performance of
\textit{stability selection}
varies for different choices of $\lambda_{\operatorname{min}}$ and $l$. The
FDRs are larger than the targeted level 0.05 for some $\lambda_{\operatorname{min}}$'s
when $l=0.8$ and for all $\lambda_{\operatorname{min}}$'s
when $l=1$. For other cases (some $\lambda_{\operatorname{min}}$'s when
$l=0.8$ and all $\lambda_{\operatorname{min}}$'s when $l=0.5$), the FDR
control is very conservative and the corresponding power is
consistently lower than BINCO. When the signals are weak,
\textit{stability selection}
is much more
conservative than BINCO and results in much lower power [Figures~\ref{fig31}(b)
and 4(d)]. In Table~\ref{table1} we report the
\textit{ideal
power}, the power for BINCO and the best power for
\textit{stability selection}
(among different
choices of $\lambda_{\operatorname{min}}$) under $l=0.5$, 0.8 and 1. We
also calculate the power efficiency as the ratio of the power for
the method over the
\textit{ideal power}, for
BINCO and
\textit{stability selection},
respectively. It can be seen that the power of BINCO\vadjust{\goodbreak} is close to the
\textit{ideal power}
for both levels of signal
strength, while
\textit{stability selection}
is
too conservative when the signal strength is weak. For more detailed
results, see Section A1 in the supplemental article [Li~et~al.
(\citeyear{Lietal})].\vspace*{-2pt}
\begin{remark}\label{rem3} In some cases we find that
\textit{stability selection}
fails to control FDR. We suspect this
may be due to the violation of the exchangeability assumption in
Theorem~1 of Meinshausen and B\"{u}hlmann (\citeyear{MeiBuh10}). We
examine the impact of the exchangeability assumption by simulation
and find that when it is violated,\vadjust{\goodbreak} the theoretical upper bound in
(\ref{eq3.1}) for $E(V)$ may not hold (see Section D2 in the supplemental
article [Li et al. (\citeyear{Lietal})] for further details).\vspace*{-2pt}
\end{remark}
\subsection{Further investigation of BINCO}\label{sec3.2}
Now we investigate the effects of the network structure,
dimensionality, signal strength and sample size on the performance of BINCO.
\begin{table}[b]\vspace*{-2pt}
\caption{Power comparison between BINCO and stability selection under
strong and weak signals}\label{table1}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccc@{}}
\hlin
&&&& \multicolumn{3}{c@{}}{\textbf{Stability selection}}\\[-4pt]
&&&& \multicolumn{3}{c@{}}{\hrulefill}\\
& &{\textbf{Ideal}\tnote{tt1}} & \textbf{BINCO} & $\bolds{l=1}$ & $\bolds{l=0.8}$ & $\bolds{l=0.5}$\\
\hlin
{Strong signal}& Power & 0.853 & 0.801 & 0.818\tnote{tt3}
& 0.785\tnote{tt3} & 0.706 \\
& MPE\tnote{tt2} & 1\phantom{00.0} & 0.939 & 0.959\tnote{tt3} & 0.920\tnote{tt3}& 0.828
\\[3pt
{Weak signal} &Power & 0.616 & 0.569 & 0.434\phantom{0} & 0.407\phantom{0} &
0.170 \\
& MPE\tnote{tt2} & 1\phantom{00.0} & 0.924 & 0.705\phantom{0} & 0.661\phantom{0} & 0.276 \\
\hlin
\end{tabular*}
\tabnotetext[1]{tt1}{``Ideal'' refers to the \textit{ideal power} that can be achieved when the true distribution of null edges is known.}
\tabnotetext[2]{tt2}{Method Power Efficiency (MPE)${}={}$method power${}/{}$ideal power.}
\tabnotetext[3]{tt3}{FDR control failed.}
\end{table}
\textit{Network structure}.
We consider four different network topologies:
\textit{empty network},
\textit{power-law network},
\textit{empirical network}
and
\textit{hub network}. In each network there are five disconnected components with 100 nodes
each. Below is a brief description of the network topologies:
\begin{longlist}[(1)]
\item[(1)]
\textit{Empty network}: there is no edge connecting any pair of nodes.
\item[(2)]
\textit{Power-law network}: the degree follows a power-law distribution with parameter $\gamma
=2.3$ as described in Section~\ref{sec3.1}
[Figure~\ref{fig32}(a)].\vadjust{\goodbreak}
\item[(3)]
\textit{Empirical network}: the topology is simulated according to an empirical degree
distribution of one genetic regulatory network [\citet{Schetal05}] [Figure~\ref{fig32}(b)].
\item[(4)]
\textit{Hub network}: three nodes per component have a large number of connecting edges
($>$15) and all other nodes have a small number of connecting edges
($<$5) [Figure~\ref{fig32}(c)].
\end{longlist}
We set the sample size $n=200$. The signal strength for all networks
except for the empty network is fixed at the strong level as in
Section~\ref{sec3.1}.
\begin{table}
\tabcolsep=0pt
\caption{Investigation of the impact of different networks on BINCO
performance}\label{table2}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
&\multicolumn{4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$} & \multicolumn {4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} =
0.10}$}\\[-4pt]
&\multicolumn{4}{c}{\hrulefill} & \multicolumn {4}{c@{}}{\hrulefill}\\
& \multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c}{\textbf{Power}} &
\multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c@{}}{\textbf{Power}} \\[-4pt]
& \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} &
\multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill} \\
\textbf{Network topology}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}\\
\hlin
Power-law & 0.046 & 0.009 & 0.810 & 0.013 & 0.096 & 0.013 & 0.845 &
0.013\\
Empirical & 0.032 & 0.019 & 0.523 & 0.040 & 0.068 & 0.034 & 0.565 &
0.040\\
Hub & 0.023 & 0.009 & 0.644 & 0.021 & 0.052 & 0.012 & 0.692 & 0.017\\
\hlin
\end{tabular*}
\end{table}
\begin{table}[b]
\caption{Comparison of BINCO power and ideal power under different
networks}\label{table3}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccc@{}}
\hline
&\multicolumn{3}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$}& \multicolumn{3}{c@{}}{\textbf{Targeted}
$\bolds{\operatorname{FDR} = 0.10}$} \\[-4pt]
&\multicolumn{3}{c}{\hrulefill}& \multicolumn{3}{c@{}}{\hrulefill} \\
\textbf{Topology} & \textbf{Power-law} & \textbf{Empirical} & \textbf{Hub} & \textbf{Power-law} & \textbf{Empirical} & \textbf{Hub} \\
\hlin
BINCO power & 0.810 & 0.523 & 0.644 & 0.845 & 0.565 & 0.692\\
Ideal power & 0.856 & 0.595 & 0.736 & 0.881 & 0.631 & 0.776\\
MPE\tnote{ttt1}& 0.946 & 0.879 & 0.875 & 0.959 & 0.895 & 0.892\\
[1ex]
\hlin
\end{tabular*}
\tabnotetext[1]{ttt1}{Method Power Efficiency (MPE)${}={}$method
power${}/{}$ideal power.}
\end{table}
For the empty network, the empirical mixture distributions of
selection frequencies monotonically decrease on a wide range of
$\lambda$ (Figure S-1) and are not recognized by BINCO as
``U-shaped.'' Thus, we reach the correct conclusion that there is no
signal in this case. In contrast, data sets from the other three
networks produce the desired ``U-shaped'' mixture distributions for
some $\lambda$ (Figure S-2).
We compare BINCO results across networks 2-4 with FDR targeted at
level $\alpha= 0.05$ and 0.1. BINCO gives slightly conservative
control on FDR and achieves reasonable power for all three networks
(Table~\ref{table2}). The comparison to the
\textit{ideal power}
shows that the network topologies investigated here have only a small
effect on BINCO's efficiency (Table~\ref{table3}).
\textit{Dimensionality}.
We investigate the impact of dimensionality on the performance of
BINCO. We consider the power-law network and let the number of nodes
$p$ vary from 500, 800 to 1000. To keep the complexity of each
component the same across different choices of $p$, we set the
component size constant, being 100, and the number of components $C
= p/100$. Again the sample size $n=200$ is used for all three cases
and the signal strength is fixed at the strong level as in Section~\ref{sec3.1}.
For all three choices of $p$, BINCO performs similarly (Table~\ref{table4}),
with slightly conservative FDR and power around 0.8. The
dimensionality does not demonstrate a significant impact on BINCO.
BINCO's result is also largely invariant when we compare networks of
differing numbers of components with $p$ fixed (such that component
size varies, see Section A3 in the supplemental article [Li et al.
(\citeyear{Lietal})]).
\begin{table}
\caption{Investigation of the impact of different dimensionality on
BINCO performance}\label{table4}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
&\multicolumn{4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$} & \multicolumn {4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} =
0.10}$}\\[-4pt]
&\multicolumn{4}{c}{\hrulefill} & \multicolumn {4}{c@{}}{\hrulefill}\\
& \multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c}{\textbf{Power}} &
\multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c@{}}{\textbf{Power}} \\[-4pt]
& \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} &
\multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill} \\
\textbf{Dimension} $\bolds{p}$&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}\\
\hlin
\phantom{0}500 & 0.046 & 0.009 & 0.810 & 0.013 & 0.096 & 0.013 & 0.845 & 0.013\\
\phantom{0}800 & 0.030 & 0.007 & 0.769 & 0.010 & 0.083 & 0.010 & 0.811 & 0.012\\
1000& 0.043 & 0.007 & 0.784 & 0.008 & 0.096 & 0.011 & 0.821 & 0.007\\
\hline
\end{tabular*} \vspace*{-2pt}
\end{table}
\begin{table}[b]\vspace*{-2pt}
\caption{Investigation of the impact of different signal strength on
BINCO performance}\label{tableS10}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
&\multicolumn{4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$} & \multicolumn {4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} =
0.10}$}\\[-4pt]
&\multicolumn{4}{c}{\hrulefill} & \multicolumn {4}{c@{}}{\hrulefill}\\
& \multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c}{\textbf{Power}} &
\multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c@{}}{\textbf{Power}} \\[-4pt]
& \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} &
\multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill} \\
\textbf{Signal strength}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}\\
\hline
Strong & 0.046 & 0.009 & 0.810 & 0.013 & 0.096 & 0.013 & 0.845 & 0.013\\
Weak & 0.032 & 0.010 & 0.579 & 0.024 & 0.063 & 0.014 & 0.617 & 0.018\\
Very weak & 0.035 & 0.026 & 0.252 & 0.040 & 0.065 & 0.037 & 0.310 &
0.039\\
\hline
\end{tabular*}
\end{table}
\textit{Signal strength}.
We consider three levels of signal strength: strong, weak and very
weak. The corresponding means and SDs of nonzero $|\rho_{ij}|$'s
are $(0.34, 0.13)$, $(0.25, 0.09)$ and $(0.21,0.07)$, respectively.
The network is the power-law network with $p=500$ and sample size is
$n=200$ for all settings.
BINCO provides good control on FDR, however, the power decreases from
0.8 to 0.3 as the signal weakens (Table~\ref{tableS10}). Comparing the power of
BINCO with the
\textit{ideal power} (Table~\ref{table33}), we
see that BINCO remains efficient and the loss in power is largely
due to reduction of signal strength.\vadjust{\goodbreak}
\begin{table}
\caption{Power comparison of BINCO power and ideal power when the
signal strength is strong, weak and very weak}\label{table33}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccc@{}}
\hline
&\multicolumn{3}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$}& \multicolumn{3}{c@{}}{\textbf{Targeted}
$\bolds{\operatorname{FDR} = 0.10}$} \\[-4pt]
&\multicolumn{3}{c}{\hrulefill}& \multicolumn{3}{c@{}}{\hrulefill} \\
\textbf{Signal strength} & \textbf{Strong} & \textbf{Weak} & \textbf{Very weak} & \textbf{Strong} & \textbf{Weak} & \textbf{Very weak}
\\
\hlin
BINCO power & 0.810 & 0.579 & 0.252 & 0.845 & 0.617 & 0.310\\
Ideal power & 0.856 & 0.615 & 0.279 & 0.881 & 0.651 & 0.345\\
MPE\tnote{tttt1}& 0.946 & 0.941 & 0.903 & 0.959 & 0.948 & 0.899\\% [1ex] adds vertical space
\hlin
\end{tabular*}
\tabnotetext[1]{tttt1}{Method Power Efficiency (MPE)${} = {}$method power${} / {}$ideal
power.}\vspace*{-3pt}
\end{table}
\textit{Sample size}.
Now we consider the impact of sample size $n$ by varying it from
200, 500 and 1000, while keeping the signal strength at the ``very
weak'' level as in the previous simulation. The network structure is
again the power-law network with $p=500$.
With an increased sample size, the power of BINCO is significantly
improved from 0.3 to nearly 0.9 while the FDRs are well controlled
(Table~\ref{tableS11}).
\begin{table}[b]\vspace*{-3pt}
\caption{Investigation of the impact of different sample size on BINCO
performance}\label{tableS11}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
&\multicolumn{4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} = 0.05}$} & \multicolumn {4}{c}{\textbf{Targeted} $\bolds{\operatorname{FDR} =
0.10}$}\\[-4pt]
&\multicolumn{4}{c}{\hrulefill} & \multicolumn {4}{c@{}}{\hrulefill}\\
& \multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c}{\textbf{Power}} &
\multicolumn{2}{c}{\textbf{FDR}} & \multicolumn{2}{c@{}}{\textbf{Power}} \\[-4pt]
& \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} &
\multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill} \\
\textbf{Sample size}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}&\textbf{Mean}&\textbf{SD}\\
\hline
\phantom{0}200 & 0.035 & 0.026 & 0.252 & 0.040 & 0.065 & 0.037 & 0.310 & 0.039\\
\phantom{0}500 & 0.024 & 0.010 & 0.684 & 0.012 & 0.049 & 0.011 & 0.714 & 0.014\\
1000& 0.045 & 0.010 & 0.869 & 0.013 & 0.090 & 0.015 & 0.891 & 0.012\\
\hline
\end{tabular*}
\end{table}
In summary, BINCO has good control for FDR under a wide
range of scenarios. Its performance is shown to be robust for
networks with different topologies and dimensionalities, and its
efficiency is not influenced much even when the signal strength is
weak. As the sample size increases, the power of BINCO is improved
significantly.
\begin{remark}\label{rem4}We propose to use bootstrap over subsampling, as the
former appears to give slightly better power. Intuitively, bootstrap
contains more distinct samples [0.632$n$, \citet{Pat62}] than $[n/2]$
subsampling (0.5$n$). However, the difference we have observed is
rather small. For example, we compare the power over 20 independent
samples between bootstrap and $[n/2]$ subsampling under the power-law
network setting. For $\operatorname{FDR}=0.05$, the power is 0.810 for bootstrap and
0.801 for subsampling (compare Tables~\ref{table3} and~\ref{table1}); while for $\operatorname{FDR}=0.1$,
the power is 0.845 for bootstrap and 0.835 for subsampling [compare
Tables~\ref{table3} and~S-7 from Li et al.\vadjust{\goodbreak} (\citeyear{Lietal})]. This observation is in agreement with the
conclusions of several others [Menshausen and B\"{u}hlmann
(\citeyear{MeiBuh10}), \citet{Fre77}, B\"{u}hlmann and Yu (\citeyear{BuhYu02})].
\end{remark}
\section{A real data application}\label{sec4}
We apply the BINCO method to a microarray expression data set
of breast cancer (BC) [\citet{Loietal}] to build a gene expression network related to the disease. The
data
(\href{http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE6532}{http://www.ncbi.nlm.}
\href{http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE6532}{nih.gov/geo/query/acc.cgi?acc=GSE6532})
contains measurements of expression levels of 44,928 probes in tumor
tissue samples from 414 BC patients based on the Affymetrix Human
Genome U133A, U133B and U133 plus 2.0 Microarray platforms.
We preprocess the data as follows. First, a global normalization is
applied by centering the median of each array to zero and scaling
the
\textit{Median Absolute Deviation}
(MAD) to
one. Probes with standard deviation (SD) greater than the
$25\%$-trimmed mean of all SDs are selected. We further focus on a
subset of $1257$ probes for genes from cell cycle and DNA-repair
related \mbox{pathways}
(\href{http://peiwang.fhcrc.org/internal/papers/DNArepair\_CellCircle\_related.csv/view}{http://peiwang.fhcrc.org/internal/papers/DNArepair\_CellCircle\_}\break
\href{http://peiwang.fhcrc.org/internal/papers/DNArepair\_CellCircle\_related.csv/view}{related.csv/view}),
as these pathways have been shown to play significant
roles in BC tumor initiation and development. Clinical information
including age, tumor size, ER-status (positive or negative) and
treatment status (tamoxifen treated or not) is incorporated in the
analysis as ``fake genes'' since we are also interested in
investigating whether gene expressions are associated with these
clinical characteristics. Finally, we standardize each expression
level to have mean zero and SD one. The resulting data set has $p =
1261$ genes/probes (including four clinical variables) and $n=414$
tumor samples.
\begin{figure}
\includegraphics{589f06.eps}
\caption{The empirical selection frequency distribution of all edges
(dots) and the estimated selection frequency distribution of null edges
(solid line). The three vertical lines are drawn at the cutoffs
$C1=0.98$, $C2=0.93$ and $C3=0.85$ for FDR at 0.05, 0.1 and 0.2,
respectively.} \label{fig41}
\end{figure}
\begin{figure}
\includegraphics{589f07.eps}
\caption{Inferred networks at $\operatorname{FDR}=0.2$ from the BC expression data. A
total of 338 edges (selection frequencies $\geq0.85$) are identified.
Among these 338 edges, those with selection frequencies $\geq0.98$
(corresponding to the set with $\operatorname{FDR}=0.05$) are colored in red, while
other edges are colored in green. Genes with degree${}>$3 are labeled by
their symbols; genes with degree${}>$4 are indicated by red nodes. In
addition, the top ten genes with consistently high connection across
perturbed data sets are labeled in blue symbols.} \label{fig42}
\end{figure}
We generate selection frequencies by applying the
\textit{space}
algorithm with randomized lasso
regularization to $B=100$ bootstrap resamples. The initial range of
the tuning parameter $\lambda$ is set to be
$\Lambda=(100,120,\ldots,580)$. We then apply the BINCO procedure and
find that the optimal values for the regularization parameters are
$\lambda=340$ and $l=0.9$. The empirical distribution of selection
frequencies of all edges and the null density estimation are given
in Figure~\ref{fig41}. When the estimated FDR is controlled at 0.05, 0.1 and
0.2, BINCO identifies 125, 222 and 338 edges, respectively. The
estimated network for $\operatorname{FDR}=0.2$ is shown in Figure~\ref{fig42}. In this figure,
two components of a large connectivity structure are observed. They
contain most of the genes that are connected by a large number of
high-selection-frequency edges. This constructed network can help to
generate a useful biological hypothesis and to design follow-up
experiments to better understand the underlying mechanism in BC. For
example, BINCO suggests with high confidence for the association
between MAP3K4 and STAT3. MAP3K4 plays a role in the signal
transduction pathways of BC cell proliferation, survival and
apoptosis [\citet{BilJoh}], and the constitutive activation
of STAT3 is also frequently detected in BC tissues and cell lines
[Hsieh, Cheng and Lin (\citeyear{HsiCheLin05})]. Interestingly,\vadjust{\goodbreak}
both MAP3K4 and STAT3 play roles in the regulation of c-Jun, a novel
candidate oncogene whose aberrant expression contributes to the
progression of breast and other human cancers [Tront, Hoffman and Liebermann (\citeyear{TroHofLie06});
\citet{Shaetal11}]. The association between
MAP3K4 and STAT3 detected by BINCO suggests their potential
cooperative roles in BC. It is also worth noting that for the four
clinical variables, the only edge with high selection frequency is
the one between age and ER-status (selection frequency $=0.96$). All
edges between clinical variables and the genes/probes are
insignificant (selection frequencies $<0.12$).
Networks built on perturbed data sets can also be used to detect hub
genes (i.e., highly connected genes), which are often of great
interest due to the central role these genes may play in genetic
regulatory networks. The idea is to look for genes that show
consistent high connection in estimated networks across perturbed
data sets. Here, we propose to detect hub genes by the ranks of their
degrees based on the estimated networks using $\lambda=340$ and
$l=0.9$. The ten genes with the largest means and the smallest SDs
of the degree rank across 100 bootstrap resamples (see Figure S-3, in
black dots) are MBD4, TARDBP, DDB2, MAP3K4, ORC3L, CDKN1B, REL, ATR,
LGMN and CDKN3. Nine out of these ten genes have been reported
relevant to BC, while the remaining one (TARDBP) is newly discovered
to be related to cancer [Postel-Vinay et al. (\citeyear{Po12})], although its role in BC is not clear at
present. The neighborhood topologies of these hub genes in the
network estimated by BINCO are illustrated in Figure~\ref{fig42}. More details of
these hub genes are given in the supplemental article [Li et al. (\citeyear{Lietal})],
Section B.\looseness=-1\vadjust{\goodbreak}
\section{Discussion}\label{sec5}
In this paper we propose the BINCO procedure to conduct
high-dimensional network inference. BINCO employs model aggregation
strategies and selects edges by directly controlling the FDR. This
is achieved by modeling the selection frequencies of edges with a
two-component mixture model, where a flexible parametric
distribution is used to model the density for the null edges. By
doing this, BINCO is able to provide a good estimate of FDR and
hence properly controls the FDR. To ensure BINCO works, we propose a
set of screening rules to identify the U-shape characteristic of
empirical selection frequency distributions. Based on our
experience, a U-shape corresponds to a proper amount of
regularization such that the FDR is well controlled and the power is
reasonable. Extensive simulation results show that BINCO performs
well under a wide range of scenarios, indicating that it can be used
as a practical tool for network inference. Although we focus on the
GGM construction problem in this paper, BINCO is applicable to a
wide range of problems where model selection is needed because it
provides a general approach to modeling the selection frequencies.
We use a mixture distribution with two components, one corresponding
to true edges and the other corresponding to null edges, to model
the selection frequency distribution. This two-component mixture
model is adequate as long as the distribution of the null component
is identifiable and can be reasonably estimated, as formalized in
the
\textit{proper condition}. Note that the
\textit{proper condition}
holds for a wide range
of commonly used (nonaggregation) selection procedures (Lemma~\ref{le1},
Remark~\ref{rem1}). To further ensure the FDR can be controlled at a
reasonable level, we propose a U-shape detection procedure and only
apply BINCO if the empirical distribution of selection frequencies
passes the detection. These rules for U-shape detection are
empirical but appear to work very well based on our extensive
simulations.
BINCO works well despite the presence of correlations between edges
(see Section D1 in the supplemental article [Li et al. (\citeyear{Lietal})]), because we
use the independence of edges only as a working assumption . It is
well known that if the marginal distribution is correctly specified,
the parameter estimates are consistent even in the presence of
correlation. This is similar to the generalized estimating
equations, where if the mean function is correctly specified, the
parameters will be consistently estimated [Liang and Zeger (\citeyear{LZ86})].
Toward this end, we use the three-parameter power beta distribution
to allow for adequate flexibility in modeling the marginal
distribution of selection frequencies.
BINCO is computationally feasible for high-dimensional data. The
major computational cost lies in generating the selection
frequencies via resampling. For each resample, the computational
cost is determined by that of the nonaggregated procedure BINCO
coupled with. In terms of
\textit{space}, it is
$O(np^2)$. The processing time for a data set with $n=200$, $p=500$,
under a given $\lambda$ and 100 bootstrap samples to generate
selection frequencies is about 20 minutes on a PC with Pentium
dual-core CPU at 2.8~GHz and 1~G ram. These selection frequencies can
be simultaneously generated through parallel computing for different
$\lambda$'s and weights. Fitting the mixture model takes much less
time, which is about 2 minutes for the above example on the same
computer.
Although we use GGM as our motivating example, BINCO works well even
if the multivariate normality assumption does not hold. Note that
the multivariate normality assumption only concerns the
interpretation of the edges. Under GGM, the presence of an edge
means conditional dependency of the corresponding nodes given all
other nodes. Without the normality assumption, one can only conclude
nonzero partial correlation between the two nodes given the rest of the
nodes. The
\textit{space} method used in this paper is to
estimate the concentration network (where an edge is drawn between
two nodes if the corresponding partial correlation is nonzero) and
has been shown to work well under nonnormal cases such as
multivariate-$t$ distributions [Peng et al. (\citeyear{PenZhoZhu09})]. We also generate data from nonnormal
distributions and found that BINCO works well in this situation (see
Section D4 in the supplemental article [Li et al. (\citeyear{Lietal})]).
BINCO is an aggregation-based procedure. In principle, it can be
coupled with any selection procedure. In this sense, it has a wide
range of applications as long as the features are defined (e.g.,
edges as in this paper, variables or canonical correlations as in
the example below) and the selection procedure is reasonably good,
for example, producing probabilities that satisfy the condition in
Lemma~\ref{le1}.
One application beyond GGM could be on the multi-attribute network
construction where the links/edges are defined based on canonical
correlations [Waaijenborg, Verselewel de Witt Hamer and Zwinderman (\citeyear{Wa08}),
Katenka and Kolaczyk (\citeyear{KatKol}),
Witten, Tibshirani and Hastie (\citeyear{WitTibHas09})]. Another interesting
extension may be on the time-varying network construction [\citet{Koletal10}] where appropriate
incorporation of the time-domain structure across aggregated models
will be important. These are beyond the scope of this paper and will
be pursued in future research.
The R package BINCO is available through CRAN.
\begin{appendix}
\section*{\texorpdfstring{Appendix: Proof of Lemma~\lowercase{\protect\ref{le1}}}{Appendix: Proof of Lemma 1}}\label{app}
\begin{pf}
Suppose as the sample size $n$ increases, an edge selection procedure
$A(\lambda)$ gives selection probabilities $\{\tilde{p}^{(n)}_{ij}\}$
(with respect to resample space) which uniformly satisfy
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\setcounter{equation}{0}
\begin{equation}\label{equA1}
\tilde{p}^{(n)}_{ij}\rightarrow1\qquad \mbox{if }(i,j)\in E
\end{equation}
and
\begin{equation}\label{equA2}
\limsup\tilde{p}^{(n)}_{ij}\leq M<1\qquad \mbox{if }(i,j)\in
E^c.
\end{equation}
Suppose $B$ is large such that $\frac{B+1}{B}M<1$. Let $X$ be a random
sample from the set of selection frequencies $\{X_{ij}^{\lambda}\}$
generated by applying $A(\lambda)$ on $B$ resamples, that is,
$\operatorname{Pr}(X=X_{ij}^{\lambda})=1/N_{\Omega}$, $(i,j)\in\Omega$. Also
suppose $X$ has density $f_{ij}^{\lambda}$ if $X=X_{ij}^{\lambda}$.
Then the mixture model (\ref{eq2.6}) becomes
\begin{eqnarray}
f^{\lambda}(x)&=& (1-\pi)f_0^{\lambda}(x)+\pi
f_1^{\lambda}(x)
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&=&\sum_{(i,j)\in E^c}
\frac{1}{N_{\Omega}}f_{ij}^{\lambda}(x)+\sum
_{(i,j)\in
E}\frac{1}{N_{\Omega}}f_{ij}^{\lambda}(x)
\end{eqnarray}
with $(1-\pi)f_0^{\lambda}(x)=\sum_{(i,j)\in E^c}\frac{1}{N_{\Omega
}}f_{ij}^{\lambda}(x)$ and $\pi f_1^{\lambda}(x)=\sum_{(i,j)\in
E}\frac{1}{N_{\Omega}}f_{ij}^{\lambda}(x)$.\vadjust{\goodbreak}
Because of the i.i.d. nature of resamples given the data,
$f_{ij}^{\lambda}$ is a binomial density with $\tilde{p}_{ij}^{(n)}$
as the probability of success, that is, $f_{ij}^{\lambda}(x) ={B
\choose k} (\tilde{p}_{ij}^{(n)})^k(1-\tilde{p}_{ij}^{(n)})^{B-k}$
for $x=k/B,k=0,1,\ldots,B$. This binomial density is monotone
decreasing for $x$ greater than its mode $\mu_{ij}=\frac{[(B+1)\tilde
{p}_{ij}^{(n)}]}{B}$ or $\frac{[(B+1)\tilde{p}_{ij}^{(n) }]-1}{B}$. By
(\ref{equA2}), given $V_1=\frac{B+1}{B}M <1$ and $\varepsilon>0$ such that
$V_1+\varepsilon<1$, $\exists N$ such that for all $n>N$ $\operatorname
{max}_{(i,j)\in E^c}(\mu_{ij})<V_1+\varepsilon$ and hence for any null
edge $(i,j)\in E^c$, $f_{ij}^{\lambda}(x)$ is monotone decreasing on
$[V_1+\varepsilon,1]$, which implies C2 since $f_0^{\lambda}(x)=\frac
{1}{(1-\pi)N_{\Omega}}\sum_{(i,j)\in E^c}f_{ij}^{\lambda}(x)$.
Also,\vspace*{2pt}
(\ref{equA1}) implies, for $(i,j)\in E$, $f_{ij}^{\lambda}(x)\rightarrow0$
uniformly for $x<1$, which implies C1 for any $V_2<1$. Taking $V_2$
such that $V_1<V_2<1$ satisfies the
\textit{proper condition}
and completes the proof.
\end{pf}
\end{appendix}
\section*{Acknowledgments}
We thank anonymous reviewers and editors for helpful comments that
significantly improved this paper. We also thank Ms. Noelle Noble
for technical editing.
\begin{supplement}[id=suppA]
\stitle{Supplement to ``Bootstrap inference for network construction
with an \mbox{application} to a breast cancer microarray study''\\}
\slink[doi]{10.1214/12-AOAS589SUPP}
\sdatatype{.pdf}
\sfilename{aoas589\_supp.pdf}
\sdescription{This supplement contains additional
simulation results, details of the hub genes detected by BINCO on
the breast cancer data, and examples of $p_{ij}$ and
$\tilde{p}_{ij}$ being close.}
\end{supplement}
|
2,877,628,089,508 | arxiv | \section{INTRODUCTION}\label{sec:intro}
The TRENDS ({\bf T}a{\bf R}getting b{\bf EN}chmark-objects with {\bf D}oppler {\bf S}pectroscopy) high-contrast imaging survey is a dedicated ground-based program that uses adaptive optics and related technologies to directly detect and study faint companions orbiting nearby stars \citep{crepp_12b}. TRENDS differs from other high-contrast campaigns primarily through its selection of targets. Rather than observing nearby young stars, we select older targets that show clear evidence for the existence of a low-mass companion as the result of years of precise radial velocity (RV) measurements. This evidence manifests as an acceleration in the RV time series which we refer to as a Doppler ``trend". The tradeoff between age and {\it a priori} knowledge of an orbiting companion results in a higher detection efficiency at the expense of mass sensitivity. However, recent advances in high-contrast imaging hardware and techniques will soon permit the detection of massive planets (super-Jupiters) around stars as old as $\approx$1 Gyr at infrared wavelengths \citep{crepp_johnson_11,hinkley_11_PASP,macintosh_12,skemer_12}. Operating primarily from Keck Observatory, the TRENDS program is sensitive to brown dwarf companions of any age for essentially all nearby ($d\lesssim50$ pc) targets \citep{montet_13}.
TRENDS survey discoveries to date include several benchmark high mass ratio binary stars (HD~53665, HD~67017, HD~71881), a triple star system (HD~8375), and a ``Sirius-like" white dwarf companion orbiting HD~114174 \citep{crepp_12b,crepp_13a,crepp_13b}. By connecting the properties of directly imaged companions to that of their primary star (such as metallicity and age), these objects serve as useful test subjects for theoretical models of cool dwarf atmospheres \citep{liu_10}. Further, the combination of Doppler observations and high-contrast imaging constrains the companion mass and orbit, essential information that brown dwarfs discovered in the field or at wide separations by seeing-limited instruments do not provide. In this paper, we report the direct imaging discovery of an old and cold brown dwarf orbiting the nearby G3V star HD~19467 (Table 1). The companion's subtle gravitational influence on HD~19467 was initially noticed as a Doppler acceleration spanning more than a decade. We show that HD~19467~B is almost certainly a T-dwarf, based on its intrinsic brightness and near-infrared colors.
\begin{table}[!ht]
\centerline{
\begin{tabular}{lc}
\hline
\hline
\multicolumn{2}{c}{HD~19467 Properties} \\
\hline
\hline
right ascension [J2000] & 03 07 18.57 \\
declination [J2000] & -13 45 42.42 \\
$B$ & 7.65 \\
$V$ & 7.00 \\
$J$ & $5.801\pm0.020$ \\
$H$ & $5.447\pm0.036$ \\
$K_s$ & $5.401\pm0.026$ \\
d [pc] & $30.86\pm0.60$ \\
$\mu_{\alpha}$ [mas/yr] & $-7.81\pm0.63$ \\
$\mu_{\delta}$ [mas/yr] & $-260.77\pm0.71$ \\
\hline
Mass [$M_{\odot}$] & $0.95\pm0.02$ \\
Radius [$R_{\odot}$] & $1.15\pm0.03$ \\
Luminosity [$L_{\odot}$] & $1.34\pm0.08$ \\
$\log R'_{HK}$ & $-4.98\pm0.01$ \\
Gyro Age [Gyr] & $4.3^{+1.0}_{-1.2}$ \\
SME Age [Gyr] & $9\pm1$ \\
$\mbox{[Fe/H]}$ & $-0.15\pm0.02$ \\
log g [cm $\mbox{s}^{-2}$] & $4.40\pm0.06$ \\
$T_{\rm eff}$ [$K$] & $5680\pm40$ \\
Spectral Type & G3V \\
v sini [km/s] & $1.6\pm0.5$ \\
\hline
\end{tabular}}
\caption{(Top) Coordinates, apparent magnitudes, distance, and proper motion of HD~19467. Magnitudes are from the 2-Micron All Sky Survey (2MASS) catalog of point sources \citep{cutri_03,skrutskie_06}. The parallax-based distance from {\it Hipparcos} uses the refined data reduction of \citealt{van_leeuwen_07}. (Bottom) Host star physical properties are estimated from SME using HIRES template spectra and theoretical isochrones \citep{valenti_fischer_05}. We estimate a gyrochronological age based upon empirical relations incorporating $B-V$ color and $R'_{HK}$ value \citep{mamajek_hillenbrand_08}.}
\label{tab:star_props}
\end{table}
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=3.2in]{f1.eps}
\caption{Relative RV measurements of HD~19467. We have directly imaged the substellar companion responsible for the long-term Doppler acceleration.}
\end{center}\label{fig:image}
\end{figure*}
\section{OBSERVATIONS}
\subsection{High-Resolution Spectroscopy}
\begin{table*}[!ht]
\centerline{
\begin{tabular}{cccc}
\hline
\hline
Date & BJD & RV & Uncertainty \\
$\mbox{[UT]}$ & -2,450,000 & [m~s$^{-1}$] & [m~s$^{-1}$] \\
\hline
\hline
1996-10-09 & 366.013 & 15.85 & 1.47 \\
1996-12-01 & 418.938 & 24.71 & 2.17 \\
1997-01-13 & 461.838 & 29.22 & 1.30 \\
1997-09-23 & 715.098 & 21.34 & 4.20 \\
1997-09-24 & 716.106 & 20.57 & 4.31 \\
1997-12-04 & 786.842 & 18.50 & 2.38 \\
1997-12-04 & 786.855 & 22.63 & 1.51 \\
1997-12-24 & 806.901 & 17.71 & 1.50 \\
1998-01-24 & 837.743 & 9.40 & 1.41 \\
1998-01-26 & 839.742 & 14.62 & 1.49 \\
1998-07-17 & 1012.120 & 23.09 & 1.40 \\
1998-07-18 & 1013.121 & 14.95 & 1.31 \\
1998-09-13 & 1070.112 & 15.81 & 4.26 \\
1998-09-16 & 1072.980 & 24.93 & 4.36 \\
1998-12-24 & 1171.774 & 21.27 & 1.48 \\
1999-08-19 & 1410.126 & 16.21 & 1.51 \\
1999-12-31 & 1543.844 & 13.69 & 1.60 \\
2000-01-08 & 1551.790 & 14.98 & 1.47 \\
2000-01-09 & 1552.841 & 9.86 & 1.73 \\
2000-02-08 & 1582.730 & 20.06 & 1.70 \\
2000-12-04 & 1882.801 & 25.05 & 1.62 \\
2000-12-22 & 1900.779 & 18.57 & 1.48 \\
2001-08-12 & 2134.078 & 15.24 & 1.58 \\
2001-11-29 & 2242.903 & 14.71 & 1.42 \\
2002-08-29 & 2516.019 & 13.55 & 1.62 \\
2002-10-28 & 2575.896 & 5.25 & 1.72 \\
2003-07-14 & 2835.129 & 15.17 & 1.84 \\
2003-10-13 & 2926.084 & 15.13 & 4.43 \\
\hline
2004-08-22 & 3240.040 & 6.29 & 1.24 \\
2005-02-26 & 3427.786 & 5.72 & 1.26 \\
2006-09-05 & 3984.035 & 0.13 & 1.07 \\
2011-09-02 & 5807.034 & -7.30 & 1.15 \\
2011-09-03 & 5808.104 & -2.49 & 1.46 \\
2011-09-04 & 5809.087 & -1.44 & 1.18 \\
2011-12-08 & 5903.778 & 3.46 & 1.43 \\
2012-08-12 & 6152.110 & -6.90 & 1.37 \\
2012-10-09 & 6210.014 & -5.92 & 1.43 \\
2013-08-14 & 6519.084 & -8.24 & 1.19 \\
2013-08-25 & 6530.024 & -10.85 & 1.15 \\
2013-09-12 & 6548.034 & -6.20 & 1.38 \\
\hline
\hline
\end{tabular}}
\caption{Doppler RV measurements for HD~19467.}
\vspace{1in}
\end{table*}
\subsubsection{Radial Velocity Measurements}
We obtained RV data for HD~19467 using HIRES at Keck \citep{vogt_94,marcy_butler_92}. First epoch RV observations were acquired on 1996 October 09 UT. Several years of measurements revealed that the star exhibits a persistent acceleration, and that $S_{HK}$ magnetic activity values do not correlate with the RV drift (Fig. 1). Precise Doppler measurements taken over an 16.9 year time frame are listed in Table 2. A horizontal line denotes the location of an RV offset resulting from the Summer 2004 HIRES detector upgrade which we include as a free parameter. A linear fit to the time series yields an acceleration of $-1.37\pm0.09\:\rm{m\:s}^{-1}\:\rm{yr}^{-1}$.
The RV time series also shows significant variations in addition to the systemic acceleration. Fourier analysis based on data acquired through the year 2012 had previously identified a periodic signal at $\approx1.6$ years. However, three more recent observations reveal the $\approx5$ m$\;$s$^{-1}$ signal to be spurious. The level of astrophysical noise (jitter) nominally expected from this type of main-sequence star is $2.4\pm0.4$ m$\;$s$^{-1}$ given its $\log R'_{HK}$ value and $B-V$ color \citep{isaacson_fischer_10}. Using Monte Carlo techniques that randomly scramble measurements in the Doppler time-series, we find that residual RV scatter seen in Fig. 2 (when comparing a linear fit with two-body Keplerian orbital models) results in a false-alarm probability well above the $\approx1\%$ threshold nominally used for Doppler discoveries \citep{marcy_05}. We consider the additional signal to be spurious and most likely caused by stellar activity, rather than an exoplanet, although further measurements are warranted.
\subsubsection{Star Properties}
Stellar template spectra (non-iodine measurements) were analyzed using the LTE spectral synthesis code {\it Spectroscopy Made Easy} (SME) described in detail in \citet{valenti_96,valenti_fischer_05}. The estimated physical properties of HD~19467 derived from spectral fitting are shown in Table 1. HD~19467 is listed in the SIMBAD database as an G1V star from medium resolution spectroscopy \citep{gray_06}. We find a best fitting spectral type of G3 using higher resolution ($R\approx55,000$) spectroscopy, and a luminosity class of dwarf (V) as we discuss in what follows.
HD~19467 is a nearby field star not obviously associated with any moving group or cluster. To facilitate our characterization of its companion, we estimate its gyrochronological age using the technique of \citealt{barnes_07}. The stellar rotation period is found empirically to be $P_r=24.9\pm2.5$ days from the measured $\log R'_{HK}$ and $B-V$ values \citep{wright_04}, which corresponding to a gyrochronological age of $4.30^{+0.96}_{-1.23}$ Gyr; this result is based upon updated coefficients that correlate the rotation period and $B-V$ color to age as determined by \citet{mamajek_hillenbrand_08}. Uncertainty in the age is dominated by intrinsic scatter in the empirical relation.
We have also attempted to calculate a system age using the iterative version of SME that self-consistently incorporates results from the LTE spectral analysis with Yonsei-Yale theoretical isochrones \citep{valenti_fischer_05}. Unfortunately, our code does not converge properly for HD~19467~A. Upon iterating, the age diverges to the end of the grid at 13.7 Gyr. Using only a single iteration, we find a much older age of $9\pm1$ Gyr compared to the gyrochronology method. G3 dwarfs may still reside on the main-sequence at this age, and the low metallicity ([Fe/H]$=-0.15\pm0.04$) of the host star does not suggest youth, but the unusual behavior of the SME iterative code casts doubt on its reliability for this particular source. As such, values listed in Table 1 (age, mass, radius, luminosity) are tabulated using the non-iterative version of SME. We note that HD~19467~B is too faint to cause any substantive spectral contamination.
Comparing to other Sun-like stars in the solar neighborhood, HD~19467 resides only $\Delta M_V=0.28$ mag above the median Hipparcos-based main-sequence at visible wavelengths \citep{wright_05}. We thus adopt a luminosity class of V. For subsequent analysis, we also adopt the gyrochronological age, noting however that the subsolar metallicity, $[Fe/H]=-0.15\pm0.04$, indicates an age older than the Sun (4.6 Gyr). In Section 3.2, we show that the model-dependent mass of HD~19467~B is still within the brown dwarf regime even for ages up to 10 Gyr.
\begin{figure}[!t]
\begin{center}
\includegraphics[height=3.2in]{f2_no_ws.eps}
\caption{High-contrast image of HD~19467~B taken with NIRC2 AO at Keck Observatory. Stellar speckles have been removed using PSF subtraction. The companion is 100,000 times fainter than its host star in the $K$-band.}
\end{center}\label{fig:image}
\end{figure}
\subsection{High-Contrast Imaging}
First-epoch high-contrast images of HD~19467 were acquired with the K' filter on 2011 August 30 UT using NIRC2 (instrument PI: Keith Matthews) and the Keck II AO system \citep{wizinowich_00}. The bright ($K_s=5.401\pm0.026$) star was placed behind the 300 mas diameter coronagraphic spot. We used angular differential imaging (ADI) to enable point-spread-function (PSF) subtraction \citep{marois_06}. Images were processed using the same techniques applied in previous TRENDS discoveries to determine photometric and astrometric quantities (see \citealt{crepp_13b} for details).
We originally noticed HD~19467~B using raw frames viewed by the NIRC2 graphical user interface, which enables basic data operations such as image subtraction. The companion is fainter than the sky background in K-band under median seeing, but sufficiently separated from the star ($\theta=1.6"$) such that it is detectable by-eye using two subtracted exposures having a small amount of (parallactic) angular diversity. Figure 2 shows a fully processed image of the companion taken on 2012 January 7 UT. HD~19467 was observed at three subsequent epochs to acquire photometric information in complementary filters and assess whether the faint source shares a common parallactic and proper-motion with the star.
Photometric measurements are summarized in Table 3. HD~19467~B is $\Delta K_s=12.57\pm0.09$ magnitudes fainter than HD~19467~A and has blue colors, $J-K_s=-0.36\pm0.34$ mag, and $J-H=-0.29\pm0.15$ mag. Our astrometric observations consist of four epochs taken over an 1.1 year baseline (Table~4). The proper-motion of HD~19467 is $260.9\pm0.7$ mas $\mbox{yr}^{-1}$. Meanwhile, the size of a NIRC2 pixel is $9.963\pm0.006$ mas as projected onto the sky \citep{ghez_08}. Comparing our relative astrometry measurements to the expected motion (vector sum of parallax and proper-motion) of an unrelated distant background source (i.e., null hypothesis), we find that HD~19467~B is clearly associated with HD~19467~A (Fig. 3). HD~19467~B has a projected separation of $51.1\pm1.0$ AU (2012-10-04 UT) and appears to exhibit clockwise orbital motion at a level of $22\pm6$ mas yr$^{-1}$.
\begin{table}[!ht]
\centerline{
\begin{tabular}{lc}
\hline
\hline
\multicolumn{2}{c}{Imaged Companion: HD~19467~B} \\
\hline
\hline
$\Delta J$ & $11.81\pm0.10$ \\
$\Delta H$ & $12.46\pm0.10$ \\
$\Delta K_s$ & $12.57\pm0.09$ \\
$J$ & $17.61\pm0.11$ \\
$H$ & $17.90\pm0.11$ \\
$K_s$ & $17.97\pm0.09$ \\
$M_J$ & $15.16\pm0.12$ \\
$M_H$ & $15.45\pm0.12$ \\
$M_{K_s}$ & $15.52\pm0.10$ \\
$m_{\rm dyn}$ [$M_J$] & $>51.9^{+3.6}_{-4.3}$ \\
$m_{\rm model}$ [$M_J$] & $56.7^{+4.6}_{-7.2}$ \\
\hline
\end{tabular}}
\caption{Photometric results and companion physical properties. The mass constraint (lower-limit) from dynamics ($m_{\rm dyn}$) using RV and imaging measurements is consistent with the model-dependent mass estimate from photometry ($m_{\rm model}$). The listed model-dependent mass is based upon the gyrochronological age of the primary star.}
\label{tab:comp_props}
\end{table}
\section{HD~19467~B PHYSICAL PROPERTIES}
\subsection{Dynamical Mass}
The measured RV acceleration may be combined with the companion projected separation to determine a lower-limit to its mass using dynamics \citep{torres_99,liu_02}. The straight-line fit to the RV time series of $-1.37\pm0.09$ m$\:$s$^{-1}\:$yr$^{-1}$ results in a minimum mass of $m\geq51.9^{+3.6}_{-4.3}M_J$, consistent with a non-Hydrogen-fusing object for near-edge-on orbits (Table 3). It will be possible to place an upper-limit on the companion mass and constrain the six orbital elements when the RV's and astrometry both show curvature.
\subsection{Model-Dependent Mass Estimate from Photometry}
The absolute magnitude of HD~19467~B (Table 3) is found using our measured magnitude difference and precise Hipparcos parallax of $32.40\pm0.62$ mas \citep{van_leeuwen_07}. A value of $M_{K_s}=15.52\pm0.10$ corresponds to a mass of $56.7^{+4.6}_{-7.2}M_J$ and effective temperature $T_{\rm eff}=1050\pm40$ K according to \citealt{baraffe_03} (COND) evolutionary models using our derived gyrochronological age. Assuming the host star is instead $9\pm1$ Gyr old, as indicated by an isochronal analysis [see notes from $\S$2.1], we find a model-dependent mass still within the brown dwarf regime, $m=67.4^{+0.9}_{-1.5}M_J$.
Figure 4 shows a plot of the companion measured absolute magnitude and $J-K$ color compared to T-dwarfs characterized using spectroscopy from \citealt{leggett_10}. HD~19467~B's brightness and spectral energy distribution are consistent with an $\approx$T5-T7 dwarf, affirming the interpretation of a cold substellar object. The $J-H=-0.29\pm0.16$ color is also consistent with a T-dwarf classification of $\approx$T5-T8 \citep{cushing_11,kirkpatrick_13,dupuy_liu_12}. Moderate resolution spectroscopy obtained with an integral field spectrograph will establish a more robust spectral-type designation [e.g., \citealt{hinkley_11_PASP}].
\subsection{Physical Separation and Period Range}
We can constrain the system orbital separation and period using the instantaneous RV acceleration and the mass estimate of HD~19467~B from photometry \citep{howard_10}. Such analysis indicates that HD~19467~B is presently $72.5\pm3.3$ AU from its host star, (self-consistently) further than the projected separation ($51.1\pm1.0$ AU). Assuming the true anomaly is near extreme values, i.e., apastron or periastron, we find that the orbital period lies between 320--1900 years for eccentricities between $0 \leq e \leq0.5$ using a host star mass of $M_*=0.95\pm0.02M_{\odot}$ (Table 1). Using the Vis-viva equation, which relates orbit velocity to the instantaneous separation and semimajor axis, we find that the maximum projected sky-motion is $28.0\pm0.9$ mas yr$^{-1}$ for the same eccentricity range, consistent with our observation that the astrometric position of HD~19467~B has changed by $22\pm6$ mas yr$^{-1}$. Our numerical simulations suggest that a unique solution for the orbit and mass of HD~19467~B may be determined as early as $\approx$15 years from now with sufficient observing cadence and continued Doppler and imaging monitoring \citep{crepp_12AAS}. Improved astrometric precision can of course facilitate a more rapid (shortened) orbit characterization timescale.
\begin{table*}[t]
\centerline{
\begin{tabular}{lccccccc}
\hline
\hline
Date [UT] & JD-2,450,000 & Filter & $\Delta t$ [min.] & $\Delta \pi$ [$^{\circ}$] & $\rho$ [mas] & P. A. [$^{\circ}$] & Proj. Sep. [AU] \\
\hline
\hline
2011-08-30 & 5,804.1 & $K'$ & 20.8 & 12.3 & $1662.7\pm4.9$ & $243.14\pm0.19$ & $51.4\pm1.0$ \\
2012-01-07 & 5,933.8 & $J$ & 5.0 & 3.6 & ----- & ----- & ----- \\
2012-01-07 & 5,933.8 & $H$ & 7.5 & 4.4 & $1665.7\pm7.0$ & $242.25\pm0.26$ & $51.4\pm1.0$ \\
2012-01-07 & 5,933.8 & $K'$ & 7.5 & 4.6 & $1657.3\pm7.2$ & $242.39\pm0.38$ & $51.2\pm1.0$ \\
2012-08-26 & 6,166.1 & $K'$ & 8.0 & 4.5 & $1661.8\pm4.4$ & $242.19\pm0.15$ & $51.3\pm1.0$ \\
2012-10-04 & 6,205.0 & $K_s$ & 10.0 & 5.4 & $1653.1\pm4.1$ & $242.13\pm0.14$ & $51.1\pm1.0$ \\
\hline
\hline
\end{tabular}}
\caption{Summary of high-contrast imaging observations, including integration time ($\Delta t$) and parallactic angle rotation ($\Delta \pi$), and resulting astrometric measurements. The primary star was not visible through the coronagraphic mask on 2012 January 07 UT in J-band due to the lower Strehl ratio. Proper-motion and parallactic motion analysis uses a weighted average for H, K' astrometry from this epoch.}
\label{tab:astrometry}
\end{table*}
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=3.4in]{f3.eps}
\caption{Relative astrometry results showing angular offset of the companion from the primary star at each epoch. The location of HD~19467~A defines the coordinate system origin. The solid curve shows the path that an infinitely distant background object would traverse from 2011 August 30 UT through 2012 October 04 UT. Dashed curves indicate uncertainty in the system proper motion and parallax. HD 19467~B is associated with its parent star ($42\sigma$). We detect systemic orbital motion of $22\pm6$ mas yr$^{-1}$ in a clockwise direction.}
\end{center}\label{fig:astrometry}
\end{figure*}
\begin{figure}[!t]
\begin{center}
\includegraphics[height=4.6in]{f4.eps}
\caption{HD~19467~B absolute J-band magnitude and J-K color (dashed lines) compared to field T-dwarfs (filled circles) \citet{leggett_10}. HD~19467~B appears to be consistent with an $\approx$T5-T7 dwarf.}
\end{center}\label{fig:image}
\end{figure}
\section{SUMMARY AND DISCUSSION}
We have acquired 40 precise Doppler observations of a nearby ($d=30.86\pm0.60$ pc) G3V star, HD~19467, over an 16.9 year time baseline. A long-term RV drift of $-1.37\pm0.09$ m$\:$s$^{-1}\:$yr$^{-1}$ indicates a distant companion with substellar minimum mass. Follow-up AO observations acquired with NIRC2 at Keck as part of the TRENDS high-contrast imaging program detect the companion responsible for the Doppler acceleration in four different filters and epochs. Astrometric analysis demonstrates unambiguous association of the source with its parent star over a 1.1 year time frame. We detect orbital motion in a clockwise direction at $3.7\sigma$.
HD~19467~B is intrinsically faint, with a contrast ratio ($\Delta K_s=12.57\pm0.09$ mag) comparable to the planets orbiting HR~8799 \citep{marois_10}. Placing the object on an H-R diagram, we find that its absolute magnitude ($M_J=15.16\pm0.12$) and blue colors ($J-K_s=-0.34\pm0.14$, $J-H=-0.29\pm0.16$) are most consistent with well-characterized field brown dwarfs in the $\approx$T5-T7 range \citep{leggett_10}. Theoretical evolutionary models indicate a mass of $m=56.7^{+4.6}_{-7.2}M_{Jup}$ using a gyrochronological age of $4.3^{+1.0}_{-1.2}$ Gyr \citep{baraffe_03,mamajek_hillenbrand_08}. Isochronal analysis suggests a much older age of $9\pm1$ Gyr, resulting in a mass of $m=67.4^{+0.9}_{-1.5}M_J$. Both photometric mass estimates are consistent with the $m\geq51.9^{+3.6}_{-4.3}M_J$ lower-limit derived using dynamics. Assuming a common origin between host star and companion, we may infer a low metallicity for HD~19467~B of [Fe/H]$=-0.15\pm0.04$. Given that the primary star is slightly evolved, residing $\Delta M_V=0.28$ mag above the median Hipparcos main-sequence, a subsolar metallicity implies an age older than that of the Sun ($4.6$ Gyr).
Substellar benchmark objects for which it is possible to simultaneously constrain the mass, age, and chemical composition are scarce but extremely valuable for calibrating theoretical atmospheric models and theoretical evolutionary models \citep{potter_02}. Many nearby T(Y)-dwarfs have been discovered as field objects by surveys that scan large regions of the sky at (red) optical, near-infrared, and mid-infrared wavelengths \citep{cushing_11,bihain_13,liu_13}. A fraction of these objects are members of multiple systems, but their projected physical separations are large, since the bright glare of host stars often precludes the detection of ultracold dwarfs orbiting in close proximity. Thus, the age of field brown dwarfs is generally highly uncertain and their utility for calibrating theoretical models limited. Of the coldest directly imaged T-dwarfs orbiting Sun-like stars [see \citealt{liu_11} for a list], HD~19467~B is the first with a measured Doppler acceleration, and so will be among the first to have a dynamically measured mass.
HD~19467~B appears to be a warmer version of GJ~758~B \citep{thalmann_09,janson_11}, except with much bluer colors. Both companions are old T-dwarfs orbiting nearby, well-characterized, main-sequence G- stars with precise parallax measurements. The difference in colors may be indicative of different cloud structures, atmospheric chemistry, or surface gravity \citep{liu_13}. HD~19467~B also represents a T-dwarf analogue to the directly imaged L4 companion to HR~7672~A \citep{liu_02}, which likewise has numerous Doppler measurements that have recently yielded a precise dynamical mass and orbit solution \citep{crepp_12a}. With an apparent magnitude of $J=17.61\pm0.11$ and angular separation of $1.653"\pm0.004"$, it should be possible to obtain high resolution, high signal-to-noise ratio near-infrared spectra of HD~19467~B comparable in quality to that obtained for the outer-planets orbiting HR~8799 \citep{bowler_10,konopacky_13}. Thus, HD~19467~B is an important benchmark object that will complement our understanding of low temperature dwarfs by exploring regions of parameter space corresponding to old age and subsolar metallicity as substellar objects evolve across the H-R diagram in time.
\section{ACKNOWLEDGEMENTS}
The TRENDS high-contrast imaging program is supported by NASA Origins of Solar Systems grant NNX13AB03G. JAJ is supported by generous grants from the David and Lucile Packard Foundation and the Alfred P. Sloan Foundation. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. 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. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium.
\begin{small}
\bibliographystyle{jtb}
|
2,877,628,089,509 | arxiv | \section{Introduction}\label{s:intro}
In this note we study new
traveling-wave solutions, that we call \emph{convective waves},
of the recently-introduced Richard-Gavrilyuk model (RG) of inclined (incompressible) shallow-water flow \cite{R,RG1,RG2}, of a type not present in the classical Saint-Venant equations (SV) from which (RG) descends.
The Saint-Venant equations are the industry standard in hydroengineering applications such as dam or spillway design \cite{BM,JNRYZ}, having been used --- apparently successfully --- unchanged for nearly a century \cite{Je}. However, the phenomena they model are sufficiently complicated that the limits of their applicability are difficult to determine. See, for example, the discussion of roll wave stability for \eqref{sv} in \cite{JNRYZ}, showing the delicacy of that question. And, in the case of roll waves at least, it has been known since the experimental work of Brock \cite{Br1,Br2} that the (explicit) roll wave solutions of \eqref{sv} deviate in shape from experimentally observed profiles, exhibiting an ``overshoot'' phenomenon near shock discontinuities. This inconsistency has recently been resolved by Richard and Gavrilyuk \cite{R,RG1,RG2} by incorporating small-scale vorticity in the modeling, converting the Saint-Venant equations (SV) to the extended Richard-Gavrilyuk model (RG)- {\it the first such advance in nearly 100 years.} See Fig. \ref{figure1} showing experimental inaccuracy of Saint-Venant waves near breaking, pointed out in \cite{Br1,Br2} as compared to near-exact fit of (RG) roll waves.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{pics/figure1.eps}
\end{center}
\caption{Numerics vs. experiment. Left panel: Saint-Venant. Right panel: Richard-Gavrilyuk. }
\label{figure1}
\end{figure}
The introduction of the (RG) model raises a number of interesting new questions: how does the existence theory for traveling waves differ between (SV) and (RG); can the rigorous stability theories developed in \cite{JNRYZ,YZ} for (SV) be adapted to the more complicated (RV)? and, most important, does (RV) predict new physical phenomena not captured by (SV)? These questions
are
addressed for hydraulic shocks and roll waves in \cite{RYZ} and \cite{RYZ2}. Here, we point out a new type of traveling wave occurring in (RG) but not (SV), of {\it convective-wave} solutions propagating with constant speed equal to the (everywhere constant) fluid velocity, but with fluid height and small-scale vorticity varying.
These may be understood as relaxation profiles for contact discontinuities of an associated equilibrium system, in the same way that hydraulic shock solutions are relaxation profiles for equilibrium shocks \cite{Bre,W,L,YZ,RYZ,SYZ}. Different from usual contact profiles, these are seen to be of degenerate type appearing in an infinite-dimensional family of possible nearby shapes. Evidently, this corresponds at linearized level to an infinite family of zero-eigenvalue modes, implying neutral stability at best, or orbital stability within this encompassing infinite-dimensional family. However, remarkably, we are able to factor out this degeneracy at the spectral level. Furthermore, for the subclass of such waves that are
both asymptotically constant at infinity and piecewise smooth with a finite number\footnote{Possibly zero.} of discontinuitites,
we establish by a generalized Sturm-Liouville argument (similar to \cite{SYZ,SZ}) that their spectral stability is completely determined by the spectral stability of their limiting endstates, hence reduced to an explicit stability condition.
Numerical time-evolution experiments confirm this conclusion, showing that Riemann data is resolved into an asymptotic pattern consisting of a hydraulic shock plus a convective contact wave, as predicted by the associated equilibrium system. They also show that small initial perturbations of a given convective wave leads in large-time to {\it another} significantly different convective wave, in agreement with the above-mentioned infinite-dimensional degneracy. Proving this observed nonlinear stability remains an interesting open question; see discussion in Section~\ref{s:disc}.
\medskip
The content of the present contribution is organized as follows. In Section~\ref{s:rg} we introduce the (RG) model. Besides its main basic properties, we provide there some educated guesses about expected nonlinear dynamics, based on intuitive analogies with more standard relaxation systems. Section~\ref{s:rg} mixes elementary observations with formal and heuristic arguments. For comparison, it is preceded in Section~\ref{s:sv} by a similar discussion for the (SV) model. Rigorous mathematical analysis is contained in Sections~\ref{s:convective}, \ref{s:stab-smooth} and~\ref{s:discont}, where we investigate the structure of convective waves and their spectral stability. They contain our main theorems, Theorem~\ref{th:main} that elucidates spectral stability of convective waves with smooth profiles, Theorem~\ref{th:conv} that studies convective stabilization by spatial weights, and Theorem~\ref{th:disc} that extends the analysis to discontinuous profiles. In Section~\ref{s:num} we provide various numerical experiments, validating some of the proved facts, confirming some of our intuitions but also trailblazing in the wild. We conclude with a few perspectives in Section~\ref{s:disc}.
\bigskip
\noindent {\bf Acknowledgment:} The authors would like to warmly thank Sergey Gavrilyuk and Pascal Noble for enlightening discussions about the (RG) model. L.M.R. and Y.Z. also express their gratitude to Indiana University for its hospitality during part of the preparation of the present contribution. L.M.R. would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme \emph{Dispersive hydrodynamics}
\section{The Saint Venant equations}\label{s:sv}
The Saint-Venant equations take the form of a $2\times 2$ {\it hyperbolic relaxation system}
\begin{align}\label{sv}
h_t + (hU)_x&=0,\\
(hU)_t
+(hU^2+ p(h) )_x
&=\hat gh-C_f|U|U,\\
\end{align}
where $h$ and $u$ denote fluid height and (vertically averaged) velocity at distance $x$ along an inclined ramp,
\be\label{piso}
p(h)= g'\frac{h^2}{2},
\ee
and $g'=g\cos \theta$, $\hat g=g\sin \theta$, where $g$ is the gravitational constant, $\theta$ is the angle from horizontal of the ramp, and $C_f$ is a coefficient of turbulent bottom friction, modeled according to Ch\'ezy's law as proportional to velocity squared.
See, e.g., \cite{Je,Dr,BM,R,RG1,RG2}.
The lefthand (first-order) side of \eqref{sv}--\eqref{piso} may be recognized as the equations of isentropic compressible gas dynamics with $\gamma$-law pressure, where $h$ plays the role of density and $\gamma=2$, from which we may deduce hyperbolicity \cite{Da,Bre}, with characteristics
\begin{align}\label{svchar}
\alpha_1&=U- a,&\alpha_2&=U+ a,
\end{align}
where $a$ is sound speed, given by
\be\label{svsound}
a=\sqrt {p_h} =\sqrt{g'H_0}.
\ee
The formal equilibrium system obtained by setting the righthand (zero-order) side to zero is the scalar, Burgers-type equation
\begin{align}\label{burgers}
h_t +q(h)_x&=0,&
\textrm{where }q(h):= \sqrt{\frac {\hat g h^3} {C_f}},
\end{align}
with characteristic speed
\be\label{echar}
\alpha_*=q'(h)=
\frac32 \sqrt{\frac {\hat g h} {C_f}},
\ee
where $u(h)= \sqrt{\frac {\hat g h} {C_f} }$ is determined by the equilibrium condition. In situations of hydrodynamic stability, or stability of constant-height equilibrium flow $(h,u)\equiv (H_0, U_0)$, $\hat g H_0=C_f U_0^2$, \eqref{burgers} is expected to approximately govern near-equilibrium behavior, in particular exhibiting ``hydraulic shock'', or ``bore'' type solutions \cite{Je,W,L,YZ,SYZ}. In situations of hydrodynamic instability, one observes, rather, pattern-formation and appearance of periodic ``roll wave'' solutions; see, e.g., \cite{C,Dr,Br1,Br2,BM,JNRYZ}.
As derived by Jeffreys \cite{Je,W}, the condition for hydrodynamic stability is
\be\label{hydro}
F:= \frac{U_0 }{a}< 2,
\ee
where
$F$ is the {\it Froude number}, a dimensionless constant relating fluid velocity to sound speed.
Condition \eqref{hydro} corresponds to the {\it subcharacteristic} (or ``interlacing'') {\it condition}
\be\label{sc}
\alpha_1<\alpha_*<\alpha_2
\ee
of Whitham \cite{W,L}, a standard necessary condition for hydrodynamic stability of relaxation systems. For, writing $q(H_0)$ as $q=H_0 u(H_0)$, where $u(h)= \sqrt{\frac {\hat g H_0} {C_f} }$, we have
\begin{align}\label{alphastar}
\alpha_*&= \frac{dq}{dH_0}= U_0 + a_*,&\textrm{where }
a_*=H_0 u'(H_0)= \frac{1}{2} U_0.
\end{align}
Comparing to \eqref{svchar}, we find that \eqref{sc} is equivalent to $a_*< a$, yielding \eqref{hydro}, independent of the choice of pressure function $p$ in \eqref{sv}. For the choice \eqref{piso} arising in the Saint Venant model (SV), $F$ is independent of height $H_0$ (or, equivalently, of $U_0=u(H_0)$), reducing to
\be\label{Fsv}
F= \frac{U_0 }{\sqrt{g'H_0} }= \sqrt{\frac {\tan \theta}{C_f }}.
\ee
\section{The Richard-Gavrilyuk equations}\label{s:rg}
The Richard-Gavrilyuk equations are a $4\times 4$ relaxation system \cite[pp 383--384]{R}
\begin{align}\label{rg4}
h_t + (hU)_x&=0,\\
(hU)_t + (hU^2+ p )_x&=\hat gh-C|U|U,\\
\big(hE)_t + \big(hUE + Up)_x &= (\hat gh -C_f U|U|)U,\\
(h\varphi)_t + (h\varphi U)_x&=0,
\end{align}
with
\begin{align}\label{gaslaws}
p&= \frac12 g'h^2+ (\Phi+\varphi)h^3,&
E&=\frac12 U^2+e,&
e&=\frac12(g'h + (\Phi+\varphi)h^2),
\end{align}
where $h$ and $U$ are fluid height and velocity; $\Phi$ and $\varphi$ are large- and small-scale enstrophies associated with vorticity; $g'$, $\hat g$, $C_t$, $C_f$ are physical parameters; and
\be\label{C}
C=C_f \frac{\varphi}{\Phi+\varphi} + C_t \frac{\Phi}{\Phi+\varphi}.
\ee
For reference, we recall here the physical meaning of variables $\Phi$ and $\varphi$,
as given in \cite{RG1,RG2}. These are so-called \emph{enstrophies}, consisting of squared vorticity. They comprise a splitting of total enstrophy/vorticity, with $\varphi$ corresponding to small-scale vorticity near the bottom, and $\Phi$ to large-scale vorticity near shocks. For both roll waves and hydraulic shocks, $\varphi$ is necessarily constant along profiles. For smooth hydraulic shock profiles, $\Phi$ is constant as well, in agreement with the physical derivation of $\Phi$ \cite{RYZ}; for
roll waves\footnote{We choose to restrain the term \emph{roll waves} to periodic waves similar to the ones of (SV), in particular discontinuous, despite the fact that there are periodic traveling waves of convective type, including but not restricted to discontinuous ones.}
and discontinuous hydraulic shock profiles, $\Phi$ builds up near the component subshock discontinuity \cite{RYZ,RYZ2} again in agreement with the derivation.
Let us point out that there are actually two slightly different Richard-Gavrilyuk models. We choose to work with the original one, from \cite{RG1}. The second version, from \cite{RG2}, is obtained from \eqref{rg4} by interchanging the roles of $C$ and $C_f$ and is presented in \cite{RG2} as leading to qualitatively and quantitatively similar numerical results, with a structure seemingly closer to the one of (SV).
\subsection{Reduced $3\times 3$ equations}\label{s:reduced}
When setting $\varphi\equiv \text{\rm constant}$ in \eqref{rg4}, we may eliminate the fourth equation, obtaining the reduced $3\times 3$ Richard-Gavrilyuk (RG3) model
\begin{align}\label{rg3}
h_t + (hU)_x&=0,\\
(hU)_t + (hU^2+ p )_x&=\hat gh-C|U|U,\\
\big(hE)_t + \big(hUE + Up)_x &= (\hat gh -C_f U|U|)U,\\
\end{align}
with $\varphi\equiv \text{\rm constant}$ considered as an additional model parameter.
This gives a large subclass of interesting solutions, including in particular all traveling waves $(h,U,\Phi, \varphi)(t,x)=(\bar h,\bar U,\bar \Phi, \bar \varphi)(x-ct)$ with\footnote{Or more generally with $\bar U$ taking the value $c$ on a discrete subset of the domain of smoothness of wave profiles.} $\bar U\neq c$ --- notably, roll waves, and hydraulic shocks \cite{R,RG1,RG2,RYZ,RYZ2}. For, combining \eqref{rg4}(i) and (iv), we obtain for smooth solutions the convection equation
\be\label{varphieq}
\varphi_t+ u\varphi_x=0
\ee
in place of \eqref{rg4}(iv). Thus, for smooth portions of a traveling-wave solution, $(\bar U-c)\bar \varphi_x=0$, giving $\bar \varphi\equiv \text{\rm constant}$ so long as $\bar U\neq c$. Similarly, the first and fourth Rankine-Hugoniot conditions for \eqref{rg4} at a discontinuity of speed $c$ are $c[h]=[hU]$ and $c[h\varphi]= [hU\varphi]$, where $[\cdot]$ denotes jump across the discontinuity. Combining these and using $h> 0$ yields
$[\varphi]=0$ so long as $U$ is not equal to $c$ on both sides of the discontinuity.
It follows that $\varphi$ remains constant also across discontinuities with speed $c$ not equal to fluid velocity $U$.
\subsubsection*{Relation to gas dynamics}\label{s:gas}
Just as the lefthand (first-order derivative) part of the Saint-Venant equations (SV) correspond to isentropic gas dynamics with pressure function of a polytropic gas law with $\gamma=2$, the lefthand (first-order derivative) part of the reduced (RG3) model \eqref{rg3} corresponds to full gas dynamics with pressure law
\be\label{palt}
p(h,e)= 2he - \frac12g'h^2,
\ee
obtained by using \eqref{gaslaws}(iii) to eliminate total enstrophy $\Phi +\varphi$ in \eqref{gaslaws}(i):
similar to but not exactly the nonisentropic polytropic gas law with $\gamma=2$ $p_{poly}(h,e)=2he$.
By this observation, we may read off the characteristics of the first-order (homogeneous) part of \eqref{rg3} using standard gas-dynamical formulae \cite{Da,Bre,BFZ} as
\begin{align}\label{3chars}
\tilde \alpha_1&= U-\tilde a,&
\tilde \alpha_2&= U,&
\tilde \alpha_3&= U+\tilde a,
\end{align}
where $a$ is sound speed, given by
\be\label{a}
\tilde a=\sqrt {p_h + \frac{p p_e}{h^2}}=\sqrt{6e-2g'h}= \sqrt{g'h+ 3(\Phi+\varphi)h^2}.
\ee
We note in passing that total enstrophy $S(h,e):=\Phi+\varphi= \frac {2e-g'h}{h^2}$
serves as a specific entropy for the first-order, gas-dynamical part of the equations
\cite{R,RG1,RG2},
satisfying for temperature $T= \frac{1}{S_e}=\frac{h^2}{2}$ the thermodynamic law $ \dD e=-p \dD\tau + T\dD S$, $\tau=1/h$, or
\begin{align}\label{therm}
\dD e&=\frac{p}{h^2} \dD h + T\dD S,&\textrm{that is } S_h&= -\frac{p}{h^2 }S_e,&
\textrm{or }e_h&=\frac{p}{h^2}.
\end{align}
This gives as a consequence, again by standard gas-dynamical facts \cite{Da,BFZ} that the first-order part of the equation for $S=(\Phi+\varphi)$ is the simple convection equation
\be\label{Scon}
S_t+US_x=0
\ee
\br\label{conrmk}
Written in terms of convectional derivatives $\dot g:=(\partial_t + U\partial_x)g$, the gas-dynamics equations become
\begin{align}\label{con1}
\dot h&=- hU_x,&
\dot U&=- p_x/h,&
\dot e&=- (p/h)U_x,
\end{align}
whence $\dot S= S_h \dot h + S_e \dot e \equiv 0$ if and only if $S_h h + S_e p/h=0$, or \eqref{therm}. Thus, any convected function $S(h,e)$ that is monotone in $e$ may serve as a specific entropy for gas dynamics; we may easily verify the role of $(\Phi+\varphi)$ a posteriori by direct computation
$(\partial_t+U\partial_x)(\Phi+\varphi)=0$.
\er
\br\label{conrmk2}
Alternatively, as in \cite{R,RG1,RG2}, expressing $p=\hat p(h,S)=\frac{g'h^2}{2}+ Sh^3$ using \eqref{gaslaws}(i) and
rewriting \eqref{con1} in standard fashion as $\dot h =- hU_x$, $\dot U =- \hat p_x/h$, $\dot S =0$, we obtain a decoupled $S$-equation and isentropic gas dynamics with pressure $\hat p(\cdot, S)$, yielding convective characteristics $0, \pm \tilde a$, with $\tilde a=\sqrt{ \hat p_h}$, hence (by \eqref{therm}(iii)) more directly recovering \eqref{a}.
Note that the sound speed $\tilde a=\sqrt{\hat p_h}$ for (RG) agrees with the sound speed $a=\sqrt{p_h}$ for (SV) only in the \emph{zero-entropic}, or zero-vorticity case $S\equiv 0$ in which (SV) was originally derived.
\er
\subsubsection*{Equilibrium system}\label{s:equilibrium}
The formal equilibrium system obtained by setting zero order derivative terms to zero is the same Burgers type equation \eqref{burgers}
as for (SV), with
\be\label{eqrel}
(u(h), \Phi(h))= \Big(\sqrt{\frac {\hat g h} {C_f} }, 0\Big),
\ee
and the same characteristic speed $\alpha_*=q'(H_0)=\frac 32 \sqrt{ \frac{\hat g H_0}{C_f}}$ given in \eqref{alphastar}. As noted in \cite{R,RG1,RG2}, (strong) hydrodynamic stability, or (strong) spectral stability of constant equilibrium flows for \eqref{rg3} holds provided
\be\label{hydroRGF}
\tilde F:=\frac{U_0}{\tilde a_0}<2,
\ee
a generalization to (RG) of condition \eqref{hydro} of Jeffreys \cite{Je}, and
\be\label{hydroRGC}
C_f\geq C_t.
\ee
See Appendix~\ref{app:split} for detailed computations.
Here, condition \eqref{hydroRGF} corresponds, by the same argument that was used for (SV), to the subcharacteristic condition $\tilde \alpha_1<\alpha_*<\tilde \alpha_3$ of Whitham \cite{W}, while \eqref{hydroRGC} corresponds to dissipativity of the zero-order derivative forcing terms on the righthand side of \eqref{rg3} in the convective, entropy mode $S$. Specifically, writing the complete reduced, inhomogeneous equations \eqref{rg3} in convective derivative form, we obtain
\be\label{inhomrg3}
\dot h + hU_x =0,\quad
\dot U + p_x/h= \frac{\hat g - C |U|U}{h},\quad
\dot S =\Big(1-\frac{\varphi}{S}\Big) \frac{ (C_t-C_f)|U|^3}{h^3},
\ee
and thus,
where $\Phi\neq0$ and $U\neq 0$, $\dot S \leq 0$ if and only if $C_t \leq C_f$, with strict inequality unless $C_t=C_f$.
\subsection{Full $4\times 4$ system}\label{s:full}
Augmenting \eqref{rg3} with the simplified $\varphi$ equation \eqref{varphieq}, we find for smooth solutions that the full (RG) equations, written in convective derivative form,
are the reduced equations \eqref{inhomrg3} together with $\dot \varphi=0$, hence hyperbolic with characteristics
\begin{align*}
\tilde \alpha&:= U-\tilde a,&0,&&0,&&U+\tilde a,
\end{align*}
where sound speed $\tilde a$ is again as in \eqref{a}. Moreover, the decoupled convective $\varphi$ mode is always neutrally stable, hence hydrodynamic stability is again equivalent to the conditions \eqref{hydroRGF}--\eqref{hydroRGC} for the reduced system \eqref{rg3}, {\it but is now at best nonstrict} in the sense that the dispersion relations for the linearization about a constant equilibrium state $(h_0,U_0, \Phi_0, \varphi_0)$ include the neutral, $\varphi$ mode $\lambda_4(k)=-iU_0k$, $k\in \R$.
\subsubsection*{Equilibrium system}\label{s:eq4}
Likewise, setting the righthand side (zero-order derivative part) of \eqref{rg4} to zero gives the same equilibrium relations \eqref{eqrel}
as for the reduced model \eqref{rg3}. However, with the addition of \eqref{rg4}(iv), the formal equilibrium model becomes now a $2\times 2$ {\it system}
\begin{align}\label{eq2}
h_t + q(h)_x&0, \\
(h\varphi)_t + (hu(h) \varphi)_x&=0,
\end{align}
or, for smooth solutions, $ h_t + q(h)_x=0$, $\varphi_t + u(h) \varphi_x=0$, with characteristics $\alpha_{*,1}= \frac{3}{2} u(h)$ and $\alpha_{2,*}=u(h)$ in characteristic modes $h$ and $\varphi$. And, since $\alpha_{*,2}=u(h)$ is {\it linearly degenerate} (independent of the associated mode $\varphi$), system \eqref{eq2}, besides Burgers shock and rarefaction waves in mode $h$, admits the new feature of {\it contact discontinuities} \cite{Bre,Da} in the mode $\varphi$, traveling with characteristic speed $c=\alpha_{*,2}=U_0$ equal to fluid velocity.
Indeed, one readily finds that solutions of a Riemann problem for \eqref{eq2}, joining arbitrary left and right states $(h_L, \varphi_L)$ and $(h_R, \varphi_R)$ consist of a contact discontinuity given by a jump in $\varphi$ from $\varphi_L$ to $\varphi_R$ with $h=h_L$ held fixed, and propagating at speed $c=u(h_L)=h_L^{1/2}$, followed by a scalar shock or rarefaction wave in $h$ with $\varphi=\varphi_R$ held fixed, and propagating with speed $\geq \alpha_*(h_L)> u(h_L)$ in the rarefaction case $h_L<h_R$ and $c=(q(h_R)-q(h_L))/(h_R-h_L)> (q(h_L)-q(0))/h_L= u(h_L)$ in the shock case $h_L \geq h_R$.
Thus, following the formalism of \cite{W,L}, in situations of hydrodynamic stability, one might expect, at least for near-equilibrium (small data) flow, that long-time asymptotic behavior of (RG) should be governed by a regularized version of that of \eqref{eq2}, that is, a superposition of relaxation profiles for a Burgers-type shock (or rarefaction) in the genuinely nonlinear $h$-mode and a contact discontinuity in the linearly degenerate $\varphi$ mode, arranged in order of increasing speed so as to form a noninteracting pattern of waves connecting equilibrium states. See for example, the corresponding analyses for the $2n\times 2n$ Jin-Xin model in \cite{HPW,Zh} under the assumption of strict hydrodynamic stability. We note that, unlike shock profiles, the contact profiles as constructed in \cite{HPW,Zh} are not traveling waves, but approximately self-similar solutions with diffusive, error-function scaling.
In the present, degenerate case of {\it neutral hydrodynamic stability}, the conclusion is less clear. It is straightforward to see that the Riemann problem \cite{Bre,Da}
for \eqref{eq2} has a unique solution consisting of a contact discontinuity in $\varphi$, with $h$ held fixed, followed by a shock or rarefaction in $h$, with $\varphi$ held fixed. And, for the shock case relevant to hydrodynamic engineering, that height $h$ is decreasing from left to right, there is a unique relaxation profile, as shown in \cite{RYZ}, connecting equilibrium states with a common value of $\varphi$.
However, as we show below, there is a large class of traveling convective-wave solutions with $U\equiv c$ that are candidates for relaxation profiles of a contact discontinuity. Moreover, as the $\varphi$ equation is decoupled from the rest of the system, it lacks the ``effective diffusion'' present in \cite{HPW,Zh}, so that one cannot expect the type of diffusive contact profiles studied there. There is also the interesting question what occurs in the case of hydrodynamic {\it instability} $F_*>2$ in terms of the linearly degenerate $\varphi$ field.
We investigate these questions in the remainder of the paper.
\section{Convective traveling waves}\label{s:convective}
The above discussion in Section \ref{s:full} motivates the study of constant-speed solutions of \eqref{rg4} with $U\equiv c=\text{\rm constant}$, in particular those connecting equilibrium states at $x=\pm \infty$.
Let us start by considering general solutions with $U\equiv c$, without regard to asymptotic states at infinity. Under this assumption, system \eqref{rg4} reduces for smooth solutions to $(h,\varphi)_t+ c(h,\varphi)_x=0$, together with
\begin{align*}
\d_x\left(g'\frac{h^2}{2} + (\Phi+ \varphi)h^3\right)&=\hat gh-C|c|c\,,\\
\frac12\,h^3\,\left(\d_t\Phi+c\,\d_x\Phi\right)
+c\,\d_x\left(g'\frac{h^2}{2} + (\Phi+ \varphi)h^3\right)&=\left(\hat gh-C_f|c|c\right)\,c\,,
\end{align*}
which imply
\[
\frac12 h^3\,\left(\d_t\Phi+c\,\d_x\Phi\right)
\,=\,c\,(C_t-C_f)\,\frac{\Phi}{\Phi+\varphi}\,.
\]
From the latter one readily deduces the following lemma.
\begin{lemma}
If $C_t\neq C_f$ and $c>0$, solutions with $U$ constant that are global and bounded (forward in time) satisfy
\[
\lim_{t\to\infty} \|\Phi(t,\cdot)\|_{L^\infty}=0\,.
\]
Furthermore if $C_t> C_f$ such solutions satisfy $\Phi\equiv 0$ and are unstable.
\end{lemma}
This motivates to restrict further to $(U,\Phi)\equiv (c,0)$. This amounts to
\begin{align}\label{twave}
(h,U,\Phi, \varphi)&=(\bar h, \bar U, \bar \Phi, \bar \varphi)(x-ct)\,,&
\textrm{with }(\bar U, \bar \Phi)\equiv(c,0)\,,
\end{align}
and
\be\label{scalarode}
\left(g'\frac{\bar h^2}{2} + \bar\varphi \bar h^3\right)' = \hat g\bar h -C_f |c|c.
\ee
That is, (i) {\it any such smooth solution} is a traveling wave, and (ii) every solution of \eqref{scalarode} yields a corresponding smooth traveling-wave solution of \eqref{rg4}, convected with fluid velocity. We denote these as {\it convective-wave} solutions.
\subsection{Piecewise smooth solutions} We now generalize the discussion to (piecewise smooth) discontinuous solutions with $(U,\Phi)\equiv (c,0)$. The foregoing arguments extend directly to smooth parts of the solutions. In turn at a discontinuity traveling with speed $s$, we find that the associated Rankine-Hugoniot conditions reduce to
\begin{align}\label{redRH1}
(s-c)[h]&=0\,,&
(s-c)[h\varphi]&=0\,,
\end{align}
together with
\be\label{redRH2}
[p]=\left[g'\frac{h^2}{2}+\varphi h^3\right]=0.
\ee
That is, we find that $s=c$, i.e., discontinuities are likewise convected with fluid velocity, so that solutions are again traveling waves, and that \eqref{scalarode} again holds, now in distributional sense. Thus, {\it convective-wave solutions}, whether smooth or piecewise smooth, {\it are completely described by \eqref{scalarode}.}
\subsection{Asymptotically constant solutions}
We now specialize the discussion to convective-waves with asymptotic limits $(h_\pm,\varphi_\pm)$ at $\pm\infty$. Note that from \eqref{scalarode} stems
\[
h_\pm=h_0:= \frac{C_f c^2}{\hat g}\,.
\]
Setting $\delta:=\bar h-h_0$, we obtain finally
\be\label{fode}
\left(g'\frac{(h_0+\delta)^2}{2} + \bar\varphi (h_0+\delta)^3\right)' = \hat g\delta.
\ee
For $\delta $ integrable but otherwise arbitrary\footnote{We deliberately omit to specify positivity constraints.} (smooth or not), this has nontrivial asymptotically-constant solutions, given by
\be\label{weird}
\left(g'\frac{(h_0+\delta)^2}{2} + \bar\varphi (h_0+\delta)^3\right)(x) = \kappa+ \hat g \int_{-\infty}^x \delta(y)\dD y,
\ee
$\kappa$ an arbitrary constant.
The convective-wave solutions so constructed have limiting asymptotic states
$(h_0, c, 0,\varphi_-)$ and $(h_0, c, 0, \varphi_+)$, with $(\varphi_-,\varphi_+)$ arbitrarily tunable by adjusting $\kappa$ and $\int_{-\infty}^{+\infty}\delta(x)\dD x$.
\br\label{nosv}
For the Saint-Venant equations \eqref{sv}, solutions with $U\equiv c= \text{\rm constant}$ must likewise be traveling waves moving with speed $c$, but now satisfying
$(p(h))'= \hat g h- C_f c^2$ with $p(h)= g' h^2/2$, or
$ h'= \big(\frac{g'}{\hat g}- \frac{C_f c^2}{g'h}\big), $ an ODE with no nontrivial bounded solutions other than the single unstable equlibrium $h_0=\frac{C_fc^2}{\hat g}$.
Thus, no such asymptotically constant $U\equiv c$ waves exist (except for the constant equilibrium). Likewise, for the reduced model \eqref{rg3}, with $\varphi$ constant, we obtain $(U,\Phi)\equiv (c, 0)$ and $(p(h))'= \hat g h- C_f c^2$, $p(h)=g'h^2/2 + \varphi h^3$, leading to $ h'= \big(\frac{\hat g h- C_f c^2}{g'h+ 3\varphi h^2}\big): $
again an ODE with a single unstable equilibrium, hence no nontrivial asymptotically constant waves.
\er
\br\label{rk:varphi}
Traveling waves with $\bar U$ not constant, have $\bar\varphi$ constant. Therefore the role of convective waves in a possible large-time traveling-wave resolution is precisely to convey $\varphi$-variations. Consistently the present analysis shows that one may indeed find an infinite-dimensional family of convective-wave fronts connecting any equilibria differing only by their $\varphi$-component.
\er
\br\label{rk:Phizero}
When motivating the restriction to solutions with $\Phi\equiv0$, we have excluded the case $C_t=C_f$ as somehow exceptional. Indeed, it is straightforward to check that in this special case, the freedom in convective-wave profiles is even larger. For instance, one may pick $h_0$, $\delta$ and $\kappa$ as above, but also pick $\bar\Phi$ arbitrarily and obtain a convective-wave profile by solving in $\bar \varphi$
\[
\left(g'\frac{(h_0+\delta)^2}{2} + (\bar\varphi+\bar \Phi) (h_0+\delta)^3\right)(x) = \kappa+ \hat g \int_{-\infty}^x \delta(y)\dD y\,.
\]
\er
\subsection{Periodic solutions and beyond}
Evidently, in the same way, we may construct spatially periodic solutions analogous to roll waves by integrating \eqref{fode} with $\delta$ periodic, and zero mean.
Note that this includes cases when such periodic waves are smooth. In contrast, the arguments in Remark \ref{nosv} show that there are no such convective periodic traveling waves for \eqref{sv} or \eqref{rg3}.
One may generalize the present construction so as to unify it with the asymptotically-constant case by noticing that we only need $\delta$ and an anti-derivative of $\delta$ to be bounded so as to obtain a bounded convective-wave.
\br\label{rk:stab}
The freedom in the construction includes the possibility to prescribe any discrete set as the set of discontinuous points for $\bar h$, discontinuities in $\bar\varphi$ being included in those. See the related discussion in \cite{JNRYZ,DR2}. However we stress that in principle it could well be that among this tremendously huge number of convective-wave solutions only a few types are stable. To exemplify this possibility, we point out that it follows from the analysis in \cite{DR1,DR2} that such a dramatic reduction does occur for scalar balance laws. A consequence of our spectral analysis is that for (RG) such a reduction does not occur, at least for asymptotically constant waves with a finite number of discontinuities.
\er
\section{Spectral stability of smooth solutions}\label{s:stab-smooth}
We now investigate spectral stability of convective waves, starting with smooth asymptotically constant solutions.
\br
Note that unlike what happens in more standard traveling-wave analyses smoothness and localization of wave profiles is not determined by profile equations but could be tuned arbitrarily by the prescription of $\delta=h-h_0$. Since we believe that the only important disctinction from the point of view of applications is whether $\delta$ is continuous or not, in the present section, devoted to the smooth case, we assume that $\delta\in\mathcal{C}^\infty$ and, in the asymptotically constant case, that $\delta$ and all its derivatives are exponentially localized. The reader interested in relaxing this assumption may adapt arguments in \cite{Pego-Weinstein} to the situation at hand.
\er
\subsection{Eigenvalue equations}
To begin with we investigate the eigenvalue problem.
Linearizing about a convective-wave solution
\begin{align*}
(h,U,\varphi,\Phi)(t,x)&=(\bar h,\bar U,\bar \varphi,\bar \Phi)(x-ct),&(\bar U,\bar\Phi)\equiv (c,0),
\end{align*}
we obtain a linear evolution whose eigenvalue equations are
\be\label{mateval}
\lambda A^0 W+ (AW)'=EW,
\ee
where $W=(h,U, \Phi, \varphi)$,
\begin{align}
A^0&=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\[0.25em]
c & \bar{h} & 0 & 0\\[0.25em]
\frac{c^2}{2}+\frac{3\bar{\varphi} \bar{h}^2}{2}+g'\bar{h} & c\bar{h} & \frac{\bar{h}^3}{2} & \frac{\bar{h}^3}{2}\\[0.25em]
\bar{\varphi} & 0 & 0 & \bar{h} \end{array}\right)\\[0.25em]
A^1&=\left(\begin{array}{cccc} c & \bar{h} & 0 & 0\\[0.25em]
c^2+3\bar{\varphi} \bar{h}^2+g'\bar{h} & 2c\bar{h} & \bar{h}^3 & \bar{h}^3\\[0.25em] \frac{c\left(c^2+9\bar{\varphi} \bar{h}^2+4g'\bar{h}\right)}{2} & \frac{\bar{h}\left(3c^2+3\bar{\varphi} \bar{h}^2+2g'\bar{h}\right)}{2} & \frac{3c\bar{h}^3}{2} & \frac{3c\bar{h}^3}{2}\\[0.25em]
c\bar{\varphi} & \bar{h}\bar{\varphi} & 0 & c\bar{h} \end{array}\right)\\[0.25em]
E&=\left(\begin{array}{cccc} 0 & 0 & 0 & 0\\[0.25em]
\hat{g} & -2C_fc & -\frac{c^2\left(C_T-C_f\right)}{\bar{\varphi} } & 0\\[0.25em]
c\hat{g} & \hat{g}\bar{h}-3C_fc^2 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right)\\[0.25em]
A=A^1-cA^0&=\left(\begin{array}{cccc} 0 & \bar{h} & 0 & 0\\[0.25em]
\bar{h}\left(g'+3\bar{h}\bar{\varphi} \right) & c\bar{h} & \bar{h}^3 & \bar{h}^3\\[0.25em] c\bar{h}\left(g'+3\bar{h}\bar{\varphi} \right) & \bar{h}\left(\frac{c^2}{2}+\frac{3\bar{\varphi} \bar{h}^2}{2}+g'\bar{h}\right) & ch^3 & c\bar{h}^3\\[0.25em]
0 & \bar{h}\bar{\varphi} & 0 & 0 \end{array}\right)
\end{align}
and $A$ has a kernel of dimension $2$. Taking the inner product with the left kernel of $A$ thus gives two algebraic relations between the variables, by which we may reduce the eigenvalue problem to a $2\times 2$ ODE.
Specifically, subtracting $\bar{\varphi}$ times the first equation from the fourth equation in \eqref{mateval} yields
\[
\lambda \varphi + \bar \varphi_x U=0,
\]
which gives one of the algebraic relations. From this we may solve for $\varphi$ in terms of $U$:
\be
\label{eqvarphi}
\varphi=-\lambda^{-1} \bar \varphi_x U.
\ee
This corresponds to left zero-eigenvector $\ell_1=(-\bar{\varphi},0,0,1)$ of $A$.
Likewise, for $\ell_2$ equal to the other left zero-eigenvector of $A$,
we obtain a relation
\[
\ell_2 (\lambda A^0 +A'-E)w= 0.
\]
Taking $\ell_2=\left(c^2-3\bar\varphi \bar h^2-2g'\bar{h} , -2c ,2 , 0 \right)$ and using \eqref{eqvarphi}, this equation yields
\be
\label{eqPhi1}
\Phi=-\frac{2\bar\varphi \left(C_fc^2-\hat{g}\bar{h}+\bar{h}^3\bar{\varphi}_{x}+3\bar{h}^2\bar{h}_{x}\bar{\varphi} +g'\bar{h}\bar{h}_{x}\right)}{2(C_f-C_T)c^3+\bar{h}^3\lambda \bar{\varphi} }U
\ee
or
\be
\label{eqPhi2}
\Phi=-\frac{2\bar\varphi \left(C_fc^2-\hat{g}\bar{h}+\left(\bar{h}^3\bar{\varphi} +\frac{1}{2}g'\bar{h}^2\right)'\right)}{2(C_f-C_T)c^3+\bar{h}^3\lambda \bar{\varphi} }U
\ee
By \eqref{scalarode}, \eqref{eqPhi2} implies
\be\label{eqPhi}
\Phi\equiv0.
\ee
The full eigenvalue equations \eqref{mateval} thus reduce to the $2\times 2$ ODE
\be
\left(\left[\begin{array}{cc} 0 & \bar{h}\\ \bar{h}\left(g'+3\bar{h}\bar{\varphi} \right) & c\bar{h}-\frac{1}{\lambda}\bar{h}^3\bar{\varphi}_x \end{array}\right]\left[\begin{array}{c} h\\ U \end{array}\right]\right)'=\left(\begin{array}{cc} -\lambda & 0\\ \hat{g}-c\lambda & -2C_fc-\bar{h}\lambda \end{array}\right)\left[\begin{array}{c} h\\ U \end{array}\right],
\ee
or
\begin{align}
\label{2by2}
&\left[\begin{array}{cc} 0 & \bar{h}\\ \bar{h}\left(g'+3\bar{h}\bar{\varphi} \right) & c\bar{h}-\frac{1}{\lambda}\bar{h}^3\bar{\varphi}_x \end{array}\right]\left[\begin{array}{c} h\\ U \end{array}\right]'\\
=&\left[\begin{array}{cc} -\lambda & -\bar{h}_{x}\\ \hat{g}-g'\bar{h}_{x}-6\bar{\varphi}\bar{h}\bar{h}_{x}-3\bar{h}^2\bar{\varphi}_{x}-c\lambda & -2C_fc-c\bar{h}_{x}+\frac{3}{\lambda}\bar{h}^2\bar{h}_x\bar{\varphi}_x+\frac{1}{\lambda}\bar{h}^3\bar{\varphi}_{xx}-\bar{h}\lambda \end{array}\right]\left[\begin{array}{c} h\\ U \end{array}\right].
\end{align}
Finally, to adapt Sturm-Liouville type arguments from \cite{SYZ}, we observe that equation \eqref{2by2} may for any $\lambda\neq 0$ be reduced to second-order scalar form
\begin{align}\label{Ueq_aux}
h&=\frac{(\bar{h}U)'}{-\lambda},\\
\label{Ueq}
U''&+f_1 U'+(f_2\lambda^2+f_3\lambda+f_4)U=0,
\end{align}
where
\begin{align} \label{fs}
&f_1=\frac{4\,\bar{\varphi}_{x}\,\bar{h}^2+12\,\bar{h}_{x}\,\bar{\varphi} \,\bar{h}-\hat{g}+3\,g'\,\bar{h}_{x}}{\bar{h}(g'+3\,\bar{h}\,\bar{\varphi}) },
&f_2&=-\frac{1}{\bar{h}(g'+3\,\bar{h}\,\bar{\varphi})}<0,\\
&f_3=-\frac{2\,C_f\,c}{\bar{h}^2\,\left(g'+3\,\bar{h}\,\bar{\varphi} \right)}<0,
& f_4&=\frac{(\bar{\varphi}\bar{h}^3)''+g'\,\bar{h}_{xx}\,\bar{h}+g'\,{\bar{h}_{x}}^2-\hat{g}\,\bar{h}_{x}}{\bar{h}^2\,\left(g'+3\,\bar{h}\,\bar{\varphi} \right)}=0
\end{align}
where $f_4=0$ is obtained from differentiating \eqref{weird} twice.
{\it System \eqref{Ueq} with \eqref{eqvarphi}, \eqref{eqPhi} and \eqref{Ueq_aux} is, evidently, equivalent to \eqref{mateval} for all $\lambda\neq 0$}.
\subsection{Removal of degenerate modes}\label{s:degen}
We now show how the computations of the former subsection may be put in an a more functional-analytic framework so as to yield corresponding reductions at the spectral level, that is, for resolvent problems, as required by abstract semigroup theory.
The linearized dynamics obeys
\be\label{eq:linearized}
A_0\d_t W+\d_x(A\,W)\,=\,E\,W
\ee
with $A_0$, $A$ and $E$ as above. By applying the invertible
\[
\bp
\begin{matrix} 1&0&0&0\\
-c\,\bar{h}^{-1}&\bar{h}^{-1}&0&0\end{matrix}\\
\bar{h}^{-3}\ell_2-\bar{h}^{-1}\ell_1\\
\bar{h}^{-1}\ell_1
\ep
\]
one shows that \eqref{eq:linearized} is equivalently written as
\[
\d_t\bp h\\U\\\Phi\\\varphi\ep\,=\,L\bp h\\U\\\Phi\\\varphi\ep
\]
where
\[
L\bp h\\U\\\Phi\\\varphi\ep
\,=\,\bp
-A_{red}\d_x\bp h\\U\ep
+E_{red}\bp h\\U\ep
+\bp 0\\-\bar{h}^{-1}\,\d_x\left(\bar{h}^3\,(\Phi+\varphi)\right)
-\frac{c^2\left(C_T-C_f\right)}{\bar{\varphi}\bar{h}}\Phi
\ep\\
-\frac{2(C_f-C_T)c^3}{\bar{h}^3\bar{\varphi}}\Phi\\
-\bar{\varphi}_x\,U
\ep
\]
with
\begin{align*}
A_{red}&=\bp0&\bar{h}\\
g'+3\bar{h}\bar{\varphi}&0\ep\,,&
E_{red}&=\bp 0&-\bar{h}_x\\
\bar{h}^{-1}\,
\left(\hat{g}-\d_x\left(\bar{h}\left(g'+3\bar{h}\bar{\varphi} \right)\right)\right)
&-2C_fc\,\bar{h}^{-1}\ep\,.
\end{align*}
The formal operator $L$ may be turned into a densely-defined closed operator on a Banach space $X$, with domain $D$, for various pairs $(X,D)$ of the form $X=X_0\times X_1$, $D=X_1\times X_1$, including those arising from $X_0=(W^{k,p}(\R))^2$ and $X_1=(W^{k+1,p}(\R))^2$ for any $(k,p)\in\N\times [1,\infty)$, or from $X_0=(BUC^{k}(\R))^2$ and $X_1=(BUC^{k+1}(\R))^2$ for any $k\in\N$ (where $BUC^\ell$ denotes functions whose derivatives up to order $\ell$ are bounded and uniformly continuous).
By definition, $\lambda$ does not belong to the spectrum of $L$ if and only if for any $F\in X$ there exists a unique $W\in D$ such that $(\lambda-L)W=F$, and the corresponding solution operator $(\lambda-L)^{-1}$ is bounded on $X$.
\begin{definition}\label{def:spec}
An asymptotically-constant smooth convective-wave will be called spectrally stable (on the functional space $X$) provided that the spectrum of the corresponding $L$ is included in $\{\,\lambda\,;\,\, \Re(\lambda)\leq 0\,\}$. It is called strongly spectrally stable provided that the foregoing spectrum is included in $\{\,\lambda\,;\,\, \Re(\lambda)< 0\,\}\,\cup\,\{0\}$.
\end{definition}
The definition of strong spectral stability is motivated by the fact that in any case $0$ belongs to the spectrum.
For the above-mentioned functional framework, one checks readily that $\lambda$ does not belong to the spectrum of $L$, if and only if $\lambda\neq0$,
\be\label{eq:spec-algebraic}
\lambda\notin -\overline{\frac{2(C_f-C_T)c^3}{\bar{h}^3\bar{\varphi}}(\R)}
\ee
and for any $G\in X_0$, there exists a unique $V\in X_1$ such that $L_{red}(\lambda)V=G$ with a solution operator $(L_{red}(\lambda))^{-1}$ bounded on $X_0$, where
\begin{align*}
&L_{red}(\lambda)\bp \tau\\U\ep
:= A_{red}\d_x\bp \tau\\U\ep
+\bp 1&0\\0&\lambda\ep(\lambda-E_{red})\bp \lambda^{-1}&0\\0&1\ep\bp \tau\\U\ep
+\bp 0\\-\bar{h}^{-1}\,\d_x\left(\bar{h}^3\,\bar{\varphi}_x\,U\right)\ep\\
&=\bp1&\d_x(\bar{h}\cdot)\\
\d_x\left(\left(g'+3\bar{h}\bar{\varphi} \right)\cdot\right)
-\bar{h}^{-1}\,
\left(\hat{g}-\bar{h}_x\left(g'+3\bar{h}\bar{\varphi} \right)\right)
&\lambda^2+\lambda\,2C_fc\,\bar{h}^{-1}
-\bar{h}^{-1}\,\d_x\left(\bar{h}^3\,\bar{\varphi}_x\,\cdot\right)\ep\bp \tau\\U\ep.
\end{align*}
This achieves the reduction from the original spectrum problem to the invertibility of a reduced operator $L_{red}(\lambda)$, whose kernel equation is precisely \eqref{2by2} for $(\tau,U)=(\lambda\,h,U)$. Note that, as the original $\lambda-L$, $L_{red}(\lambda)$ is a closed densely defined operator depending analytically on $\lambda$.
A few remarks are in order to highlight what is the nature of the gain when going from $\lambda-L$ to $L_{red}(\lambda)$.
\br
The original operator $L$ combines purely algebraic parts, that is, parts given as multiplication operators, with differential parts. The key point is that multiplication operators are Fredholm of index $0$ only when they are invertible so that the spectrum of a multiplication operator is reduced to its essential spectrum. Moreover the latter is determined by all the values of the function by which we multiply\footnote{See for instance \eqref{eq:spec-algebraic}.}, instead of being read on asymptotic limits at $\pm\infty$. This is this kind of degeneracy that is responsible for the fact that there is an uncountable family of zero eigenmodes of \eqref{mateval}, obtained through $(U,\Phi)\equiv (0,0)$ and
\[
( \bar h(g'+3\bar h\bar \varphi)h + \bar h^3 \varphi)'=\hat g h,
\]
the linear equivalent of the nonlinear solutions determined by $(\bar U,\bar \Phi)\equiv (c,0)$ and \eqref{weird}. By factoring out the algebraic part, we effectively remove this kind of degeneracy. However, it must be remembered that the full system does possess these degenerate modes, a fact with implications for nonlinear stability and asymptotic behavior ; see Section~\ref{s:disc} for further discussion. The reduced operators $L_{red}(\lambda)$ are one-dimensional\footnote{This is crucial since first-order operators possess elliptic properties only in dimension $1$.} non-characteristic differential operators with asymptotic limits at $\pm\infty$ (reached sufficiently fast) so that, as we detail below, in a large region of the spectral plane --- purely determined by asymptotic limits ---, these operators are Fredholm of index $0$, thus their invertibility is equivalent to their one-to-one character, hence to \eqref{Ueq} possessing no non trivial solution. Incidentally we point out that the failure of the non-characteristic requirement would bring other kinds of degeneracies; see the detailed discussion in \cite{DR2}.
\er
\br
The crucial part of the reduction is the elimination of the algebraic parts. But we also had to perform a suitable scaling, including $h\mapsto \lambda h=:\tau$, so as to preserve regularity near $\lambda=0$. This scaling shares similarities with classical ``flux-type'' transformations, considered for instance in \cite{PZ}, and that may be thought as spectral counterparts to the widely-used anti-derivative trick --- dating back at least to \cite{Matsumura-Nishihara,Goodman_bis} --- introduced to remove the (non-degenerate) eigenvalue $0$. For more advanced, multi-dimensional, versions of the ``flux-type'' transformations we refer the reader to \cite{HLyZ2,BHLyZ}. More generally, the need for such types of scaling also arise when connecting at spectral level Eulerian and mass-Lagrangian formulations or quantum systems and their hydrodynamic formulations ; see \cite[Section~5.2]{BMR} on the former and \cite[Section~3.2]{Audiard-Rodrigues} on the latter.
\er
\subsection{Standard spectral stability} \label{s:smooth_as_const}
We now investigate invertibility properties of $L_{red}(\lambda)$. The goal is in a relevant region of the spectral plane to reduce it to invertibility properties of its spatial asymptotic limits
\begin{align*}
L_{red}^{\pm}(\lambda)\bp \tau\\U\ep
&:=\bp1&h_0\d_x\\
\left(g'+3h_0\varphi_{\pm}\right)\d_x
-h_0^{-1}\,\hat{g}
&\lambda^2+\lambda\,2C_fc\,h_0^{-1}\ep\bp \tau\\U\ep\,.
\end{align*}
Our strategy uses classical arguments to show that this reduction holds when \eqref{Ueq} possesses no non trivial solution, and then adapts arguments from \cite{SYZ} to show that the latter does hold in the spectral region of interest. On the former classical arguments we give little detail and rather refer the reader to the already classical \cite{Z-H,Z3,MZ,Sandstede,KapitulaPromislow-stability} for detailed comprehensive exposition and to the recent \cite{Blochas-Rodrigues} for a self-contained worked-out case that could hopefully be used as a gentle entering gate.
As a preliminary we point out that the invertibility of $L_{red}^{\pm}(\lambda)$ for any $\lambda$ such that $\Re(\lambda)>0$ (respectively for any $\lambda\neq0$ such that $\Re(\lambda)\geq0$) is equivalent to
\begin{align*}
F_{\pm}:=\sqrt{\frac{\hat{g}}{C_f(g'+3h_0\varphi_\pm)}}&\leq2\,,&
(\textrm{resp. } F_{\pm}<2)\,.
\end{align*}
The foregoing claim follows from the computations involved in the stability analysis for constant states, already worked out in \cite{R,RG1,RG2}, and provided in Appendix~\ref{app:split}.
\begin{theorem}\label{th:main}
Consider an asymptotically-constant smooth profile with limiting values $(h_0, c, 0,\varphi_-)$ and $(h_0, c, 0, \varphi_+)$, $c>0$, reached exponentially fast. This wave is spectrally stable if and only if
\begin{align*}
C_f&\geq C_t\,,&
F_+&\leq 2\,,&
F_-&\leq 2\,.
\end{align*}
and it is strongly spectrally stable if and only if
\begin{align}\label{hydrodynamicstability}
C_f&\geq C_t\,,&
F_+&<2\,,&
F_-&<2\,.
\end{align}
\end{theorem}
\begin{proof}
The role of the condition on $(C_f,C_t)$ has been elucidated in the foregoing subsection. The necessity of the conditions on $F_+$ and $F_-$ stems directly from the classical fact that invertibility of both $L_{red}^{\pm}(\lambda)$ is necessary to invertibility of $L_{red}(\lambda)$. The foregoing claim is relatively easy to prove by building quasi-modes ; see Lemma~2 in the Appendix to \cite[Chapter~5]{He}, or Proposition~2.1 in \cite[Section~2.1]{DR2}. Knowing that $L_{red}^{\pm}(\lambda)$ are invertible when $\Re(\lambda)\geq 0$, $\lambda\neq0$, it is classical to deduce that for such $\lambda$, $L_{red}(\lambda)$ is invertible if and only if it is one-to-one. However the proof of the latter is longer and more involved and we simply refer to the already quoted literature. Let us only mention that the classical proof builds an inverse from Green functions, and that those are obtained by gluing together solutions to the differential equation encoding the kernel equation for $L_{red}(\lambda)$, considered on half-lines. Hence the strong connection between the structure of the kernel equation and invertibility.
We also observe that the sufficiency of the condition for spectral stability may be derived from the sufficiency of the condition for strong spectral stability through a limiting argument that we omit.
Thus, the remaining, and only non classical, task is to prove that when $C_f\geq C_t$, $F_+<2$, and $F_-<2$, and $\Re(\lambda)\geq 0$, $\lambda\neq0$, the kernel of $L_{red}(\lambda)$ is reduced to the null function, or, in other words, that \eqref{Ueq} possesses no non trivial solution (in the relevant functional space). To prove this, we modify the strategy from \cite{SYZ}.
The argument is closed by homotopy. For $\epsilon\in[0,1]$, consider the convective-wave $(\bar{h}_\epsilon,\bar{U}_\epsilon,\bar{\Phi}_\epsilon,\bar{\varphi}_\epsilon)$ built from \eqref{weird} with the same $(c,h_0,\kappa)$ but $\delta$ replaced with $\epsilon\,\delta$. At $\epsilon=0$, the wave is constant with $\bar{h}\equiv h_0$, $\bar{\varphi}_\epsilon\equiv\varphi_-$. The situation is favorable at $\epsilon=0$ thanks to $F_-<2$, and Froude conditions are seen to hold for any $\epsilon\in [0,1]$ by monotonicity in $\varphi_+$ of the criterion. Our task is to show that there is no transition from stability to instability as we vary $\epsilon$. It follows from general theory that transitions may occur only in two ways:
\begin{enumerate}
\item At some $\epsilon_0$ there is a $\lambda\in i{\mathbb R}$, $\lambda\neq0$, such that the $\epsilon_0$-version of \eqref{Ueq} possesses a non trivial solution (in the relevant functional space).
\item For some $\epsilon_0$, for any $\epsilon>\epsilon_0$ sufficiently close to $\epsilon_0$, there exists $\lambda_\epsilon$ converging to $0$ when $\epsilon$ goes to $\epsilon_0$ such that $\Re(\lambda_\epsilon)>0$ and the corresponding $\epsilon$-version of \eqref{Ueq} possesses a non trivial solution (in the relevant functional space).
\end{enumerate}
We begin by excluding the first kind of transition. For the sake of readability, we omit to mark the $\epsilon_0$ dependence. The first observation is that under the present assumptions, exponential dichotomy holds: when $\Re(\lambda)\geq0$, $\lambda\neq0$, solutions to the differential equation \eqref{Ueq} that do not blow exponentially decay exponentially, with rates matching decaying rates near $\pm\infty$ of asymptotic equations
\begin{align*}
U''-\frac{\hat{g}}{\bar{h}(g'+3\,h_0\,\varphi_\pm)} U'
-\lambda\left(\frac{1}{h_0(g'+3\,h_0\,\varphi_\pm)}\lambda
+\frac{2\,C_f\,c}{h_0^2\,\left(g'+3\,h_0\,\varphi_\pm\right)}\right)U=0\,.
\end{align*}
This implies that any such solution provides via the Liouville-type transformation
\be\label{Ltrans}
w(x)=e^{\frac{1}{2}\int_0^x f_1(y)\dD y}U(x),
\ee
an element $w$ of the kernel of the operator $\mathcal{L}(\lambda)$ acting on $L^2$ with domain $H^2$ through
\be
\label{weq}
\mathcal{L}(\lambda)(w):=w''+(f_2\lambda^2+f_3\lambda+f_4-\frac{1}{4}f_1^2-\frac{1}{2}f_1')w\,.
\ee
To conclude this part, as in \cite[Lemma~3.1]{SYZ}, we derive from $f_3<0$ that when $\lambda\in i{\mathbb R}$, $\lambda\neq0$, the kernel of $\mathcal{L}(\lambda)$ is trivial. Namely, taking the $L^2$ inner product of $i\,w$ against equation $\mathcal{L}(\lambda)(w)=0$ for $\lambda=i\alpha$, $\alpha\in\R^*$, we obtain $0=\langle w, f_3 w\rangle$, hence $w=0$.
It only remains to exclude the second kind of transition. Here we depart from \cite{SYZ}. Our first observation is that similarly classical arguments, building on limits of spatial decay rates, show that this transition may only occur if at $\epsilon_0$, the $\epsilon$-version of \eqref{Ueq} possesses at $\lambda=0$ a non zero solution $U$, bounded near $+\infty$ and decaying exponentially near $-\infty$. We now exclude this possibility, and again omit to mark any dependence on $\epsilon_0$ when doing so. Since $f_1(+\infty)>0$, the only bounded solutions to \eqref{Ueq} with $\lambda=0$, that is, to
\be\label{0eq}
U''+f_1 U'=0\,,
\ee
are solutions with $U'\equiv0$, and those do not decay to $0$ at $-\infty$ if they are not constantly zero. Hence the conclusion of this part and of the proof.
\end{proof}
\br\label{blockrmk}
For comparison, we mention that in \cite{SYZ} the argument blocking the emergence of eigenvalues through $\lambda=0$ builds on the fact that there is a non trivial element in the kernel but that its sign combined with Sturm-Liouville theory ensures that the multiplicity of $\lambda=0$ is fixed (equal to $1$), hence no crossing is possible.
\er
\subsection{Convective spectral stability}\label{s:consubs}
Let us observe that in the foregoing subsection we have not specified which of the functional spaces introduced in \eqref{s:degen} we were picking. The reason is that this choice is immaterial to conclusions of Theorem~\ref{th:main}. The conclusion would be dramatically different if we were allowing to introduce spatial weights among possible Sobolev norms.
With this in mind, we revisit briefly and at a deeper level the hydrodynamic stability conditions $F_+\leq 2$, $F_-\leq 2$, and its role in spectral stability. Though this condition is clearly necessary in standard Sobolev norms, it is also well-known that stability properties may sometimes be changed by working in exponentially weighted norms, taking account of the fact that instabilities may be convected into a traveling wave and stabilized by near-field dynamics. The upshot is that a wave may be stable with respect to sufficiently exponentially localized perturbations even when it is unstable with respect to perturbations in standard norms, a phenomenon known as {\it convective stability} or {\it convective stabilization}. On this topic, we refer to \cite{Sat} for pioneering work, to \cite[Chapter~3]{KapitulaPromislow-stability} for a comprehensive discussion at spectral level and to \cite{GR,FRYZ,Blochas} for convective analogs to \cite{DR1,DR2,YZ,SYZ,Blochas-Rodrigues}.
In order to state a convective analog to Theorem~\ref{th:main}, we extend Definition~\ref{def:spec}, to \emph{convective} spectral stability when spectral stability is obtained in a weighted topology with a smooth weight lower-bounded away from zero, and to \emph{extended convective} spectral stability when it is met for some smooth positive weight.
The weight $e^{\frac{1}{2}\int_0^x f_1(y)\dD y}$, already used in \eqref{Ltrans}, ensures invertibility properties for $L_{red}^\pm(\lambda)$ for any $\lambda\neq 0$ such that $\Re(\lambda)\geq0$ independently of the position of $F_\pm$ with respect to $2$. Once this is done the rest of the proof of Theorem~\ref{th:main} applies almost without change. Note that the above weight goes to zero as $x\to\infty$ and an inspection of spatial decay at $+\infty$ shows that this cannot be fixed by any other lower-bounded weight ; see Appendix~\ref{app:split} for some details. However $e^{\frac{1}{2}\int_0^x \chi(y)\,f_1(y)\dD y}$ with $\chi$ smooth, equal to $1$ in a neighborhood of $-\infty$ and to $0$ in a neighborhood of $+\infty$, provides a lower-bounded weight that restores invertibility of $L_{red}^-(\lambda)$ in the zone of interest.
The foregoing discussion leads to the following theorem.
\begin{theorem}\label{th:conv}
Consider an asymptotically-constant smooth profile with limiting values $(h_0, c, 0,\varphi_-)$ and $(h_0, c, 0, \varphi_+)$, $c>0$, reached exponentially fast.
\begin{enumerate}
\item Its extended convective spectral stability requires $C_f\geq C_t$, and this is sufficient to obtain strong extended convective spectral stability.
\item Its convective spectral stability is equivalent to $C_f\geq C_t$ and $F_+\leq 2$, and its strong convective spectral stability is equivalent to $C_f\geq C_t$ and $F_+< 2$.
\end{enumerate}
\end{theorem}
\br
The outcome of the analysis is consistent with the classical rule of thumb that only instabilities traveling towards the wave may be stabilized with a classical weight. To carry out this consistent check, we first recall \cite{MZ} that transition to hydrodynamic instability is closely related to the Chapman-Enskogg expansion \eqref{eq2}, with instabilities occurring in neutral modes corresponding to the characteristic modes of \eqref{eq2}. As the $\varphi$ mode decouples to all orders, it remains always neutral, hence strict, or exponential, instability should it arise occurs in the $h$-mode, convected with characteristic speed $\alpha^2_*$ strictly greater than the speed $\alpha^1_*=U$ of the $\varphi$ mode. Thus, for a {\it hydrodynamically unstable state to the left} of an asymptotically-constant convective wave solution, instabilities are convected inward toward the wave, at speed greater than the speed $c=U$ of the wave. Such instabilities are thus subject to convective stabilization, recovering stability with respect to sufficiently localized perturbations. For a {\it hydrodynamically unstable state to the right} of an asymptotically-constant convective wave solution, on the other hand, instabilities are convected outward from the wave, and cannot be so stabilized (in a classical way).
\er
There is some hope to directly use convective stability at the nonlinear level, since involved topologies behaves nicely with respect to nonlinear estimates. Nonlinear results mentioned above fit in this frame.
In turn, our motivation to also investigate extended convective stabilization and our expectations concerning its role in the nonlinear dynamics is somewhat subtle. Generally speaking, we expect that a wave that could be stabilized in an extended convective sense could enter as a block in a stable multi-wave pattern. For the present case, we are particularly interested in $2$-wave patterns, with the second (rightmost) wave (corresponding to the $h$ mode of \eqref{eq2}) being a {\it Lax shock}. In particular, then, characteristics are propagated into the shock from either
side. This observation will be important in Section \ref{s:num} below, in interpreting the results of our numerical time-evolution experiments.
Let us stress however that all our present discussion about nonlinear dynamics is somewhat speculative. Indeed, due to the presence of an infinitely degenerate neutrally stable mode, even in the smooth stable case, the present problem does not fit under the application of any result that we know of converting spectral stability into nonlinear stability.
\subsection{Stability of periodic solutions} \label{s:periodic}
Among the wide variety of smooth convective waves, besides the asymptotically-constant ones, those with a periodic profile also fit in a reasonably developed spectral stability framework. The latter is much more recent though. See for instance \cite{G,OZ1,JNRZ13,R_linearKdV} for a few significant contributions to periodic-wave nonlinear and spectral analyses, and \cite{R_HDR,R_Roscoff} for some detailed accounts. We restrain from tackling such a stability analysis, but we would like to point out why the asymptotically-constant analysis is not readily adaptable to the periodic problem.
A significant part of the asymptotically-constant study is deduced from an energy estimate carried out for the self-adjoint operator in \eqref{weq} obtained through the Liouville transformation $w=e^{\frac{1}{2}\int_0^x f_1(y)\dD y}U$ of \eqref{Ltrans}. Yet, in the periodic case, such a transformation is not available, unless $f_1$ is mean-free over a period, since otherwise $e^{\frac{1}{2}\int_0^x f_1(y)\dD y}$ is not periodic and grows without bound near one infinity, decays to zero near the other infinity.
We point out that, for discontinuous periodic waves, the stability framework of \cite{JNRYZ,DR2} could also be adapted to the present context. Yet we shall not carry out this analysis either.
Thus, in the periodic case, we do not obtain analytic results either for smooth or discontinuous profiles.
\section{Stability of discontinuous solutions}\label{s:discont}
We now consider briefly the case of piecewise-smooth convective waves, with a finite number of discontinuities, that are asymptotically constant. We prove that these waves are also spectrally stable when the limiting endstates are so. Our proof proceeds by a limiting argument from the stability of asymptotically-constant smooth waves. By taking a limit we lose the strong stability part of Theorem~\ref{th:main}.
The choice of such an argument also reflects that we have not been able to extend directly the arguments of the smooth analysis to the discontinuous case. To provide some insights on how these fail, let us give some details on what we are able to obtain in this way. The reduction to a nice eigenvalue problem and the proof of the impossibility that transitions to instability occur through the emergence of eigenvalues from $\lambda=0$ may indeed be adapted with little changes. The important part that we are not able to extend is the exclusion of transitions to instability through nonzero purely imaginary eigenvalues. In the smooth case, this eventually relies on an energy\footnote{There is some freedom in the energy estimate used but we have failed to adapt any of those.} estimate and discontinuities introduce, in the estimate, jump contributions that we have not been able to manage. Thus, by using real symmetry, the extension of the smooth-case proof only yields that transition to instability may only occur through the passage of pairs of nonzero complex conjugates eigenvalues. We are also able to extend the argument used to show the impossibility of a passage through $0$ so as to compute explicitly an instability index, counting modulo $2$ the number of eigenvalues on $(0,\infty)$; see \cite{GZ,Pego-Weinstein,BJRZ,BMR,JNRYZ} for similar computations. The upshot of the latter --- consistent with the rest of the present arguments --- is that this number is always even\footnote{We recall that we show by different arguments that this number is always $0$.}.
The rest of the section is devoted to the proof of the claimed spectral stability formalized in the following theorem.
\begin{theorem}\label{th:disc}
Consider an asymptotically-constant piecewise-smooth profile with limiting values at infinities $(h_0, c, 0,\varphi_-)$ and $(h_0, c, 0, \varphi_+)$, $c>0$, reached exponentially fast, and exhibiting a finite number of discontinuities. This wave is spectrally stable if and only if
\begin{align*}
C_f&\geq C_t\,,&
F_+&\leq 2\,,&
F_-&\leq 2\,.
\end{align*}
\end{theorem}
Some details on the notion of spectral stability used in the foregoing statement are given in the following subsection.
\subsection{The discontinuous eigenvalue problem}
For the sake of simplicity, we prove Theorem~\ref{th:disc} only for profiles with a single discontinuity, which without loss of generality we fix at $0$. We stress however that the adaptation to the general case is mostly notational.
Following \cite{JNRYZ,DR2}, we linearize in $(h,U,\varphi,\Phi,\psi)$ the system obtained from inserting the solution \emph{ansatz}
\[
(t,x)\longmapsto
(\bar h,\bar U,\bar \varphi,\bar \Phi)(x-ct-\psi(t))
+(h,U,\varphi,\Phi)(t,x-ct-\psi(t))
\]
in \eqref{rg4}, with $(\bar h,\bar U,\bar \varphi,\bar \Phi)$ and $(h,U,\varphi,\Phi)$ smooth on $\R^*$. This yields interior equations, on $\R^*$,
\[
A^0\d_tW+\d_x(AW)=EW,
\]
for $W=(h,U, \Phi, \varphi)-\psi\,(\bar h_x,\bar U_x,\bar \varphi_x,\bar \Phi_x)$, where $A^0$, $A$ and $E$ are as in \eqref{mateval}, supplemented with linearized Rankine-Hugoniot conditions. Concerning the latter we first observe that the linearized jump conditions associated with the first and fourth equations of \eqref{rg4} reduce to
\begin{align*}
[U]&=0\,,&\psi'=U(\cdot,0)\,,
\end{align*}
provided that $\bar\varphi$ does jump, which we will assume from now on. Once those are enforced the two remaining jump conditions reduce to a single equation
\[
[(g'\bar h+3\bar\varphi\bar h^2)\,h+\bar h^3\,(\Phi+\varphi)]\,=\,0\,.
\]
Incidentally, we point out that the foregoing reductions of Rankine-Hugoniot conditions also occur at the nonlinear level.
We have left aside the case when $\bar \varphi$ is actually continuous. This may indeed indeed happen but the required changes in the argument are relatively straightforward. Indeed, the comparison to smooth problems turns out to be even slightly simpler in this case since the discontinuity of $\bar \varphi$ is the main obstacle to directly extend to weak solutions on $\R$ the algebraic manipulations carried out in the smooth case.
Now, performing algebraic manipulations as in Subsection~\ref{s:degen} on the corresponding spectral problems, including the introduction of
\[
\tau:=\lambda\,(h-\psi\,\bar h_x)
\]
(as a function on $\R^*$) and the elimination of $\psi$, reduces the question of the spectral stability of the wave under consideration to $C_f\geq C_t$ and for any $\lambda$ with $\Re(\lambda)>0$, the problem
\begin{align*}
L_{red}(\lambda)\bp \tau\\U\ep&=G&\textrm{on }\R^*\,,\\
\bp
[U]\\
[(g'\bar h+3\bar\varphi\bar h^2)\,\tau+(\hat g\bar h-\bar h^3\bar\varphi_x)\,U]
\ep&=G_0\,,
\end{align*}
is boundedly invertible.
Classical arguments, that may be thought as variations on the one used in Section~\ref{s:stab-smooth}, show that spectral stability (respectively strong spectral stability) is then equivalent to $C_f\geq C_t$, $F_\pm\leq 2$
and the non existence, for $\lambda$ such that $\Re(\lambda)>0$
of a nonzero $U$ (in the relevant functional space) solving
\begin{align*}
U''+f_1 U'+(f_2\lambda^2+f_3\lambda+f_4)U&=0&\textrm{on }\R^*\,,\\
[-\bar h(g'\bar h+3\bar\varphi\bar h^2)\,U
]&=0\,,&
[U]&=0\,.
\end{align*}
In the foregoing, $f_1$, $f_2$, $f_3$ and $f_4$ are as in \eqref{Ueq}, in particular $f_4\equiv 0$.
\subsection{Limit of continuous waves}
Let $\delta:=\bar h-h_0$. Consider a mollified sequence $\delta^\varepsilon$, arising from convolution with a rescaling of a smooth compactly supported kernel, and the corresponding profiles $(\bar h_\varepsilon,c,\bar\varphi_\varepsilon,0)$ with the same limiting height $h_0$. In particular, $\bar h_\varepsilon$ is bounded uniformly in $\varepsilon$ and $\bar h_\varepsilon-h_0$ converges to $\bar h-h_0$ in $L^p(\R)$ for any $1\leq p<\infty$, and in $L^\infty(\R\setminus[-x_0,x_0])$ for any $x_0>0$. It follows from \eqref{weird} that similar conclusions also hold for $\bar\varphi_\varepsilon$. Moreover the foregoing convergences also hold in exponentially weighted topologies (with weights adapted to the convergence rate of $\delta$).
At this stage, we would like to reformulate eigenvalue problems for both smooth $(\bar h_\varepsilon,c,\bar\varphi_\varepsilon,0)$ and discontinuous $(\bar h,c,\bar\varphi,0)$ so as to reach a form for which continuity with respect to $\varepsilon$ stem from the above convergences and standard arguments. Once this is done, it follows that if for some $\lambda_0$ with $\Re(\lambda_0)>0$ the discontinuous $\lambda_0$-eigenvector system possesses a non trivial solution, this is also true when $\varepsilon>0$ is sufficiently small for the smooth $\lambda_\varepsilon$-eigenvector system associated with some $\lambda_\varepsilon$ converging to $\lambda_0$ when $\varepsilon$ goes to zero. Hence the conclusion.
Such a convenient formulation is obtained by setting (for instance)
\begin{align*}
w_1&=U\,,&
w_2&=\bar h_\varepsilon(g'\bar h_\varepsilon+3\bar\varphi_\varepsilon\bar h_\varepsilon^2)\,U'
-C_f\,|c|\,c\,U\,.
\end{align*}
Indeed, the $\lambda$-eigenvalue problem then takes the form
\begin{align*}
w_1'&=\frac{C_f\,|c|\,c}{\bar h_\varepsilon(g'\bar h_\varepsilon+3\bar\varphi_\varepsilon\bar h_\varepsilon^2)}\,w_1
+\frac{1}{\bar h_\varepsilon(g'\bar h_\varepsilon+3\bar\varphi_\varepsilon\bar h_\varepsilon^2)}\,w_2&\textrm{on }\R^*\,,\\
w_2'&=(\lambda^2\bar h_\varepsilon+\lambda\,2\,C_f\,c)\,w_1&\textrm{on }\R^*\,,\\
[w_1]&=0\,,&
[w_2]=0\,.
\end{align*}
\section{Numerical time-evolution experiments}\label{s:num}
We conclude our study with a series of numerical time-evolution experiments carried
out with the use of the numerical package CLAWPACK \cite{C1,C2}.
Many of the numerical experiments are chosen according to the principle that convective waves are mostly forced by $\varphi$-variations, with $h$-variations being slaved to those through \eqref{weird}, while $(U,\Phi)$ are held constant, $\Phi$ being identically zero. The main reason supporting the discrepancy in our perception of respective roles of $h$ and $\varphi$ is that among the traveling waves we have identified only convective waves may carry out variations in $\varphi$.
Let us also point out that for other kinds of waves we expect, from both modeling considerations and analysis in \cite{RYZ,RYZ2}, $\Phi$ to remain very small everywhere except near discontinuities. Consistently, in many of our simulations $\Phi$ remains very small throughout the whole time evolution. When this is the case, we omit the panels for variable $\Phi$.
\subsection{Experiments}
\subsubsection*{Figures~\ref{figure2} and~\ref{figure3}}
We begin with a time evolution starting from a Riemann type data, joining two stable equilibria, with initial variations purely in the $\varphi$-variable.
Explicitly, in Figure~\ref{figure2}, we show the result of the simulation of \eqref{rg4}-\eqref{C} for $g'=10\cos(\frac{\pi}{10})$, $\hat{g}=10\sin(\frac{\pi}{10})$, $C_t=0.9$, and $C_f=1$ and with initial data $h\equiv h_0=1$, $U\equiv c=\sqrt{h_0\hat{g}/C_f}$, $\Phi\equiv 0$ and $\varphi=\varphi_L\mathbbm{1}_{x\le 50}+\varphi_R\mathbbm{1}_{50<x}$ where $\varphi_L=0.2$ and $\varphi_R=0.5$. From left to right, we show the solutions at $t=0$, $1$, $10$, and $95$, respectively, with panels for variable $\Phi$ omitted.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure2.eps}
\end{center}
\caption{Contact discontinuity (stable constant state). The simulation shows emergence of a traveling convective wave.}
\label{figure2}
\end{figure}
Up to numerical smoothing effects, Figure~\ref{figure2} exhibits the emergence of a convective wave profile traveling at speed $c$, with $\varphi$ keeping its initial step-like shape, and $h$ constant equal to $h_0$ at the right of the discontinuity. The variable $U$ undergoes significant variations before spreading out back to constant $c$.
To confirm the observation, in Figure~\ref{figure3}, we compare the solution at time $t=95$ with the only convective wave profile such that $\bar\varphi=\varphi_L\mathbbm{1}_{x\le 217}+\varphi_R\mathbbm{1}_{217<x}$ and $\bar h(x)=h_0$ when $x>217$. We compute a numerical approximation of the latter by first solving the reduced Rankine-Hugoniot condition \eqref{redRH2} so as to determine $h_L$, the left-value at the jump of the height component, and then solving backwards from the jump the ODE \eqref{weird}. More explicitly, $h_L$ is obtained by solving the cubic equation $\varphi_L h_L^3+g'h_L^2/2=\varphi_R h_R^3+g'h_R^2/2$, with $h_R=h_0$, giving $h_L=1.0292\ldots $, whereas the non-constant part of $h$ is then deduced by solving
\be
\label{fode_varphi_constant}
h'=\frac{\hat{g}(h-h_0)}{g'h + 3\varphi_L h^2}\quad\textrm{on }(-\infty,217],\qquad\qquad
h(217)=h_L\,.
\ee
Up to numerical smoothing near the discontinuity, the matching is very convincing.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure3.eps}
\end{center}
\caption{Left: Blow up of \ref{figure2} at $t=95$. Right: comparison of analytically computed convective profile (black curve) and simulated convective profile (dotted curve).}
\label{figure3}
\end{figure}
\subsubsection*{Figure~\ref{figure4}}
In Figure~\ref{figure4}, we also provide a time evolution starting from a Riemann type data, joining two equilibria, with initial variations purely in the $\varphi$-variable, but this time the endstates are unstable.
Explicitly, in Figure~\ref{figure4}, we simulate with $g'=10\cos(\frac{\pi}{6})$, $\hat{g}=10\sin(\frac{\pi}{6})$, $C_t=0.04$, and $C_f=0.05$ and with initial data $h\equiv h_0=1$, $U\equiv c=\sqrt{h_0\hat{g}/C_f}$, $\Phi\equiv 0$ and $\varphi=\varphi_L\mathbbm{1}_{x\le 50}+\varphi_R\mathbbm{1}_{50<x}$ where $\varphi_L=0.3$ and $\varphi_R=0.1$.
In simulations associated with Figures~\ref{figure2} and~\ref{figure3}, the limiting constant states have been chosen to satisfy the hydrodynamical stability condition \eqref{hydro_RG}. When the condition is not satisfied, shocks can form in front of the convective wave. That is, consistent with the discussion of Section \ref{s:full}, in the absence of hydrodynamic stability, the simple time-asymptotic structure predicted by the formal equilibrium system breaks down, and more complicated patterns are expected to emerge. This is what we observe on Figure~\ref{figure4}.
Note that, as expected, variations of $\Phi$ are located near discontinuities that are not conveyed by convective waves.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure4.eps}
\end{center}
\caption{Contact discontinuity (unstable constant states). The simulation shows emergence of a traveling convective wave together with shocks in front.}
\label{figure4}
\end{figure}
\subsubsection*{Figures~\ref{figure5} and~\ref{figure6}}
In Figures~\ref{figure5} and~\ref{figure6}, we directly test nonlinear stability of smooth convective waves. Our goal is two-fold. On one hand we want to provide an example on which a relatively small perturbation is resolved in large time into another convective wave with very significantly different shape compared to the initial unperturbed profile. On the other hand, we want to exemplify that perturbations in $\varphi$ have a much stronger impact than those in $h$.
In both figures, parameters are chosen as $g'=10\cos(\frac{\pi}{10})$, $\hat{g}=10\sin(\frac{\pi}{10})$, $C_t=0.8$, and $C_f=1$, and the reference convective wave is obtained by solving \eqref{weird} in $\bar \varphi$ with $h_0=1$, $\bar U\equiv c=\sqrt{h_0\hat{g}/C_f}$, $\bar\Phi\equiv 0$, $\bar h=h_0+\delta$ where
\[
\delta=0.02\times\mathbbm{1}_{|x-3|<1}e^{-\frac{1}{1-(x-3)^2}}\,,
\]
and $\bar{\varphi}(-\infty)=4$, $\kappa=g'h_0^2/2+4h_0^3$.
In figure \ref{figure5}, we display the result of a simulation of \eqref{rg4}-\eqref{C} with the same parameters but initial data $(\bar h+h_{perturbation},\bar U,\bar \Phi,\bar\varphi)$ where
\[
h_{perturbation}=-0.01\times\mathbbm{1}_{|x-6|<1}e^{-\frac{1}{1-(x-6)^2}}\,.
\]
From left to right, we show the solution at $t=0$, $0.2$, $1$, and $6$, respectively, panels for variable $\Phi$ being omitted. The simulation shows convergence to a traveling convective wave profile having mild difference in all variables when compared with a suitable translate of the unperturbed initial profile $(\bar h,\bar U,\bar \Phi,\bar\varphi)$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure5.eps}
\end{center}
\caption{Convective wave with perturbation on $h$. The simulation shows convergence to the unperturbed traveling convective wave.}
\label{figure5}
\end{figure}
In figure \ref{figure6}, the computation is completely similar but with initial data
$(\bar h,\bar U,\bar \Phi,\bar\varphi+\varphi_{perturbation})$ where
\[
\varphi_{perturbation}=0.1\times\mathbbm{1}_{|x-5|<1}e^{-\frac{1}{1-(x-5)^2}}\,.
\]
The numerical outcome shows convergence to another convective wave, whose profile seems to have $\varphi$-component equal to $\bar\varphi+\varphi_{perturbation}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure6.eps}
\end{center}
\caption{Convective wave with perturbation on $\varphi$. The simulation shows convergence to another convective wave profile whose $\varphi$ component shows mild differences with the \emph{perturbed} initial $\varphi$.
\label{figure6}
\end{figure}
\subsubsection*{Figures~\ref{figure7} and~\ref{figure8}}
Next, we provide experiments of more direct practical interest, where we investigate interactions of hydraulic jumps arising from a dam break with downward bottom enstrophy. The dam-break experiment is a common test-case for (SV) codes. The $(h,U)$ part of the initial data is chosen to mimic the instantaneous removal of a dam wall separating two fluids at equilibrium with higher height upward. In our numerical tests we add some bottom vorticity ahead of the dam-break location.
We run these numerical experiments with unstable equilibria, but, moreover, we tune parameters to observe first a convectively stable hydraulic jump and then one that is unstable even in the convective sense. When doing so, we use analytic insights from \cite{RYZ}, where we completely determine conditions for convective stability.
Numerical simulations in Figure~\ref{figure7} have been carried out with $g'=5\sqrt{3}$, $\hat{g}=5$, $C_t=0.04$, and $C_f=0.05$ and initial data given by $\Phi\equiv 0$,
\begin{align*}
h&=h_L\mathbbm{1}_{x\le 5}+h_R\mathbbm{1}_{x>5}\,,\qquad
U\,=U_L\mathbbm{1}_{x\le 5}+U_R\mathbbm{1}_{x>5}\,,\\
\varphi&=0.3\mathbbm{1}_{x\le 10}+0.1\mathbbm{1}_{10<x\le 20}+0.5\mathbbm{1}_{20<x\le 30}+0.2\mathbbm{1}_{30<x\le 40}+0.6\mathbbm{1}_{40<x}\,,
\end{align*}
where $h_L=1$, $h_R=0.2$, $U_L=\sqrt{\hat{g}h_L/C_f}=10$ and $U_R=\sqrt{\hat{g}h_R/C_f}=2\sqrt{5}$. From left to right, we show the solution at $t=0$, $5$, $10$, and $15$, respectively.
Figure~\ref{figure7} exhibits at the end the superposition of a convective wave which carries variations in $\varphi$ and travels at speed $U_L=10$ and a (newly discovered) non-monotone hydraulic shock travels at speed $(25-\sqrt{5})/2$ (hence moving faster than the convective wave). Both the non-monotone hydraulic shock and the convective wave look very stable. The convective wave profile has the same structure (up to numerical smoothing) than the initial $\varphi$ component but a different shape, variations occurring over a shorter spatial interval. Incidentally we point out that this is the experiment reported in Figure~\ref{figure7} that has motivated our investigation of convective stability in the present paper and in its companion \cite{RYZ}, and the corresponding revisitation in \cite{FRYZ} of the Saint-Venant analysis of \cite{YZ,SYZ}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure7.eps}
\end{center}
\caption{Dam-break data in $(h,U)$ with multiple jumps in $\varphi$. The simulation shows asymptotically a non-monotone hydraulic shock to the right with a convection wave trailing behind.}
\label{figure7}
\end{figure}
Figure~\ref{figure8} report simulation of \eqref{rg4}-\eqref{C} with the same choice of parameters and initial data except that $h_R=0.5$ and, accordingly, $U_R=\sqrt{\hat{g}h_R/C_f}=5\sqrt{2}$. The final outcome is however significantly different. As is apparent at $t=10$, on intermediate times we do observe similar phenomena, but afterwards the hydraulic shock starts to be subject to instability and gradually develops a second shock as visible at $t=25$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure8.eps}
\end{center}
\caption{Dam-break data in $h/U$ with multiple jumps in $\varphi$. The simulation
shows first show emergence of a non-monotone hydraulic shock to the right with a
convection wave trailing behind. The hydraulic shock is convectively unstable and a second shock emerges after $t=15.00$. See the plot at $t=25$.}
\label{figure8}
\end{figure}
\subsubsection*{Figures~\ref{figure9} and~\ref{figure10}}
We conclude our numerical investigations with spatially periodic data, a case for which we have very few a priori analytic knowledge. Though we are mostly interested in localized (hence non-periodic) perturbations of periodic backgrounds, for numerical reasons we provide numerical simulations with periodic boundary conditions. Our goal is to observe whether a pure $\varphi$ periodic perturbation on top of a constant equilibria is resolved into a stable periodic convective wave.
This is exactly what is happening in the experiment reported in Figure~\ref{figure9}. The latter has been carried out for $g'=10\cos(\frac{\pi}{10})$, $\hat{g}=10\sin(\frac{\pi}{10})$, $C_t=0.9$, and $C_f=1$, with initial data $h\equiv h_0=1$, $\varphi=2+\sin(\pi x)$, $U\equiv c=\sqrt{h_0\hat{g}/C_f}$, $\Phi\equiv 0$, and periodic boundary conditions over $[0,10]$. From left to right, we show the solutions at $t=0$, $0.5$, $2$, and $5$, respectively, $\Phi$ panels being omitted.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure9.eps}
\end{center}
\caption{Periodic initial data in $\varphi$. The simulation shows convergence to a periodic traveling convective wave.}
\label{figure9}
\end{figure}
Figure~\ref{figure10} shows the result of the simulation of \eqref{rg4}-\eqref{C} for $g'=10\cos(\frac{\pi}{6})$, $\hat{g}=10\sin(\frac{\pi}{6})$, $C_t=0.04$, and $C_f=0.05$, with initial data $h\equiv h_0=1$, $\varphi=2+\sin(\pi x)$, $U\equiv c=\sqrt{h_0\hat{g}/C_f}$, $\Phi\equiv 0$, and periodic boundary conditions on $[0,10]$. From left to right, we show the solutions at $t=0$, $5$, $10$, and $40$, respectively.
The corresponding final outcome seems well described as a time-quasiperiodic solution resulting from the superposition of two waves traveling at different speeds\footnote{The question of whether the quasiperiodic object is actually a periodic one depends on whether or not the two speeds are rationally related, a question undecidable from numerical observations.}, namely one smooth periodic convective wave and one roll wave. In the end, variations of $U$ and $\Phi$ are conveyed by the roll-wave part whereas $\varphi$ variations are supported by the convective part. On the $h$ part one does see the quasiperiodic superposition, with the position of the roll-wave part being noticeable on discontinuities.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.42]{pics/figure10.eps}
\end{center}
\caption{Periodic initial data in $\varphi$. The simulation shows superposition of a traveling convective wave and a discontinuous roll wave.}
\label{figure10}
\end{figure}
\subsection{Numerical conclusions}\label{s:numc}
As noted in Section \ref{s:eq4}, it is readily seen that equilibrium system \eqref{eq2} admits a unique solution of the Riemann problem between two equilibrium states, consisting of a shock followed by a contact discontinuity. The result of our experiments suggest that in the hydrodynamically stable case that \eqref{hydro_RG} is satisfied both for the initial equilibrium states and the states predicted by the equilibrium Riemann solution, Riemann data for the full system \eqref{rg4} yields time-asymptotic behavior conforming to this prediction, but with relaxation profiles substituted for (equilibrium) shock and contact discontinuities. However, in the hydrodynamically unstable case that \eqref{hydro_RG} fails, either at endstates or intermediate states in the Riemann solution for the equilibrium system, more complicated but still coherent time-asymptotic patterns can emerge, involving nonmonotone shock profiles and various kinds of composite waves involving combinations of shock and roll waves.
Though we did not investigate it systematically, our brief study of the periodic case suggests that still more complicated possibilities may occur, with approximate superposition of periodic waves moving with different speeds. We conjecture that this may indeed result in both time-periodic and quasiperiodic solutions that are not traveling waves. The rigorous study of their existence would be an extremely interesting direction for further study.
\section{Discussion and open problems}\label{s:disc}
We have shown the appearance in the Richard-Gavrilyuk model (RG) of a new type of convective waves not present in the standard Saint-Venant model (SV), and characterized them as linearly degenerate waves analogous to those occurring in the entropy field for gas dynamics. Indeed, they correspond to one of the two enstrophy fields introduced in (RG), namely bottom vorticity $\varphi$.
Together with hydraulic shocks, these waves play an important role in time-asymptotic behavior
from asymptotically constant initial data,
corresponding to relaxation profiles of contact discontinuities in the associated equlibrium system \eqref{eq2} in the same way that hydraulic shocks both in (SV) and in (RG) correspond to relaxation profiles of Lax-type equilibrium shocks \cite{YZ,RYZ}.
However, different from contact relaxation profiles studied in \cite{HPW,Zh}
under Kawashima-type dissipativity conditions, these do not spread diffusively into a universal Gaussian-like profile, but appear to persist unchanged for all time; accordingly, there appear to be an infinitude of different ultimate time-asymptotic profiles.
Our numerical time evolution experiments indicate that both components ---
hydraulic shocks and
asymptotically constant
convective waves --- are
stable when their endstates are,
with arbitrary data resolving eventually into a noninteracting convective wave/hydraulic shock pattern. Under the hydrodynamic stability conditions \eqref{hydrodynamicstability} on endstates, we have shown analytically, by Sturm-Liouville considerations, that
asymptotically-constant convective waves are spectrally stable. The related problem of existence and spectral stability of hydraulic shocks is studied in \cite{RYZ}.
Interestingly, stability appear also to hold sometimes even for contact-shock components arising from asymptotically-constant data with endstates violating the hydrodynamic stability conditions \eqref{hydrodynamicstability}.
This
can be understood as an example of {\it convective stability}, whereupon instabilities arising at the end state are convected into the wave and unltimately stabilized. Indeed, one may verify that instabilities arising through violation of \eqref{hydrodynamicstability}
at the left endstate
correspond to the genuinely nonlinear $h$-field, which travels transverse to contact waves, and ultimately into the leading Lax shock, thus stabilizing. This phenomenon may be captured by working in a weighted norm, amounting to precisely the Sturm-Liouville transformation studied earlier. Thus, convective spectral stability of convective (in different sense) wave solutions follows for
asymptotically-constant
profiles by the same Sturm-Liouville argument as in the hydrodynamically stable case.
For both hydraulic shocks and convective-wave solutions, nonlinear stability remains a very interesting open question. For, independent of Froude number $\tilde F$ and strict hydrodynamic stability in the $h$ equilibrium mode, the fact that the convective $\varphi$ mode is a characteristic mode of both relaxation system (RG) and equilibrium system \eqref{eq2} means that Kawashima's genuine coupling condition is violated for this model,
leading, independent of $\tilde F$, to {\it neutral instability} in the $\varphi$ equilibrium mode, corresponding with the infinite-dimensional kernel of the linearized operator about the wave, as seen in Section \ref{s:stab-smooth}. Thus, nonlinear
asymptotic
stability, if valid, is in the weak sense of
stability of the infinite-dimensional family of convective waves in its entirety and not of individual waves, or of some of their finite-dimensional orbits under the action of symmetry group.
It may also be that waves are merely \emph{metastable}, with slowly growing instabilities occurring through resonance with this nondecaying infinite-dimensional kernel.
The resolution of this issue of nonlinear stability is an extremely interesting direction for future study.
Every example we know of among the relatively few analytical results analyzing the large-time dynamics near an infinite-dimensional family of (relative) equilibria --- including amazingly impressive analyses near homogeneous distributions of Vlasov-Poisson systems \cite{Mouhot-Villani,Bedrossian-Masmoudi-Mouhot} and shear flows of incompressible Euler systems \cite{Bedrossian-Masmoudi} --- requires an extremely fine understanding of the nonlinear structure of the system at hand.
Indeed, it appears to be a challenging problem even for the simplest case of stability of
{\it constant, equilibrium solutions}, similar to the study \cite{LZ} of compressible Navier-Stokes equations with zero heat conductivity,
or, even closer, to the analysis of a class of degenerate hyperbolic systems with relaxation in \cite{Mascia-Natalini}. For comparison with the latter, we point out that considering convective stabilization by spatial weights of discontinuous waves could also help preventing the formation of new singularities (hence relaxing the linear degeneracy assumption of \cite{Mascia-Natalini} for the marginally stable mode), as exemplified in \cite{Blochas-Wheeler}.
For
periodic convective waves, even spectral stability appears to be more complicated.
Another very interesting direction for further investigation is the numerical study of
the latter.
In particular, considering the example of (acoustic, or genuinely nonlinear) periodic roll waves for (SV), which arise if and only if the Froude number $F$ is $>2$, violating hydrodynamic stability, one may ask what is the role of the Froude number
in stability of periodic convective waves. Though it does not affect existence of periodic convective waves as in the case of periodic acoustic waves, it might yet potentially affect stability. Whether or not this is the case is a natural question that would be interesting to resolve. The related problem of existence and spectral stability of roll waves is studied in \cite{RYZ2}.
Even more generally, our numerical experiments of spatially periodic solutions suggest that the (RG) model also supports time-quasiperiodic solutions, seemingly obtained by superposing convective waves and roll waves, whose study is widely open from any point of view.
|
2,877,628,089,510 | arxiv | \section{Introduction}
The future of large-scale quantum communications will almost certainly involve distribution and manipulation of entangled pairs of photons within a quantum network; such a quantum network is likely to include small clusters of quantum processors (perhaps in a local network of quantum computers) which may require shared entanglement, and could then be connected to other network clusters, potentially via satellite communications \cite{Boone2015,Pirandola2016,Laurat2018}. However, despite the undeniably useful non-classical properties which entanglement-based quantum systems offer (such as for quantum key distribution \cite{Ekert1991quantum,Singh2014,Cao2013,Yin2017}, quantum secret sharing \cite{Nadeem2015,Wang2017,Kogias2017}, quantum repeaters \cite{Dur1999,VanLoock2006,Dias2017,Furrer2018}, quantum computing \cite{Bennett2000,Arrighi2006,Briegel2009,Saxena2019} and quantum teleportation \cite{Teleportation1993,Barrett2004,Ursin2004,Herbst2015}), entanglement is a highly fragile resource, and breaks down rapidly in the presence of noise and losses \cite{Shaham2015}.
One particularly useful proposal for circumventing the intrinsic fragility of distributing entangled photons around a quantum network is by performing entanglement swapping (ES). In ES, there exist two parties, Alice and Bob, each of whom begin the protocol with a separately entangled pair of photons, $AB$ and $CD$ respectively. They each send half of their entangled pair (i.e. modes $B$ and $D$) to a central location, where these propagating modes are mixed at a 50:50 beam-splitter, before subsequently being measured, as described in the schematic of Fig. \ref{Fig:ES_Protocol}. This mixing and measurement step is key to ES, as it causes projection of the states in modes $B$ and $D$, thus projecting modes $A$ and $C$ into an entangled state.
This distributed entanglement, now shared between modes $A$ and $C$, can then be used for further quantum communications and quantum computational purposes. Moreover, performing ES to share entanglement enhances the secrecy and security of the post-entangled state shared between Alice and Bob; if an adversary, Eve, were to measure modes $B$ and $D$, she gains no useful information on states $A$ and $C$, and in fact by carrying out this measurement Eve has actually \textit{assisted} Alice and Bob in sharing an entangled state \cite{Xu2015}.
This in fact is a form of measurement-device independence \cite{Hwang2017,Zhou2018,Jeon2019}, and is a direct consequence of the monogamy of entanglement: if Alice and Bob share an entangled state of high fidelity, then Eve must be disentangled from this state. As a result of this, Alice and Bob need not trust the source of entanglement. However, Eve, as the source of entanglement, could simply deny service, and there is no way to circumvent this form of attack.
ES, first performed experimentally in 1993 \cite{Zukowski1993}, has since been executed with discrete variables (DVs) \cite{Sciarrino2002,Jennewein2001,Kaltenbaek2009}, and also with continuous variables (CVs) \cite{Jia2004,Takei2005}. CV systems typically pose the
advantage of high success probability, whereas DVs are often robust against lossy channels; hence, an advantage could be gained from using both CV and DV \end{multicols}
\begin{figure}[t]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Schematic to represent the six channel system (in which the initial hybrid entangled states are denoted as $\ket{\Psi}^{\text{HE}}_{AB}$ and $\ket{\Psi}^{\text{HE}}_{CD}$) undergoing entanglement swapping. Modes $A$ and $C$ are the discrete variable post-entangled modes, and $B$ and $D$ are the propagating continuous variable modes. Losses are modelled through leakage of modes $B$ and $D$ into modes $E_{B}$ and $E_{D}$ respectively. The lossy modes $B$ and $D$ then meet at a 50:50 beam-splitter ($BS^{1/2}_{B,D}$) before subsequently being measured via a projective vacuum measurement and balanced homodyne detection ($D_B$ and $D_D$, respectively).}\label{Fig:ES_Protocol}}
{\includegraphics[width=11cm]{ES_Protocol.png}}
\end{figure}
\begin{multicols}{2}
\noindent states in what is referred to as a hybrid entanglement scheme \cite{Jeong2014}, and will be investigated in this work. The practicality of using CVs is also worth noting; they are compatible with current standard optical telecommunication technologies, and so could be suitable for large-scale communication protocols \cite{Orieux2016,Diamanti2015}. Both, DVs and CVs, have been extensively researched when used in entanglement swapping protocols together in the form of hybrid entanglement \cite{Brask2010, Lim2016}, and have also been demonstrated experimentally \cite{Takeda2015}.
In this work we present a new ES protocol based on hybrid entangled cat states. We show that we can produce a Bell state of high fidelity when allowing for low levels of photonic losses in the propagating CV (cat state superposition) modes, as well as allowing differences in the losses between the two propagating modes.
This paper is organised as follows: in Section \ref{Sec:ES_Protocol} we introduce our proposed ES protocol, as well as our model for photonic losses in the propagating modes, and the detection methods used; we extend this model for loss, and include averaged unequal losses between modes $B$ and $D$, in Section \ref{Sec:Unequal_Loss}; in Section \ref{Sec:Results} we show that, following our proposed ES protocol, Alice and Bob can share a Bell state of high fidelity, when allowing for low levels of equal and unequal losses, as well as investigating homodyne measurement imperfections; we discuss the practicality of our protocol for distributing entanglement in a future quantum network in Section \ref{Sec:Quantum_Network}; our conclusions are given in Section \ref{Sec:Conclusions}.
\section{The Entanglement Swapping Protocol}\label{Sec:ES_Protocol}
\noindent A space-time schematic describing the process of our proposed ES protocol, from initial hybrid entangled state generation through to detection, is given in Fig. \ref{Fig:ES_Protocol}, in which each arrow indicates a channel/mode. Within Fig. \ref{Fig:ES_Protocol}, the arrows illustrating modes $A$ and $C$ indicate a movement in space; this is to represent the possibility that these modes may be sent on for further uses in quantum communications protocols or quantum computing, or perhaps even to a potential customer (or indeed separated customers), requiring an entangled pair of qubits.
We now discuss each stage of our proposed protocol, starting with the production of the initial hybrid entangled states, before moving on to discuss how we model photonic losses in the propagating modes, and our subsequent detection methods.
\subsection{Hybrid States with Superposed Cats}
Within this ES protocol, we propose the use of so-called \textit{hybrid entangled} states. These quantum states are described as having entanglement shared between DV and CV degrees of freedom.
\begin{figure}[H]
\centering
{\includegraphics[width=\linewidth]{Cat_States.png}}
\caption{The position and momentum phase space ($\hat{x}$ and $\hat{p}$ respectively, where $\hat{x}=\frac{1}{2}(\hat{a}+\hat{a}^{\dagger})$ and $\hat{p}=\frac{\text{i}}{2}(\hat{a}^{\dagger}-\hat{a})$), in which we have two even cat states, ${\mathcal{N}^{+}_{\alpha}(\ket{\alpha}+\ket{-\alpha})}$ (left) and the phase-rotated ${\mathcal{N}^{+}_{\alpha}(\ket{\text{i}\alpha}+\ket{-\text{i}\alpha})}$ (right).}
\label{Fig:Cat_States}
\end{figure}
Extensive experimental research has been carried out preparing various hybrid entangled states, and one such commonly investigated state is the $\ket{\psi}_{AB}=\frac{1}{\sqrt{2}}(\ket{0}_{A}\ket{\alpha}_{B}+\ket{1}_{A}\ket{-\alpha}_{B})$ state, which shows bipartite hybrid entanglement between a qubit and a CV mode (commonly referred to as an entangled \enquote{Schr{\"o}dinger cat state}). This can be prepared in a multitude of ways, such as via use of polarisation photons, probabilistic heralded single photon measurements and Hadamard gates \cite{Laurat14}, by relying on Kerr non-linearities \cite{Gerry1999, Jeong2005, Loock2006,Nemoto2004}, or by exploiting entangled polarisation qubits with a series of beam-splitters (BSs) and auxiliary modes \cite{Li2018}. In fact, previous work has investigated this particular hybrid state in the same ES protocol as followed in this work, demonstrating that this simple hybrid state is theoretically resilient to low levels of photonic losses \cite{Parker2017a,Parker2018}.
\subsection{Stage 1: \textit{Preparation of Hybrid Entanglement}}
In this work we consider hybrid entangled cat states, in which the CV components in the superposition are themselves cat states, as shown in Fig. \ref{Fig:Cat_States}.
Alice's input quantum state to our proposed ES protocol is then described mathematically as follows:
\begin{align}
\ket{\Psi}^{\text{HE}}_{AB}=\frac{\mathcal{N}^{+}_{\alpha}}{\sqrt{2}}\big[&\ket{0}_{A}\big(\ket{\alpha}_{B}+\ket{-\alpha}_{B})\nonumber\\
+&\ket{1}_{A}(\ket{\text{i}\alpha}_{B}+\ket{-\text{i}\alpha}_{B}\big)\big],
\label{eq:Alice_Hybrid_State}
\end{align}
in which $\mathcal{N}^{\pm}_{\alpha}=1/\sqrt{2\pm2e^{-2\alpha^2}}$ is the normalisation of an even (or odd) cat state. We importantly note that, unless stated otherwise, $\alpha$ is real throughout this work. Furthermore, Bob's hybrid state of modes $C$ and $D$ $\left(\ket{\Psi}^{\text{HE}}_{CD}\right)$ is identical to that of Alice's in modes $A$ and $B$, as described in Eq. \ref{eq:Alice_Hybrid_State}, and so our total quantum state at this stage is:
\begin{align}
\ket{\Psi}^{\text{HE}}_{ABCD}=\ket{\Psi}^{\text{HE}}_{AB}\otimes\ket{\Psi}^{\text{HE}}_{CD}.
\end{align}
The hybrid states utilised in the protocol discussed here have increased complexity and thus require additional preparation steps, compared to the usual hybrid states. We therefore propose a sequence of quantum operations capable of producing such complex hybrid states, as the first stage of our protocol, as follows:
\begin{enumerate}
\item Begin with an initial product state of:
\begin{align}
\ket{\Psi}_{AB}=\mathcal{N}^{+}_{\alpha}[\ket{0}_{A}(\ket{\alpha}_{B}+\ket{-\alpha}_{B})].
\end{align}
\item Apply a Hadamard gate ($\hat{H}$) to mode $A$, such that $\ket{0}_{A}\xrightarrow{\hat{H}}\frac{1}{\sqrt{2}}(\ket{0}_A+\ket{1}_A)$. The state after this is then:
\begin{align}
\ket{\Psi}_{AB}=\frac{\mathcal{N}^{+}_{\alpha}}{\sqrt{2}}[&\ket{0}_{A}(\ket{\alpha}_{B}+\ket{-\alpha}_{B})\nonumber\\
+&\ket{1}_{A}(\ket{\alpha}_{B}+\ket{-\alpha}_{B})].
\end{align}
\item Finally, perform a conditional (controlled) $\frac{\pi}{2}$ rotation on mode $B$, such that when the qubit in mode $A$ is $\ket{1}_A$ then this rotation is performed on mode $B$. This can be shown as:
\begin{align}
\mathcal{N}^{+}_{\alpha}\ket{\alpha}_{B}+\ket{-\alpha}_{B}\xrightarrow{\hat{R}_{\frac{\pi}{2}}}\mathcal{N}^{+}_{\alpha}\ket{\text{i}\alpha}_{B}+\ket{-\text{i}\alpha}_{B}.
\end{align}
The final hybrid entangled state produced from this sequence of quantum logic gates is then that of Eq. \ref{eq:Alice_Hybrid_State}.
\end{enumerate}
This process can be shown as a quantum circuit as per Fig. \ref{Fig:Hybrid_State_Circuit}:
\begin{figure}[H]
\centering
{\includegraphics[width=\linewidth]{Hybrid_State_Circuit.png}}
\caption{A quantum circuit which could be used to prepare the initial hybrid entangled states ($\ket{\Psi}^{\text{HE}}_{AB}$ and $\ket{\Psi}^{\text{HE}}_{CD}$, as given in Eq. \ref{eq:Alice_Hybrid_State}) for our proposed entanglement swapping protocol, using a Hadamard gate ($\hat{H}$) and a controlled $\frac{\pi}{2}$ rotation gate $(\hat{R}_{\frac{\pi}{2}})$.}
\label{Fig:Hybrid_State_Circuit}
\end{figure}
\noindent This circuit therefore demonstrates that these particular hybrid states can be produced using standard quantum operations.
\subsection{Stage 2: \textit{Lossy Optical Modes}}
In any practical demonstration of this protocol, there will be intrinsic photonic losses that occur in the propagating modes (such as within optical fibres), and so we model these losses to investigate the tolerance of our protocol to this (see Stage 2 of Fig. \ref{Fig:ES_Protocol}).
To model photonic losses we combine the lossy modes ($B$ and $D$) with beam-splitters of transmission $T$, along with input vacuum states in modes $E_{B}$ and $E_{D}$ ($\ket{0}_{E_{B}}$ and $\ket{0}_{E_{D}}$), for losses in modes $B$ and $D$ respectively, and then trace out the loss modes as lost to the environment. Therefore, by decreasing the value of $T$ from unity we introduce greater levels of photonic loss in the system.
For now, we will consider the case in which the two beam-splitters ($BS^{T_B}_{B,E_{B}}$ and $BS^{T_D}_{D,E_{D}}$) induce equal amounts of loss between modes $B$ and $D$, thus $T_B = T_D$. In Sec. \ref{Sec:Unequal_Loss} we discuss the more realistic circumstance in which $T_B\neq T_D$.
To demonstrate how we mathematically account for photonic losses, consider an arbitrary coherent state $\ket{\beta}_i$ of complex amplitude $\beta$ in mode $i$, which we combine with a BS of transmission $T_i$, along with a vacuum state in mode $j$:
\begin{align}
BS^{T_i}_{i,j}\ket{\beta}_{i}\ket{0}_{j}&=\nonumber\\
\exp\Big[\sqrt{T_i}\beta&\hat{a}^{\dagger}-\sqrt{T_i}\beta^{*}\hat{a}\Big]\nonumber\\
\times\exp\Big[&\sqrt{1-T_i}\beta\hat{b}^{\dagger}-\sqrt{1-T_i}\beta^{*}\hat{b}\Big]\ket{0}_{i}\ket{0}_{j}\nonumber\\
&=\ket*{\sqrt{T_i}\beta}_{i}\ket*{\sqrt{1-T_i}\beta}_{j},
\label{eq:BS_Vacuum_CoherentState}
\end{align}
where $\hat{a}^{\dagger}$ ($\hat{b}^{\dagger}$) and $\hat{a}$ ($\hat{b}$) are the creation and annihilation operators for modes $A$ and $B$, respectively.
The total quantum state of the ES protocol after inducing photonic losses is then described as:
\begin{align}
\ket{\Psi}_{ABE_{B}CDE_{D}}=&\nonumber\\
BS^{T_B}_{B,E_{B}}\ket{\Psi}^{\text{HE}}_{AB}&\ket{0}_{E_{B}}\otimes BS^{T_D}_{D,E_{D}}\ket{\Psi}^{\text{HE}}_{CD}\ket{0}_{E_{D}},
\label{Eq:Lossy_State}
\end{align}
in which,
\begin{align}
\ket{\Psi}_{ABE_B}=&\frac{\left(\mathcal{N}^{+}_{\alpha}\right)^2}{2}\times\nonumber\\
\Big[\Big(\ket{0}_{A}(&\ket*{\sqrt{T_B}\alpha}_{B}\ket*{\sqrt{1-T_B}\alpha}_{E_B}\nonumber\\
&+\ket*{-\sqrt{T_B}\alpha}_{B}\ket*{-\sqrt{1-T_B}\alpha}_{E_B})\nonumber\\
+\ket{1}_{A}(&\ket*{\text{i}\sqrt{T_B}\alpha}_{B}\ket*{\text{i}\sqrt{1-T_B}\alpha}_{E_B}\nonumber\\
&+\ket*{-\text{i}\sqrt{T_B}\alpha}_{B}\ket*{-\text{i}\sqrt{1-T_B}\alpha}_{E_B})\Big)\Big],
\end{align}
and the above state is identical to that describing modes $C$, $D$ and $E_D$.
\subsection{Stage 3: \textit{50:50 Beam-Splitter}}
Following the introduction of loss into the propagating modes ($B$ and $D$), we then mix these modes at a 50:50 BS (as per Stage 3 of our protocol, outlined in Fig. \ref{Fig:ES_Protocol}). Consider now two arbitrary coherent states, of complex amplitudes $\alpha$ and $\eta$ in modes $i$ and $j$ respectively - the 50:50 BS ($BS^{1/2}_{i,j}$) convention we use here can be shown mathematically as:
\begin{equation}
BS^{1/2}_{i,j}\ket{\alpha}_{i}\ket{\eta}_{j}=\ket{\frac{\alpha-\eta}{\sqrt{2}}}_{i}\ket{\frac{\alpha+\eta}{\sqrt{2}}}_{j}.
\label{eq:5050BSCoherentStates}
\end{equation}
\subsection{Stage 4: \textit{Detection Methods}}
After mixing the lossy modes $B$ and $D$ via use of a 50:50 BS, we then need to detect these quantum states (Stage 4, Fig. \ref{Fig:ES_Protocol}) to project their respective wave-functions, thus ensuring successful ES to modes $A$ and $C$. We measure mode $B$ via a projective vacuum state measurement, and mode $D$ via balanced homodyne detection.
\subsubsection{Vacuum State Detection}
By \enquote{vacuum state measurement} we mean that to reveal the absence of a photon (therefore indicating a vacuum state) one could use a \textit{perfect} photodetector, and upon not hearing the characteristic \textit{click} of the detector, indicating the presence of one or more photons, it can be assumed that there is not a photon present \cite{Gerry2000}.
To measure the presence of a vacuum, or lack thereof, we apply a positive-operator valued measure (POVM) described by the operator \cite{Joo2009}:
\begin{align}
\hat{P}^{0}_{i}=\ket{0}_{i}\bra{0},
\label{Eq:VacuumProjector}
\end{align}
where $\ket{0}_i$ represents a vacuum state in mode $i$. This POVM measurement can be calculated by application of this vacuum projector $\hat{P}^{0}_{i}$ onto a coherent state (as is required in our proposed ES protocol). For example, consider an arbitrary coherent state of complex amplitude $\gamma$ in mode $i$ as follows:
\begin{align}
\ket{0}_i\braket{0}{\gamma}_i&=\ket{0}_{i}e^{-\frac{|\gamma|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\gamma^{n}}{\sqrt{n!}}{}_i\bra{0}\ket{n}_i\nonumber\\
&=\ket{0}_{i}e^{-\frac{|\gamma|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\gamma^{n}}{\sqrt{n!}}\delta_{n,0}=\ket{0}_{i}e^{-\frac{|\gamma|^{2}}{2}},
\end{align}
in which we have applied the Fock basis representation of the coherent state (${\ket{\gamma}_i=e^{-\frac{|\gamma|^2}{2}}\sum_{n=0}^{\infty}\frac{\gamma^n}{\sqrt{n!}}\ket{n}_i}$), and we have made use of the Kronecker delta function, in which $\delta_{n,0}=0$ for $n\neq0$ and $\delta_{n,0}=1$ for $n=0$. We also highlight here that the application of this projective vacuum state measurement introduces an exponential dampening term of $e^{-\frac{|\gamma|^2}{2}}$, which is intrinsically important to the resultant performance of our proposed ES protocol, as will be discussed later.
After performing this projective vacuum measurement, our total quantum state is:
\begin{align}
\hat{P}^0_B\ket{\Psi}_{ABE_{B}CDE_{D}}=\sqrt{\mathcal{P}_0}\ket{\Psi}_{AE_{B}CDE_{D}},
\label{eq:Vacuum_State_State}
\end{align}
in which $\mathcal{P}_0$ is the success probability of the vacuum measurement (this will be discussed further in Subsec. \ref{Subsec:Success_Probability}). For clarity, we omit the state $\ket{0}_B$ on the right-hand side of Eq. \ref{eq:Vacuum_State_State} (and in further expressions), as this is the remaining state from the vacuum measurement operator (Eq. \ref{Eq:VacuumProjector}) after projecting mode $B$, and this will be used to project the complex conjugate of mode $B$ when used to form the final density matrix.
\subsubsection{Homodyne Measurement}\label{Subsec:Homodyne_Detection}
Following the vacuum measurement of mode $B$, we proceed to measure mode $D$ via balanced homodyne detection.
\begin{figure}[H]
\centering
\scalebox{0.5}
{\includegraphics{Homodyne_Detection.png}}
\caption{The two channel system undergoing homodmyne detection, in which $B_1$ is the input signal (mode $D$ in our ES protocol of Fig. \ref{Fig:ES_Protocol}), and mode $B_2$ is the local oscillator. $I_{B_{1}-B_{2}}$ is the intensity difference between the photodetectors $D_{B_1}$ and $D_{B_2}$.}
\label{Fig:Homodyne_Detection}
\end{figure}
To perform balanced homodyne detection, the probe mode (mode $B_1$ as per Fig. \ref{Fig:Homodyne_Detection}) is mixed at a 50:50 BS with a strong coherent field $\ket{\beta e^{\text{i}\theta}}$, in which $\beta$ is real, in mode $B_2$ (also referred to commonly as the local oscillator) of equal frequency, and photodetection is used to measure the outputs of both modes $B_1$ and $B_2$ \cite{Scully1997, Barrett2005, Suzuki2006}, as shown in Fig. \ref{Fig:Homodyne_Detection}.
The intensity difference between the two photodetectors ($D_{B_1}$ and $D_{B_2}$) is then calculated using the two mode operator $\hat{I}_{\hat{B{_1}}-{\hat{B_2}}}$ as:
\begin{align}
\hat{I}_{\hat{B{_1}}-{\hat{B_2}}}=\hat{b}^{\dagger}_1\hat{b}_2+\hat{b}^{\dagger}_2\hat{b}_1.
\end{align}
Setting the local oscillator mode to $\hat{b}_2=\beta e^{\text{i}\theta}$ then yields the expectation value as:
\begin{align}
\expval{\hat{b}^{\dagger}_1\hat{b}_2+\hat{b}^{\dagger}_2\hat{b}_1}=2\beta\expval{\hat{x}_{\theta}},
\end{align}
for $\hat{x}_{\theta}=\frac{1}{2}\left(\hat{b}_1e^{-\text{i}\theta}+\hat{b}^{\dagger}_1e^{\text{i}\theta}\right)$ \cite{Gerry2005}, in which the phase of the quadrature to be measured is given by the phase $\theta$ of the local oscillator \cite{Lvovsky2009}. We adjust the phase angle such that $\hat{x}_{\theta\rightarrow 0}=\hat{x}$ and $\hat{x}_{\theta\rightarrow \pi/2}=\hat{p}$, thus giving the position and momentum operators, respectively, as:
\begin{align}
\hat{x}=\frac{1}{2}(\hat{a}+\hat{a}^{\dagger}) \quad \text{and}\quad \hat{p}=\frac{\text{i}}{2}(\hat{a}^{\dagger}-\hat{a}),
\end{align}
in which the coherent state expectation values are:
\begin{align}
\expval{\hat{x}}=\frac{1}{2}(\alpha+\alpha^*) \quad \text{and}\quad \expval{\hat{p}}=\frac{1}{2\text{i}}(\alpha^* -\alpha),
\end{align}
for $\alpha=\alpha_x +\text{i}\alpha_y$ and $\alpha^*=\alpha_x -\text{i}\alpha_y$.
Finally, the probability amplitude of a projective homodyne measurement on an arbitrary coherent state $\ket{\alpha e^{\text{i}\phi}}$ is determined by projecting with an $\hat{x}_{\theta}$ eigenstate, in which $\hat{x}_{\theta}\ket{x_{\theta}}=x_{\theta}\ket{x_{\theta}}$ \cite{QuantumNoise}:
\begin{align}
\braket{{x_{\theta}}}{{{\alpha e^{\text{i}\varphi}}}}=\quad\quad\quad\quad\quad&\nonumber\\
\frac{1}{2^{-\frac{1}{4}}\pi^{\frac{1}{4}}}\exp\bigg[-(x_{\theta})^{2}&+2e^{\text{i}(\varphi-\theta)}\alpha x_{\theta}\nonumber\\
&-\frac{1}{2}e^{2\text{i}(\varphi-\theta)}\alpha^{2}-\frac{1}{2}\alpha^{2}\bigg],
\label{Eq:HomodyneGeneral}
\end{align}
\noindent where the subscript on $x_{\theta}$ is indicative of the angle in which the homodyne measurement is performed, and this angle can be accurately chosen through the phase of the local oscillator. We recognise here that homodyne detection is a routine and very accurate measurement technique used widely in optics as a means of measuring phase-dependent quantum phenomena \cite{Noh1991,Smithey1993,Smithey1993a,Lipfert2015}.
The outcome of a homodyne measurement is a value $x_\theta$ of the continuous quadrature variable $\hat{x}_\theta$, and the resultant homodyne measurement value is given by a probability distribution that comes from the modulus squared of the wave-function (as given in Eq. \ref{Eq:HomodyneGeneral}). For this work, we therefore define an \textit{ideal} homodyne measurement as the case when the resultant homodyne measurement value is at the maximum of the probability distribution, as indicated by the position and momentum phase space diagram of Fig. \ref{Fig:Cats_Phase_Space} (note that the states in this diagram have amplitudes of $\sqrt{2}\alpha$, which is a result of the 50:50 BS operation prior to this measurement).
\begin{figure}[H]
\centering
{\includegraphics[width=\linewidth]{Cats_Phase_Space.png}}
\caption{The position and momentum phase space occupied by two cat states $\mathcal{N}^{+}_{\alpha}(\ket*{\sqrt{2}\alpha}+\ket*{-\sqrt{2}\alpha})$ and $\mathcal{N}^{+}_{\alpha}(\ket*{\sqrt{2}\text{i}\alpha}+\ket*{-\sqrt{2}\text{i}\alpha})$, in which an $\hat{x}_{\frac{\pi}{4}}$ homodyne measurement yields a probability distribution exhibiting two peaks, centred at $x_{\frac{\pi}{4}}=\pm\alpha$.}
\label{Fig:Cats_Phase_Space}
\end{figure}
Therefore, upon performing the homodyne measurement we have the total quantum state of:
\begin{align}
\hat{\Pi}_{\text{HD}}\ket{\Psi}_{AE_{B}CDE_{D}}=\ket{\Psi}_{AE_{B}CE_{D}},
\label{eq:homodyne_operator}
\end{align}
in which $\hat{\Pi}_{\text{HD}}=\ket*{x_{\frac{\pi}{4}}}_{D}\bra*{x_{\frac{\pi}{4}}}$ is the homodyne measurement projector, of measurement angle $\frac{\pi}{4}$. Again, as with the measurement of mode $B$ (Eq. \ref{eq:Vacuum_State_State}), we omit mode $D$ on the right-hand side of this expression, as this is removed when forming the final state density matrix.
We select the angle of measurement $\theta=\frac{\pi}{4}$ such that we \textit{quantum erase} information between certain peaks in the probability amplitude. To clarify, the purpose of the homodyne measurement in this scenario is to \textit{indistinguishably} detect the coherent states, thus causing the output state to exhibit entanglement; the homodyne measurement outcome heralds the entangled state that is produced.
Considering the position and momentum of the cat states in Fig. \ref{Fig:Cats_Phase_Space}, were we to measure along the $\hat{x}_{\frac{\pi}{2}}$ axis, for example, then we would only erase information between the two states along the momentum axis ($\ket*{\sqrt{2}\text{i}\alpha}$ and $\ket*{-\sqrt{2}\text{i}\alpha}$). This means that the output probability distribution would have three peaks, thus causing the remaining quantum state to be less entangled than for the circumstance in which we quantum erase information between the states by an $\hat{x}_{\frac{\pi}{4}}$ measurement, as shown in Fig. \ref{Fig:Cats_Phase_Space}. Of course, ideally one would perform this measurement with an angle such that there is only a single peak in the resultant probability distribution, however this is impossible for the case at hand.
A homodyne measurement along the $\hat{x}_{\frac{\pi}{4}}$ axis, as shown in Fig. \ref{Fig:Cats_Phase_Space} has two peaks centred at $x_{\frac{\pi}{4}}=\pm\alpha$, and so we use this result as our homodyne measurement outcome in establishing the final quantum state in this ES protocol. We emphasise that a homodyne measurement is a continuous quadrature measurement, and so realistically there is a range of outcome values in which $x_\theta$ may take (hence, as per Fig. \ref{Fig:Cats_Phase_Space}, $x_{\frac{\pi}{4}}=\pm\alpha$ are the average \textit{perfect} outcomes). Therefore, to investigate \textit{imperfect} homodyne detection we need to allow for a resolution bandwidth about the expected measurement outcome value. The mathematical method for this was derived in \cite{Laghaout2013}, and we use this in determining the acceptance value of how large the resolution bandwidth can be, whilst still producing entangled qubits of reasonable fidelity.
Following the derivation detailed in \cite{Laghaout2013}, the imperfect homodyne measurement operator becomes:
\begin{align}
\hat{\Pi}_{HD}(x_0,\Delta x)=\int_{x_{0}-\frac{\Delta x}{2}}^{x_{0}+\frac{\Delta x}{2}}\ket{x_\theta}\bra{x_\theta}\text{d}x_{\theta},
\label{eq:ImperfectHomodyneProjector}
\end{align}
in which $x_0$ is the expected measured value and $\Delta x$ is the resolution bandwidth around this measured value. Intuitively, in the limit of $\Delta x\rightarrow 0$ we should approach the perfect homodyne measurement scenario as before. This, in fact, is how we determine success probability for the homodyne measurement, however this will be discussed in detail in Subsec. \ref{Subsec:Homodyne_Success Prob}.
For successful ES, as per this protocol, it is essential that one performs a vacuum state measurement on one propagating mode, and a homodyne measurement on the other - it was found that if two homodyne measurements, or two vacuum measurements, are performed on modes $B$ and $D$ then the resultant quantum state is a linear combination of all possible two qubit states, which is a product state and therefore exhibits no entanglement, and as such is of little use for further quantum communication/computation purposes.
We also note that we post-select the state prior to this measurement, conditional on the required vacuum projection outcome on mode $B$. That is to say, if we hear the photon detector \textit{click} then we do not perform homodyne detection on mode $D$, but instead restart the entanglement swapping protocol again.
\section{Modelling Unequal Photonic Losses} \label{Sec:Unequal_Loss}
In a practical setting, the two propagating modes $B$ and $D$ will not exhibit the exact same levels of photonic losses. For example, different lengths of optical fibres correspond to varying levels of loss - shorter fibre intrinsically exhibits lower losses. In fact, optical fibres of equal length may even exhibit differing photonic losses. There are also potential errors when coupling optical fibre to components, such as the 50:50 BS used to mix modes $B$ and $D$, or fibre splices within these modes. Hence, even if the lengths of fibre are identical, slightly different optical coupling in the two modes could give a small mismatch in losses. We could even consider the free-space ES scenario, in which the path lengths of modes $B$ and $D$ differ, and as such would exhibit varying levels of loss.
As such, the two BSs used to model this loss theoretically ($BS^{T_B}_{B,E{_B}}$ and $BS^{T_D}_{D,E{_D}}$) will not have equal transmission coefficients - that is to say that $T_B\neq T_D$. We therefore now determine whether allowing for a small difference in the losses experienced in these modes, which we denote $\upsilon$, impacts the quality of the entangled state shared between Alice and Bob post-protocol.
\begin{figure}[H]
\centering
{\includegraphics[width=\linewidth]{UnequalLoss_GaussianFunction_Upsilon0_10.png}}
\caption{The one-sided (positive) Gaussian distribution for the unequal loss function ($f(\upsilon,\Upsilon)$), as a function of the (non-averaged) loss mismatch value $\upsilon$, of width $\Upsilon=0.10$.}
\label{Fig:UnequalLoss_GaussianFunction_Upsilon0_10}
\end{figure}
To avoid transmission coefficients exceeding unity (which is clearly unphysical), we parametrise our unequal loss parameter such that $T_B=T$ and $T_D=T-\upsilon$, in which $0\leq T\leq1$ and $0\leq\upsilon\leq1$. When allowing for this \enquote{loss mismatch}, our total quantum state, after the lossy BSs, is then:
\begin{align}
\ket{\Psi(\upsilon)}_{ABE_{B}CDE_{D}}=&\frac{\left(\mathcal{N}^{+}_{\alpha}\right)^2}{2}\times\nonumber\\
\Big[\Big(\ket{0}_{A}(\ket*{\sqrt{T}\alpha}_{B}&\ket{\sqrt{\gamma}\alpha}_{E_B}\nonumber\\
+&\ket*{-\sqrt{T}\alpha}_{B}\ket{-\sqrt{\gamma}\alpha}_{E_B})\nonumber\\
+\ket{1}_{A}(\ket*{\text{i}\sqrt{T}\alpha}_{B}&\ket{\text{i}\sqrt{\gamma}\alpha}_{E_B}\nonumber\\
+&\ket*{-\text{i}\sqrt{T}\alpha}_{B}\ket{-\text{i}\sqrt{\gamma}\alpha}_{E_B})\Big)\Big]\otimes\nonumber
\end{align}\vspace{-0.6cm}
\begin{align} \Big[\Big(\ket{0}_{C}(&\ket*{\sqrt{T-\upsilon}\alpha}_{D}\ket{\sqrt{\gamma+\upsilon}\alpha}_{E_D}\nonumber\\
+&\ket*{-\sqrt{T-\upsilon}\alpha}_{D}\ket{-\sqrt{\gamma+\upsilon}\alpha}_{E_D})\nonumber\\
+\ket{1}_{C}(&\ket*{\text{i}\sqrt{T-\upsilon}\alpha}_{D}\ket{\text{i}\sqrt{\gamma+\upsilon}\alpha}_{E_D}\nonumber\\
+&\ket*{-\text{i}\sqrt{T-\upsilon}\alpha}_{D}\ket{-\text{i}\sqrt{\gamma+\upsilon}\alpha}_{E_D})\Big)\Big],
\end{align}
where $\gamma=1-T$. Of course, in the limit of $\upsilon=0$ we return to the equal loss scenario.
As opposed to selecting a specific value for this loss mismatch, it is logical to explore an average over $\upsilon$ by means of a one-sided (positive) Gaussian distribution (see Fig. \ref{Fig:UnequalLoss_GaussianFunction_Upsilon0_10}), in which the width associated with this distribution corresponds to our \textit{ensemble-averaged} loss mismatch value, which we denote $\Upsilon$. Considering this value as an ensemble average means that an experimentalist performing this ES protocol could have in mind a threshold of $\Upsilon$ for which they would know to not allow the mismatch in the loss to fall below.
The function for this Gaussian profile is calculated as:
\begin{align}
f(\upsilon,\Upsilon)=\sqrt{\frac{2}{\pi\Upsilon^2}}e^{-\frac{\upsilon^2}{2\Upsilon^2}},
\end{align}
which is normalised for $\int_{0}^{\infty}f(\upsilon,\Upsilon)\text{d}\upsilon=1$.
Therefore, the final state for our overall ES protocol, when accounting for unequal (averaged) losses between the propagating CV modes ($B$ and $D$), and after having performed vacuum state detection and a homodyne measurement, is:
\begin{align}
\bar{\rho}_{AE_{B}CE_{D}}(\Upsilon)=\int_{0}^{\infty}f(\upsilon,\Upsilon)\rho_{AE_{B}CE_{D}}(\upsilon)\text{d}\upsilon,
\label{eq:Final_Density_Matrix}
\end{align}
where $\rho_{AE_{B}CE_{D}}(\upsilon)=\ket{\Psi(\upsilon)}_{AE_{B}CE_{D}}\bra{\Psi(\upsilon)}$.
Lastly, we trace out the lossy modes ($E_B$ and $E_D$) as lost to the environment, using the coherent state to trace, thus giving our final density matrix as:
\begin{align}
\Tr_{E_{B},E_{D}}\left[\bar{\rho}_{AE_{B}CE_{D}}(\Upsilon)\right]=\bar{\rho}_{AC}(\Upsilon),
\label{Eq:Final_State}
\end{align}
which we use to determine all subsequent results presented throughout this work.
Due to the length of many of the mathematical expressions within this protocol, we do not include these in this work. We direct the reader to the work of \cite{Parker2018} for an in-depth discussion, including all mathematical detail, of each step of this proposed protocol.
\section{Results and Discussion}\label{Sec:Results}
Having performed successful ES, as per the protocol outlined in the prior sections, the state which Alice and Bob share (in the ideal, no loss, perfect measurements limits) is a phase-rotated Bell state:
\begin{align}
\ket{\Phi^{+}(\alpha)}_{AC}=\frac{1}{\sqrt{2}}\left(e^{-\text{i}\alpha^2}\ket{00}_{AC}+e^{+\text{i}\alpha^2}\ket{11}_{AC}\right),
\label{Eq:Bell_Outcome}
\end{align}
and the orthogonal Bell state to this is:
\begin{align}
\ket{\Phi^{-}(\alpha)}_{AC}=\frac{1}{\sqrt{2}}\left(e^{-\text{i}\alpha^2}\ket{00}_{AC}-e^{+\text{i}\alpha^2}\ket{11}_{AC}\right).
\label{Eq:Orthogonal_Bell_Outcome}
\end{align}
This phase in the ideal outcome state (Eq. \ref{Eq:Bell_Outcome}) could be corrected for via a suitable quantum operation (i.e. a phase-space rotation), or tracked through the protocol which the post-entangled state is being used for, such that the outcome state would be the maximally entangled $\ket{\Phi^+}_{AC}=\frac{1}{\sqrt{2}}(\ket{00}_{AC}+\ket{11}_{AC})$ Bell state.
\subsection{Fidelity}
As the aim and purpose of our proposed ES protocol is to produce a specific Bell state of the highest quality (and therefore highest level of entanglement), the most useful measure of the quantum state shared between Alice and Bob is fidelity.
We calculate the fidelity ($F$) using the standard formula of:
\begin{align}
F=\bra{\Phi^{+}(\alpha)}\bar{\rho}_{AC}(\Upsilon)\ket{\Phi^{+}(\alpha)},
\end{align}
in which $\ket{\Phi^{+}(\alpha)}=\frac{1}{\sqrt{2}}\left(e^{-\text{i}\alpha^2}\ket{00}+e^{+\text{i}\alpha^2}\ket{11}\right)$ is the desired protocol outcome, and $\bar{\rho}_{AC}(\Upsilon)$ is the final density matrix of our ES protocol, given in Eq. \ref{Eq:Final_State}. In the limit in which our protocol outcome is identical to $\ket{\Phi^{+}(\alpha)}$ the fidelity is $F=1$, and as the closeness of these quantum states the fidelity drops from unity and approaches zero.
Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne} shows the fidelity against the phase-rotated Bell state, and the state orthogonal to this Bell state ($\ket{\Phi^{-}(\alpha)}$), as a function of the coherent state amplitude $\alpha$ of our final density matrix (Eq. \ref{eq:Final_Density_Matrix}), but in the limit of $\Upsilon=0$ (i.e. the losses experienced in modes $B$ and $D$ are equal).
Firstly, we can see in Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne} that the plot of fidelity against the $\ket{\Phi^{+}(\alpha)}$ state plateaus at unity for $T=1$ and $\alpha\geq2.3$. This confirms that in the ideal limit (i.e. no loss) we can indeed produce the maximally entangled $\ket{\Phi^{+}(\alpha)}$ Bell state, following our ES protocol. What is also evident in Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne} is that in the limit of very large $\alpha$, and for non-unity $T$ the fidelity of both plots (for $\ket{\Phi^{+}(\alpha)}$ and $\ket{\Phi^{-}(\alpha)}$) tends to $F=0.50$; in fact, this 50:50 mixture of both Bell states is a mixed state, and exhibits no entanglement, and so is undesirable as the protocol outcome.
Moreover, we can also see in Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne} that as we increase the level of losses in modes $B$ and $D$, the fidelity against the desired Bell state is lower for all $\alpha$ - correspondingly, the fidelity therefore increases for the orthogonal Bell state, indicating the increased level of mixing.
If we consider the plot of Fig. \ref{Fig:CatES_Fidelity_UnequalLoss}, in which we now allow for an averaged loss mismatch between modes $B$ and $D$, we can see that increasing this loss mismatch value ($\Upsilon$) merely causes the fidelity plots to plateau to $F=0.50$ more rapidly as a function of $\alpha$. In fact, this result is positive for the performance of this protocol - allowing for unequal losses between modes $B$ and $D$ does nothing more to impact the outcome of our protocol than simply increasing the level of equal losses between these modes.
Lastly, noticeable in both the equal loss and unequal loss plots (Figs. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne} and \ref{Fig:CatES_Fidelity_UnequalLoss} respectively) is that there is a double peak present, for all values of $T$ and $\Upsilon$ investigated (although we recognise that the second peak becomes less pronounced as $T$ decreases and/or $\Upsilon$ increases).
Mathematically, this is a direct consequence from the numerous exponential terms that are present in the final density matrix describing the resultant two qubit matrix - we do not explicitly present this density matrix, as each of the matrix terms contains a vast number of complicated exponential terms, however, in Appendix \ref{AppendixA} we include the final quantum state (prior to tracing out the lossy modes), for the equal loss scenario. We also direct the reader to \cite{Parker2018} for the mathematical detail of each stage of this ES protocol, including the final state generated.
These exponential terms present in the equal loss case of Eq. \ref{eq:FinalState_EqualLoss} (which are also present in the unequal loss scenario) can be seen as somewhat competing with each other; instead of seeing a simple peak followed by a decay due to exponential dampening, as a result of the introduction of loss into our system, we see a dip which is a consequence of exponential interference. This dip becomes less pronounced as the level of loss increases (for example, in the $T=0.95$, $\Upsilon=0.10$ plot of Fig. \ref{Fig:CatES_Fidelity_UnequalLoss}) - this is due to the exponential dampening effect, dependent on the level of loss, having a stronger impact on the final density matrix compared to the smaller effect of exponential interference.
Of course, this too is a positive result, as it means that we have a wide acceptance window of the coherent state amplitudes to prepare our initial states in, whilst still giving a tolerable fidelity. The maximum fidelity value varies as a function of $\alpha$, dependent on the level of loss considered: for $T=0.97$ the second peak gives the maximum fidelity, and for $T\leq0.96$, the first of these peaks gives the best outcome state, as shown in Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne}.
Finally, we also plot the fidelity against the desired $\ket{\Phi^{+}(\alpha)}$ Bell state, as a function of both $\alpha$ and the averaged unequal loss value $\Upsilon$, for $T=1$, in Fig. \ref{Fig:3DFidelity_Cats_UnequalLoss_T100}.
The plot of Fig. \ref{Fig:3DFidelity_Cats_UnequalLoss_T100} does not quite reach unity, even at the peak $\alpha$ value. The reason for this is that the numerical calculation cannot be evaluated for $\Upsilon$ very close to 0, as the Gaussian function (Fig. \ref{Fig:UnequalLoss_GaussianFunction_Upsilon0_10}) then becomes a
delta function (i.e. no longer a continuous spectrum as per Fig. \ref{Fig:UnequalLoss_GaussianFunction_Upsilon0_10}, but instead is
zero everywhere except for $\upsilon=0$). Intrinsically, we expect that in the limit of $\Upsilon = 0$ we return to the results shown in the equal loss scenario (Fig. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne}), and so we would indeed then expect the plot of Fig. \ref{Fig:3DFidelity_Cats_UnequalLoss_T100} to reach, and plateau at, unity.
From Figs. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne}, \ref{Fig:CatES_Fidelity_UnequalLoss} and \ref{Fig:3DFidelity_Cats_UnequalLoss_T100}, we can conclude that we do not desire an averaged loss mismatch value of $\Upsilon>0.10$, as this gives an unacceptably low fidelity ($F\leq0.80$) for all $\alpha$. We consider an acceptable fidelity result to be $F\geq0.80$, as we could use any of the multitude of entanglement purification protocols available \cite{Pan2001,Pan2003,Feng2000,Chun2015,Zhang2017} to increase this fidelity to a sufficient level for further quantum communication/computation uses (i.e. to $F\geq0.95$).
We acknowledge here that this limit we have for unequal losses $\Upsilon\leq0.10$ to give an acceptable protocol output fidelity is not detrimental to the usefulness of our ES protocol: as previously stated, we refer to $\Upsilon$ as an average for unequal losses, as this variable is intended to reflect the practical perspective of running this protocol as an experiment, in which one would perhaps have a range of optical fibres, each of differing length, for example. It then follows that an averaged loss mismatch value of $\Upsilon>0.10$ represents quite a large range of optical fibre lengths, and so it would be possible for an experimentalist to avoid unequal losses greater than this limit proposed in any case.
\end{multicols}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket{\Phi^{+}(\alpha)}$ (Eq. \ref{Eq:Bell_Outcome}) Bell state (solid lines), and the orthogonal $\ket{\Phi^{-}(\alpha)}$ (Eq. \ref{Eq:Orthogonal_Bell_Outcome}) Bell state (dashed lines), as a function of the coherent state amplitude $\alpha$, for the final state generated via our entanglement swapping protocol (Eq. \ref{Eq:Final_State}), for varying levels of equal losses between modes $B$ and $D$.}\label{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne}}
{\includegraphics[width=11cm]{CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket{\Phi^{+}(\alpha)}$ (Eq. \ref{Eq:Bell_Outcome}) Bell state, as a function of the coherent state amplitude $\alpha$, for the final state generated via our entanglement swapping protocol (Eq. \ref{Eq:Final_State}), for varying levels of equal loss, and averaged unequal loss ($\Upsilon$).}\label{Fig:CatES_Fidelity_UnequalLoss}}
{\includegraphics[width=11cm]{CatES_Fidelity_UnequalLoss.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket{\Phi^{+}(\alpha)}$ (Eq. \ref{Eq:Bell_Outcome}) Bell state, as a function of the coherent state amplitude $\alpha$ and the averaged unequal loss value $\Upsilon$, for the final state generated via our entanglement swapping protocol (Eq. \ref{Eq:Final_State}), for $T=1$.}
\label{Fig:3DFidelity_Cats_UnequalLoss_T100}}
{\includegraphics[width=11cm]{3DFidelity_Cats_UnequalLoss_T100_2.png}}
\end{figure}
\begin{multicols}{2}
\subsection{Imperfect Homodyne Detection}\label{Subsec:Imperfect_Homodyne_Detection}
Thus far in the results and discussion, we have only analysed the scenario in which the homodyne measurement outcomes are the average \textit{perfect} case. As briefly mentioned in Subsec. \ref{Subsec:Homodyne_Detection}, to allow for imperfections in the homodyne measurement outcome, we follow the method outlined in \cite{Laghaout2013}, and consider a resolution bandwidth ($\Delta x$) around the average measurement outcome. The final state of our protocol when analysing homodyne imperfections is given in Appendix \ref{AppendixA} Subsec. \ref{AppendixB}, as a density matrix in Eq. \ref{eq:CatES_ImperfectHomodyne_FinalRho}.
Analysis of homodyne measurement imperfections are vital within this work, as no practical homodyne detector is able to measure with a bandwidth equal to $\Delta x \approx 0$. Hence, in this section we investigate the tolerance our protocol has to increasing this measurement bandwidth, whilst still producing a final state of acceptable fidelity. As we reduce $\Delta x$ we are effectively allowing for fewer possible measurement outcomes when performing the protocol practically, and this is how we determine success probability of the homodyne measurement, as discussed in Subsec. \ref{Subsec:Homodyne_Success Prob}.
Firstly, we plot the scenario of no loss (Fig. \ref{Fig:CatES_FidelityANDOrthogonal_EqualLoss_T100_ImperfectHomodyne}), to investigate the impact on output fidelity when increasing $\Delta x$ to investigate the impact in the most idealised circumstance. From Fig. \ref{Fig:CatES_FidelityANDOrthogonal_EqualLoss_T100_ImperfectHomodyne} we can see that the fidelity against the ideal Bell state ($\ket*{\Phi^{+}(\alpha)}$) begins to oscillate as we increase $\Delta x$. Intuitively we can conclude that these oscillations are present as a result of the numerous competing exponentials, which are dependent on $\Delta x$ and also $\alpha$ within our final state (see Eq. \ref{eq:CatES_EqualLoss_ImperfectHomodyne_State}).
Interestingly, this oscillatory behaviour is not present when observing higher values of loss ($T=0.95$) as per Figs. \ref{Fig:CatES_Fidelity_EqualLoss_T95_ImperfectHomodyne} and \ref{Fig:CatES_ORTHOGONALFidelity_EqualLoss_T95_ImperfectHomodyne}. This is because the dampening exponentials introduced when accounting for losses, after having traced out the lossy modes, have a stronger influence on the final density matrix compared to the competing exponentials which produce the oscillations.
Promisingly, we find that introducing a relatively high level of detection imperfection ($\Delta x \leq 0.25$) does not significantly impact the resultant state fidelity against the ideal Bell state, even in the higher loss scenario. In fact, it is still evident that there is a peak $\alpha$ value as noticed in the previous fidelity plots discussed, and so if performing this practically one could take into account the known homodyne detector resolution bandwidth, and select the value of $\alpha$ which is likely to achieve the highest fidelity result.
Finally, we note that we do not show results for averaged unequal losses as these scale similarly to merely increasing the level of equal loss.
\subsection{Homodyne Measurement Success Probability}\label{Subsec:Homodyne_Success Prob}
Another important quantity to consider, in evaluating the performance of a protocol, is the success probability of the measurement schemes. As previously discussed, evaluating the homodyne detection resolution bandwidth $\Delta x$ allows us to investigate the success probability of this measurement; increasing $\Delta x$ means allowing for more outcome results of this measurement, and so we expect to see the success probability increase as $\Delta x$ becomes larger.
Contrastingly, allowing for higher values of $\Delta x$ impacts the resultant fidelity of the output state of the protocol, as discussed in Subsec. \ref{Subsec:Imperfect_Homodyne_Detection} - therefore we expect to witness a trade-off between the success of the protocol and the quality (fidelity) of the outcome. This is useful to understand, because in some cases the customer might wish to suffer a lower success rate to obtain high fidelity pairs.
To calculate the homodyne success probability we determine the modulus square of the normalised probability amplitudes (given by using the projector of Eq. \ref{eq:ImperfectHomodyneProjector} onto the coherent states present in mode $D$) to
give us the probability distribution we integrate over, as:
\begin{align}
\abs{{}_D\braket*{x_{\frac{\pi}{4}}}{\Psi}_{A\varepsilon_{B}CD\varepsilon_{D}}}^2,
\label{eq:CatState_ProbabilityDistributionFunction}
\end{align}
which then gives the success probability as:
\begin{align}
&\mathcal{P}_{Hom.}(\%)(\Upsilon)=\int_{0}^{\infty}f(\upsilon,\Upsilon)\mathcal{N_\upsilon}\nonumber\\
&\times\Bigg(\int_{\frac{\mathcal{T}^{+}}{2}|\alpha|-\frac{\Delta x}{2}}^{\frac{\mathcal{T}^{+}}{2}|\alpha|+\frac{\Delta x}{2}}\abs{{}_D\braket*{x_{\frac{\pi}{4}}}{\Psi}_{A\varepsilon_{B}CD\varepsilon_{D}}}^2\text{d}x_{\frac{\pi}{4}}\nonumber\\
&+\int_{-\frac{\mathcal{T}^{+}}{2}|\alpha|-\frac{\Delta x}{2}}^{-\frac{\mathcal{T}^{+}}{2}|\alpha|+\frac{\Delta x}{2}}\abs{{}_D\braket*{x_{\frac{\pi}{4}}}{\Psi}_{A\varepsilon_{B}CD\varepsilon_{D}}}^2\text{d}x_{\frac{\pi}{4}}\Bigg)\text{d}\upsilon\times 100,
\label{eq:Homodyne_CatES_SuccessProb_UnequalLoss}
\end{align}
where, $\mathcal{T}^{+}=\sqrt{T}+\sqrt{T+\upsilon}$ and,
\begin{align}
\mathcal{N}_{\upsilon}=1/\Big(4+8e^{-\frac{|\mathcal{T}^+\alpha|^2}{4}}+24e^{-\frac{|\mathcal{T}^+\alpha|^2}{2}}+8e^{-\frac{|\mathcal{T}^+\alpha|^2}{4/3}}\nonumber\\
+4e^{-|\mathcal{T}^+\alpha|^2}+8e^{-(2+\text{i})|\mathcal{T}^+\alpha|^2}+8e^{-(2-\text{i})|\mathcal{T}^+\alpha|^2}\Big)^{\frac{1}{2}},
\end{align}
is the normalisation.
Although we show the full expression above for clarity, we do not present the results for equal and averaged unequal losses for homodyne success probability, as these scale as expected - incremental increases in the level of loss (be that equal or unequal) slightly lowers the success probability.
Fig. \ref{Fig:CatES_Homodyne_SuccessProb} shows the plot of the homodyne success probability $\mathcal{P}_{Hom.}(\%)$ as a function of $\alpha$, for increasing $\Delta x$. Here we can see that for $\Delta x = 5.0$ the success probability is 100\% for all $\alpha$ - this is because the resolution bandwidth at such a high value covers the entire spectrum of the cat state probability distribution. To further understand the plot of Fig. \ref{Fig:CatES_Homodyne_SuccessProb} it is useful to look at the probability distribution of the cat state equation (Eq. \ref{eq:CatState_ProbabilityDistributionFunction}) for varying $\alpha$, as per Fig. \ref{Fig:ProbDist_CatState_varyingalpha_Comparison}.
In Fig. \ref{Fig:ProbDist_CatState_varyingalpha_Comparison}, $\alpha=0$ gives the vacuum state probability distribution, as expected. Contrastingly, if we look to higher values of $\alpha$, then we see that for $\alpha=2.0$ the two peaks of the probability distribution are almost entirely separated; this is the ideal scenario for successful homodyne detection, in which we have two outcome peaks to be measured (i.e. $x_{\frac{\pi}{4}}=\pm\alpha$ - see Fig. \ref{Fig:Cats_Phase_Space}).
Moreover, when considering $\alpha=1.0$ in Fig. \ref{Fig:ProbDist_CatState_varyingalpha_Comparison}, we can see that the width of this peak is broader than that of $\alpha=0$. This is as a result of the vacuum states present in mode $D$, which are not ideal states for the homodyne measurement to project to, as measurement of these will not give an entangled output. These vacuum states are exponentially dampened as a function of $\alpha$, and so it is not until $\alpha > 2.0$ when the contribution by the vacuum is reduced to a negligible amount.
Finally, we highlight that we indeed witness a trade-off between homodyne success probability and the output state fidelity. The success probability is optimum for very large $\Delta x$, however the output fidelity of the protocol would be very low in this scenario. Ideally, we would opt to use $\alpha$ values in the range of $1.0 \leq \alpha \leq 2.0$ and a measurement bandwidth of $\Delta x \leq 0.25$, which gives a success probability of around $10-15 \%$ in the no loss regime.
\end{multicols}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket*{\Phi^{+}(\alpha)}$ (Eq. \ref{Eq:Bell_Outcome}) Bell state (solid lines in plot) and the orthogonal $\ket*{\Phi^{-}(\alpha)}$ (Eq. \ref{Eq:Orthogonal_Bell_Outcome}) Bell state (dotted lines in plot) as a function of $\alpha$ for the final state generated via our entanglement swapping protocol (Eq. \ref{eq:CatES_ImperfectHomodyne_FinalRho}), for $T=1$ and varying homodyne measurement bandwidth $\Delta x$.}
\label{Fig:CatES_FidelityANDOrthogonal_EqualLoss_T100_ImperfectHomodyne}}
{\includegraphics[width=11cm]{CatES_FidelityANDOrthogonal_EqualLoss_T100_ImperfectHomodyne.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket*{\Phi^{+}(\alpha)}$ (Eq. \ref{Eq:Bell_Outcome}) Bell state as a function of $\alpha$ for the final state generated via our cat state entanglement swapping protocol (Eq. \ref{eq:CatES_ImperfectHomodyne_FinalRho}), for $T=0.95$ and varying homodyne measurement bandwidth $\Delta x$.}
\label{Fig:CatES_Fidelity_EqualLoss_T95_ImperfectHomodyne}}
{\includegraphics[width=11cm]{CatES_Fidelity_EqualLoss_T95_ImperfectHomodyne.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Fidelity against the $\ket*{\Phi^{-}(\alpha)}$ (Eq. \ref{Eq:Orthogonal_Bell_Outcome}) Bell state as a function of $\alpha$ for the final state generated via our cat state entanglement swapping protocol (Eq. \ref{eq:CatES_ImperfectHomodyne_FinalRho}), for $T=0.95$ and varying homodyne measurement bandwidth $\Delta x$.}
\label{Fig:CatES_ORTHOGONALFidelity_EqualLoss_T95_ImperfectHomodyne}}
{\includegraphics[width=11cm]{CatES_ORTHOGONALFidelity_EqualLoss_T95_ImperfectHomodyne.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Success probability ($\mathcal{P}_{Hom.}(\%)$) of the homodyne measurement (Eq. \ref{eq:Homodyne_CatES_SuccessProb_UnequalLoss}), as a function of $\alpha$, for varying homodyne measurement bandwidth $\Delta x$ and $T=1$.}
\label{Fig:CatES_Homodyne_SuccessProb}}
{\includegraphics[width=11cm]{CatES_Homodyne_SuccessProb.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Probability distribution $f(x_{\frac{\pi}{4}})$ of the cat state equation (given by Eq. \ref{eq:CatState_ProbabilityDistributionFunction}), as a function of $x_{\frac{\pi}{4}}$, for $\alpha=0$, $\alpha=1.0$ and $\alpha=2.0$ (for no loss).}
\label{Fig:ProbDist_CatState_varyingalpha_Comparison}}
{\includegraphics[width=11cm]{ProbDist_CatState_varyingalpha_Comparison.png}}
\end{figure}
\begin{figure}[H]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=5cm}}]{figure}[\FBwidth]
{\caption{Success probability ($\mathcal{P}_0(\%)$) of the vacuum measurement, as a function of the coherent state amplitude $\alpha$, for varying levels of equal losses between modes $B$ and $D$. Note that we truncate the $y$-axis of the plot so that the range is from $20\%\rightarrow100\%$.}
\label{Fig:catesvacuumsuccessprob}}
{\includegraphics[width=11cm]{CatES_Vacuum_SuccessProb.png}}
\end{figure}
\begin{multicols}{2}
\subsection{Vacuum Measurement Success Probability}\label{Subsec:Success_Probability}
We now discuss the success probability associated with the vacuum state measurement. Fig. \ref{Fig:catesvacuumsuccessprob} shows a plot of the vacuum measurement success probability ($\mathcal{P}_0$) as a function of the coherent state amplitude $\alpha$. We only plot up to $\alpha=2.5$, as this is the region we are primarily concerned with in our protocol, due to the fact that this is where our fidelity values peak as a function of $\alpha$. Noticeably, we can see in Fig. \ref{Fig:catesvacuumsuccessprob} that for $\alpha=0$, $\mathcal{P}_0=100\%$. Naturally, this is entirely expected, because at this value of $\alpha=0$ all of these coherent states are in fact vacuum states, and so intrinsically any vacuum measurement here will always be performed with $100\%$ success probability. More interestingly, we see that, for all values of loss covered in Fig. \ref{Fig:catesvacuumsuccessprob}, all plots plateau at $\mathcal{P}_0=25\%$.
We also point out that although we have only considered equal losses between modes $B$ and $D$ in Fig. \ref{Fig:catesvacuumsuccessprob}, the results are effectively identical when considering the same levels of unequal losses between these modes (as we noticed in the fidelity plots previously discussed).
Given the fidelity results presented in Figs. \ref{Fig:CatES_FidelityANDOrthogonal_UnknownT_EqualLoss_PerfectHomodyne}, \ref{Fig:CatES_Fidelity_UnequalLoss} and \ref{Fig:3DFidelity_Cats_UnequalLoss_T100}, we can conclusively state that, for the levels of equal and unequal losses evaluated, we would desire coherent state amplitudes of $1.0\leq\alpha\leq2.0$, so as to achieve the peak fidelity results. In this region, however, the success probability begins to drop, and in fact at the higher end of this limit ($\alpha\approx2.0$), we can see in Fig. \ref{Fig:catesvacuumsuccessprob} that the success probability is around $\mathcal{P}_0=25\%$.
This trade-off between fidelity and success probability is a common occurrence in quantum communication schemes, and is something we must accept. Although success probability is undoubtedly significant when considering the practical implementation of our proposed ES protocol, we argue that high fidelity is far more important to aim for as opposed to better success probabilities; producing fewer pairs of higher fidelity quantum states is more useful for further quantum communication purposes, as opposed to producing a greater number of lower fidelity states.
\section{Application in a Quantum Network}\label{Sec:Quantum_Network}
Thus far, we have shown that one could theoretically produce a highly entangled Bell state of fidelity $F\geq0.80$, when allowing for photonic transmissions of $T=0.95$ in the propagating modes (corresponding to losses of $5\%\equiv0.22$ dB). Currently, the highest-performing low-loss fibre available exhibits losses of $0.149$ dB/km \cite{Kawaguchi2015}, and so $5\%$ loss in our protocol corresponds to maximum distances between Alice and Bob of 3.0 km (assuming the measurement apparatus to detect modes $B$ and $D$ is located precisely in the middle of Alice and Bob), whilst still being able to share an entangled Bell state of fidelity $F\geq0.80$.
Of course, this limits our protocol to being used in short-distance entanglement distribution networks. Nevertheless, this scheme could be highly suitable for sharing entanglement between adjacent quantum computers within a future quantum network, possibly in a local area network (LAN), such as a university campus or research centre.
However, if we allow for higher levels of loss, and therefore lower resultant fidelity, we can intrinsically distribute entanglement to two parties further distanced apart. If we set our fidelity acceptance threshold to $F\geq0.70$, then the losses we can tolerate in modes $B$ and $D$ are $T=0.88$ (equivalent to 0.56 dB/km), allowing Alice and Bob to be located around 7.5 km apart if using ultra low-loss fibre of 0.149 dB/km. Moreover, if we allow our fidelity acceptance window to drop even further to $F\geq0.60$, then we can allow for losses of $T=0.84$, which corresponds to 0.76 dB, thus allowing Alice and Bob to share an entangled state whilst separated by an overall distance of 10.5 km.
Again, this still makes our proposed entanglement swapping protocol suitably only for relatively short LAN-type distances, however this comes at the advantage of being able to distribute highly entangled Bell states.
A Bell state of fidelity $F\leq0.95$ is impractical for further uses with quantum computing, communications or information processing, and so allowing for further losses within our protocol means that a subsequent entanglement purification scheme would be required. Nevertheless, this is a common requirement for entanglement swapping protocols, and proposed quantum repeater networks use entanglement purification nodes as a fundamental part of the protocol.
Research into increasing the level of entanglement shared between two distant parties has been carried out extensively, and there exist a multitude of potential protocols which can increase Bell state fidelity from $F=0.60$ to $F\geq0.95$ \cite{Pan2001,Pan2003,Feng2000,Chun2015,Zhang2017}. Of course, in this circumstance this then requires one to use a higher number of lower fidelity pairs to produce one very high fidelity pair, although we argue that for the intended purpose of this protocol it is logical to sacrifice high bit rates to ensure that the entangled pairs delivered are of the best quality.
\section{Conclusions}\label{Sec:Conclusions}
We summarise the key findings to this work as follows:
\begin{itemize}
\item Following our proposed ES protocol, we can produce a phase-rotated Bell state $\ket{\Phi^{+}(\alpha)}$ with fidelity of $F\geq0.80$ for equal losses between modes $B$ and $D$ of $T\geq0.95$ with an averaged loss mismatch value of $\Upsilon\geq0.05$.
\item Introduction of averaged unequal losses between the propagating modes does not significantly impact the protocol results, and in fact has no more of an effect than merely increasing the levels of equal losses between these channels.
\item We witness a double peak in all fidelity plots, as a function of the coherent state amplitude $\alpha$, and so we have a broader range of acceptable $\alpha$ values than in our previously proposed (coherent state superposition) protocol of \cite{Parker2017a}, in which we witness only a single sharp peak.
\item The protocol investigated in this work is also slightly more tolerant to induced losses than the work of \cite{Parker2017a}.
\item For the range of $\alpha$ which delivers a fidelity of $F\geq0.80$, in the higher loss limits of ($T=0.95$ and $\Upsilon=0.05$) the success probability of the vacuum measurement is $\mathcal{P}_0\approx40\%$.
\item We can allow for a fairly broad homodyne measurement bandwidth of $\Delta x \leq 0.50$, whilst still giving an entangled output state of respectable fidelity against the ideal case.
\end{itemize}
We reiterate that the phase present in the Bell state produced through our protocol, $\ket{\Phi^{+}(\alpha)}$, is not of detriment
to the usefulness of this protocol in distributing entanglement to two parties; however in a practical implementation, this would require the customer to be informed of the exact phase present each time. As this phase is fixed by the value of $\alpha$ chosen, all post-entangled pairs received by the two parties will have the same phase.
Moreover, we conclude that this protocol is tolerant only to low levels of photonic losses, and so is more suitable for distributing entangled quantum states over a local area network between parties located 5-10 km apart, when followed by a suitable entanglement purification scheme. These entangled pairs of photons could then be used for further quantum communication purposes, or by adjacent quantum computing processors, which may require entanglement for communications or information processing \cite{Bennett2000}.
\section{Acknowledgements}
The authors would like to think Prof. Andrew Lord for useful discussions. We acknowledge support from EPSRC (EP/M013472/1).
\end{multicols}
|
2,877,628,089,511 | arxiv | \section{INTRODUCTION}
FermiQCD~\cite{DiPierro1,DiPierro2,DiPierro3} is a library for fast
development of parallel applications for Lattice Quantum Field
Theories and Lattice Quantum Chromodynamics~\cite{DiPierro4}. It was
designed both to be easy to use, with a syntax very similar to common
mathematical notation, and, at the same time, optimized for PC
clusters.
FermiQCD takes a top-down approach: the top level functions were
designed first, followed by optimized implementations of those
functions. The most critical parts are optimized in assembler using
SSE/SSE2 instructions. All FermiQCD algorithms are parallel but
parallelization is hidden from the high level programmer. At the
lowest level, parallelization is implemented in MPI and/or in PSIM.
PSIM is a parallel emulator that allows running, testing and debugging
of parallel code on a single processor PC without requiring MPI.
All components are implemented as separate, although related, classes.
For example, in FermiQCD lattices and fields are objects while actions
and inverters are static methods of the corresponding classes.
FermiQCD components range from low level linear algebra, fitting and
statistical functions (including the Bootstrap and a Bayesian fitter
based on Levenberg-Marquardt minimisation) to high level parallel
algorithms specifically designed for lattice quantum field theories
such as the Wilson~\cite{Wilson} and $O(a^2)$-improved gauge actions,
the Clover fermionic action, the Asqtad~\cite{asqtad} action for KS
fermions, and the Domain Wall action~\cite{kaplan}.
One can create a new action by creating a new class and plugging it
into the rest the library. All other components, such as the
inverters, will work with it. For example FermiQCD provides three
inverters, MinRes, BiCGStab and UML. The first two are general and
work with any action and any type of field, the third (UML) is highly
optimized for KS and ASQTAD actions.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.5\textwidth]{BlockDiagram2.eps}
\end{center}
\caption{Components of FermiQCD.}
\label{fig:cpts}
\end{figure}
Figure~\ref{fig:cpts} shows a schematic representation of FermiQCD's
components. The lower components are referred to as Matrix Distributed
Processing and they define the language used in FermiQCD. The upper
components are the algorithms. The top components represent examples,
applications and other tools. These tools include converters for the
most gauge field formats: MILC, NERSC, UKQCD, CANOPY, and some binary
formats.
\section{SYNTAX OVERVIEW AND PROGRAM EXAMPLE}
All FermiQCD algorithms are implemented on top of an Object Oriented Linear Algebra package with a Maple-like syntax. For example
\begin{eqnarray}
A&=&\mathrm{trace}(\gamma^2 (\gamma^0-\mathbf{1})/2) \\
B&=&e^{i\theta\lambda_3}
\end{eqnarray}
where $\gamma^\mu$ are the Dirac Gamma matrices in Euclidean space and
$\lambda_3$ is one of the generators of $SU(3)$, are implemented in
FermiQCD as
\begin{Verbatim}[numbersep=3pt]
Complex A = trace(Gamma[i]*(Gamma[0]-1)/2);
Matrix B = exp(I*theta*Lambda[3]);
\end{Verbatim}
A $4D$ $16\times8^3$ lattice is declared (with obvious generalization
to arbitrary dimensions and sizes) as
\begin{Verbatim}[numbersep=3pt]
int L[]={16,8,8,8};
mdp_field lattice(4,L);
\end{Verbatim}
An $SU(n)$ gauge field $U_\mu(x)$ is declared and initialized by
\begin{Verbatim}[numbersep=3pt]
gauge_field U(lattice,n); set_cold(U);
\end{Verbatim}
The following sets $\beta=6.0$ and performs 10
heatbath~\cite{heatbath} steps with the Wilson gauge action
\begin{Verbatim}[numbersep=3pt]
coefficients gauge; gauge["beta"]=6.0;
WilsonGaugeAction::heatbath(U,gauge,10);
\end{Verbatim}
Any field can be saved: {\tt U.save("filename");} loaded: {\tt
U.load("filename");} and translated: {\tt U.shift(mu);}. A field can
also be transformed locally. Here is how to implement a global gauge
transformation $G$ of $U$
\begin{Verbatim}[numbersep=3pt]
Matrix G=exp(2*I*Lambda[2]);
mdp_site x(lattice);
forallsites(x)
for(int mu=0; mu<U.ndim; mu++)
U(x,mu)=G*U(x,mu)*inv(G);
\end{Verbatim}
{\tt U(x,mu)} is an $n\times n$ matrix and {\tt x} is an object that
represents a lattice site. {\tt forallsites(x)} is a parallel loop.
Each processing node loops over the lattice sites stored by the node.
A Wilson fermionic field is declared as
\begin{Verbatim}[numbersep=3pt]
fermi_field psi(lattice,n);
\end{Verbatim}
and a gauge invariant shift can be implemented as
\begin{Verbatim}[numbersep=3pt]
psi_up(x)=U(x,mu)*psi(x+mu);
psi_dw(x)=hermitian(U(x-mu,mu))*psi(x-mu);
\end{Verbatim}
Notice that {\tt x+mu} reads as $x+\hat\mu$ and {\tt x-mu} reads as $x-\hat\mu$ where $\mu=0,1,2,3$ is one of the possible lattice directions.
Multiplication by the fermionic matrix is invoked as follows
\begin{Verbatim}[numbersep=3pt]
coefficients quark; quark["kappa"]=0.1245; quark["c_{SW}"]=0.0;
if(quark["c_{SW}"]!=0) compute_em_field(U);
default_fermi_action=FermiCloverActionFast::mul_Q;
mul_Q(psi_out,psi_in,U,quark);
\end{Verbatim}
The chromo-electromagnetic field is required by the clover term and
computed only if required. It is stored inside a gauge field object.
FermiQCD includes three different equivalent implementations of the
above algorithm, declared in the following classes:
FermiCloverActionSlow, FermiCloverActionFast, FermiCloverActionSSE2.
The second is optimized in C++, the third is optimized in assembler.
The inverse multiplication
$\psi_\mathrm{out}=Q^{-1}[U]\psi_\mathrm{in}$ is invoked with the
following call
\begin{Verbatim}[numbersep=3pt]
mul_invQ(psi_out,psi_in,U,quark,1e-20,1e-12);
\end{Verbatim}
where {\tt 1e-20} is the target absolute precision for the numerical inversion and {\tt 1e-12} is the target relative precision.
An ordinary quark propagor is declared and generated by
\begin{Verbatim}[numbersep=3pt]
fermi_propagator S(lattice,n);
generate(S,U,quark,1e-20,1e-12);
\end{Verbatim}
and it can be used, for example, to build a meson propagator
$C_{\pi}(t)$ by summing the following expression over $x$ and over the
spin components {\tt a}, {\tt b}
\begin{Verbatim}[numbersep=3pt]
Cpi[x(TIME)]+=real(trace(S(x,a,b)*hermitian(S(x,a,b))));
\end{Verbatim}
Everything works similary for the other actions and other types of fields.
Figure~\ref{fig:prog} shows a complete parallel program for generating
{\tt nconfig} gauge configurations in $SU(5)$ on a $16\times 8^4$
lattice, saving them and computing the average plaquette and the pion
propagator on each configuration.
\begin{figure}
\begin{center
\begin{Verbatim}[numbers=left, numbersep=3pt, xleftmargin=2em,
fontsize=\small, baselinestretch=0.97]
#include "fermiqcd.h"
int main(int argc, char **argv) {
mdp.open_wormholes(argc, argv); // START
define_base_matrices("FERMIQCD"); // set Gamma convention
int n=5, nconfig=100;
int L[] = {16,8,8,8};
mdp_lattice lattice(4,L); // declare lattice
gauge_field U(lattice, n); // declare fields
fermi_propagator S(lattice,n); // declare propagator
mdp_site x(lattice); // declare a site var
coefficients gauge; gauge["beta"]=6.0; // set parameters
coefficients quark; quark["kappa"]=0.1234; quark["c_{SW}"]=0.0;
default_fermi_action=FermiCloverActionFast::mul_Q;
mdp_array<float,1> Cpi(L[TIME]); // declare and zero Cpi
for(int t=0; t<L[TIME]; t++) Cpi(t)=0;
set_hot(U);
for(int k=0; k < nconfig; k++) {
WilsonGaugeAction::heatbath(U,gauge,10); // do heatbath
mdp << average_plaquette(U) << endl; // print plaquette
U.save(string("gauge")+tostring(k)); // save config
if(quark["c_{SW}"]!=0) compute_em_field(U);
generate(S,U,quark,1e-20,1e-12); // make propagator
forallsites(x) // contract pion
for(int a=0; a<4; a++) // source spin
for(int b=0; b<4; b++) // sink spin
Cpi[x(TIME)]+=real(trace(S(x,a,b)*hermitian(S(x,a,b))));
mpi.add(Cpi.address(),Cpi.size()); // parallel add
for(int t=0; t<L[TIME]; t++)
mdp << t << " " << Cpi(t) << endl; // print output
}
mdp.close_wormholes(); // STOP
return 0;
}
\end{Verbatim}
\end{center}
\caption{A complete parallel program in FermiQCD.}
\label{fig:prog}
\end{figure}
\section{BENCHMARKS}
Table~\ref{tab:bmarks} shows typical running times for the FermiQCD
inverters applied to different actions. Times are in microsecond per
lattice site per step. Notice that MinRes involves one {\tt mul\_Q}
per step, BiCGStab involves two, and the UML inverter also involves
two but only applied to sites of even parity. These times were
computed on one 3.2GHz Pentiutm 4 (typical computations in parallel
with a Myrinet network show a drop in efficiency of 20-30\% when
scaling up to 16-32 processors).
\begin{table}
{\footnotesize \begin{center}
\begin{tabular}{|l|l|cccc|} \hline
Action & Inverter & float & float (SSE)& double & double (SSE2) \\ \hline
Wilson & Min Res & 8.83 & 1.79 & 6.84 & 2.07 \\
Wilson & BiCGStab & 17.8 & 3.16 & 13.8 & 4.42 \\
Clover & Min Res & 9.76 & 1.98 & 12.08 & 2.82 \\
Clover & BiCGStab & 19.63 & 4.71 & 24.95 & 6.08 \\
KS & Min Res & 1.42 & 0.78 & 1.71 & 1.01 \\
KS & BiCGStab & 2.95 & 1.63 & 3.56 & 2.11 \\
KS & UML & 1.89 & 1.14 & 2.08 & 1.34 \\
Asqtad & Min Res & 3.73 & 2.47 & 4.29 & 5.24 \\
Asqtad & BiCGStab & 7.65 & 5.02 & 8.79 & 6.61 \\
Asqtad & UML & 1.14 & 3.14 & 5.24 & 3.81 \\ \hline
\end{tabular}
\end{center}}
\caption{Running times for FermiQCD inverters using different
actions.}
\label{tab:bmarks}
\end{table}
\section{CONCLUSIONS}
FermiQCD is now a stable and mature product and the project mailing
list currently numbers more than 30 members. The Wilson and Asqtad
inverters are as fast if not faster than any other software package
available for PC clusters. FermiQCD is an Open Source project and
users can contribute to its improvement by creating new classes and
adding functionality. Some of our objectives include the addition of
an optimized gauge action, optimized Domain Wall fermions, HMC for
dynamical fermions, compatibility with the ILDG format~\cite{ildg},
support for the SciDAC QMP API, and a GUI for visual development.
FermiQCD and additional documentation can be downloaded from:
{\tt www.fermiqcd.net}
\section*{Acknowledgements}
We wish to acknowledge the Fermilab theory group, the University of
Southmpton, and the University of Iowa for their contribution to the
development of FermiQCD.
|
2,877,628,089,512 | arxiv | \section{Introduction} To determine the equation of state (EoS)
of isospin-asymmetric nuclear matter (ANM) has been a longstanding
goal shared by both nuclear physics and astrophysics\,\cite{EPJA}.
Usually one uses the so-called empirical parabolic law for the
energy per nucleon, i.e.,
$E(\rho,\delta)=E_0(\rho)+E_{\rm{sym}}(\rho)\delta^2+\mathcal{O}(\delta^4)$
where $\rho=\rho_{\rm{n}}+\rho_{\rm{p}}$ and
$\delta=(\rho_{\rm{n}}-\rho_{\rm{p}})/\rho$ are the nucleon density
and isospin asymmetry of the system in terms of the neutron and
proton densities $\rho_{\rm{n}}$ and $\rho_{\rm{p}}$, respectively.
The isospin quadratics of the ANM EoS has been verified to high
accuracies from symmetric ($\delta=0$) up to pure neutron
($\delta=1$) matter by most of the available nuclear many-body
theories using various interactions, see, e.g., ref.\,\cite{Bom91}.
Nevertheless, it has been shown consistently in a number of studies
that for some physical quantities relevant for understanding
properties of neutron stars, such as the proton fraction at $\beta$
equilibrium, core-crust transition density and the critical density
for the direct URCA process to happen, even a very small coefficient
$E_{\rm{sym,4}}(\rho)$ of the isospin quartic term in the EoS can
make a big difference \cite{Sjo74}.
Here we concentrate on examining the isospin quadratics of the
kinetic EoS. For many purposes in both nuclear physics and
astrophysics, such as simulating heavy-ion collisions\,\cite{LCK08}
and determining critical formation densities of different charge
states of $\Delta$ resonances in neutron stars\,\cite{Cai14}, one
has to know separately the kinetic and potential parts of the EoS.
While neither any fundamental physical principle nor the empirical
parabolic law of the EoS requires the kinetic and potential parts of
the EoS to be quadratic in $\delta$ individually, in practice
especially in most phenomenological models the free Fermi gas (FFG)
EoS is often used for the kinetic part and then the generally less
known potential EoS is explored by comparing model predictions with
experimental data. It is well known that the FFG model predicts a
kinetic symmetry energy of $E^{\rm{kin}}_{\rm{sym}}(\rho_0)\approx
12.3$ MeV and a negligibly small quartic term of
$E^{\rm{kin}}_{\rm{sym},4}(\rho_0)=E^{\rm{kin}}_{\rm{sym}}(\rho_0)/27\approx
0.45$ MeV at $\rho_0=0.16/\rm{fm}^3$. However, nuclear interactions,
in particular the short-range repulsive core and tensor force, lead
to a high (low) momentum tail (depletion) in the single-nucleon
momentum distribution above (below) the nucleon Fermi
surface\,\cite{Mig57,bethe,pan92,Pan99}. Much progress has been made
recently both theoretically and experimentally in quantifying
especially the nucleon high momentum tails (HMT) in ANM, see, e.g.,
refs.\,\cite{Wei15,Wei15a,Hen15,Hen14,Egi06}. In this work, using
isospin-dependent nucleon HMT constrained by recent high-energy
electron scattering and medium-energy nuclear photodisintegration
experiments as well as the state-of-the-art calculations of the
deuteron wave function and the EoS of pure neutron matter (PNM) near
the unitary limit within several modern microscopic many-body
theories, we show that the kinetic ANM EoS has a significant
quartic term of $E^{\rm{kin}}_{\rm{sym},4}(\rho_0)=7.18\pm
2.52\,\rm{MeV}$ that is about 16 times the FFG model prediction.
\section{Isospin dependence of single-nucleon momentum
distribution with a high momentum tail in neutron-rich matter}
Guided by well-known predictions of microscopic nuclear many-body
theories, see, e.g., reviews in ref.\,\cite{Ant88}, and recent
experimental findings\,\cite{Wei15,Wei15a,Hen15,Hen14}, we describe
the single-nucleon momentum distribution in ANM using
\begin{equation}\label{MDGen}
n^J_{\v{k}}(\rho,\delta)=\left\{\begin{array}{ll}
\Delta_J+\beta_J{I}\left(\displaystyle{|\v{k}|}/{k_{\rm{F}}^J}\right),~~&0<|\v{k}|<k_{\rm{F}}^J,\\
&\\
\displaystyle{C}_J\left({k_{\rm{F}}^{J}}/{|\v{k}|}\right)^4,~~&k_{\rm{F}}^J<|\v{k}|<\phi_Jk_{\rm{F}}^J.
\end{array}\right.
\end{equation}
Here, $J=\rm{n,p}$ is the isospin index,
$k_{\rm{F}}^J=k_{\rm{F}}(1+\tau_3^J\delta)^{1/3}$ is the transition
momentum\,\cite{Hen14} where $k_{\rm{F}}=(3\pi^2\rho/2)^{1/3}$ and
$\tau_3^{\rm{n}}=+1$, $\tau_3^{\rm{p}}=-1$.
\begin{figure}[h!]
\centering
\includegraphics[width=8.cm]{CaiFig1.eps}
\caption{A sketch of the single-nucleon momentum distribution with a high momentum tail.}
\label{mom-dis}
\end{figure}
The main features of $n^J_{\v{k}}(\rho,\delta)$ are depicted in Fig.
\ref{mom-dis}. The $\Delta_J$ measures the depletion of the Fermi sphere
at zero momentum with respect to the FFG model prediction while the
$\beta_J$ is the strength of the momentum dependence
$I(|\v{k}|/k_{\rm{F}}^J)$\,\cite{Bel61,Czy61,Sar80} of the depletion
near the Fermi surface. The jump $Z^J_{\rm{F}}$ of the momentum distribution at $k_{\rm{F}}^J$, namely, the ``renormalization
function", contains information about the nucleon effective E-mass and its isospin dependence\cite{Jeu76}. Specifically,
$Z^J_{\rm{F}}=n_{k_{\rm{F}}^J-0}^J-n_{k_{\rm{F}}^J+0}^J={M}/{{M}_{\rm{E}}^{J,\ast}}$,
where $ {{M}^{J,\ast}_{\rm{E}}}/{M}\equiv[1-{\partial
V}/{\partial\omega}]^{-1}$ with $V$ and $\omega$ being the real part of the single-particle potential
and energy\,\cite{Mig57,Lut60}, respectively.
The amplitude ${C}_J$ and cutoff coefficient
$\phi_J$ determine the fraction of nucleons in the HMT via
\begin{equation}\label{xPNM}
x_J^{\rm{HMT}}=3C_{{J}}\left(1-\frac{1}{\phi_{{J}}}\right).
\end{equation}
The normalization condition $
[{2}/{(2\pi)^3}]\int_0^{\infty}n^J_{\v{k}}(\rho,\delta)\d\v{k}=\rho_J={(k_{\rm{F}}^{J})^3}/{3\pi^2}
$ requires that only three of the four parameters, i.e., $\beta_J$,
${C}_J$, $\phi_J$ and $\Delta_J$, are independent. Here we choose
the first three as independent and determine the $\Delta_J$ from
\begin{equation}\label{DeltaJ}
\Delta_J=1-\frac{3\beta_J}{(k_{\rm{F}}^{J})^3}\int_0^{k_{\rm{F}}^J}{I}\left(\frac{k}{k_{\rm{F}}^J}\right)k^2\d
k-3{C}_J\left(1-\frac{1}{\phi_J}\right).
\end{equation}
Hinted by the finding within the self-consistent Green function
(SCGF) theory\,\cite{Rio09} and the Brueckner-Hartree-Fock
(BHF) theory \,\cite{Yin13} the depletion $\Delta_J$ has an almost linear
dependence on $\delta$ in the opposite directions for neutrons and
protons, we expand all four parameters in the form
$Y_J=Y_0(1+Y_1^J\delta)$. Then, the total kinetic energy per nucleon
in ANM
\begin{equation}\label{kinE}
E^{\rm{kin}}(\rho,\delta)=\frac{1}{\rho}\frac{2}{(2\pi)^3}\sum_{J=\rm{n,p}}\int_0^{\phi_Jk_{\rm{F}}^J}\frac{\v{k}^2}{2M}n_{\v{k}}^J(\rho,\delta)\d\v{k}
\end{equation}
would obtain a linear term in $\delta$ of the form
\begin{align}\label{Ekin1}
&E_1^{\rm{kin}}(\rho)=\frac{3}{5}\frac{k_{\rm{F}}^2}{2M}\Bigg[\frac{5}{2}C_0\phi_0(\phi_1^{\rm{n}}+\phi_1^{\rm{p}})
\notag\\
&+\frac{5}{2}C_0(\phi_0-1)(C_1^{\rm{n}}+C_1^{\rm{p}})
+\frac{1}{2}\Delta_0(\Delta_1^{\rm{n}}+\Delta_1^{\rm{p}})\notag\\
&+\frac{5\beta_0(\beta_1^{\rm{n}}+\beta_1^{\rm{p}})}{2k_{\rm{F}}^5}
\int_0^{k_{\rm{F}}}I\left(\frac{k}{k_{\rm{F}}}\right)k^4\d k\Bigg]
\end{align}
where $M$ is the nucleon mass. To ensure that the
$E_1^{\rm{kin}}(\rho)$ vanishes as required by the neutron-proton
exchange symmetry of the EoS, we require that
$\Delta_1^{\rm{n}}=-\Delta_1^{\rm{p}}$,
$\beta_1^{\rm{n}}=-\beta_1^{\rm{p}}$,
${C}_1^{\rm{n}}=-{C}_1^{\rm{p}}$ and
$\phi_{1}^{\rm{n}}=-\phi_1^{\rm{p}}$, i.e., more compactly
$Y_J=Y_0(1+Y_1\tau_3^J\delta)$.
\section{Constraining the parameters of the single-nucleon
momentum distribution} It is well known that the nucleon HMT from
deuteron to infinite nuclear matter scales, see, e.g.,
refs.\,\cite{Fan84,Pie92,Cio96}, leading to constant per nucleon
inclusive $(\rm{e},\rm{e}^{\prime})$ cross sections for heavy nuclei
with respect to deuteron for the Bjorken scaling parameter
$x_{\rm{B}}$ between about 1.5 and 1.9, see, e.g.,
ref.\,\cite{Arr12} for a recent review. Systematic analyses of these
inclusive experiments and data from exclusive two-nucleon knockout
reactions induced by high-energy electrons or protons have firmly
established that the HMT fraction in symmetric nuclear matter (SNM)
is about $x^{\rm{HMT}}_{\rm{SNM}}=28\%\pm4$\% and that in PNM is
about
$x_{\rm{PNM}}^{\rm{HMT}}=1.5\%\pm0.5\%$\,\cite{Hen15,Hen14,Egi06,Hen14b}.
The ${C}/{|\mathbf{k}|^4}$ shape of the HMT for both SNM and PNM is
strongly supported by recent findings theoretically and
experimentally. The HMT for deuteron from variational many-body
calculations using several modern nuclear forces decrease as
$|\mathbf{k}|^{-4}$ within about 10\% and in quantitative agreement
with that from analyzing the $\rm{d}(\rm{e}, \rm{e}^{\prime}\rm{p})$
cross section in directions where final state interaction suffered
by the knocked-out proton is small\,\cite{Hen15}. The extracted
magnitude $C_{\rm{SNM}}=C_0$ of the HMT in SNM at $\rho_0$ is
${C}_0\approx 0.15\pm0.03$\,\cite{Hen15} (properly rescaled
considering the factor of 2 difference in the adopted normalizations
of $n_{\mathbf{k}}$ here and that in refs.\,\cite{Hen15,Hen14b}).
Rather remarkably, a very recent evaluation of medium-energy
photonuclear absorption cross sections has also presented clear and
independent evidence for the ${C}/{|\mathbf{k}|^4}$ behavior of the
HMT and extracted a value of ${C}_0\approx
0.172\pm0.007$\,\cite{Wei15} for SNM at $\rho_0$ in very good
agreement with that found in ref.\,\cite{Hen15}. In the following,
we use $C_0\approx0.161\pm0.015$ from taking the average of the
above two constraints. With this $C_0$ and the value of
$x^{\rm{HMT}}_{\rm{SNM}}$ given earlier, the HMT cutoff parameter in
SNM is determined to be
$\phi_0=(1-x_{\rm{SNM}}^{\rm{HMT}}/3{C}_0)^{-1}=2.38\pm0.56$.
Very interestingly, the ${1}/{|\mathbf{k}|^4}$ behavior of the HMT
nucleons is identical to that in two-component (spin-up and -down)
cold fermionic atoms first predicted by Tan\,\cite{Tan08} and then
quickly verified experimentally\,\cite{Ste10}. Tan's general
prediction is for all two-component fermion systems having an s-wave
contact interaction with a scattering length $a$ much larger than
the inter-particle distance $d$ which has to be much longer than the
interaction range $r_{\rm{e}}$. At the unitary limit when
$|k_{\rm{F}}a|\rightarrow \infty$, Tan's prediction is universal for
all fermion systems. Since the HMT in nuclei and SNM is known to be
dominated by the tensor force induced neutron-proton pairs with the
$a\approx 5.4$ fm and $d\approx 1.8$ fm at $\rho_0$, as noted in
refs.\,\cite{Hen15,Wei15}, Tan's stringent conditions for unitary
fermions is obviously not satisfied in normal nuclei and SNM. The
observed identical ${1}/{|\mathbf{k}|^4}$ behavior of the HMT in
nuclei and cold atoms may have some deeper physical reasons
deserving further investigations. Indeed, a very recent study on the
$A(\rm{e,e}'\rm{p})$ and $A(\rm{e,e}'\rm{pp})$ scattering has shown
that the majority of the short range correlation (SRC)-susceptible
n-p pairs are in the $^3\rm{S}_1$ state\,\cite{Col15}. On the other
hand, because of the unnaturally large neutron-neutron scattering
length $a_{\rm{nn}}(^1\rm{S}_0)=-18.8$ fm, it is known that PNM is
closer to the unitary limit\,\cite{Sch05}. The EoS of PNM can thus
be expanded as\,\cite{Bak99}
\begin{equation}\label{EPNMTan}
E_{\rm{PNM}}(\rho)\simeq\frac{3}{5}\frac{(k^{\textrm{PNM}}_{\textrm{F}})^2}{2M}\left[\xi-\frac{\zeta}{k^{\textrm{PNM}}_{\textrm{F}}a_{\textrm{nn}}}
-\frac{5\nu}{3(k^{\textrm{PNM}}_{\textrm{F}}a_{\textrm{nn}})^2}\right],
\end{equation}
where $k_{\rm{F}}^{\rm{PNM}}=2^{1/3}k_{\rm{F}}$ is the transition
momentum in PNM, $\xi\approx0.4\pm0.1$ is the Bertsch
parameter\,\cite{BertPara}, $\zeta\approx\nu\approx1$ are two
universal constants\,\cite{Bul05}.
\begin{figure}[h!]
\centering
\includegraphics[width=8.5cm]{CaiFig2.eps}
\caption{(Color Online) The EoS of PNM obtained from Eq.
(\ref{EPNMTan}) (dashed red band) and that from next-leading-order
(NLO) lattice calculation\,\cite{Epe09a} (blue solid points), chiral
perturbative theories\,\cite{Tew13} (green band), quantum Monte
Carlo simulations (QMC)\,\cite{Gez13,Gez10} (magenta band and purple
stars), and effective field theory\,\cite{Sch05}.}\label{TanPNMLow}
\end{figure}
Shown in Fig. \ref{TanPNMLow} is a comparison of the EoS of PNM
obtained from Eq. (\ref{EPNMTan}) (dashed red band) with several
state-of-the-art calculations using modern microscopic many-body
theories. At densities less than about 0.01\,$\textrm{fm}^{-3}$, as
shown in the inset, the Eq. (\ref{EPNMTan}) is consistent with the
prediction by the effective field theory\,\cite{Sch05}. In the range
of 0.01\,$\textrm{fm}^{-3}$ to about $0.02\,\textrm{fm}^{-3}$, it
has some deviations from predictions in ref.\,\cite{Sch05} but
agrees very well with the NLO lattice simulations\,\cite{Epe09a}. At
higher densities up to about $\rho_0$, it overlaps largely with
predictions by the chiral perturbation theories\,\cite{Tew13} and
the quantum Monte Carlo simulations\,\cite{Gez13,Gez10}.
In addition, recent studies on the spin-polarized neutron matter within
the chiral effective field theory including two-, three-,
and four-neutron interactions indicate that properties of PNM
is similar to the unitary Fermi gas at least upto $\rho_0$ far beyond the scattering-length regime
of $\rho\lesssim\rho_0/100$\,\cite{Kru15}.
Overall, the above comparison and studies clearly justify the use of Eq. (\ref{EPNMTan}) to
calculate the PNM EoS up to about $\rho_0$.
Both the HMT and EoS can be experimentally measured independently and calculated simultaneously within the same model.
Tan has proven in great detail that the two are directly related by the so-called adiabatic sweep theorem \cite{Tan08}.
It is valid for any two-component Fermi systems under the same conditions as the Eq. \ref{EPNMTan} near the unitary limit. For PNM, it can be written as
\begin{equation}\label{ast}
C_{\rm{n}}^{\rm{PNM}}\cdot(k_{\rm{F}}^{\rm{PNM}})^4=-4\pi
M\cdot\frac{\d (\rho E_{\rm{PNM}})}{\d(a^{-1})}.
\end{equation}
While the results shown in Fig. \ref{TanPNMLow} justify the use of Eq. \ref{EPNMTan} for the EoS of PNM up to about $\rho_0$,
indeed, to our best knowledge there is currently no proof that the Eq. \ref{ast} is also valid in the same density range as the Eq. \ref{EPNMTan}.
Thus, it would be very interesting to examine the validity range of Eq. \ref{ast} using the same models as those used to calculate the EoS. In this work, we assume
that the Eqs. \ref{EPNMTan} and \ref{ast} are both valid in the same density range. Then, the strength of the HMT in PNM can be readily obtained as
\begin{equation}
C_{\rm{n}}^{\rm{PNM}}\approx 2\zeta/5\pi+4\nu/(3\pi
k_{\rm{F}}^{\rm{PNM}}a_{\rm{nn}}(^1\rm{S}_0))\approx0.12.
\end{equation}
Noticing that $C_{\rm{n}}^{\rm{PNM}}=C_0(1+C_1)$, we can then infer that
$C_1=-0.25\pm0.07$ with the $C_0$ given earlier. Next, after
inserting the values of $x_{\rm{PNM}}^{\rm{HMT}}$ and
$C_{\rm{n}}^{\rm{PNM}}$ into Eq. ({\ref{xPNM}), the high momentum
cutoff parameter for PNM is determined to be
$\phi_{\rm{n}}^{\rm{PNM}}\equiv
\phi_0(1+\phi_1)=(1-x_{\rm{PNM}}^{\rm{HMT}}/3C_{\rm{n}}^{\rm{PNM}})^{-1}=1.04\pm0.02$.
It is not surprising that the $\phi_{\rm{n}}^{\rm{PNM}}$ is very close to
unity since only about 1.5\% neutrons are in the HMT in
PNM. Subsequently, using the $\phi_0$ determined earlier, we get
$\phi_1=-0.56\pm0.10$.
The two parameters $\beta_0$ and $\beta_1$ in
$\beta_J=\beta_0(1+\beta_1\tau_3^J\delta)$ depend on the function
$I(|\v{k}|/k_{\rm{F}}^J)$ which is still model dependent. To
minimize the model assumptions and evaluate the dominating terms in
the kinetic EoS, in the following we shall first use a
momentum-independent depletion of the Fermi sea as in most studies
in the literature. The HMT parameters $C_J$ and $\phi_J$ evaluated
above remain the same. Then, we examine the maximum correction to
each term in the kinetic EoS by using the largest values of
$\beta_0$ and $\beta_1$ allowed and a typical function
$I(|\v{k}|/k_{\rm{F}}^J)$. Not surprisingly, the corrections are all
small.
\section{Isospin dependence of kinetic EoS of ANM} The kinetic
EoS can be expanded in $\delta$ as
\begin{equation}
E^{\rm{kin}}(\rho,\delta)=E_0^{\rm{kin}}(\rho)+E_{\rm{sym}}^{\rm{kin}}(\rho)\delta^2+E_{\rm{sym,4}}^{\rm{kin}}(\rho)\delta^4+\mathcal{O}(\delta^6).
\end{equation}
The coefficients evaluated from Eq. (\ref{kinE}) using the $n^J_{\v{k}}(\rho,\delta)$ in Eq. (\ref{MDGen}) with $\beta_J=0$ are
\begin{align}
E^{\rm{kin}}_0(\rho)=&\frac{3}{5}E_{\rm{F}}(\rho)\left[
1+{C}_0\left(5\phi_0+\frac{3}{\phi_0}-8\right)\right],\label{E0kin}\\
E_{\rm{sym}}^{\rm{kin}}(\rho)=&\frac{1}{3}E_{\rm{F}}(\rho)\Bigg[1+{C}_0\left(1+3{C}_1\right)\left(5\phi_0+\frac{3}{\phi_0}-8\right)\notag\\
&\hspace*{-1.5cm}+3{C}_0\phi_1\left(1+\frac{3}{5}{C}_1\right)\left(5\phi_0-\frac{3}{\phi_0}\right)+\frac{27{C}_0\phi_1^2}{5\phi_0}\Bigg],\label{Esymkin}\\
E_{\rm{sym,4}}^{\rm{kin}}(\rho)=&\frac{1}{81}E_{\rm{F}}(\rho)\Bigg[1+{C}_0(1-3{C}_1)\left(5\phi_0+\frac{3}{\phi_0}
-8\right)\notag\\
&\hspace*{-1.5cm}+3{C}_0\phi_1(9{C}_1-1)\left(5\phi_0-\frac{3}{\phi_0}\right)\notag\\
&\hspace*{-1.5cm}+\frac{81{C}_0\phi_1^2(9\phi_1^2-9{C}_1\phi_1-15\phi_1+15{C}_1+5)}{5\phi_0}
\Bigg].\label{Esymkin4}
\end{align}
In the FFG where there is no HMT, $\phi_0=1$, $\phi_1=0$ and thus
$5\phi_0+3/\phi_0-8=0$, the above expressions reduce naturally to the well
known results of $E^{\rm{kin}}_0(\rho)=3E_{\rm{F}}(\rho)/5$,
$E_{\rm{sym}}^{\rm{kin}}(\rho)=E_{\rm{F}}(\rho)/3$, and
$E_{\rm{sym,4}}^{\rm{kin}}(\rho)/E_{\rm{sym}}^{\rm{kin}}(\rho)=1/27$
where $E_{\rm{F}}(\rho)=k_{\rm{F}}^2/2M$ is the Fermi energy.
For the interacting nucleons in ANM with the momentum distribution and its parameters given earlier, we found that
$E_0^{\rm{kin}}(\rho_0)=40.45\pm8.15\,\rm{MeV}$,
$E_{\rm{sym}}^{\rm{kin}}(\rho_0)=-13.90\pm11.54\,\rm{MeV}$ and
$E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)=7.19\pm2.52\,\rm{MeV}$,
respectively. Compared to the corresponding values for the FFG, it
is seen that the isospin-dependent HMT increases significantly the average kinetic
energy $E_0^{\rm{kin}}(\rho_0)$ of SNM but decreases the kinetic
symmetry energy $E_{\rm{sym}}^{\rm{kin}}(\rho_0)$ of ANM to a negative value qualitatively consistent with
findings of several recent studies of the kinetic EoS considering short-range nucleon-nucleon correlations using both phenomenological
models and microscopic many-body theories\,\cite{CXu11,Vid11,Lov11,Car12,Rio14,Car14}.
However, it was completely unknown before if the empirical isospin parabolic law is still valid for the kinetic EoS of ANM
when the isospin-dependent HMTs are considered. Very surprisingly and interestingly, our calculations here show clearly
that it is broken seriously. More quantitatively, the ratio
$|E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)/E_{\rm{sym}}^{\rm{kin}}(\rho_0)|$
is about $52\%\pm26$\% that is much larger than the FFG value of
$3.7$\%. We also found that the large quartic term is mainly due to the
isospin dependence of the HMT cutoff described by the $\phi_1$ parameter.
For example, by artificially setting $\phi_1=0$, we obtain
$E^{\rm{kin}}_{\rm{sym}}(\rho_0)=14.68\pm2.80$ MeV and
$E^{\rm{kin}}_{\rm{sym},4}(\rho_0)=1.12\pm0.27$ MeV which are all
close to their FFG values.
Considering short-range nucleon-nucleon correlations but assuming
that the isospin parabolic approximation is still valid, some
previous studies have evaluated the kinetic symmetry energy
$E_{\rm{sym}}^{\rm{kin}}$ by taking the difference between the
kinetic energies of PNM and SNM, i.e., subtracting the
$E_{\rm{PNM}}^{\rm{kin}}$ by $E_0^{\rm{kin}}$. This actually
approximately equals to
$E_{\rm{sym}}^{\rm{kin}}(\rho_0)+E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)=-6.71\pm9.11$\,MeV
in our current work. This value is consistent quantitatively with
the $E_{\rm{sym}}^{\rm{kin}}(\rho_0)$ found in ref.\,\cite{Hen14b}
using the parabolic approximation.
\section{Corrections due to the momentum-dependent depletion of
the Fermi sea} To estimate corrections due to the momentum
dependence of the depletion very close to the Fermi surface, i.e., a
finite $\beta_J$, we consider a widely used single-nucleon momentum
distribution parameterized in ref.\,\cite{Cio96} based on
calculations using many-body theories. For
$|\v{k}|\lesssim2\,\rm{fm}^{-1}$, it goes like $\sim
e^{-\alpha|\v{k}|^2}$ with $\alpha\approx 0.12\,\rm{fm}^2$. At
$\rho_0$ since $\alpha k_{\rm{F}}^2\approx0.21$,
$e^{-\alpha|\v{k}|^2}\approx1-\alpha|\v{k}|^2+\mathcal{O}(|\v{k}|^4)$
is a good approximation in the range of $0<|\v{k}|<k_{\rm{F}}^J$.
Thus, we adopt a quadratic function
${I}(|\v{k}|/k_{\rm{F}}^J)=(|\v{k}|/k_{\rm{F}}^J)^2$. The constants
in the parameterization of ref.\,\cite{Cio96} are absorbed into our
parameters $\Delta_J$ and $\beta_J$. Then Eq. (\ref{DeltaJ}) gives
us $\Delta_J=1-3\beta_J/5-3{C}_J\left(1-1/\phi_J\right)$.
Specifically, we have $\beta_0=(5/3)[1-\Delta_0-3C_0(1-\phi_0^{-1})]=(5/3)[1-\Delta_0-x_{\rm{SNM}}^{\rm{HMT}}]$
for SNM. Then using the predicted value of $\Delta_0\approx 0.88\pm0.03$\,\cite{Yin13,Pan99,Fan84} and the experimental value
of $x_{\rm{SNM}}^{\rm{HMT}}\approx 0.28\pm 0.04$, the value of $\beta_0$ is estimated to be about $-0.27\pm0.08$.
Similarly, the condition $\beta_J=\beta_0(1+\beta_1\tau_3^J\delta)<0$, i.e., $n_{\v{k}}^J$ is
a decreasing function of momentum towards $k_{\rm{F}}^J$, indicates that $|\beta_1|\leq1$.
\begin{figure}[h!]
\centering
\includegraphics[width=7.5cm]{CaiFig3.eps}
\caption{(Color Online) Corrections to the $E_{\rm{sym}}^{\rm{kin}}(\rho_0)$ and $E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)
$ as functions of $\beta_1$ with $\beta_0=-0.35$.}
\label{fig_betaJeffect}
\end{figure}
First of all, a finite value of $\beta_J$ is expected to affect the
``renormalization function" $Z^J_{\rm{F}}$. For SNM, we have
$Z_{\rm{F}}^0=1+2\beta_0/5-C_0-x_{\rm{SNM}}^{\rm{HMT}}= 0.45\pm0.07$
($0.56\pm0.04$) in the presence (absence) of $\beta_0$. For ANM,
however, the $Z^J_{\rm{F}}$ depends on the less constrained value of
$\beta_1$. It is worth noting that the latter also determines the
neutron-proton effective E-mass splitting which has significant
effects on isovector observables in heavy-ion
collisions\,\cite{BALi15}, and a study is underway to further
constrain the value of $\beta_1$ using data from heavy-ion
reactions.
Contributions from a finite $\beta_J$ to the first three terms of
the kinetic EoS are
\begin{align}
\delta
E_{0}^{\rm{kin}}(\rho)=&\frac{3}{5}E_{\rm{F}}(\rho_0)\cdot\frac{4\beta_0}{35},\\
~~\delta
E_{\rm{sym}}^{\rm{kin}}(\rho)=&\frac{1}{3}E_{\rm{F}}(\rho_0)\cdot\frac{4\beta_0(1+3\beta_1)}{35},\\
\delta
E_{\rm{sym,4}}^{\rm{kin}}(\rho)=&\frac{1}{81}E_{\rm{F}}(\rho_0)\cdot\frac{4\beta_0(1-3\beta_1)}{35}.
\end{align}
With the largest magnitude of $\beta_0=-0.35$, we examine in Fig. \ref{fig_betaJeffect}
the corrections to the $E_{\rm{sym}}^{\rm{kin}}(\rho_0)$ and
$E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)$ as functions of $\beta_1$ in its
full range allowed. In this case the maximum effects of the finite
$\beta_J$ are revealed. It is seen that the correction on the
$E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)$ is negligible while the
correction on the $E_{\rm{sym}}^{\rm{kin}}(\rho_0)$ is less than
2\,MeV. Considering the corrections due to the finite
$\beta_0$ and $\beta_1$ and their uncertainties, we finally obtain
$E_0^{\rm{kin}}(\rho_0)=39.77\pm8.13\,\rm{MeV}$,
$E_{\rm{sym}}^{\rm{kin}}(\rho_0)=-14.28\pm11.59\,\rm{MeV}$ and
$E_{\rm{sym,4}}^{\rm{kin}}(\rho_0)=7.18\pm2.52\,\rm{MeV}$,
respectively. We notice here that the $\delta^6$ term was also consistently
evaluated and was found to be negligibly small at $\rho_0$.
\section{Summary and Discussions} In summary, using an isospin-dependent
single-nucleon momentum distribution including a high (low) momentum
tail (depletion) with its shape parameters constrained by the latest
results of several relevant experiments and the state-of-the-art
predictions of modern microscopic many-body theories, we found for
the first time that the kinetic EoS of interacting nucleons in ANM
is not parabolic in isospin asymmetry. It has a significant quartic
term of $7.18\pm2.52\,\rm{MeV}$ while its quadratic term is
$-14.28\pm11.60\,\rm{MeV}$ at saturation density of nuclear matter.
To this end, it is necessary to point out the limitations of our
approach and a few physical implications of our findings. Since we
fixed the parameters of the nucleon momentum distribution (Eq.
(\ref{MDGen})) by using experimental data and/or model calculations
at the saturation density, the possible density dependence of these
parameters is not explored in this work. The density dependence of
the various terms in the kinetic EoS is thus only due to that of the
Fermi energy as shown in Eqs.(\ref{E0kin})-(\ref{Esymkin4}). In this
limiting case, the slope of the kinetic symmetry energy, i.e.,
$L^{\rm{kin}}=3\rho_0\partial
E_{\rm{sym}}^{\rm{kin}}(\rho)/\partial\rho|_{\rho=\rho_0}=-27.81\pm
23.08$ MeV while that of the FFG is about 25.04 MeV.
The SRC-reduced kinetic symmetry energy with respect to the FFG
prediction has been found to affect significantly not only our
understanding about the origin of the symmetry energy but also
several isovector observables, such as the free neutron/proton and
$\pi^-/\pi^+$ ratios in heavy-ion collisions
\cite{Hen14b,Li15,Yong15}. However, to our best knowledge, an
investigation on possible effects of a large isospin quartic term on
heavy-ion collisions has never been done while its effects on
properties of neutron stars have been studied extensively
\cite{Sjo74}. Of course, effects of the quartic and quadratic terms
should be studied together within the same approach. To extract from
nuclear reactions and neutron stars information about the EoS of
neutron-rich matter, people often parameterize the EoS as a sum of
the kinetic energy of a FFG and a potential energy involving unknown
parameters upto the isospin-quadratic term only. Our findings in
this work indicate that it is important to include the
isospin-quartic term in both the kinetic and potential parts of the
EoS. Moreover, to accurately extract the completely unknown
isospin-quartic term $E^{\rm{pot}}_{\rm{sym,4}}(\rho)\delta^4$ in
the potential EoS it is important to use the kinetic EoS of
quasi-particles with reduced kinetic symmetry energy and an enhanced
quartic term due to the isospin-dependence of the HMT. Most relevant
to the isovector observables in heavy-ion collisions, such as the
neutron-proton ratio and differential flow, is the nucleon isovector
potential. Besides the so-called Lane potential $\pm 2\rho
E^{\rm{pot}}_{\rm{sym}}(\rho)\delta$ where the
$E^{\rm{pot}}_{\rm{sym}}(\rho)$ is the potential part of the
symmetry energy and the $\pm$ sign is for neutrons/protons, the
$E^{\rm{pot}}_{\rm{sym,4}}(\rho)\delta^4$ term contributes an
additional isovector potential $\pm 4 \rho
E^{\rm{pot}}_{\rm{sym,4}}(\rho)\delta^3$. In neutron-rich systems
besides neutron stars, such as nuclear reactions induced by rare
isotopes and peripheral collisions between two heavy nuclei having
thick neutron-skins, the latter may play a significant role in
understanding the isovector observables or extracting the sizes of
neutron-skins of the nuclei involved. We plan to study effects of
the isospin-quartic term in the EoS in heavy-ion collisions using
the isospin-dependent transport model \cite{LCK08} in the near
future.
\section{Acknowledgement} We would like to thank L.W. Chen, O. Hen, X.H. Li, W.G.
Newton, E. Piasetzky, A. Rios, I. Vida$\tilde{\textrm{n}}$a and L.B.
Weinstein for helpful discussions. This work is supported in part by
the US National Science Foundation under Grant No. PHY-1068022 and the U.S. Department of Energy Office of Science under Award Number DE-SC0013702.
|
2,877,628,089,513 | arxiv | \section{Introduction}
The predominant cause of stellar collisions is thought
to be random encounters between stars in dense clusters. These collisions,
where stars pass within a single stellar radius of each other, may
result in a wide range of outcomes, most notably the formation of
close and eccentric compact binaries \citep[and references therein]{sha99,sha02,dav02,iva10}.
Because the collision rate per star, $\Gamma$, due to random encounters
depends directly on the local number density of stars $\Gamma\propto n$,
such interactions are common only in the densest stellar environments
such as globular clusters (GCs). Outside of these clusters, the densities
are lower by a factor as great as $10^{7}$: random collisions effectively do not occur in
the field. In this paper, we describe a dynamical instability that
occurs in evolved systems containing three or more star
\footnote{We focus here on triple systems, but the scenario is also applicable
to higher multiplicity systems
}. This mechanism, which we term the triple evolution dynamical instability
(hereinafter TEDI),
leads to close encounters and collisions between stars even in low
density stellar environments.
The TEDI occurs due to an interplay between stellar mass loss and
orbital instability. Consider a hierarchical triple system composed
of an inner binary coupled to an outer binary, where the outer binary
consists of a tertiary companion in orbit about the center of mass
of the inner pair. When the orbits of the inner and outer pair are
well separated, the system can remain stable indefinitely. Stellar
evolution can disrupt this architecture. When the more massive component
of the inner binary evolves, it begins to shed mass. As a result,
the orbits expand in proportion to the ratio between the initial and
final mass in the enclosed system \citep{had+63,egg06}. Since the
relative mass loss in the inner binary is greater than that in the
outer binary, the inner orbit encroaches on the outer orbit. This
relative orbital expansion can trigger a dynamical instability, which
results in the dissolution of the system (see Fig. \ref{fig:scenarios}).
The subsequent evolution of such unstable systems is very similar
to that of binary-single star encounters in globular clusters, and
often leads to close encounters between any of the three components.
The TEDI provides a unique pathway to physical collisions during these
close approaches: as stars evolve off of the main sequence, they not
only lose mass, but also expand radially by a factor of a few hundred
on the Asymptotic Giant Branch (AGB). The simultaneity of mass loss
(and thus instability) with stellar bloating greatly increases the
cross section for collisions. For this reason, as we now demonstrate,
the TEDI is the dominant route to stellar collisions in the Universe,
producing nearly 30 times more collisions than random gravitational
encounters in GCs.
The significance of mass-loss induced instability in triple systems
has been proposed in several studies \citep{kis+94,ibe+99,per10,fre+11,por+11}.
However, the frequency of this phenomena, and its importance for stellar
collisions and the consequences thereof, have not been explored.
Our primary goal in this paper is to estimate the rate of collisions per galaxy
induced by the TEDI mechanism, $\Gamma_{\rm col}$. This calculation requires the estimation of the following
quantities:
\begin{itemize}
\item \textbf{$N_{\star}$} - The number of stars in the Galaxy
\item \textbf{$f_{{\rm evolve}}$} - The observed fraction of evolved stars
\item \textbf{$f_{{\rm triple}}$} - The fraction of stars in hierarchical
triple systems
\item \textbf{$f_{{\rm des}}$} - The fraction of hierarchical systems that
become unstable as the stars evolve
\item \textbf{$f_{\rm col}$} - The fraction of unstable triple systems that
end with a collision between two of the stars
\end{itemize}
Thus $\Gamma_{\rm col}=N_{\star}\times f_{\rm evolve}\times f_{\rm triple}\times f_{\rm des}\times f_{\rm col}$.
Observations provide estimates of the first two variables, and less certain estimates for the third (triple fraction). Attaining
estimates for $f_{{\rm des}}$ and $f_{\rm col}$ occupies the remainder of this paper, as well as addressing the uncertainties in the triple fraction, $f_{{\rm triple}}$. The latter is done with both observational estimates and theoretical population synthesis models to provide independent estimates.
The main difficulties in doing so are due to the limited data existing on
triple systems and the distribution of their properties, and the large
phase space of possible evolutionary routes. Given these caveats,
we can only hope to provide an order of magnitude estimate of the
importance of the TEDI. Though we provide the formal statistical uncertainties
for the resulting fractions we obtain, the
systematic uncertainties due to our limited knowledge of the distribution
of triple system properties are likely larger than the statistical ones.
The outline of the paper is as follows. In Section \ref{sec:TEDI} we
quantify the criterion for the TEDI. We then determine the fraction
of systems that undergo the TEDI, $f_{\rm des}$, in Section \ref{sec:basic-estimate},
followed by an estimate of the frequency, $f_{\rm col}$, of collisions in the destabilized
systems (\S\ref{sec:coll}). In Section \ref{sec:rate}
we use these results to provide an estimate of the total collision
rate in the galaxy. We describe the various consequences of the TEDI
in Section \ref{sec:outcomes}, and briefly point out the influence of secular
dynamics on triple stellar evolution (\S\ref{sec:koz}). We close
the paper with the discussion and summary in Section \ref{sec:summary}.
\section{The triple evolution dynamical instability (TEDI) }
\label{sec:TEDI} A triple system is dynamically unstable when $Q<Q_{{\rm st}}$,
where $Q=a_{{\rm out}}(1-e_{{\rm out}})/a_{{\rm in}}$ and $Q_{{\rm st}}$
is (we adopt \citeauthor{val+08} 2008; but see also \citeauthor{mar08} 2008 for a different definition) : \begin{equation}
Q_{{\rm st}}=3(1+m_{3}/M_{12})^{1/3}(1-e_{{\rm out}})^{-1/6}\left(\frac{7}{4}+\frac{1}{2}\cos{i}-\cos^{2}i\right)^{1/3},\label{eq:stability}\end{equation}
where $a_{{\rm in}}$ and $a_{{\rm out}}$ are the semi-major axes
(SMAs) of the inner and outer binaries in the triple system, $e_{{\rm out}}$
is the outer binary eccentricity, and $m_{3}$ and $M_{12}=m_{1}+m_{2}$,
are the masses of the outer component and the inner binary, respectively.
The mutual inclination between the inner and outer binaries is $i$.
As the primary (with mass $m_{1}$) in the inner binary evolves off
the main sequence (MS), it loses mass. We consider only systems where
the stellar-wind driven mass loss occurs adiabatically, i.e. on a
timescale that is long compared to the outer orbital period (prompt
mass loss could occur in supernovae and also lead to instability)
\footnote{The triple supernova instability scenario is limited only
to high mass systems. Moreover, in the majority of cases, supernova
explosions lead to the prompt unbinding of the system (especially
if they are accompanied by a natal kick), making longer term chaotic
evolution, and hence physical collisions, less likely than in the
case of adiabatic mass loss. The contribution of this process to the
over all collision rate in the Galaxy is therefore likely to be small,
and not affect our main results. Nevertheless, this process, and the
effects of natal kicks, could be especially important for collisions
with neutron stars and black holes, and merits further study.}. We further assume that mass loss is isotropic, and that
mass transfer is negligible. The expansion of the inner orbit due
to mass loss will be larger than the expansion of the outer orbit
because the relative mass loss in the inner system is larger. Thus the SMAs evolve as:
\begin{equation}\frac{a_{{\rm in,f}}}{a_{{\rm in,i}}}=\frac{m_{{\rm 1,i}}+m_{2}}{m_{{\rm 1,f}}+m_{2}}>\frac{m_{{\rm 1,i}}+m_{2}+m_{3}}{m_{{\rm 1,f}}+m_{2}+m_{3}}=\frac{a_{{\rm out,f}}}{a_{{\rm out,i}}},\label{eq:expansion}\end{equation}
\citep{had+63} where the subscripts correspond to the system components
and the evolutionary stage of the system ($i$ and $f$ for the initial
and the final, post-mass loss system, respectively). The expansion
of each of the SMAs in direct proportion to the mass loss derives
from the conservation of the quantity $GMa(1-e^{2})$, where $M$
is the total system mass. Note that slow, isotropic mass loss is required
for $e$ to remain constant.
The ratio between the SMAs of the outer and inner binaries therefore
increases, by a factor of \begin{equation}\frac{{a_{{\rm in,f}}}/{a_{{\rm out,f}}}}{{a_{{\rm in,i}}}/{a_{{\rm out,i}}}}=\left(\frac{m_{{\rm 1,i}}+m_{2}}{m_{{\rm 1,f}}+m_{2}}\right)\left(\frac{m_{{\rm 1,f}}+m_{2}+m_{3}}{m_{{\rm 1,i}}+m_{2}+m_{3}}\right).\label{eq:sma-ratio-increase}\end{equation}
The TEDI occurs when this increase drives a stable triple system
across the threshold defined in Eq. (\ref{eq:stability}). The subsequent
dynamical evolution of an unstable system is chaotic. The system may
evolve through various configurations, including exchanges between
the stars in the inner and outer binaries, eccentricity excitation,
and collisions (see Fig. \ref{fig:scenarios} for an example). The
final outcome is typically the ejection of one star and/or a physical
collision. Because this three body problem is genuinely chaotic (typical
Lyapunov exponent of $1/2$; \citealp{iva+91,hei+99,val+06} and references
therein), the outcome is strongly dependent on small variations in
the initial conditions.
\includegraphics[scale=0.5]{fig1_new}
\begin{figure}[h!]
\begin{centering}
\par\end{centering}
\centering{}\caption{\label{fig:scenarios}{Three-body simulations of the dynamical evolution
of triple systems undergoing adiabatic mass loss.} The separations
($r$) between each of the three components in each system are plotted
as a function of time. Also shown are the mass (dashed-dotted) and
the stellar radius (dashed) of the primary, $M_{1}$. Thick lines
show the pairs including the primary. Top (1a): A collision between
a MS star and an AGB star (the subsequent evolution, assuming point
masses and no collision is also shown). Middle (1b): the formation
of an eccentric close WD-MS binary, similar to the Sirius system.
Bottom (1c): the WD primary is ejected after several exchanges, leaving
behind an eccentric MS-MS binary. In all cases the systems started with masses of $M_1 = 5\, M_\odot,\, M_2 =2.0\, M_\odot,\, M_3 = 2.25\, M_\odot,\, a_{\rm in} = 15\,{\rm AU}, a_{\rm out} = 53\, \rm{ AU},\, e_{\rm out }=0.1,\, i= 0.1$.}
\end{figure}
\section{Estimate of the fraction of systems destabilized due to the TEDI}
\label{sec:basic-estimate} To determine whether a given system is
unstable, we simultaneously follow its orbital and stellar evolution,
and check whether or not it violates the stability criterion at any point during its evolution.
We estimate the fraction of destabilized systems in two ways:
(1) we determine the fraction of observed triple systems \citep{tok08}
with known orbital parameters that become unstable, and (2) we calculate the unstable fraction of
a synthetic population of triples modeled on binary
statistics to account for potential biases in the observed sample.
We first describe the procedure to identify systems that should become
unstable, and then calculate the instability rates for the observed
and the synthetic samples.
\subsection{Identifying Unstable Systems}
\label{sec:unstable}
In order to determine whether a given triple system is susceptible
to the TEDI during its evolution, we make use of single and binary
population synthesis stellar evolution codes (SSE and BSE, described
in detail in \citealp{hur+00,hur+02}), to which we couple an analytic
calculation of the orbital evolution of the wider third component
using Equation (\ref{eq:expansion}).
The specific procedure is as follows.
\begin{enumerate}
\item The inner binary in the triple is evolved using the BSE code up to
a Hubble time (13.7 Gyrs), assuming Solar metallicity.
\item The third companion is evolved using the SSE code up to a Hubble time
(13.7 Gyrs), assuming Solar metallicity.
\item The mass-dependent orbital evolution of the inner binary and outer
single star are combined to form a time series evolution of all three
stars. The inner orbit of the binary is determined by the BSE code.
The outer orbit is updated at every output time step, $t_i$ of the BSE using
Eq. (2), where the initial and final time steps are the preceding
and current time steps \[
\frac{a_{{\rm out}}(t_i)}{a_{{\rm out}}(t_{i-1})}=\frac{m_{{\rm triple}}(t_{i-1})}{m_{{\rm triple}}(t_i)}.\]
We require that mass transfer between the inner binary and the third
companion is always negligible, so that we may apply the adiabatic
formula above. If the outer binary separation is smaller than $15$
AU, we assume non-negligible mass transfer is possible either
following a common-envelope phase for the inner-binary, or stellar
evolution of the tertiary. We therefore consider all triples with
outer separations smaller than $15$ AU to be stable to the TEDI.
\item We calculate the stability coefficient, $Q$ at every step in the
evolution, and using Eq. (1) find all systems that become unstable.
Note that if the inner binary merges, or any of the stars explode
as supernovae prior to destabilization, the system is considered stable
(in the context of adiabatic mass loss). All other destabilized triples
comprise our sample of unstable triples whose
size is denoted by $N_{{\rm des}}$.
\item For all destabilized systems we find the radius, $R_{{\rm des}}$,
of the mass losing evolving star at the point of destabilization,
and note the types and radii of the other components.
\item The fraction of destabilized systems is defined as $f_{{\rm des}}={N_{{\rm des}}/N_{{\rm triple}}}$.
\end{enumerate}
\subsection{Observed Triples}
The observed triple sample \citep{tok08} provides estimated masses
and orbital periods for each star, but does not provide their mutual
inclinations and eccentricities. We randomly assign each system relative inclinations
and eccentricities drawn from realistic distributions (see \citealt{fab+07}).
We employ a distribution of
eccentricities that is function of period, in line with
observed binaries \citep{duq+91}. For periods shorter than 1000 days,
the eccentricity is chosen from a Rayleigh distribution {[}$dp\propto e\exp(-\beta e^{2})de${]}
with $\left\langle e^{2}\right\rangle ^{1/2}=\beta^{-1/2}=0.33$.
For periods longer than 1000 days, the eccentricity is chosen from
an Ambartsumian distribution ($dp=2ede$), which corresponds to a uniform
distribution on the energy surface in phase space. The inclinations
are chosen from a random distribution that is uniform in $\cos i$.
We repeat the random sampling of these properties 30 times for each
system.
We only evolve triples that are initially stable, since unstable systems are unlikely to be
observable due to the instability timescale. If our random assignment of $e$ and $i$ produce
an unstable system, we draw new values from the distributions.
We follow the procedure discussed in Section \ref{sec:unstable} to determine
whether a given system becomes unstable. We find that the fraction
of all observed triple systems with primary mass $m_{1}>1$ M$_{\odot}$
destabilized within a Hubble time is $f_{{\rm des,o}}=0.035\pm0.006$,
where the error bars correspond to the 1 $\sigma$ statistical uncertainty.
In Fig. \ref{fig:unstable-eccs}, we illustrate the
sensitivity of the results to the eccentricity distribution: we calculate
the fraction of unstable systems for a complete grid of eccentricities.
For this sample we conservatively assume retrograde orbits, $i=180^{\circ}$,
which are the most stable configurations (see Eq. \ref{eq:stability}).
\begin{figure}
\begin{centering}
\includegraphics[scale=0.35]{fig2_2}
\par\end{centering}
\centering{}\caption{\label{fig:unstable-eccs} The fraction of destabilized systems, $f_{\rm des}$,
is shown for a grid of inner and outer eccentricities applied to the
observed sample of triple systems (smoothed and interpolated; original
bin size is $\Delta e=0.05$).}
\end{figure}
\subsection{Synthetic Sample of Triples}
We complement our use
of the existing observed triple sample, which may introduce unknown
biases, with a synthetic samples of triples. The method and
the assumptions used to produce the synthetic samples are based on
a method used by \citep{fab+07} for low mass triples and \citep{per09}
for high mass triples (with primary mass larger than $3$ M$_{\odot}$).
We assume that the orbital and stellar characteristics of triple systems
are equivalent to two (uncorrelated) binary systems, with the inner
binary acting like a single mass when choosing the third component.
The study by \citet{kra+11} suggests that this assumption is in good
agreement with observations (though this was only checked for a small
sample of triples). We choose initial orbital periods such that both
the inner and outer binaries follow the observed distribution. The
inner and outer periods are chosen from a distribution that is Gaussian
for low mass binaries with $M<3M_{\odot}$ \citep{rag+10} and log
flat for massive binaries corresponding to $f(r)\propto1/r$ i.e.,
$\ddot{O}$pik's law \citep{kob+07}
\footnote{We assume A stars share a similar distribution as that of F/G/K stars
in the \citep{rag+10} sample
}
The mass ratio of the inner binary, $q=m_{1}/m_{2}$, is chosen from
a Gaussian distribution (mean of $0.6$ and dispersion of $0.1$)
for low mass primaries and a power law for massive stars ($f(q)\propto q^{-0.4}$);.
The mass of the tertiary companion, $m_{3}$, is chosen such that
$q=m_{3}/(m_{1}+m_{2})$ follows the same mass ratio distribution.
Thus the mass of the third star is correlated with the combined mass
of the inner binary.
The inner and outer SMAs are computed from the masses and periods
assuming non-interacting Keplerian orbits. The inclination and eccentricity
distributions are chosen in the same manner as the observed sample.
We produce a sample of triples with primary masses in the range $1\le m_{p}\le20$,
with 1000 systems per primary mass bin (with bin size $\Delta M=0.3$
M$_{\odot}$, for a total number of $6\times10^{4}$ systems).
Note that since the triple outer component is randomly chosen
from a distribution with respect to the inner binary mass, the term
primary mass refers to the inner binary primary mass (which is not
necessarily the most massive component in the triple).
In Fig. \ref{fig:TEDI-triples}a we show the fraction of destabilized
triples from this sample as a function of primary mass. Fig. \ref{fig:TEDI-triples}b
shows the typical radius of the mass losing star at the time of destabilization
($R_{{\rm des}}$). The change in the assumed binary properties between
the low mass and the high mass stars introduces a sharp change in
the typical $R_{{\rm des}}$, since higher mass stars have more compact
triple configurations on average, and can be destabilized at an earlier
phase of their evolution, at which point the evolved star has a smaller
radius.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.35]{figS1av3}
\par\end{centering}
\begin{centering}
\includegraphics[scale=0.35]{figS1bv3}
\par\end{centering}
\centering{}\caption{\label{fig:TEDI-triples} Top: The fraction of destabilized triples
as a function of the inner binary primary component mass. The down
turn at about $8$ M$_{\odot}$ is due to supernova explosions that
occur before the system can be destabilized due to mass loss. Bottom:
The median radius of the mass losing stars at the onset of instability,
as a function of mass. Separate lines indicate the low mass ($m<3$
M$_{\odot}$) and the high mass stars $(m>3$ M$_{\odot}$), for which
different triple distributions are assumed (see text). }
\end{figure}
Using the fraction of destabilized triples as a function of primary
mass, $f_{{\rm des,s}}(m)$, shown in Fig \ref{fig:TEDI-triples},
we can estimate the total fraction of destabilized triples. The statistical
uncertainty for these fractions is small (1-2 \%; not shown). The
fraction of observed stars in triple systems
ranges from 11\% for Solar like (F, G, K) stars \citep{rag+10,ras+10}
to possibly 50\% for more massive B stars \citep{eva11}. We use a
Salpeter initial mass function,
$g(m)=dn/dm\propto m^{-2.35}$, and assume the triple fraction of
massive stars is about $5$ times larger than that of lower mass stars
\citep{kob+07,eva11}. Given these distributions, we estimate the
total fraction of destabilized triple systems in our synthetic sample
to be:
\begin{eqnarray}
&f_{{\rm des,s}}=\int_{1}^{3}f_{{\rm des,s}}(m)\frac{dn}{dm}dm\nonumber \\
&+5\times\int_{3}^{30}f_{{\rm des,s}}(m)\frac{dn}{dm}dm=0.010+0.043=0.053.
\end{eqnarray}
The overall fraction of destabilized triples in our synthetic
sample is higher than but consistent with the results for the observed sample (0.053, compared
with 0.035) (see also Fig. \ref{fig:unstable-eccs}).
Our estimates suggest that a few percent of all evolved triple systems undergo the
TEDI.
We now turn to direct integrations of three-body systems to both validate the analytic instability criterion and to determine the fraction
of unstable systems that produce a collision, $f_{\rm col}$.
\subsection{Three-body Simulations}
\label{sec:N-body}
Because TEDI systems begin in stable configurations and adiabatically
evolve into unstable configurations, they may not be directly comparable
to unstable triples explored in previous studies \citep{val+06}.
Therefore we conduct a series of three-body simulations including
mass loss both to check that our destabilized systems are truly unstable,
and to estimate the frequency of collisions based on the statistics
of close approaches during the chaotic evolutionary phase. In the appendix, we provide a
semi-analytic estimate of the collision rate.
The integrations are done using a modified version
of the Hermite integrator described in \citet{hut+95}. The integrator
uses a variable time step constrained to be $10^{-2}$ of the minimum
collision time of any two stars. Reducing the time-step by a factor
of 10 does not change the outcome. In runs without mass loss, energy
is conserved to about one part in $10^{10}$ over the 10 Myr run time.
We only consider point masses, and include no tidal effects, or mass
transfer. We use a constant mass loss rate, determined by the total
mass loss in the system as found in the stellar evolution calculation,
divided by a constant mass loss timescale, $\tau_{{\rm loss}}=0.5$
Myr (the typical lifetime of the highly evolved stars, during which
most of the mass loss occurs). After the mass loss epoch, $\tau_{{\rm loss}}$,
we continue the simulation, at constant mass up to 10 Myrs to check
for longer term stability. In cases where our stellar evolution calculation
shows that instability occurred only after both the primary and secondary
in the inner binary evolved, we simulated the system beginning only
at the second stage of mass loss (i.e. from the secondary), after
the primary became a white dwarf.
We simulated 100 realizations of each observed triple system found
to be unstable in Section \ref{sec:basic-estimate}; each realization
differed only in the initial orbital phase of the stars. Since we
do not account for mass transfer, we simulated only systems in which
the inner binary separation was larger than 15 AU (these comprise
nearly half of the observed sample). We ran $300$ different system
configurations, resulting in a total of $3\times10^{4}$ runs.
To identify systems that become unstable, we require that either one
star is ejected from the system, or an exchange occurs between the
tertiary and one of the inner binary components. We find that
approximately $0.33$ of the systems found to be unstable in our simplified
calculations become unstable in our simulations over a 0.5 Myr run.
Although more triples are likely to show exchanges and escapes over
longer time scales (for example we find approximately $55$\% were destabilized
within 10 Myrs), we adopt the short timescale simulation result, since our primary concern is with collisions, which are
much more likely if they occur while the primary is on the AGB.
We take the most conservative estimate possible by assuming that the results from the three body integrations
are more representative of the entire sample. We thus reduce the fraction of unstable observed
systems found in Section \ref{sec:basic-estimate} ($f_{\rm des,o}=0.035$) by a factor of $0.33$; we
adopt as the fraction of destabilized systems $f_{{\rm des}}=0.035\times0.33\simeq0.012$.
\section{Stellar collisions during the TEDI (calculating $f_{\rm col}$)}
\label{sec:coll} Having calculated the expected fraction of triple
systems that are unstable to the TEDI, we now estimate the fraction
of such systems that might suffer physical collisions. We show in the appendix that applying the results of previous simulations for both single star - binary encounters and unstable triple systems
gives the fraction of unstable systems that undergo a collision as $0.49\pm0.03$. The analytically derived collision rate for unstable triples is significantly higher than that for random encounters because an unstable triple typically undergoes multiple close approaches before the system dissolves, and because the stellar radius at the time of collision is so large. Because the analytic results are based on initially unstable systems covering a limited part of the the three body problem phase-space, we determine a more conservative collision rate based on our three body integrations (see below).
\subsection{Collisions in three-body simulations}
We define collisions in our three body integrations as those cases where the
closest approach between the primary and another star is smaller than
the typical stellar radius of the evolving star, $R_{{\rm des}}$
where $R_{{\rm des}}$ is defined to be the stellar radius at the
point of instability, when the system first satisfies $Q<Q_{{\rm st}}$.
We only classify close approaches as collisions if they occur within
the 0.5 Myr mass loss timescale when the star is likely bloated. Figure
\ref{fig:close-approach} shows the cumulative distribution of closest
approaches amongst the unstable triples that we simulated. The distribution
of radii of the evolved stars that go unstable ($R_{{\rm des}}$
defined before) is also shown. Most stars involved
in the collisions are highly evolved AGB stars, with radii in the
range of tens to hundreds of AU.
We find the collision rate between the evolving star and either of
the other stellar components to be $f_{{\rm col}}\simeq0.09\pm0.03$
amongst the unstable systems, where the error bars are the 1 $\sigma$
statistical uncertainty. Note that this fraction is smaller than that obtained from the simplified semi-analytic approximation
discussed in the appendix. However, detailed analysis of the distribution
of closest approaches between stars during the triple evolution, suggests
that the simple linear approximation in Eq. (\ref{eq:collision-prob})
is inaccurate; we find that the closest approach statistics depend more complexly on the
various orbital properties of the system. We therefore adopt the collision
fraction found in our simulations for our subsequent calculations.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.45]{fig3}
\par\end{centering}
\centering{}\caption{\label{fig:close-approach}{Stellar radii of evolved stars, and closest
approaches between stars in destabilized systems.} Dashed red lines indicate the distribution
of closest approaches between the evolved star and another member
of an evolving triple during the mass loss phase
(taken to be $\tau_{{\rm loss}}=0.5$ Myr). Solid blue lines show
the distribution of the stellar radii of the evolved mass losing star
at the point of instability. Inset shows the cumulative distributions.
Results were obtained from the three-body simulations of systems that
became unstable.}
\end{figure}
The sample chosen for the three body integrations excluded close inner binaries (with SMAs smaller than 15 AU),
because we did not account for mass transfer. Physical collisions are even more likely for these systems due to the monotonic
dependence of the close approach probability on the inner binary separation
(seen both in our integrations and in Eq. \ref{eq:collision-prob}.) Thus the neglected systems likely have a higher collision probability.
Because we adopt the same collision fraction for more compact triples,
these rates should be considered a lower bound. Compact systems comprise
about half of our observed sample, so the neglect of these systems
in our calculations would change our results by at most a factor of
two.
Note that we cannot correctly follow
the full evolution of systems in which tidal effects dominate the
dynamics at any point. Nevertheless, for the short timescale of the
TEDI, tidal effects become important when the closest approach is
comparable to the stellar radius, and can be considered a sub-type
of collision. Moreover, because most collisions occur when the physical
radius is comparable to the tidal disruption radius, our neglect of
tidal effects should not significantly alter our calculated collision
rates. Determining the configurations of post-collision systems is beyond the scope of this work.
\section{Galactic Collision rate due to the TEDI}
\label{sec:rate} In the previous sections we provided estimates
for the fraction of destabilized systems, $f_{\rm des}$, and the fraction
of colliding systems, $f_{\rm col},$ among these systems. We can now
make use of these results to estimate the overall rate of stellar
collisions due to the TEDI. Taking the estimate of the collision
rate together with the calculated fraction of destabilized triples,
we find that the collision rate between evolved and companion stars/compact
objects in a Milky Way like galaxy, over the age of the Universe,
$t_{{\rm H}}$ is: \begin{eqnarray}
& \Gamma_{{\rm col}} & =\frac{N_{\star}\times f_{{\rm evolve}}\times f_{{\rm triple}}\times f_{{\rm des}}\times f_{{\rm col}}}{t_{{\rm H}}}=\\
& = & 1.2\times10^{-4}\left(\frac{N_{{*}}}{1.5\times10^{11}}\right)\nonumber \\
& \times & \left(\frac{f_{{\rm evolve}}}{0.1}\right)\left(\frac{f_{{\rm triple}}}{0.1}\right)\left(\frac{f_{{\rm des}}}{0.012}\right)\left(\frac{f_{{\rm col}}}{0.09}\right)\,{\rm {yr}^{-1},\label{eq:collision-rate}}\nonumber \end{eqnarray}
where $N_{\star}$ is the number of stars in the
Galaxy (\citealp{bin+08}; taking the mean stellar mass to be $0.3$
M$_{\odot}$; \citealp{kro+01}), $f_{{\rm evolve}}$ is the observed
fraction of evolved stars (the fraction of WDs, \citealp{bel+02}
can serve as a proxy for their total number), $f_{{\rm triple}}$
is the fraction of triple systems (where we conservatively adopt the
triple fraction of $f_{{\rm triple}}=0.1$, found for F/G/K stars,
\citealp{rag+10}, for stars of all masses). We adopt $f_{{\rm des}}$
(the fraction of triple systems that become unstable) and $f_{{\rm col}}$
(the fraction of unstable systems in which a collision occurs) from
our calculations in the previous sections.
For comparison, the collision rate in the GC M15, one of the densest
in the galaxy ($n_{\rm c}>10^{6}$ M$_{\odot}$ pc$^{-3}$; \citealp{umb+08}),
was estimated to be $\Gamma=2\pm1\times10^{-7}$ yr$^{-1}$ (\citealp{umb+08};
see also refs. \citealp{sha99,lee+10}). The total collision rate
in Galactic GCs, which is dominated by such clusters (i.e. over 100 GCs exist in the Galaxy, but the dominant contribution to collsions come from the most dense, massive clusters), is therefore
$\Gamma_{{\rm gal}}=4\times10^{-6}\,(N_{{\rm GC}}/20)$ yr$^{-1}$,
where ${\rm N}_{{\rm {\rm GC}}}\approx20$ is an estimate of the number
of similar GCs in the galaxy as massive and dense as M15 \citep{gne+97}.
Collisions due to destabilized triples are therefore about $30\,(N_{{\rm GC}}/20)$
times more frequent than collisions due to random encounters in GCs.
The number of extragalactic GCs is typically proportional to the host
galaxy mass \citep{bro+06}, suggesting a similar ratio of field to
GC collisions in other galaxies.
Note that the collsion rate due to TEDI \emph{inside} GCs scales with the total number of stars in GCs and does not depend on their specific density. Taking approximately 150 GCs exiting in the Galaxy with about $10^6$ stars in each cluster, we find the total collsion rate in GCs due to TEDI to be at most 0.1 percent of the total Galactic rate, and only about 3 percente of the collsion rate in GC due to random encounters. In other words, inside GCs, sellar collsions are dominated by random encounters rather than by triple stellar evolution.
\section{Consequences of the TEDI}
\label{sec:outcomes}
The TEDI scenario has a wide variety of possible outcomes.
Many of these are very similar to those resulting from binary-single
and single-single close encounters between stars and/or compact objects
that occur in globular clusters. However, the TEDI scenario provides a pathway to these
exotic scenarios previously reserved for dense stellar populations.
In the following we discuss various outcomes of the TEDI and
highlight their unique aspects, either in terms of their qualitative
difference from typical encounters in clusters, or their specific relations
to systems in the field. We first briefly overview several
potential outcomes, and then discuss in more detail a specific example,
providing a novel formation scenario for Sirius like eccentric WD
binaries.
\textbf{\emph{\underbar{Stellar collisions in the field:}}}\
As discussed above, the TEDI produces a high rate of stellar collisions
\emph{in the field}, most of which involve AGB
stars; these were previously thought to comprise a
negligible fraction of all collisions. These encounters might be observed
as intermediate luminosity optical transients (with total energy ranging
between $10^{46}-10^{48}$ ergs; these could be detectable in current
and future optical transient surveys; \citealp{kul+09}), possibly
similar to those suggested to occur due to mergers or through tidal/off
axis collisions in eccentric binaries \citep{kas+10,smi11}. Beside
potentially producing immediate energetic transient events, the TEDI
can produce a wide variety of collision products, including merged
stars and blue stragglers as well as peculiar binaries.
Numerical simulations of close encounters with stars bloated to $100 \, R_{\odot}$
show that the resulting collisional and/or
tidal captures lead to the formation of highly eccentric binaries
\citep{bai+99,yam+08}. Such collisions might also lead to a common
envelope phase and the formation of a close binary. The TEDI scenario
therefore predicts the existence of a sub-population of close, but
eccentric, evolved binaries in the field. The TEDI can explain the
puzzling orbits of WD binaries such as the nearby Sirius system (see
below), as well as highly eccentric Barium binary systems (expected
to form through mass transfer in a close binary). These systems pose
a problem for standard binary stellar evolution models, which produce
only \emph{circular} close binaries \citep{bon+08,kar+00,izz+10} (but see \citeauthor{der+12} (2012), and references therein, for some alternative suggestions).
The peculiar close WD-blue
straggler binary (KOI-74) found by Kepler, which may have formed in a highly
eccentric evolved binary system \citep{dis11}, could easily be produced by the TEDI. Similarly, TEDI
evolution can lead to the formation of eccentric WD-WD binaries, which
could provide unique sources for future gravitational wave detection
missions, and were predicted to exist only in GCs \citep{wil+07}.
Our findings suggest that there could be a few times more such sources
in the field than estimated to exist in GCs.
\textbf{\emph{\underbar{Neutron stars/black hole binaries: }}} Mass
loss from primaries more massive than $8$ M$_{\odot}$ which undergo the TEDI, might form a close
eccentric binary before the supernova explosion of the evolving massive
star. Since such binaries have a higher probability of surviving the
SN explosion \citep{hil83}, the TEDI might aid in the retention of
neutron stars and black holes in binaries, and the later formation
of X-ray and pulsar binaries. Such triples could later evolve,
during mass loss/mass transfer from the other triple components. Possible
outcomes include instability and thus collisions with a neutron star/black
hole, or even exchanges. See \citet{fre+11} and \citet{por+11} for
the latter possibility, but note that these studies did not account
for the possibility of a physical collision during the chaotic evolutionary
phase.
\textbf{\emph{\underbar{Envelope stripping through collisions:}}} A
collision between a neutron star and its evolving companion could
result in the tidal disruption of the latter. The remnants of this
star might form an accretion disk surrounding the neutron star, which
could provide the necessary mass and angular momentum to create a
recycled pulsar \citep{ver+87}. If the third companion is ejected
during the course of the instability, the pulsar would appear as an
isolated recycled pulsar in the field. If the third companion is retained
it will be observed as a wide and likely eccentric recycled binary
pulsar in the field. The latter scenario provides an additional channel
for the formation of the pulsar J1903+0327 \citep{cha+08,fre+11,por+11}.
Another possibility is the formation of a recycled pulsar with a stripped
companion on an eccentric orbit, if the companion's core stays intact.
There are many observed peculiar stars and compact remnants
thought to form through envelope stripping by a close companion, such
as hot sub-dwarf stars \citep{han+02}, helium white dwarfs \citep{mar+95},
type Ib/c supernova progenitors \citep{yoo+10}. However, cases are
found where these peculiar objects are observed to be single or to
have a wide companion, configurations that are inconsistent with the
close binary progenitor scenario. Such cases might be explained
through a TEDI induced collision.
Further studies are required in order to quantitatively explore each scenario.
\section{Formation of the Sirius WD-binary system}
One of the closest WD binaries to Earth is the Sirius system composed
of a MS star of mass $2\, M_{\odot}$ and a WD companion of about
a Solar mass \citep{lie+05}. The SMA of 20 AU and and high eccentricity
of $0.59$ make this binary quite peculiar, as standard stellar evolution
scenarios suggest that the progenitor of such a close binary would
have circularized during stellar evolution of the system \citep{bon+08,izz+10}.
However, the Sirius system could have formed through the TEDI; alternative scenarios are also in development (\citeauthor{der+12} 2012; P. Eggleton, private communication 2012).
To assess the likelihood that Sirius formed via this mechanism, we
ran 6500 three-body simulations of the dynamical evolution of initially
stable triple systems undergoing mass loss. For these systems, we follow the exact
mass loss and radius evolution as obtained from the SSE code. The
primary starts out with $M=5.05\, M_{\odot}$, and ends as a $0.98\, M_{\odot}$
WD. We evolve the systems analytically until the primary
has reached about $4$ M$_{\odot}$ and begins the rapid mass loss
phase. We assume that the secondary and tertiary components have undergone
no evolution. We explore a range of inner binary separations from
$15-30$ AU, outer binary separations from $3.5-5.5$ times the inner
separation, eccentricities from $0-0.25$, and companion masses from
$0.6-5.5\, M_{\odot}$. One component is always constrained to be
$2\, M_{\odot}$, the mass of Sirius A. The minimum separation of
the inner binary is chosen to avoid mass transfer. To account for
variations in the chaotic evolution at late times, we run each set
of parameters starting from five different initial orbital phases.
We limit the initial inclination of the systems to $i=0.1$ radian.
In Fig. \ref{fig:Sirius}, we show examples of final binary configurations
of unstable systems where one of the stars was ejected and a Sirius
like binary was left behind. Also shown is the observed SMA and eccentricity
of the Sirius system. The Sirius binary falls within one of three
populated regions of final configurations, suggesting that Sirius
could have formed through the TEDI process. The inner cluster of eccentric
systems is formed via ejection of the wider tertiary companion, while
the outer cluster of eccentric systems formed via an exchange followed
by the ejection of the initial inner binary companion to the WD. The
moderately eccentric systems are composed of bound triple systems
that underwent only mild eccentricity excitation.
We note that if Sirius and other objects formed through this mechanism, it might
be possible to identify its ejected third star, as a
very wide companion (e.g. \citealp{jia+10}) or through similar methods
used to locate stars that originally formed close to the Sun \citep{por+09,bob+11}.
\begin{figure}
\centering{}\includegraphics[scale=0.45]{figS3} \caption{\label{fig:Sirius} The initial and final eccentricities and semi-major
axes of triple systems evolved using our three-body simulations. Circles
represent the initial orbital configuration of a given pair of stars
in the triple. Note that each circle corresponds not only to multiple
realizations of a given system, but also to different masses for the
third component. Lines connect the initial and final configurations,
which are marked by stars. Each initial configuration can correspond
to multiple final outcomes due to the chaotic nature of the instability.
Final configurations similar to the observed Sirius system (marked
with a red triangle), or even more eccentric are a typical outcome
for systems with initial SMAs in the range 15-30 AU. Also shown are
binaries formed via an exchange between the inner and outer companions,
followed by ejection of the initial inner companion.}
\end{figure}
\section{Triple stellar evolution coupled with secular Kozai cycles}
\label{sec:koz} Triple systems in which the inner and outer orbits
have a high mutual inclination may secularly evolve through Kozai
cycles, in which the inclination and eccentricity of the inner binary
can periodically/quasi-periodically change by a large amplitude \citep{koz62,lid62},
and even flip from prograde to retrograde orbits (\citealp{nao+10};
and vice versa). It was already suggested that the coupling of secular
Kozai evolution with dissipative processes such as tidal friction
or GW emission could play an important role in the evolution of the
inner binaries (e.g. \citealp{maz+79,egg+06b,per+09,tho11} and references
therein). Mass loss evolution in triple systems could also couple
with this secular Kozai process, and change its outcomes. Our simulations
suggest that mass loss does not quench Kozai cyles. The high eccentricities
that could be induced through the Kozai mechanism
can therefore lead to close encounters between the components of the
inner binary during peri-center approach prior to destabilization
during the AGB stage. Kozai evolution could lead to close interactions
between triple components at a late epoch in the evolution, even if
the original triple did not interact while on the main sequence (see also \citeauthor{sha+12} 2012).
The coupled stellar-dynamical evolution TEDI stable, Kozai susceptible,
systems is beyond the scope of this paper, and will be explored elsewhere.
\section{Discussion and summary}
\label{sec:summary} In this paper we have studied the triple evolution
dynamical instability (TEDI), in which mass loss in an evolving triple
system leads to orbital instability. The chaotic evolution that ensues
gives rise to close encounters between the stars and/or exchanges
between them. Such evolution typically ends in a collision between
two of the stars or the ejection of one of the components.
This instability can lead to the dynamical formation of eccentric compact
binaries, and produces a high stellar collision rate between stars
in the field. The TEDI scenario results in a collision
rate in the field of approximately $10^{-4}$ yr$^{-1}$ per Milky-Way
Galaxy, about 30 times higher than the collision rate in all Galactic
GCs. This high rate is attributable to two features of the TEDI. First, the chaotic orbital evolution
results in binary-single star like resonant encounters in the field,
which provide numerous opportunities for collisions. Secondly, the
cross section for physical collisions is much larger than in typical
random encounters. The mass loss that precipitates the instability
reaches its maximum at the AGB phase, and therefore the instability
typically occurs when the stellar radius is at its peak, a few hundred
Solar radii (as seen in Fig. \ref{fig:close-approach}). The cross
section for collisions is therefore increased proportionally. In contrast,
random collisions in GCs are heavily dominated by MS and slightly
evolved stars \citep{lom+06,umb+08} with radii of $1-10$ Solar radii.
The discrepancy between the peak collision phases is due to the $10^{5}-10^{6}$
year duration of the AGB phase. This timescale is a tiny fraction
of a star's life, and thus short compared to the random encounter
timescale, even in the densest environment. In contrast, the close
encounters during the TEDI always coincide with the AGB phase. The
TEDI model therefore requires not only a revision of the rate of stellar
collisions, but also a reconsideration of their nature; most stellar
collisions involve AGB stars.
In addition to enhancing the number of stellar collisions,
the TEDI suggests that various types of exotic binaries collisionally formed inside GCs (X-ray
binaries, cataclysmic variables, single and binary recycled pulsars,
eccentric WD binaries, including eccentric inspiraling WD-WD GW sources,
and stripped stars; \citealt{fab+75,poo+03,bai+99,lom+06,wil+07} ) can similarly \emph{collisionally} form through TEDI \emph{outside} of dense clusters; adding an additional evolutionary channel for the formation of such binaries in the field.
In particular, the TEDI provides a novel explanation for the origin of
WD binaries observed in puzzling configurations, such as the well known
Sirius system. The exotic stellar systems produced through
these pathways have an outsize impact on our understanding of fundamental
physics, and the evolution of stars in the field. They serve as the
main sources for gravitational wave emission, and as unique probes
of the structure of compact objects.
This study focused on the TEDI scenario for triple
stellar systems. However, a system comprised of a stellar binary and
a planet on a circumstellar orbit around an evolving star can also
suffer a similar fate \citep{per10,per11}. The evolution of such
systems has recently been described by \citet{kra+12}. In addition,
the TEDI scenario can be extended to higher multiplicity systems (note
that $1/4$ of all multiple systems are quadruples or higher order
multiples \citealp{rag+10}).
Such systems can produce an even wider variety of outcomes than triple
systems, and in general mimic almost any type of outcome that is expected
to occur following close encounters in GCs.
\acknowledgments We thank J. Hurley for providing the openly accessible
SSE and BSE stellar evolution codes, S. Tremaine for helpful discussion
on wide binaries in the context of this work, as well as F. Rasio and P. Eggelton for helpful
comments on an earlier version of this manuscript. We also thank the referee, Christopher Tout, for helpful
suggestions which much improved the clarity of this work. HBP acknowledges
support from the BIKURA (FIRST) Israel Science Foundation and the
CfA fellowship through the Harvard-Smithsonian Center for Astrophysics.
KMK is supported by the Institute for Theory and Computation Fellowship
through the Harvard College Observatory.
|
2,877,628,089,514 | arxiv | \section{Introduction}
Violation of the combined charge-parity symmetry ($CP$ violation) in the standard model (SM) arises from a single irreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix~\cite{Cabibbo,KM}. A main objective of the Belle experiment at KEK, Japan, is to over-constrain the unitarity triangle of the CKM matrix related to $B_{u,d}$ decays. This permits a precision test of the CKM mechanism for $CP$ violation as well as the search for new physics (NP) effects. Mixing-induced $CP$ violation in the $B$ sector has been clearly established by Belle~\cite{jpsiks_Belle1,jpsiks_Belle2} and BaBar~\cite{jpsiks_BABAR1,jpsiks_BABAR2} in the $\bar b \rightarrow \bar c c \bar s$ induced decay $\ensuremath{B^{0}} \rightarrow J/\psi \ensuremath{K^{0}}$. There are many other modes that may provide additional information on various $CP$ violating parameters.
\begin{figure}
\centering
\includegraphics[height=120pt,width=!]{pipi_tree.eps}
\includegraphics[height=120pt,width=!]{pipi_peng.eps}
\put(-444,110){(a)}
\put(-218,110){(b)}
\caption{Leading-order Feynman diagrams for \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ decays. (a) depicts the dominant first-order amplitude (tree) while (b) shows the second-order loop (penguin) diagram. In the penguin diagram, the subscript $x$ in $V_{xb}$ refers to the flavor of the intermediate-state quark $(x=u,c,t)$.}
\label{fig_hh}
\end{figure}
Decays that proceed predominantly through the $\bar b \rightarrow \bar u u \bar d$ transition are sensitive to the interior angle of the unitarity triangle $\ensuremath{\phi_{2}} \equiv \arg(-V_{td}V^{*}_{tb})/(V_{ud}V^{*}_{ub})$\footnote{Another notation, $\alpha \equiv \arg(-V_{td}V^{*}_{tb})/(V_{ud}V^{*}_{ub})$, also exists in literature.}. This paper describes a measurement of $CP$ violation parameters in \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ decays, whose dominant amplitudes are shown in Fig.~\ref{fig_hh}. Belle, BaBar and LHCb have reported time-dependent $CP$ asymmetries in related modes including $\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{-}}$~\cite{pipi_Belle,pipi_BaBar,pipi_LHCb}, $(\rho \pi)^{0}$~\cite{rhopi_Belle,rhopi_BaBar}, $\rho^{+} \rho^{-}$~\cite{rhorho_Belle,rhorho_BaBar} and $\ensuremath{a_{1}^{\pm}} \pi^{\mp}$~\cite{a1pi_Belle,a1pi_BaBar}.
The decay of the \ensuremath{\Upsilon(4S)}\ can produce a \ensuremath{B^{0} \bar{B}^{0}}\ pair in a coherent quantum-mechanical state, from which one meson (\ensuremath{B^{0}_{\rm Rec}}) may be reconstructed in the $\ensuremath{\pi^{+}} \ensuremath{\pi^{-}}$ decay mode. This decay mode does not determine whether the \ensuremath{B^{0}_{\rm Rec}}\ decayed as a \ensuremath{B^{0}}\ or as a \ensuremath{\bar{B}^{0}}. The $b$ flavor of the other $B$ meson (\ensuremath{B^{0}_{\rm Tag}}), however, can be identified using information from the remaining charged particles and photons. This dictates the flavor of \ensuremath{B^{0}_{\rm Rec}}\ as it must be opposite that of the \ensuremath{B^{0}_{\rm Tag}}\ flavor at the time \ensuremath{B^{0}_{\rm Tag}}\ decays. The proper time interval between \ensuremath{B^{0}_{\rm Rec}}\ and \ensuremath{B^{0}_{\rm Tag}}, which decay at time $t_{\rm Rec}$ and $t_{\rm Tag}$, respectively, is defined as $\ensuremath{\Delta t} \equiv t_{\rm Rec} - t_{\rm Tag}$ measured in the \ensuremath{\Upsilon(4S)}\ frame. For the case of coherent \ensuremath{B^{0} \bar{B}^{0}}\ pairs, the time-dependent decay rate for a $CP$ eigenstate when \ensuremath{B^{0}_{\rm Tag}}\ possesses flavor $q$, where \ensuremath{B^{0}}\ has $q=+1$ and \ensuremath{\bar{B}^{0}}\ has $q=-1$, is given by
\begin{equation}
{\cal P}(\ensuremath{\Delta t}, q) = \frac{e^{-|\ensuremath{\Delta t}|/\ensuremath{\tau_{B^{0}}}}}{4\ensuremath{\tau_{B^{0}}}} \biggl\{1 + q \biggl[\ensuremath{{\cal A}_{CP}} \cos \ensuremath{\Delta m_{d}} \ensuremath{\Delta t} + \ensuremath{{\cal S}_{CP}}\sin \ensuremath{\Delta m_{d}} \ensuremath{\Delta t}\biggr]\biggr\}.
\end{equation}
Here, \ensuremath{\tau_{B^{0}}}\ is the \ensuremath{B^{0}}\ lifetime and \ensuremath{\Delta m_{d}}\ is the mass difference between the two mass eigenstates of the neutral $B$ meson. This time dependence assumes $CPT$ invariance, no $CP$ violation in the mixing, and that the difference in decay rates between the two mass eigenstates is negligible. The parameter \ensuremath{{\cal A}_{CP}}\ measures the direct $CP$ violation, while \ensuremath{{\cal S}_{CP}}\ is a measure of the amount of mixing-induced $CP$ violation.
In the limit that only the dominant tree amplitude contributes, no flavor-dependent direct $CP$ violation is expected and \ensuremath{{\cal S}_{CP}}\ is $\sin2\ensuremath{\phi_{2}}$. However, in the \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ final state and other $\bar b \rightarrow \bar u u \bar d$ self-conjugate modes, the value of \ensuremath{\phi_{2}}\ is shifted by an amount $\Delta \ensuremath{\phi_{2}}$, due to the presence of additional penguin contributions that interfere with the dominant tree contribution (see Fig.~\ref{fig_hh}). Thus, the observable mixing-induced $CP$ parameter becomes $\ensuremath{{\cal S}_{CP}} = \sqrt{1 - \ensuremath{{\cal A}_{CP}}^2}\sin (2 \ensuremath{\phi_{2}} + 2 \Delta \ensuremath{\phi_{2}})$.
Despite penguin contamination, it is still possible to determine $\ensuremath{\phi_{2}}$ in $\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{-}}$ with an $SU(2)$ isospin analysis~\cite{theory_su2} by considering the set of $B \rightarrow \pi\pi$ decays into the three possible charge states for the pions. Here, the two pions in $\ensuremath{B^{+}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{0}}$ decays must have a total isospin of $I=1$ or $I=2$, since $I_{3} = 1$. For the penguin contributions, only $I=0$ or $I=1$ is possible because the gluon is an isospin singlet carrying $I=0$. However, $I=1$ is forbidden by Bose-Einstein statistics; thus, strong loop decays cannot contribute and hence $\ensuremath{B^{+}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{0}}$ decays only through the tree diagram in the limit of negligible electroweak penguins.
The complex $\ensuremath{B^{0}} \rightarrow \pi\pi$ and $\ensuremath{\bar{B}^{0}} \rightarrow \pi\pi$ decay amplitudes obey the relations
\begin{equation}
A_{+0} = \frac{1}{\sqrt{2}}A_{+-} + A_{00}, \;\;\;\; \bar{A}_{-0} = \frac{1}{\sqrt{2}}\bar{A}_{+-} + \bar{A}_{00},
\label{eq_iso}
\end{equation}
respectively, where the subscripts refer to the combination of the pion charges. The decay amplitudes can be represented as the triangles shown in Fig.~\ref{fig_iso}. As $\ensuremath{B^{+}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{0}}$ is a pure tree mode, these triangles share the same base, $A_{+0}=\bar{A}_{-0}$, and $\Delta \ensuremath{\phi_{2}}$ can be determined from the difference between the two triangles. These triangles and \ensuremath{\phi_{2}}\ can be fully determined from the branching fractions, ${\cal B}(\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{-}})$, ${\cal B}(\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{0}}\piz)$ and ${\cal B}(\ensuremath{B^{+}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{0}})$, and the $CP$ violation parameters, $\ensuremath{{\cal A}_{CP}}(\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{+}} \ensuremath{\pi^{-}})$, $\ensuremath{{\cal S}_{CP}}(\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{+}}\ensuremath{\pi^{-}})$ and $\ensuremath{{\cal A}_{CP}}(\ensuremath{B^{0}} \rightarrow \ensuremath{\pi^{0}}\piz)$. This method has an eightfold discrete ambiguity in the determination of \ensuremath{\phi_{2}}, which arises from the four triangle orientations about $A_{+0}$ and the two solutions of \ensuremath{\phi^{\rm eff}_{2}}\ in the measurement of \ensuremath{{\cal S}_{CP}}.
\begin{figure}
\centering
\includegraphics[height=100pt,width=!]{iso_anal.eps}
\caption{Complex isospin triangles from which $\Delta \ensuremath{\phi_{2}}$ can be determined.}
\label{fig_iso}
\end{figure}
Belle, BaBar and LHCb have reported measurements~\cite{pipi_Belle,pipi_BaBar,pipi_LHCb}, summarized in Table~\ref{tab_hh_prev}, of the $CP$ violation parameters reported here. The previous Belle measurements were based on a sample of 535 million \ensuremath{B \bar{B}}\ pairs and are superseded by the analysis presented here.
\begin{table}
\centering
\caption{Summary of $CP$ violation parameters obtained by Belle~\cite{pipi_Belle}, BaBar~\cite{pipi_BaBar} and LHCb~\cite{pipi_LHCb}. For all parameters, the first uncertainty is statistical and the second is systematic. The Belle value for \ensuremath{{\cal A}_{CP}}\ is marginally consistent ($1.9\sigma$) with the BaBar and LHCb measurements.}
\begin{tabular}
{@{\hspace{0.5cm}}c@{\hspace{0.25cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}}}
\hline \hline
Parameter & Belle & BaBar & LHCb\\
& ($535\times10^6$ \ensuremath{B \bar{B}}\ pairs) & ($467\times10^6$ \ensuremath{B \bar{B}}\ pairs) & (0.7 fb$^{-1}$)\\
\hline
$\ensuremath{{\cal A}_{CP}}(\ensuremath{B^{0} \to \pi^{+}\pi^{-}})$ & $+0.55 \pm 0.08 \pm 0.05$ & $+0.25 \pm 0.08 \pm 0.02$ & $+0.11 \pm 0.21 \pm 0.03$\\
$\ensuremath{{\cal S}_{CP}}(\ensuremath{B^{0} \to \pi^{+}\pi^{-}})$ & $-0.61 \pm 0.10 \pm 0.04$ & $-0.68 \pm 0.10 \pm 0.03$ & $-0.56 \pm 0.17 \pm 0.03$\\
\hline \hline
\end{tabular}
\label{tab_hh_prev}
\end{table}
In Sec.~\ref{Data Set And Belle Detector}, we briefly describe the data set and Belle detector. We explain the selection criteria used to identify signal candidates and suppress backgrounds in Sec.~\ref{Event Selection}, followed by the fit method used to extract the signal component in Sec.~\ref{Event Model}. In Sec.~\ref{Fit Result}, the results of the fit are presented along with a discussion of the systematic uncertainties in Sec.~\ref{Systematic Uncertainties}. Finally, our conclusions are given in Sec.~\ref{Conclusion}.
\section{Data Set And Belle Detector}
\label{Data Set And Belle Detector}
This measurement of the $CP$ violation parameters in \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ decays is based on the final data sample containing $772 \times 10^{6}$ \ensuremath{B \bar{B}}\ pairs collected with the Belle detector at the KEKB asymmetric-energy \ensuremath{e^{+} e^{-}}\ ($3.5$ on $8~{\rm GeV}$) collider~\cite{KEKB}. At the \ensuremath{\Upsilon(4S)}\ resonance ($\sqrt{s}=10.58$~GeV), the Lorentz boost of the produced \ensuremath{B \bar{B}}\ pairs is $\beta\gamma =0.425$ nearly along the $+z$ direction, which is opposite the positron beam direction. We also use a $100 \; {\rm fb}^{-1}$ data sample recorded at 60 MeV below the \ensuremath{\Upsilon(4S)}\ resonance, referred to as off-resonance data, for continuum ($e^+ e^- \to q\bar{q}$, where $q = d,\ u,\ s,\ c$) background studies.
The Belle detector is a large-solid-angle magnetic
spectrometer that consists of a silicon vertex detector (SVD),
a 50-layer central drift chamber (CDC), an array of
aerogel threshold Cherenkov counters (ACC),
a barrel-like arrangement of time-of-flight
scintillation counters (TOF), and an electromagnetic calorimeter (ECL)
comprising CsI(Tl) crystals located inside
a superconducting solenoid coil that provides a 1.5~T
magnetic field. An iron flux-return located outside of
the coil is instrumented to detect $K_L^0$ mesons and to identify
muons (KLM). The detector
is described in detail elsewhere~\cite{Belle}.
Two inner detector configurations were used. A 2.0-cm-radius beampipe
and a three-layer silicon vertex detector (SVD1) were used for the first sample
of $152 \times 10^6 B\bar{B}$ pairs, while a 1.5-cm-radius beampipe, a four-layer
silicon detector (SVD2) and a small-cell inner drift chamber were used to record
the remaining $620 \times 10^6 B\bar{B}$ pairs~\cite{svd2}. We use a GEANT-based
Monte Carlo (MC) simulation to model the response of the detector and to determine
its acceptance~\cite{GEANT}.
\section{Event Selection}
\label{Event Selection}
The decay channel \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ is reconstructed from two oppositely charged tracks. Charged tracks are identified using a loose requirement on the distance of closest approach with respect to the interaction point (IP) along the beam direction, $|dz| < 4.0 \; {\rm cm}$, and in the transverse plane, $dr < 0.4 \; {\rm cm}$. Additional SVD requirements of at least two $z$ hits and one $r-\phi$ hit~\cite{ResFunc} are imposed on all charged tracks so that a good quality vertex of the reconstructed $B$ candidate can be determined. Using information obtained from the CDC, ACC and TOF, particle identification (PID) is determined from a likelihood ratio ${\cal L}_{i/j} \equiv {\cal L}_{i}/({\cal L}_{i} + {\cal L}_{j})$. Here, ${\cal L}_{i}$ (${\cal L}_{j}$) is the likelihood that the particle is of type $i$ ($j$). To suppress background due to electron misidentification, ECL information is used to veto particles consistent with the electron hypothesis. The PID ratios of the two charged tracks \ensuremath{{\cal L}^{\pm}_{K/\pi}}, are used in the fit model to discriminate among the three possible two-body channels: $\ensuremath{B^{0}} \to \ensuremath{\pi^{+}} \ensuremath{\pi^{-}}$, $\ensuremath{B^{0}} \to \ensuremath{K^{+}} \ensuremath{\pi^{-}}$ and $\ensuremath{B^{0}} \to \ensuremath{K^{+}} \ensuremath{K^{-}}$.
Reconstructed $B$ candidates are identified with two nearly uncorrelated kinematic variables: the beam-energy-constrained mass $\ensuremath{M_{\rm bc}} \equiv \sqrt{(E^{\rm CMS}_{\rm beam})^{2} - (p^{\rm CMS}_{B})^{2}}$ and the energy difference $\ensuremath{\Delta E} \equiv E^{\rm CMS}_{B} - E^{\rm CMS}_{\rm beam}$, where $E^{\rm CMS}_{\rm beam}$ is the beam energy and $E^{\rm CMS}_{B}$ ($p^{\rm CMS}_{B}$) is the energy (momentum) of the $B$ meson, all evaluated in the $e^+ e^-$ center-of-mass system (CMS). The $B$ candidates that satisfy $\ensuremath{M_{\rm bc}} > 5.24 \; {\rm GeV}/c^{2}$ and $-0.20 \; {\rm GeV} < \ensuremath{\Delta E} < 0.15 \; {\rm GeV}$ are retained for further analysis.
The dominant background in the reconstruction of \ensuremath{B^{0}_{\rm Rec}}\ arises from continuum production. Since continuum events tend to be jetlike, in contrast to spherical \ensuremath{B \bar{B}}\ decays, continuum background can be distinguished from \ensuremath{B \bar{B}}\ signal using event-shape variables, which we combine into a Fisher discriminant \ensuremath{{\cal F}_{b \bar b/q \bar q}}~\cite{nazi_stuff}. The \ensuremath{B \bar{B}}\ training sample is taken from signal MC, while the \ensuremath{q \bar{q}}\ training sample is from the off-resonance data sample. The Fisher discriminant is then constructed from the variables described in Ref.~\cite{a1pi_Belle}. The variable providing the strongest discrimination against continuum is the cosine of the angle between the \ensuremath{B^{0}_{\rm Rec}}\ thrust direction (TB) and the thrust of the tag side (TO) $|\cos \theta_{\rm TB, TO}|$. The thrust is defined as the vector that maximizes the sum of the longitudinal momenta of the particles. For a \ensuremath{B \bar{B}}\ event, the pair is nearly at rest in the CMS, so the thrust axis of \ensuremath{B^{0}_{\rm Rec}}\ is uncorrelated with the thrust axis of \ensuremath{B^{0}_{\rm Tag}}. In a \ensuremath{q \bar{q}}\ event, on the other hand, the decay products align along two nearly back-to-back jets, so the two thrust axes tend to be collinear. Before training, a loose requirement of $|\cos \theta_{\rm TB, TO}| < 0.9$ is imposed that retains 90\% of the signal while rejecting 50\% of the continuum background. The range of the Fisher discriminant $-3 < \ensuremath{{\cal F}_{b \bar b/q \bar q}} < 2$ encompasses all signal and background events.
Backgrounds from charm ($b \to c$) decays are found to be negligible and are thus not considered, while charmless ($b \to u,d,s$) decays of the $B$ meson may contribute, though rarely in the same region of \ensuremath{M_{\rm bc}}\ and \ensuremath{\Delta E}\ where signal is present.
As the \ensuremath{B^{0}_{\rm Rec}}\ and \ensuremath{B^{0}_{\rm Tag}}\ are almost at rest in the \ensuremath{\Upsilon(4S)}\ CMS, the difference in decay time between the two $B$ candidates, $\Delta t$, can be determined approximately from the displacement in $z$ between the final state decay vertices as
\begin{equation}
\ensuremath{\Delta t} \simeq \frac{(z_{\rm Rec} - z_{\rm Tag})}{\beta \gamma c} \equiv \frac{\Delta z}{\beta \gamma c}.
\end{equation}
The vertex of reconstructed $B$ candidates is determined from the charged daughters, with a further constraint coming from the known IP. The IP profile is smeared in the plane perpendicular to the $z$ axis to account for the finite flight length of the $B$ meson in that plane. To obtain the \ensuremath{\Delta t}\ distribution, we reconstruct the tag side vertex from the tracks not used to reconstruct \ensuremath{B^{0}_{\rm Rec}}~\cite{ResFunc}. Candidate events must satisfy the requirements $|\ensuremath{\Delta t}| < 70 \; {\rm ps}$ and $h_{\rm Rec, Tag} < 500$, where $h_{\rm Rec, Tag}$ is the multitrack vertex goodness-of-fit, calculated in three-dimensional space without using the IP profile constraint~\cite{jpsiks_Belle2}. To avoid the necessity of also modeling the event-dependent observables that describe the \ensuremath{\Delta t}\ resolution in the fi
~\cite{Punzi}, the vertex uncertainty is required to satisfy the loose criteria $\sigma^{\rm Rec, Tag}_z < 200 \; \mu {\rm m}$ for multitrack vertices and $\sigma^{\rm Rec, Tag}_z < 500 \; \mu {\rm m}$ for single-track vertices.
The flavor tagging procedure is described in Ref.~\cite{Tagging}. The tagging information is represented by two parameters, the \ensuremath{B^{0}_{\rm Tag}}\ flavor $q$ and the flavor-tagging quality $r$. The parameter $r$ is continuous and determined on an event-by-event basis with an algorithm trained on MC simulated events, ranging from zero for no flavor discrimination to unity for an unambiguous flavor assignment. To obtain a data-driven replacement for $r$, we divide it into seven regions and determine a probability of mistagging $w$ for each $r$ region using high statistics control samples. Due to a nonzero probability of mistagging $w$, the $CP$ asymmetry in data is thus diluted by a factor $1-2w$ instead of the MC-determined $r$. The measure of the flavor tagging algorithm performance is the total effective tagging efficiency $\epsilon_{\rm eff} = \epsilon_{\rm Tag}(1-2w)^2$, rather than the raw tagging efficiency $\epsilon_{\textrm{Tag}}$, as the statistical significance of the $CP$ parameters is proportional to $(1-2w)\sqrt{\epsilon_{\rm Tag}}$. These are determined from data to be $\epsilon_{\rm eff} = 0.284\pm0.010$ and $\epsilon_{\rm eff} = 0.301\pm0.004$ for the SVD1 and SVD2 data, respectively~\cite{jpsiks_Belle2}.
About 1\% of events have more than one $B$ candidate. For these events, the candidate containing the two highest momentum tracks in the lab frame is selected.
Differences from the previous Belle analysis~\cite{pipi_Belle} include an improved tracking algorithm that was applied to the SVD2 data sample and the inclusion of the event shape \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ into the fit rather than the optimization of selection criteria for this variable. As the latter strategy results in a large increase of the continuum background level, a reduced fit region in \ensuremath{M_{\rm bc}}\ and \ensuremath{\Delta E}\ is chosen in order to reduce this background without significant loss of signal events. According to MC simulation, these changes increase the detection efficiency by 19\% over the previous analysis at a cost of continuum levels rising 4.7 times higher in the signal region defined by the previous analysis.
\section{Event Model}
\label{Event Model}
The $CP$ violation parameters are extracted from a seven-dimensional unbinned extended maximum likelihood fit to \ensuremath{M_{\rm bc}}, \ensuremath{\Delta E}, \ensuremath{{\cal F}_{b \bar b/q \bar q}}, \ensuremath{{\cal L}^{\pm}_{K/\pi}}, \ensuremath{\Delta t}\ and $q$ from a data sample divided into seven bins ($l = 0..6$) in the flavor-tag quality $r$ and 2 SVD configurations $s$. Seven categories are considered in the event model: \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ signal, \ensuremath{B^{0} \to K^{+}\pi^{-}}, \ensuremath{\bar{B}^{0} \to K^{-}\pi^{+}}\ and \ensuremath{B^{0} \to K^{+}K^{-}}\ peaking backgrounds, continuum, charmless neutral and charged $B$ decays. For most categories, the linear correlations between fit variables are small, so the probability density function (PDF) for each category $j$ is taken as the product of individual PDFs for each variable: ${\cal P}^{l,s}_{j}(\ensuremath{M_{\rm bc}},\ensuremath{\Delta E},\ensuremath{{\cal F}_{b \bar b/q \bar q}},\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}},\ensuremath{\Delta t},q) = {\cal P}^{l,s}(\ensuremath{M_{\rm bc}}) \times {\cal P}^{l,s}(\ensuremath{\Delta E}) \times {\cal P}^{l,s}(\ensuremath{{\cal F}_{b \bar b/q \bar q}}) \times {\cal P}^{l,s}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}) \times {\cal P}^{l,s}(\ensuremath{\Delta t},q)$ in each $l,s$ bin, unless stated otherwise.
\subsection{Peaking models}
The four peaking shapes, including the signal, are determined from reconstructed MC events. The PDFs for \ensuremath{M_{\rm bc}}\ and \ensuremath{\Delta E}\ are taken to be the sum of three Gaussian functions, where the two tail Gaussians are parametrized relative to the core, which incorporates calibration factors that correct for the difference between data and MC simulation. These factors calibrate the mean and width of the core Gaussian component. The PDF for \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ is taken to be the sum of three Gaussians in each flavor-tag bin $l$, where the shape parameters are identical for all peaking channels. Calibration factors that correct for the shape differences between data and MC are incorporated into the core mean and width. These factors for \ensuremath{M_{\rm bc}}\ are determined directly in the fit, while for \ensuremath{\Delta E}\ and \ensuremath{{\cal F}_{b \bar b/q \bar q}}, these factors are determined from a large-statistics control sample of \ensuremath{B^{+} \to \bar{D}^{0} [K^{+}\pi^{-}] \pi^{+}}\ decays. The \ensuremath{{\cal L}^{\pm}_{K/\pi}}\ shape is modeled with a two-dimensional histogram that has been corrected for the difference between data and MC in PID as determined from an independent study with inclusive $D^{*+} \to D^{0} [\ensuremath{K^{-}} \ensuremath{\pi^{+}}] \ensuremath{\pi^{+}}_{\rm slow}$ decays. The PDF of \ensuremath{\Delta t}\ and $q$ for \ensuremath{B^{0} \to \pi^{+}\pi^{-}} is given by
\begin{eqnarray}
{\cal P}^{l,s}_{\ensuremath{\pi^{+}}\ensuremath{\pi^{-}}}(\ensuremath{\Delta t}, q) &\equiv& \frac{e^{-|\ensuremath{\Delta t}|/\ensuremath{\tau_{B^{0}}}}}{4\ensuremath{\tau_{B^{0}}}} \biggl\{1-q\ensuremath{\Delta w}^{l,s}+q(1-2w^{l,s})\times \nonumber \\
& & \biggl[\ensuremath{{\cal A}_{CP}}\cos \ensuremath{\Delta m_{d}} \ensuremath{\Delta t} + \ensuremath{{\cal S}_{CP}} \sin \ensuremath{\Delta m_{d}} \ensuremath{\Delta t}\biggr]\biggr\} \otimes R^{s}_{\ensuremath{B^{0} \bar{B}^{0}}}(\ensuremath{\Delta t}),
\end{eqnarray}
which accounts for $CP$ dilution from the probability of incorrect flavor tagging $w^{l,s}$ and the wrong tag difference $\ensuremath{\Delta w}^{l,s}$ between \ensuremath{B^{0}}\ and \ensuremath{\bar{B}^{0}}, both of which are determined from flavor-specific control samples using the method described in Ref~\cite{Tagging}. The physics parameters \ensuremath{\tau_{B^{0}}}\ and \ensuremath{\Delta m_{d}}\ are fixed to their respective current world averages~\cite{PDG}. This PDF is convolved with the \ensuremath{\Delta t}\ resolution function for neutral $B$ particles $R^{s}_{\ensuremath{B^{0} \bar{B}^{0}}}$, as in Ref.~\cite{jpsiks_Belle2}. We consider the \ensuremath{\Delta t},$q$ distributions for the flavor-specific \ensuremath{B^{0} \to K^{+}\pi^{-}}\ and \ensuremath{\bar{B}^{0} \to K^{-}\pi^{+}}\ peaking backgrounds separately with
\begin{equation}
{\cal P}^{l,s}_{\ensuremath{K^{\pm}}\ensuremath{\pi^{\mp}}}(\ensuremath{\Delta t}, q) \equiv \frac{e^{-|\ensuremath{\Delta t}|/\ensuremath{\tau_{B^{0}}}}}{4\ensuremath{\tau_{B^{0}}}} \biggl\{1-q\ensuremath{\Delta w}^{l,s} \mp q(1-2w^{l,s}) \cos \ensuremath{\Delta m_{d}} \ensuremath{\Delta t}\biggr\} \otimes R^{s}_{\ensuremath{B^{0} \bar{B}^{0}}}(\ensuremath{\Delta t}).
\end{equation}
For the \ensuremath{B^{0} \to K^{+}K^{-}}\ peaking background, the \ensuremath{\Delta t},$q$ PDF is taken to be the same as that for \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ signal, but as \ensuremath{B^{0} \to K^{+}K^{-}}\ has not yet been observed, the $CP$ parameters are set to zero. To account for the outlier \ensuremath{\Delta t}\ events not described by the \ensuremath{\Delta t}\ resolution function, a broad Gaussian PDF is introduced for every category,
\begin{equation}
{\cal P}^{l,s}_{\rm Out}(\ensuremath{\Delta t}, q) \equiv \frac{1}{2}G(\ensuremath{\Delta t}; 0, \sigma^{s}_{\rm Out}).
\end{equation}
\subsection{Continuum model}
The parametrization of the continuum model is based on the off-resonance data; however, all the shape parameters of \ensuremath{M_{\rm bc}}, \ensuremath{\Delta E}, \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ and \ensuremath{{\cal L}^{\pm}_{K/\pi}}\ are floated in the fit. As continuum is the dominant component, extra care is taken to ensure that this background shape is understood as precisely as possible, incorporating correlations above 2\%. The PDF for \ensuremath{M_{\rm bc}}\ is an empirical ARGUS function~\cite{ARGUS}, while \ensuremath{\Delta E}\ is modeled by a linear fit in each flavor-tag bin with a slope parametrized by $p^{l,s}_{0}$ and $p^{s}_{1}$, depending linearly on \ensuremath{{\cal F}_{b \bar b/q \bar q}},
\begin{equation}
{\cal P}^{l,s}_{\ensuremath{q \bar{q}}}(\ensuremath{\Delta E}|\ensuremath{{\cal F}_{b \bar b/q \bar q}}) = 1 + (p^{l,s}_{0} + p^{s}_{1}\ensuremath{{\cal F}_{b \bar b/q \bar q}})\ensuremath{\Delta E}.
\end{equation}
The \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ shape is observed to shift depending on the PID region, so the PDF is a sum of two Gaussian functions in two PID regions, $\ensuremath{{\cal L}^{\pm}_{K/\pi}} \leq 0.5$ and (\ensuremath{{\cal L}^{+}_{K/\pi}}\ or $\ensuremath{{\cal L}^{-}_{K/\pi}}) > 0.5$. A small correlation between the \ensuremath{{\cal L}^{\pm}_{K/\pi}}\ shape and flavor-tag $q$ is also observed due to the $s\bar s$ component of continuum. As an example, consider the case where two jets are produced in which one contains a \ensuremath{K^{+}}\ and the other contains a \ensuremath{K^{-}}. If a \ensuremath{B^{0}_{\rm Rec}}\ candidate is successfully reconstructed with the \ensuremath{K^{+}}, it inhabits the flavor-specific $\ensuremath{K^{+}}\ensuremath{\pi^{-}}$ sector of \ensuremath{{\cal L}^{\pm}_{K/\pi}}. Then the accompanying \ensuremath{K^{-}}\ could then be used as part of the flavor-tagging routine, which leads to a preferred flavor tag of \ensuremath{\bar{B}^{0}}. This enhances the \ensuremath{{\cal L}^{\pm}_{K/\pi}}\ distribution in the $\ensuremath{K^{+}}\ensuremath{\pi^{-}}$ region and depletes it in the $\ensuremath{K^{-}}\ensuremath{\pi^{+}}$ region for $q=-1$. To account for this effect, we model \ensuremath{{\cal L}^{\pm}_{K/\pi}}\ with an effective asymmetry $A^{l,s}_{\ensuremath{q \bar{q}}}$ that modifies the two-dimensional PID histogram model $H^{l,s}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}})$, in each $l,s$ bin depending on the flavor tag,
\begin{equation}
{\cal P}^{l,s}_{\ensuremath{q \bar{q}}}(\ensuremath{{\cal L}^{\pm}_{K/\pi}},q) = \frac{1+qA^{l,s}_{\ensuremath{q \bar{q}}}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}})}{2}H^{l,s}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}),
\end{equation}
where
\begin{eqnarray}
A^{l,s}_{\ensuremath{q \bar{q}}}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}) &=& +a_{0}^{l,s}|\ensuremath{{\cal L}^{-}_{K/\pi}}-\ensuremath{{\cal L}^{+}_{K/\pi}}|^{a^{s}_{1}} \hspace{20pt} \textrm{if } \ensuremath{{\cal L}^{-}_{K/\pi}}-\ensuremath{{\cal L}^{+}_{K/\pi}} \geq 0 \nonumber\\
&=& -a_{0}^{l,s}|\ensuremath{{\cal L}^{-}_{K/\pi}}-\ensuremath{{\cal L}^{+}_{K/\pi}}|^{a^{s}_{1}} \hspace{20pt} \textrm{if } \ensuremath{{\cal L}^{-}_{K/\pi}}-\ensuremath{{\cal L}^{+}_{K/\pi}} < 0
,
\label{manta_ray}
\end{eqnarray}
which we hereafter refer to as the ``manta ray'' function. The \ensuremath{\Delta t}\ model,
\begin{equation}
P^{l,s}_{\ensuremath{q \bar{q}}}(\ensuremath{\Delta t}) \equiv \biggl[(1 - f_{\delta}) \frac{e^{-|\ensuremath{\Delta t}|/\tau_{\ensuremath{q \bar{q}}}}}{2\tau_{\ensuremath{q \bar{q}}}} + f_{\delta} \; \delta( \ensuremath{\Delta t} - \mu^{s}_{\delta})\biggr] \otimes R^{s}_{\ensuremath{q \bar{q}}}(\ensuremath{\Delta t}),
\end{equation}
contains a lifetime and prompt component to account for the charmed and charmless contributions, respectively. It is convolved with a sum of two Gaussians,
\begin{equation}
R^{s}_{\ensuremath{q \bar{q}}}(\ensuremath{\Delta t}) \equiv (1-f^{s}_{\rm tail})G(\ensuremath{\Delta t}; \mu^{s}_{\rm mean}, S^{s}_{\rm main}\sigma) + f^{s}_{\rm tail}G(\ensuremath{\Delta t}; \mu^{s}_{\rm mean}, S^{s}_{\rm main}S^{s}_{\rm tail}),
\end{equation}
which uses the event-dependent \ensuremath{\Delta t}\ error constructed from the estimated vertex resolution $\sigma \equiv (\sqrt{\sigma^{2}_{\rm Rec}+\sigma^{2}_{\rm Tag}})/\beta \gamma c$ as a scale factor of the width parameters $S^{s}_{\rm main}$ and $S^{s}_{\rm tail}$.
\subsection{\mbox{\boldmath${\ensuremath{B \bar{B}}}$} model}
The charmless $B$ background shape is determined from a large sample of MC events based on $b \to u,d,s$ transitions that is further subdivided into neutral and charged $B$ samples. A sizeable correlation of 18\% is found between \ensuremath{M_{\rm bc}}\ and \ensuremath{\Delta E}\ and is taken into account with a two-dimensional histogram. The PDF for \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ is taken to be the sum of three Gaussians in each flavor-tag bin $l$, similar to the peaking model. Here, we are able to fix the shape parameters from the peaking model except for the core mean and width. A similar correlation between the flavor tag and \ensuremath{{\cal L}^{\pm}_{K/\pi}}, similar to that in continuum, is also observed. Due to \ensuremath{B^{0} \bar{B}^{0}}\ mixing in the neutral $B$ background, this effect is correlated with $\ensuremath{\Delta t}$ and $q$. For the neutral $B$ background, the PDF is given by
\begin{eqnarray}
{\cal P}^{l,s}_{\ensuremath{B^{0} \bar{B}^{0}}}(\ensuremath{{\cal L}^{\pm}_{K/\pi}},\ensuremath{\Delta t}, q) &=& H^{l,s}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}) \times \nonumber \\
&& \frac{e^{-|\ensuremath{\Delta t}|/\tau_{\ensuremath{B^{0} \bar{B}^{0}}}}}{4\tau_{\ensuremath{B^{0} \bar{B}^{0}}}} \biggl\{1 + q A^{l,s}_{\ensuremath{B^{0} \bar{B}^{0}}}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}) \cos \ensuremath{\Delta m_{d}} \ensuremath{\Delta t}\biggr\} \otimes R^{s}_{\ensuremath{B^{0} \bar{B}^{0}}}(\ensuremath{\Delta t}), \nonumber \\
\end{eqnarray}
and the charged $B$ background PDF is given by
\begin{equation}
{\cal P}^{l,s}_{\ensuremath{B^{+} B^{-}}}(\ensuremath{{\cal L}^{\pm}_{K/\pi}},\ensuremath{\Delta t}, q) = \frac{1+qA^{l,s}_{\ensuremath{B^{+} B^{-}}}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}})}{2}H^{l,s}(\ensuremath{{\cal L}^{+}_{K/\pi}},\ensuremath{{\cal L}^{-}_{K/\pi}}) \frac{e^{-|\ensuremath{\Delta t}|/\tau_{\ensuremath{B^{+} B^{-}}}}}{2\tau_{\ensuremath{B^{+} B^{-}}}} \otimes R^{s}_{\ensuremath{B^{+} B^{-}}}(\ensuremath{\Delta t}),
\end{equation}
where $A^{l,s}_{\ensuremath{B \bar{B}}}$
are manta ray functions
for each \ensuremath{B \bar{B}}\ category and $R_{\ensuremath{B^{+} B^{-}}}$ is the \ensuremath{\Delta t}\ resolution function for charged $B$ events. As reconstructed background $B$ candidates may borrow a track from the tag side, the average \ensuremath{\Delta t}\ lifetime tends to be smaller and is taken into account with the effective lifetime, $\tau_{\ensuremath{B \bar{B}}}$.
\subsection{Full model}
The total likelihood for $559797$ \ensuremath{B^{0} \to h^{+}h^{-}}\ candidates in the fit region is
\begin{equation}
{\cal L} \equiv \prod_{l,s} \frac{e^{-\sum_{j}N^{s}_{j}\sum_{l,s}f^{l,s}_{j}}}{N_{l,s}!} \prod^{N_{l,s}}_{i=1} \sum_{j}N^{s}_{j}f^{l,s}_{j}{\cal P}^{l,s}_{j}(\ensuremath{M_{\rm bc}}^{i}\ensuremath{\Delta E}^{i},\ensuremath{{\cal F}_{b \bar b/q \bar q}}^{i},{\cal L}_{K/\pi}^{+ \; i},{\cal L}_{K/\pi}^{- \; i},\ensuremath{\Delta t}^{i},q^{i}),
\label{eq_likelihood}
\end{equation}
which iterates over $i$ events, $j$ categories, $l$ flavor-tag bins and $s$ detector configurations. The fraction of events in each $l,s$ bin, for category $j$, is denoted by $f^{l,s}_{j}$. The fraction of signal events in each $l,s$ bin, $f^{l,s}_{\rm Sig}$, is calibrated with the \ensuremath{B^{+} \to \bar{D}^{0} [K^{+}\pi^{-}] \pi^{+}}\ control sample. Free parameters of the fit include the \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ and \ensuremath{B^{0} \to K^{+}K^{-}}\ yields, $N^{s}_{\ensuremath{q \bar{q}}}$ and $N^{s}_{\ensuremath{B^{0} \bar{B}^{0}}}$. The individual \ensuremath{B^{0} \to K^{+}\pi^{-}}\ and \ensuremath{\bar{B}^{0} \to K^{-}\pi^{+}}\ yields are parametrized in terms of their combined yield $N_{K\pi}$ and the $CP$ violating parameter $\ensuremath{{\cal A}_{CP}}^{K\pi}$, which are both free in the fit: $N_{K^{\pm}\pi^{\mp}} = N_{K\pi}(1 \mp \ensuremath{{\cal A}_{CP}}^{K\pi})/2$. The remaining $N^{s}_{\ensuremath{B^{+} B^{-}}}$ yields are fixed to $N^{\rm SVD1}_{\ensuremath{B^{+} B^{-}}} = (0.269\pm0.010)N^{\rm SVD1}_{\ensuremath{B^{0} \bar{B}^{0}}}$ and $N^{\rm SVD2}_{\ensuremath{B^{+} B^{-}}} = (0.268\pm0.004)N^{\rm SVD2}_{\ensuremath{B^{0} \bar{B}^{0}}}$ as determined from MC simulation. In addition, all shape parameters of the continuum model with the exception of the \ensuremath{\Delta t}\ parameters are allowed to vary in the fit. In total, there are 116 free parameters in the fit: 10 for the peaking models, 104 for the continuum shape and 2 for the \ensuremath{B \bar{B}}\ background.
To determine the component yields and $CP$ violation parameters, in contrast to the previous Belle analysis~\cite{pipi_Belle}, we fit all variables simultaneously. The previous analysis applied a two-step procedure where the event-dependent component probabilities were calculated from a fit without \ensuremath{\Delta t}\ and $q$. These were then used as input in a fit to \ensuremath{\Delta t}\ and $q$ to set the fractions of each component to determine the $CP$ parameters. Our procedure has the added benefit of further discrimination against continuum with the \ensuremath{\Delta t}\ variable and makes the treatment of systematic uncertainties more straightforward, at a cost of analysis complexity and longer computational time. A pseudoexperiment study indicates a 10\% improvement in statistical uncertainty of the $CP$ parameters over the previous analysis method.
\section{Results}
\label{Fit Result}
From the fit to the data, the following $CP$ violation parameters are obtained:
\begin{eqnarray}
\ensuremath{{\cal A}_{CP}}(\ensuremath{B^{0} \to \pi^{+}\pi^{-}}) &=& +0.33 \pm 0.06 \textrm{ (stat)} \pm 0.03 \textrm{ (syst)},\nonumber\\
\ensuremath{{\cal S}_{CP}}(\ensuremath{B^{0} \to \pi^{+}\pi^{-}}) &=& -0.64 \pm 0.08 \textrm{ (stat)} \pm 0.03 \textrm{ (syst)},
\end{eqnarray}
where the first uncertainty is statistical and the second is the systematic error (Sec.~\ref{Systematic Uncertainties}). Signal-enhanced fit projections are shown in Figs.~\ref{fig_data_pipi1} and \ref{fig_data_pipi2}. The effects of neglecting the correlation between \ensuremath{M_{\rm bc}}\ and \ensuremath{\Delta E}\ in the peaking models can be seen there as the slight overestimation of signal; however, pseudoexperiments show that this choice does not bias the $CP$ violation parameters. These results are the world's most precise measurements of time-dependent $CP$ violation parameters in \ensuremath{B^{0} \to \pi^{+}\pi^{-}}. The statistical correlation coefficients between the $CP$ violation parameters is $+0.10$. The peaking event yields including signal are $N(\ensuremath{B^{0} \to \pi^{+}\pi^{-}}) = 2964 \pm 88$, $N(\ensuremath{B^{0} \to K^{+}\pi^{-}}) = 9205 \pm 124$ and $N(\ensuremath{B^{0} \to K^{+}K^{-}}) = 23 \pm 35$, where the uncertainties are statistical only. From the yields obtained in the fit, the relative contributions of each component are found to be $0.5\%$ for \ensuremath{B^{0} \to \pi^{+}\pi^{-}}, $1.6\%$ for \ensuremath{B^{0} \to K^{+}\pi^{-}}, $97.7\%$ for continuum and $0.2\%$ for \ensuremath{B \bar{B}}\ background. For the $CP$ violating parameter $\ensuremath{{\cal A}_{CP}}^{K\pi}$, we obtain a value of $-0.061 \pm 0.014$, which is consistent with the latest Belle measurement~\cite{hphm_Belle}.
Our results confirm $CP$ violation in this channel as reported in previous measurements and other experiments~\cite{pipi_Belle,pipi_BaBar,pipi_LHCb}, and the value for \ensuremath{{\cal A}_{CP}}\ is in marginal agreement with the previous Belle measurement. As a test of the accuracy of the result, we perform a fit on the data set containing the first $535 \times 10^6$ \ensuremath{B \bar{B}}\ pairs, which corresponds to the integrated luminosity used in the previous analysis. We obtain $\ensuremath{{\cal A}_{CP}} = +0.47 \pm 0.07$ which is in good agreement with the value shown in Table~\ref{tab_hh_prev}, considering the new tracking algorithm and the 19\% increase in detection efficiency due to improved analysis strategy. In a separate fit to only the new data sample containing $237 \times 10^6$ \ensuremath{B \bar{B}}\ pairs, we obtain $\ensuremath{{\cal A}_{CP}} = +0.06 \pm 0.10$. Using a pseudoexperiment technique based on the fit result, we estimate the probability of a statistical fluctuation in the new data set causing the observed shift in central value of \ensuremath{{\cal A}_{CP}}\ from our measurement with the first $535 \times 10^6$ \ensuremath{B \bar{B}}\ pairs to be 0.5\%.
To test the validity of the \ensuremath{\Delta t}\ resolution description, we perform a separate fit with a floating \ensuremath{B^{0}}\ lifetime; the result for $\tau_{\ensuremath{B^{0}}}$ is consistent with the current world average~\cite{PDG} within $2\sigma$. As a further check of the \ensuremath{\Delta t}\ resolution function and the parameters describing the probability of mistagging, we fit for the $CP$ parameters of our control sample \ensuremath{B^{+} \to \bar{D}^{0} [K^{+}\pi^{-}] \pi^{+}}; the results are consistent with the expected null asymmetry. Finally, we determine a possible fit bias from a MC study in which the peaking channels and \ensuremath{B \bar{B}}\ backgrounds are obtained from GEANT-simulated events, and the continuum background is generated from our model of off-resonance data. The statistical errors observed in this study agree with those obtained from our fit to the data.
\begin{figure}
\centering
\includegraphics[height=180pt,width=!]{mbc_pipi.eps}
\includegraphics[height=180pt,width=!]{de_pipi.eps}
\put(-227,155){(a)}
\put(-40,155){(b)}
\includegraphics[height=180pt,width=!]{pidp_pipi.eps}
\includegraphics[height=180pt,width=!]{pidm_pipi.eps}
\put(-227,155){(c)}
\put(-40,155){(d)}
\includegraphics[height=180pt,width=!]{fd_pipi.eps}
\put(-40,155){(e)}
\caption{(color online) Projections of the fit to the data enhanced in the \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ signal region. Points with error bars represent the data and the solid black curves or histograms represent the fit results. The signal enhancements, $\ensuremath{M_{\rm bc}} > 5.27 \textrm{ GeV}/c^2$, $|\ensuremath{\Delta E}| < 0.04 \textrm{ GeV}$, $\ensuremath{{\cal F}_{b \bar b/q \bar q}} > 0$, $\ensuremath{{\cal L}^{\pm}_{K/\pi}} < 0.4$ and $r > 0.5$, except for the enhancement of the dimension being plotted are applied to each projection. (a), (b), (c), (d) and (e) show the \ensuremath{M_{\rm bc}}, \ensuremath{\Delta E}, \ensuremath{{\cal L}^{+}_{K/\pi}}, \ensuremath{{\cal L}^{-}_{K/\pi}}\ and \ensuremath{{\cal F}_{b \bar b/q \bar q}}\ projections, respectively. Blue hatched curves show the \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ signal component, green dotted curves show the \ensuremath{B^{0} \to K^{\pm}\pi^{\mp}}\ peaking background component, dashed red curves indicate the total background, and purple dash-dotted curves show the \ensuremath{B \bar{B}}\ background component.}
\label{fig_data_pipi1}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=200pt,width=!]{dtq.eps}
\put(-42,173){(a)}
\put(-42,53){(b)}
\caption{(color online) Background subtracted time-dependent fit results for \ensuremath{B^{0} \to \pi^{+}\pi^{-}}. (a) shows the \ensuremath{\Delta t}\ distribution for each \ensuremath{B^{0}_{\rm Tag}}\ flavor $q$. The solid blue and dashed red curves represent the \ensuremath{\Delta t}\ distributions for \ensuremath{B^{0}}\ and \ensuremath{\bar{B}^{0}}\ tags, respectively. (b) shows the asymmetry of the plot above them, $(N_{\ensuremath{B^{0}}} - N_{\ensuremath{\bar{B}^{0}}})/(N_{\ensuremath{B^{0}}} + N_{\ensuremath{\bar{B}^{0}}})$, where $N_{\ensuremath{B^{0}}}$ ($N_{\ensuremath{\bar{B}^{0}}}$) is the measured signal yield of \ensuremath{B^{0}}\ (\ensuremath{\bar{B}^{0}}) events in each bin of \ensuremath{\Delta t}.}
\label{fig_data_pipi2}
\end{figure}
Using Eq.~(\ref{eq_iso}) and input from other Belle publications~\cite{hphm_Belle,pi0pi0_Belle}, an isospin analysis is performed to constrain the angle \ensuremath{\phi_{2}}. A goodness-of-fit $\chi^{2}$ is constructed for the five amplitudes shown in Fig.~\ref{fig_iso}, accounting for the correlations between our measured physics observables used as input. The $\chi^{2}$ is then converted into a $p$ value (CL) as shown in Fig.~\ref{fig_CL}. The region $23.8^{\circ} < \ensuremath{\phi_{2}} < 66.8^{\circ}$ is disfavored and the constraint on the shift in \ensuremath{\phi_{2}}\ caused by the penguin contribution is $|\Delta\ensuremath{\phi_{2}}| < 44.8^{\circ}$ at the $1\sigma$ level, including systematic uncertainties.
\begin{figure}
\centering
\includegraphics[height=160pt,width=!]{phi2_CL.eps}
\includegraphics[height=160pt,width=!]{dphi2_CL.eps}
\put(-268,40){(a)}
\put(-45,40){(b)}
\caption{Difference 1-CL, plotted for a range of \ensuremath{\phi_{2}}\ (a) and (b) $|\Delta \ensuremath{\phi_{2}}|$ values as shown by the solid curve. The dashed lines indicate the $1\sigma$ exclusion level.}
\label{fig_CL}
\end{figure}
\section{Systematic Uncertainties}
\label{Systematic Uncertainties}
Systematic errors from various sources are considered and estimated with independent internal studies and cross-checks. These are summarized in Table~\ref{tab_syst}. Uncertainties affecting the vertex reconstruction include the IP profile, charged track selection based on track helix errors, helix parameter corrections, \ensuremath{\Delta t}\ and vertex goodness-of-fit selection, \ensuremath{\Delta z}\ bias and SVD misalignment. The fit model uncertainties including the fixed physics parameters \ensuremath{\tau_{B^{0}}}\ and \ensuremath{\Delta m_{d}}, parameters describing the difference between data and MC simulation, \ensuremath{\Delta t}\ resolution function parameters, as well as the flavor-tagging performance parameters $w$ and \ensuremath{\Delta w}, are varied by $\pm 1 \sigma$. The parametric and nonparametric shapes describing the background are varied within their uncertainties. For nonparameteric shapes ({\it i.e.}, histograms), we vary the contents of the histogram bins by $\pm 1\sigma$. The fit bias is determined from the difference between the generated and fitted physics parameters using pseudoexperiments. Finally, a large number of MC pseudoexperiments are generated and an ensemble test is performed to obtain possible systematic biases from interference on the tag side arising between the CKM-favored $b \bar d \to (c \bar u d) \bar d$ and doubly CKM-suppressed $\bar b d \to (\bar u c \bar d) d$ amplitudes in the final states used for flavor tagging~\cite{tsi}.
\begin{table}
\footnotesize
\centering
\caption{Systematic uncertainties of the measured physics parameters.}
\begin{tabular}
{@{\hspace{0.5cm}}c@{\hspace{0.25cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}} @{\hspace{0.25cm}}c@{\hspace{0.5cm}}}
\hline \hline
Category & $\delta\ensuremath{{\cal A}_{CP}}(\ensuremath{\pi^{+}}\ensuremath{\pi^{-}})$ $(10^{-2})$ & $\delta\ensuremath{{\cal S}_{CP}}(\ensuremath{\pi^{+}}\ensuremath{\pi^{-}})$ $(10^{-2})$\\
\hline
IP profile & 0.13 & 1.19\\
\ensuremath{B^{0}_{\rm Tag}}\ track selection & 0.30 & 0.33\\
Track helix errors & 0.00 & 0.01\\
\ensuremath{\Delta t}\ selection & 0.01 & 0.03\\
Vertex quality selection & 0.37 & 0.23\\
$\Delta z$ bias & 0.50 & 0.40\\
Misalignment & 0.40 & 0.20\\
\ensuremath{\tau_{B^{0}}}\ and \ensuremath{\Delta m_{d}} & 0.12 & 0.09\\
Data/MC shape & 0.15 & 0.19\\
\ensuremath{\Delta t}\ resolution function & 0.83 & 2.02\\
Flavor tagging & 0.40 & 0.31\\
Background Parametric shape & 0.15 & 0.28\\
Background Nonparametric shape & 0.37 & 0.57\\
Fit bias & 0.54 & 0.86\\
Tag-side interference & 3.18 & 0.17\\\hline
Total & 3.48 & 2.68\\
\hline \hline
\end{tabular}
\label{tab_syst}
\end{table}
\section{Conclusion}
\label{Conclusion}
We report an improved measurement of the $CP$ violation parameters in \ensuremath{B^{0} \to \pi^{+}\pi^{-}}\ decays, confirming $CP$ violation in this channel as reported in previous measurements and other experiments~\cite{pipi_Belle,pipi_BaBar,pipi_LHCb}. These results are based on the full Belle data sample after reprocessing with a new tracking algorithm and with an optimized analysis performed with a single simultaneous fit, and they supersede those of the previous Belle analysis~\cite{pipi_Belle}. They are now the world's most precise measurement of time-dependent $CP$ violation parameters in \ensuremath{B^{0} \to \pi^{+}\pi^{-}}, disfavoring the range $23.8^{\circ} < \ensuremath{\phi_{2}} < 66.8^{\circ}$, at the $1\sigma$ level.
\section*{ACKNOWLEDGMENTS}
We thank the KEKB group for the excellent operation of the
accelerator; the KEK cryogenics group for the efficient
operation of the solenoid; and the KEK computer group,
the National Institute of Informatics, and the
PNNL/EMSL computing group for valuable computing
and SINET4 network support. We acknowledge support from
the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) of Japan, the Japan Society for the
Promotion of Science (JSPS), and the Tau-Lepton Physics
Research Center of Nagoya University;
the Australian Research Council and the Australian
Department of Industry, Innovation, Science and Research;
Austrian Science Fund under Grant No. P 22742-N16;
the National Natural Science Foundation of China under
Contract No.~10575109, No. 10775142, No. 10875115 and No. 10825524;
the Ministry of Education, Youth and Sports of the Czech
Republic under Contract No.~MSM0021620859;
the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft
and the VolkswagenStiftung;
the Department of Science and Technology of India;
the Istituto Nazionale di Fisica Nucleare of Italy;
the BK21 and WCU program of the Ministry of Education, Science and
Technology; the National Research Foundation of Korea Grants No.\
2010-0021174, No. 2011-0029457, No. 2012-0008143, No. 2012R1A1A2008330;
the BRL program under NRF Grant No. KRF-2011-0020333;
the GSDC of the Korea Institute of Science and Technology Information;
the Polish Ministry of Science and Higher Education and
the National Science Center;
the Ministry of Education and Science of the Russian
Federation and the Russian Federal Agency for Atomic Energy;
the Slovenian Research Agency;
the Basque Foundation for Science (IKERBASQUE) and the UPV/EHU under
program UFI 11/55;
the Swiss National Science Foundation; the National Science Council
and the Ministry of Education of Taiwan; and the U.S.\
Department of Energy and the National Science Foundation.
This work is supported by a Grant-in-Aid from MEXT for
Science Research in a Priority Area (``New Development of
Flavor Physics'') and from JSPS for Creative Scientific
Research (``Evolution of Tau-lepton Physics'').
|
2,877,628,089,515 | arxiv | \section{Introduction}
According to Alfv\'en's theorem, magnetic fields lines are frozen into highly conducting plasmas and are advected with the plasmas as they move under the influence of external forces. This freeze-in effect operates near black holes when the accreting plasma falls onto the hole, and thus it is natural for a black hole to have field lines that thread its event horizon. If the horizon-threading field lines are open and lightly loaded with plasma so that the local magnetization\footnote{Here, $B$ is the strength of the magnetic field, $\rho$ is the rest-mass density of the plasma, and $c$ is the speed of light. In this paper, we use Lorentz--Heaviside units for electromagnetic quantities.} is much larger than unity
\begin{equation}
\sigma \equiv \frac{B^2}{\rho c^2} \gg 1
\end{equation}
in the region close to the horizon, then their motion is controlled by gravity, and they are forced to rotate if the black hole has nonzero spin.
Forced rotation of field lines was first studied by \cite[hereafter BZ]{blandford77} who solved a force-free magnetosphere model in the limit that the black hole dimensionless spin $a_\star \equiv Jc/GM^2 \ll 1$ (here $J \equiv$ spin angular momentum, $M \equiv$ mass). BZ found that the field behaves as if it were anchored in a star rotating with frequency
\begin{equation}
\Omega_F \equiv \frac{1}{2}\Omega_H = \frac{a_\star}{8} \frac{c^3}{G M} + O(a_\star^3),
\end{equation}
where $\Omega_H$ is the rotation frequency of the event horizon. Field line rotation produces an outward-directed energy current at the horizon. In the force-free limit this is known as the BZ effect, whereas
if the field lines are more heavily loaded it is also sometimes called the magnetohydrodynamic (MHD) Penrose process \citep{takahashi1990}. The BZ effect is a favored mechanism for powering extragalactic radio jets.
In recent decades, numerical general relativistic magnetohydrodynamics (GRMHD) simulations have been used to study black hole accretion and the BZ mechanism (see \citet{Davis2020} and \citet{Komissarov2021} for reviews).
In GRMHD models with a trapped magnetic flux $\Phi$, a low density region forms around an axis parallel to the accretion flow angular momentum vector as plasma falls down the field lines into the hole or is expelled to larger radius. This low density region, with $\sigma \gg 1$, contains horizon-threading field lines moving with rotation frequency $\Omega_F$ and an associated, outward-directed energy current \citep[Poynting flux;][]{mckinney04}. In what follows we will refer to this region as the {\em jet}. It is difficult for numerical codes to robustly evolve parts of the simulation domain with low density and high $\sigma$, like in the jet, so semi-analytic magnetosphere models are often invoked to study these regions \citep[see, e.g.,][]{Ogihara2021}.
The jet is bounded by an accretion flow that pins magnetic flux in the hole. We will refer to the accretion flow as a {\em disk}, although it may have sub-Keplerian rotation. At the boundary layer between the jet and the disk, the density contrast is large. The plasma velocity can also change dramatically, with maximal shear occurring when the black hole and disk rotate in opposite directions (a \emph{retrograde} disk).
The jet--disk boundary layer has large shear and strong currents. It can suffer instabilities that lead to mass loading onto the jet's open field lines. It may also be an important particle acceleration site (see the reviews of (see the reviews of \citealt{Ostrowski1999, Rieger2019} for particle acceleration in relativistic shear layers). This paper considers the jet--disk boundary layer in the relativistic regime, within $\sim 20\;GM/c^2$ of the event horizon.
In Section~\ref{sec:estimates} we provide simple estimates for shear at the jet--disk boundary layer. In Section~\ref{sec:grmhd} we describe the GRMHD simulations we use to study the jet--disk boundary layer, and in Section~\ref{sec:results}, we explore the dynamics of the boundary layer by using tracer particles to both analyze the flow of matter through state space and investigate mass loading into the jet. Along the way we discuss the disk structure for retrograde accretion. In Section~\ref{sec:discussion} we consider model limitations, convergence, and possible extensions. Section~\ref{sec:summary} provides a summary and a guide to the main results.
\section{Scaling and Estimates}
\label{sec:estimates}
We now define the physical parameters that describe accretion systems, identify their ranges for the systems we consider, and provide an analytic estimate for flow dynamics at the jet--disk boundary layer.
\subsection{Parameters}
We consider radiatively inefficient accretion flows (RIAFs; \citealt{Reynolds96}) where radiative cooling is negligible, motivated by EHT observations of M87* and Sgr A*, which have accretion rate $\dot{m} \equiv \dot{M}/\dot{M}_{\mathrm{Edd}} \ll 1$ ($\dot{M}_{\mathrm{Edd}}$ is the Eddington accretion rate) and are therefore near or in this regime. RIAFs are geometrically thick disks, with ratio of scale height $H$ to local radius $R$ of order $1$.
In general, the angular momentum of accreting matter far from the horizon may be tilted with respect to the black hole's spin angular momentum. Although there are plausible scenarios that produce zero tilt, there is at present no way of rejecting models with strong or even maximal ($180$ degree) tilt. In this paper we restrict attention to systems where the orbital angular momentum of the accreting plasma is parallel or anti-parallel to the black hole spin vector (prograde or zero tilt and retrograde or maximal tilt, respectively). Disks with intermediate tilt are the subject of ongoing study \citep{Fragile2007, McKinney2013, moralesteixeira2014, Liska2018, White2019}.
In addition to $a_\star$, $\dot{m}$, and tilt, black hole accretion flows are characterized by $\Phi$, the trapped magnetic flux measured through the contour formed by the black hole's equator. Accretion of flux with a consistent sign eventually increases $|\Phi|$ until the accumulated magnetic flux is large enough that magnetic pressure $B^2 \sim (\Phi/(G M/c^2)^2)^2$ balances accretion ram pressure $\rho c^2$. Since $\dot{M} \sim \rho c (G M/c^2)^2$, when the dimensionless flux $\phi \equiv \Phi/\sqrt{G^2 M^2 \dot{M}/c^3}$ approaches a critical value $\phi_c \sim 15$ (\citealt{Tchekhovskoy2011maddefn}, but we use the normalization of \citealt{porth19}), the field can push aside infalling plasma and escape.
The unstable equilibrium with $\phi \sim \phi_c$ is known as a magnetically arrested disk \citep[MAD, see][]{BisnovatyiKogan1974, Igumenschchev2003, Narayan2003}, in contrast to accretion flows with $\phi \ll \phi_c$, which are said to follow standard and normal evolution \citep[SANE, see][]{Narayan2012, Sadowski2013}. Notice that $\phi$ is determined by the nonlinear evolution of the flow and is not trivially related to the initial conditions, although initial conditions have been identified that lead to SANE or MAD outcomes over finite integration times. We will consider both SANE and MAD accretion flows.
\subsection{Shear at the Jet-Disk Boundary}
\label{sec:jet-disk-shear}
Changes in velocity across the jet--disk boundary may drive Kelvin-Helmholtz instability. What is the expected velocity difference? The jet and disk are unsteady and strongly nonaxisymmetric in the numerical GRMHD models that motivate this calculation. In the interest of producing a model that can be studied analytically, we nevertheless treat the system as axisymmetric and steady, and because this is already a drastic approximation, we use a nonrelativistic fluid model for simplicity.
The jet can be idealized as a steady flow anchored in an object rotating with angular velocity $\Omega_F$. For a steady, axisymmetric, nonrelativistic MHD wind with plasma angular velocity $\Omega$ and generalized specific angular momentum $L$, angular velocity changes with cylindrical radius $R$ like
\begin{equation}\label{eq:mhdwind}
\Omega = \Omega_F \dfrac{1}{1 + M_{\rm A}^2} + \dfrac{L}{R^2}\dfrac{M_{\rm A}^2}{1 + M_{\rm A}^2}
\end{equation}
\citep[e.g.,][]{ogilvie2016} where $M_{\rm A}^2 \equiv v_p^2 / v_{\rm A}^2$ is the {Alfv\'en~} Mach number, defined as the ratio of the poloidal plasma velocity to the {Alfv\'en~} velocity $\bv_{\rm A} = {\bf B}/\sqrt{\rho}$. Since $\sigma \gg 1$, $v_{\rm A} \simeq c$.
Particles flow inward at the horizon and outward at large radius, and therefore a steady state can be achieved only if plasma is loaded onto field lines at intermediate radius. We assume this occurs, perhaps through turbulent diffusion or through pair production (in numerical GRMHD models plasma is added via numerical floors; see \citealt{wong2021drizzle} for a study of drizzle pair production in this region), and that there is a stagnation point at $r \sim {\rm few} \times G M/c^2$ between an inner, inflow {Alfv\'en~} point ($M_{\rm A}^2 = 1$) and an outer, outflow {Alfv\'en~} point. The outer {Alfv\'en~} point is close to the light cylinder $r_l \sin\theta = c/\Omega_F$.
Equation \ref{eq:mhdwind} implies that for $M_{\rm A}^2 \ll 1$, $\Omega \sim \Omega_F$, and for $M_{\rm A}^2 \gg 1$ the specific angular momentum of the wind is conserved. Inside of the light cylinder, in the limit that $a_\star \ll 1$, rotation is controlled by the rotation frequency of the hole $\Omega_H$, like $\Omega_F \approx \Omega_H / 2 \approx a_\star / 8$, so
\begin{align}
\Omega \approx
\begin{dcases}
\dfrac{\Omega_H}{2} & r < r_l \\
\dfrac{2 c^2}{\Omega_H} \dfrac{1}{(r \sin\theta)^2} & r > r_l \\
\end{dcases}
\end{align}
The jet--disk boundary is at $\theta_{\rm JD}$, so the outer light cylinder radius is $r_l = (8/a_\star) (G M/c^2)/(\sin \theta_{\rm JD}) + \mathcal{O}(a_\star)$. Taking $\sin\theta_{\rm JD} \simeq 1/\sqrt{2}$, then $r_l \simeq (11/a) (G M/c^2)$.
The disk rotates with approximately constant angular velocity $\Omega = s \Omega_K$ on spherical surfaces; here $\Omega_K = (G M)^{1/2} r^{-3/2}$ is the Keplerian angular velocity and $0 < s < 1$ measures how sub-Keplerian the accretion flow is. Numerical simulations suggest $s \lesssim 1/2$ for MADs \citep[e.g.,][]{Narayan2012} and $\sim 1$ for SANEs.
The toroidal component of the velocity difference across the jet--disk boundary is thus
\begin{equation}\label{eq:veldiff}
\Delta v_\phi \simeq r\sin\theta_{\rm JD} (\Omega_F - s \Omega_K).
\end{equation}
Without a model for flow along the field lines it is not possible to constrain the other components of the velocity difference. For retrograde accretion with $a_\star < 0$, the two angular frequencies in Equation~\ref{eq:veldiff} have the same sign and the magnitude of the velocity jump is at least of order the orbital speed. The velocity difference is approximately $c$ at $r = r_l$. For prograde accretion with $a_\star > 0$, the shear vanishes at $r = 4 (s/a_\star)^{2/3} (G M/c^2)$, and as in the retrograde case, the velocity difference is $\sim c$ at $r = r_l$.
\subsection{Stability of the Jet-Disk Boundary}
The jet--disk boundary is associated with sharp changes in density and magnetic field. The jet contains a laminar $\sigma > 1$ plasma, analogous to a pulsar wind, that rotates with the black hole. The disk contains a turbulent $P_{\mathrm{gas}}/B^2 \sim 1$ plasma whose angular momentum need not be related to the spin of the central hole. The relative orientation of the shear, jet magnetic field, and disk magnetic field may vary as turbulence in the disk produces varying conditions at the boundary.
Is the jet--disk boundary linearly stable? If we model the boundary layer as an infinitely thin current-vortex sheet, then we expect to capture the main features of the linear theory; finite thickness $H$ tends to suppress instability for modes with wavelength smaller than or of order $H$ and fastest growth is at wavelength $\sim H$. The current-vortex sheet can be subject to Kelvin--Helmholtz instability (KHI) as well as the plasmoid instability (\citealt{Loureiro2007}). High resolution axisymmetric models of black hole accretion flows (\citealt{Ripperda2020}; \citealt{Nathanail2020}) see evidence for plasmoid instability at the jet--disk boundary, but we do not, perhaps due to inadequate resolution. We therefore focus on KHI. It is well known that magnetic fields weaken the KHI because they resist corrugation of the vortex sheet. Do magnetic fields stabilize the jet--disk boundary?
A general linear theory of the plane-parallel, {\em relativistic}, ideal current-vortex sheet does not exist. \cite{Osmanov2008} consider the special case where magnetic field is oriented parallel to the velocity shear and the density, pressure, and field strength are continuous across the sheet. They do not consider the large density contrast that is an important feature of the jet--disk boundary problem.
The linear theory of the plane-parallel, compressible, {\em nonrelativistic}, ideal current-vortex sheet is better understood. The general (arbitrary field orientation on either side of the sheet) incompressible case was considered by \cite{Axford1960}; \cite{Shivamoggi1981} considers aligned and transverse fields; \cite{Sen1964} and \cite{Fejer1964} consider a general, arbitrarily oriented field on either side of the sheet. The stability of a finite-width layer has been considered in a well-known analysis by \cite{MiuraPritchett1982}, but an analytic dispersion relation is not available. Since the general, nonrelativistic problem is relatively tractable we provide a brief discussion and use it to obtain a qualitative understanding of stability of the jet--disk boundary.
Consider a plane-parallel, nonrelativistic, current-vortex sheet. The flow velocity and magnetic field are constant away from the sheet, which we position at $z = 0$. Let $i=J$ denote the low density (jet) side and $i=D$ the high density (disk) side. In equilibrium, ${v_{\rm A}}_z$ vanishes and total pressure is continuous across $z = 0$.
Now consider a perturbation of the form $f(z) \exp(i k_x x + i k_y y + i \omega t)$ with $f(z)=\exp(\kappa z)$, where $\kappa$ is in general complex.
The general dispersion relation is
\begin{gather}\label{eq:nonrelDR}
\lambda_J m_D+\lambda_D m_J=0 \\
\lambda_i=\rho_i\left[{\left(\omega-{\boldsymbol{k}} \cdot \bv_i\right)}^2-{\left({\bv_{\rm A}}_i\cdot {\boldsymbol{k}}\right)}^2\right] \\
m_i=\sqrt{k^2+\frac{{\left(\omega - {\boldsymbol{k}} \cdot \bv_i\right)}^4}{{{c_s}_i}^2{\left({\bv_{\rm A}}_i\cdot {\boldsymbol{k}} \right)}^2 - {c_s}_i^2{\left(\omega-{\boldsymbol{k}} \cdot \bv_i\right)}^2}} \label{eq:midefn}
\end{gather}
\citep{Sen1964, Fejer1964}. Here, $c_s \equiv$ sound speed, $c_m^2 \equiv v_{\rm A}^2 + c_s^2$ is the magnetosonic speed, and $\bv$ is the plasma velocity. The exponential factor $\kappa$ can be $m_i$ or $-m_i$ (see Equation~\ref{eq:midefn}) depending on the boundary condition and whether $z>0$ or $z<0$.
The general dispersion relation cannot be solved analytically. In the case of interest to us, however, $\rho_J \ll \rho_D$, ${c_s}_D \sim {v_{\rm A}}_D$, and ${c_s}_J \sim {c_s}_D$. Furthermore, physics provides a hint to the mathematical solution: the field in the jet is stiff (the {Alfv\'en~} speed is large due to the low density), motivating us to look for instability in modes with ${\boldsymbol{k}} \cdot {\bv_{\rm A}}_J = 0$. This is enough to make analytic progress. Taking $\rho_J/\rho_D \sim \epsilon^2 \ll 1$ and assuming that ${\boldsymbol{k}} \cdot {\bv_{\rm A}}_D \sim \epsilon$, we can solve the dispersion relation to lowest order in $\epsilon$. The relevant mode has
\begin{equation}\label{eq:KH_NR_LOW_DISP}
\omega^2 = ({\boldsymbol{k}} \cdot {\bv_{\rm A}}_D)^2 - \frac{\rho_J}{\rho_D} \, {\left[{\boldsymbol{k}} \cdot (\bv_J - \bv_D)\right]}^2,
\end{equation}
which suggests that the current-vortex sheet is unstable when ${\boldsymbol{k}} \cdot {\bv_{\rm A}}_D$ is sufficiently small, which we have confirmed by numerically solving the full dispersion relation.
In Equation (\ref{eq:KH_NR_LOW_DISP}) the nonrelativistic current-vortex sheet is unstable for small $\rho_J$. This is precisely the limit where one might worry about relativistic corrections: if $B_J^2/\rho_J > 1$, then the inertia of the jet is dominated by the magnetic field. In a fully relativistic analysis (Y.~Du et al., in prep.)~the current-vortex sheet has a near-identical dispersion relation in the limit $\rho_J \rightarrow 0$, except that $\rho_J/\rho_D$ in the above dispersion relation is replaced by $B_J^2/\rho_D$.
Evidently the current-vortex sheet is not generically unstable at large density contrast: a particular configuration of magnetic fields is needed for instability. The disk contains a turbulent magnetic field that is constantly changing strength and orientation, while the jet has a steadier field. This suggests a picture in which turbulent mixing driven by the KHI is episodic and occurs when jet and disk magnetic fields are aligned or anti-aligned. Mixing as a result of nonlinear development of the KHI will then only occur when there exist modes with growth times that are small compared to the correlation time of the turbulent eddies.
\subsection{Dissipation at the Jet-Disk Boundary}
The jet--disk boundary would appear to be a fertile setting for particle acceleration: particles that cross the boundary from the disk plasma frame to the jet plasma frame gain energy in a process akin to Fermi acceleration. This has been investigated by, e.g., \citet{Berezhko1981, Jokipii1990, Ostrowski1990} (see \citealt{Rieger2019} for a review), usually in the context of extragalactic radio jets kiloparsecs from the central source. \citet{Sironi2021Reconnection} performed 2D particle-in-cell simulations of the shear layer between a relativistic, magnetically-dominated electron--positron jet and a weakly magnetized ion--electron plasma and showed that the non-linear evolution of Kelvin--Helmholtz instabilities leads to magnetic reconnection, which can in turn drive particle acceleration. The formation of magnetic islands at the jet--disk boundary \citep[see, e.g.,][]{Ripperda2020, Nathanail2020} can also lead to particle acceleration; this process has been extensively investigated in kinetic simulations of current sheets.
To schematically address this question, we adopt a turbulent resistivity model for dissipation in the jet--disk boundary with magnetic diffusivity $\eta \simeq \alpha\, W \, \Delta v$, where $\alpha$ is the inverse of the magnetic Reynolds number, the width of the boundary layer is $W$ $\sim f R$ ($f < 1$; here, $R \equiv$ cylindrical radius), and $\Delta v \sim c$, so that $\eta \simeq \alpha f c R$. Next, we assume that the boundary is steady, axisymmetric, and follows $R = R_0 (z/z_0)^\beta$, with the jet intersecting the horizon at $(R_0, z_0)$. We assume that the magnetic flux in the jet $\Phi \simeq \pi B R^2$ is approximately independent of $R$ and thus take $B \simeq \Phi (z/z_0)^{-2\beta}/(\pi {R_0}^2)$.
If the magnetic field in the disk is similar in magnitude to that in the jet but randomly oriented, the dissipation rate per unit volume in the boundary layer is $\Lambda \sim \alpha B^2 (c/(f R))$, and the total dissipated power per unit height $z$ is independent of $f$:
\begin{equation}
\frac{d P}{dz} = \frac{1}{\pi}\alpha c \frac{\Phi^2}{R_0^3} \left( \frac{z}{z_0} \right)^{-3 \beta} \left( 1 + \beta^2 \frac{R_0^2}{z_0^2} \left(\frac{z}{z_0}\right)^{-2 + 2 \beta} \right)^{1/2}.
\end{equation}
Notice that this scales asymptotically as $z^{-1 - 2 \beta}$ for $\beta \ge 1$, so nearly all dissipation occurs close to the black hole. Integrating over $z$, the dissipated power is
\begin{equation}
P = \frac{\alpha c^5\Phi^2}{\pi (G M)^2} F(\beta, z_0/R_0),
\end{equation}
where $F$ is a dimensionless function of order unity. The power differs only by a factor of $a_\star^2/\alpha$ from the Blandford--Znajek power (e.g., \citealt{Tchekhovskoy2011}). To sum up: a fraction $\sim \alpha/a_\star^2$ of the jet power can be dissipated in the jet--disk boundary close to the black hole; this provides additional motivation for a numerical study.
\section{Simulating Black Hole Accretion}
\label{sec:grmhd}
We now study the jet--disk boundary layer using GRMHD simulations.
\begin{figure*}[th!]
\centering
\includegraphics[width=.95 \textwidth]{grmhd_initial.pdf}
\caption{Initial distribution of plasma and magnetic field for representative retrograde SANE (left) and MAD (right) simulations. Both black holes have $a_\star=-0.94$. The initial plasma density and magnetic field are axisymmetric. The central black hole is plotted at the center left of each panel. Color encodes log$_{10}$ of plasma density, and magnetic field lines, which are purely poloidal, are overplotted in black. Notice that the domain of the MAD plot is $10$x larger than the SANE simulation domain.}
\label{fig:grmhd_initial_condition}
\end{figure*}
\subsection{Numerical Setup}
We integrate the equations of GRMHD using the {\tt{}iharm3D} code, a descendent of the second order conservative shock capturing scheme {\tt{}harm} \citep{Gammie2003}. Written in a coordinate basis, the governing equations of GRMHD are
\begin{align}
\partial_t \left( \sqrt{-g} \rho_0 u^t \right) &= -\partial_i \left( \sqrt{-g} \rho_0 u^i \right), \label{eqn:massConservation}\\
\partial_t \left( \sqrt{-g} {T^t}_{\nu} \right) &= - \partial_i \left( \sqrt{-g} {T^i}_{\nu} \right) + \sqrt{-g} {T^{\kappa}}_{\lambda} {\Gamma^{\lambda}}_{\nu\kappa}, \\
\partial_t \left( \sqrt{-g} B^i \right) &= - \partial_j \left[ \sqrt{-g} \left( b^j u^i - b^i u^j \right) \right], \label{eqn:fluxConservation} \\
\partial_i \left( \sqrt{-g} B^i \right) &= 0, \label{eqn:monopoleConstraint}
\end{align}
where the plasma is defined by its rest mass density $\rho_0$, its four-velocity $u^\mu$, and $b^\mu$ is the magnetic field four-vector following \citet{mckinney04}. Here, $g \equiv {\rm det}(g_{\mu\nu})$ is the determinant of the covariant metric, $\Gamma$ is a Christoffel symbol, and $i$ and $j$ denote spatial coordinates.
In Equations~\ref{eqn:fluxConservation} and~\ref{eqn:monopoleConstraint}, we express components of the electromagnetic field tensor $F^{\mu\nu}$ as $B^i \equiv {^\star\!} F^{it}$ for notational simplicity. The stress--energy tensor ${T^\mu}_\nu$ contains contributions from both the fluid and the electromagnetic field:
\begin{align}
T^{\mu}_{\nu} &= \left( \rho_0 + u + P + b^{\lambda}b_{\lambda}\right)u^{\mu}u_{\nu} \nonumber \\
& \quad + \left(P + \frac{b^{\lambda}b_{\lambda}}{2} \right)g^{\mu}_{\nu} - b^{\mu}b_{\nu},
\end{align}
where $u$ is the internal energy of the fluid and the fluid pressure $P$ is related to its internal energy through an adiabatic index $\hat{\gamma}$ with $P \equiv \left(\hat{\gamma} - 1\right) u$. The {\tt{}iharm3D} code has been extensively tested and converges at second order on smooth flows \citep{Gammie2003}. A comparison of contemporary GRMHD codes can be found in \citet{porth19}.
Our model has several limitations. First, we treat the accreting plasma as a nonradiating ideal fluid of protons and electrons. We do not consider effects due to anisotropy and conduction (\citet{Sharma2006, Johnson2007}, but see \citet{foucart2017} for an evaluation of the limits of this approximation). We also neglect radiation. This approximation may be inappropriate in systems with high mass accretion rates, like M87 \citep{Dibi2012, Ryan2017}, but it is sensible in systems with low $\dot{m}$ like Sgr A* (but see \citealt{Yoon2020}, who show a different result under the assumption that the ions and electrons are perfectly coupled). The equations of nonradiative GRMHD are invariant under rescalings of both length and density, so our numerical results can be scaled to the desired $M$ and $\dot{M}$.
The {\tt{}iharm3d} code evolves plasma on a logically Cartesian grid. For these simulations, we use FMKS coordinates, which are a modified version of the conventional horizon-penetrating Kerr--Schild coordinates. We provide a detailed description of FMKS in Appendix~\ref{sec:fmks}. We use outflow boundary conditions for the radial direction, and we use a reflecting boundary condition at poles that mirrors the elevation components of the magnetic field and fluid velocity across the one-dimensional border.
We have added a passive tracer particle capability to {\tt{}iharm3D} to track mass loading into the jet. Each tracer particle is introduced with probability proportional to the coordinate particle density $\sqrt{-g} \rho u^t$, where $\rho$ is the rest-mass density, $g$ is the determinant of the covariant metric, and $u^t$ is the time component of the four-velocity. Initial positions are uniformly distributed in the coordinate basis in each zone. Particles are advected with the fluid according to
\begin{align}
\dd{x^i}{t} = \dfrac{u^i}{u^t},
\end{align}
where $x^i$ are the spatial components of the tracer particle's position and $u^\mu$ is the fluid four velocity.
The computational cost of evolving the tracer particles alongside the fluid scales linearly with the number of particles; we use $\approx 2^{25}$ particles, and this noticeably increases simulation cost. We therefore use completed GRMHD simulations to identify an epoch of interest, restart the fluid simulation at the beginning of the epoch, initialize the particles, and re-evolve the fluid to the end of the epoch.
The {\tt{}iharm3D} code has several limitations. It is not robust when $\sigma \gg 1$ \citep[e.g., in the strong cylindrical explosion test in][]{komissarov1999} or when the ratio of the gas pressure to the magnetic pressure $\beta \equiv 2 P_{\mathrm{gas}} / B^2 \ll 1$. Numerical stability is ensured by imposing artificial ceilings on $\sigma$ and $1/\beta$ in each zone at each timestep, which are enforced by resetting the density or internal energy density to a floor value that depends on position but not on time. This has a minimal effect on the flow (as can be checked by varying the ceilings), but it does inject particles in the nearly-evacuated funnel region, where $\sigma$ is large and $\beta$ is small.
\begin{figure}[th]
\centering
\includegraphics[width=\linewidth]{grmhd_models.pdf}
\caption{
Logarithmic plots over three decades of density in the poloidal plane for $a_\star=-0.5$ MAD and SANE models. Each image shows time- and azimuth- averaged density (left panels) and timeslices at azimuth $\phi = 0$ (right panels). The density is particularly variable in the MAD models, where the timeslice is not well approximated by the average state. The density is less variable in the SANE models, where the timeslice and average state are comparatively similar.
}
\label{fig:azimuthal_snapshot_model_compare}
\end{figure}
\begin{figure*}[th!]
\centering
\includegraphics[width=.95 \textwidth]{grmhd_2d.pdf}
\caption{Azimuthal slice from an individual timeslice of the $a_\star = 0.94$ retrograde MAD simulation. Left panel: log density of plasma near the black hole. Center panel: log internal energy of the plasma $u = \rho T$. Right panel: plasma magnetization $\sigma = b^2/\rho$. The high $\sigma$, low density conical regions around the poles are the jet funnel. The disk is the low $\sigma$, high density region near the midplane. The intermediate region between the funnel and the disk and with $\sigma \approx 1$ is the corona. The disordered accretion near the horizon is accentuated by streams of infalling plasma that are characteristic of MAD accretion.
}
\label{fig:azimuthal_snapshot_cartoon}
\end{figure*}
The fluid sector is initialized with a perturbed Fishbone--Moncrief torus solution \citep{Fishbone1976}, which is parametrized by the inner disk edge radius $r_{\mathrm{in}}$ and pressure maximum radius $r_{\mathrm{max}}$. The thermal energy is perturbed to seed the instabilities that jump start accretion (including the magnetorotational instability). The SANE models have $r_{\mathrm{in}}=6$ and $r_{\mathrm{max}}=12$ in a domain that extends from within the horizon to $r_{\mathrm{out}}=50 M$. The MAD models have $r_\mathrm{in}=20 M$ and $r_\mathrm{max}=41 M$ in a domain that extends to $r_\mathrm{out}=1000 M$. Our MAD disks are larger than our SANE disks. Figure~\ref{fig:grmhd_initial_condition} shows the initial conditions for plasma and magnetic field in representative SANE and MAD simulations.
The initial magnetic field is described by the toroidal component of the vector four-potential $A_\phi(r, \theta)$. For SANE disks
\begin{align}
A_\phi = \mathrm{max}\left[\dfrac{\rho}{\rho_{\mathrm{max}}} - 0.2, 0\right],
\end{align}
where $\rho_{\mathrm{max}}$ is the maximum initial plasma density.
For MAD disks the initial field is concentrated towards the inner edge of the disk and forced to taper at large $r$ according to
\begin{align}
A_{\phi} = \mathrm{max}\left[\dfrac{\rho}{\rho_{\mathrm{max}}} \left( \dfrac{r}{r_0} \sin \theta \right)^3 e^{-r/400} - 0.2, 0 \right],
\end{align}
where $r_0$ is chosen to be the inner boundary of the simulation domain~\citetext{B.~R.~Ryan, priv.~comm.}.
\subsection{Simulations}
Table~\ref{table:grmhd_models} provides a summary of the models we consider. Our simulations are similar to the retrograde ones generated for the EHT simulation library in \citetalias{PaperV}, except that: our simulations are evolved twice as long to mitigate natural stochasticity in matter entrainment; and a subset of our simulations are rerun at multiple resolutions.
We focus on four retrograde simulations with $a_\star = -0.5$ or $-0.94$. By convention, negative spins means that the black hole spin is anti-parallel to the angular momentum of the accretion flow (i.e., tilt is $180$deg). For each spin, we consider MAD and SANE models. We set the magnetic flux (and thus MAD or SANE state) by varying the field structure in the initial conditions.
Each simulation was run for at least 20,000 $GM/c^3$ and has an initial transient phase during which the initial torus relaxes, and magnetic winding and a combination of Rayleigh--Taylor and Kelvin--Helmholtz instabilities operate. The transient phase is followed at each radius by a turbulent quasi-equilibrium, with equilibrium radius, defined as the largest radius where $d\dot{M}/dr \simeq 0$, increasing as $r_{\mathrm{eq}} \sim t^{2/3}$ (see, e.g., \citealt{penna2010,dexter2020inflow} for a discussion). Beyond $r_{\mathrm{eq}}$, the flow is strongly dependent on initial conditions, so we consider information only from $r < r_{\mathrm{eq}}$. GRMHD models may be in equilibrium at large radii near the poles if there are strong outflows {\em and} the outflow structure is independent of the structure of the surrounding unequilibrated disk.
Our MAD simulations are run with bulk fluid adiabatic index $\Gamma = 13/9$, and our SANE simulations are run with $\Gamma = 4/3$ to be in agreement with \citetalias{PaperV} and \citet{porth19}.
\begin{deluxetable*}{ lllllllc }
\tablecaption{GRMHD Simulation Parameters} \label{table:grmhd_models}
\tablehead{
\colhead{id} &
\colhead{flux} &
\colhead{$a_\star$} &
\colhead{$r_{\mathrm{in}}$} &
\colhead{$r_{\mathrm{max}}$} &
\colhead{$r_{\mathrm{out}}$} &
\colhead{resolution} &
\colhead{notes}
}
\startdata
Sa-0.5 & SANE & $-0.5$ & $6$ & $12$ & $50$ & 288x128x128 & medium disk \\
Sa-0.94 & SANE & $-0.94$ & $6$ & $12$ & $50$ & 288x128x128 & medium disk \\
Ma-0.5 & MAD & $-0.5$ & 20 & 41 & $1000$ & 384x192x192 & large disk \\
Ma-0.94\_192 & MAD & $-0.94$ & 20 & 41 & $1000$ & 192x96x96 & large disk \\
Ma-0.94\_288 & MAD & $-0.94$ & 20 & 41 & $1000$ & 288x128x128 & large disk \\
Ma-0.94$^\dagger$ & MAD & $-0.94$ & 20 & 41 & $1000$ & 384x192x192 & large disk, multiple realizations, tracer particles \\
Ma-0.94\_448 & MAD & $-0.94$ & 20 & 41 & $1000$ & 448x224x224 & large disk \\
\enddata
\tablecomments{Retrograde GRMHD fluid simulations parameters. Flux labels the relative strength of the magnetic flux at the horizon, $a_\star$ describes the spin of the black hole, $r_{\mathrm{in}}$ and $r_{\mathrm{max}}$ are parameters for the initial Fishbone--Moncrief torus, $r_{\mathrm{out}}$ is the outer edge of the simulation domain, resolution gives the $N_r \times N_\theta \times N_\phi$ number of grid zones in the simulation.
$^\dagger$ The 384x192x192 MAD $a_\star=-0.94$ simulation was run using a different perturbed initial condition, and passive tracer particles were tracked for a part of its evolution.}
\end{deluxetable*}
\section{Results}
\label{sec:results}
We begin by discussing characteristic differences between MAD and SANE accretion flows before considering each of our simulations in detail. We explore the properties of fluid flow at small radii and within the jet, and then we relate outbursts in the MAD flows to magnetic flux ejection events. We explore qualitative features of the jet--disk boundary layer, including the development of Kelvin--Helmholtz instability. Finally, we use tracer particles to study mass entrainment across the jet--disk boundary layer.
\subsection{Overview}
It is convenient to divide low-luminosity black hole accretion flows into three regions: (1) the matter-dominated disk of plasma near the midplane, which on average flows inward, (2) the magnetically dominated, polar Poynting jet, and (3) the virial temperature intermediate region that contains the jet--disk boundary layer and the corona (here defined as the region with $\beta \sim 1$).
In a region extending from the event horizon out to somewhat beyond the innermost stable circular orbit (ISCO), the inflow plunges supersonically onto the hole and fluctuates strongly. Notice that the jet we consider here (at horizon scales) is dynamically distinct from the jet at large radius.
\begin{figure}[th]
\centering
\includegraphics[width=\linewidth]{grmhd_tracers_from_above.pdf}
\caption{
Tracer particle position in the MAD, $a_\star=-0.94$ model, projected onto the equatorial plane. Particle color varies linearly with local rest-mass density. The event horizon is a gray sphere. The inner region of the accretion flow is chaotic and characterized by plasma streams that break off the main disk at large radius. Plasma streams experience large magnetic torques ($u_\phi$ may change sign) as they plunge toward the horizon.
}
\label{fig:grmhd_tracer_fallingin}
\end{figure}
\begin{figure*}[th]
\centering
\includegraphics[width=.48\textwidth]{sigma_data_1896.pdf}
\includegraphics[width=.48\textwidth]{sigma_data_1943.pdf}
\caption{
Interaction between disk and jet magnetic field lines.
Magnetic field lines that intersect the disk at small radii are shown for two sequential timeslices of the plasma evolution. Field lines are sampled according to magnetization in the midplane. The colored surface shows the logarithm over two decades of density in the midplane of the simulation, and the event horizon is plotted as a black circle in the center of the plane. Left panel: the same timeslice as shown in Figure~\ref{fig:grmhd_tracer_fallingin}, rotated $45^\circ$ counter-clockwise. Magnetic field lines emanating from the high density region towards the left of the figure trace an accretion stream and are disk-dominated. Magnetic field lines that wind the opposite direction make up a flux tube and are being pulled clockwise with the hole as it spins. The two sets of field lines are about to collide.
Right panel: same simulation approximately $50\, GM/c^3$ later. Disk-threading and funnel-threading magnetic field lines have interacted, and a much stronger flux tube passes through the midplane in the low density region to the right of the hole.
}
\label{fig:grmhd_flux_tubes}
\end{figure*}
SANE and MAD accretion flows exhibit qualitatively different behavior. SANE models are relatively tame: plasma falls uniformly from the ISCO to the event horizon, the boundary of the accretion disk remains well defined, and the time-averaged accretion state is a fair approximation of an individual timeslice. In contrast, MAD accretion is choppy and tends to proceed in isolated, thin plasma streams that begin far from the hole and plunge onto it. MAD accretion is punctuated by violent eruptions that release excess trapped magnetic flux. Although the flux ejection events are not understood in detail, their structure suggests a Rayleigh--Taylor interaction between the disk and hole \citep[see, e.g.,][]{Marshall2018}.
For MAD flows, the time average is often not a good approximation to a single timeslice. These differences are particularly apparent in Figure~\ref{fig:azimuthal_snapshot_model_compare}, which shows log density for sample SANE and MAD models and compares the time-averaged solution (left) to representative timeslices (right).
In SANE models it is easy to separate the high-density disk from the low-density jet region. In contrast, in MAD models, identifying the location of the jet--disk boundary is a challenge.
In Figure~\ref{fig:azimuthal_snapshot_cartoon}, we show a typical timeslice on a poloidal slice of an $a_\star = -0.94$ MAD model, where the strength of the magnetic flux near the horizon prevents steady disk-accretion.
Here, accretion occurs when plasma streams break from the bulk disk at large radius and plunge onto the hole.
These streams are not confined to the midplane as they fall. Figure~\ref{fig:grmhd_tracer_fallingin} shows the projected locations of tracer particles in the same MAD $a_\star=-0.9373$ flow of Figure~\ref{fig:azimuthal_snapshot_cartoon} but viewed from above. The color of each particle corresponds to the linear density of particles in a three-dimensional voxel of space centered at the particle and is used to visualize the complicated vertical structure of the flow.
The figure shows one accretion stream connecting the disk and the hole in the bottom right and the launch of two new streams in the upper right.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{grmhd_histogram_all.pdf}
\caption{
Distribution of matter in the angular momentum and radial velocity versus radius ( $u_\phi-r$ and $v^r-r$) planes for the four fiducial simulations. The vertical gray line marks the ISCO. The colorscale is linear and shows the distribution of matter at each radius. In the SANE models the plasma lies on a well defined curve associated with Keplerian rotation as it accretes. In the MAD models plasma is perturbed away from the disk even before it enters the plunging region.
}
\label{fig:grmhd_statehistogram}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width= \linewidth]{grmhd_flowlines.pdf}
\caption{Timeslice of a MAD, $a_\star = -0.94$ model. Brightness shows plasma density, color saturation encodes value of $u_\phi$, and flow lines describe the poloidal motion of the plasma. The jet--disk boundary is visible as the surface where $u_\phi$ changes sign. Eddies tend to form at the jet--disk boundary as infalling, positive $u_\phi$ matter interacts with outflowing, negative $u_\phi$ matter. The sign of $u_\phi$ in the funnel is set by the sign of black hole spin.
}
\label{fig:grmhd_shear_layer}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width= \linewidth]{grmhd_models_averaged_2d.pdf}
\caption{
Density--weighted poloidal profile of $u_\phi$ for each of the four fiducial models after time and azimuthal averaging. The black circle at the origin marks the extent of the event horizon. All simulations have a similar structure: a parabolic jet (boundary defined by $u_\phi = 0$) and a peak in $u_\phi$ away from the pole.
}
\label{fig:grmhd_funnel_profiles_2d}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width= \linewidth]{grmhd_funnel_profile_all.pdf}
\caption{
Profile of $u_\phi$ versus elevation at $r=2, 5, 10,$ and $20\;GM/c^2$ for each of the models in Figure~\ref{fig:grmhd_funnel_profiles_2d}. Notice that $u_\phi < 0$ implies angular momentum aligned with the black hole. The average $u_\phi$ of plasma at small radii is smaller in MAD models than SANE models. The latitude of the shear layer within which $u_\phi$ changes sign increases with radius, corresponding to a narrowing jet. The (average) shear layer is wider for MAD models because their jet--disk boundary fluctuates over a wider range in latitude. As matter flows out in the jet, magnetic torques increase $u_\phi$.
}
\label{fig:grmhd_funnel_profiles}
\end{figure}
\subsection{Counterrotation and the disk}
As the black hole rotates, trapped magnetic field lines wind around the polar axis and produce a Poynting jet via the BZ mechanism.
In the jet--disk boundary layer, however, the jet field lines (that rotate with the hole) are mixed with disk field lines (that rotate against the hole in retrograde models). This interaction leads to an exchange of angular momentum via magnetic and fluid stresses. Some of the infalling plasma then acquires negative $u_\phi$, i.e., its specific angular momentum aligns with the black hole spin.
Exchange of angular momentum in the jet--disk boundary layer is more noticeable in MAD models, where accretion occurs in streams and where the magnetic field tends to be stronger. In MAD models, the inhomogeneous flow magnifies the effects of magnetic torques, since some equator-crossing field lines are lightly loaded (in contrast to SANE models, in which the equator-crossing field lines pass through a dense disk). Moreover, the more concentrated magnetic flux tubes in the MAD models can result in stronger torques \citep[see][]{porth2020}: when matter in the accretion stream with $u_\phi > 0$ interacts with a flux tube with $u_\phi < 0$, the plasma is rapidly braked and its angular momentum is reversed. Figure~\ref{fig:grmhd_flux_tubes} shows an example of this interaction as counterrotating field lines collide with the corotating field lines near the horizon. During these events, the front edge of an accretion stream commonly erodes and accelerates radially outwards.
The stronger angular momentum transfer in MAD flows produces more disorder in the inner region of the accretion flows.
This difference between MAD and SANE models can be seen in Figure~\ref{fig:grmhd_statehistogram}, which plots the time-integrated distributions of rest-mass over $u_\phi, r$ and $v^r, r$. The infalling matter accelerates within the plunging region (close to the ISCO) in both MAD and SANE flows, but the widths of the distributions of $u_\phi$ and $v_r$ at a given radius differ sharply: the MAD models have larger width because they experience larger fluctuations.
\subsection{Jet wall shape}
\label{sec:funnel_profiles}
In general, it is challenging to identify the jet--disk boundary since there is no clear criterion that distinguishes matter in the jet from matter in the disk (although proxy surfaces derived from magnetization or the Bernoulli parameter have been used in the past). Nevertheless, it is straightforward to find the surface where $u_\phi = 0$. Since $u_\phi$ has a definite sign in the jet, this surface may be a reasonable tracer of the boundary.
Figure~\ref{fig:grmhd_shear_layer} shows an azimuthal timeslice of plasma density and angular momentum in the MAD $a_\star=-0.94$ simulation and overplots the flow of the plasma. The lines change color at the $u_\phi=0$ surface, which broadly separates outgoing matter from infalling matter. The extended jet--disk boundary is turbulent and mixes mass, angular momentum, and energy between the two regions. Figure~\ref{fig:grmhd_funnel_profiles_2d} plots time and azimuth averaged $u_\phi$ for each of six models. We fit the $u_\phi = 0$ surface (within $r < 30\, GM/c^2$) to $z = a x^b$ and plot it as a dashed line. Recall that the boundary produced from ($\phi,t$)-averaged data may not be a good approximation to the boundary at fixed $\phi, t$, especially for MAD models. The parameters for the fit are reported in Table~\ref{table:funnel_wall_parameters}.
\begin{table}[t!]
\caption{Funnel wall ($u_\phi = 0$ surface) fit parameters}
\begin{center}
\tabcolsep=0.09cm
\begin{tabularx}{0.8\columnwidth}{ c c c }
\hline\hline
\hspace{.75cm}id &\hspace{.75cm} $a$\hspace{.75cm} & \hspace{.75cm}$b$\hspace{.75cm} \\
\hline
\hspace{.75cm}Sa-0.5 & $0.22$ & $1.8$ \\
\hspace{.75cm}Sa-0.94 & $0.18$ & $1.8$ \\
\hspace{.75cm}Ma-0.5 & $0.07$ & $2$ \\
\hspace{.75cm}Ma-0.94 & $0.1$ & $2$ \\
\hline
\end{tabularx}
\label{table:funnel_wall_parameters}
\end{center}
\tablecomments{Best fit parameters of $z = A x^b$ model for the location of the zero angular momentum surface in the GRMHD models.}
\end{table}
In Figure~\ref{fig:grmhd_funnel_profiles} we plot $\langle u_\phi\rangle$, where the brackets indicate an average over time and azimuth versus elevation at four radii in each of the simulations.
In MAD flows, we see that the average $u_\phi$ of matter in the midplane at $\theta = \pi/2$ decreases with radius; this makes sense since horizon-scale accretion flow is much choppier in MADs.
The average $u_\phi$ of the plasma tends to increase with radius in both the disk and in the funnel. The point where $u_\phi$ changes sign corresponds to the location of the jet--disk boundary layer and roughly tracks the shape of the jet. In our SANE simulations, the boundary layer is resolved by $\gtrsim 16$ zones at all radii, and the jet spans approximately $10$ zones at $r=20\,GM/c^2$ and approximately $40$ zones at $r=2\,GM/c^2$. The boundary layer in our MAD simulations spans approximately $\gtrsim 30$ zones at all radii, and the jet is resolved by between $20$ and $60$ zones at $r=20\,GM/c^2$ and $r=2\,GM/c^2$ respectively.
\begin{figure*}[t]
\centering
\includegraphics[width=.90 \textwidth]{grmhd_tracer_entrainment_times.pdf}
\caption{Histogram showing when tracer particles are entrained into the jet over a brief interval in the MAD $a_\star = -0.94$ model. Entrainment is conservatively defined to only include particles that begin in the disk region and end at large radius with positive $v^r$. This definition discounts particles that spend time in the mixing region but ultimately fall onto the hole. In this MAD model and by these criteria, entrainment is evidently a stochastic process that is characterized by periods of increased entrainment corresponding to times when instabilities form and break at horizon scales.
}
\label{fig:grmhd_tracer_entrainment_times}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.98\linewidth]{grmhd_proj3d_compiled.pdf}
\caption{
Logarithm over two decades of density on $r\approx 1.5 M$ slices for the MAD $a_\star = -0.94$ model at five times separated by $\Delta t = 25 M$. Matter in the jet near the poles flows clockwise from above (left on the page), and matter in the midplane flows counterclockwise (right on the page). The boundary between the funnel and the midplane results in the development of an unstable shear layer. A Kelvin--Helmholtz roll develops in the shear layer over the sequence of panels.
}
\label{fig:grmhd_projection}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{grmhd_equirectangular.pdf}
\caption{
Left panels: log over two decades of density in the $\theta-\phi$ plane for shells at $r=1.5, 3, 40\, GM/c^2$.
Right panels: same shells as left showing logarithm over two decades of $u_\phi$ with $u_\phi >0$ blue and red otherwise. These plots are from the central time slice of Figure~\ref{fig:grmhd_projection}, for the MAD $a_\star=-0.94$ model. The flow becomes increasingly chaotic at smaller radii; however, the shear layer between the disk and funnel persists, and the funnel region consistently has $u_\phi < 0$, indicating corotation with the hole.
}
\label{fig:grmhd_equirectangular}
\end{figure}
\subsection{Mass entrainment}
\label{sec:mass-entrainment}
The shear layer at the jet--disk boundary is episodically unstable in our models. As instabilities develop, plasma from the disk is transported across the boundary, reverses direction, and is entrained into the jet. We use tracer particles to study mass entrainment and track matter that passes through the mixing region. The computational cost of tracking tracer particles in the global flow over the course of the entire simulation makes a full study prohibitively expensive. We instead perform a single high-resolution, high-cadence study that focuses on the evolution of approximately $3.2\times 10^6$ particles within the inner region of the accretion flow over a $500\, GM/c^3$ interval. We chose to consider a range of time in the MAD $a_\star=-0.94$ model because it corresponded to an active period when multiple KHI knots are easily identifiable.
Entrained particles satisfy two criteria: they begin with $v^r < 0$ and $u_\phi > 0$, and they leave the simulation at the outer boundary with $u_\phi < 0$. In the mixing layer tracer particles may repeatedly transition between the disk and jet; we define entrainment to have happened for a tracer particle when its $u_\phi$ and $v^r$ change sign for the last time.
Because this definition of entrainment depends on the worldline of a fluid parcel, it is not immediately analogous to any quantity that can be directly computed from the raw fluid data.
Figure~\ref{fig:grmhd_tracer_entrainment_times} shows the computed mass entrainment rate over time. We find that entrainment events occur in bursts lasting $\sim 100\, GM/c^3$. Mass loading occurs at an average rate $\sim 10^{-2} \dot{M}$.
Note that our definition produces a measurement that does not count mass that has been injected by the numerical floor prescription in the funnel: the tracer particles are initialized once, so the application of floors during the subsequent evolution does not increase the number of the tracer particles. We discard the beginning epoch of tracer data to avoid including the floors' effect on the transient tracer particle initial condition.
In both SANE and MAD models, mass entrainment is driven by instabilities in the boundary between the accreting plasma and the matter in the jet.
Figure~\ref{fig:grmhd_projection} plots log plasma density on shells of constant radius over time and shows the development of an instability: as the high density midplane disk region moves to the right, it interacts with the low density funnel plasma moving to the left and forms Kelvin--Helmholtz rolls. Figure~\ref{fig:grmhd_equirectangular} plots density and specific angular momentum in the central frame of Figure~\ref{fig:grmhd_projection} in the $\theta-\phi$ plane at three different radii. Evidently, the KH roll is well resolved.
We observe that Kelvin--Helmholtz rolls develop in all simulations regardless of the accretion flow parameters; however, it is especially apparent in the MAD flows which have a more turbulent boundary layer. Mass entrainment thus proceeds in part through the Kelvin--Helmholtz instability at the jet--disk boundary. Still, the full structure of the jet--disk boundary layer is complicated, and braked accretion streams near the event horizon also contribute to mass loading.
We also use the tracer particles to visualize the flow of matter through phase space. Figure~\ref{fig:grmhd_tracer_stateimgs} shows the time-averaged flow of tracer particles in the radius--specific angular momentum plane. Plasma density is represented by the density and thickness of the white flow lines. Color denotes particle speed in phase space and helps differentiate between the disk/plunging region and the jet.
The flow at $r < 20$ can be divided into the three triangular regions shown in Figure \ref{fig:grmhd_tracer_stateimgs}. Region A contains particles that are falling towards the event horizon and gradually losing angular momentum. It contains the plunging region (where the figure is brightest), the disk, and the characteristic MAD accretion streams seen in Figure~\ref{fig:grmhd_tracer_fallingin}. Region B is the disk wind. Region C is the jet. Particles enter the jet from Region A, are torqued until their angular momentum has the same sign as the black hole, and then are accelerated outward. Particles gain angular momentum as they accelerate away from the hole, as expected in a sub-Alfv\'enic wind.
\begin{figure*}[t]
\centering
\includegraphics[width=0.85\textwidth]{grmhd_tracers_rsph_u_phi.pdf}
\caption{Time-averaged flow of tracer particles through the $r-u_\phi$ state space. The gray hatched region at the left of the figure lies within the horizon. The background shows a false-color representation of the average speed of the particles through the two-dimensional state space and helps to visually differentiate the disk (region A), disk wind (region B), and jet (region C).
The density of white lines is proportional to the density of particles in state space; for the purposes of visualization, the density is capped for regions in the disk that have large density. Average particle flow follows the thin white lines. As particles are entrained in the jet they cross $u_\phi=0$ and are then torqued and accelerate outwards.
}
\label{fig:grmhd_tracer_stateimgs}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
We have studied a set of retrograde MAD and SANE black hole accretion models. We found that the angular momentum of plasma in both the jet and parts of the jet--disk boundary layer is aligned with the spin of the hole.
We also found that the boundary layer region, in which $u_\phi$ transitions between its value in the midplane and its value in the jet, was wider in the MAD models than in SANE models. This is unsurprising, since MAD flows tend to be more chaotic near the horizon where much of the jet--disk interaction occurs, so the time-averaged boundary location is spread out.
The existence of a shear layer is not restricted to retrograde models, as noted in \S\ref{sec:jet-disk-shear}, but we have focused on retrograde models because the shear is strongest there.
As noted in \S\ref{sec:mass-entrainment}, the jet--disk boundary is sufficiently resolved to see the development of Kelvin--Helmholtz rolls; this strongly suggests that numerical diffusion does not control the entrainment rate. Nevertheless increasing the simulation resolution may expose new structures, such as the plasmoids seen in recent high resolution axisymmetric models \citep{Nathanail2020, Ripperda2020}.
To assess the effect of resolution we studied six different realizations of a MAD $a_\star = -0.94$ model at four resolutions: two at 192 radial zones, one at 288, two at 384, and one at 448 (resolution in other coordinates is scaled proportionately). We include multiple realizations at the same resolution to assess the error bars on measurements associated with turbulent fluctuations. We consider convergence in two time-averaged quantities: the profiles of $u_\phi$ presented in Figure~\ref{fig:grmhd_funnel_profiles} and the total mass in the jet near the hole as measured from the GRMHD.
The time-averaged specific angular momentum profile $\left\langle u_\phi\right\rangle(r,\theta)$ is remarkably consistent across all resolutions everywhere except in the zones adjacent to the polar boundary, where we do not necessarily expect agreement because of our treatment of the boundary condition. In the shear region, the profiles are consistent to $5\%$ and exhibit no discernible trend with resolution.
We compute the total mass in the jet near the hole by integrating the GRMHD density variable within a volume $V$
\begin{align}
M_{\mathrm{j}}(t) \equiv \int\limits_{V} \rho \; \sqrt{-g} \, \mathrm{d} r \, \mathrm{d} \theta \, \mathrm{d} \phi,
\end{align}
where we have chosen $V$ to be the region with $u_\phi < 0$ and $v^r > 0$ at $2 < r < r_* = 20$. Note that $M_{\mathrm{j}}(t)$ has contributions from both mass entrainment and numerical floors.
The time-dependent variation in the entrainment rate (see Figure~\ref{fig:grmhd_tracer_entrainment_times}), causes $M_{\mathrm{j}}(t)$ to fluctuate, so evaluations of the time-averaged $\langle M_{\mathrm{j}}(t)\rangle_t$ are subject to noise. We find that $M_{\mathrm{j}}(t)$ has a correlation time $\approx 200 \, GM/c^3$ in the MAD, $a_\star = -0.94$ model. The full model duration is $20,000 \, GM/c^3$, but the first $5,000 \, G M/c^3$ is an unequilibrated transient, so we have $N \sim 80$ independent samples over the full model; therefore, we expect fractional errors of order $N^{-1/2} \sim 10\%$. We find that $\langle M_{\mathrm{j}}(t)\rangle = 140, 130, 160$, and $130$ for simulations with radial resolution 192, 288, 384, and 448 respectively, which is consistent with the expected error. We also note that the widths of the jet and boundary-layer regions (in zones) reported in \S\ref{sec:funnel_profiles} scales linearly with the simulation resolution.
There may be additional mixing processes that occur on unresolved scales, so the consistency of $M_{\mathrm{j}}$ across resolutions does not prove that we have accurately accounted for mass mixing between the jet and disk. Future convergence studies should probe not only longer timescales to reduce the fluctuation noise but also higher resolution.
We also note that since the equilibration time increases with radius, the long-term average jet--disk interaction may be poorly represented at large radii where the disk is still strongly dependent on initial conditions. We have chosen to overstep this issue by only reporting fits and statistics from equilibrated parts of our simulations.
\citet{Chatterjee2019agnjet} also studied mass loading in their study of black hole jet launching. They performed multiple long-time, large-scale ($r_{\mathrm{max}} \gtrsim 10^5\;GM/c^2$) 2D GRMHD simulations and found that additional mass entrainment occurred at large radii. As noted above, the details of the jet--disk interaction at such large radii may be influenced by the choice of initial condition.
\section{Summary}
\label{sec:summary}
We have studied a set of three-dimensional GRMHD simulations of retrograde SANE and MAD black hole accretion disks at $a_\star = -0.5$ and $-0.94$, with a focus on the jet--disk boundary near the horizon. We have found that:
\vspace{0.5em}
1. Plasma in the jet rotates with the hole and not the disk. This generates a jet--disk boundary with strong currents and vorticity. \vspace{0.2em}
2. In MAD models accretion occurs through narrow plasma streams near the horizon. These streams erode as they interact with the counterrotating jet, loading the jet with plasma. \vspace{0.2em}
3. In both MAD and SANE models, disk plasma is entrained in the jet in well-resolved Kelvin--Helmholtz rolls. \vspace{0.2em}
4. The entrainment rate is $\sim 0.01 \, \dot{M}$ for the MAD, $a_\star = -0.94$ model that we are able to study in detail. \vspace{0.2em}
5. The entrainment rate and boundary layer structure are insensitive to resolution over the range in resolution we are able to study. \vspace{0.2em}
6. In retrograde MAD models accretion near the horizon fluctuates strongly: individual timeslices do not look like time- and azimuth- averaged data. Relatedly, the jet in MAD models wobbles significantly. The fluctuations create a complicated interface between jet and disk.
\vspace{0.8em}
This study has considered a limited range of models and could be extended by comparing a broader range of black hole spins and tilts between the hole and the accretion flow. Understanding the behavior of jet plasma and the jet--disk boundary layer may be crucial in developing a robust model of the connection between black hole spin and motion in the jet, which can now be resolved in time and space by the Event Horizon Telescope.
\acknowledgements
The authors would like to thank the Event Horizon Telescope collaboration, especially Jason Dexter, Ramesh Narayan, and Andrew Chael, as well as Eliot Quataert and Patrick Mullen, for stimulating discussions. The authors also thank Hector Olivares and the anonymous referee for their insightful comments and suggestions that improved the clarity of the manuscript. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.
This work was supported by NSF grants AST 17-16327 and OISE 17-43747. GNW was supported in part by a Donald and Shirley Jones Fellowship and a research fellowship from the University of Illinois. BSP was supported in part by the US Department of Energy through Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the US Department of Energy (Contract No.~89233218CNA000001). CFG was supported in part by a Richard and Margaret Romano Professorial scholarship.
\software{NumPy \citep{numpy}, OpenCV \citep{opencv}, Matplotlib \citep{matplotlib}}
\pagebreak
|
2,877,628,089,516 | arxiv | \section{Introduzione. Scelta metodologica}
\begin{scriptsize}
\begin{flushright}
« L'aver avuto un passato è sempre stato, nella storia delle istituzioni italiane, un motivo sufficiente per domandare un pezzetto d'avvenire che poi si sarebbe sommato, accrescendolo, al passato, e avrebbe fornito i titoli per richiedere una maggiore quota di futuro »\\
Luigi Besana \emph{L'Accademia scientifica nel periodo della sua formazione e costituzione}, Torino, 1978
\end{flushright}
\end{scriptsize}
Lo scopo di questo articolo è di offrire una breve ricognizione dello stato della fisica italiana nel primo trentennio del XIX secolo. Cogliendo lo spunto metodologico di Pietro Redondi \cite{redondiCulturaScienzaDall1978}, seguiremo un'analisi basata su tre ambiti: un'area geografica, che include il Regno di Sardegna e il Lombardo-Veneto, ossia due entità che giocarono un ruolo importante negli sviluppi della comunità scientifica nella penisola italiana tra la fine del XVIII secolo e il 1861; un ambito biografico, con un tentativo di analisi quantitativa dell'articolazione interna alla comunità scientifica dell'epoca basato sulla teoria delle reti e infine un taglio generale che escluda le personalità più in vista (Spallanzani, Volta, Avogadro, Lagrange, per illustrare il calibro). L'ultima assunzione è forse la più discutibile e l'analizzerò in dettaglio successivamente. Le prime due necessitano comunque di una precisazione. Ho "tagliato via", con l'accetta, importanti aree della penisola: la Repubblica di San Marco, lo Stato pontificio, il Granducato di Toscana, il Regno delle due Sicilie, i ducati di Parma e Piacenza, il Ducato di Modena, la Repubblica di Lucca. Queste sono aree che, sottolineiamolo, hanno apportato un contributo alla storia della comunità scientifica, un contributo che è rimasto nel tessuto sociale e tecnico dell'Italia unita, in termini di personale impegnato nella ricerca, accademie, associazioni, istituti di ricerca (molta parte del granulare sistema universitario italiano ha sede in queste zone, per esempio). Ma la natura di questo lavoro impone una scelta chiara e definita ad un ambito di ricerca che sarebbe altrimenti vasto quanto una Nazione intera ed attraverserebbe decine e centinaia di contributi. Per quanto riguarda invece l'ambito "biografico", è mia intenzione evitare di cadere in una discussione di storia della scienza dove manca la "storia" ed invece è preponderante la "scienza": dunque una storia di idee. Intendo invece seguire la da un lato l'impostazione epistemologica tratteggiata da Boris Hessen nel Congresso di Londra del 1930\footnote{B. Hessen, \emph{Science at the cross Road}, Frank Coss \& Co. 1971, pg, 166-167} in cui venne messa in luce la stretta interdipendenza delle condizioni ambientali - economiche, politiche, sociali - con l'elaborazione scientifica; dall'altro, una trattazione che sappia tessere una cornice narrativa in grado di presentare il periodo storico come autonomo - nei limiti della concezione di "autonomia" per una periodizzazione che è comunque arbitraria ultima analisi e tratta di fenomeni che si estendono, per cause ed effetti, ben al di là della finestra temporale individuata - per favorirne la ricezione e rendere più facile coglierne l'importanza.
\\
L'idea di effettuare un'operazione di sottrazione, spostando sul margine le figure che invece i contemporanei e una buona parte della storiografia successiva hanno invece messo in primo piano, coglie le mosse da un'intuizione dello stesso Redondi che cita il Gramsci dei "corpi catalitici":
\begin{quote}
[\ldots] Si tratterebbe insomma di uno studio dei «corpi catalitici» nel campo storico-politico italiano, elementi catalitici, che non lasciano traccia di sé ma hanno avuto una insostituibile e necessaria funzione strumentale nella creazione del nuovo organismo storico\cite{gramsciQuaderniDalCarcere2014}.
\end{quote}
Questo permette di mettere in luce il contributo di uno strato molto ampio che la storiografia posteriore ha esaltato, spesso in funzione teleologica, avente come obiettivo l'unificazione nazionale e la conseguente costruzione di una Scienza nazionale. Quella dell''unità è un'aspirazione certamente presente nel periodo pre-Risorgimentale, che nell'intellettualità diffusa della penisola italica data indietro fino a Dante, Petrarca e Boccaccio; non può però diventare l'unica chiave di lettura acriticamente acquisita. Per usare le parole dello stesso Redondi:
\begin{quote}
[\ldots] non c'è alcun dubbio che gli scienziati italiani del secolo XIX guardavano la scienza con una radicata «coscienza del sapere nazionale», come disse Cesare Correnti nel 1872. Non neghiamo infatti l'esistenza di una ricerca di identità nazionale che si fece avvertire nei diversi scienziati sparsi negli Stati italiani e in esilio. Ciò che neghiamo è che quella consapevolezza implichi per lo storico della scienza in Italia una nozione non problematica, un dato acquisito. In altre parole, neghiamo una prospettiva risorgimentista che ricavi dal motto «scienza e nazionale» degli scienziati ottocenteschi una categoria storiografica di italianità capace di riunire sotto di sè le idee scientifiche presenti in Italia.
\end{quote}
Aggiungiamo che questo è tanto più problematico per un periodo come quello in esame, che rischia di essere schiacciato dalla presenza di alcuni ingombranti \emph{vicini}: alle sue spalle, la Rivoluzione francese e l'ascesa di Napoleone, sul piano storico, e la potente opera di sistematizzazione della meccanica razionale sul piano scientifico, esemplificata da formulazioni come il cosiddetto \emph{demone di Laplace}; dall'altro, in avanti, il Risorgimento, che per l'Italia ha rappresentato una stagione fecondissima di impegno civile e crescita della coscienza nazionale, ma che il fascismo ha, in seguito, in una prospettiva nazionalista, eretto a canone per l'intero XIX secolo, rileggendo tutti gli eventi da quell'osservatorio privilegiato. Il primo trentennio del XIX secolo, invece, necessita di un approccio "leggero", dove si possa portare in luce anche le sfumature più tenui. Il periodo in esame si presta infatti ad una molteplicità di analisi: prendiamo per esempio il primo intervento diretto francese nella penisola nel 1796. Dovremmo considerarlo una invasione o piuttosto la risposta ad una guerra "esterna" portata avanti contro un nemico anch'esso esterno e spesso in rotta con le "aspirazioni nazionali" dei patrioti italiani, per definirlo un "intervento amico"? Le Repubbliche sorte per volontà dei Francesi sono un esempio di "colonialismo" e di oppressione, oppure un occasione di autogoverno? Lo sviluppo scientifico degli Stati italiani è stato "autonomo" oppure si è trattato di uno sviluppo di temi e situazioni importate dall'esterno, dovuta all'arretratezza congenita delle comunità scientifiche locali? Risulta evidente che non esista una risposta unica ma che ogni questione è attraversata da contraddizioni, spesso laceranti, che hanno influenzato con forza un'epoca che si pone a cavallo di importanti snodi della storia europea e mondiale. Una continua tensione fra diversi centri di forza che rappresentano un punto di interesse da preservare nell'analisi del periodo.
\newpage
\section{La situazione italiana alla fine dell'ultima decade del XVIII secolo}
\begin{scriptsize}
\begin{flushright}
« All who served the revolution have plowed the sea »\\
Simon Bolivar, towards the end of his life, about 1830.
\end{flushright}
\end{scriptsize}
\begin{wrapfigure}{r}{0.5 \textwidth}
\centering
\includegraphics[width=0.5 \textwidth]{1italy1796}
\caption{Situazione politica della penisola italiana alla vigilia del primo intervento francese dell'aprile 1796}
\end{wrapfigure}
La mappa mostra la frammentazione politica della penisola italiana nel 1796, prima della spedizione francese comandata dal generale Bonaparte. Oltre alla differenza di governi ed istituzioni, dobbiamo pensare alle differenze in termini di linguaggio, leggi, valute e sistemi di pesi e misure, che complicavano enormemente gli scambi sia fisici che di conoscenza.
\\
Nel 1796 il Direttorio francese, nella sua lotta contro gli Asburgo, decise di approntare una spedizione in Italia con lo scopo di assicurarsi la tranquillità al confine sud-est. Tra la primavera del 1796 e l'aprile del 1797 Napoleone, al comando dell'Armata d'Italia, sconfisse la coalizione tra il Regno di Sardegna e l'Impero asburgico, arrivando pericolosamente vicino a Vienna (a soli 250 km, per la precisione\footnote{\cite{bantiEtaContemporaneaDalle2009}, pg. 95}). Con il trattato di Campoformio (o Campoformido nel dialetto veneto dell'epoca) venne sancita la devoluzione della maggior parte dei territori della ex-Repubblica di San Marco all'Austria in cambio della cessione del Lombardo-Veneto e della fine dell'ingerenza di Casa Asburgo nello scacchiere italiano (ricordiamo che gli Asburgo erano direttamente coinvolti nelle vicende toscane, in cui la casa regnante era un ramo cadetto degli Asburgo stessi, gli Asburgo - Lorena), direttamente e indirettamente. I territori così "liberati" vennero riorganizzati nelle cosiddette "Repubbliche sorelle", entità politico-amministrative autonome in cui era comunque evidente la funzione di tutela della Francia (in quello che potremmo modernamente considerare un "protettorato" francese).
\\
Le cosiddette "repubbliche sorelle" sono un gruppo di repubbliche instaurate con il supporto dell'esercito francese di occupazione nel corso del triennio 1796 - 1799. Il Direttorio francese, e Napoleone in qualità di comandante dell'Armata francese in Italia, cercò di instaurare governi amici che sollevassero i francesi dall'amministrazione diretta delle terre occupate, rendendo più semplice e agevole la permanenza su suolo straniero. Le repubbliche non furono mai realmente indipendenti e si appoggiarono al corpo di occupazione francese per i compiti di polizia, esercito e spesso anche per compiti amministrativi come l'esazione delle tasse. Nonostante ciò, le repubbliche del 1796/1799 rappresentano il primo tentativo di unitarietà nel governo della penisola italica.
\begin{wrapfigure}{r}{0.5 \textwidth}
\centering
\includegraphics[width=0.5 \textwidth]{3cisalpineRepublic1799.jpg}
\caption{Repubbliche dell'Italia settentrionale, 1799 \cite{CisalpineRepublic}}
\end{wrapfigure}
\section{Il panorama geografico e politico}
\subsection{Il Regno di Sardegna}
Qui è doverosa una ulteriore precisazione di carattere geografico ma che avrà importanti ripercussioni anche sul futuro di una parte importante dell'Italia. Se infatti ho utilizzato qui la dizione di "Regno di Sardegna", è stato per intendere il Piemonte, che del Regno di Sardegna rappresentava il cuore dell'economia e delle istituzioni politiche e culturali, laddove la Sardegna stava in un rapporto di subordinazione che si esprimeva anche in campo scientifico, mancando nell'isola istituzioni e personale impegnato in ricerche autonome. Tale situazione rende difficoltoso, in questa sede, dare menzione di contributi che non appaiono in evidenza nella storiografia consultata, ma che pur dobbiamo supporre siano esistiti, in qualche forma. Nel proseguo mi limiterò quindi a delineare i caratteri economici e scientifici del Piemonte.
\\
Nel trentennio preso in esame, la situazione economica del Piemonte è caratterizzata da una molteplicità di fattori frenanti il suo sviluppo che possiamo raggruppare in tre classi:
\begin{itemize}
\item la \emph{dominazione francese}, che pur dotando la regione di istituzioni e politiche ricalcate su quelle francesi, spesso all'avanguardia confrontate con quelle sabaude, ne asserviva le risorse ad obiettivi stranieri, non favorendo una ricaduta positiva sul territorio delle azioni intraprese;
\item la \emph{concentrazione di capitali} nelle mani di un ceto nobile interessato alla rendita ed alla speculazione fondiaria (per l'agricoltura) e in commercianti interessati alla vendita di materie prime e semilavorati nei mercati europei avanzati;
\item la \emph{politica protezionista} dello Stato centrale, che proteggeva soprattutto la produzione agricola da merci a basso costo provenienti dalla Russia, ma che non incentivava uno sfruttamento razionale delle terre, che si trovavano quindi ad essere coltivate con strumenti e tecniche non efficienti.
\end{itemize}
Lo spartiacque della Restaurazione, con il suo colpire ogni movimento riformista, non incentivò un cambiamento in queste dinamiche, sebbene lo stesso Prospero Balbo cercasse di trovare un punto di equilibrio tra le istanze (che potremmo definire) \emph{di mercato} e un controllo statale che assicurasse la stabilità sociale (anche se soprattutto da una prospettiva aristocratica e non a favore delle classi subalterne)\footnote{cfr. M. Ciardi, \emph{La fine dei privilegi}, L. Olschki editore, 1995, Firenze, pg. 133}. I moti del 1821 riacutizzarono le paure della Corte, che fece retrocedere i tentativi di Balbo, così che nel 1830 si poteva definire ancora il Piemonte <<uno Stato ad involuto indirizzo agricolo>>\footnote{\emph{Ibidem}, pg. 135}.
\subsubsection{Il Privilegio. Il ruolo del brevetto agli esordi dello sviluppo industriale piemontese}
Il brevetto è un protagonista importante dello sviluppo industriale, ma la sua forma attuale discende da un'evoluzione storica complessa che ha attraversato differenti fasi in differenti aree geografiche. In questa sede, la sua importanza deriva dal ruolo che giocò nello sviluppo della rivoluzione industriale in Piemonte. Esso rappresenta uno snodo importante nel rapporto tra scienza, tecnica ed evoluzione dell'imprenditoria, con ricadute in ogni settore. In Inghilterra esso era regolato fin dal 1623 dallo <<Statuto dei Monopoli>> emanato da Giacomo II nel 1623 e si basava su una semplice richiesta di registrazione\footnote{\emph{Ib.}, pg. 146} da parte del proponente. In Francia, e questa sarà la linea del Piemonte nel periodo in esame, esso era legato a due condizioni: novità e utilità. La prassi era quella di sottomettere la richiesta ad un parere dell'Accademia delle Scienze, che aveva le competenze e il prestigio necessari per l'istruttoria.
\\
Alla fine del XVIII secolo, però, la pressione dell'innovazione divenne tale che l'Accademia non fu più in grado di reggere il ritmo delle proposte e nel 1791 questo esame venne rimosso\footnote{\emph{Ib.}, pg. 147}. A seguito dell'occupazione francese, questa modifica si propagò anche nel Piemonte annesso, fino a quando Vittorio Emanuele I, ritornato in possesso del Regno, il 21 Maggio 1814 restaurò la legislazione precedente. Nonostante le difficoltà dello sviluppo piemontese, nel corso degli anni successivi il numero di richieste aumentò e si dovette mettere mano ad una riforma, che portò all'approvazione di una nuova legge il 28 Febbraio 1826, \emph{Regie patenti, colle quali sono stabilite nuove regole per la concessione delle Privative industriali}. Tale legge riformava la concessione del Privilegio in senso "moderno", legandolo ad un esame delle caratteristiche tecniche, affidato all'Accademia delle Scienze di Torino. La caratteristica importante ai fini di questa analisi lega la nuova legge alla consuetudine del Regno di Sardegna di porre sotto Privilegio, e spesso finanziare, quelle invenzioni che avessero una ricaduta positiva nel ramo d'industria indicato. Tale principio era inteso così importante che si legava la concessione alla dimostrazione che l'attività del proponente fosse in buona salute e già avviata da tempo, per <<andare di riparo di alcuni abusi, come per es. quando il ramo d'industria privilegiato fosse dal concessionario tenuto in attività per uno o per pochi anni, e non più; nel qual caso il privilegio resterebbe senza effetto con danno dell'industria comune\footnote{Regie Patenti, 28 Febbario 1826, art. 10, in M. Ciardi, \emph{La fine dei privilegi}, L. Olschki editore, 1995, Firenze, pg. 151}>>. Con questa legge l'Accademia delle Scienze di Torino si dotava inoltre di un regolamento per disciplinare l'analisi tecnica delle richieste; Amedeo Avogadro, in qualità di membro dell'Accademia, si interessò personalmente di molti dei procedimenti, la cui analisi delle carte ci permette di trarre molte utili informazioni sul rapporto tra scienza, tecnica e sviluppo industriale come veniva inteso all'epoca. Nel corso della prima relazione a cui prese parte, riguardante delle macchine per la filatura della seta e di altri tessuti, la commissione riporta:
\begin{quote}
<<I deputati non dubitano che l'uso di queste macchine abbrevierà e diminuirà di molto la man d'opera per le operazioni che ne formano l'oggetto, le quali sin qui generalmente, e in Piemonte, e per quanto essi credono anche nell'estero, si fanno almeno in gran parte a mano \ldots [Questa riduzione permette di] diminuire con gran vantaggio del pubblico, il prezzo delle materie preparate, come già si è sperimentato dall'uso delle macchine da pochi anni introdotte per la filatura del cotone>>\footnote{\emph{Ib.}, pg. 147}.
\end{quote}
La connessione tra introduzione della macchina e riduzione del prezzo è già evidente e definita.
\\
L'impianto della Regie Patenti trae origine da una più antica attidudine dello Stato sabaudo nei confronti di quella che potremmo definire con un anacronismo "politica industriale", legata al ruolo dello Stato nella gestione dell'economia. Nel corso del XVIII secolo si venne delineando un organismo, il Consiglio di Commercio, un organismo consultivo in materia economica, con poteri di vigilanza sull'intero Regno\footnote{Luisa Dolza, \emph{<<A vantaggio di pochi e danno evidente di molti>>: Amedeo Avogadro, la tecnica e i privilegi in piemonte nella prima metà dell'Ottocento}, pg. 2, in \cite{ciardiFisicoSublimeAmedeo2007a}}. Le fonti mostrano come il ruolo di questo Consiglio fosse di analisi ed indirizzo delle innovazioni proposte a tutela della redditività dell'industria ma anche della stabilità sociale ad essa collegata. Il privilegio era quindi utilizzato come leva per favorire lo sviluppo di un settore, ma veniva accordato in un quadro di garanzie per le realtà locali coinvolte (soprattutto nel caso di un proponente estero, in linea con la politica protezionista allora egemone nel pensiero economico europeo). In questo era molto importante, per esempio, la formazione delle maestranze, che veniva esplicitamente richiesta come condizione per la concessione\footnote{Luisa Dolza, \emph{<<A vantaggio di pochi e danno evidente di molti>>: Amedeo Avogadro, la tecnica e i privilegi in piemonte nella prima metà dell'Ottocento}, pg. 4, in \cite{ciardiFisicoSublimeAmedeo2007a}}. Spesso inoltre non era tanto il privilegio ad essere accordato, ma finanziamenti per ricompensare l'inventore e permettere la commercializzazione dell'invenzione. Vi era anche un'attenta analisi del ruolo distorsivo del monopolio nell'influenzare lo sviluppo di un ramo d'industria\footnote{\emph{Id.}, pg. 3}.
\\
Con la Restaurazione e "l'effervescenza dovuta alla rivoluzione industriale", che aumenta notevolmente il carico di lavoro del Consiglio e la complessità dell'analisi dei manufatti, si delinea una situazione di tensione tra Consiglio di Commercio ed Accademia delle Scienze, situazione che verrà risolta con la succitata legge del 1826, che trasferirà la competenza interamente nelle mani dell'Accademia\footnote{Luisa Dolza, \emph{<<A vantaggio di pochi e danno evidente di molti>>: Amedeo Avogadro, la tecnica e i privilegi in piemonte nella prima metà dell'Ottocento}, pg. 6, in \cite{ciardiFisicoSublimeAmedeo2007a}}.
\subsection{Il Lombardo-Veneto}
\subsubsection{Le riviste scientifiche}
Una componente importante dello sviluppo scientifico del Lombardo-Veneto durante questo trentennio passa attraverso le riviste scientifiche pubblicate in loco. Fra questi, all'indomani della Restaurazione (1815/1816) ebbero una certa importanza il "Giornale di fisica, chimica, storia naturale, medicina ed arti dè professori Brugnatelli, Brunacci e Configliachi, compilato dal Dottore Gaspare Brugnatelli", su cui Avogadro pubblicherà una serie di lavori, e la "Biblioteca Italiana". Quest'ultima era stata creata su impulso del governo austriaco per mostrare pubblicamente la sua volontà di riconciliazione, ma fin da subito mostrò invece la reale intenzione di mantenere la discussione intellettuale nell'alveo di una politica filo-austriaca che risultò indigeribile per una certa parte dell'intellettualità lombarda\footnote{cfr. M. Ciardi, \emph{La fine dei privilegi}, L. Olschki editore, 1995, Firenze, pg. 62}. Dall'opposizione a questa esperienza nacque "Il Conciliatore", pubblicato a cavallo tra il 1818 e il 1819, diretto da Silvio Pellico, tra i quali collaborò anche il Romagnosi. Al Conciliatore parteciperà attivamente l'astronomo reale sabaudo Giovanni Plana, professore di matematica presso l'Università di Torino e importante punto di riferimento matematico per Avogadro.
La "Biblioteca" nasceva con un forte intento divulgativo, proponendosi di recensire le nuove pubblicazioni di carattere medico, fisico e letterario ma anche contributi originali di memorie tecniche e scientifiche\footnote{\emph{Ib.}, pg. 64}. Si noti come la fraseologia impiegata riporti spesso l'utilizzo dei termini "Italia", "Italiani" e si riportino contributi di giornali e riviste pubblicate in altri luoghi della penisola, non direttamente sotto controllo austriaco\footnote{\emph{Ib.}, pg. 64}.
\section{Il panorama epistemologico}
\subsection{Il Piemonte}
\subsubsection{Empirismo e ricerca teoretica nello stato Sabaudo}
In questo clima la ricezione del programma di ricerca di Laplace all'interno dei confini italiani e segnatamente nel Regno di Sardegna, che per vicende storiche e posizione geografiche meglio degli altri si prestava ad una stretta connessione con lo sviluppo della scienza dell'ingombrante Vicino, assume la forma dell'evoluzione della cattedra di Fisica Sublime all'Università di Torino. "Sublime" è un aggettivo che indicava, in campo matematico, la branca del calcolo integrale e differenziale\footnote{M. Ciardi, \emph{La fine dei privilegi}, L. Olschki editore, 1995, Firenze, pg. 71}. In Fisica, esso denotava quella che chiamiamo "meccanica razionale", cioè la matematizzazione della fisica che assumeva in quel momento storico la funzione di universalizzazione dell'approccio newtoniano e rappresentava un "ponte" tra scienze teoriche e sperimentali. In quest'ottica la disciplina si caratterizzava come centrale per lo sviluppo del Piemonte e così era considerata dai protagonisti dell'epoca, Prospero Balbo e Amedeo Avogadro. Nei progetti di riforma che seguirono la Restaurazione, a cui un personaggio come Balbo non aderì mai fino in fondo, pur provenendo da posizioni monarchiche e non rivoluzionarie, alle scienze toccava un posto centrale per mantenere lo Stato (piemontese) all'avanguardia nella competizione economica e militare internazionale\footnote{\emph{Ib.}, pg. 72}.
\subsection{Il Lombardo-Veneto}
\subsubsection{La ricezione di Newton e Leibniz nella cultura scientifica milanese}
Il clima culturale del Lombardo-Veneto nei primi decenni del XIX secolo affonda le sue radici nell'inedito sviluppo che la regione seppe avere sotto il dominio austriaco, dove spinte politiche ed economiche seppero saldarsi per creare le condizioni di uno sviluppo massiccio\footnote{P. Redondi, \emph{Cultura e scienza dall'illuminismo al positivismo}, in AA.VV., \emph{Storia d'Italia. Annali}, vol III, pg. 686}. L'elaborazione epistemologica risente in grande misura del tentativo di accordare le conseguenze più durature degli approcci filosofici al problema della scienza moderna di Newton e Leibnitz. Sul primo infatti si basa la decisa opera di \emph{meccanizzazione} della fisica, dove cioè ogni problema veniva inserito in un quadro di interazioni reciproche la cui intensità si basava su leggi di potenza (l'inverso del quadrato della distanza in analogia con l'attrazione gravitazionale), con la conseguenza che il movimento della materia (delle sue componenti) usurpava il centro dell'analisi rispetto alle sue qualità, come era stato invece nella fisica aristotelica\footnote{\emph{Ib.}, pg. 689}. La cultura illuministica aveva poi interpretato questa conseguenza in chiave decisamente antimetafisica, limitando l'analisi delle cause del movimento per concentrarsi invece soltanto su di esso, operazione esplicitamente annunciata da D'Alembert:
\begin{quote}
<<Conseguentemente a questa riflessione ho per così dire distolto la vista dalle cause motrici per non esaminare altro che il solo movimento che esse producono\footnote{J. B. Le Rond D'Alembert, \emph{Traitè de dynamique}, 2nd ed. Paris, 1758, pp. XVI sg., in P. Redondi, \emph{Cultura e scienza dall'illuminismo al positivismo}, in AA.VV., \emph{Storia d'Italia. Annali}, vol III, pg. 690}.>>
\end{quote}
Questo non poteva non porre problemi di ordine religioso in una società in cui gli organismi ecclesiastici avevano un peso notevole, e questo accadeva anche nella scienza\footnote{A titolo d'esempio, tra i maggiori protagonisti del dibattito epistemologico della seconda metà del XVIII secolo a Milano troviamo Giuseppe Boscovich e Paolo Frisi, rispettivamente un gesuita e un barnabita}. Con l'intento di recuperare una cornice trascendente veniva utilizzato Leibnitz, con il suo rifiuto della teoria del discreto della materia e l'accento sulla sua infinita divisibilità\footnote{P. Redondi, \emph{Cultura e scienza dall'illuminismo al positivismo}, in AA.VV., \emph{Storia d'Italia. Annali}, vol III, pg. 689}. Di conseguenza, non era \emph{l'impulso} ad essere la quantità principale in gioco ma la \emph{forza}, cosa che lascia spazio ad un'ipotesi di unificazione in senso anche metafisico\footnote{\emph{Ib.}, pg. 691}. Si ricreava così la possibilità per la metafisica di giocare un ruolo positivo interno all'epistemologia, in una maniera che permetteva di recuperare anche lo stesso Galileo, nonostante i famosi suoi problemi. Dulcis in fundo, una teoria del continuo poteva essere geometrizzata e sottoposta ad un'operazione di ordinamento e gerarchizzazione analoga all'assiomizzazione della geometria, che permetteva quindi un approccio deduttivo basato su una base empirica\footnote{\emph{Ib.}, pg. 689}.
\\
Sul versante più decisamente antimetafisico, sempre prendendo le mosse da Newton, si muovevano coloro i quali che aderivano più decisamente allo sviluppo in senso fisco-matematico della meccanica, in accordo con la tendenza maggioritaria nella comunità scientifica francese\footnote{\emph{Ib.}, pg. 693}. Questo si esplicava nell'utilizzo dell'analisi matematica e dei suoi procedimenti all'interno di un'elaborazione che includesse il dato sperimentale senza però assegnarli una posizione decisamente superiore. In questo clima vennero quindi accolte le sollecitazioni all'utilizzo della matematica in maniera rigorosa e produttiva, estendendo gli ambiti in cui l'analisi poteva essere utilizzata per risolvere problemi. Qui si trova però anche il limite di un approccio che, pur non negando l'importanza della teoria, aveva comunque, per ragioni anche extrascientifiche - il riformismo teresiano e l'impulso allo sviluppo produttivo - la necessità di trovare una giustificazione nella sua capacità di \emph{risolvere problemi}. Per questo troviamo un esito essenzialmente "operativista" alla scienza nel Lombardo-Veneto, che nella dialettica con il dato "concreto" non seppe uscire da una riflessione binaria empirismo/idealismo\footnote{\emph{Ib.}, pg. 697}. Il pragmatismo con cui la cultura scientifica lombardo-veneta affrontò il secolo permise comunque al pensiero scientifico di circolare e farsi portatore di un impegno civile che avrebbe poi dato i suoi frutti nella stagione del Risorgimento\footnote{\emph{Ib.}, pg. 698}.
\subsubsection{Lo sviluppo del dibattito epistemologico nel primo trentennio del XIX secolo}
Il pensiero di Ambrogio Fusinieri si inserisce nella polemica ottocentesca tra Boscovich e Frisi esprimendo una certa sintesi delle posizioni antimeccanicistiche e vitaliste che attraversavano la critica alla fisica-matematica ed al programma di ricerca di Laplace. Il nocciolo fisico delle sue teorie si basa su un dinamismo interno alla materia, generato da una forza repulsiva che specchia l'esistenza di una forza attrattiva di carattere gravitazionale fra le molecole\footnote{P. Redondi, \emph{Cultura e scienza dall'illuminismo al positivismo}, in AA.VV., \emph{Storia d'Italia. Annali}, vol III, pg. 721}. Tale forza repulsiva, che si osservava quando la materia si trovata in "forma attenuta", cioè riscaldata e/o sotto forma di lamine sottili e pellicole (come le bolle di sapone, fenomeno già studiato da Newton a suo tempo), portava alla conclusione di una forza interna responsabile di un processo di allontanamento infinito delle componenti della materia, un processo di suddivisione infinitesimale che trovava in Leibniz una giustificazione matematica\footnote{\emph{Ib.}, pg. 722}. Questa ipotesi/osservazione era comune ad una serie di fenomeni che riguardavano calore, elettricità e trasformazioni fisiche e giustificava l'idea di una fondamentale \emph{unitarietà} dei fenomeni. Qui sta l'importanza di Fusinieri nel panorama scientifico dell'epoca: la sua ricerca di un principio unitario, che esprimeva una tensione epistemologica e teologica a una spiegazione generale dei fenomeni fisici, e la sua concezione empirista della ricerca scientifica. Entrambi i tratti - unitarietà e sperimentalismo - lo pongono come interprete della sua epoca, segnata da fermenti romantici e positivisti in critica con la concezione illuministica e razionalista ben esemplificata da Laplace e dai "francesi" (con cui Fusinieri battaglierà a lungo, nella persona di Poisson, in merito alla fisica dei fluidi\footnote{\emph{Ib.}, pg. 723}). Tale apparente modernità lo poneva all'avanguardia rispetto ad una fisica-matematica che, con il suo approccio matematizzante ed il suo distacco dall'esperienza, non riusciva a dare conto delle relazioni tra fenomeni diversi e si trovava a postulare apparenti contraddizioni (l'esistenza degli atomi come centri del moto posti a distanze molto grandi rispetto all'entità delle forze gravitazionali supposte esistenti fra loro, il vuoto interno alla materia su cui si propagavano forse senza mediatori), ma la sua epistemologia tutta fondata su un empirismo "semplice" non andava oltre, come criterio metodologico, al richiamo all'esperienza come giudice della validità "a priori" di una ipotesi scientifica. Mancava un'analisi del ruolo della matematica come strumento predittivo delle relazioni tra entità, cosa che il positivismo in elaborazione non mancava di dare\footnote{\emph{Ib.}, pg. 726}. Egli scrive:
\begin{quote}
[\ldots] In luogo di abbandonarsi a tali finzioni [l'esistenza di fluidi imponderabili come il calorico, il vuoto fra gli atomi, la loro natura indivisibile], le quali non possono che impedire i veri progressi della scienza, se deve limitarsi a quanto i fenomeni presentano, scegliendo i più semplici e i più generali e classificando i fatti. In ciò solo può consistere il vero fondamento del sapere\footnote{A. Fusinieri, \emph{Riflessioni generali contro la teoria degli atomi e contro quella degli imponderabili}, in \emph{Annali delle scienze del regno lombardo-veneto}, vol. V, 1835, pp.gg. 149-153, in \footnote{\emph{Id.}, pg. 726}}.
\end{quote}
L'esito, per lo più infelice, di queste elaborazioni avrebbe condotto ad un certo ristagno nella comunità scientifica italiana fino alla vigilia del Risorgimento, che avrebbe rivitalizzato il dibattito attraverso una partecipazione in prima persona del personale scientifico nella penisola italica.
\newpage
\section{Il panorama biografico. Un abbozzo di analisi delle relazioni tra i soggetti in esame}
Questa sezione è un tentativo di analisi quantitativa del network fra gli scienziati nel periodo in esame creato dalla comune appartenenza a due istituzioni, la Reale Accademia delle Scienze di Torino e l'Accademia Nazionale delle Scienze detta dei XL.
L'immagine di apertura rappresenta il totale delle interazioni reciproche fra i membri delle due istituzioni nel periodo in esame.
\begin{figure}[h]
\centering
\includegraphics[width= \textwidth]{Grafiche_ConnessioniRiunioniScienziatiTorinoModena}
\caption{La comunità scientifica ristretta agli/alle appartenenti alle due istituzioni nel primo trentennio del XIX secolo}
\end{figure}
Nell'immagine qui sotto ho invece ridotto il numero delle connessioni fra i membri interni alla stessa Accademia all'arco di cerchio che connette i punti, in modo da concentrarsi sulle interconnessioni fra le due comunità.
\begin{figure}[h]
\centering
\includegraphics[width= \textwidth]{Grafiche_ConnessioniTorinoModena}
\caption{Connessioni fra scienziati mediate dall'appartenenza alla medesima istituzione}
\end{figure}
I dati sono stati recuperati dall'elenco dei membri dell'Accademia dei XL su Wikipedia\footnote{\url{https://it.wikipedia.org/wiki/Soci_dell\%27Accademia_nazionale_delle_scienze}, consultato per l'ultima volta in data 14 giugno 2021}, estrapolandone i membri iscritti tra il 1800 e il 1830, mentre per la Reale Accademia delle Scienze di Torino sono stati ripresi dal volume \emph{Il primo secolo della R. Accademia delle scienze di Torino. Notizie storiche e bibliografiche. (1783-1883)}, edito nel 1883 tra gli Atti dell'Accademia e digitalizzato nel 2011 dalla University of Toronto\footnote{Accessibile all'indirizzo: \url{https://teca.accademiadellescienze.it/book/ilprimosecolodel00acca} grazie all'Internet Archive}.
\\
Si tratta di una rete composta da 74 membri, la gran parte (53) iscritta all'Accademia dei XL, che aveva nello Statuto l'indicazione a non tenere in considerazione la provenienza geografica dei suoi iscritti, laddove l'Accademia di Torino discriminava fra "soci nazionali" (cioè del Regno di Sardegna) e "soci stranieri" (come sono, nel nostro caso, Pietro Paoli e Franz Xaver von Zach, rispettivamente di Pisa e Bratislava).
\\
Nel proseguo, ho assunto che la partecipazione alla stessa istituzione presupponga la presenza di un link fra i soggetti e la presenza contemporanea in due (o più) istituzioni costituisca un link tra le due comunità. La rete è assunta come indiretta, per rappresentare lo scambio fra soggetti ed istituzioni nei due sensi. Un piccolo caveat - fra i molti che si possono immaginare nell'affrontare una trattazione di questo tipo: ho cercato di restringere i soggetti ai soli membri delle classi di Scienze fisiche (fisica e chimica); non sempre questo restituisce risultati in linea con le aspettative contemporanee, dove gli ambiti di ricerca sono chiaramente delineati. Non era raro trovare, durante l'epoca studiata, molti medici e quelli che definiremmo "scienziati naturali" interessarsi attivamente a ricerche più propriamente di fisica e chimica.
\\
Come si può osservare dalla figura, sono pochi i membri che condividono la doppia iscrizione, segno di una regionalizzazione della comunità che restituisce un network dove pochi nodi hanno la responsabilità di funzionare come \emph{bridge} fra i sottoinsiemi. Il grado medio <\emph{k}> è pari a 3,2, mentre si osserva che gli \emph{hub} Avogadro, Michelotti, Carena e Lavini hanno un grado molto maggiore (rispettivamente 12, 12, 4, 4).
\newpage
La figura successiva mostra una struttura ad albero della rete. Si noti la struttura "a lobi" del network, tenuto insieme da soli quattro hub, di diversa importanza.
\begin{figure}[h]
\centering
\includegraphics[width= \textwidth]{Grafo_TorinoMmodena}
\caption{Grafo ad albero della rete}
\end{figure}
Nel seguito ho invece posto a zero il collegamento fra i membri della stessa istituzione per concentrarci sul sottoinsieme dei bridge (l'uso dei colori distingue fra le due comunità).
\begin{figure}[h]
\centering
\includegraphics[width= \textwidth]{Grafiche_GrafoTorinoModena}
\caption{Grafo ad albero della rete, con l'isolamento del sottoinsieme più connesso}
\end{figure}
La fragilità della comunità scientifica del periodo rispecchia la profonda regionalizzazione allora imperante e le difficoltà poste dall'ambiente circostante nel portare avanti una ricerca organica. La presenza delle Accademie costituiva al tempo stesso una \textbf{opportunità} di connessione ma un \textbf{limite} alla condivisione, nella misura in cui all'interno delle Accademie si aveva la possibilità di essere aggiornati sui lavori altrui (con i limiti che i rituali e la natura dei membri ponevano, in un'epoca in cui lo "scienziato di professione" stava solo iniziando a prendere forma) ma il numero e la diffusione locale delle istituzioni portava ad una frammentarietà che inibiva la formazione di gruppi di lavoro "internazionali".
\\
\newpage
L'ultima figura mostra come un evento esterno porti al rafforzamento della struttura: la Prima Riunione degli Scienziati Italiani a Pisa nel 1839. Ho ricavato i dati dall'elenco dei partecipanti alla Riunione raccolti nel volume \emph{Prima riunione dè naturalisti, medici ed altri scienziati italiani tenuta in Pisa nell'ottobre 1839}, stampato nell'ottobre 1839 dalla Tipografia Nistri e digitalizzato dal Museo Galileo\footnote{\url{https://bibdig.museogalileo.it/Teca/Viewer?an=301346}, url consultato l'ultima volta il 14 giugno 2021}.
\begin{figure}[h]
\centering
\includegraphics[width= \textwidth]{Grafiche_ConnessioniTorinoModenaPisa1839}
\caption{I cambiamenti nella comunità con l'introduzione delle Riunioni degli scienziati italiani nel 1839}
\end{figure}
Si può osservare un irrobustimento degli hub già presenti (Avogadro non risulta presente alla Riunione), con Michelotti che passa da un grado di 12 a 19 e Carena da 4 a 10 e la formazione di nuovi, come Configliachi (per altro Presidente della Sezione di Scienze Naturali della Riunione), Carlini, Lavini, Amici, Tommasini e Targioni - Tozzetti (che un certo peso ebbe anche nella definizione della nascente letteratura romantica).
\section{Conclusioni}
Il periodo in esame permette alcune riflessioni sullo sviluppo scientifico della penisola italiana e sulle sue ripercussioni sull'attualità. Le profonde differenze regionali, in termini di sviluppo interno e di connessione con l'esterno (Regno Unito e Francia in primis, ma nel volgere di qualche decennio anche l'area di lingua germanica avrebbe fatto il suo ingresso unificato sul proscenio internazionale) hanno giocato un ruolo preponderante nel determinare gli orientamenti della comunità scientifica "protoitaliana". Questo insieme di circostanze ambientali si riflettono tutt'ora nella varietà e vitalità delle istituzioni scientifiche: basti pensare che entrambe le Accademie citate sono ancora operanti, così come le decine di università che esistevano all'epoca. Il rapporto dialettico fra centri e periferie continua ad essere un fattore determinante anche nella ricaduta della ricerca scientifica sui settori produttivi dell'economia, con una differenza marcata fra Nord e Sud del Paese ed una differenziazione notevole all'interno degli stessi distretti produttivi\footnote{\url{https://www.istat.it/it/files/2020/07/Livelli-di-istruzione-e-ritorni-occupazionali.pdf}, url consultato in data 14 giugno 2021}. La "debolezza" della comunità scientifica, brevemente messa in luce nelle ultime pagine, ispira una serie di questioni e richieste di approfondimenti sul suo stato attuale.
\\
Seppur centrale nello sviluppo della comunità scientifica, dunque, rimane la netta sensazione che l'epoca in esame sia stata letta con una certa difficoltà dalla storiografia di parte. Ho accennato brevemente alle difficoltà incontrate nel reperire fonti secondarie con un fuoco diretto sull'argomento; tale difficoltà è accentuata dalle letture successive del periodo \emph{in funzione} dei periodi successivi: il Risorgimento, soprattutto, e la lettura che del Risorgimento ne fece il fascismo. Come ricorda il prof. Barbero in una conferenza del 15 maggio 2021 a Lucca\footnote{\url{https://www.youtube.com/watch?v=umUFW0n5LFA}, url consultato il 14 giugno 2021} riferendosi all'esperienza delle Repubbliche sorelle filofrancesi, prevale nella storiografia la necessità di riconnettere lo sviluppo dell'Italia unita con il passato mitico piuttosto che con il vicino prossimo; ecco quindi il recupero del Regno longobardo d'Italia e il vagheggiare del ritorno dell'Impero sui "colli fatali" di Roma, piuttosto che il riferimento ad uno Stato unitario sorto ad appena cinquant'anni di distanza negli stessi luoghi e con personale politico probabilmente ancora in vita. Per quel poco che posso testimoniare, è più facile trovare un'analisi del ruolo degli scienziati italiani a Curtatone e Montanara piuttosto che uno studio comparato delle Accademie italiane nei primi anni del XIX secolo. Eppure tale studio critico sarebbe interessante anche per valutare correttamente, fuori dall'impatto emotivo, pur necessario e da valorizzare, l'origine dei fattori che hanno plasmato gli appuntamenti decisivi a cui l'intera comunità scientifica della Penisola è stata chiamata e a cui aderirà anche con entusiasmo. E forse potremmo capire meglio parte di questo entusiasmo se ci concentrassimo sullo studio delle condizioni di vita di quegli stessi scienziati dieci o venti anni prima.
\newpage
\nocite{*}
\bibliographystyle{ieeetr}
|
2,877,628,089,517 | arxiv | \section{Introduction}
The first multimessenger detection of a neutron star (NS) merger was jointly detected in gravitational waves (GWs) and $\gamma$-rays \citep{GW170817_LVC,GW170817_GBM,GW170817_INTEGRAL,GW170817-GRB170817A}. The aftermath was detected in follow-up observations across the electromagnetic (EM) spectrum \citep[e.g][]{GW170817_afterglow_Xray_discovery,GW170817_afterglow_discovery_radio,GW170817_afterglow_optical_discovery, xray_afterglow_fading,GW170817_kilonova_optical,GW170817_kilonova_UV,GW170817_kilonova_infrared}. We describe the outstanding science possible with observations of NS mergers, followed by the capabilities necessary for discovery.
\section{Science}
This science is predicated on astrophysical observations of these phenomena; therefore, we first discuss the astrophysics of NS mergers and follow with the broader science they enable.
\subsection{Astrophysics}
The detections and associations of GW170817, GRB 170817A and the kilonova, AT2017gfo, confirmed decades old predictions from theory, observation, and simulation on the nature of short gamma-ray bursts (SGRBs) and the origin of heavy (r-process) elements \citep[e.g.][]{NSBH_rprocess_Lattimer, Blinnikov1984, Eichler_1989, Paczynski1991, Narayan1992, jets_neutrino_antineutrino, NSM_Rosswog_1998, Li_1998, NSM_Bloom_1999, Popham_1999, Fryer_1999, SGRBs_fong}. We first discuss the (mostly) distinct SGRB and kilonovae transients, followed by astrophysical highlights informed by studies of these sources.
\subsubsection{Short Gamma-Ray Bursts}
Gamma-Ray Bursts (GRBs) originate from collimated, relativistic outflows. Their prompt emission are the most luminous (EM) events in the universe, but we do not understand where or how the $\gamma$-rays are produced \citep[see, e.g.][for reviews]{GRBs_synchrotron,GRBs_photospheric}. Prompt emission is followed by afterglow from synchrotron radiation as the jet interacts with surrounding material \citep{GRB_afterglow_synchrotron}, observations of which enable studies of the local environment and energetic efficiencies \citep[e.g.][]{SGRBs_fong}. GRB 170817A was underluminous compared to the existing sample of SGRBs, and the temporal behavior of its afterglow had never been previously observed \citep[e.g.][]{GW170817_GBM,GW170817-GRB170817A,GW170817_afterglow_Xray_discovery,GW170817_structured_jet_margutti}. This led to significant debate on the structure of the non-thermal emitting region \citep[e.g.][]{GW170817_kilonova_UV,GW170817_afterglow_Xray_discovery,GW170817_structured_jet_margutti, GW170817_cocoon_Mooley,GRB170817A_Eichler,GW170817_afterglow_latettime_2018,GW170817_VLBI_Mooley,ghirlanda2019compact} and the origin of the prompt emission \citep[e.g.][]{GW170817_cocoon_Kasliwal,GW170817-GRB170817A,GRB170817A_prompt_example,KM_1}.
Multimessenger studies of NS mergers will determine the origin and implications of their non-thermal emission (prompt, afterglow, and more) \citep[e.g.][]{GW170817-GRB170817A,GRB170817A_prompt_example,SGRB_precursors,SGRB_EE,SGRB_magnetar_rowlinson_2013,SGRB_precursors,tsang2012resonant,nonthermal_Murase} and their relation to the different possible central engines \citep[e.g.][]{SGRB_magnetar_lu_2015, SGRB_magnetar_metzger_2008}, the structure of SGRB jets and their interaction with the kilonova material \citep[e.g.][]{GW170817_kilonova_UV}, and how they relate to the intrinsic parameters of the progenitor system. These studies may soon be augmented by TeV detections of SGRBs \citep{TeV_GRB_1,TeV_GRB_2,CTA_1,CTA_2,TeV_add_2,TeV_add_3}. Well characterized events will uncover the fraction of SGRBs from binary neutron star (BNS) and neutron star black hole (NSBH) mergers, and the differences between the two. The merger time from GWs and the onset time of $\gamma$-rays in joint GW-GRB detections will constrain the bulk Lorentz factor of the jet (which determines the observable region of the jet due to Doppler beaming) and the size of the $\gamma$-ray emitting region \citep[as in][]{GW170817-GRB170817A}, and provide new information to uncover the origin of the prompt emission \citep[e.g.][]{GW170817-GRB170817A,GRB170817A_prompt_example}.
\subsubsection{Kilonovae}
Prior searches for kilonovae following well-localized SGRBs resulted in a few claims of detection \citep{KN_130603B,KN_060614,KN_050709,KN_magnetar,KN_150101B} and interesting non-detections \citep[e.g.][]{kilonova_diversity_observed}; the only unambiguous, well-studied kilonova is the one that followed GW170817: AT2017gfo. This event met broad expectations, and a general picture has emerged: a bright blue kilonova from the relatively lanthanide-poor material in the polar regions followed by a transition to a fainter, progressively redder kilonova from lanthanide-rich material in the planar region \citep[e.g.][]{GW170817_kilonova_theory}. However, the ultraviolet emission at 12 hours post merger was brighter than some expectations \citep[e.g.][]{metzger2010electromagnetic,GW170817_kilonova_UV} and earlier optical and ultraviolet observations are key to determining the origin of this emission \citep{GW170817_kilonova_models_early}.
Detections of kilonovae following GW triggers will reveal the relationship between kilonova color and luminosity to properties of the progenitor, including merger type (BNS or NSBH), the short-term remnant object (BH, metastable NS, stable NS), the mass and spin of the progenitors, constrain the NS equation of state (EOS), and ultimately tie the observed diversity to specific physical processes \citep[see][for a review]{review_kilonovae_metzger}.
\subsubsection{Neutron Star-Black Hole Systems}
NSBH mergers are also predicted to result in both SGRBs and kilonova, so long as the NS is disrupted outside of the innermost stable circular orbit. However, there is no known stellar system that is definitely a NSBH binary. Since NSBH mergers have larger detectable volumes than BNS mergers \citep[e.g.][]{LVCK_prospects} we should expect a detection in the next few years, unless they are significantly less common. The first classification of a NSBH merger will likely require multimessenger observations (unless the GW signal is particularly loud). The BH can be inferred from the GW mass measures and the NS from an associated SGRB or predominantly red kilonova. Such an observation would prove NSBH systems exist by observing one merge.
\subsubsection{Stellar Formation and Evolution}
Population synthesis studies bring together models of stellar formation, stellar evolution, binary evolution, and supernovae to make predictions on the properties of compact binaries. The formation channels of compact binary systems that merge in a Hubble time include the evolution of field binaries \citep[see][for a review]{BNS_review_rasio, dominik_2013} or dynamical capture. Multimessenger observations can classify GW triggers as NS mergers, measure mass and spin distributions, the offset from their host galaxies, the local volumetric merger rates of these systems and the source evolution of those rates. These observations with population synthesis studies can differentiate between formation channels, constrain the fraction of stellar systems that result in NS mergers \citep[see, e.g.][for results from GW170817]{GW170817_LVC_progenitor}, the initial offset distribution for the compact objects, the fraction of supernovae that result in NS or BH remnants and this relation to progenitor mass \citep[see, e.g.][]{merger_rates_review_Sadowski}.
\subsection{Cosmology}
$H_0$ sets the age and size of the universe. It is one of six parameters in the base $\Lambda$CDM concordance cosmological model. It can be measured directly in the nearby (late) universe from type Ia supernovae studies \citep[e.g.][]{H0-Riess_2018}, or inferred from observations of the distant (early) universe from studies using observations of the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), and an assumed cosmological model \citep[e.g.][]{Planck_2018_cosmo}. Comparing $H_0$ values from observations of the early universe against those from observations of the late universe provides a stringent test of cosmological models. These measures disagree with 3.8$\sigma$ significance \citep{Riess_H0_38} with no obvious systematic origin \citep{H0_Riess_2016,Planck_2018_cosmo}.
Mergers of compact objects are standard sirens with an intrinsic luminosity predicted from GR which can be combined with an associated redshift to measure $H_0$ \citep{H0-Schutz-1986,schutz_2002}. The precision is limited by the correlated inclination and distance determination from GW measures, which can be ameliorated for events with an associated SGRB \citep[e.g.][]{dalal_2006,guidorzi_2017_H0,GW170817_VLBI_Mooley,hotokezaka_2018_H0}. Over the next decade standard siren cosmology could resolve the current $H_0$ controversy and independently calibrate the cosmological ladder \citep[see][for a review]{freedman2001final}. Further, as the GW interferometers improve and detect NS mergers deep into the universe, those with (EM-determined) redshift will create the most precise Hubble Diagram spanning from the local universe to deep into the deceleration era \citep{GW_IFO_3rd_Gen_EU}. Combining standard siren cosmology with CMB+BAO observations could resolve the neutrino mass hierarchy question and improve constraints on the effective number of neutrino species and the EOS of dark energy \citep{Planck_2018_cosmo}. Further adding information from forthcoming transition era cosmology experiments (e.g. LSST, EUCLID, WFIRST) gives sub-percent precision cosmology throughout the universe, allowing studies on multi-parameter extensions to $\Lambda$CDM (e.g. $\Omega_k$, $w_0$, and $w_a$ simultaneously).
\subsection{Fundamental Physics}
The $\sim$seconds difference in the arrival times of gravity and light across cosmological baselines enable observations of NS mergers to probe some fundamental aspects of physics far greater than any other method. Detections with GWs and GRBs in $\sim$keV-MeV energy (which do not undergo significant extinction, dispersion, nor absorption) provide the best measure of the speed of gravity \citep{GW170817-GRB170817A}, and relative violations of the Weak Equivalence Principle \citep[e.g.][]{Review_Tests_of_GR_overall_Will_2014,GW170817-GRB170817A, WEP_2,KM_1} and Lorentz Invariance (LIV) \citep{LIV_SME,GW170817-GRB170817A,KM_1}. GeV-TeV (EM) observations of SGRBs have the greatest discovery space, and current best constraints, on general LIV \citep{LIV_090510_limit_2009}. These observations test the Special Theory of Relativity which has been woven into all of modern physics, the Einstein Equivalence Principle that all metric theories of gravity must obey, the quantization of spacetime itself (and therefore quantum gravity), and measure fundamental constants of the universe \citep[see e.g.][]{2015CQGra..32x3001B}.
\subsection{The Origin of Heavy Elements}
For many decades, astronomers have debated whether NS mergers or Core-Collapse Supernovae (CCSN) are the site of heavy (above the iron peak), rapid neutron capture (r-process) elements \citep[see][for a summary of the arguments]{qian_2000_summary}. Even with the well-sampled AT2017gfo we are unsure if all three abundance peaks were synthesized. With a greater understanding of the r-process production in NS mergers from future kilonovae observations and improved measures on the rates of such events from GW observations, we can determine the total and relative heavy elemental abundances produced from these events. This can be compared to the observed solar system abundances to determine the fraction of heavy elements that come from NS mergers, as opposed to CCSN \citep[see e.g.][]{rprocess_1_siegel, rprocess_2_cote, rprocess_3_drout}. CCSN are thought to track the cosmic stellar formation rate, as are BNS and NSBH mergers modulo their inspiral times. An improved understanding of the source evolution of NS mergers will determine the heavy element abundance through cosmic time.
\subsection{Relativistic Jets and Particle Acceleration}
Many astrophysical sources, such as blazars, microquasars, and protostars, arise from collimated outflows referred to as jets; GRB jets are an ultrarelativistic version. Multimessenger observations provide direct measures of the central engine, which when combined with EM observations of the SGRB will uncover key information about jets. We will learn if SGRB jet formation requires an event horizon \citep{EH_Eichler} (i.e. a BH central engine), or if they can be created by magnetar central engines \citep[see, e.g.][]{review_grb_central_engine,TeV_GRB_3}. Multimessenger observations can determine required energetics efficiencies that may delineate between models of jet formation \citep[e.g.][]{jets_blandford_znajek,jets_neutrino_antineutrino,jet_process_gw170817, jet_form_2, TeV_GRB_1}. The structure of GRB jets can be studied from observations of the non-thermal emission, and move our understanding beyond the on-axis top-hat jet models rejected by GRB 170817A. Relativistic jets may be either hadron or magnetically dominated, which could be determined from (non-)detections of neutrinos \citep[e.g.][]{Zhang_Kumar,TeV_GRB_1,TeV_GRB_2,KM_2,KM_3,UHECR_A} with sufficiently sensitive detectors. The origin of cosmic rays beyond the knee or second knee energies is unknown, and neutron star mergers have been suggested among the promising candidate sources \citep{UHECR_A, UHECR_B}.
\subsection{Neutron Star Equation of State}
NSs are the densest known matter in the universe, far beyond anything achievable in terrestrial laboratories. The NS EOS characterizes the density-pressure relationship \citep[see][for a review]{NS_EOS_review_Lattimer}. From an assumed EOS, several observables can be predicted, such as a mass-radius relation \citep[e.g.][]{NS_EOS_review_Ozel}. In BNS mergers the remnant object just after merger can be a stable NS, a metastable NS, or a BH, with the NS EOS determining which case for a given merger. These options can be resolved by observations of associated SGRBs or kilonovae with specific characteristics \citep[e.g.][]{SGRB_EE, SGRB_magnetar_metzger_2008, SGRB_magnetar_rowlinson_2013, metzger2014optical, siegel2016electromagnetic, nonthermal_Murase, GW170817-GRB170817A,NS_EOS_metzger,NS_EOS_Radice_2017, NS_EOS_margalit_2017, NS_EOS_radice_2018} and eventually directly confirmed with GW or MeV neutrino observations \citep{MeV_neutrino_1,MeV_neutrino_2}. Multimessenger observations of NSBH mergers provide additional constraints as the orbital radius for NS disruption is a strong function of the mass and spin of the BH and the NS radius \citep{NS_EOS_foucart_2012}. Multimessenger observations of NS mergers will provide information, complementary to EM-only or GW-only constraints, to determine the EOS of supranuclear matter, and possibly constrain the phase diagram of quantum chromodynamics \citep[e.g.][]{NS_EOS_most_2018, NS_EOS_bauswein_2018}.
\section{Recommendations}
NS mergers emit across decades in energy in several messengers which enables greater understanding but requires vast observational resources. Prompt $\gamma$-ray, afterglow (TeV to radio), and TeV-PeV neutrino observations will provide new insight into SGRBs, relativistic jets, and particle acceleration. Ultraviolet, optical, and infrared (UVOIR) observations uncover and characterize kilonovae and the origin of heavy elements. Probing fundamental physics requires GW and prompt GRB detections. Cosmological studies require GW distance and EM redshift determinations. All observations help constrain the NS EOS. Much of this science is only possible with a population of events. Information on the inspiral, coalescence, or jet before the coasting phase (obtained from GW, $\gamma$-ray and neutrino detectors) can only be done from serendipitous observations, requiring all-sky monitors with high duty cycles and large fields of view.
Some detections also enable other observations. Despite collimation, joint GW-GRB detections are expected to be reasonably common because of preferential selection effects \citep[combining information from][]{GW_inclination,SGRBs_fong} and the increase of GW detections from GRB observations \citep[e.g.][]{joint_williamson, joint_blackburn}. Joint GW-GRB detections further constrain the region of interest for follow-up. Arcsecond localizations are necessary for host-galaxy identification and redshift determination and are only possible from follow-up observations. The earliest follow-up signal that can be detected is the GRB afterglow; however, based on the prompt vs afterglow brightness of GRB 170817A, we do not expect off-axis afterglow detections (with no detectable prompt emission) to be common. When there is no detectable SGRB, the first EM emission for a blue kilonova appears to be in the UV and in optical or infrared for red kilonovae (on longer timescales).
Kilonovae will be the dominant EM counterpart for nearby NS mergers as they are omnidirectional, but SGRBs will dominate for distant events as they can be significantly brighter. While we split our recommendations between the next decade and beyond, regardless of time period, \textit{we recommend vigorous funding for upgraded GW interferometers}.
\subsection{The Next Ten Years}
With the full design network of current GW detectors, localizations sufficient for existing optical facilities will be relatively common \citep{LVCK_prospects}. The funded LIGO A+ upgrade (nominally available in 2024), will detect a NS merger roughly once a week. Follow-up ground observations with current or expected missions (e.g. HAWC, CTA, VLA, SKA, ZTF, LSST) reliably cover TeV, optical, near-IR, and radio. As kilonovae transition to redder emission over time, the IR-only emission is sufficiently late ($\sim$days) that these wavelengths can be covered by JWST and WFIRST observations. \textit{We endorse the allocation of appropriate observing time and target of opportunity programs for pointed telescopes and directly recommend LSST follow-up of NS mergers}. Megaton-class MeV neutrino detectors could detect particularly nearby NS mergers. Initial upgrades to TeV-PeV neutrino detectors are on-going; \textit{we endorse the full IceCube Gen-2} \citep{IceCube-Gen2}.
The critical wavelengths detectable only in space are keV-MeV $\gamma$-rays, X-rays, and UV. As such, we recommend the extension of the \textit{Fermi} mission (because it detects more prompt SGRBs than all other active mission combined) and the \textit{Neil Gehrels Swift Observatory} (primarily for fast-response X-ray and UV coverage). To capture the full range of possible kilonova colors, \textit{we recommend wide-field UV (space-based) and NIR (ground-based) facilities}. To enable time-domain astronomy, \textit{we recommend improvements to real-time communication for space-based missions.}
Beyond the required observational capabilities, \textit{we strongly recommend greater NSF-NASA cooperation} as both agencies have assets critical for multimessenger science. \textit{Specific recommendations include the funding of beam studies to understand the nuclear processes that currently limit kilonova models \citep[e.g.][]{review_kilonovae_metzger, FRIB_rprocess}, robust funding of multimessenger simulation and theory studies via the TCAN program (created in response to Astro2010), and the consideration of resolving grand problems through NASA-NSF partnership (akin to the DRIVE initiative from the 2013 Solar and Space Science Decadal).} \textit{Technical improvements include improvements to real-time reporting, automated multimission and multimessenger searches, and prompt reporting of initial parameter estimation from GW detections (e.g. masses) to enable follow-up prioritization.}
\subsection{Future Large-scale Missions}
The proposed LIGO-Voyager upgrade has a nominal timeline of $\sim$2030 and it would detect several NS mergers per week with some at cosmological distances ($z>0.1$). \textit{Proposed large-scale missions in the Astro2020 Decadal must be designed for the 2030s, not the current era.} Proposed third generation GW interferometers \citep{GW_IFO_3rd_Gen_US,GW_IFO_3rd_Gen_EU} with best-case timelines of mid-to-late 2030s could detect dozens of NS mergers per day with some beyond the transition era. Fewer upgraded interferometers will result in poorer GW localizations for distant events. To enable all NS merger science \textit{we recommend future GW interferometers aim for broadband sensitivity improvements}. With these upgrades a significant fraction of SGRBs will have associated GW emission. Therefore, \textit{we recommend a large-scale $\gamma$-ray observatory that will detect far more SGRBs than any prior mission (through improved $\sim$keV-MeV sensitivity and broad sky coverage) and localize them to sufficient accuracy for sensitive follow-up observations.} To enable a full study of these events, including redshift determination, they need to be localized to arcsecond precision. With such a $\gamma$-ray instrument, \textit{we would need well-matched follow-up facilities such as a sensitive, fast-reponse, high spatial resolution X-ray telescope and sensitive UVOIR and radio telescopes.}
\pagebreak
\bibliographystyle{abbrv}
|
2,877,628,089,518 | arxiv |
\section{Conclusion}
In this paper, we show that phrase retrieval models also learn passage retrieval without any modification.
By drawing connections between the objectives of DPR and DensePhrases, we provide a better understanding of how phrase retrieval learns passage retrieval, which is also supported by several empirical evaluations on multiple benchmarks.
Specifically, phrase-based passage retrieval has better retrieval quality on top $k$ passages when $k$ is small, and this translates to an efficient use of passages for open-domain QA.
We also show that DensePhrases can be fine-tuned for more coarse-grained retrieval units, serving as a basis for any retrieval unit.
We plan to further evaluate phrase-based passage retrieval on standard information retrieval tasks such as MS MARCO.
\section*{Acknowledgements}
We thank Chris Sciavolino, Xingcheng Yao, the members of the Princeton NLP group, and the anonymous reviewers for helpful discussion and valuable feedback.
This research is supported by the James Mi *91 Research Innovation Fund for Data Science and gifts from Apple and Amazon.
It was also supported in part by the ICT Creative Consilience program (IITP-2021-0-01819) supervised by the IITP (Institute for Information \& communications Technology Planning \& Evaluation) and National Research Foundation of Korea (NRF-2020R1A2C3010638).
\section*{Ethical Considerations}
Models introduced in our work often use question answering datasets such as Natural Questions to build phrase or passage representations.
Some of the datasets, like SQuAD, are created from a small number of popular Wikipedia articles, hence could make our model biased towards a small number of topics.
We hope that inventing an alternative training method that properly regularizes our model could mitigate this problem.
Although our efforts have been made to reduce the computational cost of retrieval models, using passage retrieval models as external knowledge bases will inevitably increase the resource requirements for future experiments.
Further efforts should be made to make retrieval more affordable for independent researchers.
\section{A Unified View of Dense Retrieval}
\label{sec:analysis}
As shown in the previous section, phrase-based passage retrieval is able to achieve competitive passage retrieval accuracy, despite that the models were not explicitly trained for that.
In this section, we compare the training objectives of DPR and DensePhrases in detail and explain how DensePhrases learns passage retrieval.
\subsection{Training Objectives}\label{sec:objectives}
Both DPR and DensePhrases set out to learn a similarity function~$f$ between a passage or phrase and a question.
Passages and phrases differ primarily in characteristic length,
so we refer to either as a retrieval unit $x$.\footnote{Note that phrases may overlap, whereas passages are usually disjoint segments with each other.}
DPR and DensePhrases both adopt a dual-encoder approach with inner product similarity as shown in Eq.~\eqref{eqn:passage-ret} and \eqref{eqn:phrase-ret}, and they are initialized with BERT~\citep{devlin2019bert} and SpanBERT~\citep{joshi2020spanbert}, respectively.
These dual-encoder models are then trained with a negative log-likelihood loss for discriminating positive retrieval units from negative ones:
\begin{equation}\label{eqn:loss}
\mathcal{L} = -\log \frac{e^{f(x^+, q)}}{e^{f(x^+, q)} + \sum\limits_{x^- \in \mathcal{X}^-} e^{f(x^-, q)}},
\end{equation}
where $x^+$ is the positive phrase or passage corresponding to question $q$, and $\mathcal{X}^-$ is a set of negative examples.
The choice of negatives is critical in this setting and both DPR and DensePhrases make important adjustments.
\paragraph{In-batch negatives}~
In-batch negatives are a common way to define $\mathcal{X}^-$, since they are available at no extra cost when encoding a mini-batch of examples. Specifically, in a mini-batch of $B$ examples, we can add $B-1$ in-batch negatives for each positive example.
Since each mini-batch is randomly sampled from the set of all training passages, in-batch negative passages are usually \textit{topically negative}, i.e., models can discriminate between $x^+$ and $\mathcal{X}^-$ based on their topic only.
\paragraph{Hard negatives}~
Although topic-related features are useful in identifying broadly relevant passages, they often lack the precision to locate the exact passage containing the answer in a large corpus.
\citet{karpukhin2020dense}~propose to use additional hard negatives which have a high BM25 lexical overlap with a given question but do not contain the answer.
These hard negatives are likely to share a similar topic and encourage DPR to learn more fine-grained features to rank $x^+$ over the hard negatives.
Figure \ref{fig:dpr-densephrases} (left) shows an illustrating example.
\paragraph{In-passage negatives}~
While DPR is limited to use positive passages $x^+$ which contain the answer, DensePhrases is trained to predict that the positive phrase $x^+$ \textit{is} the answer.
Thus, the fine-grained structure of phrases allows for another source of negatives, \textit{in-passage negatives}.
In particular, DensePhrases augments the set of negatives $\mathcal{X}^-$ to encompass all phrases within the same passage that do not express the answer.\footnote{Technically, DensePhrases treats start and end representations of phrases independently and use start (or end) representations other than the positive one as negatives.}
See Figure~\ref{fig:dpr-densephrases} (right) for an example.
We hypothesize that these in-passage negatives achieve a similar effect as DPR's hard negatives: They require the model to go beyond simple topic modeling since they share not only the same topic but also the same context.
Our phrase-based passage retriever might benefit from this phrase-level supervision, which has already been shown to be useful in the context of distilling knowledge from reader to retriever \citep{izacard2021distilling,yang2020retriever}.
\subsection{Topical vs. Hard Negatives}
To address our hypothesis, we would like to study how these different types of negatives used by DPR and DensePhrases affect their reliance on topical and fine-grained entailment cues.
We characterize their passage retrieval based on two metrics (losses): $\mathcal{L}_\text{topic}$ and $\mathcal{L}_\text{hard}$.
We use Eq.~\eqref{eqn:loss} to define both $\mathcal{L}_\text{topic}$ and $\mathcal{L}_\text{hard}$, but use different sets of negatives $\mathcal{X}^-$.
For $\mathcal{L}_\text{topic}$, $\mathcal{X}^-$ contains passages that are topically different from the gold passage---In practice, we randomly sample passages from English Wikipedia.
For $\mathcal{L}_\text{hard}$, $\mathcal{X}^-$ uses negatives containing topically similar passages, such that $\mathcal{L}_\text{hard}$ estimates how accurately models locate a passage that contains the exact answer among topically similar passages.
From a positive passage paired with a question, we create a single hard negative by removing the sentence that contains the answer.\footnote{While $\mathcal{L}_\text{hard}$ with this type of hard negatives might favor DensePhrases, using BM25 hard negatives for $\mathcal{L}_\text{hard}$ would favor DPR since DPR was directly trained on BM25 hard negatives. Nonetheless, we observed similar trends in $\mathcal{L}_\text{hard}$ regardless of the choice of hard negatives.}
In our analysis, both metrics are estimated on the Natural Questions development set, which provides a set of questions and (gold) positive passages.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.55]{figures/analysis-syn.pdf}
\caption{
Comparison of DPR and DensePhrases on NQ (dev) with $\mathcal{L}_\text{topic}$ and $\mathcal{L}_\text{hard}$.
Starting from each model trained with in-batch negatives (in-batch), we show the effect of using hard negatives (+BM25), in-passage negatives (+in-passage), as well as training on multiple QA datasets (+multi. dataset).
The $x$-axis is in log-scale for better visualization. For both metrics, lower numbers are better.
}\label{fig:metrics}\vspace{-0.1cm}
\end{figure}
\paragraph{Results}~
Figure~\ref{fig:metrics} shows the comparison of DPR and DensePhrases trained on NQ with the two losses.
For DensePhrases, we compute the passage score using $\tilde{f}(p,q)$ as described in Eq.~\eqref{eqn:pp-ret}.
First, we observe that in-batch negatives are highly effective at reducing $\mathcal{L}_\text{topic}$ as DensePhrases trained with only in-passage negatives has a relatively high $\mathcal{L}_\text{topic}$.
Furthermore, we observe that using in-passage negatives in DensePhrases (+in-passage) significantly lowers $\mathcal{L}_\text{hard}$, even lower than DPR that uses BM25 hard negatives (+BM25).
Using multiple datasets (+multi. dataset) further improves $\mathcal{L}_\text{hard}$ for both models.
DPR has generally better (lower) $\mathcal{L}_\text{topic}$ than DensePhrases, which might be due to the smaller training batch size of DensePhrases (hence a smaller number of in-batch negatives) compared to DPR.
The results suggest that DensePhrases relies less on topical features and is better at retrieving passages based on fine-grained entailment cues.
This might contribute to the better ranking of the retrieved passages in Table~\ref{tab:openqa}, where DensePhrases shows better MRR@20 and P@20 while top-20 accuracy is similar.
\paragraph{Hard negatives for DensePhrases?}~
We test two different kinds of hard negatives in DensePhrases to see whether its performance can further improve in the presence of in-passage negatives.
For each training question, we mine for a hard negative passage, either by BM25 similarity or by finding another passage that contains the gold-answer phrase, but possibly with a wrong context.
Then we use all phrases from the hard negative passage as additional hard negatives in $\mathcal{X}^-$ along with the existing in-passage negatives.
As shown in Table~\ref{tab:hard-negatives}, DensePhrases obtains no substantial improvements from additional hard negatives, indicating that in-passage negatives are already highly effective at producing good phrase (or passage) representations.
\input{tables/hard_negatives}
\section{Encoding Phrase Representations}\label{sec:apdx1}
\label{sec:appendix}
\section{Background}\label{sec:background}
\paragraph{Passage retrieval}~
Given a set of documents $\mathcal{D}$, passage retrieval aims to provide a set of relevant passages for a question $q$.
Typically, each document in $\mathcal{D}$ is segmented into a set of disjoint passages and we denote the entire set of passages in $\mathcal{D}$ as $\mathcal{P} = \{p_1, \dots, p_M\},$ where each passage can be a natural paragraph or a fixed-length text block.
A passage retriever is designed to return top-$k$ passages $\mathcal{P}_k \subset \mathcal{P}$ with the goal of retrieving passages that are relevant to the question.
In open-domain QA, passages are considered relevant if they contain answers to the question. However,
many other knowledge-intensive NLP tasks (e.g., knowledge-grounded dialogue) provide human-annotated evidence passages or documents.
While traditional passage retrieval models rely on sparse representations such as BM25~\citep{robertson2009probabilistic}, recent methods show promising results with dense representations of passages and questions, and enable retrieving passages that may have low lexical overlap with questions.
Specifically, \citet{karpukhin2020dense}~introduce DPR that has a \ti{passage encoder} $E_p(\cdot)$ and a \ti{question encoder} $E_q(\cdot)$ trained on QA datasets and retrieves passages by using the inner product as a similarity function between a passage and a question:
\begin{equation}\label{eqn:passage-ret}
f(p, q) = E_p(p)^\top E_q(q).
\end{equation}
For open-domain QA where a system is required to provide an exact answer string $a$, the retrieved top $k$ passages $\mathcal{P}_k$ are subsequently fed into a reading comprehension model such as a BERT model~\citep{devlin2019bert}, and this is called the retriever-reader approach~\citep{chen2017reading}.
\input{tables/openqa}
\paragraph{Phrase retrieval}~
While passage retrievers require another reader model to find an answer, \citet{seo2019real} introduce the phrase retrieval approach that encodes phrases in each document and performs similarity search over all phrase vectors to directly locate the answer.
Following previous work~\citep{seo2018phrase,seo2019real}, we use the term `phrase' to denote any contiguous text segment up to $L$ words (including single words), which is not necessarily a linguistic phrase and we take phrases up to length $L=20$.
Given a phrase $s^{(p)}$ from a passage $p$, their similarity function $f$ is computed as:
\begin{equation}\label{eqn:phrase-ret}
\begin{split}
f(s^{(p)}, q) = {E}_{s}(s^{(p)})^\top {E}_{q}(q),
\end{split}
\end{equation}
where $E_s(\cdot)$ and $E_q(\cdot)$ denote the \ti{phrase encoder} and the \ti{question encoder}, respectively.
Since this formulates open-domain QA purely as a maximum inner product search (MIPS), it can drastically improve end-to-end efficiency.
While previous work~\cite{seo2019real,lee2020contextualized} relied on a combination of dense and sparse vectors, \citet{lee2021learning} demonstrate that dense representations of phrases alone are sufficient to close the performance gap with retriever-reader systems.
For more details on how phrase representations are learned, we refer interested readers to \citet{lee2021learning}.
\section{Improving Coarse-grained Retrieval}
\label{sec:coarse}
While we showed that DensePhrases implicitly learns passage retrieval, Figure~\ref{fig:metrics} indicates that DensePhrases might not be very good for retrieval tasks where topic matters more than fine-grained entailment, for instance, the retrieval of a single evidence document for entity linking.
In this section, we propose a simple method that can adapt DensePhrases to larger retrieval units, especially when the topical relevance is more important.
\paragraph{Method}~
We modify the query-side fine-tuning proposed by \citet{lee2021learning}, which drastically improves the performance of DensePhrases by reducing the discrepancy between training and inference time.
Since it is prohibitive to update the large number of phrase representations after indexing, only the query encoder is fine-tuned over the entire set of phrases in Wikipedia.
Given a question $q$ and an annotated document set $\mathcal{D}^*$, we minimize:
\begin{equation}\label{eqn:qsft}
\mathcal{L}_\text{doc} = -\log \frac{\sum_{\substack{s \in \tilde{\mathcal{S}}(q), {d(s) \in \mathcal{D}^*}}} e^{f(s, q)}}{\sum_{s \in \tilde{\mathcal{S}}(q)} e^{f(s, q)}},
\end{equation}
where $\tilde{\mathcal{S}}(q)$ denotes top $k$ phrases for the question $q$, out of the entire set of phrase vectors.
To retrieve coarse-grained text better, we simply check the condition $d(s) \in \mathcal{D}^*$, which means $d(s)$, the source document of $s$, is included in the set of annotated gold documents $\mathcal{D}^*$ for the question.
With $\mathcal{L}_\text{doc}$, the model is trained to retrieve any phrases that are contained in a relevant document.
Note that $d(s)$ can be changed to reflect any desired level of granularity such as passages.
\paragraph{Datasets}\label{sec:kilt}~
We test DensePhrases trained with $\mathcal{L}_\text{doc}$ on entity linking~\citep{hoffart2011robust,guo2018robust} and knowledge-grounded dialogue~\citep{dinan2019wizard} tasks in KILT~\citep{petroni2021kilt}.
Entity linking contains three datasets: AIDA CoNLL-YAGO (AY2)~\citep{hoffart2011robust}, WNED-WIKI (WnWi)~\citep{guo2018robust}, and WNED-CWEB (WnCw)~\citep{guo2018robust}.
Each query in entity linking datasets contains a named entity marked with special tokens (i.e., \ttt{[START\_ENT]}, \ttt{[END\_ENT]}), which need to be linked to one of the Wikipedia articles.
For knowledge-grounded dialogue, we use Wizard of Wikipedia (WoW)~\citep{dinan2019wizard} where each query consists of conversation history, and the generated utterances should be grounded in one of the Wikipedia articles.
We follow the KILT guidelines and evaluate the document (i.e., Wikipedia article) retrieval performance of our models given each query. We use R-precision, the proportion of successfully retrieved pages in the top R results, where R is the number of distinct pages in the provenance set.
However, in the tasks considered, R-precision is equivalent to precision@1, since each question is annotated with only one document.
\paragraph{Models}\label{sec:kilt-model}~
DensePhrases is trained with the original query-side fine-tuning loss (denoted as $\mathcal{L}_\text{phrase}$) or with $\mathcal{L}_\text{doc}$ as described in Eq.~\eqref{eqn:qsft}.
When DensePhrases is trained with $\mathcal{L}_\text{phrase}$, it labels any phrase that matches the title of gold document as positive.
After training, DensePhrases returns the document that contains the top passage.
For baseline retrieval methods, we report the performance of TF-IDF and DPR from~\citet{petroni2021kilt}.
We also include a multi-task version of DPR and DensePhrases, which uses the entire KILT training datasets.\footnote{We follow the same steps described in \citet{petroni2021kilt} for training the multi-task version of DensePhrases.}
While not our main focus of comparison, we also report the performance of other baselines from \citet{petroni2021kilt}, which uses generative models (e.g., RAG~\citep{lewis2020retrieval}) or task-specific models (e.g., BLINK~\citep{wu2020scalable}, which has additional entity linking pre-training).
Note that these methods use additional components such as a generative model or a cross-encoder model on top of retrieval models.
\input{tables/kilt}
\paragraph{Results}~
Table~\ref{tab:kilt} shows the results on three entity linking tasks and a knowledge-grounded dialogue task.
On all tasks, we find that DensePhrases with $\mathcal{L}_\text{doc}$ performs much better than DensePhrases with $\mathcal{L}_\text{phrase}$ and also matches the performance of RAG that uses an additional large generative model to generate the document titles.
Using $\mathcal{L}_\text{phrase}$ does very poorly since it focuses on phrase-level entailment, rather than document-level relevance.
Compared to the multi-task version of DPR (i.e., DPR$^\clubsuit$), DensePhrases-$\mathcal{L}_\text{doc}$$^\clubsuit$ can be easily adapted to non-QA tasks like entity linking and generalizes better on tasks without training sets (WnWi, WnCw).
\section{Discussion}
\label{sec:discussion}
\paragraph{Diverse retrieval}
\begin{itemize}
\item prev works on retrieving diverse passages~\citep{min2021joint}
\end{itemize}
\paragraph{Multi-vector encoding}
\begin{itemize}
\item how our model is related to multi-vector
\end{itemize}
\section{Experiments}
\label{sec:experiment}
We demonstrate the performance and versatility of phrase-based passage retrieval in two different settings: First, we examine whether the empirical benefits in retrieval observed in Section \ref{sec:prelim_exp} translate to gains in the full open-domain QA setting (\Cref{sec:openqa})
Second, we apply our system to document-level retrieval for entity-linking and knowledge-grounded dialogue in the KILT dataset \citep{petroni2021kilt} to show that it can be easily adapted to a new task and retrieval unit (\Cref{sec:kilt}).
\begin{comment}
\subsection{Open-domain QA}\label{sec:openqa}
Recently, \citet{izacard2020leveraging} have shown that a generative decoder can benefit from accessing a large quantity of (independently encoded) top-$k$ passages, achieving state-of-the-art performance on open-domain QA. However, processing up to 100 passages of 100 word length with a T5 model takes vast computational resources.%
\footnote{According to \citet{izacard2020leveraging}, it takes up to 425 GPU hours for training their largest model.}
In this case study, we ask the question if comparable performance can be achieved with smaller passages as the input, drastically reducing the computational requirements for this \textit{Fusion-in-Decoder} (FiD) approach.
\paragraph{Implementation Details}~
We evaluate each model on the two open-domain QA datasets used in \Cref{sec:prelim_exp}: Natural Questions~\citep{kwiatkowski2019natural} and TriviaQA~\citep{joshi2017triviaqa}.
For the experiments, we replace the retriever of FiD to DensePhrases-multi.
We use T5-base models to process up to top-50 passages, mainly due to the resource constraint.
DensePhrases can also retrieve sentences out-of-the-box by locating sentences that contain the phrases.
Hence, we train another T5-base model on top-100 sentences, which consumes much smaller GPU memory compared to using 100 passages.
\paragraph{Results}~
As shown in Table~\ref{tab:openqa-em}, using DensePhrases as a passage retriever achieves competitive performance with DPR + T5-base models.
Its better retrieval quality at top-$k$ indeed translates to better open-domain QA accuracy, achieving +6.4\% gain compared to DPR + T5-base when $k=5$.
To obtain comparable performance with using $k=100$ passages from DPR, DensePhrases only requires $k=25 \sim 50$ passages.
Although reading $k^*=100$ sentences requires much smaller memory than reading $k=100$ passages, its performance isn't as good as using passages, indicating that reading the passages is better than reading the sentences for the generative readers despite of its computational cost.
\end{comment}
\subsection{Coarse-grained Retrieval}\label{sec:kilt}
While DensePhrases has been directly used to retrieve phrases for open-domain QA and slot filling tasks~\citep{lee2021learning}, we further train DensePhrases to better retrieve coarse-grained retrieval units such as documents.
Specifically, we use entity linking and knowledge-grounded dialogue tasks from KILT~\citep{petroni2021kilt} where each model needs to provide a correct document in English Wikipedia.
We use document-level supervision as described in \Cref{sec:coarse} to train DensePhrases.
\paragraph{Results}~
In the last row of Table~\ref{tab:openqa}, we show the result of applying passage-level supervision during query-side fine-tuning.
Table~\ref{tab:kilt} shows the results on three entity linking tasks and a knowledge-grounded dialogue task.
\jinhyuk{results pending}
\input{tables/kilt}
\begin{comment}
\subsection{Efficient Phrase Retrieval}\label{sec:efficient-result}
The multi-vector encoding models as well as ours are notoriously large since they contain multiple vector representations for every passage in the entire corpus.
In Figure~\ref{fig:efficient}, we plot the changes in top-5 retrieval accuracy with respect to the size of the phrase index in DensePhrases.
By tuning the of phrase filtering properly, the number of vectors per each passage can be reduced without hurting the performance significantly.
The performance decreases with the smaller number of vectors per passage, which aligns with the findings of multi-vector encoding models~\citep{humeau2020polyencoders,luan2021sparse}.
While applying OPQ can reduce the size of the phrase index in DensePhrases from 320GB to 84GB, the top-5 retrieval accuracy also deteriorates a lot.
Applying quantization-aware query-side fine-tuning can recover the performance by mitigating the effect of quantization loss.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.40]{figures/efficient}
\caption{
Top-$k$ accuracy on Natural Questions (dev) for different index sizes in DensePhrases.
We can easily control the size of the phrase index of DensePhrases by tuning the filtering threshold.
\jinhyuk{exact numbers will be changed}
}
\label{fig:efficient}
\end{figure}
\end{comment}
\section{Introduction}
Dense retrieval aims to retrieve relevant contexts from a large corpus, by learning dense representations of queries and text segments.
Recently, dense retrieval of passages~\citep{lee2019latent,karpukhin2020dense,xiong2021approximate} has been shown to outperform traditional sparse retrieval methods such as TF-IDF and BM25 in a range of knowledge-intensive NLP tasks~\cite{petroni2021kilt}, including open-domain question answering (QA)~\citep{chen2017reading}, entity linking~\citep{wu2020scalable}, and knowledge-grounded dialogue~\cite{dinan2019wizard}.
\begin{figure}[!t]
\centering
\includegraphics[scale=1.05]{figures/overview}
\vspace{-0.2cm}
\caption{
Comparison of passage representations from DPR~\citep{karpukhin2020dense} and DensePhrases~\citep{lee2021learning}. Unlike using a single vector for each passage, DensePhrases represents each passage with multiple phrase vectors and the score of a passage can be computed by the maximum score of phrases within it.
}
\label{fig:overview} \vspace{-0.4cm}
\end{figure}
One natural design choice of these dense retrieval methods is the retrieval unit. For instance, the dense passage retriever (DPR)~\cite{karpukhin2020dense} encodes a fixed-size text block of 100 words as the basic retrieval unit.
On the other extreme, recent work~\cite{seo2019real,lee2021learning} demonstrates that phrases can be used as a retrieval unit.
In particular, \newcite{lee2021learning} show that learning dense representations of phrases alone can achieve competitive performance in a number of open-domain QA and slot filling tasks.
This is particularly appealing since the phrases can directly serve as the output, without relying on an additional reader model to process text passages.
In this work, we draw on an intuitive motivation that every single phrase is embedded within a larger text context and ask the following question: If a retriever is able to locate phrases, can we directly make use of it for passage and even document retrieval as well?
We formulate \emph{phrase-based passage retrieval}, in which the score of a passage is determined by the maximum score of phrases within it (see Figure~\ref{fig:overview} for an illustration).
By evaluating DensePhrases~\cite{lee2021learning} on popular QA datasets, we observe that it achieves competitive or even better passage retrieval accuracy compared to DPR, without any re-training or modification to the original model (Table~\ref{tab:openqa}).
The gains are especially pronounced for top-$k$ accuracy when $k$ is smaller (e.g., 5), which also helps achieve strong open-domain QA accuracy with a much smaller number of passages as input to a generative reader model~\cite{izacard2020leveraging}.
To better understand the nature of dense retrieval methods, we carefully analyze the training objectives of phrase and passage retrieval methods.
While the in-batch negative losses in both models encourage them to retrieve topically relevant passages, we find that phrase-level supervision in DensePhrases provides a stronger training signal than using hard negatives from BM25, and helps DensePhrases retrieve correct phrases, and hence passages.
Following this positive finding, we further explore whether phrase retrieval can be extended to retrieval of coarser granularities, or other NLP tasks.
Through fine-tuning of the query encoder with document-level supervision, we are able to obtain competitive performance on entity linking~\citep{hoffart2011robust} and knowledge-grounded dialogue retrieval~\citep{dinan2019wizard} in the KILT benchmark~\citep{petroni2021kilt}.
Finally, we draw connections to multi-vector passage encoding models~\citep{khattab2020colbert,luan2021sparse}, where phrase retrieval models can be viewed as learning a dynamic set of vectors for each passage.
We show that a simple phrase filtering strategy learned from QA datasets gives us a control over the trade-off between the number of vectors per passage and the retrieval accuracy.
Since phrase retrievers encode a larger number of vectors, we also propose a quantization-aware fine-tuning method based on Optimized Product Quantization~\citep{ge2013optimized}, reducing the size of the phrase index from 307GB to 69GB (or under 30GB with more aggressive phrase filtering) for full English Wikipedia, without any performance degradation.
This matches the index size of passage retrievers and makes dense phrase retrieval a practical and versatile solution for multi-granularity retrieval.
\section{DensePhrases as a Multi-Vector Passage Encoder}\label{sec:efficient}
In this section, we demonstrate that DensePhrases can be interpreted as a multi-vector passage encoder, which has recently been shown to be very effective for passage retrieval~\cite{luan2021sparse,khattab2020colbert}.
Since this type of multi-vector encoding models requires a large disk footprint, we show that we can control the number of vectors per passage (and hence the index size) through filtering. We also introduce quantization techniques to build more efficient phrase retrieval models without a significant performance drop.
\begin{comment}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.40]{figures/efficient}
\caption{
Top-$k$ accuracy on Natural Questions (dev) for different index sizes in DensePhrases.
We can easily control the size of the phrase index of DensePhrases by tuning the filtering threshold.
\jinhyuk{exact numbers will be changed}\jinhyuk{change style}
}
\label{fig:efficient}
\end{figure}
\end{comment}
\subsection{Multi-Vector Encodings}\label{sec:multivec}
Since we represent passages not by a single vector, but by a set of phrase vectors (decomposed as token-level start and end vectors, see \citet{lee2021learning}), we notice similarities to previous work, which addresses the capacity limitations of dense, fixed-length passage encodings.
While these approaches store a fixed number of vectors per passage~\citep{luan2021sparse,humeau2020polyencoders} or all token-level vectors~\citep{khattab2020colbert}, phrase retrieval models store a dynamic number of phrase vectors per passage, where many phrases are filtered by a model trained on QA datasets.
Specifically, \citet{lee2021learning} trains a binary classifier (or a phrase filter) to filter phrases based on their phrase representations.
This phrase filter is supervised by the answer annotations in QA datasets, hence denotes candidate answer phrases.
In our experiment, we tune the filter threshold to control the number of vectors per passage for passage retrieval.
\subsection{Efficient Phrase Retrieval}\label{sec:efficient-phrase}
The multi-vector encoding models as well as ours are prohibitively large since they contain multiple vector representations for every passage in the entire corpus.
We introduce a vector quantization-based method that can safely reduce the size of our phrase index, without performance degradation.
\paragraph{Optimized product quantization}~
Since the multi-vector encoding models are prohibitively large due to their multiple representations, we further introduce a vector quantization-based method that can safely reduce the size of our phrase index, without performance degradation.
We use Product Quantization (PQ)~\citep{jegou2010product} where the original vector space is decomposed into the Cartesian product of subspaces.
Using PQ, the memory usage of using $N$ number of $d$-dimensional centroid vectors reduces from $Nd$ to $N^{1/M}d$ with $M$ subspaces while each database vector requires $\log_2 N$ bits.
Among different variants of PQ, we use Optimized Product Quantization (OPQ)~\citep{ge2013optimized}, which learns an orthogonal matrix $R$ to better decompose the original vector space.
See \citet{ge2013optimized} for more details on OPQ.
\begin{figure}[!t]
\includegraphics[scale=0.55]{figures/size_top5.pdf}
\centering
\caption{
Top-5 passage retrieval accuracy on Natural Questions (dev) for different index sizes of DensePhrases.
The index size (GB) and the average number of saved vectors per passage (\#~vec~/~$p$) are controlled by the filtering threshold $\tau$.
For instance, \#~vec~/~$p$ reduces from 28.0 to 5.1 with higher $\tau$, which also reduces the index size from 69GB to 23GB.
OPQ: Optimized Product Quantization~\citep{ge2013optimized}.
}\label{fig:efficient}\vspace{0.0cm}
\end{figure}
\paragraph{Quantization-aware training}~
While this type of aggressive vector quantization can significantly reduce memory usage, it often comes at the cost of performance degradation due to the quantization loss.
To mitigate this problem, we use quantization-aware query-side fine-tuning motivated by the recent successes on quantization-aware training~\citep{jacob2018quantization}.
Specifically, during query-side fine-tuning, we reconstruct the phrase vectors using the trained (optimized) product quantizer, which are then used to minimize Eq.~\eqref{eqn:qsft}.
\subsection{Experimental Results}
In Figure~\ref{fig:efficient}, we present the top-5 passage retrieval accuracy with respect to the size of the phrase index in DensePhrases.
First, applying OPQ can reduce the index size of DensePhrases from 307GB to 69GB, while the top-5 retrieval accuracy is poor without quantization-aware query-side fine-tuning.
Furthermore, by tuning the threshold $\tau$ for the phrase filter, the number of vectors per each passage (\#~vec~/~$p$) can be reduced without hurting the performance significantly.
The performance improves with a larger number of vectors per passage, which aligns with the findings of multi-vector encoding models~\citep{khattab2020colbert,luan2021sparse}.
Our results show that having 8.8 vectors per passage in DensePhrases has similar retrieval accuracy with DPR.
\section{Phrase Retrieval for Passage Retrieval}
\label{sec:prelim_exp}
Phrases naturally have their source texts from which they are extracted.
Based on this fact, we define a simple phrase-based passage retrieval strategy, where we retrieve passages based on the phrase-retrieval score:
\begin{equation}\label{eqn:pp-ret}
\tilde{f}(p, q) := \mathop{\text{max}}_{s^{(p)} \in \mathcal{S}(p)}{E}_{s}(s^{(p)})^\top {E}_{q}(q),
\end{equation}
where $\mathcal{S}(p)$ denotes the set of phrases in the passage $p$.
In practice, we first retrieve a slightly larger number of phrases, compute the score for each passage, and return top $k$ unique passages.\footnote{In most cases, retrieving $2k$ phrases is sufficient for obtaining $k$ unique passages. If not, we try $4k$ and so on.}
Based on our definition, phrases can act as a basic retrieval unit of any other granularity such as sentences or documents by simply changing $\mathcal{S}(p)$ (e.g., $s^{(d)} \in \mathcal{S}(d)$ for a document $d$).
Note that, since the cost of score aggregation is negligible, the inference speed of phrase-based passage retrieval is the same as for phrase retrieval, which is shown to be efficient in~\citet{lee2021learning}.
In this section, we evaluate the passage retrieval performance (Eq. \eqref{eqn:pp-ret}) and also how phrase-based passage retrieval can contribute to end-to-end open-domain QA.
\subsection{Experiment: Passage Retrieval}\label{sec:prelim_psg}
\paragraph{Datasets}~
We use two open-domain QA datasets: Natural Questions~\citep{kwiatkowski2019natural} and TriviaQA~\citep{joshi2017triviaqa}, following the standard train/dev/test splits for the open-domain QA evaluation.
For both models, we use the 2018-12-20 Wikipedia snapshot.
To provide a fair comparison, we use Wikipedia articles pre-processed for DPR, which are split into 21-million text blocks
and each text block has exactly 100 words.
Note that while DPR is trained in this setting, DensePhrases is trained with natural paragraphs.\footnote{We expect DensePhrases to achieve even higher performance if it is re-trained with 100-word text blocks. We leave it for future investigation.}
\paragraph{Models}~
For DPR, we use publicly available checkpoints\footnote{\href{https://github.com/facebookresearch/DPR}{https://github.com/facebookresearch/DPR}.} trained on each dataset (DPR$^\diamondsuit$) or multiple QA datasets (DPR$^\spadesuit$),
which we find to perform slightly better than the ones reported in \citet{karpukhin2020dense}.
For DensePhrases, we train it on Natural Questions (DensePhrases$^\diamondsuit$) or multiple QA datasets (DensePhrases$^\spadesuit$) with the code provided by the authors.\footnote{DPR$^\spadesuit$ is trained on NaturalQuestions, TriviaQA, CuratedTREC~\citep{baudivs2015modeling}, and WebQuestions~\citep{berant2013semantic}. DensePhrases$^\spadesuit$ additionally includes SQuAD~\citep{rajpurkar2016squad}, although it does not contribute to Natural Questions and TriviaQA much.}
Note that we do not make any modification to the architecture or training methods of DensePhrases and achieve similar open-domain QA accuracy as reported.
For phrase-based passage retrieval, we compute Eq.~\eqref{eqn:pp-ret} with DensePhrases and return top $k$ passages.
\paragraph{Metrics}~
Following previous work on passage retrieval for open-domain QA, we measure the top-$k$ passage retrieval accuracy (Top-$k$), which denotes the proportion of questions whose top $k$ retrieved passages contain at least one of the gold answers.
To further characterize the behavior of each system, we also include the following evaluation metrics: mean reciprocal rank at $k$ (MRR@$k$) and precision at $k$ (P@$k$).
MRR@$k$ is the average reciprocal rank of the first relevant passage (that contains an answer) in the top $k$ passages.
Higher MRR@$k$ means relevant passages appear at higher ranks.
Meanwhile, P@$k$ is the average proportion of relevant passages in the top $k$ passages.
Higher P@$k$ denotes that a larger proportion of top $k$ passages contains the answers.
\paragraph{Results}~
As shown in Table~\ref{tab:openqa}, DensePhrases achieves competitive passage retrieval accuracy with DPR, while having a clear advantage on top-1 or top-5 accuracy for both Natural Questions (+6.9\% Top-1) and TriviaQA (+8.1\% Top-1).
Although the top-20 (and top-100, which is not shown) accuracy is similar across different models, MRR@20 and P@20 reveal interesting aspects of DensePhrases---it ranks relevant passages higher and provides a larger number of correct passages.
Our results suggest that DensePhrases can also retrieve passages very accurately, even though it was not explicitly trained for that purpose.
For the rest of the paper, we mainly compare the DPR$^\spadesuit$ and DensePhrases$^\spadesuit$ models, which were both trained on multiple QA datasets.
\subsection{Experiment: Open-domain QA}
Recently, \citet{izacard2020leveraging} proposed the Fusion-in-Decoder (FiD) approach where they feed top 100 passages from DPR into a generative model T5~\citep{raffel2020exploring} and achieve the state-of-the-art on open-domain QA benchmarks.
Since their generative model computes the hidden states of all tokens in 100 passages, it requires large GPU memory and \citet{izacard2020leveraging} used 64~Tesla~V100~32GB for training.
In this section, we use our phrase-based passage retrieval with DensePhrases to replace DPR in FiD and see if we can use a much smaller number of passages to achieve comparable performance, which can greatly reduce the computational requirements.
We train our model with 4 24GB RTX GPUs for training T5-base, which are more affordable with academic budgets.
Note that training T5-base with 5 or 10 passages can also be done with 11GB GPUs.
We keep all the hyperparameters the same as in \citet{izacard2020leveraging}.\footnote{We also accumulate gradients for 16 steps to match the effective batch size of the original work.}
\paragraph{Results}~
As shown in Table~\ref{tab:openqa-em}, using DensePhrases as a passage retriever achieves competitive performance to DPR-based FiD and significantly improves upon the performance of original DensePhrases (NQ $=41.3$ EM without a reader).
Its better retrieval quality at top-$k$ for smaller $k$ indeed translates to better open-domain QA accuracy, achieving +6.4\% gain compared to DPR-based FiD when $k=5$.
To obtain similar performance with using 100 passages in FiD, DensePhrases needs fewer passages ($k=25$ or $50$), which can fit in GPUs with smaller RAM.
\input{tables/openqa_em}
\section{Related Work}
Text retrieval has a long history in information retrieval, either for serving relevant information to users directly or for feeding them to computationally expensive downstream systems.
While traditional research has focused on designing heuristics, such as sparse vector models like TF-IDF and BM25, it has recently become an active area of interest for machine learning researchers. This was precipitated by the emergence of open-domain QA as a standard problem setting \citep{chen2017reading} and the spread of the retriever-reader paradigm \citep{
yang-etal-2019-end, nie-etal-2019-revealing}. The interest has spread to include a more diverse set of downstream tasks, such as fact checking \citep{thorne-etal-2018-fever}, entity-linking \citep{wu2020scalable} or dialogue generation \citep{dinan2019wizard}, where the problems require access to large corpora or knowledge sources. Recently, REALM \citep{guu2020realm} and RAG (retrieval-augmented generation) \citep{lewis2020retrieval} have been proposed as general-purpose pre-trained models with explicit access to world knowledge through the retriever. There has also been a line of work to integrate text retrieval with structured knowledge graphs \citep{sun-etal-2018-open, sun-etal-2019-pullnet, min2020knowledge}.
We refer to \citet{lin2020pretrained} for a comprehensive overview of neural text retrieval methods.
|
2,877,628,089,519 | arxiv | \section{Introduction}
\begin{figure}[here]
\includegraphics[scale=0.43]{figure1.eps}
\caption{\label{figure1} (Color online) Magnetic sites of (a) the trillium lattice and (b) the distorted windmill lattice. In both cases three corner-shared equilateral triangles meet at each spin. The unit cell of the trillium lattice is only 4 sites and the next smallest spin loop is 5 sided. The unit cell of the distorted windmill lattice is 12 sites and the smallest spin loop is 4 sided. Atoms within the first unit cell are depicted\cite{jmol} in boldface (black) and labeled ((b) as in Table I). }
\end{figure}
Geometrically frustrated (GF) magnetic systems offer a rich avenue to the search
for emergent or collective phases of matter.
Frustrated magnetism arises when spins between nearby magnetic sites cannot
form a unique alignment to minimize their magnetic interactions,
and leads to the possibility of a macroscopically degenerate classical ground state.
Materials with antiferromagnetic nearest neighbor interactions between local moments
lying at the sites of corner-sharing tetrahedra or triangles comprise a considerable
fraction of the GF magnetic systems known to date.
Most magnetic materials select a particular ordering wavevector, representative of their magnetic structure, upon the onset of magnetic order. In contrast,
in frustrated magnets, it is not uncommon to find within the mean field
approximation a set of degenerate continuously connected ``ordering wavevectors"
which span the three-dimensional (3-D) space.
For example, the disordered ground state of the classical Heisenberg model on the pyrochlore lattice \cite{reimersmc, moessenrchalker}
features such a wavevector manifold.
Partially ordered magnets lie between these two extremes, forming degenerate wavevector (2-D) surfaces or (1-D) lines in a 3-D space.
The name of ``partially ordered'' comes from the fact that
there is a long-range order along the direction normal to this degenerate manifold.
In other words, in such systems, to very low temperatures, a system may appear
disordered/ordered to measurements probing parallel/perpendicular directions to this line or surface.
A growing number of frustrated spin models have been shown to exhibit such a partial order.\cite{hopkinson,canals,bergman,unpublishedus}
Experimental evidence for partial order has been observed in single-crystal neutron scattering measurements
of the correlated metals, MnSi{\cite{Pflei}} under pressure, and CeCu$_{5.9}$Au$_{0.1}$\cite{Schroder}, although in these materials it is unlikely to arise from considerations as simple as here considered.
Our interest in this problem arose from the realization that partial order has been found using large-$N$ theory on both the trillium and distorted windmill lattices.
The key question we address in this paper is whether these partially ordered states can be captured
beyond the large-$N$ theory.\cite{noteref3} If not, to what extent can the results of large-$N$ theory
be regained due to thermal fluctuations?
We answer this question by carrying out large scale classical Monte Carlo simulations of the AF Heisenberg
model on the trillium and distorted windmill lattices.
Large-$N$ theory and MC simulations of the classical Heisenberg model give consistent results for both highly frustrated systems (ex: on the pyrochlore lattice a disordered ground state is found), and for unfrustrated lattices, (ex: choosing the correct ordering wavevector).\cite{note1a} Despite this trend, the finding of a partially ordered ground state in large-$N$ theory does {\it{not}} necessarily translate into a partially ordered ground state within MC simulations.
We find that the classical Heisenberg model on both the trillium and distorted windmill lattices shows a first order phase transition to a coplanar magnetically ordered state featuring neighboring spins rotated by 120$^\circ$.
We further show that this model on the trillium lattice does not have a macroscopic ground state degeneracy.
The chosen 120$^\circ$ coplanar state is a unique ground state, and the partial order is an artificial effect of the spherical approximation
constraint\cite{berlin} used in the large-$N$ theory\cite{hopkinson}.
On the other hand, the selection of a particular coplanar ordering on the distorted windmill lattice
proceeds due to an order by disorder mechanism.
In the cooperative paramagnetic state (for an intermediate temperature regime $T_c<T<\theta_{CW}$ extending an order of
magnitude below the intercept of a Curie-Weiss fit, $\theta_{CW}$),
we find that the large-$N$ description remains quantitatively valid for both lattices, despite the inability of large-$N$ theory to capture the true nature of the classical ground state.
That is, calculations of the angle-resolved static structure factor by large-$N$ and MC techniques show
at most a 5\% difference. We will henceforth refer to this temperature window as the cooperative
paramagnetic regime as it features reasonably strong but short range spin correlations, finite temperature remnants of an avoided partial order.
In the next section, we present the structures of the two lattices, and discuss the approaches used
to study the magnetic properties of the classical Heisenberg model on these lattices.
We show the results of MC simulations on the trillium and distorted windmill lattices in Sec. III and IV, respectively.
In particular, we focus on the questions addressed above:
``is there a transition to a magnetically ordered state
at finite temperatures'' and
``if so, do we find remnants of the partial order above the transition temperature as a result of thermal fluctuations''.
We also address the mechanism of ordering in these lattices after an enumeration of their ground states.
The possible relevance of these results to real materials and the conclusion are discussed in Sec. V. A heuristic derivation of the degeneracy of the model on the distorted windmill lattice is presented in Appendix A.
\section{Model, lattice structures, and approaches}
We study the classical $O(N)$ model on the trillium and distorted windmill
lattices given by the following Hamiltonian
\begin{equation}
H=J \sum_{\langle ij \rangle} \mathbf{S}_i \cdot \mathbf{S}_j,
\end{equation}
where $J>0$ is the antiferromagnetic exchange coupling constant, the sum runs over the nearest
neighbors only, and $\mathbf{S}=(S^1,\ldots,S^N)$ is an $N$-component
classical spin.
It was shown in Ref.~\onlinecite{hopkinson} that
the magnetic lattice of MnSi forms a three-dimensional network of corner-sharing equilateral triangles
with the cubic P2$_1$3 symmetry shown in Fig.~1 (a), which we have named the trillium lattice.
The trillium lattice is common to many systems including the antiferromagnetically correlated Ce local moments of CeIrSi.
The magnetic lattice of $\beta$-Mn, the distorted windmill lattice (P4$_1$32 symmetry),
bears a remarkable qualitative resemblance to the trillium lattice as shown in Fig.~1 (b). In particular, both lattice structures feature three corner-shared equilateral triangles joined at a common site.
The coordinates of each site within a unit cell for both the trillium lattice \cite{hopkinson} and the distorted windmill lattice\cite{nakamura1} have been previously presented. For completeness, these are listed in Table \ref{TableI}
\begin{table}[hbtp]
\begin{tabular}{|l|l|}
\hline
label&coordinates\\
\hline
1&$(u,u,u)$\\
\hline
2&$(\frac{1}{2}+u,\frac{1}{2}-u,1-u)$\\
\hline
3&$(1-u,\frac{1}{2}+u,\frac{1}{2}-u)$\\
\hline
4&$(\frac{1}{2}-u,1-u,\frac{1}{2}+u)$\\
\hline
$a$&$(\frac{1}{8},y,\frac{1}{4}+y)$\\
\hline
$b$&$(\frac{5}{8},\frac{1}{2}-y,\frac{3}{4}-y)$\\
\hline
$c$&$(\frac{1}{4}-y,\frac{7}{8},\frac{1}{2}+y)$\\
\hline
$d$&$(1-y,\frac{3}{4}+y,\frac{3}{8})$\\
\hline
$e$&$(\frac{3}{8},1-y,\frac{3}{4}+y)$\\
\hline
$f$&$(\frac{1}{4}+y,\frac{1}{8},y)$\\
\hline
$g$&$(\frac{3}{4}-y,\frac{5}{8},\frac{1}{2}-y)$\\
\hline
$h$&$(\frac{1}{2}+y,\frac{1}{4}-y,\frac{7}{8})$\\
\hline
$i$&$(\frac{7}{8},\frac{1}{2}+y,\frac{1}{4}-y)$\\
\hline
$j$&$(\frac{3}{4}+y,\frac{3}{8},1-y)$\\
\hline
$k$&$(y,\frac{1}{4}+y,\frac{1}{8})$\\
\hline
$l$&$(\frac{1}{2}-y,\frac{3}{4}-y,\frac{5}{8})$\\
\hline
\end{tabular}
\caption{\label{TableI}
The coordinates of the trillium and distorted windmill lattice sites in the first unit cell as shown in Fig. 1, are expressed in terms of lattice parameters $u$, and $y$ respectively, where we set the corresponding lattice constants of the cubic lattices to unity.
}
\end{table}
To carry out a quantitative comparison between large-$N$ theory and $N=3$ Monte Carlo results,
we have carried out large scale classical Monte Carlo simulations of the $N$=3 Heisenberg model on the trillium and distorted windmill lattices for lattices with $n_s=n\times L\times L\times L$ spins, where $n$ is the number of sites (or spins) in the unit cell and $L$ is the number of spins along each dimension of a cube. On the trillium lattice ($n$=4), we have considered $L=\{6,9,12,18\}$.
On the distorted windmill lattice ($n=12$), we have considered $L=\{6,8,12\}$.
The standard Metropolis algorithm has been used in which we attempt to update
a spin within a small angular range $\delta$ around its original direction.
We choose $\delta$ in such a way that around 50\% of attempted spin updates
are accepted. Starting with a random configuration, we usually perform
$2 \cdot 10^5$ Monte Carlo steps (MCS) for equilibration and $10^6$ MCS for
measurements with one MCS consisting of $\sim n_s/T$ single spin updates,
where $T$ is the temperature.
Within large$-N$ theory, one writes the partition function for the $N$-component spin model\cite{stanley} and solves for the spin-spin correlations \cite{garcan, isakov, hopkinson} in the limit $N\rightarrow\infty$.
That is we rewrite the Hamiltonian as
\begin{equation}
H = {T \over 2} \sum_{i,j} M_{ij} \mathbf{S}_i \cdot \mathbf{S}_j,
\end{equation}
where $M_{ij}$ is the interaction matrix and the spins are subject to the constraint
$\mathbf{S}_i^2=N$. The corresponding partition function is given by
\begin{equation}
Z = \int {\cal D} \phi {\cal D} \lambda \ e^{-S({\phi},\lambda)}
\label{eq:partfunc}
\end{equation}
with the action
\begin{equation}
S({\phi},\lambda) = \sum_{i,j} \left [ {1 \over 2}
M_{ij} {\phi}_i \cdot {\phi}_j + {\lambda_i \over 2} \delta_{ij}
({\phi}_i \cdot {\phi}_i - N) \right ],
\label{eq:action}
\end{equation}
where ${\phi}_i=(\phi^1_i, ..., \phi^N_i)$ is an $N$-component real
vector field and $\lambda_i$ the Lagrange multiplier for the
constraint ${\phi}_i \cdot {\phi}_i = N$. To proceed, we take
the $N \rightarrow \infty$ limit and set a uniform $\lambda_i = \lambda_0$.
The locations $i = (l,\mu)$ of spins can be labeled by
those of the cubic unit cell $l = 1,\ldots, n_c$ and the lattice
sites $\mu=1,\ldots,n$ within the unit cell ($n_c=L\times L\times L$ is
the total number of the unit cells in the lattice).
The Fourier transform with respect to the positions of the unit cells leads to
\begin{equation}
S({\phi}) = \sum_{\bf q} \sum_{\mu,\nu} {1 \over 2}
A^{\mu \nu}_{\bf q} {\phi}_{{\bf q},\mu} \cdot {\phi}_{{\bf q},\nu}
\end{equation}
with $A^{\mu \nu}_{\bf q} = M^{\mu \nu}_{\bf q} + \delta_{\mu \nu} \lambda_0$.
Performing Gaussian integrations over the ${\phi}$ fields in
Eq.~\ref{eq:partfunc}, one finds that
$\lambda_0$ is determined by the saddle point equation,
\begin{equation}
n n_c=\sum_q \sum_{\rho=1}^n\frac{1}{\beta \epsilon_q^\rho+\lambda_0},
\end{equation}
where $\beta\epsilon_q^\rho$ are the eigenvalues of the $n \times n$
interaction matrix $M^{\mu \nu}_{\bf q}$ (which has been shifted such that Min$(\epsilon_q^\rho)=0)$ and $\beta$ the inverse temperature.
Note that the sum over $q$ is carried out for finite size periodic lattices to effectively compare with the Monte Carlo results.
Plugging $\lambda_0$ back into Eqs.~\ref{eq:partfunc} and \ref{eq:action},
we readily deduce the spin-spin correlation functions. For example,
these can be found by calculating the second derivative of $Z$ with respect to an auxiliary field which couples to the spin and can be added to Eq. \ref{eq:action}.
The static structure factor is found to be
\begin{equation}
S(\mathbf{q}) \propto
\sum_{\kappa,\kappa'=1}^n\langle S_\mathbf{q}^{\kappa'}S_{-\mathbf{q}}^{\kappa}\rangle
=\sum_{\kappa,\kappa',\rho=1}^n
\frac{U_{\kappa'\rho}U^*_{\kappa\rho}}{\beta\epsilon_q^\rho+\lambda_0},
\end{equation}
where $U$ is the matrix that diagonalizes the interaction matrix
$M^{\mu \nu}_{\bf q}$, $\kappa$ and $\rho$ are the sublattice indices.
Note that, for simplicity, we will normalize spins to $\sqrt{N}$ in the
large-$N$ theory.\cite{prv} To compare with MC results ($N=3$), one needs to fix the
energy scale. Here we rescale $J\rightarrow J/3$ in $\epsilon_{q}^{\rho}$.
\section{The trillium lattice}
Within large-$N$ theory, we reported\cite{hopkinson} that the classical antiferromagnetic (AF) Heisenberg model on the trillium lattice has a partially ordered ground state, with a surface of degenerate wavevectors following
\begin{equation}
\cos^2\left(\frac{q_x}{2}\right)+\cos^2\left(\frac{q_y}{2}\right)
+\cos^2\left(\frac{q_z}{2}\right)=\frac{9}{4},\label{equation7}
\end{equation}
where we set the lattice constant $a=1$.
However, in Ref. \onlinecite{hopkinson} we relaxed the hard spin constraint of $\mathbf{S}_i^2=1$ to a soft constraint, $\sum^{n}_{i=1} \mathbf{S}_i^2 = n$, where $n$ is the number of sites in the unit cell.
Using energy minimization on spin clusters of classical spins (with $N$=3), we effectively imposed the hard spin constraint show
that the lowest energy state actually exhibits a coplanar magnetic order with wavevector ($\frac{2\pi}{3}$,0,0),\cite{andothers}
.
We speculated that the soft constraint of conventional large-$N$ theory is crucial to the realization of partial order as $T \rightarrow 0$\cite{hopkinson}.
\begin{figure}
\includegraphics[width=3in]{figure2.eps}
\caption{\label{figure2b} (Color online) Average energy per spin (a), heat capacity (b) and magnetic order parameter (c) vs. temperature for the Heisenberg model on the trillium lattice. These results are independent of the lattice parameter $u$ which determines the position of the sites in a cubic unit cell. In this and all other plots, error bars are smaller
than the symbol size if not visible.}
\end{figure}
To find out whether a finite temperature transition to a magnetically ordered state occurs,
we have carried out large scale MC simulations. The energy as a function of temperature, as seen in
Fig.~\ref{figure2b} (a), exhibits a smooth behavior before an abrupt jump, which indicates a first order transition to an ordered state.
The heat capacity as a function of temperature, shown in Fig.~\ref{figure2b} (b), clearly shows a peak corresponding to the onset of magnetic order on the trillium lattice.
Because of strong hysteresis effects, the location of this jump appears to
scale to lower temperatures as the size of the system increases. We have
not attempted to determine the precise location of this strongly first order
transition. Our best estimate for the transition temperature is $T_c=0.21(1)J$.
In Fig.~\ref{figure2b} (c), we plot the magnetic order parameter of this transition,
which is defined as the structure factor at the ordering wavevector,\cite{specifically} $\mathbf{Q} = (\frac{2 \pi}{3}, 0,0)$:
\begin{equation}
\langle m \rangle^2={S(\mathbf{q}=\mathbf{Q}) \over n_s}.
\end{equation}
We see that this order parameter exhibits a sharp onset at the transition temperature as would be expected for a first order phase transition.
It was anticipated that the MC would find a transition to a 120$^\circ$ coplanar magnetic state with
the wavevector ($\frac{2\pi}{3},0,0$), where the spins on each triangle form at 120$^\circ$ to each other,
since it was shown (by minimization) that this is the lowest energy state. Thus this ground state does not have a macroscopic degeneracy\cite{hopkinson}
and the observed transition is not expected to result from an order by disorder mechanism.
An intuitive way to understand the uniqueness of this ground state is to start with a coplanar state
with spins labeled $\alpha$, $\beta$ and $\gamma$ where the letters
$\alpha$, $\beta$ and $\gamma$ denote 120$^\circ$ rotated spins on every triangle.
It is then natural to ask whether
one can generate degenerate states by rotating $\beta$ and $\gamma$ spins around the axis of spin $\alpha$.
The high connectivity of this lattice in comparison with the kagome and hyperkagome lattices\cite{lawler} where such degeneracies naturally arise, prevents any such rotations which are not consistent with the crystal symmetries of the lattice.
Therefore, the rotation of $\beta$ and $\gamma$ spins around $\alpha$ spin axis cannot generate distinctly different states.
This argument itself is not sufficient to prove that
the ground state is unique because, in principle, there might be other
ground states that are not connected to the $(\frac{2\pi}{3},0,0)$ state
by simple spin rotations. However, our previous minimization showed that the state with
the wavevector $(\frac{2\pi}{3},0,0)$ is the lowest energy state. Combining our current MC
result, we conclude that the ground state of Heisenberg model on the trillium lattice is unique.
In the disordered state at temperatures above this ordering transition, however, the spins fluctuate, and one might expect to recover features found by large-$N$ theory with the soft constraint.
Now the question is
{\it{to what extent is the partial order of large-$N$ theory recovered due to temperature fluctuations?}}
\begin{figure}
\includegraphics[scale=0.4]{figure3.eps}
\caption{\label{figure3} Contour plots of the intensity of the structure factor in the hhk ((a) and (b)) and hk0 ((c) and (d)) planes for the trillium lattice in the cooperative paramagnetic regime (for $u=0.138$, $L=12$ and $T=0.25J$) show prominant features near these surfaces. Classical Monte Carlo ((a) and (c)) agrees well with large-$N$ results ((b) and (d)). The maximal intensity is shown in white, and the axes run from $-4\pi$ to $4\pi$ along k(001) (vertical), h(110) (horizontal), k(010) (vertical), h(100)(horizontal). Notice the prominent features near the zero energy surfaces given by Eq. \ref{equation7} of the large-$N$ theory. }
\end{figure}
\begin{figure}
\includegraphics[width=3in]{figure4.eps}
\caption{\label{figure5} (Color online) A quantitative comparison of the angle-dependent structure factor is shown along three high symmetry directions between the Monte Carlo (Heisenberg model) and large-$N$ results for classical spins in the cooperative paramagnetic phase on the trillium lattice ($u=0.138,L=12$). The solid line is the large-$N$ result, while the symbols are from MC simulations.}
\end{figure}
Magnetic susceptibility (not shown) has been calculated within both approaches and shows good agreement until very close to the ordering temperature $T_c\approx0.21J$, well below the Curie-Weiss temperature $\Theta_\text{CW}=2J$, and in the zero temperature limit.
Qualitative comparisons between the large-$N$ theory and Monte Carlo results for the structure factor are shown in Fig.~\ref{figure3} in the cooperative paramagnetic phase at temperatures slightly above the onset of magnetic order. We see excellent qualitative agreement between these two approaches. Within the cooperative paramagnetic window we see that the static structure factor on the trillium lattice peaks around the surface of a sphere-like shape determined by Eq. \ref{equation7}, although part of this sphere is
obliterated by geometric (rather than energetic) effects.
Within both approaches the geometric factor (that is cancellations between spin contributions within the unit cell) imposes that there is very little weight within the first Brillouin zone. In Fig.~\ref{figure5} a quantitative comparison along several high symmetry directions between these two approaches is presented. There is extremely good agreement at all wavevectors.
Note that the cooperative paramagnetic regime is smoothly connected to the partial order within large-$N$ approaches.
Therefore excellent agreement between the angle dependent spin-spin correlation functions implies that remnants of partial order would be expected to appear at finite temperatures above the transition temperature as a result of thermal fluctuations.
\begin{figure}
\includegraphics[scale=0.67]{figure5.eps}
\caption{\label{figure2} (Color online) Average energy per spin (a), heat capacity (b), and magnetic order parameter (c) vs. temperature for the Heisenberg model on the idealized $\beta$-Mn lattice.}
\end{figure}
\section{Distorted windmill lattice}
Mean field calculations\cite{canals} on the distorted windmill lattice also revealed a partially ordered ground state, with a line of degenerate wavevectors along the $(qqq)$ direction. In this paper we show that Monte Carlo simulations of the AF Heisenberg model on this lattice find an ordered ground state as on the trillium lattice. However, the physical origin of the two phase transitions differs. While the ordering in the trillium lattice can be understood by minimization, being the result of energetically different states, the ordering in the distorted windmill lattice cannot. Rather, it must proceed via an order by disorder mechanism.
As on the trillium lattice it is interesting to ask whether the partial ordering features obtained by large-$N$ (or mean field) theory can be found at finite T, where one might expect the spins to fluctuate strongly.\cite{note}
Again, on the distorted windmill lattice, MC simulations find
a jump in the energy as a function of temperature indicative of a first order phase transition. Nonetheless, this transition is qualitatively different from that seen on the trillium lattice, as it does not show strong hysteresis
effects and much variation with system size. This is reflected in a sharp transition from one smooth energy vs. temperature curve to another at the transition temperature as seen in Fig.~\ref{figure2} (a).
The heat capacity as a function of temperature, shown in Fig.~\ref{figure2} (b), clearly shows a peak corresponding to the onset of magnetic order.
The variation of the order parameter shows an abrupt onset at the transition temperature. Here the order parameter has been defined in terms of the structure factor at the ordering wave vector,\cite{specifically}
$\mathbf{Q}_0 = (0,0,0)$ as
\begin{equation}
\langle m \rangle^2={S(\mathbf{q}=\mathbf{Q}_0) \over n_s}.
\end{equation}
As in the case of the trillium lattice, we have not attempted to determine
the precise location of this first order transition. Our best estimate for
the transition temperature is $T_c=0.155(5)J$.
To understand this spin ordering we have carried out an energetic minimization of spin clusters with $N=3$.
The minimization of spin clusters on the distorted windmill lattice with $qqq$ symmetry is found to admit only two types of magnetic spin substructures: one of which can repeat as it is from one unit cell to the next, and another which undergoes a 120$^\circ$ spin rotation in progressing by one unit cell along the ($111$) direction. Thus one has topologically distinct ground state sectors, unit cell structures which are shown in Table \ref{TableII} and labeled $s_{q=0}$ and $s_{q=2\pi/3}$ respectively,
where the ordering wavevector in $s_{q=2\pi/3}$ is ${\bf Q}_{sp}=(\frac{2\pi}{3},\frac{2\pi}{3},\frac{2\pi}{3})$.
Curiously, as each spin structure features coplanar 120$^\circ$ rotated spins on each triangle, generic ground state spin structures are found to interchange between these two orderings. The only constraint arising for a $s_{q=2\pi/3}$ spin structure is that the spin structure must complete three times an integral number of $q=\frac{2\pi}{3}$ unit cells within its boundaries.
There are a macroscopic number of degenerate ground states which can be generated by this mixing of
these two states, all of which exhibit a coplanar order.
This tendency to form a coplanar spin structure is reminiscent of the nematically ordered state recently found for the classical Heisenberg model on the hyperkagome lattice.\cite{usprl} We estimate that the degeneracy of the lattice grows exponentially in the linear lattice size, L, roughly as $e^{0.69L}/6$,
as shown in Appendix A.
\begin{figure}
\includegraphics[width=3in]{figure6.eps}
\caption{\label{energy_0vssp} (Color online) For lattice size $L=6$, (a) Energy and (b) Heat capacity vs. temperature
for the AF Heisenberg model on the distorted windmill lattice with different initial spin configurations.
The filled squares, circles, and triangles represent the $\mathbf{Q}_\text{0}$,
$\mathbf{Q}_\text{sp}$, and $\mathbf{Q}_\text{mix}$ spin configurations, respectively. The system remains in its initial configuration below the
transition. Note that starting with any random spin configurations leads to the $\mathbf{Q}_\text{0}$ state.
(a) shows that the $\mathbf{Q}_\text{sp}$ and $\mathbf{Q}_\text{mix}$ states have the same energy as the $\mathbf{Q}_\text{0}$ state as $T \rightarrow 0$,
and have lower energy than the $\mathbf{Q}_\text{0}$ state just below the transition temperature. However,
(b) indicates that the $\mathbf{Q}_\text{sp}$ state has lower entropy below the transition than the $\mathbf{Q}_\text{mix}$ state and that state in turn has lower
entropy than $\mathbf{Q}_\text{0}$ state.
This is evidence that the selection of the $\mathbf{Q}_\text{0}$ state is of entropic origin, i.e., order
by disorder.}
\end{figure}
\begin{table}[hbtp]
\begin{tabular}{|l|l|l|l|l|l|}
\hline
label&near. neighb. 1&near. neighb. 2&$s_{q=0}$&$s_{q=2\pi/3}$\\
\hline
$a$&$c_{-\hat{y}},d_{-\hat{x}-\hat{y}},f,k$&$j_{-\hat{x}},l$&$\alpha$&$\alpha$\\
\hline
$b$&$g,h,j,l$&$d_{-\hat{y}},f$&$\alpha$&$\alpha$\\
\hline
$c$&$a_{+\hat{y}},d_{-\hat{x}},e,l$&$h_{-\hat{x}+\hat{y}},i_{-\hat{x}+\hat{z}}$&$\beta$&$\alpha$\\
\hline
$d$&$a_{+\hat{x}+\hat{y}},c_{+\hat{x}},g,i$&$b_{+\hat{y}},f_{+\hat{y}}$&$\gamma$&$\alpha$\\
\hline
$e$&$c,f_{+\hat{y}+\hat{z}},h_{+\hat{y}},l$&$g_{+\hat{z}},k_{+\hat{z}}$&$\alpha$&$\beta$\\
\hline
$f$&$a,e_{-\hat{y}-\hat{z}},h_{-\hat{z}},k$&$b,d_{-\hat{y}}$&$\beta$&$\beta$\\
\hline
$g$&$b,d,i,l$&$e_{-\hat{z}},k$&$\beta$&$\beta$\\
\hline
$h$&$b,e_{-\hat{y}},f_{+\hat{z}},j$&$c_{+\hat{x}-\hat{y}},i_{-\hat{y}+\hat{z}}$&$\gamma$&$\beta$\\
\hline
$i$&$d,g,j_{-\hat{z}},k_{+\hat{x}}$&$c_{+\hat{x}-\hat{z}},h_{+\hat{y}-\hat{z}}$&$\alpha$&$\gamma$\\
\hline
$j$&$b,h,i_{+\hat{z}},k_{+\hat{x}+\hat{z}}$&$a_{+\hat{x}},l_{+\hat{x}}$&$\beta$&$\gamma$\\
\hline
$k$&$a,f,i_{-\hat{x}},j_{-\hat{x}-\hat{z}}$&$e_{-\hat{z}},g$&$\gamma$&$\gamma$\\
\hline
$l$&$b,c,e,g$&$a,j_{-\hat{x}}$&$\gamma$&$\gamma$\\
\hline
\end{tabular}
\caption{\label{TableII}
Connections and ground state candidates of the classical Heisenberg model
on the distorted windmill ($\beta$-Mn) lattice. When the lattice parameter, $y=\frac{9-\sqrt{33}}{16}$,
then the nearest neighbors 1 and 2 are at equivalent distances. The final column shows the two spin structures
minimization finds. Here $\{\alpha,\beta,\gamma\}$ refer to 120$^\circ$ rotated spins.
In moving to the next unit cell, the $s_{q=0}$ structure is unchanged, while the
$s_{q=2\pi/3}$ structure
replaces $\alpha\rightarrow\beta\rightarrow\gamma\rightarrow\alpha$. }
\end{table}
Let us refer to spin configurations which order with ordering wavevectors,
$\mathbf{Q}_\text{0}$ and $\mathbf{Q}_\text{sp}$, as $\mathbf{Q}_\text{0}$ and $\mathbf{Q}_\text{sp}$ states respectively.
To understand the selection of the $\mathbf{Q}_\text{0}$ state
over other states, we carried out MC simulations
with a $\mathbf{Q}_\text{sp}$ state
and a $\mathbf{Q}_\text{mix}$ state that has three planes with $s_{q=0}$ spin configurations and three
planes with $s_{q=2\pi/3}$ spin configurations for $L=6$
as initial states, which allows the system to remain in these states.
We found that the energy of the $\mathbf{Q}_\text{sp}$ state
just below the transition temperature is lower than the energy of
the $\mathbf{Q}_\text{mix}$ state, and the energy of the $\mathbf{Q}_\text{mix}$ state
is lower than the energy of $\mathbf{Q}_\text{0}$, as shown in
Fig.~\ref{energy_0vssp} (a).
However, the latter state is always selected as a ground state if an initial state has a random configuration.
This implies that the selection between $\mathbf{Q}_\text{0}$ and other states is due to entropy. To confirm
our intuition, we computed the specific heats of the $\mathbf{Q}_\text{0}$,
$\mathbf{Q}_\text{mix}$, and $\mathbf{Q}_\text{sp}$ states, and found that the specific heat of $\mathbf{Q}_\text{0}$ state is larger than that of $\mathbf{Q}_\text{mix}$
or $\mathbf{Q}_\text{sp}$ states below the transition temperature as shown in Fig.~\ref{energy_0vssp} (b).
This confirms there are more low lying modes for the $\mathbf{Q}_\text{0}$ state, which leads to the selection of the $\mathbf{Q}_\text{0}$
state over the other states of the degenerate ground state manifold.
\begin{figure}
\includegraphics[scale=0.4]{figure7.eps}
\caption{\label{figure4} Contour plots of the intensity of the structure factor in the hhk ((a) and (b)) and hk0 ((c) and (d)) planes for the $\beta$-Mn lattice in the cooperative paramagnetic regime (for the lattice parameter $y=(9-\sqrt{33})/16$, $L=8$ and $T=0.2J$). Classical Monte Carlo ((a) and (c)) agrees well with large-$N$ results ((b) and (d)). The maximal intensity is shown in white. Axes run from $(-4\pi,4\pi)$ in $k$ and $h$, where $k(001)$ describes the vertical axis of (a) and (b), and $h(110)$ the horizontal; $k(010)$ describes the vertical axis of (c) and (d) and $h(100)$ the horizontal.}
\end{figure}
Now let us study the cooperative paramagnetic state above the transition temperature.
Magnetic susceptibility (not shown) has been calculated within both approaches and shows good agreement until very close to the ordering temperature $T_c\approx0.155J$, well below the Curie-Weiss temperature $\Theta_\text{CW}=2J$, and in the zero temperature limit.
Qualitative comparisons between the large-$N$ theory and Monte Carlo results for the structure factor are shown in Fig.~\ref{figure4} in the cooperative paramagnetic phase at temperatures slightly above the onset of magnetic order. We see excellent qualitative agreement between these two approaches. On the distorted windmill lattice the structure factor intensity is concentrated along the degenerate lines of the large-$N$ result.
Within both approaches
there is very little weight in the first Brillouin zone due to the geometric factor.
In Fig.~\ref{figure7} a quantitative comparison along several high symmetry directions between these two approaches is presented. There is extremely good agreement at all wavevectors.
\begin{figure}
\includegraphics[width=3in]{figure8.eps}
\caption{\label{figure7} (Color online) A quantitative comparison of the angle-dependent structure factor is shown along three high symmetry directions between the Monte Carlo (Heisenberg model) and large-$N$ results for classical spins in the cooperative paramagnetic phase on $\beta$-Mn lattice ($L=8$). The solid line is the large-$N$ result, while the symbols are from MC simulations.}
\end{figure}
\begin{figure*}
\includegraphics[scale=0.55]{figure9.eps}
\caption{\label{figure6}
Contour plots of the intensity of the large-$N$ theory structure factor
in the hhk plane for different values of $u$. The vertical axis is $k(001)$, and the horizontal axis is $h(110)$, where $k$ and $h$ both run from $(-4\pi,4\pi)$.
Varying the parameter $u$, does not change the energy within either the large-$N$ theory (here $N=3$, $L=24$ and $T=1/4.5 J$), nor in classical Monte Carlo simulations. It does, however, change the geometric contribution to the structure factor quite dramatically. While near $u=0.125$ (center) one appears to see few vestiges of the degenerate spheres of the treatment, for both large (right) and small (left) values of $u$, one sees that these are quite well captured. Curiously, most real systems seem to lie closer to the middle graph.}
\end{figure*}
\section{ Discussion and summary }
A magnetic material where the magnetic properties can be described by the AF Heisenberg model on the trillium lattice
is CeIrSi.
It shows\cite{heying} a Curie-Weiss susceptibility above 100 K with a magnetic moment of 2.56$\mu_B$/Ce atom
and a $\theta_{CW}=$ -24 K. At low temperatures, $\chi^{-1}$ shows a gradual downturn,\cite{heying,note3}
a characteristic common to many triangle-based frustrated magnets.
X-ray scattering shows the lattice structure of this material to have a cubic P2$_1$3 symmetry\cite{heying},
with a positional parameter $x_{Ce}$=0.6183.\cite{note4}
Then the Ce sites, which hold magnetic moments consistent with $4f^1$ electrons,
form a trillium lattice of corner-shared triangles. The (non-zeroed) interaction matrix is identical to that presented in Eq. 5
of Ref.~\onlinecite{hopkinson} if one takes the lattice parameter $u_{Ce}$=$x_{Ce}-\frac{1}{2}$.\cite{note2}
Curiously the energetics of the AF Heisenberg
model do not depend on the lattice parameter $u$ directly. As shown in Fig.~\ref{figure6}, one expects that the
realization of this model with different values of $u$ will again access a cooperative
paramagnetic regime, but with differing weights over the surface of the MF degenerate spheres.
To our knowledge Ref.~\onlinecite{heying} is the only study of the physics
of CeIrSi. In this work the magnetic susceptibility and x-ray spectra
of polycrystalline powdered samples have been measured, giving evidence that
this material remains disordered to low temperatures.
It would be very interesting to see whether or not neutron scattering
measurement on a single crystal of CeIrSi may show evidence of partial order
at low temperatures giving way to long range order, as we predict for
an antiferromagnetic Heisenberg model on the trillium lattice.
The relevance of our study to $\beta$-Mn and MnSi is less obvious, as both materials are metallic. However, the magnetic sites of each lattice features one of the three-dimensional corner-shared triangle lattices structures here studied, with $\beta$-Mn being the only known material to form in the distorted windmill structure. While the origin of the unusual non-Fermi liquid resistivity seen under pressure in MnSi\cite{doironpedra} ($\Delta \rho \approx AT^{\frac{3}{2}}$) is not at present understood, the same temperature exponent is observed\cite{stewart} in $\beta$-Mn, where it is expected to result from antiferromagnetic spin fluctuations. We believe it is worthwhile to investigate the possible link between the magnetic fluctuations and the non-Fermi liquid behavior in these materials. In this vein, it is interesting to note that powder neutron scattering down to 1.4 K\cite{nakamura} in $\beta$-Mn shows no signature of magnetic order. We are hopeful that this study might provide the motivation for single crystal neutron scattering to be carried out on $\beta$-Mn.
In summary, we have used large-$N$ theory for O($N$) vector spins and classical MC simulations to study the AF Heisenberg model on two three-dimensional corner-shared triangle lattices, each site of which belongs to
three equilateral triangles.
The large-$N$ studies suggested that the geometrical frustration present would lead to a partially ordered state on both lattices.
However, through the minimization of finite size spin clusters, we found the ground state manifolds on these two lattices to be quite different,
despite the local similarity between these corner sharing triangle lattice structures.
In both cases, we found that there is a first order transition to a magnetically ordered state using MC methods.
We further showed that the trillium lattice exhibits a unique ground state with a spiral ordering,
while the distorted windmill has a macroscopic ground state degeneracy.
Magnetic ordering of the classical AF Heisenberg model on the distorted windmill lattice is therefore seen to arise via an order by disorder mechanism. The degeneracy of this model on the trillium lattice is seen to be an artificial effect of the soft-constraint of the
large-$N$ theory.
Despite the above noted differences at low temperatures between large-$N$ and MC results,
in the cooperative paramagnetic phase above the transition temperature,
we find a remarkable resemblance between the respective spin-spin correlations.
This leads us to ask whether the salient features of the large-$N$ theory, the angular and directional dependences of the spin-spin correlations found in
the partially ordered state obtained by large-$N$ theory are present at finite temperatures above the transition temperature.
As true partial order exhibits long range order along particular directions only
as $T \rightarrow 0$, it is not possible to have partial order at any finite temperatures,
since the spin-spin correlation decay exponentially at finite temperatures.
This being said, the qualitative directional dependence characteristic of a partially ordered state survives above the transition temperature
(note that the ground state is smoothly connected to the cooperative paramagnetic phase in large-$N$ theory)
allowing us to conclude that a ``disguised" partial order has been recovered in the cooperative paramagnetic phase.
\section*{Acknowledgments}
This work is supported by NSERC of Canada, Canada Research Chair, the
Canadian Institute for Advanced Research (J. M. H., S. V. I., H. Y. K.),
and the Swiss National Science Foundation (S. V. I.).
|
2,877,628,089,520 | arxiv | \section{Introduction}
We tackle long-range video understanding, specifically anticipating not-yet observed but upcoming actions. When developing intelligent systems, one needs not only to recognize what is \emph{currently} taking place -- but also predict what will happen \emph{next}. Anticipating human actions is essential for applications such as smart surveillance, autonomous driving, and assistive robotics.
While action anticipation is a niche (albeit rapidly growing) area, the key issues that arise are germane to long-range video understanding as a whole. How should temporal or sequential relationships be modelled? What temporal extent of information and context needs to be processed? At what temporal scale should they be derived, and how much semantic abstraction is required? The answers to these questions are not only entangled with each other but also depend very much on the videos being analyzed. Here, one needs to distinguish between clipped actions, e.g.\@\xspace of UCF101~\cite{Soomro101}, versus the multiple actions in long video streams, e.g.\@\xspace of the Breakfast~\cite{kuehne2014language}. In the former, the actions and video clips are on the order of a few seconds, while in the latter, it is several minutes. As such, temporal modelling is usually not necessary for simple action recognition~\cite{huang2018makes}, but more relevant for understanding complex activities~\cite{richard2016temporal,sener2018unsupervised}.
Temporal models that are built into the architecture~\cite{ding2018weakly,farha2019ms,huang2016connectionist,richard2017weakly} are generally favoured because they allow frameworks to be learned end-to-end. However, this means that the architecture also dictates the temporal extent that can be accounted for. This tends to be short, either due to difficulties in memory retention or model size. As a result, the context for anticipation can only be drawn from a limited extent of recent observations, usually on the order of seconds~\cite{lan2014hierarchical,vondrick2016anticipating,miech2019leveraging}. This, in turn, limits the temporal horizon and granularity of the prediction.
One way to ease the computational burden, especially under longer temporal extents, is to use higher-level but more compact feature abstractions, e.g.\@\xspace by using detected objects, people~\cite{lfb2019} or sub-activity labels~\cite{abu2018will,Ke_2019_CVPR} based on the outputs of video segmentation algorithms~\cite{richard2017weakly}. Such an approach places a heavy load on the initial task of segmentation and is often difficult to train end-to-end. Furthermore, since labelling and segmenting actions from video are difficult tasks, their errors may propagate onwards when anticipating future actions.
Motivated by these questions of temporal modelling, extent, scaling, and level of semantic abstraction, we propose a general framework for encoding long-range video. We aim for flexibility in frame input, i.e.\@\xspace ranging from low-level visual features to high-level semantic labels, and the ability to meaningfully integrate recent observations with long-range context in a computationally efficient way. To do so, we split video streams into snippets of equal length and max-pool the frame features within the snippets. We then create ensembles of multi-scale feature representations that are aggregated bottom-up based on scaling and temporal extent. Temporal aggregation~\cite{kline1995computing} is a form of summarization used in database systems. Our framework is loosely analogous as it summarizes the past observations through aggregation, so we name it ``temporal aggregates''. We summarize our main contributions as follows:
\begin{itemize}
\item We propose a simple and flexible single-stage framework of multi-scale temporal aggregates for videos by relating recent to long-range observations.
\item Our representations can be applied to several video understanding tasks; in addition to action anticipation, it can be used for recognition and segmentation with minimal modifications and is able to achieve competitive results.
\item Our model has minimal constraints regarding the type of anticipation (dense or next action), type of the dataset (instructional or daily activities), and type of input features (visual features or frame-level labels).
\item We conduct experiments on Breakfast~\cite{kuehne2014language}, 50Salads~\cite{stein2013combining} and EPIC-Kitchens~\cite{damen2018scaling}.
\end{itemize}
\section{Related Works}
\textbf{Action recognition} has advanced significantly with deep networks in recent years. Notable works include two steam networks~\cite{simonyan2014two,wang2016temporal}, 3D convolutional networks~\cite{tran2015learning,carreira2017quo}, and RNNs~\cite{donahue2015long,yue2015beyond}. These methods have been designed to encode clips of a few seconds and are typically applied to the classification of \emph{trimmed} videos containing a single action~\cite{Soomro101,kay2017kinetics}. In our paper, we work with long \emph{untrimmed} sequences of complex activities. Such long videos are not simply a composition of independent short actions, as the actions are related to each other with sequence dynamics. Various models for complex activity understanding have been addressed before~\cite{ding2018weakly,farha2019ms,sener2018unsupervised}; these approaches are designed to work on instructional videos by explicitly modelling their sequence dynamics. These models are not flexible enough to be extended to daily activities with loose orderings. Also, when only partial observations are provided, e.g.\@\xspace for anticipation, these models cannot be trained in a single stage.
\textbf{Action anticipation} aims to forecast actions before they occur. Prior works in immediate anticipation were initially limited to movement primitives like \emph{reaching}~\cite{Koppula15pami} or interactions such as \emph{hugging}~\cite{vondrick2016anticipating}. \cite{mahmud2017joint} presents a model for predicting both the next action and its starting position. \cite{damen2018scaling} presents a large daily activities dataset, along with a challenge for anticipating the next action one second before occurrence. \cite{miech2019leveraging} proposes next action anticipation from recent observations. Recently, \cite{furnari2019rulstm} proposed using an LSTM to summarize the past and another LSTM for future prediction. These works assume near-past information, whereas we make use of long-range past.
\textbf{Dense anticipation} predicts actions multiple steps into the future. Previous methods~\cite{abu2018will,Ke_2019_CVPR} to date require having already segmented temporal observations. Different than these, our model can perform dense anticipation in a single stage without any pre-segmented nor labelled inputs.
The \textbf{role of motion and temporal dynamics} has been well-explored for video understanding, though the focus has been on short clips~\cite{lin2019tsm,carreira2017quo,huang2018makes}. Some works use longer-term temporal contexts, though still in short videos ~\cite{li2017temporal,nonlocalNetVLAD}. Recently, Wu~\etal\cite{lfb2019} proposed integrating long-term features with 3D CNNs in short videos and showed the importance of temporal context for action recognition. Our model is similar in spirit to~\cite{lfb2019} in that we couple the recent with the long-range past using attention. One key difference is that we work with ensembles of multiple scalings and granularities, whereas~\cite{lfb2019} work at a single frame-level granularity. As such, we can handle long videos up to tens of minutes, while they are only able to work on short videos. Recently, Feichtenhofer~\etal\cite{feichtenhofer2019slowfast} proposed SlowFast networks, which, similar to our model, encode time-wise multi-scale representations. These approaches are limited to short videos and cannot be extended to minutes-long videos due to computational constraints.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{img/overview_full_new.pdf}
\caption{Model overview: In this example we use 3 scales for computing the ``spanning past'' features $\mathbf{S}_{K_1}, \mathbf{S}_{K_2}, \mathbf{S}_{K_3}$, and 2 starting points to compute the ``recent past'' features, $\mathbf{R}_{i_1}, \mathbf{R}_{i_2}$, by max-pooling over the frame features in each snippet. Each recent snippet is coupled with all the spanning snippets in our Temporal Aggregation Block (TAB). An ensemble of TAB outputs is used for dense or next action anticipation.}
\label{fig:overview_new}
\end{figure}
\section{Representations}
We begin by introducing the representations, which are inputs to the building blocks of our framework, see Fig.~\ref{fig:overview_new}. We had two rationales when designing our network. First, we relate recent to long-range observations, since some past actions directly determine future actions. Second, to represent recent and long-range past at various granularities, we pool snippets over multiple scales.
\subsection{Pooling}
For a video of length $T$, we denote the feature representation of a single video frame indexed at time $t$ as $f_t \in \mathbb{R}^{D}, 1\leq t \leq T$. $f_t$ can be derived from low-level features, such as IDT~\cite{wang2013action} or I3D~\cite{carreira2017quo}, or high-level abstractions, such as sub-activity labels derived from segmentation algorithms. To reduce computational load, we work at a snippet-level and define a snippet feature $\textbf{F}_{ij;K}$ as the concatenation of max-pooled features from $K$ snippets, where snippets are partitioned consecutively from frames $i$ to $j$:
\begin{align}\label{eqn:maxpool}
\begin{split}
\textbf{F}_{ij;K} & = [F_{i,i+k}, F_{i+k+1,i+2k}, ..., F_{j-k+1,j}], \; \text{where} \\
( F_{p,q})_d & = \max\limits_{p \leq t \leq q} \{ f_{t}\}_d\,, \; 1\leq d \leq D,\;\; k=\sfrac{(j-i)}{K}.
\end{split}
\end{align}
\noindent Here, $F_{p,q}$ indicates the maximum over each dimension $d$ of the frame features in a given snippet between frames $p$ and $q$, though it can be substituted with other alternatives. In the literature, methods representing snippets or segments of frames range from simple sampling and pooling strategies to more complex representations such as learned pooling~\cite{lin2018nextvlad} and LSTMs~\cite{ostyakov2018label}. Especially for long snippets, it is often assumed that a learned representation is necessary~\cite{girdhar2017actionvlad,lee20182nd}, though their effectiveness over simple pooling is still controversial~\cite{wang2016temporal}. The learning of novel temporal pooling approaches goes beyond the scope of this work and is an orthogonal line of development. We verify established methods (see Sec.~\ref{sec:ablations}) and find that a simple max-pooling is surprisingly effective and sufficient.
\subsection{Recent vs. Spanning Representations}
Based on different start and end frames $i$ and $j$ and number of snippets $K$, we define two types of snippet features: \emph{``recent''} features $\{\mathcal{R}\}$ from recent observations, and \emph{``spanning''} features $\{\mathcal{S}\}$ drawn from the entire video. The recent snippets cover a couple of seconds (or up to a minute, depending on the temporal granularity) before the current time point while spanning snippets refer to the long-range past and may last up to ten minutes. For \emph{``recent''} snippets, the end frame $j$ is fixed to the current time point $t$, and the number of snippets is fixed to $K_R$. The recent snippet features $\mathcal{R}$ can be defined as a feature bank of snippet features with different start frames $i$, i.e.\@\xspace
\begin{align}\label{eqn:recentsnippet}
\begin{split}
\mathcal{R} & = \{\mathbf{F}_{i_{1}t;K_R}, \mathbf{F}_{i_{2}t;K_R}, ..., \mathbf{F}_{i_Rt;K_R}\} = \{\mathbf{R}_{i_1}, \mathbf{R}_{i_2}, ...,
\mathbf{R}_{i_R}\},
\end{split}
\end{align}
where $\mathbf{R}_i \in \mathbb{R}^{D \times K_R} $ is a shorthand to denote $\mathbf{F}_{i\,t;K_R}$, since endpoint $t$ and number of snippets $K_R$ are fixed. In Fig.~\ref{fig:overview_new} we use two starting points to compute the ``recent'' features and represent each with $K_R\!=\!3$ snippets (\tikz{\snippetBox{boxYellow}{0}\snippetBox{boxGreen}{1.25}\snippetBox{boxRed}{2.5}} \& \tikz{\snippetBox{boxGreen}{0}\snippetBox{boxRed}{1.25}\snippetBox{boxRed}{2.5}}).
For \emph{``spanning''} snippets, $i$ and $j$ are fixed to the start of the video and current time point,~i.e.\@\xspace $i\!=\!0, j\!=\!t$. Spanning snippet features $\mathcal{S}$ are defined as a feature bank of snippet features with varying number of snippets $K$, i.e.\@\xspace
\begin{align}\label{eqn:spanningsnippet}
\begin{split}
\mathcal{S} & = \{\mathbf{F}_{0\,t;K_1}, \mathbf{F}_{0\,t;K_2}, ..., \mathbf{F}_{0\,t;K_S}\} = \{\mathbf{S}_{K_1}, \mathbf{S}_{K_2}, ...,
\mathbf{S}_{K_S}\},
\end{split}
\end{align}
\noindent where $\mathbf{S}_K \in \mathbb{R}^{D \times K} $ is a shorthand for $\mathbf{F}_{0\,t;K}$. In Fig.~\ref{fig:overview_new} we use three scales to compute the ``spanning'' features with $K = \{7,5,3\}$ (\tikz{\snippetBox{boxBlue}{0}\snippetBox{boxOrange}{1.25}\snippetBox{boxYellow}{2.5}\snippetBox{boxYellow}{3.75}\snippetBox{boxYellow}{5.0}\snippetBox{boxGreen}{6.25}\snippetBox{boxRed}{7.5}},\tikz{\snippetBox{boxBlue}{0}\snippetBox{boxOrange}{1.25}\snippetBox{boxYellow}{2.5}\snippetBox{boxGreen}{3.75}\snippetBox{boxRed}{5.0}} \&\tikz{\snippetBox{boxBlue}{0}\snippetBox{boxYellow}{1.25}\snippetBox{boxRed}{2.5}}).
Key to both types of representations is the ensemble of snippet features from multiple scales. We achieve this by varying the number of snippets $K$ for the spanning past. For the recent past, it is sufficient to keep the number of snippets $K_R$ fixed, and vary only the start point $i$, due to redundancy between $\mathcal{R}$ and $\mathcal{S}$ for the snippets that overlap. For our experiments, we work with snippets ranging from seconds to several minutes.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{img/overview_blocks.pdf}
\caption{Model components: Non-Local Blocks (NLB) compute interactions between two representations via attention (Sec.~\ref{sec:nlb}). Two such NLBs are combined in a Coupling Block (CB), which calculates the attention-reweighted spanning and recent representations (Sec.~\ref{sec:cb}). We couple each recent with all spanning representations via individual CBs and combine their outputs in a Temporal Aggregation Block (TAB) (Sec.~\ref{sec:tab}). The outputs of multiple such TABs are combined to perform anticipation, Fig.~\ref{fig:overview_new}.}
\label{fig:overview}
\end{figure*}
\section{Framework}
In Fig.~\ref{fig:overview} we present an overview of the components used in our framework, which we build in a bottom-up manner, starting with the recent and spanning features $\mathcal{R}$ and $\mathcal{S}$, which are coupled with non-local blocks (NLB) (Sec.~\ref{sec:nlb}) within coupling blocks (CB) (Sec.~\ref{sec:cb}). The outputs of the CBs from different scales are then aggregated inside temporal aggregation blocks (TAB) (Sec.~\ref{sec:tab}). Outputs of different TABs can then be chained together for either next action anticipation or dense anticipation (Secs.~\ref{sec:instructional_act},~\ref{sec:daily_act}).
\subsection{Non-Local Blocks (NLB)}\label{sec:nlb}
We apply non-local operations to capture relationships amongst the spanning snippets and between spanning and recent snippets. Non-local blocks~\cite{wang2018non} are a flexible way to relate features independently from their temporal distance and thus capture long-range dependencies. We use the modified non-local block from~\cite{lfb2019}, which adds layer normalization~\cite{ba2016layer} and dropout~\cite{srivastava2014dropout} to the original one in~\cite{wang2016temporal}. Fig.~\ref{fig:overview} (left) visualizes the architecture of the block, the operation of which we denote as $\text{NLB}(\cdot,\cdot)$.
\subsection{Coupling Block (CB)}\label{sec:cb}
Based on the NLB, we define attention-reweighted spanning and recent outputs:
\begin{align}
\textbf{S}'_{K} = NLB(\textbf{S}_{K},\textbf{S}_{K})
\quad \text{and} \quad
\textbf{R}'_{i,K} = NLB(\textbf{S}'_{K}, \textbf{R}_i)\label{coupledAttn}.
\end{align}
\noindent $\textbf{R}'_{i,K}$ is coupled with either $\textbf{R}_i$ or $\textbf{S}'_K$ via concatenation and a linear layer. This results in the fixed-length representations $\textbf{R}''_{i,K}$ and $\textbf{S}''_{i,K}$, where $i$ is the starting point of the recent snippet and $K$ is the scale of the spanning snippet.
\subsection{Temporal Aggregation Block (TAB)}\label{sec:tab}
The final representation for recent and spanning past is computed by aggregating outputs from multiple CBs. For the same recent starting point $i$, we concatenate $\textbf{R}''_{i,K_1}, ...,$ $\textbf{R}''_{i,K_S}$ for all spanning scales and pass the concatenation through a linear layer to compute $\textbf{R}'''_i$. The final spanning representation $\textbf{S}'''_{i}$ is a max over all $\textbf{S}''_{i,K_1}, ..., \textbf{S}''_{i,K_S}$. We empirically find that taking the max outperforms other alternatives like linear layers and/or concatenation for the spanning past (Sec. \ref{sec:ablations}). TAB outputs, by varying recent starting points $\{i\}$ and scales of spanning snippets $\{K\}$, are multi-granular video representations that aggregate and encode both the recent and long-range past. We name these \textbf{temporal aggregate representations}. Fig.\ref{fig:overview_new} shows an example with 2 recent starting points and 3 spanning scales. These representations are generic and can be applied in various video understanding tasks (see Sec.~\ref{sec:pred}) from long streams of video.
\subsection{Prediction Model}\label{sec:pred}
\textbf{Classification:} For single-label classification tasks such as next action anticipation, temporal aggregate representations can be used directly with a classification layer (linear + softmax). A cross-entropy loss based on ground truth labels $Y$ can be applied to the predictions $\hat{Y}_{i}$, where $Y$ is either the current action label for recognition, or the next action label for next action prediction (see Fig.~\ref{fig:next_action_pred}).
When the individual actions compose a complex activity (e.g.\@\xspace ``take bowl'' and ``pour milk'' as part of ``making cereal'' in Breakfast~\cite{kuehne2014language}), we can add an additional loss based on the complex activity label $Z$. Predicting $Z$ as an auxiliary task helps with anticipation. For this we concatenate $\textbf{S}'''_{i_1}, \ldots, \textbf{S}'''_{i_R}$ from all TABs and pass them through a classification layer to obtain $\hat{Z}$. The overall loss is a sum of the cross entropies over the action and complex activity labels:
\begin{align}\label{eq:loss_next_action}
\mathcal{L}_{\text{cl}} = \mathcal{L}_{\text{comp}} + \mathcal{L}_{\text{action}}
= - \sum_{n=1}^{N_Z} Z_n \log ( \hat{Z} )_n - \sum_{r=1}^{R} \sum_{n=1}^{N_Y} Y_n \log ( \hat{Y}_{i_r} )_n,
\end{align}
where $i_r$ is one of the R recent starting points, and $N_Y$ and $N_Z$ are the total number of actions and complex activity classes respectively. During inference, the predicted scores are summed for a final prediction, i.e.\@\xspace $\hat{Y}\!=\! \max_n(\sum_{r = 1}^R \hat{Y}_{i_r})_n\label{inference_next}$.
We frame sequence segmentation as a classification task and predict frame-level action labels of complex activities. Multiple sliding windows with fixed start and end times are generated and then classified into actions using Eq.\ref{eq:loss_next_action}.
\begin{figure}[t]
\centering
\includegraphics[width=0.89\columnwidth]{img/next_action_corrected_wide.pdf}
\caption{Prediction models for classification (left) and sequence prediction (right).}
\label{fig:next_action_pred}
\end{figure}
\textbf{Sequence prediction:} The dense anticipation task predicts frame-wise actions of the entire future sequence. Previously,~\cite{abu2018will} predicted future segment labels via classification and regressed the durations. We opt to estimate both via classification. The sequence duration is discretized into $N_D$ intervals and represented as one-hot encodings $D\in \{0,1\}^{N_D}$. For dense predictions, we perform multi-step estimates. We first estimate the current action and complex activity label, as per Eq.~\ref{loss_next_action}. The current duration $D$ is then estimated via a classification layer applied to the concatenation of recent temporal aggregates $\textbf{R}'''_{i_1}, ..., \textbf{R}'''_{i_R}$.
For future actions, we concatenate all recent and spanning temporal aggregates $\textbf{R}'''_{i_1}, ..., \textbf{R}'''_{i_R}$ and $\textbf{S}'''_{i_1}, ..., \textbf{S}'''_{i_R}$ and the classification layer outputs $\hat{Y}_{i_1}, ..., \hat{Y}_{i_R}$, and pass the concatenation through a linear layer before feeding the output to a one-layer LSTM. The LSTM predicts at each step $m$ an action vector $\hat{Y}^{m}$ and a duration vector $\hat{D}^{m}$ (see Fig.~\ref{fig:next_action_pred}). The dense anticipation loss is a sum of the cross-entropies over the current action, its duration, future actions and durations, and task labels respectively:
{
{
\begin{align}\label{loss_next_action}
\mathcal{L}_{\text{dense}}\!=\!
\mathcal{L}_{\text{cl}}\!-\!\sum_{n=1}^{N_D} D_n \log ( \hat{D} )_n\!
-\!\frac{1}{M}\sum_{m=1}^{M}\left(
\sum_{n=1}^{N_Y} Y^{m}_{n}\log(\hat{Y}^{m})_{n} \!+\!
\sum_{n=1}^{N_D} D^{m}_{n}\log(\hat{D}^{m})_{n}\right)
\end{align}
}
}
During inference we sum the predicted scores (post soft-max) for all starting points $i_r$ to predict the current action as $\max_n(\sum_{r = 1}^R \hat{Y}_{i_r})_n$.
The LSTM is then applied recurrently to predict subsequent actions and durations.
\section{Experiments}
\subsection{Datasets and Features}\label{sec:datasets}
We experiment on Breakfast~\cite{kuehne2014language}, 50Salads~\cite{stein2013combining} and EPIC-Kitchens~\cite{damen2018scaling}. The sequences in each dataset reflect realistic durations and orderings of actions, which is crucial for real-world deployment of anticipation models. Relevant datasets statistics are given in Table~\ref{tab:my_label}. One notable difference between these datasets is the label granularity; it is very fine-grained for EPIC, hence their 2513 action classes, versus the coarser 48 and 17 actions of Breakfast and 50Salads. As a result, the median action segment duration is 8-16x shorter.
Feature-wise, we use pre-computed Fisher vectors~\cite{abu2018will} and I3D~\cite{carreira2017quo} for Breakfast, Fisher vectors for 50Salads, and appearance, optical flow and object-based features provided by~\cite{furnari2019rulstm} for EPIC. Results for Breakfast and 50Salads are averaged over the predefined 4 and 5 splits respectively. Since 50Salads has only a single complex activity (making salad) we omit complex activity prediction for it. For EPIC, we report results on the test set. Evaluation measures are class accuracy (Acc.) for next action prediction and mean over classes~\cite{abu2018will} for dense prediction. We report Top-1 and Top-5 accuracies to be consistent with~\cite{miech2019leveraging,furnari2019rulstm}.
Hyper-parameters for spanning $\{K\}$, recent scales $K_R$ and recent starting points $\{i\}$ are given in Table~\ref{tab:my_label}. We cross validated the parameters on different splits of 50Salads and Breakfast; on EPIC, we select parameters with the validation set~\cite{furnari2019rulstm}.
\begin{table}[t!]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Dataset& \begin{tabular}[c]{@{}c@{}}video duration\\ median, mean $\pm$std\end{tabular} & \# classes& \# segments & $\{i\}$(in seconds) & $K_R$ & $\{K\}$\\
\hline
Breakfast(@15fps) & 15.1s, 26.6s $\pm$36.8 & 48 & 11.3K & $\{t-10,t-20,t-30\}$ & 5 & $\{10,15,20\}$\\
50Salads(@30fps) & 29.7s, 38.4s $\pm$31.5 & 17 & 0.9K & $\{t-5,t-10,t-15\}$ & 5 & $\{5,10,15\}$\\
EPIC(@60fps) & 1.9s, 3.7s $\pm$ 5.6 & 2513 & 36.6K& $\{t\!-\!1.6,t\!-\!1.2,t\!-\!0.8,t\!-\!0.4\}$ & 2 & $\{2,3,5\}$\\
\hline
\end{tabular}}
\caption{Dataset details and our respective model parameters.}
\label{tab:my_label}
\end{table}
\subsection{Component validation}\label{sec:ablations}
We verify each component's utility via a series of ablation studies summarized in Table~\ref{tab:componentAblations}. As our main motivation was to develop a representation for anticipation in long video streams, we validate on Breakfast for next action anticipation. Our full model gets a performance of 40.1\% accuracy averaged over actions.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{4pt}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Pooling type & \cellcolor{magenta!15} frame sampling & \cellcolor{magenta!15}GRU & \cellcolor{magenta!15}BiLSTM & \multicolumn{2}{l|}{\cellcolor{magenta!15}mean-pooling} & \cellcolor{magenta!15}max-pooling \\ \cline{1-1}
Acc. & \cellcolor{magenta!15}32.1 & \cellcolor{magenta!15}37.9 & \cellcolor{magenta!15}38.7 & \multicolumn{2}{l|}{\cellcolor{magenta!15}36.6} & \cellcolor{magenta!15}\textbf{40.1} \\ \hline\hline
Influence of & \multicolumn{5}{l|}{Changes in components} & Acc. (Drop) \\ \hline
Non-Local Blocks (NLB) & \multicolumn{5}{l|}{\cellcolor{blue!10}replace all NLBs with concatenation + linear layer} & \cellcolor{blue!10}33.7 (6.4\% ) \\ \hline
\multirow{3}{*}{Coupling Blocks (CB)}
& \multicolumn{5}{l|}{\cellcolor{green!10}only couple the $\textbf{S}_{K}$ and $\textbf{S}_{K}$ in CBs} & \cellcolor{green!10}35.1 (5.0\% )\\ \cline{2-7}
& \multicolumn{5}{l|}{\cellcolor{green!10}only couple the $\textbf{R}_i$ and $\textbf{R}_i$ in CBs } & \cellcolor{green!10}34.2 (5.9\% )\\ \cline{2-7}
& \multicolumn{5}{l|}{\cellcolor{green!10}replace CBs with concatenation + linear layer} & \cellcolor{green!10}33.4 (6.7\% )\\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Temporal Aggregation\\ Blocks (TAB)\end{tabular}}
& \multicolumn{5}{l|}{\cellcolor{gray!20}a single CB is used in TABs} & \cellcolor{gray!20}38.0 (2.1\% ) \\ \cline{2-7}
& \multicolumn{5}{l|}{\cellcolor{gray!20}three CBs are used in a single TAB} & \cellcolor{gray!20}37.7 (2.4\% ) \\ \cline{2-7}
& \multicolumn{5}{l|}{\cellcolor{gray!20}a single a CB is used without any TABs} & \cellcolor{gray!20}32.1 (8.0\% ) \\ \hline
\end{tabular}}\\
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{0.1pt}
\begin{tabular}{|cll|ccccccc|}
\hline
\multirow{6}{*}{\begin{tabular}[c]{@{}c@{}}Recent \&\\ Spanning \\ Repr.\end{tabular}} & \multicolumn{1}{c}{\multirow{2}{*}{\textbf{(a)}}} & starting points $i$ & $i_1=t-10$ & $i_2=t-20$ & $i_3=t-30$ & $i_4=0$ & \textbf{$\mathbf{\{i_1, i_2,i_3\}}$} &&\\
& \multicolumn{1}{c}{} & Acc. & 36.9& 37.7& 37.2& 35.1 & 40.1 &&\\ \cline{2-10}
& \multirow{2}{*}{\textbf{(b)}} & spanning scales $K$ &$\{5\}$ & $\{10\}$& $\{15\}$& $\{20\}$ & $\{10,15\}$ & $\{$10,15,20$\}$ & $\{$5,10,15,20$\}$ \\
&& Acc. & 37.4& 38.0& 37.5& 37.4 & 39.0 & \textbf{40.1} & 40.2 \\ \cline{2-10}
& \multirow{2}{*}{\textbf{(c)}} & recent scales $K_R$ &$1$ & $3$ & \textbf{5} & $10$ &&&\\
&& Acc. & 38.7& 39.5& \textbf{40.1}& 38.6 &&&\\\hline
\end{tabular}}
\caption{Ablations on the influence of different model components.}
\label{tab:componentAblations}
\end{table}
\textbf{Video Representation:} Several short-term feature representations have been proposed for video, e.g.\@\xspace~3D convolutions~\cite{tran2015learning}, or combining CNNs and RNNs for sequences~\cite{yue2015beyond,donahue2015long}. For long video streams, it becomes difficult to work with all the raw features. Selecting representative features can be as simple as sub-sampling the frames~\cite{feichtenhofer2019slowfast,xiao2020audiovisual}, or pooling~\cite{wang2016temporal}, to more complex RNNs~\cite{yue2015beyond}. Current findings in the literature are not in agreement. Some propose learned strategies~\cite{miech2017learnable,lee20182nd}, while others advocate pooling~\cite{wang2016temporal}. Our experiments align with the latter, showing that max-pooling is superior to both sampling (+8\%) and the GRU (+2.2\%) or bi-directional LSTM~\cite{conneau2017supervised} (+1.4\%). The performance of GRU and BiLSTM are comparable to average-pooling, but require much longer training and inference time. For us, max-pooling works better than average pooling; this contradicts the findings of~\cite{wang2016temporal}. We attribute this to the fact that we pool over minutes-long snippets and it is likely that mean- smooths away salient features that are otherwise preserved by max-pooling. We conducted a similar ablations on EPIC, where we observed a 1.3\% increase with max- over mean-pooling.
\textbf{Recent and Spanning Representations:}
In our ablations, unless otherwise stated, an ensemble of 3 spanning scales $K\!=\!\{10,15,20\}$ and 3 recent starting points $i\!=\!\{t\!-\!10,t\!-\!20,t\!-\!30\}$ are used. Table~\ref{tab:componentAblations} \textbf{(a)} compares single starting points for the recent snippet features versus an ensemble. With a single starting point, points too near to and too far from the current time decrease the performance. The worst individual result is with $i_4 = 0$, i.e.\@\xspace using the entire sequence; the peak is at $i_2 = t - 20$, though an ensemble is still best. In Table~\ref{tab:componentAblations} \textbf{(b)}, we show the influence of spanning snippet scales. These scales determine the temporal snippet granularity; individually, results are not significantly different across the scales, but as we begin to aggregate an ensemble, the results improve. The ensemble with 4 scales is best but only marginally better than 3, at the expense of a larger network, so we choose $K\!=\!\{10,15,20\}$. In Table~\ref{tab:componentAblations} \textbf{(c)}, we show the influence of recent snippet scales, we find $K_R = 5$ performs best.
\textbf{Coupling Blocks:} Previous studies on simple video understanding have shown the benefits of using features from both the recent and long-range past~\cite{li2017temporal,lfb2019}. A na\"ive way to use both is to simply concatenate, though combining the two in a learned way, e.g.\@\xspace via attention, is superior (+6.4\%). To incorporate attention, we apply NLBs~\cite{wang2018non}, which is an adaptation of the attention mechanism that is popularly used in machine translation. When we replace our CBs with concatenation and a linear layer, there is a drop of 6.7\%. When we do not use coupling but separately pass the $\textbf{R}_i$ and $\textbf{S}_{K}$ through concatenation and a linear layer, there is a drop of 7.5\%. We find also that coupling the recent $\textbf{R}_i$ and long range $\textbf{S}_{K}$ information is critical. Coupling only recent information (-5.9\%) does not keep sufficient context, whereas coupling only long-range past (-5\%) does not leave sufficient representation for the more relevant recent aspects.
\textbf{Temporal Aggregation Blocks} (TAB) are the most critical component. Omitting them and classifying a single CB's outputs significantly decreases accuracy (-8\%). The strength of the TAB comes from using ensembles of coupling blocks as input (single, -2.1\%) and using the TABs in an ensemble (single, -2.4\%).
\textbf{Additional ablations:} When we omit the auxiliary complex activity prediction, i.e.\@\xspace removing the $Z$ term from Eq.~\ref{loss_next_action} (``no Z''), we observe a slight performance drop of 1.1\%. In our model we max pool over all $\textbf{S}''_{i,K_1}, ..., \textbf{S}''_{i,K_S}$ in our TABs. When we replace the max-pooling with concatenation + linear, we reach an accuracy of 37.4.
We also try to disentangle the ensemble effect from the use of multi-granular representations. When we fix the spanning past scales $K$ to $\{15,15,15\}$ and all the starting points to $i=t-20$, we observe a drop of 1.2\% in accuracy which indicates the importance of our multi-scale representation.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{15pt}
\begin{tabular}{|l|l|l|l|c|c|c|}
\hline
Method & Input & Segmentation Method and Feature & Breakfast & 50Salads \\\hline
\cite{vondrick2016anticipating} & FC7 features & - & 8.1 & 6.2\\
\cite{miech2019leveraging} & R(2+1)D & - & 32.3 & \\\hline
\rowcolor{red!15}RNN \cite{abu2018will} & segmentation & ~\cite{richard2017weakly}, Fisher & \textbf{30.1} & 30.1 \\
\rowcolor{red!15}CNN \cite{abu2018will} & segmentation & ~\cite{richard2017weakly}, Fisher & 27.0 & 29.8 \\
\rowcolor{red!15}ours no $Z$ & Fisher & - & 29.2 & \textbf{31.6}\\
\rowcolor{red!15}ours & Fisher & - & 29.7 & \\\hline
\rowcolor{blue!15}ours & I3D & - & 40.1 & 40.7 \\
\rowcolor{blue!15}ours & segmentation & ours, I3D & 43.1 & \\
\rowcolor{blue!15}ours & segmentation + I3D & ours, I3D & 47.0 & \\\hline
\rowcolor{green!15}ours & frame GT & - & 64.7 & 63.8\\
\rowcolor{green!15}ours & frame GT + I3D & - & 63.1 & \\ \hline
\end{tabular} }
\caption{Next action anticipation comparisons on Breakfast and 50Salads, given different frame inputs frame inputs, GT action labels, Fisher vectors and I3D features.}
\label{tab:SOA_next_50Salads}
\end{table}
\subsection{Anticipation on Procedural Activities - Breakfast \& 50 Salads} \label{sec:instructional_act}
\subsubsection{Next Action Anticipation}\label{sec:next_action}
predicts the action that occurs 1 second from the current time $t$. We compare to the state of the art in Table \ref{tab:SOA_next_50Salads} with two types of frame inputs: spatio-temporal features (Fisher vectors or I3D) and frame-wise action labels (either from ground truth or via a separate segmentation algorithm) on Breakfast. Compared to previous methods using only visual features as input, we outperform CNN (FC7) features~\cite{vondrick2016anticipating} and spatio-temporal features R(2+1)D~\cite{miech2019leveraging} by a large margin (+32.3\% and +8.1\%). While the inputs are different, R(2+1)D features were shown to be comparable to I3D features~\cite{tran2018closer}. Since~\cite{miech2019leveraging} uses only recent observations, we conclude that incorporating the spanning past into the prediction model is essential.
Our method degrades when we replace I3D with the weaker Fisher vectors (40.1\% vs 29.7\%). Nevertheless, this result is competitive with methods using action labels~\cite{abu2018will} (30.1\% with RNN) derived from segmentation algorithms~\cite{richard2017weakly} using Fisher vectors as input. For fair comparison, we report a variant without the complex activity prediction (``no $Z$''), which has a slight performance drop (-0.5\%). If we use action labels as inputs instead of visual features, our performance improves from 40.1\% to 43.1\%; merging labels and visual features gives another 4\% boost to 47\%. In this experiment we use segmentation results from our own framework,
(see Sec.~\ref{sec:video_seg}). However, if we substitute ground truth instead of segmentation labels, there is still a 17\% gap. This suggests that the quality of the segmentation matters. When the segmentation is very accurate, adding additional features does not help and actually slightly deteriorates results (see Table~\ref{tab:SOA_next_50Salads} ``frame GT'' vs. ``frame GT + I3D'').
In Table \ref{tab:SOA_next_50Salads}, we also report results for 50Salads. Using Fisher vectors we both outperform the state of the art by 1.8\% and the baseline with CNN features~\cite{vondrick2016anticipating} by 25.4\%. Using I3D features improves the accuracy by 9.1\% over Fisher vectors.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{5pt}
\begin{tabular}{|l|llllllll||llllllll|}
\hline
& \multicolumn{8}{c||}{Breakfast} & \multicolumn{8}{c|}{50salads} \\ \hline
Obs. & \multicolumn{4}{c|}{20\%} & \multicolumn{4}{c||}{30\%} & \multicolumn{4}{c|}{20\%} & \multicolumn{4}{c|}{30\%} \\ \hline
Pred.& 10\% & 20\% & 30\% & \multicolumn{1}{l|}{50\%}& 10\% & 20\% & 30\% & 50\% & 10\% & 20\% & 30\% & \multicolumn{1}{l|}{50\%}& 10\% & 20\% & 30\% & 50\% \\ \hline
\textbf{A} & \multicolumn{8}{l||}{\cellcolor{green!15} \textbf{Labels} (GT)} & \multicolumn{8}{l|}{\cellcolor{green!15} \textbf{Labels} (GT)}\\ \hline
\rowcolor{green!15}RNN\cite{abu2018will} & 60.4 & 50.4 & 45.3 & 40.4 & 61.5 & 50.3 & 45.0 & 41.8 & 42.3 & 31.2 & 25.2 & 16.8 & 44.2 & 29.5 & 20.0 & 10.4 \\
\rowcolor{green!15}CNN\cite{abu2018will} & 58.0 & 49.1 & 44.0 & 39.3 & 60.3 & 50.1 & 45.2 & 40.5 & 36.1 & 27.6 & 21.4 & 15.5 & 37.4 & 24.8 & 20.8 & 14.1 \\
\rowcolor{green!15}Ke\cite{Ke_2019_CVPR} & 64.5 & \textbf{56.3} & \textbf{50.2} & \textbf{44.0} & 66.0 & 55.9 & \textbf{49.1} & \textbf{44.2} & 45.1 & 33.2 & 27.6 & 17.3 & \textbf{46.4} & \textbf{34.8} & \textbf{25.2} & 13.8 \\
\rowcolor{green!15}ours & \textbf{65.5} & 55.5 & 46.8 & 40.1 & \textbf{67.4} & \textbf{56.1} & 47.4 & 41.5 & \textbf{47.2} & \textbf{34.6} & \textbf{30.5} & \textbf{19.1} & 44.8 & 32.7 & 23.5 & \textbf{15.3} \\ \hline
\textbf{B}& \multicolumn{8}{l||}{\cellcolor{red!15} \textbf{Features} (Fisher) }& \multicolumn{8}{l|}{\cellcolor{red!15} \textbf{Features} (Fisher)} \\ \hline
\rowcolor{red!15}CNN\cite{abu2018will}& 12.8 & 11.6 & 11.2 & 10.3 & 17.7 & 16.9 & 15.5 & 14.1 &&&&&&&&\\
\rowcolor{red!15} ours& \textbf{15.6} & \textbf{13.1} & \textbf{12.1} & \textbf{11.1} & \textbf{19.5} & \textbf{17.0} & \textbf{15.6} & \textbf{15.1} & {25.5} & {19.9} & {18.2} & {15.1} & {30.6} &{22.5} & {19.1} & {11.2} \\ \hline
\textbf{C}& \multicolumn{8}{l||}{\cellcolor{red!15} \textbf{Labels} (Fisher + \cite{richard2017weakly} (Acc. 36.8/42.9)) }& \multicolumn{8}{l|}{\cellcolor{red!15} \textbf{Labels} (Fisher + \cite{richard2017weakly} (Acc. 66.8/66.7)) }\\ \hline
\rowcolor{red!15}RNN\cite{abu2018will} & 18.1 & \textbf{17.2} & 15.9 & 15.8 & 21.6 & 20.0 & 19.7 & 19.2 & 30.1 & 25.4 & 18.7 & 13.5 & 30.8 & 17.2 & 14.8 & 9.8\\
\rowcolor{red!15}CNN\cite{abu2018will} & 17.9 & 16.4 & 15.4 & 14.5 & 22.4 & 20.1 & 19.7 & 18.8 & 21.2 & 19.0 & 16.0 & 9.9& 29.1 & 20.1 & 17.5 & 10.9 \\
\rowcolor{red!15}Ke\cite{Ke_2019_CVPR} & 18.4 & \textbf{17.2} & 16.4 & \textbf{15.8} & 22.8 & \textbf{20.4} & 19.6 & \textbf{19.8} & 32.5 & \textbf{27.6} & 21.3 & \textbf{16.0} & \textbf{35.1} & \textbf{27.1} & 22.1 & 15.6 \\
\rowcolor{red!15}ours & \textbf{18.8} & 16.9 & \textbf{16.5}& 15.4& \textbf{23.0}& 20.0& \textbf{19.9}& 18.6 & \textbf{32.7} & 26.3 & \textbf{21.9} & 15.6 & 32.3 & 25.5 & \textbf{22.7} & \textbf{17.1} \\ \hline
& \multicolumn{8}{l||}{\cellcolor{red!15} Concatenate B and C}& \multicolumn{8}{l|}{\cellcolor{red!15} Concatenate B and C}\\ \hline
\rowcolor{red!15} ours& \textbf{25.0} & \textbf{21.9} & \textbf{20.5} & \textbf{18.1} & \textbf{23.0} & \textbf{20.5} & 19.8 &\textbf{19.8} & \textbf{34.7} & 25.9 & \textbf{23.7} & 15.7 & 34.5 & 26.1 & 19.0 & 15.5 \\ \hline
\textbf{D}& \multicolumn{8}{l||}{\cellcolor{blue!15} \textbf{Features} (I3D)} &&&&&&&&\\
\cellcolor{blue!15} ours &\cellcolor{blue!15} 24.2 &\cellcolor{blue!15} 21.1 &\cellcolor{blue!15} 20.0 &\cellcolor{blue!15} 18.1 &\cellcolor{blue!15} 30.4 &\cellcolor{blue!15} 26.3 &\cellcolor{blue!15} 23.8 &\cellcolor{blue!15} 21.2 &&&&&&&& \\ \cline{1-9}
\textbf{E}& \multicolumn{8}{l||}{\cellcolor{blue!15} \textbf{Labels} (I3D + our seg. (Acc. 54.7/57.8)) } &&&&&&&&\\
\cellcolor{blue!15} ours &\cellcolor{blue!15} \textbf{37.4} &\cellcolor{blue!15} 31.2 &\cellcolor{blue!15} 30.0 &\cellcolor{blue!15} 26.1 &\cellcolor{blue!15} 39.5 &\cellcolor{blue!15} 34.1 &\cellcolor{blue!15} 31.0 &\cellcolor{blue!15} \textbf{27.9} &&&&&&&&\\ \cline{1-9}
& \multicolumn{8}{l||}{\cellcolor{blue!15} Concatenate D and E} &&&&&&&&\\
\cellcolor{blue!15} ours&\cellcolor{blue!15} 37.1 &\cellcolor{blue!15} \textbf{31.8} &\cellcolor{blue!15} \textbf{30.1} &\cellcolor{blue!15} \textbf{27.1} &\cellcolor{blue!15} \textbf{39.8} &\cellcolor{blue!15} \textbf{34.2} &\cellcolor{blue!15} \textbf{31.9} &\cellcolor{blue!15} 27.8 &&&&&&&&\\\hline
\end{tabular} }
\caption{Dense anticipation mean over classes on Breakfast and 50salads, given different frame inputs frame inputs, GT action labels, Fisher vectors and I3D features.}
\label{tab:SOA_dense_breakfast_50salad}
\end{table}
\subsubsection{Dense Anticipation}\label{sec:dense_ant}
predicts frame-wise actions; accuracies are given for specific portions of the remaining video (Pred.) after observing a given percentage of the past (Obs.). We refer the reader to the supplementary for visual results. Competing methods \cite{abu2018will} and \cite{Ke_2019_CVPR} have two stages; they first apply temporal video segmentation and then use outputs~\cite{richard2017weakly}, i.e.\@\xspace frame-wise action labels, as inputs for anticipation. We experiment with both action labels and visual features.
For Breakfast (Table~\ref{tab:SOA_dense_breakfast_50salad}, left), when using GT frame labels, we outperform the others, for shorter prediction horizons. For 50Salads (Table~\ref{tab:SOA_dense_breakfast_50salad}, right), we outperform the state of the art for the observed 20\%, and our predictions are more accurate on long-range anticipation (Pred. 50\%). We outperform~\cite{abu2018will} when we use visual features as input (B Features (Fisher)). When using the segmentations (from~\cite{richard2017weakly}, which has a frame-wise temporal segmentation accuracy of 36.8\% and 42.9\% for the observed 20\% and 30\% of video respectively), we are comparable to state of the art~\cite{Ke_2019_CVPR}. We further merge visual features with action labels for dense anticipation. With Fisher vectors and the frame labels obtained from \cite{richard2017weakly}, we observe a huge performance increase in performance compared to only using the frame labels (up to +7\%) in Breakfast. In 50Salads, this increase is not significant nor consistent. This may be due to the better performing segmentation algorithm on 50Salads (frame-wise accuracy of 66.8\% and 66.7\% for 20\% and 30\% observed respectively). We observe further improvements on Breakfast once we substitute Fisher vectors with I3D features and segmentations from our own framework (I3D + ours seg.). Similar to next action anticipation, performance drops when using only visual features as input (I3D is better than Fisher vectors). When using I3D features and the frame label outputs of our segmentation method, we obtain our model's best performance, with a slight increase over using only frame label outputs.
\subsection{How much spanning past is necessary?}\label{sec:spanning_past_inf}
We vary the duration of spanning snippets (Eq.~\ref{eqn:spanningsnippet}) with start time $i$ as fractions of the current time $t$; $i\!=\!0$ corresponds to the full sequence, i.e.\@\xspace 100\% of the spanning past, while $i\!=\!t$ corresponds to none, i.e.\@\xspace using only recent snippets since the end points $j$ remain fixed at $t$. Using the entire past is best for Breakfast (Fig.~\ref{fig:pastImportance} left). Interestingly, this effect is not observed on EPIC (Fig.~\ref{fig:pastImportance} right). Though we see a small gain by 1.2\% until 40\% past for the appearance features (rgb), beyond this, performance saturates. We believe this has to do with the fine granularity of labels in EPIC; given that the median action duration is only 1.9s, one could observe as many as 16 actions in 30 seconds. Given that the dataset has only 28.5K samples split over 2513 action classes, we speculate that the model cannot learn all the variants of long-range relationships beyond 30 seconds. Therefore, increasing the scope of the spanning past does not further increase the performance. Based on experiments on the validation set, we set the spanning scope to 6 seconds for EPIC for the rest of the paper.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{img/past_importance_new2.pdf}
\caption{Effect of spanning scope on instructional vs.\ daily activities. For EPIC we report Top-5 Acc. on the validation set with rgb, flow and object features and late fusion.}
\label{fig:pastImportance}
\end{figure}
\subsection{Recognition and Anticipation on Daily Activities - EPIC}\label{sec:daily_act}
The \textbf{anticipation} task of EPIC requires anticipating the future action $\tau_{\alpha}\!=\!1$s before it starts. For fair comparison to the state of the art~\cite{furnari2019rulstm} (denoted by ``RU''),
we directly use features (appearance, motion and object) provided by the authors. We train our model separately for each feature modality with the same hyper-parameters and fuse predictions from the different modalities by voting. Note that for experiments on this dataset we do not use the entire past for computing our spanning snippet features (see Section~\ref{sec:spanning_past_inf}).
Results on hold-out test data of EPIC are given in Table \ref{tab:epic_ex_test} for seen kitchens (S1) with the same environments as the training data and unseen kitchens (S2) of held out environments.
We outperform state of the art, RU~\cite{furnari2019rulstm}, in the Top-1 and Top-5 action accuracies by 2\% and 2.7\% on S1 and by 1.8\% and 2.3\% on S2 using the same features suggesting superior temporal reasoning abilities of our model. When we add verb and noun classification to our model as auxiliary tasks to help with anticipation, ``ours v+n'', our performance improves for action and especially for noun and verb scores. For challenge results see supplementary.
For \textbf{recognition}, we classify pre-trimmed action segments. We adjust the scope of our spanning and recent snippets according to the action start and end times $t_s$ and $t_e$. Spanning features are computed on a range of $[t_s - 6, t_e + 6]$; the first recent snippet scope is fixed to $[t_s,t_e]$ and the rest to $[t_s-1,t_e+1], [t_s-2,t_e+2]$ and $[t_s-3,t_e+3]$. Remaining hyper-parameters are kept the same. In Table~\ref{tab:epic_ex_test}, we compare to state of the art; we outperform all other methods including SlowFast networks with audio data~\cite{xiao2020audiovisual} (+5.4\% on S1, +2.2\% on S2 for Top-1) and LFB~\cite{lfb2019}, which also uses non-local blocks (+8.6\% on S1, +5\% on S2 for Top-1) and RU~\cite{furnari2019rulstm} by approximately +7\% on both S1 and S2. Together with the anticipation results we conclude that our method generalizes to both anticipation and recognition tasks and is able to achieve state-of-the-art results on both, while~\cite{furnari2019rulstm} performs very well on anticipation but poorly on recognition.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{7pt}
\begin{tabular}{|clccc|ccc|ccc|ccc|}
\hline
& \multicolumn{7}{c|}{\textbf{Action Anticipation}} & \multicolumn{6}{c|}{\textbf{Action Recognition}} \\ \hline
& \multicolumn{4}{c|}{Top-1 Accuracy\%} & \multicolumn{3}{c|}{Top-5 Accuracy\%} & \multicolumn{3}{c|}{Top-1 Accuracy\%} & \multicolumn{3}{c|}{Top-5 Accuracy\%} \\ \hline
& & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action \\\hline\rowcolor{red!10}
\multirow{7}{*}
&\cite{miech2019leveraging} & 30.7 & 16.5 & 9.7 & 76.2 & 42.7 & 25.4 & - & - & - & - & - & - \\\rowcolor{red!10}
& TSN~\cite{damen2018scaling} & 31.8 & 16.2 & 6.0 & 76.6 & 42.2 & 28.2 & 48.2 & 36.7 & 20.5 & 84.1 & 62.3 & 39.8 \\\rowcolor{red!10}
& RU~\cite{furnari2019rulstm} & {33.0} & {22.8} & 14.4 & {79.6} & {50.9} & 33.7 & 56.9 & 43.1 & 33.1 & 85.7 & 67.1 & 55.3 \\\rowcolor{red!10}
\textbf{S1}& LFB~\cite{lfb2019} & - & - & - & - & - & - & 60.0 & 45.0 & 32.7 & 88.4 & {71.8} & 55.3 \\\rowcolor{red!10}
&\cite{xiao2020audiovisual} & - & - & - & - & - & - & {65.7} & 46.4 & 35.9 & 89.5 & 71.7 & 57.8 \\\rowcolor{red!10}
& ours & 31.4 & 22.6 & {16.4} & 75.2 & 47.2 & \textbf{36.4} & 63.2 & {49.5} & {41.3} & 87.3 & 70.0 & {63.5} \\\rowcolor{red!10}
& ours v+n & \textbf{37.9} & \textbf{24.1} & \textbf{16.6} & \textbf{79.7} & \textbf{54.0} & 36.1 & \textbf{66.7} & \textbf{49.6} & \textbf{41.6} & \textbf{90.1} & \textbf{77.0} & \textbf{64.1} \\\hline\rowcolor{blue!10}
\multirow{7}{*}
&\cite{miech2019leveraging} & 28.4 & 12.4 & 7.2 & {69.8} & 32.2 & 19.3 & - & - & - & - & - & - \\\rowcolor{blue!10}
& TSN~\cite{damen2018scaling} & 25.3 & 10.4 & 2.4 & 68.3 & 29.5 & 6.6 & 39.4 & 22.7 & 10.9 & 74.3 & 45.7 & 25.3 \\\rowcolor{blue!10}
& RU~\cite{furnari2019rulstm} & 27.0 & 15.2 & 8.2 & 69.6 & {34.4} & 21.1 & 43.7 & 26.8 & 19.5 & 73.3 & 48.3 & 37.2 \\\rowcolor{blue!10}
\textbf{S2}& LFB~\cite{lfb2019} & - & - & - & - & - & - & 50.9 & 31.5 & 21.2 & 77.6 & 57.8 & 39.4 \\\rowcolor{blue!10}
&\cite{xiao2020audiovisual} & - & - & - & - & - & - & \textbf{55.8} & {32.7} & 24.0 & \textbf{81.7} & {58.9} & 43.2 \\\rowcolor{blue!10}
& ours & {27.5} & \textbf{16.6} & {10.0} & 66.8 & 32.8 & {23.4} & 52.0 & 31.5 & {26.2} & 76.8 & 52.7 & {45.7} \\\rowcolor{blue!10}
& ours v+n & \textbf{29.5} & 16.5 & \textbf{10.1} & \textbf{70.1} & \textbf{37.8} & \textbf{23.4} & 54.6 & \textbf{33.5} & \textbf{27.0} & 80.4 & \textbf{61.0} & \textbf{46.4}\\ \hline
\end{tabular}}
\caption{Action anticipation and recognition on EPIC tests sets S1 and S2}
\label{tab:epic_ex_test}
\end{table}
\subsection{Temporal Video Segmentation}\label{sec:video_seg}
We compare our performance against the state of the art, MS-TCN (I3D)~\cite{farha2019ms}, in Table~\ref{video_segmentation} on Breakfast. We test our model with 2s and 5s windows. We report the frame-wise accuracy (Acc), segment-wise edit distance (Edit) and F1 scores at overlapping thresholds of 10\%, 25\% and 50\%. In the example sequences, in the F1 scores and edit distances in Table~\ref{video_segmentation}, we observe more fragmentation in our segmentation for 2s than for 5s. However, for 2s, our model produces better accuracies, as the 5s windows are smoothing the predictions at action boundaries. Additionally we provide our model's upper bound, ``ours I3D GT.seg.'', for which we classify GT action segments instead of sliding windows. The results indicate that there is room for improvement, which we leave as future work. We show that we are able to easily adjust our method from its main application and already get close to the state of the art with slight modifications.
\begin{table}[t]
\centering
\begin{minipage}[b]{0.45\linewidth}
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|llllll|}
\hline
& \multicolumn{3}{l}{F1@\{10, 25, 50\}} & Edit & Acc. \\ \hline
MS-TCN (I3D) ~\cite{farha2019ms} & 52.6 & \multicolumn{1}{l}{48.1} & 37.9 & \textbf{61.7} & \textbf{66.3} \\ \hline
ours I3D 2s & 52.3 & 46.5 & 34.8 & 51.3 & 65.3 \\
ours I3D 5s & \textbf{59.2} & \textbf{53.9} & \textbf{39.5} & 54.5 & 64.5 \\\hline
ours I3D GT.seg. & - & - & - & - & \textbf{75.9} \\ \hline
\end{tabular} }
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.99\columnwidth]{img/seg_drawing_novideo.pdf}\\
\end{minipage}
\caption{Exemplary segmentation and comparisons on Breakfast. }
\label{video_segmentation}
\end{table}
\section{Discussion \& Conclusion}
This paper presented a temporal aggregate model for long-range video understanding. Our method computes recent and spanning representations pooled from snippets of video that are related via coupled attention mechanisms. Validating on three complex activity datasets, we show that temporal aggregates are either comparable or outperform the state of the art on three video understanding tasks: action anticipation, recognition and temporal video segmentation.
In developing our framework, we faced questions regarding temporal extent, scaling, and level of semantic abstraction. Our experiments show that max-pooling is a simple and efficient yet effective way of representing video snippets; this is the case even for snippets as long as two minutes. For learning temporal relationships in long video, attention mechanisms relating the present to long range context can successfully model and anticipate upcoming actions. The extent of context that is beneficial, however, may depend on the nature of activity (instructional vs.~daily) and label granularity (coarse vs.~fine) of the dataset.
We found significant advantages to using ensembles of multiple scales, both in recent and spanning snippets. Our aggregates model is flexible and can take as input either visual features or frame-wise action labels. We achieve competitive performance with either form of input, though our experiments confirm that higher levels of abstraction such as labels are more preferable for anticipation. Nevertheless, there is still a large gap between what can be anticipated with inputs from current segmentation algorithms in comparison to ground truth labels, leaving room for segmentation algorithms to improve.\\
\noindent{\textbf{Acknowledgments} This work was funded partly by the German Research Foundation (DFG) YA 447/2-1 and GA 1927/4-1 (FOR2535 Anticipating Human Behavior) and partly by National Research Foundation Singapore under its NRF Fellowship Programme [NRF-NRFFAI1-2019-0001] and Singapore Ministry of Education (MOE) Academic Research Fund Tier 1 T1251RES1819.}
\clearpage
\bibliographystyle{splncs04}
\section{More on Datasets and Features}
We provide more statistics about the datasets used in our paper to show a broader comparison about their scale and label granularity. \\
\noindent\textbf{The Breakfast Actions dataset \cite{kuehne2014language}} contains 1712 videos of 10 high level tasks like ``making coffee'', ``making tea'' and so on. There are in total 48 different actions, such as ``pouring water'' or ``stirring coffee'', with on average 6 actions per video. The average duration of the videos is 2.3 minutes. There are 4 splits and we report our results averaged over them. We use two types of frame-wise features: Fisher vectors computed as in \cite{abu2018will} and I3D features~\cite{carreira2017quo}.
\noindent\textbf{The 50Salads dataset~\cite{stein2013combining}} includes 50 videos and 17 different actions for a single task, namely making mixed salads. When training on this dataset, we therefore omit task prediction in our model. On average, 50Salads has 20 actions per video due to repetitions. The average video duration is 6.4 minutes. There are 5 splits, and we again average our results over them. We represent the frames using Fisher vectors as in \cite{abu2018will}.
\noindent\textbf{The EPIC-Kitchens dataset~\cite{damen2018scaling}} is a large first-person video dataset which contains 432 sequences and 39,594 action segments recorded by participants performing non-scripted daily activities in their kitchen. The average duration of the videos is 7.6 minutes ranging from 1 minute to 55 minutes. An action is defined as a combination of a verb and a noun, e.g. ``boil milk``. There are in total 125 verbs, 351 nouns and 2513 actions. The dataset provides a training and test set which contains 272 and 160 videos, respectively. The test set is divided into two splits: Seen Kitchens (S1) where sequences from the same environment are in the training data, and Unseen Kitchens (S2) where complete sequences of some participants are held out for testing. The labels for the test set are not shared, as there is an action anticipation challenge\footnote{ \texttt{\url{https://competitions.codalab.org/competitions/20071}}} and action recognition challenge\footnote{ \texttt{\url{https://competitions.codalab.org/competitions/20115}}}. We use the RGB, optical flow and object-based features provided by Furnari and Farinella~\etal~\cite{furnari2019rulstm}. The minimum and maximum snippet durations, over which we apply pooling, are 0.4s and 115.3s for 50Salads, 0.1s and 64.5s for Breakfast, and 1.2s and 3.0s for EPIC.
\noindent\textbf{Implementation Details : } We train our model using the Adam optimizer~\cite{kingma2014adam} with batch size 10, learning rate $10^{-4}$ and dropout rate 0.3. We train for 25 epochs and decrease the learning rate by a factor of 10 every $10^{\text{th}}$ epoch. We use 1024 dimensions for all non-classification linear layers for the Breakfast Actions and 50Salads datasets and 512 dimensions for the EPIC-Kitchens dataset. The LSTMs in dense anticipation have one layer and 512 hidden units. We use intervals of $20$ seconds for Breakfast and 50Salads for discretizing the durations in dense anticipation.
\section{Model Validation}
\begin{table}[!]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{6pt}
\begin{tabular}{|llllllllllll|}
\hline
& cereal & coffee & f.egg & juice & milk & panc. & salat & sand. & s.egg & tea & mean\scriptsize{$\pm$std}\\\hline
TM &\cellcolor[HTML]{4A9BD0}77.8 &\cellcolor[HTML]{C7E0F0}50.8 &\cellcolor[HTML]{ACD1EA}57.2 &\cellcolor[HTML]{ACD1EA}57.2 &\cellcolor[HTML]{EBF4FA}40.1 &\cellcolor[HTML]{ECF4FA}39.6 &\cellcolor[HTML]{A9CFE9}57.9 &\cellcolor[HTML]{C0DCEF}52.4 &\cellcolor[HTML]{A2CCE7}59.4 &\cellcolor[HTML]{B9D8ED}54.2 &\cellcolor[HTML]{B7D7EC}54.6\scriptsize{$\pm$10.8} \\ \hline
LUT &\cellcolor[HTML]{ABD0E9}57.5 &\cellcolor[HTML]{A0CAE6}\textbf{59.9} &\cellcolor[HTML]{B0D3EB}56.2 &\cellcolor[HTML]{A5CDE8}58.8 &\cellcolor[HTML]{B1D4EB}56.1 &\cellcolor[HTML]{ACD1E9}57.3 &\cellcolor[HTML]{B5D6EC}55.1 &\cellcolor[HTML]{CBE2F2}49.6 &\cellcolor[HTML]{9AC7E5}\textbf{61.2} &\cellcolor[HTML]{9FCAE6}60.1 &\cellcolor[HTML]{ACD1EA}57.2\scriptsize{$\pm$3.1} \\ \hline
LSTM &\cellcolor[HTML]{4095CD}\textbf{79.8} &\cellcolor[HTML]{D5E8F4}47.2 &\cellcolor[HTML]{BEDBEE}52.9 &\cellcolor[HTML]{9AC7E5}61.2 &\cellcolor[HTML]{64A9D7}72.7 &\cellcolor[HTML]{5EA6D5}\textbf{73.9} &\cellcolor[HTML]{8CBFE1}\textbf{64.3} &\cellcolor[HTML]{D6E8F4}46.9 &\cellcolor[HTML]{9DC9E6}60.5 &\cellcolor[HTML]{77B4DC}\textbf{68.7} &\cellcolor[HTML]{93C3E3}62.8\scriptsize{$\pm$11.3} \\ \hline
ours &\cellcolor[HTML]{72B1DA}69.8 &\cellcolor[HTML]{B7D7EC}54.7 &\cellcolor[HTML]{94C4E3}\textbf{62.5} &\cellcolor[HTML]{85BCDF}\textbf{65.7} &\cellcolor[HTML]{63A8D6}\textbf{72.9} &\cellcolor[HTML]{83BADF}66.2 &\cellcolor[HTML]{8FC1E2}63.6 &\cellcolor[HTML]{8BBFE1}\textbf{64.6} &\cellcolor[HTML]{A8CFE9}58.0 &\cellcolor[HTML]{8DC0E1}64.1 &\cellcolor[HTML]{8CC0E1}\textbf{64.2}\scriptsize{$\pm$5.2} \\ \hline
\end{tabular}}
\caption{
Model validation using GT labels for next action anticipation on the Breakfast Actions, presented are accuracies.
We compare transition matrices (TM), lookup tables (LUT), LSTMs, and our temporal aggregates model (without complex activity prediction).
}
\label{tab:baselineComparison}
\end{table}
For validating our method's capabilities in modelling sequences, we make baseline comparisons.
The simplest approach for solving the next action anticipation task is using a transition matrix (TM)~\cite{miech2019leveraging}, which encodes the transition from one action to the next. A more sophisticated solution is building a lookup table (LUT) of varying length sequences which allows encoding the context in a more explicit manner. The problem with LUTs is that their completeness depends on the coverage of the training data, and they rapidly grow with the number of actions. So far, for next step prediction, RNNs achieve good performance \cite{abu2018will}, as they learn modelling the sequences.
For our baseline comparisons, instead of frame features, we use the frame-level ground truth labels as input to our model. We compute the TM, LUT and RNN on the ground truth segment-level labels. In Table \ref{tab:baselineComparison} we present comparisons on the Breakfast Actions for the next action anticipation per complex activity. Overall, transition matrices provide the worst results. LUTs improve the results, as they incorporate more contextual information. Both the RNN and our method outperform the other alternatives, while our method still performs better than the RNN on average. However, applying RNNs requires parsing the past into action sequences~\cite{abu2018will}, which turns the problem into separate segmentation and prediction phases. Our model, on the other hand, can be trained end-to-end, and can represent the long-range observations good enough to outperform RNNs. We show that our model is doing better than simply learning pairwise statistics of the dataset.
\begin{figure}[t!]
\centering
\includegraphics[width=0.55\columnwidth]{im_supp/br1.pdf}
\includegraphics[width=0.55\columnwidth]{im_supp/br4.pdf}
\includegraphics[width=0.55\columnwidth]{im_supp/br2.pdf}
\includegraphics[width=0.55\columnwidth]{im_supp/br5.pdf}
\includegraphics[width=0.55\columnwidth]{im_supp/br3.pdf}
\includegraphics[width=0.55\columnwidth]{im_supp/br6.pdf}
\caption{Qualitative results for dense anticipation on Breakfast Actions dataset when using the GT labels and I3D features. Best viewed in color.}
\label{fig:dense_vis}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\textwidth]{im_supp/ep1.pdf}
\includegraphics[width=0.9\textwidth]{im_supp/ep4.pdf}
\includegraphics[width=0.9\textwidth]{im_supp/ep2.pdf}
\includegraphics[width=0.9\textwidth]{im_supp/ep3.pdf}
\caption{Exemplary qualitative results for next action anticipation on EPIC-Kitchens dataset, showing the success of our method. We list our Top-5 predictions at different anticipation times, $\tau_{\alpha}$. The closer we are the better are our model's predictions. Best viewed in color.}
\label{fig:epic_vis}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\textwidth]{im_supp/attention.pdf}
\caption{Attention visualization on the Breakfast Actions dataset for next action anticipation. Rectangles are the top 5 five spanning snippets (at different granularities where K = {10,15,20}), weighted highest by the attention mechanism in the Non-Local Blocks (NLB). Best viewed in color.}
\label{fig:atten_vis}
\end{figure*}
\section{Visual Results}
In Fig.~\ref{fig:dense_vis}, we provide qualitative results from our method for dense anticipation on the Breakfast Actions dataset. We show our method's predictions after observing 30\% of the video. We compare our results when we use the GT labels and I3D features as input.
In Fig.~\ref{fig:epic_vis}, we present qualitative results from our method for next action anticipation on the EPIC-Kitchens dataset for multiple anticipation times $\tau_{\alpha}$ between $0.25$ and $2$ seconds. We show examples where our method is certain about the next action for all different times. We also show examples where our method's prediction gets more accurate when the anticipation time is closer.
In Fig.~\ref{fig:atten_vis}, we present some visualizations of regions attended by our non-local blocks. We show the five highest weighted spanning snippets (at different granularities). Our model attends different regions over the videos, for example for predicting 'fry egg' when making fried eggs, it attends regions both when pouring oil and cracking eggs. Pouring oil is an important long-range past action for frying eggs. Our method can encode long video durations while attending to salient snippets.
\section{Action Anticipation on EPIC-Kitchens}
Furnari and Farinella~\cite{furnari2019rulstm} reports prediction results at multiple anticipation times ($\tau_{\alpha}$) between $0.25$s and $2$s on EPIC. We compare in Table~\ref{tab:anticipation_ek} on the validation set and note that our prediction scores are better than~\cite{furnari2019rulstm} for all time points. Our improvements are greater when the anticipation time decreases.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\setlength{\tabcolsep}{13pt}
\begin{tabular}{|rrrrrrrrr|}
\hline
\multicolumn{1}{|c}{} & \multicolumn{8}{c|}{Top-5 ACTION Accuracy\%} \\ \hline
$\tau_a$ & 2s & 1.75s & 1.5s & 1.25s & 1.0s & 0.75s & 0.5s & 0.25s \\ \hline
RU~\cite{furnari2019rulstm} & 29.4 & 30.7 & 32.2 & 33.4 & 35.3 & 36.3 & 37.4 & 39.0 \\
\textbf{ours} & \textbf{30.9} & \textbf{31.8} & \textbf{33.7} & \textbf{35.1} & \textbf{36.4} & \textbf{37.2} & \textbf{39.5} & \textbf{41.3} \\ \hline
\end{tabular} }
\caption{Action anticipation on EPIC validation set at different anticipation times.}
\label{tab:anticipation_ek}
\end{table}
We report our results for hold-out test data on EPIC-Kitchens Egocentric Action Anticipation Challenge (2020) in Table \ref{tab:epic_ex_test} for seen kitchens (S1) with the same environments as in the training data and unseen kitchens (S2) of held out environments. The official ranking on the challenge is based on the Top-1 action accuracy. Our submission (Team ``NUS\_CVML'') is ranked first on S1 and third on S2 sets. We refer the reader to EPIC-Kitchens 2020 Challenges Report~\cite{epicreport20} for details on the competing methods.
\begin{table*}[!t]
\centering
\resizebox{\textwidth}{!}{{}
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|clccc|ccc|ccc|ccc|}
\hline
& \multicolumn{4}{c|}{Top-1 Accuracy\%} & \multicolumn{3}{c|}{Top-5 Accuracy\%} & \multicolumn{3}{c|}{ Precision (\%)} & \multicolumn{3}{c|}{Recall (\%)} \\ \hline
& & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action \\\hline\rowcolor{red!10}
\multirow{7}{*}
& \textbf{1st} (S1) & 37.87 & 24.10 & \textbf{16.64} & 79.74 & 53.98 & 36.06 & 36.41 & 25.20 & 9.64 & 15.67 & 22.01 & 10.05 \\\hline\rowcolor{blue!10}
\multirow{7}{*}
& \textbf{3rd} (S2) & 29.50 & 16.52 & \textbf{10.04} & 70.13 & 37.83 & 23.42 & 20.43 & 12.95 & 4.92 & 8.03 & 12.84 & 6.26
\\ \hline
\end{tabular}}
\caption{Action anticipation on EPIC tests sets, seen (S1) and unseen (S2)}
\label{tab:epic_ex_test}
\end{table*}
\begin{table*}[!t]
\centering
\resizebox{\textwidth}{!}{{}
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|clccc|ccc|ccc|ccc|}
\hline
& \multicolumn{4}{c|}{Top-1 Accuracy\%} & \multicolumn{3}{c|}{Top-5 Accuracy\%} & \multicolumn{3}{c|}{ Precision (\%)} & \multicolumn{3}{c|}{Recall (\%)} \\ \hline
& & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action \\\hline\rowcolor{red!10}
\multirow{7}{*}
& \textbf{2nd} (S1) & 66.56 & 49.60 & \textbf{41.59} & 90.10 & 77.03 & 64.11 & 59.43 & 45.62 & 25.37 & 41.65 & 46.25 & 26.98 \\\hline\rowcolor{blue!10}
\multirow{7}{*}
& \textbf{3rd} (S2) & 54.56 & 33.46 & \textbf{26.97} & 80.40 & 60.98 & 46.43 & 33.60 & 30.54 & 14.99 & 25.28 & 28.39 & 17.97
\\ \hline
\end{tabular}}
\caption{Action recognition on EPIC tests sets, seen (S1) and unseen (S2)}
\label{tab:epic_ex_test_rec}
\end{table*}
\section{Action Recognition Challenge on EPIC-Kitchens}
We present our results for the EPIC-Kitchens Egocentric Action Recognition Challenge 2020 in Table \ref{tab:epic_ex_test_rec} for S1 and S2. Our team ``NUS\_CVML'' is ranked second on S1 and third on S2 sets. Please see EPIC-Kitchens 2020 Challenges Report~\cite{epicreport20} for further details.
\bibliographystyle{splncs04}
|
2,877,628,089,521 | arxiv | \section{Introduction}
\label{sect:intro}
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{example_vv}
\caption{
Two real essays generated by a strong model CTEG and our TegFormer method.
In light of the final texts, we made a few marks on the English translations.
First, we use three colors to distinguish each topic, and dye the segment of text to the same color as the topic, of which the semantics are covered by that segment.
Second, we underline the segment of text where there exist a problem of logic or repeat.
Clearly, texts generated by CTEG not only suffer from logical contradiction, but also fail to cover the ``Health'' topic.
}
\label{fig:example}
\end{figure}
The task of \textbf{t}opic-to-\textbf{e}ssay \textbf{g}eneration~(TEG)~\cite{MTA-LSTM} is to let machines generate a coherent essay that covers a few given topics (in the form of keywords/keyphrases), as illustrated in Fig~\ref{fig:example}.
Due to many potential real-life applications such as automated report generation, intelligent tutoring, and robot journalism~\cite{INLG}, TEG has attracted more and more attention from both academia and industry in recent years.
As a form of creative writing~\cite{C-Text-Generation}, it is challenging for machines to produce a human-readable essay with good topic coverage and high text coherence because the number of input topics are often too small to provide sufficient semantic contexts that could be leveraged by computational models.
Recently, several papers addressing the TEG task have been published and made notable progress in automatic text generations.
In the seminal work from \cite{MTA-LSTM}, a novel MTA method~\cite{MTA-LSTM} is developed to utilize a coverage mechanism to balance the expressions of topics, and thus generate readable texts.
\cite{CTEG} introduced commonsense knowledge and adversarial training to their CTEG model~\cite{CTEG}, and enhanced the generated texts' topic coverage.
\cite{SCKTG} integrated sentiment, knowledge, and adversarial training with CVAE~\cite{CVAE-1,CVAE-2} as SCTKG~\cite{SCKTG}, and realized good topic coverage.
\cite{CKE} used various topic-related semantics via knowledge distillation~\cite{know-distill} and knowledge graph in their TEGKE approach~\cite{CKE}, and yielded high-quality and diverse essays.
To sum up, these research have tried their best to make full use of input topics either with extra knowledge (distillation/commonsense) or learning manners (adversarial training/Conditional VAE), and produced readable, topic coverage, or diverse texts.
However, as Fig.~\ref{fig:example} exhibits that given three topics, CTEG, a most competitive method, generated an essay with a logical fallacy (``\textbf{a high-school and middle school student}''), a repeated expression (``\textbf{i don't know how to do}''), and the ``\textbf{Health}'' topic not covered, which tells us that linguistic grammars are also essential for natural language generations (e.g., TEG) apart from the widely concerned semantics. Besides, methods like CTEG~\cite{CTEG} and SCTKG~\cite{SCKTG} adopt adversarial training, where the discriminator is a classifier over topics, limiting the number of topics to only 100 in Zhihu-Refined dataset\footnote{\url{https://pan.baidu.com/s/17pcfWUuQTbcbniT0tBdwFQ}\label{fn_zhihuRefined}}~\cite{CTEG}, far smaller than 5,559, the number of topics appeared in the original Zhihu dataset\footnote{\url{https://pan.baidu.com/s/1eC4gb_We33kr-ZbHn3KdIA}\label{fn_Zhihu}}~\cite{MTA-LSTM}.
This implies that adversarial training would consume huge computing resources for robust results, and probably even fail on large-scale dataset.
To ensure good topic coverage and high text coherence in TEG, we propose a novel method called \textbf{TegFormer} which extends the Transformer architecture by enriching the encoder with more domain-specific contexts and enhancing the decoder with more background linguistic knowledge.
The main contributions of this paper are as follows.
\begin{itemize}
\item We have designed the Topic-Extension layer which is plugged in the encoder seamlessly to capture the semantic interactions between the given topics and their domain-specific contexts.
It can bring in more topic-related semantics for the subsequent decoder, when the given topics are usually sparse in semantics.
\item We have devised the Embedding-Fusion module which is plugged in the decoder seamlessly to capture the attentive sum of the domain-specific token embedding and the pre-trained GPT-2's embedding.
It can empower the decoder to generate more human-friendly texts since GPT-2 learns from large-scale open data and embeddings inferred from it implicitly contain abundant semantics and grammars.
\item Extensive experimental comparisons between TegFormer and several state-of-the-art methods (including the pre-trained big models such as BART~\cite{BART} and T5~\cite{T5}) show that TegFormer performs best according to both automatic and human evaluation.
Additional ablation studies confirm the effectiveness of Topic-Extension and Embedding-Fusion in boosting the quality of the generated text.
\end{itemize}
\section{Related Work}
\paragraph{Topic-to-Essay Generation.} Automatically generating an essay with several topics (i.e., TEG) is quite a difficult task in NLP.
\cite{MTA-LSTM} use coverage vector to integrate topic information.
\cite{CTEG} employ general knowledge to enrich the input semantics and adopt adversarial training to enhance topic coverage. \cite{SCKTG} inject the sentiment information into the generator for controlling sentiment based on CVAE and adversarial training. \cite{CKE} propose comprehensive knowledge enhanced model via knowledge distillation and knowledge graph.
These methods can generate either readable, topic coverage or diverse texts.
Besides, there are other relevant generation tasks such as controllable text generations, which demand on style/field/sentiment~\cite{PP-VAE,PPLM} control, and poetry generations~\cite{Poetry-1,Poetry-2} which require neural networks to compose poems with specific logics and rhymes given the input title.
In this paper, we mainly focus on the topic-to-essay generation, whose success is up to not only semantic enhancements but also linguistic grammars.
\paragraph{Pre-trained Language Models.}
There exist two types of pretrained language models (PLMs)~\cite{ZY}: discriminative PLMs such as BERT~\cite{BERT} and generative PLMs such as GPT~\cite{GPT-2,GPT-3}, BART~\cite{BART} and T5~\cite{T5}.
The key difference lies in that there is a decoder in generative PLMs so that they can generate text sequences.
The way of transferring a pre-trained generative PLMs to downstream tasks is to construct suitable inputs and outputs, and then fine-tune the models on the task-oriented datasets.
Such a two-stage paradigm, i.e., pre-training plus fine-tuning, shows notable effects on most generation tasks.
However, there also exist some issues, such as catastrophic forgetting, the task gap between the pretraining and finetuning stage, and no pertinent designs for downstream tasks~\cite{CF-1,CF-2}.
Different from the above learning mode, TegFormer freezes the introduced GPT-2 and devises a Topic-Extension layer and an Embedding-Fusion module based on the characteristics of the TEG task.
\section{Problem Statement}
Currently, there are two representative understandings (i.e., automatic free-form texts and sequential storytelling) w.r.t. topic-to-essay generations, e.g., MTA~\cite{MTA-LSTM}, and Plan\&Write~\cite{Plan_And_Write}.
Specifically, approaches like MTA view the input topics as a word set and let the machine produce a free-form text;
while methods like Plan\&Write treat them as a word sequence and let the machine generate a story.
In this paper, due to the properties of datasets and application scenarios, we formulate the task as a transition from a topic set to a paragraph-level text instead of the seq2seq task.
Formally, given a dataset including pairs of a topic set $t^{1:N}=\{t_{1},\cdots,t_{N}\}$, and its related essay $y^{1:L}=\{y_{1},\cdots,y_{L}\}$, we want a $\bm{\theta}$-parameterized model to generate an essay which could well match its marked essay $y^{1:L}$ in terms of topic coverage and text coherence; in other words, there holds:
\begin{equation}
\hat{\bm{\theta}} = \arg \max_{\bm{\theta}}{\rm{P}}_{\bm{\theta}}(y^{1:L}|t^{1:N}),
\end{equation}
where $\hat{\bm{\theta}}$ is the optimal parameter of the expected model $\rm{P}_{\bm{\theta}}(\cdot|\cdot)$. By the way, $L$ is the length of the generated text, and $N$ denotes the number of topic words, generally far smaller than $L$ (i.e., $N \ll L$).
\begin{figure*}[!tb]
\center
\includegraphics[width=0.98\textwidth]{model}
\caption{The \textbf{TegFormer} framework: the inputs are the given topics and each topic's top-$k$ most relevent keywords, and the output in each token-generated step is a probability over vocabulary.
}
\label{fig:model}
\end{figure*}
\section{Methodology}
\subsection{The Overall Framework}
Different from previous models which usually use LSTM/GRU~\cite{LSTM,GRU} as encoders or decoders, we adopt Transformer \cite{Transformer} as our base architecture, generating sequential texts from given topic sets.
Considering the problem of sequences generated by Transformer models, which are poor in performance on topic coverage~\cite{PPLM,PLM-3,C-Text-Generation,T5}, we devise the Topic-Extension layer, i.e., a semantic interaction between given topics and its surrounded semantic contexts, which can be seamlessly plugged in the Transformer encoder (seen as \cref{fig:model}).
To alleviate the logical problems (as illustrated in \cref{fig:example}), we marry both domain and general embeddings into the Transformer decoder (seen as \cref{fig:model}).
Since our methed is based on Transformer with customized layer/module for solving the TEG task, we call it \textbf{TegFormer}.
\begin{algorithm}[!tb]
\caption{Topic Extension}
\label{alg:1}
\LinesNumbered
\KwIn{Corpus $\chi$, i.e., a set of (topics, essay) pairs; the pre-set number of topic neighbors $k$;}
\KwOut{(topic, keywords) mappings $\psi$.}
Initialize topic-keyword Co-occurrence matrix $O$=\textbf{0};
\ForEach{$x \in \chi$}{
topics, essay = $x$;
keywords = Extractor(essay); {/* The TextRank~\cite{TextRank} algorithm provided by the Python package \texttt{jieba} is used to extract keywords from essays.*/}
\ForEach{topic in topics}{
\ForEach{keyword in keywords}{
$O$[topic][keyword] += 1;
}
}
}
Keywords with the top-$k$ largest values for each topic in $O$ as $\psi$.
\end{algorithm}
Concretely, given topics $t_{1},\cdots,t_{N}$ as the input of the Transformer encoder, we remove their position encodings and convert them into:
\begin{equation}
\textbf{M} = {\rm{Encoder}}(t^{1:N}),
\end{equation}
where $\textbf{M}$ represents the output hidden embeddings of the Transformer encoder from input topics.
We then use $\textbf{M}$ and the generated tokens $y_{1},\cdots,y_{i-1}$ in former steps to unlock the output hidden states $\textbf{h}_i^{o}$ of the Transformer decoder in the current step $i$:
\begin{equation}
\textbf{h}_i^{o} = {\rm{Decoder}}(\textbf{M}, y^{1:(i-1)}).
\end{equation}
Finally, as a conventional operation, we leverage the combination of a linear and a softmax layer, to yield a probability distribution over the vocabulary, based on which the $i$-th token $y_{i}$ is sampled with:
\begin{equation}
{\rm{P}}(y_{i}|t^{1:N}, y^{1:(i-1)}) = {\rm Softmax}(\textbf{W} \times {\textbf{h}}_{i}^{o} + \textbf{b}),
\end{equation}
where $\textbf{W}$ and $\textbf{b}$ are the TegFormer's learnable parameters.
\subsection{The Topic-Extension Encoder}
\label{sect:TGET}
The number of given topics ususally range from $2$ to $7$, which is naturally not semantically sufficient for text generations.
To address this problem, we deliberately design a Topic-Extension layer, i.e., a semantic interaction between input topics and their semantic contexts, which is implanted between the self-attention (Multi-Head Attention) sublayer and the FFN (Feed Forward Network) sublayer of the Transformer encoder, illustrated in the left and middle parts of \cref{fig:model}.
In light of the topics' contexts, we mainly search for their top-$k$ most frequently-co-occurrent keywords within all sentences of the given dataset (as specified in \cref{alg:1}).
Note that \cref{alg:1} is a pre-processing step: once the extracted keywords for each topic are obtained, they would be fixed and the corresponding vocabulary indexes would be supplied to the Topic-Extension layer. Therefore, it should not add much computational burden to training or inference.
Denote the output hidden states of multi-head attention sublayer about topic $i$ in Transformer encode-layer $l$ as $\textbf{h}_i^{(l)}$, and the extracted keywords (semantic contexts) for topic $i$ as ${w}_{t_i} = \{{w}_{t_i}^{1}, \cdots, {w}_{t_i}^{k}\}$.
We then use $\textbf{e}_{j}$ to represent the input embedding of the topic $i$'s context-keyword $w^j_{t_i}$.
Based on the above, the Topic-Extension layer can be described as follows:
\begin{equation}
{q}_i^{(l)} = \textbf{W}_q^{(l)} \times \textbf{h}_i^{(l)}
\textbf{k}_j^{(l)} = \textbf{W}_k^{(l)} \times \textbf{e}_{j}
\textbf{v}_j^{(l)} = \textbf{W}_v^{(l)} \times \textbf{e}_{j},
\end{equation}
and
\begin{equation}
\alpha^{(l)}_{ij} = \frac{\exp(\textbf{q}_i^{(l)} \times \textbf{k}_j^{(l)} )}{ {\sum_{z \in {w}_{t_i}} \exp( \textbf{q}_i^{(l)} \times \textbf{k}_z^{(l)} ) } },
\end{equation}
where $\textbf{W}_q^{(l)}$, $\textbf{W}_k^{(l)}$, and $\textbf{W}_v^{(l)}$ are learnable parameters.
Next, we use an Add\&Norm layer to stimulate the final output embedding of the encoder layer $l$ about topic $i$, which is marked as $\widetilde{\textbf{h}}_i^{(l)}$:
\begin{equation}
\widetilde{\textbf{h}}_i^{(l)} = \text{LN}(\textbf{h}_i^{(l)} + {\sum_{j \in w_{t_i}} (\alpha^{(l)}_{ij} \times \textbf{v}_{j})}).
\end{equation}
Since the semantically-coherent keywords for each topic are located within the given dataset, such keywords could be treated as domain-specific semantic contexts for the given topics.
As is evidently shown in \cref{fig:model}, the Transformer encoder covers not only the self-attentions among the given topics, but also the cross-attentions between the given topics and their semantic contexts, which accomplishes the semantic enhancements of the few input topics for the subsequent Transformer decoder.
\subsection{The Embedding-Fusion Decoder}
\label{sec: EFET}
Note that the pre-trained language model~\cite{PLM-1,PLM-2} can assimilate both semantic and grammatical knowledge from a large-scale corpus~\cite{PLM-1,PLM-3}, and thus we devise an Embedding-Fusion module, i.e., an attentive (or to say, a data-driven adaptive) sum of the previous already-generated domain-specific token embeddings and their corresponding GPT-2 embeddings, which is plugged in the Transformer decoder (seen as \cref{fig:model}).
Specifically, recall the already-generated sequential tokens $\{y_{1},\cdots,y_{i-1}\}$,
their general token embeddings are achieved by a frozen GPT-2\footnote{It is from \textbf{https://huggingface.co/uer/gpt2-chinese-cluecorpussmall} and is pre-trained on a 14GB dataset named CLUECorpusSmall~\cite{CLUECorpus2020}.} and a learnable linear layer which could transform the output dimension of GPT-2 into the input dimension of the original Transformer decoder (as illustrated in the right branch of \cref{fig:model}). Meanwhile, their domain embeddings are obtained by a Finder (i.e., a token-embedding lookup table) with an up projection to the input dimension of the original Transformer decoder (as illustrated in the left branch of \cref{fig:model}).
The formulation is:
\begin{equation}
\label{LM}
\{\textbf{e}_g(y_{j})\}_{j=1}^{i-1} = \textbf{W} \times \text{LM}(y_{1},\cdots, y_{i-1}),
\end{equation}
\begin{equation}\label{Finder}
\textbf{e}_d(y_{j}) = \textbf{W}_{\text{up}} \times \text{Finder}(y_{j}),
\end{equation}
\begin{equation}
\alpha_j = \text{sigmoid}(\textbf{v}^\top \times [\textbf{e}_g(y_{j}), \textbf{e}_d(y_{j})]),
\end{equation}
and
\begin{equation}
\textbf{e}(y_{j}) = \alpha_j~\textbf{e}_g(y_{j}) + (1 - \alpha_j)~\textbf{e}_d(y_{j}),
\end{equation}
where LM is the frozen pre-trained GPT-2 model and $\text{Finder}(y_{j})$ marks a lookup function that can locate a token's data-driven to-be-learnt embedding, \textbf{W} and $\textbf{W}_{\text{up}}$ are two trainable linear layers.
Since $\textbf{e}_d(y_j)$ is a to-be-learnt embedding based on the given text dataset, and $\textbf{e}_g(y_j)$ is a pre-trained model's embedding based on the open large-scale corpus, the Embedding-Fusion module enhances the Transformer decoder with domain and general semantics, and also with implicit linguistic grammars (GPT-2).
\subsection{Training and Inference}
We freeze the parameters of GPT-2 and adopt the Label Smoothing (LS in \cref{eq_8}) trick into a KL divergence loss, which is written as:
\begin{equation}
\label{eq_8}
\begin{split}
{\rm{D}}(\bm{\theta}) = \text{KL}({\rm{P}_{\bm{\theta}}}(y_{i}|y^{1:i-1}, t^{1:N})||\text{LS}(\tilde{y}_{i}) ),
\end{split}
\end{equation}
where $\tilde{y}_{i}$ denotes the $i$-th word of the labelled essay.
As \cite{Repeat-analysis} analysed that the ``High Inflow'' problem is the major reason of the common repetition phenomenon in text generation tasks.
To solve this issue, we use top-$k$ sampling~\cite{Topk} for inference.
\section{Experiments}
\begin{table*}[!tb]
\small
\centering
\setlength{\tabcolsep}{3.0mm}{
\begin{tabular}{@{}l|rrrrrr@{}}
\toprule
\textbf{Dataset} & \textbf{Total} & \textbf{Train} & \textbf{Test} & \textbf{Topics} & \textbf{Avg-T} & \textbf{Avg-Len} \\
\midrule
\textbf{Essay} & 494,944 & 300,000 & 5,000 & 7,995 & 5.00 & 64.13 \\
\textbf{Zhihu} & 56,221 & 50,000 & 5,000 & 5,559 & 5.00 & 78.14 \\
\textbf{Zhihu-Refined} & 30,000 & 27,000 & 2,300 & 100 & 2.36 & 71.94 \\
\bottomrule
\end{tabular}}
\caption{The dataset statistics: \textbf{``Total/Train/Test''} represents the number of essays in the whole/training/testing dataset.
\textbf{``Topics''} stands for the number of the topics appeared in the whole dataset.
\textbf{``Avg-T''} denotes the average number of topics for each essay.
\textbf{``Avg-Len''} refers to the average number of words within each essay.}
\label{tab:data_stat}
\end{table*}
\subsection{Datasets}
Three public datasets have been used in our experiments as depicted in \cref{tab:data_stat}, where \textbf{Essay}\footnote{\url{https://pan.baidu.com/s/1_JPh5-g2rry2QmbjQ3pZ6w}} and \textbf{Zhihu}\textsuperscript{\ref{fn_Zhihu}} were both constructed by \cite{MTA-LSTM},
while \textbf{Zhihu-Refined}\textsuperscript{\ref {fn_zhihuRefined}} was constructed by \cite{CTEG} to replace the Zhihu corpus in their experiments.
All these datasets are paragraph-level Chinese essays.
\begin{table*}[!tb]
\small
\centering
\begin{tabular}{@{}l|l|cccc|cccc@{}}
\toprule
& & \multicolumn{4}{c|}{Automatic Evaluation} & \multicolumn{4}{c}{Human Evaluation} \\
\textbf{Dataset} & \textbf{Method} & \textbf{BLEU-2} & \textbf{Rouge-L} & \textbf{Dist-1} & \textbf{Dist-2} & \textbf{Cov.} & \textbf{Nov.} & \textbf{Log.} & \textbf{Coh.} \\
\midrule
\multirow{7}{*}{\textbf{Essay}} & MTA & 4.04 & 9.32 & 2.11 & 11.08 & 2.71 & 3.52 & 3.30 & 3.30 \\
& Plan\&Write & 4.77 & 11.89 & 3.40 & 24.34 & 2.10 & 3.22 & 2.17 & 2.58 \\
& Apex & 5.81 & 12.67 & 4.14 & 28.81 & 4.16 & 3.41 & 3.21 & 3.42 \\
& Transformer & 5.78 & 11.35 & 4.02 & 28.42 & 3.80 & 3.51 & 2.62 & 3.18 \\
& BART & 4.79 & 10.02 & 4.34 & 27.58 & 2.91 & 3.31 & 3.29 & 3.50 \\
& T5 & 5.89 & 12.91 & 4.36 & 28.04 & 4.18 & 3.43 & 3.36 & 3.48 \\
& \textbf{TegFormer} & \textbf{6.32} & \textbf{14.24} & \textbf{4.64} & \textbf{30.86} & \textbf{4.23} & \textbf{3.59} & \textbf{3.42} & \textbf{3.87} \\ \midrule
\multirow{7}{*}{\textbf{Zhihu}} & MTA & 2.12 & 7.37 & 0.92 & 4.86 & 1.85 & 2.90 & 2.20 & 2.85 \\
& Plan\&Write & 2.42 & 8.81 & 3.48 & 24.26 & 2.00 & 2.85 & 1.90 & 2.25 \\
& Apex & 3.42 & 11.30 & 4.37 & 29.67 & 2.81 & 3.21 & 3.09 & 3.27 \\
& Transformer & 2.91 & 10.60 & 4.00 & 29.39 & 2.39 & 2.85 & 2.05 & 2.25 \\
& BART & 2.55 & 9.29 & 3.49 & 27.60 & 2.52 & 2.87 & 2.98 & 3.02 \\
& T5 & 3.41 & 11.31 & 4.34 & 30.16 & 2.87 & 3.12 & 3.11 & 3.24 \\
& \textbf{TegFormer} & \textbf{3.53} & \textbf{12.21} & \textbf{4.61} & \textbf{32.67} & \textbf{2.91} & \textbf{3.45} & \textbf{3.25} & \textbf{3.45} \\
\midrule
\multirow{2}{*}{\textbf{Zhihu-Refined}} & CTEG & 3.62 & 11.92 & 5.21 & 25.91 & 2.34 & 2.70 & 2.50 & 2.65 \\
& \textbf{TegFormer} & \textbf{3.81} & \textbf{13.02} & \textbf{5.81} & \textbf{33.62} & \textbf{2.95} & \textbf{3.21} & \textbf{3.75} & \textbf{3.48} \\
\bottomrule
\end{tabular
\caption{The automatic and human evaluations for different methods on three real-world datasets. According to the t-test, the improvements brought by TegFormer over all baselines are significant at the level $p<0.05$.
}
\label{tab:main_res}
\end{table*}
\subsection{Baseline Methods}
Following are seven methods selected as baselines for comparable experiments.
\begin{itemize}
\item \textbf{MTA}~\cite{MTA-LSTM} utilizes an LSTM decoder and a topic coverage vector to balance the expression of all topic information during generation.
\item \textbf{Plan\&Write}~\cite{Plan_And_Write} applies storyline as the intermediate process of story generation. The framework uses BiLSTM as encoder and LSTM as decoder.
\item \textbf{CTEG}~\cite{CTEG} introduces memory mechanism to store commonsense knowledge and leverages adversarial training to improve generation. It also uses LSTM as backbone of its framework.
\item \textbf{Transformer}~\cite{Transformer} is implemented as a Seq2Seq baseline.
To be specific, we use a special token to separate sequential topic words as its input.
\item \textbf{BART}~\cite{BART} is a Seq2Seq task oriented pre-trained model, which tries to re-construct the original texts from the corrupted ones.
We finetune it on our datasets.
\item \textbf{T5}~\cite{T5} is also a Seq2Seq task oriented pre-trained model, which views all NLP problems as Seq2Seq tasks, and transfers tasks such as NMT~\cite{NMT}, text classification~\cite{classification}, and summarization~\cite{summarization} to Seq2Seq formats.
We finetune it on our datasets.
\item \textbf{Apex}~\cite{Apex} mixes the CVAEs model with a PI controller, where P-term is proportional to KL divergence loss and I-term is proportional to the integral of it, aiming to manipulate the diversity and accuracy of generated texts in E-commerce from the view of PID Control Algorithm~\cite{PID-1,PID-2}.
We follow~\cite{Apex} and set PI-$v$ to $2$.
\end{itemize}
Considering that the discriminator of CTEG is a multi-label classifier~(i.e., each topic is treated as a label), and thus CTEG has strict requirements on its training data, {in other words,} only the most frequent topics are involved.
To make experiments go smoothly, CTEG builds the~\textbf{Zhihu-Refined} (much smaller than \textbf{Zhihu}) dataset.
Following the conventions, we conduct experiments on \textbf{Zhihu-Refined} between TegFormer and CTEG, and make comparisons with other six baselines on \textbf{Essay} and \textbf{Zhihu}.
\subsection{Implementation Details}
We train our method for $120$ epochs and the batch size is 128. The optimizer is Adam~\cite{Adam} with learning rate set to $10^{-3}$.
We use 6-layer encoders and decoders; the hidden layer size is 512 nodes with 8 attention heads; the intermediate size of FFN is 2048. Furthermore, to minimize the influence of model size and focus the comparison on model architecture, we use baselines of similar size: T5-small (6-layer), BART-base (6-layer) and Apex (with 6-layer Transformer).
Our GPU is Tesla V100. When training Transformer~(base) or finetuning BART and T5, we split the input topics with a special token.
We set 150 as the maximum length of the generated essays.
The inference model stops when the end token~({ i.e.}, [EOS] or [SEP]) is encountered in the process of generation.
More details can be found in the code to be released to the public.
\subsection{Evaluation Metrics}
Both automatic and human evaluations are adopted to compare various methods' performance.
For the automatic evaluation, we employ the popular metrics: Dist-1/2~\cite{CTEG}, ROUGE-L~\cite{Rouge-L}, and BLEU~\cite{BLEU}.
Following the popular settings of the human evaluation~\cite{CTEG,SCKTG,CKE},
we sample $200$ experimental results with each containing the given topics and their generated essays of different models.
Then we invite $3$ annotators who don't know which model the generated essays come from, and ask them to score them from 1 to 5 in four criteria, i.e., coherence (\textbf{``Coh.''}), topic coverage (\textbf{``Cov.''}), novelty (\textbf{``Nov.''}) and logic (\textbf{``Log.''}).
Finally, each method's score on a criterion is collected by averaging the scores of the three annotators. Notice that the Spearman's rank correlations between different human annotators' scores are high above $0.7$.
\subsection{Main Results}
\cref{tab:main_res} exhibits all the experimental results on three corpus. It's clear that no matter what kind of evaluations, \textbf{TegFormer} yields the highest scores, which proves our method's effectiveness on topic-to-essay generations.
By comparing several pre-trained language models (PLMs) such as BART\cite{BART}, Transformer~\cite{Transformer}, and T5~\cite{T5}, with the TEG task oriented models such as MTA~\cite{MTA-LSTM}, CTEG~\cite{CTEG} and Apex~\cite{Apex}, PLMs could catch up with the TEG approaches;
this is mainly due to PLMs' rich semantics and grammars learned from large-scale datasets.
However, the PLMs are still inferior to TegFormer, which implies that the gap between distinct tasks (pre-trained on one task, but learned for another task) limits their potentials to the fullest.
When it comes to TegFormer and Transformer, we could conclude that the Topic-Extension layer and the Embedding-Fusion module do promote the generated essays' topic coverage and text coherence.
Besides, it's reported that CTEG defeats several previous methods on \textbf{Zhihu-Refined}~\cite{CTEG}, but this refined Zhihu dataset is quite small and not practical in real-life scenarios;
and hence we just draw a comparison between our method and CTEG on \textbf{Zhihu-Refined}.
As exhibited in \cref{tab:main_res}, our TegFormer method still outperforms CTEG on all evaluation metrics.
\subsection{Hyperparameter Analysis}
TegFormer owns two important hyper-parameters.
The first is the dimension of the token embedding in the devised Embedding-Fusion module;
as \cref{fig:dim} shows,
$\text{dim}(\text{Finder}(y_{j}))=32$ in \cref{Finder} leads to competitive results with that of larger dimensions.
The second is the number of the extended keywords for each topic in the Topic-Extension layer;
as can be seen from \cref{fig:k}, $k=8$ (i.e., taking the top-$8$ most relevant keywords for each topic) is suitable to extract the semantic contexts of their topics.
\begin{figure*}[htb]
\centering
\subfigure[The dimension of token embedding $\text{dim}(\text{Finder}(y_{j}))$.]{
\label{fig:dim}
\includegraphics[width=0.46\textwidth]{dim}
}
\subfigure[The number of extended keywords $k$.]{
\label{fig:k}
\includegraphics[width=0.46\textwidth]{k}
}
\caption{BLEU scores of our TegFormer with different number of extended keywords in the Topic-Extension layer, and different dimensions of token embeddings in the Embedding-Fusion module, on \textbf{Zhihu} and \textbf{Essay} datasets.}
\vspace{-0.3cm}
\label{fig:agg_model}
\end{figure*}
\begin{table*}[!tb]
\small
\centering
\setlength{\tabcolsep}{1.0mm}{
\begin{tabular}{@{}l|l|cccc|cccc@{}}
\toprule
& & \multicolumn{4}{c|}{Automatic Evaluation} & \multicolumn{4}{c}{Human Evaluation} \\
\textbf{Dataset} & \textbf{Method} & \textbf{BLEU} & \textbf{Rouge-L} & \textbf{Dist-1} & \textbf{Dist-2} & \textbf{Cov.} & \textbf{Nov.} & \textbf{Log.} & \textbf{Coh.} \\
\midrule
\multirow{4}{*}{\textbf{Essay}} & Transformer & 5.78 & 11.35 & 4.02 & 28.42 & 3.80 & 3.51 & 2.62 & 3.18 \\
& TegFormer$_{EF}$ & 6.22 & 13.77 & 4.43 & 29.04 & 4.21 & 3.50 & 2.91 & 3.41 \\
& TegFormer$_{TE}$ & 5.81 & 11.91 & 4.53 & 30.42 & 4.08 & 3.53 & 3.36 & 3.78 \\
& \textbf{TegFormer} & \textbf{6.32} & \textbf{14.24} & \textbf{4.62} & \textbf{30.79} & \textbf{4.23} & \textbf{3.59} & \textbf{3.42} & \textbf{3.87} \\
\midrule
\multirow{4}{*}{\textbf{Zhihu}} & Transformer & 2.91 & 10.60 & 4.00 & 29.39 & 2.39 & 2.85 & 2.05 & 2.25 \\
& TegFormer$_{EF}$ & 3.45 & 12.14 & 4.36 & 29.81 & 2.88 & 2.98 & 2.21 & 2.41 \\
& TegFormer$_{TE}$ & 3.02 & 10.72 & 4.57 & 32.43 & 2.57 & 3.36 & 2.89 & 3.26 \\
& \textbf{TegFormer} & \textbf{3.53} & \textbf{12.21} & \textbf{4.60} & \textbf{32.48} & \textbf{2.91} & \textbf{3.45} & \textbf{3.25} & \textbf{3.45} \\
\bottomrule
\end{tabular}}
\caption{The automatic and human evaluation results of ablation studies.}
\label{tab:abl_study}
\end{table*}
\subsection{Ablation Study}
We have carried out ablation studies by comparing TegFormer with its incomplete variants including Transformer (Base), TegFormer w/o Embedding-Fusion (TegFormer$_{EF}$), and TegFormer w/o Topic-Extension (TegFormer$_{TE}$).
As can be seen from \cref{tab:abl_study}, TegFormer achieves the best results according to all the evaluations.
Obvious improvements on the logic and diversity from the vanilla Transformer to TegFormer (w/o Topic-Extension) demonstrate the effectiveness of the Embedding-Fusion module.
Similarly, increased scores on topic coverage from the vanilla Transformer to TegFormer (w/o Embedding-Fusion) testify that the proposed Topic-Extension layer could further enhance the quality of generated texts.
\section{Conclusion}
In this paper, we propose the TegFormer model --- a customized Transformer with Topic-Extension and Embedding-Fusion for TEG tasks.
In the encoder, to alleviate the semantic sparsity of input topics, we have designed a built-in Topic-Extension layer which enhances the topic coverage by extending the semantic contexts of the given topics with automatically extracted keywords.
In the decoder, we have developed a plug-and-play Embedding-Fusion module which blends domain-specific and general-purpose input embeddings to improve the text coherence of the generated essays.
Comparing our proposed TegFormer with several generative pre-trained models and the SOTA approaches to TEG tasks, we have found that while either the Topic-Extension layer or the Embedding-Fusion module alone is able to increase the output quality, their ``marriage'' leads to the maximum performance boost for TEG.
\newpage
\bibliographystyle{splncs04}
|
2,877,628,089,522 | arxiv | \section{Experimental Evaluation}
In this section, we present experimental results on real-world datasets. We compare our methods with several state of the art methods, in particular, those analyzed in section 4 (see the methods in the first column of Table 3) together with {\textsc{ADMM}}, the accelerated alternating direction method of multipliers~\citep{goldstein2013}.
To the best of our knowledge, such a detailed study has not been done for parallel and distributed $l_1$ regularized solutions in terms of (a) accuracy and solution optimality performance, (b) variable selection schemes, (c) computation versus communication time and (d) solution sparsity. The results demonstrate the effectiveness of our methods in terms of total (computation + communication) time on both accuracy and objective function measures.
\subsection{Experimental Setup}
\noindent{\bf Datasets:} We conducted our experiments on two popular benchmark datasets \textsc{KDD} and \textsc{URL}\footnote{{\small See \url{http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}. We refer to kdd2010 (algebra) dataset as \textsc{KDD}.}}. \textsc{KDD} has $n=8.41M$, $m=20.21M$ and $nz=0.31B$. \textsc{URL} has $n=2.00M$, $m=3.23M$ and $nz=0.22B$.
These datasets have sufficiently interesting characteristics of having a large number of examples and features such that (1) feature partitioning, (2) $l_1$ regularization and (3) communication are important. \vspace*{0.05in}
\noindent{\bf Methods and Metrics:} We evaluate the performance of all the methods using (a) Area Under Precision-Recall Curve (AUPRC)~\citep{alekh2013} and (b) Relative Function Value Difference (RFVD) as a function of time taken. RFVD is computed as $\log(\frac{F(w^t)-F^{*}}{F^{*}})$ where $F^{*}$ is taken as the best value obtained across the methods after a long duration. We stopped each method after 800 outer iterations. We also report per node computation time statistics and sparsity pattern behavior of all the methods. \vspace*{0.05in}
\noindent{\bf Parameter Settings:} We experimented with the $\lambda$ values of $(123,13.7,4.6) \times 10^{-7}$ and $(727,242,9) \times 10^{-8}$ for the \textit{KDD} and \textit{URL} datasets respectively. These values are chosen in such a way that they are centered around the respective optimal $\lambda$ value and have good sparsity variations over the optimal solution. With respect to Algorithm 1, the working set size (WSS) per node and number of nodes ($P$) are common across all the methods. We set WSS in terms of the fraction ($r$) of the number of features per node, i.e., WSS=$r\, m/P$. Note that WSS will change with $P$ for a given fraction $r$. For \textit{KDD} and \textit{URL}, we used three $r$ values $(0.01,0.1,0.25)$ and $(0.001,0.01,0.1)$ respectively. We experimented with $P = 25, 100$.
Also, $r$ does not play a role in ADMM since all variables are optimized in each node. \vspace*{0.05in}
\noindent{\bf Platform:} We ran all our experiments on a Hadoop cluster with $200$ nodes. Each node has Intel (R) Xeon (R) E5-2450L (2 processors) running at 1.8 GHz and 192 GB RAM. (Though both the datasets can fit in this memory configuration, our intention is to test the performance in a distributed setting.) All our implementations were done in $C\#$ including our binary tree \textit{AllReduce} support~\citep{alekh2013} on Hadoop.
\subsection{Method Specific Parameter Settings}
We discuss method specific parameter setting used in our experiments and associated practical implications.
Let us begin with {\textsc{ADMM}}. We use the feature partitioning formulation of ADMM described in section 8.3 of \citet{boyd2011}.
{\textsc{ADMM}}~does not fit into the format of Algorithm 1, but the communication cost per outer iteration is comparable to the other methods that fit into Algorithm 1.
In {\textsc{ADMM}}, the augmented Lagrangian parameter ($\rho$) plays an important role in getting good performance. In particular, the number of iterations required by ADMM for convergence is very sensitive with respect to $\rho$. While many schemes have been discussed in the literature~\citep{boyd2011} we found that selecting $\rho$ using the objective function value gave a good estimate; we selected $\rho^*$ from a handful of values with ADMM run for $10$ iterations (i.e., not full training) for each $\rho$ value tried.{\footnote{These initial ``tuning" iterations are not counted against the limit of 800 we set for the number of iterations. Thus, for ADMM, the total number of iterations can go higher than 800.} However, this step incurred some computational/communication time. In our time plots shown later, the late start of ADMM results is due to this cost. Note that this minimal number of iterations was required to get a decent $\rho^*$. \vspace*{0.05in}
\noindent{\bf Choice of ${\mathbf\mu}$ and ${\mathbf k}$:} To get a practical implementation that gives good performance in our method, we deviate slightly from the conditions of Theorem 1. First, we find that the proximal term does not contribute usefully to the progress of the algorithm (see the left side plot in Figure 1). So we choose to set $\mu$ to a small value, e.g., $\mu=10^{-12}$. Second, we replace the stopping condition (\ref{appopt}) by simply using a fixed number of cycles of coordinate descent to minimize $f_p^t$. The right side plot in Figure 1 shows the effect of number of cycles, $k$. We found that a good choice for the number of cycles is $10$ and we used this value in all our experiments.
For {\it GROCK}~, {\it FPA}~~and {\it HYDRA}~~we set the constants $(L^j)$ as suggested in the respective papers. Unfortunately, we found {\it GROCK}~~to be either unstable and diverging or extremely slow. The left side plot in Figure~\ref{fig:Issues} depicts these behaviors. The solid red line shows the divergence case. {\it FPA}~~requires an additional parameter ($\gamma$) setting for the stochastic approximation step size rule. Our experience is that setting right values for these parameters to get good performance can be tricky and highly dataset dependent. The right side plot in Figure~\ref{fig:Issues} shows the extremely slow convergence behavior of {\it FPA}~. Therefore, we do not include {\it GROCK}~~and {\it FPA}~~further in our study.
\begin{figure}[t]
\hspace{-0in}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/mueffect-nodes-100.eps}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/inneriterations.eps}
\caption{\small{Left: the effect of $\mu$. Right: the effect of $k$, the number of cycles to minimize $f^t_p$. $\mu = 10^{-12}$ and $k = 10$ are good choices. $P=100$.}}
\label{fig:MuPlot}
\vspace{-0.15in}
\end{figure}
\begin{figure}[t]
\hspace{-0in}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/grockplot-data-url-lambda-2pt424689e-06-nodes-25.eps}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/fpaplot-data-kdd-lambda-4pt585581e-07-nodes-100.eps}
\caption{\small{Left: Divergence and slow convergence of {\it GROCK}~~on the \textsc{URL} dataset ($\lambda = 2.4 \times 10^{-6}$ and $P = 25$). Right: Extremely slow convergence of {\it FPA}~~on the \textsc{KDD} dataset ($\lambda = 4.6 \times 10^{-7}$ and $P = 100$).}}
\label{fig:Issues}
\vspace{-0.15in}
\end{figure}
\subsection{Performance Evaluation}
We begin by comparing the efficiency of various methods and demonstrating the superiority of the new methods that were motivated in section 4 and developed in section 5. After this we analyze and explain the reasons for the superiority. \vspace*{0.05in}
\noindent{\bf Study on AUPRC and RFVD:}
We compare the performance of all methods by studying the variation of AUPRC and RFVD as a function of time, for various choices of $\lambda$, working set size (WSS) and the number of nodes ($P$) on \textsc{KDD} and \textsc{URL} datasets. To avoid cluttering with too many plots, we provide only representative ones; but, the observations that we make below hold for others too.
Figure~\ref{fig:kddobj} shows the objective function plots for (\textit{KDD}) with $\lambda$ set to $4.6 \times 10^{-7}$. We see that {\it DBCD-Southwell}~~clearly outperforms all other methods; for example, if we set the RFVD value to $-2$ as the stopping criterion, {\it DBCD-Southwell}~~is faster than existing methods by an order of magnitude. {\it PCDN-Southwell}~~comes as the second best. The S-scheme gives significant speed improvement over the R-scheme.
As we compare the performance for two different WSS (see Figure~\ref{fig:kddobj}(a)(b)), larger WSS gives some improvement and this speed-up is significant for {\it HYDRA}~, {\textsc{PCD-R}}~and {\it DBCD-Random}~. Note that {\textsc{ADMM}}~is WSS independent since all the variables are updated. Because all variables are updated, ADMM performs slightly better than {\it HYDRA}~, {\textsc{PCD-R}}~and {\it DBCD-Random}~~when WSS is small (see Figure~\ref{fig:kddobj}(a)(c)). In this case, other methods take some time to reach optimal values, when the working set is selected randomly using the R-scheme. If we compare {\textsc{ADMM}}~and {\it DBCD-Southwell}~, {\textsc{ADMM}}~is inferior; this holds even if we leave out the initial time needed for setting $\rho$.
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 8086$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-8086-lambda-4pt585581e-07-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 25$, $WSS = 80867$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-80867-lambda-4pt585581e-07-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-2021-lambda-4pt585581e-07-nodes-100.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-20216-lambda-4pt585581e-07-nodes-100.eps}
\vspace*{-2in}
}
\caption{\small{\textsc{KDD} dataset. Relative function value difference in log scale. $\lambda = 4.6 \times 10^{-7}$}}
\label{fig:kddobj}
\vspace{-0.15in}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 1292$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-1292-lambda-8pt980331e-08-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 25$, $WSS = 12927$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-12927-lambda-8pt980331e-08-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-323-lambda-8pt980331e-08-nodes-100.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-3231-lambda-8pt980331e-08-nodes-100.eps}
\vspace*{-2in}
}
\caption{\small{\textsc{URL} dataset. Relative function value difference in log scale. $\lambda = 9.0 \times 10^{-8}$}}
\label{fig:urlobj}
\vspace{-0.15in}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 8086$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-8086-lambda-4pt585581e-07-nodes-25.eps}
}
\subfigure[$P = 25$, $WSS = 80867$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-80867-lambda-4pt585581e-07-nodes-25.eps}
}
\subfigure[$P = 100$, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-2021-lambda-4pt585581e-07-nodes-100.eps}
}
\subfigure[$P = 100$, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-20216-lambda-4pt585581e-07-nodes-100.eps}
}
\caption{\small{\textsc{KDD} dataset. AUPRC Plots. $\lambda = 4.6 \times 10^{-7}$}}
\label{fig:kddauprc}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 1292$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-1292-lambda-8pt980331e-08-nodes-25.eps}
}
\subfigure[$P = 25$, $WSS = 12927$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-12927-lambda-8pt980331e-08-nodes-25.eps}
}
\subfigure[$P = 100$, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-323-lambda-8pt980331e-08-nodes-100.eps}
}
\subfigure[$P = 100$, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-3231-lambda-8pt980331e-08-nodes-100.eps}
}
\caption{\small{\textsc{URL} dataset. AUPRC plots. $\lambda = 9.0 \times 10^{-8}$}}
\label{fig:urlauprc}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[\textsc{KDD}, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{KDD}, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-323-lambda-7pt274068e-06-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-3231-lambda-7pt274068e-06-nodes-100.eps}
}
\caption{\small{Relative function value difference in log scale. \textsc{KDD} dataset: $\lambda = 1.2 \times 10^{-5}$. \textsc{URL} dataset: $\lambda = 7.3 \times 10^{-7}$}}
\label{fig:kddurlobj}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[\textsc{KDD}, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{KDD}, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-323-lambda-7pt274068e-06-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-3231-lambda-7pt274068e-06-nodes-100.eps}
}
\caption{\small{AUPRC plots. \textsc{KDD} dataset: $\lambda = 1.2 \times 10^{-5}$. \textsc{URL} dataset: $\lambda = 7.3 \times 10^{-7}$}}
\label{fig:kddurlauprc}
\end{figure*}
Figure~\ref{fig:urlobj} shows the objective function plots for (\textit{URL}) with $\lambda$ set to $9 \times 10^{-8}$. Here again, {\it DBCD-Southwell}~~gives the best RFVD performance with order of magnitude speed-up over existing methods. {\it HYDRA}~~suffers slow convergence and {\textsc{ADMM}}~gives a decent second best performance. Interestingly, the new variable selection rule (S-scheme) did not do very well for {\it PCDN}~~for large WSS. This shows that working with quadratic approximations (like {\it PCDN-Southwell}~~does) can at times be quite inferior compared to using the actual nonlinear objective (like {\it DBCD-Southwell}~~does). On comparing the performance for two different WSS, some speed improvement is achieved as in the case of \textsc{KDD} with similar observations. All these objective function progress behaviors are consistent with the AUPRC plots (Figures~\ref{fig:kddauprc} and \ref{fig:urlauprc}) as well except in one case. For example, the AUPRC performance of {\it PCDN-Southwell}~~is quite good although it is a bit slow on the objective function.
Figures~\ref{fig:kddurlobj} and \ref{fig:kddurlauprc} show the performance plots for another choice of different $\lambda$ values for the datasets. {\it DBCD-Southwell}~~gives the best performance on \textit{URL}. On \textit{KDD}, it is the second best after {\it PCDN-Southwell}~. This happens because the $S$ scheme selects features having a large number of non-zero feature values. As a result, computation cost goes up a bit as we do more inner iterations compared to {\it PCDN-Southwell}~. Nevertheless, the performance of {\it DBCD-Southwell}~~is still very good. Overall, the results point to the choice of {\it DBCD-Southwell}~~as the preferred method as it is highly effective with an order of magnitude improvement over existing methods. It is also worth noting that our proposal of using the S-scheme with the {\it PCDN}~~method~\citep{bian2013} offers significant value.
Next we analyze the reason behind the superior performance of {\it DBCD-Southwell}~. It is very much along the motivational ideas laid out in section 4: since communication cost dominates computation cost in each outer iteration, {\it DBCD-Southwell}~~reduces overall time by decreasing the number of outer iterations.
\def{ {> 800} }{{ {> 800} }}
\begin{table}[h]
\caption{$T^P$, the number of outer iterations needed to reach RFVD$\le \tau$, for various $P$ and $\tau$ values. Best values are indicated in boldface. (Note the $\log$ in the definition of RFVD.) Working set size is set at 1\% ($r=0.1$, $WSS=rm/P$). For each dataset, the $\tau$ values were chosen to cover the region where AUPRC values are in the process of reaching the steady state value.}
\label{tab:numiter}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\multicolumn{8}{c}{} \\
\multicolumn{8}{c}{KDD, $\lambda = 4.6\times 10^{-7}$} \\
\multicolumn{8}{c}{} \\
\hline
\multicolumn{2}{|c}{} & \multicolumn{3}{|c|}{Existing methods} & \multicolumn{3}{c|}{Our methods} \\
\hline
P & $\tau$ & {\it HYDRA}~ & {\textsc{ADMM}} & {\textsc{PCD-R}} & {\it PCDN-Southwell}~ & {\it DBCD-Random}~ & {\it DBCD-Southwell}~ \\
\hline
& $-1$ & 298 & 159 & 294 & 12 & 236 & {\bf 8} \\
25 & $-2$ & ${ {> 800} }$ & 317 & ${ {> 800} }$ & 311 & ${ {> 800} }$ & {\bf 104} \\
& $-3$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & {\bf 509} \\
\hline
& $-1$ & 297 & 180 & 299 & 12 & 230 & {\bf 10} \\
100 & $-2$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & 311 & ${ {> 800} }$ & {\bf 137} \\
& $-3$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & {\bf 650} & ${ {> 800} }$ & 668 \\
\hline
\multicolumn{8}{c}{} \\
\multicolumn{8}{c}{URL, $\lambda = 9.0\times 10^{-8}$} \\
\multicolumn{8}{c}{} \\
\hline
& $0$ & 376 & 137 & 106 & 65 & 101 & {\bf 6} \\
25 & $-0.5$ & ${ {> 800} }$ & 179 & 337 & 117 & 193 & {\bf 14} \\
& $-1$ & ${ {> 800} }$ & 214 & 796 & 196 & 722 & {\bf 22} \\
\hline
& $0$ & 400 & 120 & 91 & 64 & 78 & {\bf 7} \\
100 & $-0.5$ & ${ {> 800} }$ & 176 & 313 & 116 & 182 & {\bf 16} \\
& $-1$ & ${ {> 800} }$ & 231 & 718 & 190 & 582 & {\bf 28} \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent{\bf Study on the number of outer iterations:}
We study $T^P$, the number of outer iterations needed to reach RFVD$\le \tau$. Table~\ref{tab:numiter} gives $T^P$ values for various methods in various settings. {\it DBCD-Southwell}~~clearly outperforms other methods in terms of having much smaller values for $T^P$. {\it PCDN-Southwell}~~is the second best method, followed by {\textsc{ADMM}}. The solid reduction of $T^P$ by {\it DBCD-Southwell}~~validates the design that was motivated in section 4. The increased computation associated with {\it DBCD-Southwell}~~is immaterial; because communication cost overshadows computation cost in each iteration for all methods, {\it DBCD-Southwell}~~is also the best in terms of the overall computing time. The next set of results gives the details. \vspace*{0.05in}
\noindent{\bf Computation and Communication Time:}
As emphasized earlier, communication plays an important role in the distributed setting. To study this effect, we measured the computation and communication time separately at each node. Figure~\ref{fig:comptime} shows the computation time per node on the \textit{KDD} dataset. In both cases, {\textsc{ADMM}}~incurs significant computation time compared to other methods. This is because it optimizes over all variables in each node. {\it DBCD-Southwell}~~and {\it DBCD-Random}~~come next because our method involves both line search and $10$ inner iterations. {\textsc{PCD-R}}~and {\it PCDN-Southwell}~~take a little more time than {\it HYDRA}~~because of the line search. As seen in both {\textsc{DBCD}}~and {\it PCDN}~~cases, a marginal increase in time is incurred due to the variable selection cost with the S-scheme compared to the R-scheme.
\begin{table}[h]
\caption{Computation and communication costs per iteration (in secs.) for KDD, $P=25$.}
\label{tab:timings}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c}
\hline
Method & Comp. & Comm. & Comp. & Comm.\\
\hline
& \multicolumn{2}{|c|}{WSS: $r=0.1$} & \multicolumn{2}{|c|}{WSS: $r=0.01$}\\
\hline
Hyd & 0.022 & 5.192 & 0.131 & 4.888\\
\hline
PCD-R & 0.138 & 5.752 & 0.432 & 5.817\\
\hline
PCD-S & 1.564 & 7.065 & 1.836 & 7.032\\
\hline
DBCD-R & 0.991 & 6.322 & 1.978 & 6.407\\
\hline
DBCD-S & 5.054 & 6.563 & 5.557 & 8.867\\
\hline
\end{tabular}
\end{center}
\end{table}
We measured the computation and communication time taken per iteration by each method for different $P$ and $WSS$ settings. From Table~\ref{tab:timings} (which gives representative results for one situation, \textsc{KDD} and $P=25$), we see that the communication time dominates the cost in {\it HYDRA}~~and {\textsc{PCD-R}}.
{\it DBCD-Random}~~takes more computation time than {\textsc{PCD-R}}~and {\it HYDRA}~~since we run through $10$ cycles of inner optimization. Note that the methods with S-scheme take more time; however, the increase is not significant compared to the communication cost. {\it DBCD-Southwell}~~takes the maximum computation time and is quite comparable to the communication time. Recall our earlier observation of {\it DBCD-Southwell}~~giving order of magnitude speed-up in the overall time compared to methods such as {\it HYDRA}~~and {\textsc{PCD-R}}~(see Figures 3-8). Though the computation times taken by {\it HYDRA}~, {\textsc{PCD-R}}~and {\it PCDN-Southwell}~~are lesser, they need significantly more number of iterations to reach some specified objective function optimality criterion. As a result, these methods become quite inefficient due to extremely large communication cost compared to \textsc{DBCD}. All these observations point to the fact our {\textsc{DBCD}}~method nicely trades-off the computation versus communication cost, and gives an excellent order of magnitude improvement in overall time. With the additional benefit provided by the S-scheme, {\it DBCD-Southwell}~~clearly turns out to be the method of choice for the distributed setting. \vspace*{0.05in}
\noindent{\bf Sparsity Pattern:} To study weight sparsity behaviors of various methods during optimization, we computed the percentage of non-zero weights ($\rho$) as a function of outer iterations. We set the initial weight vector to zero. Figure~\ref{fig:sparse} shows similar behaviors for all the random (variable) selection methods. After a few iterations of rise they fall exponentially and remain at the same level. For methods with the S-scheme, many variables remain non-zero for some initial period of time and then $\rho$ falls a lot more sharply. It is interesting to note that such an initial behavior seems necessary to make good progress in terms of both function value and AUPRC (Figure~\ref{fig:kddurlobj}(a)(b) and Figure~\ref{fig:kddurlauprc}(a)(b)) In all the cases, many variables stay at zero after initial iterations; therefore, shrinking ideas can be used to improve efficiency. \vspace*{0.05in}
\noindent{\bf Remark on Speed up:}
Let us consider the RFVD plots corresponding to {\it DBCD-Southwell}~~in Figures~\ref{fig:kddobj}~and~\ref{fig:urlobj}. It can be observed that the times associated with $P=25$ and $P=100$ for reaching a certain tolerance, say RFVD=-2, are close to each other. This means that using 100 nodes gives almost no speed up over 25 nodes, which may prompt the question: {\it Is a distributed solution really necessary?} There are two answers to this question. First, as we already mentioned, when the training data is huge\footnote{The \textsc{KDD} and \textsc{URL} datasets are really not huge in the {\it Big data} sense. In this paper we used them only because of lack of availability of much bigger public datasets.} and so {\it the data is generated and forced to reside in distributed nodes}, the right question to ask is not whether we get great speed up, but to ask which method is the fastest. Second, for a given dataset, if the time taken to reach a certain optimality tolerance is plotted as a function of $P$, it may have a minimum at a value different from $P=1$. In such a case, it is appropriate to choose a $P$ (as well as $WSS$) optimally to minimize training time. Many applications involve periodically repeated model training. For example, in Advertising, logistic regression based click probability models are retrained on a daily basis on incrementally varying datasets. In such scenarios it is worthwhile to spend time to tune parameters such as $P$ and $WSS$ in an early deployment phase to minimize time, and then use these parameter values for future runs.
It is also important to point out that the above discussion is relevant to distributed settings in which communication causes a bottleneck. If communication cost is not heavy, e.g., when the number of examples is not large and/or communication is inexpensive such as in multicore solution, then good speed ups are possible; see, for example, the results in~\citet{richtarik2013}.
\begin{figure}[t]
\subfigure[$WSS = 2021$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/comptimevsobjfunc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[$WSS = 20216$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/comptimevsobjfunc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\caption{\small{Per-node computation time on the \textsc{KDD} dataset ($\lambda = 1.2 \times 10^{-5}$ and $P = 100$).}}
\label{fig:comptime}
\end{figure}
\begin{figure}[t]
\subfigure[$WSS = 2021$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/itersvssparsity-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[$WSS = 20216$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/itersvssparsity-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\caption{\small{\textsc{KDD} dataset: Percentage of non-zero weights. $\lambda = 1.2 \times 10^{-5}$ and $P = 100$. }}
\label{fig:sparse}
\end{figure}
\section{Experimental Evaluation}
\label{sec:expts}
In this section, we present experimental results on real-world datasets. We compare our methods with several state of the art methods, in particular, those analyzed in Section~\ref{sec:motiv} (see the methods in the first column of Table 3) together with {\textsc{ADMM}}, the accelerated alternating direction method of multipliers~\citep{goldstein2013}.
To the best of our knowledge, such a detailed study has not been done for parallel and distributed $l_1$ regularized solutions in terms of (a) accuracy and solution optimality performance, (b) variable selection schemes, (c) computation versus communication time and (d) solution sparsity. The results demonstrate the effectiveness of our methods in terms of total (computation + communication) time on both accuracy and objective function measures.
\subsection{Experimental Setup}
\label{subsec:setup}
\noindent{\bf Datasets:} We conducted our experiments on two popular benchmark datasets \textsc{KDD} and \textsc{URL}\footnote{{\small See \url{http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}. We refer to kdd2010 (algebra) dataset as \textsc{KDD}.}}. \textsc{KDD} has $n=8.41\times 10^6$, $m=20.21\times 10^6$ and $nz=0.31\times 10^9$. \textsc{URL} has $n=2.00\times 10^6$, $m=3.23\times 10^6$ and $nz=0.22\times 10^9$.
These datasets have sufficiently interesting characteristics of having a large number of examples and features such that (1) feature partitioning, (2) $l_1$ regularization and (3) communication are important. \vspace*{0.05in}
\noindent{\bf Methods and Metrics:} We evaluate the performance of all the methods using (a) Area Under Precision-Recall Curve (AUPRC)~\citep{alekh2013} and (b) Relative Function Value Difference (RFVD) as a function of time taken. RFVD is computed as $\log(\frac{F(w^t)-F^{*}}{F^{*}})$ where $F^{*}$ is taken as the best value obtained across the methods after a long duration. We stopped each method after 800 outer iterations. We also report per node computation time statistics and sparsity pattern behavior of all the methods. \vspace*{0.05in}
\noindent{\bf Parameter Settings:} We experimented with the $\lambda$ values of $(1.23\times 10^{-5},1.37\times 10^{-6},4.6\times 10^{-7})$ and $(7.27\times 10^{-6},2.42\times 10^{-6},9\times 10^{-8})$ for the \textit{KDD} and \textit{URL} datasets respectively. These values are chosen in such a way that they are centered around the respective optimal $\lambda$ value and have good sparsity variations over the optimal solution. With respect to Algorithm 1, the working set size (WSS) per node and number of nodes ($P$) are common across all the methods. We set WSS in terms of the fraction ($r$) of the number of features per node, i.e., WSS=$r m/P$. Note that WSS will change with $P$ for a given fraction $r$. For \textit{KDD} and \textit{URL}, we used three $r$ values $(0.01,0.1,0.25)$ and $(0.001,0.01,0.1)$ respectively. We experimented with $P = 25, 100$.
Also, $r$ does not play a role in ADMM since all variables are optimized in each node. \vspace*{0.05in}
\noindent{\bf Platform:} We ran all our experiments on a Hadoop cluster with $379$ nodes and 10 Gbit interconnect speed. Each node has Intel (R) Xeon (R) E5-2450L (2 processors) running at 1.8 GHz and 192 GB RAM. (Though the datasets can fit in this memory configuration, our intention is to test the performance in a distributed setting.) All our implementations were done in $C\#$ including our binary tree \textit{AllReduce} support~\citep{alekh2013} on Hadoop.
\subsection{Method Specific Parameter Settings}
\label{subsec:par}
We discuss method specific parameter setting used in our experiments and associated practical implications.
Let us begin with {\textsc{ADMM}}. We use the feature partitioning formulation of ADMM described in Subsection 8.3 of \citet{boyd2011}.
{\textsc{ADMM}}~does not fit into the format of Algorithm 1, but the communication cost per outer iteration is comparable to the other methods that fit into Algorithm 1.
In {\textsc{ADMM}}, the augmented Lagrangian parameter ($\rho$) plays an important role in getting good performance. In particular, the number of iterations required by ADMM for convergence is very sensitive with respect to $\rho$. While many schemes have been discussed in the literature~\citep{boyd2011} we found that selecting $\rho$ using the objective function value gave a good estimate; we selected $\rho^*$ from a handful of values with ADMM run for $10$ iterations (i.e., not full training) for each $\rho$ value tried.{\footnote{These initial ``tuning" iterations are not counted against the limit of 800 we set for the number of iterations. Thus, for ADMM, the total number of iterations can go higher than 800.} However, this step incurred some computational/communication time. In our time plots shown later, the late start of ADMM results is due to this cost. Note that this minimal number of iterations was required to get a decent $\rho^*$. \vspace*{0.05in}
\noindent{\bf Choice of ${\mathbf\mu}$ and $k$:} To get a practical implementation that gives good performance in our method, we deviate slightly from the conditions of Theorem 1. First, we find that the proximal term does not contribute usefully to the progress of the algorithm (see the left side plot in Figure 1). So we choose to set $\mu$ to a small value, e.g., $\mu=10^{-12}$. Second, we replace the stopping condition (\ref{appopt}) by simply using a fixed number of cycles of coordinate descent to minimize $f_p^t$. The right side plot in Figure 1 shows the effect of number of cycles, $k$. We found that a good choice for the number of cycles is $10$ and we used this value in all our experiments.
For {\it GROCK}~, {\it FPA}~~and {\it HYDRA}~~we set the constants $(L_j)$ as suggested in the respective papers. Unfortunately, we found {\it GROCK}~~to be either unstable and diverging or extremely slow. The left side plot in Figure~\ref{fig:Issues} depicts these behaviors. The solid red line shows the divergence case. {\it FPA}~~requires an additional parameter ($\gamma$) setting for the stochastic approximation step size rule. Our experience is that setting right values for these parameters to get good performance can be tricky and highly dataset dependent. The right side plot in Figure~\ref{fig:Issues} shows the extremely slow convergence behavior of {\it FPA}~. Therefore, we do not include {\it GROCK}~~and {\it FPA}~~further in our study.
\begin{figure}[t]
\hspace{-0in}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/mueffect-nodes-100.eps}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/inneriterations.eps}
\caption{\small{Left: the effect of $\mu$. Right: the effect of $k$, the number of cycles to minimize $f^t_p$. $\mu = 10^{-12}$ and $k = 10$ are good choices. $P=100$.}}
\label{fig:MuPlot}
\vspace{-0.15in}
\end{figure}
\begin{figure}[t]
\hspace{-0in}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/grockplot-data-url-lambda-2pt424689e-06-nodes-25.eps}
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/fpaplot-data-kdd-lambda-4pt585581e-07-nodes-100.eps}
\caption{\small{Left: Divergence and slow convergence of {\it GROCK}~~on the \textsc{URL} dataset ($\lambda = 2.4 \times 10^{-6}$ and $P = 25$). Right: Extremely slow convergence of {\it FPA}~~on the \textsc{KDD} dataset ($\lambda = 4.6 \times 10^{-7}$ and $P = 100$).}}
\label{fig:Issues}
\vspace{-0.15in}
\end{figure}
\subsection{Performance Evaluation}
\label{subsec:eval}
We begin by comparing the efficiency of various methods and demonstrating the superiority of the new methods that were motivated in Section~\ref{sec:motiv} and developed in Section~\ref{sec:dbcd}. After this we analyze and explain the reasons for the superiority. \vspace*{0.05in}
\noindent{\bf Study on AUPRC and RFVD:}
We compare the performance of all methods by studying the variation of AUPRC and RFVD as a function of time, for various choices of $\lambda$, working set size (WSS) and the number of nodes ($P$) on \textsc{KDD} and \textsc{URL} datasets. To avoid cluttering with too many plots, we provide only representative ones; but, the observations that we make below hold for others too.
Figure~\ref{fig:kddobj} shows the objective function plots for (\textit{KDD}) with $\lambda$ set to $4.6 \times 10^{-7}$. We see that {\it DBCD-Southwell}~~clearly outperforms all other methods; for example, if we set the RFVD value to $-2$ as the stopping criterion, {\it DBCD-Southwell}~~is faster than existing methods by an order of magnitude. {\it PCDN-Southwell}~~comes as the second best. The S-scheme gives significant speed improvement over the R-scheme.
As we compare the performance for two different WSS (see Figure~\ref{fig:kddobj}(a)(b)), larger WSS gives some improvement and this speed-up is significant for {\it HYDRA}~, {\textsc{PCD-R}}~and {\it DBCD-Random}~. Note that {\textsc{ADMM}}~is WSS independent since all the variables are updated. Because all variables are updated, ADMM performs slightly better than {\it HYDRA}~, {\textsc{PCD-R}}~and {\it DBCD-Random}~~when WSS is small (see Figure~\ref{fig:kddobj}(a)(c)). In this case, other methods take some time to reach optimal values, when the working set is selected randomly using the R-scheme. If we compare {\textsc{ADMM}}~and {\it DBCD-Southwell}~, {\textsc{ADMM}}~is inferior; this holds even if we leave out the initial time needed for setting $\rho$.
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 8086$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-8086-lambda-4pt585581e-07-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 25$, $WSS = 80867$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-80867-lambda-4pt585581e-07-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-2021-lambda-4pt585581e-07-nodes-100.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-20216-lambda-4pt585581e-07-nodes-100.eps}
\vspace*{-2in}
}
\caption{\small{\textsc{KDD} dataset. Relative function value difference in log scale. $\lambda = 4.6 \times 10^{-7}$}}
\label{fig:kddobj}
\vspace{-0.15in}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 1292$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-1292-lambda-8pt980331e-08-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 25$, $WSS = 12927$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-12927-lambda-8pt980331e-08-nodes-25.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-323-lambda-8pt980331e-08-nodes-100.eps}
\vspace*{-2in}
}
\subfigure[$P = 100$, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-3231-lambda-8pt980331e-08-nodes-100.eps}
\vspace*{-2in}
}
\caption{\small{\textsc{URL} dataset. Relative function value difference in log scale. $\lambda = 9.0 \times 10^{-8}$}}
\label{fig:urlobj}
\vspace{-0.15in}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 8086$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-8086-lambda-4pt585581e-07-nodes-25.eps}
}
\subfigure[$P = 25$, $WSS = 80867$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-80867-lambda-4pt585581e-07-nodes-25.eps}
}
\subfigure[$P = 100$, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-2021-lambda-4pt585581e-07-nodes-100.eps}
}
\subfigure[$P = 100$, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-20216-lambda-4pt585581e-07-nodes-100.eps}
}
\caption{\small{\textsc{KDD} dataset. AUPRC Plots. $\lambda = 4.6 \times 10^{-7}$}}
\label{fig:kddauprc}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[$P = 25$, $WSS = 1292$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-1292-lambda-8pt980331e-08-nodes-25.eps}
}
\subfigure[$P = 25$, $WSS = 12927$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-12927-lambda-8pt980331e-08-nodes-25.eps}
}
\subfigure[$P = 100$, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-323-lambda-8pt980331e-08-nodes-100.eps}
}
\subfigure[$P = 100$, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-3231-lambda-8pt980331e-08-nodes-100.eps}
}
\caption{\small{\textsc{URL} dataset. AUPRC plots. $\lambda = 9.0 \times 10^{-8}$}}
\label{fig:urlauprc}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[\textsc{KDD}, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{KDD}, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-323-lambda-7pt274068e-06-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsobjfunc-data-url-workingset-3231-lambda-7pt274068e-06-nodes-100.eps}
}
\caption{\small{Relative function value difference in log scale. \textsc{KDD} dataset: $\lambda = 1.2 \times 10^{-5}$. \textsc{URL} dataset: $\lambda = 7.3 \times 10^{-7}$}}
\label{fig:kddurlobj}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[\textsc{KDD}, $WSS = 2021$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{KDD}, $WSS = 20216$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 323$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-323-lambda-7pt274068e-06-nodes-100.eps}
}
\subfigure[\textsc{URL}, $WSS = 3231$]{
\includegraphics[width=0.45\linewidth]{plots/finalplotseps/timevsauprc-data-url-workingset-3231-lambda-7pt274068e-06-nodes-100.eps}
}
\caption{\small{AUPRC plots. \textsc{KDD} dataset: $\lambda = 1.2 \times 10^{-5}$. \textsc{URL} dataset: $\lambda = 7.3 \times 10^{-7}$}}
\label{fig:kddurlauprc}
\end{figure*}
Figure~\ref{fig:urlobj} shows the objective function plots for (\textit{URL}) with $\lambda$ set to $9 \times 10^{-8}$. Here again, {\it DBCD-Southwell}~~gives the best RFVD performance with order of magnitude speed-up over existing methods. {\it HYDRA}~~suffers slow convergence and {\textsc{ADMM}}~gives a decent second best performance. Interestingly, the new variable selection rule (S-scheme) did not do very well for {\it PCDN}~~for large WSS. This shows that working with quadratic approximations (like {\it PCDN-Southwell}~~does) can at times be quite inferior compared to using the actual nonlinear objective (like {\it DBCD-Southwell}~~does). On comparing the performance for two different WSS, some speed improvement is achieved as in the case of \textsc{KDD} with similar observations. All these objective function progress behaviors are consistent with the AUPRC plots (Figures~\ref{fig:kddauprc} and \ref{fig:urlauprc}) as well except in one case. For example, the AUPRC performance of {\it PCDN-Southwell}~~is quite good although it is a bit slow on the objective function.
Figures~\ref{fig:kddurlobj} and \ref{fig:kddurlauprc} show the performance plots for another choice of different $\lambda$ values for the datasets. {\it DBCD-Southwell}~~gives the best performance on \textit{URL}. On \textit{KDD}, it is the second best after {\it PCDN-Southwell}~. This happens because the $S$ scheme selects features having a large number of non-zero feature values. As a result, computation cost goes up a bit as we do more inner iterations compared to {\it PCDN-Southwell}~. Nevertheless, the performance of {\it DBCD-Southwell}~~is still very good. Overall, the results point to the choice of {\it DBCD-Southwell}~~as the preferred method as it is highly effective with an order of magnitude improvement over existing methods. It is also worth noting that our proposal of using the S-scheme with the {\it PCDN}~~method~\citep{bian2013} offers significant value.
Next we analyze the reason behind the superior performance of {\it DBCD-Southwell}~. It is very much along the motivational ideas laid out in Section~\ref{sec:motiv}: since communication cost dominates computation cost in each outer iteration, {\it DBCD-Southwell}~~reduces overall time by decreasing the number of outer iterations.
\def{ {> 800} }{{ {> 800} }}
\begin{table}[h]
\caption{$T^P$, the number of outer iterations needed to reach RFVD$\le \tau$, for various $P$ and $\tau$ values. Best values are indicated in boldface. (Note the $\log$ in the definition of RFVD.) Working set size is set at 1\% ($r=0.1$, $WSS=rm/P$). For each dataset, the $\tau$ values were chosen to cover the region where AUPRC values are in the process of reaching the steady state value.}
\label{tab:numiter}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\multicolumn{8}{c}{} \\
\multicolumn{8}{c}{KDD, $\lambda = 4.6\times 10^{-7}$} \\
\multicolumn{8}{c}{} \\
\hline
\multicolumn{2}{|c}{} & \multicolumn{3}{|c|}{Existing methods} & \multicolumn{3}{c|}{Our methods} \\
\hline
P & $\tau$ & {\it HYDRA}~ & {\textsc{ADMM}} & {\textsc{PCD-R}} & {\it PCDN-Southwell}~ & {\it DBCD-Random}~ & {\it DBCD-Southwell}~ \\
\hline
& $-1$ & 298 & 159 & 294 & 12 & 236 & {\bf 8} \\
25 & $-2$ & ${ {> 800} }$ & 317 & ${ {> 800} }$ & 311 & ${ {> 800} }$ & {\bf 104} \\
& $-3$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & {\bf 509} \\
\hline
& $-1$ & 297 & 180 & 299 & 12 & 230 & {\bf 10} \\
100 & $-2$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & 311 & ${ {> 800} }$ & {\bf 137} \\
& $-3$ & ${ {> 800} }$ & ${ {> 800} }$ & ${ {> 800} }$ & {\bf 650} & ${ {> 800} }$ & 668 \\
\hline
\multicolumn{8}{c}{} \\
\multicolumn{8}{c}{URL, $\lambda = 9.0\times 10^{-8}$} \\
\multicolumn{8}{c}{} \\
\hline
\multicolumn{2}{|c}{} & \multicolumn{3}{|c|}{Existing methods} & \multicolumn{3}{c|}{Our methods} \\
\hline
& $0$ & 376 & 137 & 106 & 65 & 101 & {\bf 6} \\
25 & $-0.5$ & ${ {> 800} }$ & 179 & 337 & 117 & 193 & {\bf 14} \\
& $-1$ & ${ {> 800} }$ & 214 & 796 & 196 & 722 & {\bf 22} \\
\hline
& $0$ & 400 & 120 & 91 & 64 & 78 & {\bf 7} \\
100 & $-0.5$ & ${ {> 800} }$ & 176 & 313 & 116 & 182 & {\bf 16} \\
& $-1$ & ${ {> 800} }$ & 231 & 718 & 190 & 582 & {\bf 28} \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent{\bf Study on the number of outer iterations:}
We study $T^P$, the number of outer iterations needed to reach RFVD$\le \tau$. Table~\ref{tab:numiter} gives $T^P$ values for various methods in various settings. {\it DBCD-Southwell}~~clearly outperforms other methods in terms of having much smaller values for $T^P$. {\it PCDN-Southwell}~~is the second best method, followed by {\textsc{ADMM}}. The solid reduction of $T^P$ by {\it DBCD-Southwell}~~validates the design that was motivated in Section~\ref{sec:motiv}. The increased computation associated with {\it DBCD-Southwell}~~is immaterial; because communication cost overshadows computation cost in each iteration for all methods, {\it DBCD-Southwell}~~is also the best in terms of the overall computing time. The next set of results gives the details. \vspace*{0.05in}
\noindent{\bf Computation and Communication Time:}
As emphasized earlier, communication plays an important role in the distributed setting. To study this effect, we measured the computation and communication time separately at each node. Figure~\ref{fig:comptime} shows the computation time per node on the \textit{KDD} dataset. In both cases, {\textsc{ADMM}}~incurs significant computation time compared to other methods. This is because it optimizes over all variables in each node. {\it DBCD-Southwell}~~and {\it DBCD-Random}~~come next because our method involves both line search and $10$ inner iterations. {\textsc{PCD-R}}~and {\it PCDN-Southwell}~~take a little more time than {\it HYDRA}~~because of the line search. As seen in both {\textsc{DBCD}}~and {\it PCDN}~~cases, a marginal increase in time is incurred due to the variable selection cost with the S-scheme compared to the R-scheme.
\begin{table}[h]
\caption{Computation and communication costs per iteration (in secs.) for KDD, $P=25$.}
\label{tab:timings}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c}
\hline
Method & Comp. & Comm. & Comp. & Comm.\\
\hline
& \multicolumn{2}{|c|}{WSS: $r=0.1$} & \multicolumn{2}{|c|}{WSS: $r=0.01$}\\
\hline
{\it HYDRA}~ & 0.022 & 5.192 & 0.131 & 4.888\\
\hline
{\textsc{PCD-R}} & 0.138 & 5.752 & 0.432 & 5.817\\
\hline
{\it PCDN-Southwell}~ & 1.564 & 7.065 & 1.836 & 7.032\\
\hline
{\it DBCD-Random}~ & 0.991 & 6.322 & 1.978 & 6.407\\
\hline
{\it DBCD-Southwell}~ & 5.054 & 6.563 & 5.557 & 8.867\\
\hline
\end{tabular}
\end{center}
\end{table}
We measured the computation and communication time taken per iteration by each method for different $P$ and $WSS$ settings. From Table~\ref{tab:timings} (which gives representative results for one situation, \textsc{KDD} and $P=25$), we see that the communication time dominates the cost in {\it HYDRA}~~and {\textsc{PCD-R}}.
{\it DBCD-Random}~~takes more computation time than {\textsc{PCD-R}}~and {\it HYDRA}~~since we run through $10$ cycles of inner optimization. Note that the methods with S-scheme take more time; however, the increase is not significant compared to the communication cost. {\it DBCD-Southwell}~~takes the maximum computation time and is quite comparable to the communication time. Recall our earlier observation of {\it DBCD-Southwell}~~giving order of magnitude speed-up in the overall time compared to methods such as {\it HYDRA}~~and {\textsc{PCD-R}}~(see Figures 3-8). Though the computation times taken by {\it HYDRA}~, {\textsc{PCD-R}}~and {\it PCDN-Southwell}~~are lesser, they need significantly more number of iterations to reach some specified objective function optimality criterion. As a result, these methods become quite inefficient due to extremely large communication cost compared to \textsc{DBCD}. All these observations point to the fact our {\textsc{DBCD}}~method nicely trades-off the computation versus communication cost, and gives an excellent order of magnitude improvement in overall time. With the additional benefit provided by the S-scheme, {\it DBCD-Southwell}~~clearly turns out to be the method of choice for the distributed setting. \vspace*{0.05in}
\noindent{\bf Sparsity Pattern:} To study weight sparsity behaviors of various methods during optimization, we computed the percentage of non-zero weights ($\rho$) as a function of outer iterations. We set the initial weight vector to zero. Figure~\ref{fig:sparse} shows similar behaviors for all the random (variable) selection methods. After a few iterations of rise they fall exponentially and remain at the same level. For methods with the S-scheme, many variables remain non-zero for some initial period of time and then $\rho$ falls a lot more sharply. It is interesting to note that such an initial behavior seems necessary to make good progress in terms of both function value and AUPRC (Figure~\ref{fig:kddurlobj}(a)(b) and Figure~\ref{fig:kddurlauprc}(a)(b)) In all the cases, many variables stay at zero after initial iterations; therefore, shrinking ideas can be used to improve efficiency. \vspace*{0.05in}
\noindent{\bf Remark on Speed up:}
Let us consider the RFVD plots corresponding to {\it DBCD-Southwell}~~in Figures~\ref{fig:kddobj}~and~\ref{fig:urlobj}. It can be observed that the times associated with $P=25$ and $P=100$ for reaching a certain tolerance, say RFVD=-2, are close to each other. This means that using 100 nodes gives almost no speed up over 25 nodes, which may prompt the question: {\it Is a distributed solution really necessary?} There are two answers to this question. First, as we already mentioned, when the training data is huge\footnote{The \textsc{KDD} and \textsc{URL} datasets are really not huge in the {\it Big data} sense. In this paper we used them only because of lack of availability of much bigger public datasets.} and so {\it the data is generated and forced to reside in distributed nodes}, the right question to ask is not whether we get great speed up, but to ask which method is the fastest. Second, for a given dataset, if the time taken to reach a certain optimality tolerance is plotted as a function of $P$, it may have a minimum at a value different from $P=1$. In such a case, it is appropriate to choose a $P$ (as well as $WSS$) optimally to minimize training time. Many applications involve periodically repeated model training. For example, in Advertising, logistic regression based click probability models are retrained on a daily basis on incrementally varying datasets. In such scenarios it is worthwhile to spend time to tune parameters such as $P$ and $WSS$ in an early deployment phase to minimize time, and then use these parameter values for future runs.
It is also important to point out that the above discussion is relevant to distributed settings in which communication causes a bottleneck. If communication cost is not heavy, e.g., when the number of examples is not large and/or communication is inexpensive such as in multicore solution, then good speed ups are possible; see, for example, the results in~\citet{richtarik2013}.
\begin{figure}[t]
\subfigure[$WSS = 2021$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/comptimevsobjfunc-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[$WSS = 20216$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/comptimevsobjfunc-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\caption{\small{Per-node computation time on the \textsc{KDD} dataset ($\lambda = 1.2 \times 10^{-5}$ and $P = 100$).}}
\label{fig:comptime}
\end{figure}
\begin{figure}[t]
\subfigure[$WSS = 2021$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/itersvssparsity-data-kdd-workingset-2021-lambda-1pt238107e-05-nodes-100.eps}
}
\subfigure[$WSS = 20216$]{
\includegraphics[width=0.48\linewidth]{plots/finalplotseps/itersvssparsity-data-kdd-workingset-20216-lambda-1pt238107e-05-nodes-100.eps}
}
\caption{\small{\textsc{KDD} dataset: Percentage of non-zero weights. $\lambda = 1.2 \times 10^{-5}$ and $P = 100$. }}
\label{fig:sparse}
\end{figure}
\section{Introduction}
\label{sec:intro}
The design of sparse linear classifiers using $l_1$ regularization is an important problem that has received great attention in recent years. This is due to its value in scenarios where the number of features is large and the classifier representation needs to be kept compact. Big data is becoming common nowadays. For example, in online advertising one comes across datasets with about a billion examples and a billion features. A substantial fraction of the features is usually irrelevant; and, $l_1$ regularization offers a systematic way to choose the small fraction of relevant features and form the classifier using them. In the future, one can foresee even bigger sized datasets to arise in this and other applications. For such big data, distributed storage of data over a cluster of commodity machines becomes necessary. Thus, fast training of $l_1$ regularized classifiers over distributed data is an important problem.
A number of algorithms have been recently proposed for parallel and distributed training of $l_1$ regularized classifiers; see Section~\ref{sec:rw} for a review. Most of these algorithms are based on coordinate-descent and they assume the data to be feature-partitioned. They are designed for multicore and MPI platforms in which data communication costs are negligible. These platforms are usually equipped with only a small number of computing nodes. Distributed systems, e.g., Hadoop running on a cluster of commodity machines, are better for employing a large number of nodes and hence, for inexpensive handling of big data. However, in such systems, communication costs are high; current methods for $l_1$ regularization are not optimally designed for such systems. {\em In this paper we develop a distributed block coordinate descent (DBCD) method that is efficient on distributed platforms in which communication costs are high.}
The paper is organized as follows. Most methods (including the current ones and the one we propose) fit into a generic algorithm format that we describe in Section~\ref{sec:generic}. This gives a clear view of existing methods and allows us to motivate the new method. In Section~\ref{sec:rw} we discuss the key related work in some detail. The analysis of computation and communication costs in Section~\ref{sec:motiv} motivates our DBCD method. In Section~\ref{sec:dbcd} we describe the DBCD method in detail and prove its convergence. Experiments comparing our method with several existing methods on a few large scale datasets are given in Section~\ref{sec:expts}. These experiments strongly demonstrate the efficiency of one version of our method that chooses update variables greedily. This best version of the DBCD method is described in Section~\ref{sec:recom}. Section~\ref{sec:conc} contains some concluding comments.
\section{A generic algorithm}
\label{sec:generic}
The generic algorithm format allows us to explain the roles of key elements of various methods and point out how new choices for the steps can lead to a better design. Before describing it, we first formulate the $l_1$ regularization problem.
{\bf Problem formulation.}
Let $w$ be the weight vector with $m$ variables, $w_j$, $j=1,\ldots,m$, and $x_i\in R^m$ denote the $i$-th example.
Note that we have denoted vector components by subscripts, e.g., $w_j$ is the $j$-th component of $w$; we have also used subscripts for indexing examples, e.g., $x_i$ is the $i$-th example, which itself is a vector. But this will not cause confusion anywhere.
A linear classifier produces the output vector $y_i=w^Tx_i$. The loss is a nonlinear convex function applied on the outputs. For binary class label $c_i\in\{1,-1\}$, the loss is given by $\ell(y_i;c_i)$. Let us simply view $\ell(y_i;c_i)$ as a function of $y_i$ with $c_i$ acting as a parameter. We will assume that $\ell$ is non-negative and convex, $\ell\in{\cal{C}}^1$, the class of continuously differentiable functions, and that $\ell^\prime$ is Lipschitz continuous\footnote{{\small A function $h$ is Lipschitz continuous if there exists a (Lipschitz) constant $L\ge 0$ such that $\|h(a)-h(b)\|\le L \|a-b\| \;\; \forall \; a,b$.}}. Loss functions such as least squares loss, logistic loss, SVM squared hinge loss and Huber loss satisfy these assumptions. The total loss function, $f:R^m\to R$ is $f(w)=\frac{1}{n}\sum_i \ell(y_i;c_i)$.
Let $u$ be the $l_1$ regularizer given by $u(w) = \lambda \sum_j |w_j|$, where $\lambda>0$ is the regularization constant. Our aim is to solve the problem
\begin{equation}
\min_{w\in R^m} F(w) = f(w) + u(w).
\label{minF}
\end{equation}
Let $g=\nabla f$. The optimality conditions for (\ref{minF}) are:
\begin{equation}
\forall j: \;\; g_j+\lambda\; \mbox{sign} (w_j) = 0 \; \mbox{if}\;\; |w_j|>0; \;\; |g_j|\le\lambda \; \mbox{if}\;\; w_j=0.
\label{viol}
\end{equation}
Let there be $n$ training examples and let $X$ denote the $n\times m$ data matrix, whose $i$-th row is $x_i^T$. For problems with a large number of features, it is natural to randomly partition the columns of $X$ and place the parts in $P$ computing nodes. Let $\{B_p\}_{p=1}^P$ denote this partition of ${\cal{M}} = \{1,\ldots,m\}$, i.e., $B_p \subset {\cal{M}} \;\forall p$ and $\cup_p B_p = \cal{M}$.
{\em We will assume that this feature partitioning is given and that all algorithms operate within that constraint.} The variables associated with a particular partition get placed in one node. Given a subset of variables $S$, let $X_S$ be the submatrix of $X$ containing the columns corresponding to $S$. For a vector $z\in R^m$, $z_S$ will denote the vector containing the components of $z$ corresponding to $S$.
{\bf Generic algorithm.}
Algorithm 1 gives the generic algorithm.
Items such as $B_p$, $S_p^t$, $w_{B_p}$, $d_{B_p}^t$, $X_{B_p}$ stay local in node $p$ and do not need to be communicated. Step (d) can be carried out using an {\it AllReduce} operation~\citep{alekh2013} over the nodes and then $y$ becomes available in all the nodes. The gradient sub-vector $g_{B_p}^t$ (which is needed for solving~(\ref{Fapprox})) can then be computed locally as $g_{B_p}^t = X_{B_p}^T b$ where $b\in R^n$ is a vector with $\{\ell^\prime(y_i)\}$ as its components.
\begin{algorithm2e}
\caption{A generic distributed algorithm\label{GA}}
Choose $w^0$ and compute $y^0=Xw^0$\;
\For{$t=0,1 \ldots$}{
\For{$p=1,\ldots, P$}{
(a) Select a subset of variables, $S_p^t\subset B_p$\;
(b) Form $f_p^t(w_{B_p})$, an approximation of $f$ and solve (exactly or approximately):
\begin{equation}
\min f_p^t(w_{B_p}) + u(w_{B_p}) \;\;\; \mbox{s.t.} \;\;\; w_j = w^t_j \; \forall \; j\not\in B_p\setminus{S_p^t}
\label{Fapprox}
\end{equation}
to get $\bar{w}_{B_p}^t$ and set direction: $d_{B_p}^t=\bar{w}_{B_p}^t-w_{B_p}^t$\;
(c) Choose $\alpha^t$ and update: $w_{B_p}^{t+1} = w_{B_p}^t + \alpha^td_{B_p}^t$\;
}
(d) Update $y^{t+1} = y^t + \alpha^t \sum_p \Xbpd_{B_p}^t$\;
(e) Terminate if optimality conditions hold\;
}
\end{algorithm2e}
{\bf Step (a) - variable sampling.} Some choices are:
\begin{itemize}
\item
{\bf (a.1)} random selection~\citep{bradley2011, richtarik2012};
\item
{\bf (a.2)} random cyclic: over a set of consecutive iterations $(t)$ all variables are touched once~\citep{bian2013};
\item
{\bf (a.3)} greedy: always choose a set of variables that, in some sense violate (\ref{viol}) the most at the current iterate~\citep{peng2013, facchinei2013}; and,
\item
{\bf (a.4)} greedy selection using the Gauss-Southwell rule~\citep{tseng2009, yun2011}.
\end{itemize}
{\bf Step (b) - function approximation.} Most methods choose a quadratic approximation that is decoupled at the individual variable level:
\begin{equation}
f_p^t(w_{B_p}^t) = \sum_{j\in B_p} g_{j}(w^t) (w_j-w^t_j) + \frac{L_j}{2} (w_j-w^t_j)^2
\label{quad1}
\end{equation}
The main advantages of (\ref{quad1}) are its simplicity and closed-form minimization when used in~(\ref{Fapprox}). Choices for $L^j$ that have been tried are:
\begin{itemize}
\item
{\bf (b.1)} $L_j=$ a Lipschitz constant for $g_j$~\citep{bradley2011,peng2013};
\item
{\bf (b.2)} $L_j=$ a large enough bound on the Lipschitz constant for $g_j$ to suit the sampling in step (a)~\citep{richtarik2012};
\item
{\bf (b.3)} adaptive adjustment of $L_j$~\citep{facchinei2013}; and
\item
{\bf (b.4)} $L_j=H_{jj}^t$, the $j$-th diagonal term of the Hessian at $w^t$~\citep{bian2013}.
\end{itemize}
{\bf Step (c) - step size.} The choices are:
\begin{itemize}
\item
{\bf (c.1)} always fix $\alpha^t=1$~\citep{bradley2011,richtarik2012,peng2013};
\item
{\bf (c.2)} use stochastic approximation ideas to choose $\{\alpha^t\}$ so that $\sum_t(\alpha^t)^2<\infty$ and $\sum_t|\alpha^t|=\infty$~\citep{facchinei2013}; and
\item
{\bf (c.3)} choose $\alpha^t$ by line search that is directly tied to the optimization of $F$ in (\ref{minF})~\citep{bian2013}.
\end{itemize}
To understand the role of the various choices better, let us focus on the use of (\ref{quad1}) for $f_p^t$. Algorithm 1 may not converge to the optimal solution due to one of the following decisions: (i) choosing too many variables ($|S_p^t|$ large) for parallel updating in step (a); (ii) choosing small values for the proximal coefficient $L_j$ in step (b); and (iii) not controlling $\alpha^t$ to be sufficiently small in step (c). This is because each of the above has the potential to cause large step sizes leading to increases in $F$ value and, if this happens uncontrolled at all iterations then convergence to the minimum cannot occur. Different methods control against these by making suitable choices in the steps.
The choice made for step (c) gives a nice delineation of methods. With {\bf (c.1)}, one has to do a suitable mix of large enough $L_j$ and small enough $|S_p^t|$. Choice {\bf (c.2)} is better since the proper control of $\{\alpha^t\}\rightarrow 0$ takes care of convergence; however, for good practical performance, $L_j$ and $\alpha^t$ need to be carefully adapted, which is usually messy. Choice {\bf (c.3)} is good in many ways: it leads to monotone decrease in $F$; it is good theoretically and practically; and, it allows both, small $L_j$ as well as large $|S_p^t|$ without hindering convergence. Except for~\citet{bian2013}, \citet{tseng2009} and \citet{yun2011}\footnote{Among these three works, \citet{tseng2009} and \citet{yun2011} mainly focus on general theory and little on distributed implementation.}, {\bf (c.3)} has been unused in other methods because it is considered as `not-in-line' with a proper parallel approach as it requires a separate $\alpha^t$ determination step requiring distributed computations and also needing $F$ computations for several $\alpha^t$ values within one $t$. With line search, the actual implementation of Algorithm 1 merges steps {\bf (c)} and {\bf (d)} and so it deviates slightly from the flow of Algorithm 1. Specifically, we compute $\delta y = \sum_p \Xbpd_{B_p}^t$ before line search using AllReduce. Then each node can compute $f$ at any $\alpha$ locally using $y+\alpha\, \delta y$. Only a scalar corresponding to the $l_1$ regularization term needs to be communicated for each $\alpha$. This means that the communication cost associated with line search is minimal.\footnote{Later, in Section~\ref{sec:motiv} when we write costs, we write it to be consistent with Algorithm 1. The total cost of all the steps is the same for the implementation described here for line search. For genericity sake, we keep Algorithm 1 as it is even for the line search case. The actual details of the implementation for the line search case will become clear when we layout the final algorithm in Section~\ref{sec:recom}.} But truly, the slightly increased computation and communication costs is amply made up by a reduction in the number of iterations to reach sufficient optimality. So we go with the choice {\bf (c.3)} in our method.
The choice of (\ref{quad1}) for $f_p^t$ in step (b) is pretty much unanimously used in all previous works. While this is fine for communication friendly platforms such as multicore and MPI, it is not the right choice when communication costs are high. Such a setting permits more per-node computation time, and there is much to be gained by using a more complex $f_p^t$. We propose the use of a function $f_p^t$ that couples the variables in $S_p^t$. We also advocate an approximate solution of (\ref{Fapprox}) (e.g., a few rounds of coordinate descent within each node) in order to control the computation time.
Crucial gains are also possible via resorting to the greedy choices, {\bf (a.3)} and {\bf (a.4)} for choosing $S_p^t$. On the other hand, with methods based on {\bf (c.1)}, one has to be careful in using {\bf (a.3)}: apart from difficulties in establishing convergence, practical performance can also be bad, as we show in Section~\ref{sec:expts}.
{\bf Contributions.}
Following are our main contributions.
\begin{enumerate}
\item We provide a cost analysis that brings out the computation and communication costs of Algorithm 1 clearly for different methods. In the process we motivate the need for new efficient methods suited to communication heavy settings.
\item We make careful choices for the three steps of Algorithm 1, leading to the development of a distributed block coordinate descent (DBCD) method that is very efficient on distributed platforms with high communication cost.
\item We establish convergence theory for our method using the results of~\citet{tseng2009} and \citet{yun2011}. It is worth noting the following: (a) though \citet{tseng2009} and \citet{yun2011} cover algorithms using quadratic approximations for the total loss, we use a simple trick to apply them to general nonlinear approximations, thus bringing more power to their results; and (b) even these two works use only (\ref{quad1}) in their implementations whereas we work with more powerful approximations that couple features.
\item We give an experimental evaluation that shows the strong performance of DBCD against key current methods in scenarios where communication cost is significant. Based on the experiments we make a final recommendation for the best method to employ for such scenarios.
\end{enumerate}
\input{rw1.tex}
\section{DBCD method: Motivation}
\label{sec:motiv}
Our goal is to develop an efficient distributed learning method that jointly optimizes the costs involved in the various steps of the algorithm. We observed that the methods discussed in Section~\ref{sec:rw} lack this careful optimization in one or more steps, resulting in inferior performance. In this section, we present a detailed cost analysis study and motivate our optimization strategy that forms the basis for our DBCD method.
{\bf Remark.} A non-expert reader could find this section hard to read in the first reading because of two reasons: (a) it requires a decent understanding of several methods covered in Section~\ref{sec:rw}; and (b) it requires knowledge of the details of our method. For this sake, let us give the main ideas of this section in a nutshell. (i) The cost of Algorithm 1 can be written as $T^P(C_{\mbox{comp}}^P + C_{\mbox{comm}}^P)$ where $P$ denotes the number of nodes, $T^P$ is the number of outer iterations\footnote{For practical purposes, one can view $T^P$ as the number of outer iterations needed to reach a specified closeness to the optimal objective function value. We will say this more precisely in Section~\ref{sec:expts}.}, and, $C_{\mbox{comp}}^P$ and $C_{\mbox{comm}}^P$ respectively denote the computation and communication costs per-iteration. (ii) In communication heavy situations, existing algorithms have $C_{\mbox{comp}}^P \ll C_{\mbox{comm}}^P$. (iii) Our method aims to improve overall efficiency by making each iteration more complex ($C_{\mbox{comp}}^P$ is increased) and, in the process, making $T^P$ much smaller. {\it A serious reader can return to study the details behind these ideas after reading Section~\ref{sec:dbcd}.}
Following Section~\ref{sec:rw}, we
select the following methods for our study:
(1) {\it HYDRA}~~\citep{richtarik2013a}, (2) {\it GROCK}~~(Greedy coordinate-block)~\citep{peng2013}, (3) {\it FPA}~~(Flexible Parallel Algorithm)~\citep{facchinei2013}, and (4) {\it PCDN}~~(Parallel Coordinate Descent Newton method)~\citep{bian2013}.
We will use the above mentioned abbreviations for the methods in the rest of the paper.
Let $nz$ and $|S| = \sum_p |S^t_p|$ denote the number of non-zero entries in the data matrix $X$ and the number of variables updated in each iteration respectively. To keep the analysis simple, we make the homogeneity assumption that the number of non-zero data elements in each node is $nz/P$. Let $\beta (\gg 1)$ be the relative computation to communication speed in the given distributed system; more precisely, it is the ratio of the times associated with communicating a floating point number and performing one floating point operation. Recall that $n$, $m$ and $P$ denote the number of examples, features and nodes respectively. Table~\ref{tab:stepcost} gives cost expressions for different steps of the algorithm in one outer iteration. Here $c_1$, $c_2$, $c_3$, $c_4$ and $c_5$ are method dependent parameters.
Table~\ref{tab:methodcost} gives the cost constants for various methods.\footnote{As in Table~\ref{tab:methodscomp}, for ease of comparison, we also list the cost constants for our method (three variations), in Table~\ref{tab:methodcost}. The details for them will become clear in Section~\ref{sec:dbcd}.}
We briefly discuss different costs below.
\begin{table}[h]
\caption{Cost of various steps of Algorithm~\ref{GA}. $C_{\mbox{comp}}^P$ and $C_{\mbox{comm}}^P$ are respectively, the sums of costs in the computation and communication rows.}
\label{tab:stepcost}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Cost & \multicolumn{4}{|c|}{Steps of Algorithm~\ref{GA}} \\
\hline
& Step a & Step b & Step c & Step d \\
& {\small Variable sampling} & {\small Inner optimization} & {\small Choosing step size} & {\small Updating output} \\
\hline
Computation & $c_1\frac{nz}{P}$ & $c_2\frac{nz}{P}\frac{|S|}{m}$ & $c_3|S| + c_4n$ & $c_5\frac{nz}{P}\frac{|S|}{m}$ \\
\hline
Communication\protect\footnotemark & - & - & $c_4\beta logP$ & $\beta n logP$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\footnotetext{Note that the communication latency cost (time for the first byte to reach the destination) is ignored in the communication cost expressions because it is dominated by the throughput cost for large $n$. Moreover, our {\it{AllReduce}} is a non-pipelined version of the implementation in~\citet{alekh2013}.}
\begin{table}[h]
\caption{Cost parameter values and costs for different methods. $q$ lies in the range: $1 \leq q \leq \frac{m}{|S|}$. $R$ and $S$ refer to variable selection schemes for step (a); see Section~\ref{sec:dbcd}. {\it PCDN}~~uses the $R$ scheme and so it can also be referred to as {\textsc{PCD-R}}. Typically $\tau_{ls}$, the number of $\alpha$ values tried in line search, is very small; in our experiments we found that on average it is not more than $10$. Therefore all methods have pretty much the same communication cost per iteration.}
\vspace*{-0.1in}
\label{tab:methodcost}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Method & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & Computation & Communication \\
& & & & & & cost per iteration & cost per iteration \\
\hline
\multicolumn{8}{c}{{\bf Existing methods}} \\
\hline
{\it HYDRA}~ & 0 & 1 & 1 & 0 & 1 & $2\frac{nz}{P}\frac{|S|}{m} + |S|$ & $\beta n \log P$\\
\hline
{\it GROCK}~ & 1 & $q$ & 1 & 0 & $q$ & $\frac{nz}{P} + 2q\frac{nz}{P}\frac{|S|}{m} + |S|$ & $\beta n \log P$ \\
\hline
{\it FPA}~ & 1 & $q$ & 1 & 1 & $q$ & $\frac{nz}{P} + 2q\frac{nz}{P}\frac{|S|}{m} + |S| + n$ & $\beta (n+1) \log P$\\
\hline
{\it PCDN}~ & 0 & 1 & $\tau_{ls}$ & $\tau_{ls}$ & 1 & $2\frac{nz}{P}\frac{|S|}{m} + \tau_{ls}|S| + \tau_{ls}n$ & $\beta (n+\tau_{ls}) \log P$ \\
\hline
\multicolumn{8}{c}{{\bf Variations of our method}} \\
\hline
{\it PCDN-Southwell}~ & 1 & $q$ & $\tau_{ls}$ & $\tau_{ls}$ & $q$ & $\frac{nz}{P} + 2q\frac{nz}{P}\frac{|S|}{m} + \tau_{ls}|S| + \tau_{ls}n$ & $\beta (n+\tau_{ls}) \log P$ \\
\hline
{\it DBCD-Random}~ & 0 & $k$ & $\tau_{ls}$ & $\tau_{ls}$ & 1 & $(k+1)\frac{nz}{P}\frac{|S|}{m} + \tau_{ls}|S| + \tau_{ls}n$ & $\beta (n+\tau_{ls}) \log P$ \\
\hline
{\it DBCD-Southwell}~ & 1 & $kq$ & $\tau_{ls}$ & $\tau_{ls}$ & $q$ & $\frac{nz}{P} + q(k+1)\frac{nz}{P}\frac{|S|}{m} + \tau_{ls}|S| + \tau_{ls}n$ & $\beta (n+\tau_{ls}) \log P$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent{\bf{Step a:}} Methods like our {\it DBCD-Southwell}~\footnote{The DBCD and PCD methods have two variants, R and S corresponding to different ways of implementing step a; these will be discussed in Section~\ref{sec:dbcd}.}, {\it GROCK}~, {\it FPA}~~ and {\it PCDN-Southwell}~~ need to calculate the gradient and model update to determine which variables to update. Hence, they need to go through the whole data once ($c_1=1$). On the other hand {\it HYDRA}~,~{\it PCDN}~~ and {\it DBCD-Random}~~ select
variables randomly or in a cyclic order. As a result variable subset selection cost is negligible for them ($c_1=0$).\vspace{0.05in}\\
{\bf{Step b:}} All the methods except {\it DBCD-Southwell}~~ and {\it DBCD-Random}~~ use the decoupled quadratic approximation~(\ref{quad1}). For {\it DBCD-Random}~~ and {\it DBCD-Southwell}~,~ an additional factor of $k$ comes in $c_2$ since we do $k$ inner cycles of \textsc{CDN} in each iteration. {\it HYDRA}~, {\it PCDN}~~ and {\it DBCD-Random}~~ do a random or cyclic selection of variables. Hence, a factor of $\frac{|S|}{m}$ comes in the cost since only a subset $|S|$ of variables is updated in each iteration. However, methods that do selection of variables based on the magnitude of update or expected objective function decrease ({\it DBCD-Southwell}~, {\it GROCK}~, {\it FPA}~~ and {\it PCDN-Southwell}~) favour variables with low sparsity. As a result, $c_2$ for these methods has an additional factor $q$ where $1 \leq q \leq \frac{m}{|S|}$.\vspace{0.05in}\\
{\bf{Step c:}} For methods that do not use line-search, $c_3 = 1$ and $c_4 = 0$\footnote{{\small For {\it FPA}~, $c_4=1$ since objective function needs to be computed to automatically set the proximal term parameter.}}. The overall cost is $|S|$ to update the variables. For methods like {\it DBCD-Southwell}~, {\it DBCD-Random}~, {\it PCDN}~~ and {\it PCDN-Southwell}~~ that do line-search, $c_3 = c_4 = \tau_{ls}$ where $\tau_{ls}$ is the average number of steps ($\alpha$ values tried) in one line search. For each line search step, we need to recompute the loss function which involves going over $n$ examples once. Moreover, \textit{AllReduce} step needs to be performed to sum over the distributed $l_1$ regularizer term. Hence, an additional $\beta log P$ cost is incurred to communicate the local regularizer. As pointed out in~\citet{bian2013}, $\tau_{ls}$ can increase with $P$; but it is still negligible compared to $n$.\vspace{0.05in}\\
{\bf{Step d:}} This step involves computing and doing \textit{AllReduce }on updated local predictions to get the global prediction vector for the next iteration and is common for all the methods.
The analysis given above is only for $C_{\mbox{comp}}^P$ and $C_{\mbox{comm}}^P$, the computation and communication costs in one iteration. If $T^P$ is the number of iterations to reach a certain optimality tolerance, then the total cost of Algorithm 1 is: $C^P = T^P(C_{\mbox{comp}}^P + C_{\mbox{comm}}^P)$. For $P$ nodes, speed-up is given by $C^1/C^P$. To illustrate the ill-effects of communication cost, let us take the method of~\citet{richtarik2013}. For illustration, take the case of $|S|=P$, i.e., one variable is updated per node per iteration. For large $P$, $C^P\approx T^PC_{\mbox{comm}}^P = T^P\; \beta n \log P$; both $\beta$ and $n$ are large in the distributed setting. On the other hand, for $P=1$, $C_{\mbox{comm}}^P=0$ and $C^P=C_{\mbox{comp}}^P \approx \frac{nz}{m} $. Thus
${\mbox speed up} \;\; = \;\; \frac{T^1}{T^P} \frac{C^1}{C^P} \;\; = \;\; \frac{T^1}{T^P} \frac{\frac{nz}{m}}{\beta n \log P}$.
\citet{richtarik2013} show that $T^1/T^P$ increases nicely with $P$. But, the term $\beta n$ in the denominator of $C^1/C^P$ has a severe detrimental effect. Unless a special distributed system with efficient communication is used, speed up has to necessarily suffer. When the training data is huge and so the data is forced to reside in distributed nodes, {\it the right question to ask is not whether we get great speed up, but to ask which method is the fastest}. Given this, we ask how various choices in the steps of Algorithm 1 can be made to decrease $C^P$. Suppose we devise choices such that (a) $C_{\mbox{comp}}^P$ is increased while still remaining in the zone where $C_{\mbox{comp}}^P \ll C_{\mbox{comm}}^P$, and (b) in the process, $T^P$ is decreased greatly, then $C^P$ can be decreased. The basic idea of our method is to use a more complex $f_p^t$ than the simple quadratic in (\ref{quad1}), due to which, $T^P$ becomes much smaller. The use of line search, {\bf (c.3)} for step {\bf c} aids this further. We see in Table~\ref{tab:methodcost} that, {\it DBCD-Random}~~ and {\it DBCD-Southwell}~~ have the maximum computational cost. On the other hand, communication cost is more or less the same for all the methods (except for few scalars in the line search step) and dominates the cost. In Section~\ref{sec:expts}, we will see on various datasets how, by doing more computation, our methods reduce $T^P$ substantially over the other methods while incurring a small computation overhead (relative to communication) per iteration. These will become amply clear in Section~\ref{sec:expts}; see, for example, Table~\ref{tab:numiter} in that section.
\section{DBCD method}
\label{sec:dbcd}
The DBCD method that we propose fits into the general format of Algorithm 1. It is actually {\it a class of algorithms} that allows various possibilities for steps (a), (b) and (c). Below we lay out these possibilities and establish convergence theory for our method. We also show connection to other methods on aspects such as function approximation, variable selection, etc.
\subsection{Function approximation}
\label{subsec:func}
Let us begin with step (b). There are three key items involved: (i) what are some of the choices of approximate functions possible, used by our methods and others? (ii) what is the stopping criterion for the inner optimization (i.e., local problem), and, (iii) what is the method used to solve the inner optimization? We discuss all these details below. We stress the main point that, unlike previous methods, we allow $f_p^t$ to be non-quadratic and also to be a joint function of the variables in $w_{B_p}$. We first describe a general set of properties that $f_p^t$ must satisfy, and then discuss specific instantiations that satisfy these properties.
{\bf P1}. $f_p^t\in{\cal{C}}^1$; $g_p^t=\nabla f_p^t$ is Lipschitz continuous, with the Lipschitz constant uniformly bounded over all $t$; $f_p^t$ is strongly convex (uniformly in $t$), i.e., $\exists\; \mu>0$ such that $f_p^t-\frac{\mu}{2} \| w_{B_p} \|^2$ is convex; and, $f_p^t$ is gradient consistent with $f$ at $w_{B_p}^t$, i.e., $g_p^t(w_{B_p}^t) = g_{B_p}(w^t)$.
This assumption is not restrictive. Gradient consistency is essential because it is the property that connects $f_p^t$ to $f$ and ensures that a solution of (\ref{Fapprox}) will make $d_{B_p}^t$ a descent direction for $F$ at $w_{B_p}^t$, thus paving the way for a decrease in $F$ at step (c).
Strong convexity is a technical requirement that is needed for establishing sufficient decrease in $F$ in each step of Algorithm 1. Our experiments indicate that it is sufficient to set $\mu$ to be a very small positive value. Lipschitz continuity is another technical condition that is needed for ensuring boundedness of various quantities; also, it is easily satisfied by most loss functions. Let us now discuss some good ways of choosing $f_p^t$. {\em For all these instantiations, a proximal term is added to get the strong convexity required by} {\bf P1}.
{\bf Proximal-Jacobi.} We can follow the classical Jacobi method in choosing $f_p^t$ to be the restriction of $f$ to $w_{S_p^t}^t$, with the remaining variables fixed at their values in $w^t$. Let $\bar{B}_p$ denote the complement of $B_p$, i.e., the set of variables associated with nodes other than $p$. Thus we set
\begin{equation}
f_p^t(w_{B_p}) = f(w_{B_p},w_{\bpbar}^t) + \frac{\mu}{2} \| w_{B_p} - w_{B_p}^t \|^2
\label{ours}
\end{equation}
where $\mu>0$ is the proximal constant. It is worth pointing out that, since each node $p$ keeps a copy of the full classifier output vector $y$ aggregated over all the nodes, the computation of $f_p^t$ and $g_p^t$ due to changes in $w_{B_p}$ can be locally computed in node $p$. Thus the solution of (\ref{Fapprox}) is local to node $p$ and so step (b) of Algorithm 1 can be executed in parallel for all $p$.
{\bf Block GLMNET.} GLMNET~\citep{yuan2012, friedman2010} is a sequential coordinate descent method that has been demonstrated to be very promising for the sequential solution of $l_1$ regularized problems with logistic loss. At each iteration, GLMNET minimizes the second order Taylor series of $f$ at $w^t$, followed by line search along the direction generated by this minimizer. We can make a distributed version by choosing $f_p^t$ to be the second order Taylor series approximation of $f(w_{B_p},w_{\bpbar}^t)$ restricted to $w_{B_p}$ while keeping $w_{\bar{B}_p}$ fixed at $w_{\bpbar}^t$.
{\bf Block \textsc{L-BFGS}.} One can keep a limited history of $w_{B_p}^t$ and $g_{B_p}^t$ and use an $L-BFGS$ approach to build a second order approximation of $f$ in each iteration to form $f_p^t$.
{\bf Decoupled quadratic.} Like in existing methods we can also form a quadratic approximation of $f$ that decouples at the variable level. If the second order term is based on the diagonal elements of the Hessian at $w^t$, then the PCDN algorithm given in~\citet{bian2013} can be viewed as a special case of our DBCD method. PCDN~\citep{bian2013} is based on Gauss-Seidel variable selection. But it can also be used in combination with the distributed greedy scheme that we propose in Subsection 5.2 below.
{\bf Approximate stopping.}
In step (b) of Algorithm 1 we mentioned the possibility of approximately solving~(\ref{Fapprox}). This is irrelevant for previous methods which solve individual variable level quadratic optimization in closed form, but very relevant to our method. Here we propose an approximate relative stopping criterion and later, in Subsection 5.4, also give convergence theory to support it.
Let $\partial u_j$ be the set of sub-gradients of the regularizer term $u_j = \lambda |w_j|$, i.e.,
\begin{equation}
\partial u_j = [-\lambda,\lambda] \;\; \mbox{if} \; w_j=0; \;\; \lambda \; \mbox{sign} (w_j) \;\; \mbox{if} \; w_j\not=0.
\label{subg}
\end{equation}
A point $\bar{w}_{B_p}^t$ is optimal for (\ref{Fapprox}) if, at that point,
\begin{equation}
(g_p^t)_j + \xi_j = 0, \;\; \mbox{for some} \;\; \xi_j\in\partial u_j \;\; \forall \; j\in S_p^t.
\label{opt}
\end{equation}
An approximate stopping condition can be derived by choosing a tolerance $\epsilon>0$ and requiring that, for each $j\in S_p^t$ there exists $\xi_j\in\partial u_j$ such that
\begin{equation}
\delta^j = (g_p^t)_j + \xi_j, \;\; |\delta_j| \le \epsilon |d^t_j| \;\; \forall \; j\in S_p^t
\label{appopt}
\end{equation}
{\bf Method used for solving~(\ref{Fapprox}).}
Now (\ref{Fapprox}) is an $l_1$ regularized problem restricted to $w_{S_p^t}$. It has to be solved within node $p$ using a suitable sequential method. Going by the state of the art for sequential solution of such problems~\citep{yuan2010} we use the coordinate-descent method described in~\citet{yuan2010} for solving~(\ref{Fapprox}).
\subsection{Variable selection}
\label{subsec:varsel}
Let us now turn to step (a) of Algorithm 1. We propose two schemes for variable selection, i.e., choosing $S_p^t\subset B_p$.
{\bf Gauss-Seidel scheme.}
In this scheme, we form cycles - each cycle consists of a set of consecutive iterations - while making sure that every variable is touched once in each cycle. We implement a cycle as follows. Let $\tau$ denote the iteration where a cycle starts. Choose a positive integer $T$ ($T$ may change with each cycle). For each $p$, randomly partition $B_p$ into $T$ equal parts: $\{S_p^t\}_{t=\tau}^{\tau+T-1}$. Use these variable selections to do $T$ iterations. {\em Henceforth, we refer to this scheme as the $R$-scheme.}
{\bf Distributed greedy scheme.}
This is a greedy scheme which is purely distributed and so more specific than the Gauss-Southwell schemes in~\citet{tseng2009}.\footnote{{\small Yet, our distributed greedy scheme can be shown to imply the Gauss-Southwell-$q$ rule for a certain parameter setting. See the appendix for details.}} In each iteration, our scheme chooses variables based on how badly~(\ref{viol}) is violated for various $j$. For one $j$, an expression of this violation is as follows. Let $g^t$ and $H^t$ denote, respectively, the gradient and Hessian at $w^t$. Form the following one variable quadratic approximation:
\begin{eqnarray}
q_j(w_j) = g^t_j (w_j - w^t_j) + \frac{1}{2} (H_{jj}^t+\nu) (w_j - w^t_j)^2 + \nonumber \\
\lambda |w_j| - \lambda|w^t_j|
\label{quad}
\end{eqnarray}
where $\nu$ is a small positive constant. Let ${\bar q}_j$ denote the optimal objective function value obtained by minimizing $q_j(w_j)$ over all $w_j$. Since $q_j(w^t_j)=0$, clearly ${\bar q}_j\le 0$. The more negative ${\bar q}_j$ is, the better it is to choose $j$.
Our distributed greedy scheme first chooses a working set size WSS and then, in each node $p$, it chooses the top WSS variables from $B_p$ according to smallness of ${\bar q}_j$, to form $S_p^t$. {\em Hereafter, we refer to this scheme as the $S$-scheme.}
It is worth pointing out that, our distributed greedy scheme requires more computation than the Gauss-Seidel scheme. However, since the increased computation is local, non-heavy and communication is the real bottleneck, it is not a worrisome factor.
\subsection{Line search}
\label{subsec:ls}
Line search (step (c) of Algorithm 1) forms an important component for making good decrease in $F$ at each iteration. For non-differentiable optimization, there are several ways of doing line search. For our context, \citet{tseng2009} and~\citet{patriksson1998} give two good ways of doing line search based on Armijo backtracking rule. In this paper we use ideas from the former. Let $\beta$ and $\sigma$ be real parameters in the interval $(0,1)$. (We use the standard choices, $\beta=0.5$ and $\sigma=0.01$.) We choose $\alpha^t$ to be the largest element of $\{\beta^k\}_{k=0,1,\ldots}$ satisfying
\begin{eqnarray}
F(w^t + \alpha^t d^t) \le F(w^t) + \alpha^t \sigma \Delta^t, \label{ls1} \\
\Delta^t \stackrel{\text{def}}{=} (g^t)^T d^t + \lambda u(w^t+d^t) - \lambda u(w^t). \label{ls2}
\end{eqnarray}
\subsection{Convergence}
\label{subsec:conv}
We now establish convergence for the class of algorithmic choices discussed in Subections 5.1-5.3. To do this, we make use of the results of~\citet{tseng2009}. An interesting aspect of this use is that, while the results of~\citet{tseng2009} are stated only for $f_p^t$ being quadratic, we employ a simple trick that lets us apply the results to our algorithm which involves non-quadratic approximations.
Apart from the conditions in {\bf P1} (see Subection 5.1) we need one other technical assumption.
{\bf P2.} For any given $t$, $w_{B_p}$ and ${\hat w}_{B_p}$, $\exists$ a positive definite matrix $\hat{H}\ge\mu I$ (note: $\hat{H}$ can depend on $t$, $w_{B_p}$ and ${\hat w}_{B_p}$) such that
\begin{equation}
g_p^t(w_{B_p}) - g_p^t({\hat w}_{B_p}) = \hat{H} (w_{B_p} - {\hat w}_{B_p})
\label{mvt}
\end{equation}
Except {\it Proximal-Jacobi}, the other instantiations of $f_p^t$ mentioned in Subection 5.1 are quadratic functions; for these, $g_p^t$ is a linear function and so (\ref{mvt}) holds trivially. Let us turn to {\it Proximal-Jacobi}. If $f_p^t\in{\cal{C}}^2$, the class of twice continuously differentiable functions, then {\bf P2} follows directly from mean value theorem; note that, since $f_p^t-\frac{\mu}{2} \|w\|^2$ is convex, $H_p\ge \mu I$ at any point, where $H_p$ is the Hessian of $f_p^t$. Thus {\bf P2} easily holds for least squares loss and logistic loss.
Now consider the SVM squared hinge loss, $\ell(y_i;c_i) = 0.5(\max \{0, 1-y_ic_i\})^2$, which is not in ${\cal{C}}^2$. {\bf P2} holds for it because $g=\sum_i \ell^\prime (y_i;c_i) x_i$ and, for any two real numbers $z_1, z_2$, $\ell^\prime(z_1;c_i)-\ell^\prime(z_2;c_i) = \kappa (z_1,z_2,c_i) (z_1-z_2)$ where $0 \le \kappa(z_1,z_2,c_i) \le 1$.
The main convergence theorem can now be stated. Its proof is given in the appendix.
{\bf Theorem 1.} Suppose, in Algorithm 1: (i) step (a) is done via the Gauss-Seidel or distributed greedy schemes of Subection 5.2; (ii) $f_p^t$ in step (b) satisfies {\bf P1} and {\bf P2}; (iii) (\ref{appopt}) is used to terminate (\ref{Fapprox}) with $\epsilon=\mu/2$ (where $\mu$ is as in {\bf P1}); and (iv) in step (c), $\alpha^t$ is chosen via Armijo backtracking of Subection 5.3. Then Algorithm 1 is well defined and produces a sequence, $\{w^t\}$ such that any accumulation point of $\{w^t\}$ is a solution of (\ref{minF}). If, in addition, the total loss, $f$ is strongly convex, then $\{F(w^t)\}$ converges Q-linearly and $\{w^t\}$ converges at least R-linearly.\footnote{{\small See chapter 9 of~\citet{ortega1970} for definitions of Q-linear and R-linear convergence.}}
\input{Experiments.tex}
\section{Recommended DBCD algorithm}
\label{sec:recom}
In Section~\ref{sec:dbcd} we explored various options for the steps of Algorithm 1 looking beyond those considered by existing methods and proposing new ones, and empirically analyzing the various resulting methods in Section~\ref{sec:expts}. The experiments clearly show that {\it DBCD-Southwell}~~is the best method. We collect full implementation details of this method in~Algorithm~\ref{GAfinal}.
\begin{algorithm2e}
\caption{Recommended DBCD algorithm\label{GAfinal}}
Parameters:
Proximal constant $\mu>0$ (Default: $\mu=10^{-12}$)\;
WSS = \# variables to choose for updating per node (Default: WSS=$r\, m/P$, $r=0.1$)\;
$k=$ \# CD iterations to use for solving~(\ref{Fapprox}) (Default: $k=10$)\;
Line search constants: $\beta,\sigma\in (0,1)$ (Default: $\beta=0.5$, $\sigma=0.01$)\;
Choose $w^0$ and compute $y^0=Xw^0$\;
\For{$t=0,1 \ldots$}{
\For{$p=1,\ldots, P$ (in parallel)}{
(a) For each $j\in B_p$, solve~(\ref{quad}) to get $q_j$. Sort $\{ q_j: j\in B_p\}$ and choose WSS indices with least $q_j$ values to form $S_p^t$\;
(b) Form $f_p^t(w_{B_p})$ using~(\ref{ours}) and solve~(\ref{Fapprox}) using $k$ CD iterations to get $\bar{w}_{B_p}^t$ and set direction: $d_{B_p}^t=\bar{w}_{B_p}^t-w_{B_p}^t$\;
(c) Compute $\delta y^t = \sum_p \Xbpd_{B_p}^t$ using AllReduce\;
(d) $\alpha=1$\;
\While{(\ref{ls1}-\ref{ls2}) are not satisfied}{
$\alpha \leftarrow \alpha\beta$\;
Check (\ref{ls1})-(\ref{ls2}) using $y + \alpha\,\delta y$ and aggregating the $l_1$ regularization value via AllReduce\;
}
(e) Set $\alpha^t=\alpha$, $w_{B_p}^{t+1} = w_{B_p}^t + \alpha^td_{B_p}^t$ and $y^{t+1} = y^t + \alpha^t \, \delta y^t$\;
}
(f) Terminate if the optimality conditions~(\ref{viol}) hold to the desired approximate level\;
}
\end{algorithm2e}
\section{Conclusion}
\label{sec:conc}
In this paper we have proposed a class of efficient block coordinate methods for the distributed training of $l_1$ regularized linear classifiers. In particular, the proximal-Jacobi approximation together with a distributed greedy scheme for variable selection came out as a strong performer. There are several useful directions for the future. It would be useful to explore other approximations such as block GLMNET and block \textsc{L-BFGS} suggested in Subsection 5.1. Like~\citet{richtarik2013}, developing a complexity theory for our method that sheds insight on the effect of various parameters (e.g., $P$) on the number of iterations to reach a specified optimality tolerance is worthwhile. It is possible to extend our method to non-convex problems, e.g., deep net training, which has great value.
\section{Introduction}
\section{Related Work}
There is a large volume of literature addressing the problem of optimizing $L1$ regularized objective functions. The literature can be organized as papers focusing on solving the underlying optimization problem in single and parallel/distributed computing environments under various assumptions on the loss functions (e.g., convex/non-convex, smooth/non-smooth, decomposable, etc) and applications (e.g., compressed sensing and high dimensional classification/regression). While we briefly cover various related work in this vast area, our focus is to relate our work in the parallel/distributed setting with the assumptions stated in Section 1.
\subsection{$L1$ Optimization}
Many methods have been proposed to address the problem of optimizing $L1$ regularized objective functions~\cite{lee2006,andrew2007,yun2009,beck2009}. One important and popular category of methods is decomposition methods where a subset of variables are optimized at a time, and the solution is obtained by iterating over all the variables multiple times until some convergence criterion is met. All coordinate descent based methods fall under this category, and solving the sub-problem component is at the core of many parallel and distributed optimization methods. There are several key aspects that need to be considered, and choices to be made while solving the problem: (1) selection of variables~\cite{tseng2009,yun2009}, (2) approximation of the sub-problem objective function~\cite{beck2009,yun2009,yuan2012,fount2012,bryd2012}, (3) conditions to be met to ensure (sufficient) decrease in the objective function value, in deterministic or expectation sense so that convergence guarantees can be given~\cite{nesterov2010,beck2009,yun2009,tseng2009, wang2010}, and, (4) acceleration mechanisms to improve the rate of convergence~\cite{beck2009,nesterov2010}. Yuan et al~\cite{yuan2010} conducts a comparative study of various methods specifically focusing on large scale linear classification problems. Our interest in this paper is coordinate descent methods which naturally lend to and benefit from parallel/distributed learning algorithms. We have seen how a generalized coordinate descent algorithm (Algorithm 1) can be written in a parallel/distributed computing setting. So, we discuss how our work derives its parallel and where it differs from other methods next, in particular, along the aspects (1)-(3) described above.
\subsection{Parallel and Distributed Algorithms}
\noindent{\bf Parallel Coordinate Descent Newton (PCDN)~\cite{bian2013}} Bian et al~\cite{bian2013} proposed a parallel coordinate descent method where randomized block partitioning of variables is done in each (outer) iteration. Each block of variables is taken in inner iteration, and variables in each block is solved in parallel using coordinate descent Newton method followed by a line search. This line search is critical to show global convergence, and they achieve linear rate of convergence in expectation sense with expectation taken over the random partitioning of the variables. PCDN can be seen as a special case of our method; this follows from the generality of our method with block coordinate optimization and various approximation function possibilities. Furthermore, we show global convergence and local linear rate of convergence in deterministic sense.
\begin{itemize}
\item Efficient parallel coordinate descent algorithm for convex optimization problems with separable constraints: application to distributed MPC (Necoara and Clipici, 2013).
\end{itemize}
\noindent{Distributed Coordinate Descent Method (DCDM)~\cite{richtarik2013,richtarik2012} \\
\noindent{Generic Parallel Coordinate Descent Algorithms (GenCD)~\cite{Scherrer2012}} \\
\noindent{Decomposition Methods~\cite{Patriksson1997}} \\
\noindent{Parallel and Distributed Sparse Optimization (PDSO - GRock)~\cite{peng2013}}
\noindent{ADMM~\cite{boyd2010,boyd2013}}
\begin{itemize}
\item Accelerated ADMM
\item On the $O(\frac{1}{k})$ convergence and parallelization of the ADMM (Deng et al, 2013).
\end{itemize}
\noindent{Distributed Iterative Hard/Soft Thresholding Algorithms (DIHTA/DISTA)~\cite{fosson2013}}
\noindent{Flexible Parallel Algorithms (FPA)~\cite{fachhinei2013} } \\
The FPA method~\cite{facchinei2013} has some similarities with our method in terms of the approximate function optimized at the nodes. But there are several fundamental differences between the methods. (a) FPA uses stochastic approximation step sizes instead of the line search used in our method. Because of this, FPA is a non-monotone method. As we will demonstrate in section x.y, there are serious practical issues associated with FPA's step size rules. (b) For our method, the proximal term merely plays a technical role for proving convergence. For example, when the loss function $f$ is strictly convex, our method can be shown to be convergent even without the proximal term. As we will show in section x.y, our method works very well even without the proximal term. On the other hand, the proximal term plays a crucial practical role for FPA. Tuning of the proximal parameter in FPA needs to be done carefully to get good efficiency. We will show in section x.y that the tuning idea outlined in~\cite{facchinei2013} is inadequate. (c) FPA does not address the issue of heavy communication costs in distributed solution. (d) The stopping condition used to stop the inner optimization of FPA is unverifiable; our method uses a verifiable relative condition. (e) When the loss is strictly convex, our method enjoys linear rate of convergence. This is not guaranteed for FPA.
\begin{table*}
\centering
\caption{Properties of Various Methods}
{\small
\begin{tabular}{|p{1.85cm}|p{1.75cm}|p{2cm}|p{1.75cm}|p{1.75cm}|p{2cm}|p{1.75cm}|} \hline
Method & F behavior & Feature limit/selection & Proximal term & Line search & Convergence & Rate of convergence \\ \hline
Bradley~et~al~\cite{bradley2011} & Non-monotone & Limited /Random & Lipschitz & None & In expectation & \\ \hline
Richtarik and Takac~\cite{richtarik2012} & Non-monotone & No~limits /Random & Maximal bound & None & In expectation & Linear \\ \hline
GRock~\cite{peng2013} & Non-monotone & Limited /Greedy & Lipschitz & None & None & \\ \hline
PCDN~\cite{bian2013} & Descent & No~limits /Random & Hessian diagonals & Armijo & Deterministic & Linear \\ \hline
Our method & Descent & No~limits /Random or Greedy & Free & Armijo & Deterministic & Linear \\
\hline\end{tabular}
}
\end{table*}
\bibliographystyle{abbrv}
\section{Related Work}
\label{sec:rw}
Our interest is mainly in parallel/distributed computing methods. There are many parallel algorithms targeting a single machine having multi-cores with shared memory~\citep{bradley2011, richtarik2013, bian2013, peng2013}. In contrast, there exist only a few efficient algorithms to solve (\ref{minF}) when the data is distributed \citep{richtarik2013a, Ravazzi2013} and communication is an important aspect to consider. In this setting, the problem (\ref{minF}) can be solved in several ways depending on how the data is distributed across machines~\citep{peng2013,boyd2011}: (A) example (horizontal) split, (B) feature (vertical) split and (C) combined example and feature split (a block of examples/features per node). While methods such as distributed \textsc{FISTA}~\citep{peng2013} or \textsc{ADMM}~\citep{boyd2011} are useful for (A), the block splitting method~\citep{parikh2013} is useful for (C). We are interested in (B), and the most relevant and important class of methods is parallel/distributed coordinate descent methods, as abstracted in Algorithm 1. Most of these methods set $f_p^t$ in step (b) of Algorithm 1 to be a quadratic approximation that is decoupled at the individual variable level. Table~\ref{tab:methodscomp} compares these methods along various dimensions.\footnote{Although our method will be presented only in Section 5, we include our method's properties in the last row of Table~\ref{tab:methodscomp}. This helps to easily compare our method against the rest.}
Most dimensions arise naturally from the steps of Algorithm~1, as explained in Section 2. Two important points to note are: (i) except~\citet{richtarik2013a} and our method, none of these methods target and sufficiently discuss distributed setting involving communication and, (ii) from a practical view point, it is difficult to ensure stability and get good speed-up with no line search and non-monotone methods. For example, methods such as~\citet{bradley2011,richtarik2012,richtarik2013,peng2013} that do not do line search are shown to have the monotone property only in expectation and that too only under certain conditions. Furthermore, variable selection rules, proximal coefficients and other method-specific parameter settings play important roles in achieving monotone convergence and improved efficiency. As we show in Section 6, our method and the parallel coordinate descent Newton method~\citep{bian2013} (see below for a discussion) enjoy robustness to various settings and come out as clear winners.
It is beyond the scope of this paper to give a more detailed discussion, beyond Table~\ref{tab:methodscomp}, of the methods from a theoretical convergence perspective on various assumptions and conditions under which results hold. We only briefly describe and comment on them below. \vspace*{0.05in}
\noindent{\bf Generic Coordinate Descent Method~\citep{scherrer2012a,scherrer2012}} \citet{scherrer2012a} and \citet{scherrer2012} presented an abstract framework for coordinate descent methods (\textsc{GenCD}) suitable for parallel computing environments. Several coordinate descent algorithms such as stochastic coordinate descent~\citep{shwartz2011}, \textsc{Shotgun}~\citep{bradley2011} and \textsc{GROCK}~\citep{peng2013} are covered by \textsc{GenCD}. \textsc{GROCK} is a {thread greedy} algorithm~\citep{scherrer2012a} in which the variables are selected greedily using gradient information. One important issue is that algorithms such as \textsc{Shotgun} and \textsc{GROCK} may not converge in practice due to their non-monotone nature with no line search; we faced convergence issues on some datasets in our experiments with \textsc{GROCK} (see Section 6). Therefore, the practical utility of such algorithms is limited without ensuring necessary descent property through certain spectral radius conditions on the data matrix. \vspace*{0.05in}
\noindent{\bf Distributed Coordinate Descent Method~\citep{richtarik2013a}}
The multi-core parallel coordinate descent method of \citet{richtarik2012} is a much refined version of \textsc{GenCD} with careful choices for steps (a)-(c) of Algorithm 1 and a supporting stochastic convergence theory. \citet{richtarik2013a} extended this to the distributed setting; so, this method is more relevant to this paper. With no line search, their algorithm \textsc{HYDRA} (Hybrid coordinate descent) has (expected) descent property only for certain sampling types of selecting variables and $L_j$ values. One key issue is setting the right $L_j$ values for good performance. Doing this accurately is a costly operation; on the other hand, inaccurate setting using cheaper computations (e.g., using the number of non-zero elements as suggested in their work) results in slower convergence (see Section 6). \vspace*{0.05in}
\citet{neco2013} suggest another variant of parallel coordinate descent in which all the variables are updated in each iteration. \textsc{HYDRA} and \textsc{GROCK} can be considered as two key, distinct methods that represent the set of methods discussed above. So, in our analysis as well as experimental comparisons in the rest of the paper, we do not consider the methods in this set other than these two.
\noindent{\bf Flexible Parallel Algorithm (\textsc{FPA})~\citep{facchinei2013} } This method has some similarities with our method in terms of the approximate function optimized at the nodes. Though~\citet{facchinei2013} suggest several approximations, they use only (\ref{quad1}) in its final implementation. More importantly, \textsc{FPA} is a non-monotone method using a stochastic approximation step size rule. Tuning this step size rule along with the proximal parameter $L_j$ to ensure convergence and speed-up is hard. (In Section 6 we conduct experiments to show this.) Unlike our method, \textsc{FPA}'s inner optimization stopping criterion is unverifiable (for e.g., with (\ref{ours})); also, \textsc{FPA} does not address the communication cost issue. \vspace*{0.05in}
\noindent{\bf Parallel Coordinate Descent Newton (\textsc{PCD})~\citep{bian2013}}
One key difference between other methods discussed above and our \textsc{DBCD} method is the use of line search. Note that the \textsc{PCD} method can be seen as a special case of \textsc{DBCD} (see Section 5.1). In DBCD, we optimize per-node block variables jointly, and perform line search across the blocks of variables; as shown later in our experimental results, this has the advantage of reducing the number of outer iterations, and overall wall clock time due to reduced communication time (compared to \textsc{PCD}). \vspace*{0.05in}
\noindent{\bf Synchronized Parallel Algorithm~\citep{patriksson1998fp}} \citet{patriksson1998fp} proposed a Jacobi type synchronous parallel algorithm with line search using a generic cost approximation (\textsc{CA}) framework for differentiable objective functions~\citep{patriksson1998}. Its local linear rate of convergence results hold only for a class of strong monotone \textsc{CA} functions. If we view the approximation function, $f_p^t$ as a mapping that is dependent on $w^t$, \citet{patriksson1998fp} requires this mapping to be continuous, which is unnecessarily restrictive.
\noindent{\bf \textsc{ADMM} Methods} Alternating direction method of multipliers is a generic and popular distributed computing method. It does not fit into the format of Algorithm~1. This method can be used to solve (\ref{minF}) in different data splitting scenarios~\citep{boyd2011,parikh2013}. Several variants of global convergence and rate of convergence (e.g., $O(\frac{1}{k})$) results exist under different weak/strong convexity assumptions on the two terms of the objective function~\citep{deng2012,deng2013}. Recently, an accelerated version of \textsc{ADMM}~\citep{goldstein2013} derived using the ideas of Nesterov's accelerated gradient method~\citep{nesterov2012} has been proposed; this method has dual objective function convergence rate of $O(\frac{1}{k^2})$ under a strong convexity assumption. \textsc{ADMM} performance is quite good when the augmented Lagrangian parameter is set to the right value; however, getting a reasonably good value comes with computational cost. In Section 6 we evaluate our method and find it to be much faster. \vspace*{0.05in}
Based on the above study of related work, we choose \textsc{HYDRA}, \textsc{GROCK}, \textsc{PCD} and \textsc{FPA} as the main methods for analysis and comparison with our method.\footnote{In the experiments of Section~\ref{sec:expts}, we also include ADMM.} Thus, Table~1 gives various dimensions only for these methods.
\begin{sidewaystable}
\centering
\caption{Properties of selected methods that fit into the format of Algorithm~1. Methods: {\it HYDRA}~~\citep{richtarik2013a}, {\it GROCK}~~~\citep{peng2013}, {\it FPA}~~~\citep{facchinei2013}, {\it PCDN}~~~\citep{bian2013}. }
\label{tab:methodscomp}
{\small
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
{\bf Method} & {\bf Is ${\mathbf F(w^t)}$} & {\bf Are limits} & {\bf How is ${\mathbf S_p^t}$} & {\bf Basis for} & {\bf How is} & {\bf Convergence} & {\bf Convergence} \\
& {\bf monotone?} & {\bf forced on ${\mathbf |S_p^t|}$?} & {\bf chosen?} & {\bf choosing ${\mathbf L_j}$} & {\bf ${\mathbf \alpha^t}$ chosen} & {\bf type} & {\bf rate} \\
\hline
\multicolumn{8}{c}{{\bf Existing methods}} \\ \hline
{\it HYDRA}~ & No & No, if $L_j$ is & Random & Lipschitz bound for $g_j$ & Fixed & Stochastic & Linear \\
& & varied suitably & & suited to $S_p^t$ choice & & & \\ \hline
{\it GROCK}~ & No & Yes & Greedy & Lipschitz bound for $g_j$ & Fixed & Deterministic & Sub-linear \\ \hline
{\it FPA}~ & No & No & Random & Lipschitz bound for $g_j$ & Adaptive & Deterministic & None \\ \hline
{\it PCDN}~ & Yes & No & Random & Hessian~diagonal & Armijo & Stochastic & Sub-linear \\
& & & & & line search & & \\ \hline
\multicolumn{8}{c}{{\bf Our method}} \\ \hline
\textsc{DBCD} & Yes & No & Random/Greedy & Free & Armijo & Deterministic & Locally linear \\
& & & & & line search & & \\ \hline
\end{tabular}
}
\end{sidewaystable}
\section{Related Work}
There exists a large volume of literature addressing the problem of optimizing $L1$ regularized objective functions~(see for e.g., \cite{lee2006,andrew2007,yun2011,beck2009,yuan2012,yuan2010}). The aspects in which the methods proposed therein differ include: (i) applications (e.g., compressed sensing and high dimensional classification/regression) to exploit any structure, (ii) assumptions made on the loss functions (e.g., convex/non-convex, squared error/hinge loss, smooth/non-smooth, decomposable, etc), (iii) computing environment (single, parallel, distributed) and (iv) algorithmic (details given below). We are interested in building high dimensional linear classifiers with convex and smooth loss functions.
In the computing environment aspect, there are several algorithms that focus on doing parallel computations in a single machine having multi-cores (see~\cite{bradley2011}, \cite{richtarik2013}, \cite{bian2013}, \cite{peng2013}). In this scenario, memory is shared; therefore, there is no communication overhead. On the other hand, there exist only a few efficient algorithms that address the scenario when the data is distributed (see~\cite{richtarik2013a},~\cite{Ravazzi2013}). Here, communication cost plays an important role and may have to be traded-off with per node computation cost. Furthermore, the optimization problem (\ref{minF}) can be solved in different ways depending on how the data is available per node~\cite{peng2013,boyd2011}: (1) example partition with all features, (2) feature partition with all examples and (3) combined example and feature partition (a block of examples/features). Methods such as distributed FISTA~\cite{peng2013} or ADMM~\cite{boyd2011} can be used when the data is example partitioned. In the combined partitioning case, methods such as block splitting~\cite{parikh2013} are useful. We are interested in the feature partitioning setting.
In the algorithmic aspect, one important and popular category of methods is coordinate descent methods where a subset of variables are optimized at a time in an iterative fashion as given in Algorithm 1 until some convergence criterion is met. There are several key elements that need to be considered, and choices to be made for each one of them; they are: (i) selection of variables~\cite{tseng2009,yun2011}, (ii) approximation of the sub-problem objective function and its optimization for direction finding or weight update~\cite{beck2009,yun2011,yuan2012,fount2012,bryd2012}, (iii) line search (exact, approximate, no), (iv) conditions to be met to ensure (sufficient) decrease in the objective function value, in deterministic or expectation sense so that convergence guarantees can be given~\cite{nesterov2012,beck2009,yun2011,tseng2009, wang2013}, and, (5) acceleration mechanisms to improve the rate of convergence~\cite{beck2009,nesterov2012}. Yuan et al~\cite{yuan2012} studied various methods, specifically focusing on large scale linear classification problems with serial computations done in a single machine. Our interest is in comparing parallel/distributed computing methods.
Parallel coordinate descent (PCD) methods can be grouped into methods that handle smooth and non-smooth objective functions. Since the $L1$ regularized loss function can be transformed to a smooth objective function in $2d$ variables with positivity constraints, PCD methods~\cite{neco2013,patriksson1998fp} that handle smooth objective functions are also of interest. We have seen how a generalized coordinate descent algorithm (Algorithm 1) can be written in a parallel/distributed computing setting involving the various elements (i)-(iv). So, we discuss how our work is similar to and different from other methods in the various elements; while Table 1 gives the summary, more details are presented below. \\
\subsection{P(D)CD Methods with $L1$ Regularized Sub-problems}
In this class of methods, the objective function of each sub-problem is written as the sum of a function that approximates the loss function and the regularizer term; therefore, it is still non-smooth. Due to the decomposable structure of the approximating function and the regularizer term, these sub-problems can be solved in parallel, and in some cases efficiently (e.g., closed form).
\noindent{\bf Parallel Coordinate Descent Newton (PCDN)~\cite{bian2013}} Bian et al~\cite{bian2013} proposed a parallel coordinate descent method where randomized block partitioning of variables is done in each (outer) iteration. Then, each block of variables is sequentially optimized, and variables in each block are optimized in parallel using (diagonal) Hessian term in (\ref{quad}) followed by a line search. This line search is critical to achieve global convergence, and Bian et al. showed linear rate of convergence in the expectation sense (where expectation taken over the random partitioning of the variables). PCDN can be seen as a special case of the Algorithm 1 and our method; this follows from the generality of the algorithm with block coordinate optimization and various approximation function possibilities. We optimize per node block variables jointly, and perform line search across the blocks of variables; as the experimental results show, this can reduce the number of outer iterations significantly. Furthermore, we show global convergence and local linear rate of convergence in deterministic sense. \\
\noindent{\bf Distributed Coordinate Descent Method (DCDM)~\cite{richtarik2013a}} Richtarik and Takac~\cite{richtarik2013a} extended their initial work of parallel coordinate descent method~\cite{richtarik2013} suitable for multicore setting to the distributed setting. This method is closely related to ours and follows Algorithm 1, but differs from our method as indicated in Table 1. One key issue with this method is that setting the parameters $L^j$ accurately is a costly operation; on the other hand, inaccurate setting using cheaper computations results in slower convergence as our experiments show in Section~\ref{sec:motiv}. This also implies another level of trade-off in setting the parameters right to get improved efficiency. \\
\noindent{\bf Generic Coordinate Descent Method~\cite{scherrer2012a,scherrer2012}} Scherrer et al~\cite{scherrer2012a,scherrer2012} presented an abstract framework for coordinate descent methods (GenCD) suitable in parallel computing environments. In an extension, they introduced a randomized block greedy version and proved convergence results under certain descent conditions based on data properties. Several coordinate descent algorithms such as stochastic coordinate descent (SCD)~\cite{shwartz2011}, \textit{Shotgun}~\cite{bradley2011} and \textit{GRock}~\cite{peng2013} are covered by their method. Besides this, they suggested feature clustering before running the algorithm to improve efficiency, and discussed a \textit{thread greedy} selection of variables approach. We observe that GRock is a {thread greedy} algorithm in which the variables are selected greedily using gradient information. Note that feature clustering can be expensive, particularly, when data movement is needed in a distributed computing environment. Our algorithm (Algorithm 1) is similar to GenCD. However, besides the differences seen from Table 1, our method is targeted for the distributed computing environment involving communication (see for example, \textit{AllReduce} operation in step (d) of our algorithm). More importantly, algorithms such as \textit{Shotgun} and \textit{GRock} may not converge in practice due to their non-monotone nature; for example, we faced convergence issues on some datasets in our experiments with the GRock algorithm~\cite{peng2013}. Therefore, the practical utility of such algorithms is limited without ensuring necessary descent conditions.\\
\noindent{\bf Flexible Parallel Algorithms (FPA)~\cite{facchinei2013} } The FPA method~\cite{facchinei2013} has some similarities with our method in terms of the approximate function optimized at the nodes. But there are several fundamental differences between the methods. (a) FPA uses stochastic approximation step sizes instead of the line search used in our method. Because of this, FPA is a non-monotone method. As we will demonstrate in section x.y, there are serious practical issues associated with FPA's step size rules. (b) For our method, the proximal term merely plays a technical role for proving convergence. For example, when the loss function $f$ is strictly convex, our method can be shown to be convergent even without the proximal term. As we will show in section x.y, our method works very well even without the proximal term. On the other hand, the proximal term plays a crucial practical role for FPA. Tuning of the proximal parameter in FPA needs to be done carefully to get good efficiency. We will show in section x.y that the tuning idea outlined in~\cite{facchinei2013} is inadequate. (c) FPA does not address the issue of heavy communication costs in distributed solution. (d) The stopping condition used to stop the inner optimization of FPA is unverifiable; our method uses a verifiable relative condition. (e) When the loss is strictly convex, our method enjoys linear rate of convergence. This is not guaranteed for FPA. \\
\subsection{PCD Methods for Smooth Functions}
We briefly discuss two methods in this category.
\noindent{\bf Synchronized Parallel Algorithm~\cite{patriksson1998fp}} Patriksson~\cite{patriksson1998} developed a cost approximation (CA) framework of descent algorithms for nonlinear optimization problems. He showed how a Jacobi type synchronous parallel algorithm with line search fits in this framework for differentiable objective functions~\cite{patriksson1998fp} over Cartesian product sets. When the CA functions (used to solve the direction finding sub-problems in parallel) are strongly convex, and additional conditions such as Armijo rule for line search step size selection and cyclic coordinate selection of variables are imposed, he proved linear rate of convergence. \\
\noindent{\bf Parallel Block Coordinate Descent Method~\cite{neco2013}} Necora and Clipici~\cite{neco2013} proposed an iterative algorithm where they solved (\ref{quad}) for all coordinates (block wise) in parallel, using coordinate Lipschitz constants for the gradient. The weights are updated in parallel as a weighted combination of previous weight and the incremental update obtained from solving (\ref{quad}). They showed sub-linear and linear rate of convergence results without and with strong convexity assumption. \\
\subsection{ADMM Methods}
\begin{itemize}
\item Accelerated ADMM
\item On the $O(\frac{1}{k})$ convergence and parallelization of the ADMM (Deng et al, 2013).
\end{itemize}
\begin{table*}
\centering
\caption{{\scriptsize Properties of Various Methods. $*$ indicates monotone in the expectation sense for certain combination of sampling types and constants $L_i$ settings. $+$ indicates distributed versions (like our method) implemented in our experiments. $\dagger$ indicates convergence only under certain conditions on the spectral radius of data matrix. $\triangle$ represents the class of smooth function optimization and all variables are updated in parallel; variants are possible. The descent and convergence rate (sub-linear/linear) results are dependent on conditions such as (strong) convexity of the approximating functions, Lipschitz continuous gradient, line search step size and variable selection rules. The variables in the sub-problem are jointly optimized only in our method.}}
{\small
\begin{tabular}{|p{1.85cm}|p{1.75cm}|p{2cm}|p{1.75cm}|p{1.75cm}|p{2cm}|p{1.75cm}|} \hline
Method & F behavior (descent) & Feature limit/selection & Proximal term & Line search & Convergence & Rate of convergence \\ \hline
Bradley~et~al~\cite{bradley2011} & Non-monotone & Limited /Random & Lipschitz & None & In expectation & Sub-linear$^\dagger$ \\ \hline
Richtarik and Takac~\cite{richtarik2012} & Monotone$^*$ & No~limits /Random & Maximal bound & None & In expectation & Linear \\ \hline
Richtarik and Takac~\cite{richtarik2013}$^+$ & Monotone$^*$ & No~limits /Random & Maximal bound & None & In expectation & Linear \\ \hline
GRock~\cite{peng2013}$^{+}$ & Non-monotone & Limited /Greedy & Lipschitz & None & None & Sub-linear$^\dagger$ \\ \hline
PCDN~\cite{bian2013}$^+$ & Descent & No~limits /Random & Hessian diagonals & Armijo & In Expectation & Linear \\ \hline
PCD~\cite{neco2013}$^\triangle$ & Descent & Full & Lipschitz & None & Deterministic & Linear \\ \hline
SPA~\cite{patriksson1998fp}$^\triangle$ & Descent & Full & Lipschitz & Armijo & Deterministic & Linear \\ \hline
Our method & Descent & No~limits /Random or Greedy & Free & Armijo & Deterministic & Linear \\ \hline
\end{tabular}
}
\end{table*}
|
2,877,628,089,523 | arxiv |
\section{Introduction}
Let $\VM{A}(\VM{\omega}), \VM{B}(\VM{\omega})$ be $n \times n$ symmetric matrices which smoothly depend on a parameter $\VM{\omega}$ in a compact subset $\Omega \subset \mathbb{R}^d$. We impose that $\VM{B}(\VM{\omega})$ is positive definite, for $\VM{\omega} \in \Omega$, so all eigenvalues $\lambda(\VM{\omega})$ of the generalized eigenvalueproblem $\left(\VM{A}(\VM{\omega}), \VM{B}(\VM{\omega})\right)$,
$$\VM{A}(\VM{\omega}) \VM{x}(\VM{\omega}) = \lambda(\VM{\omega}) \VM{B}(\VM{\omega}) \VM{x}(\VM{\omega}),$$ are real, see, e.g, \cite{Saad2011} Ch. 9. The objective is to calculate in an efficient way, an accurate global approximation of the minimal eigenvalue $\lambda_1(\VM{\omega})$ over the whole parameterspace. Standard eigensolvers like the Lanczos' method \cite{Lanczos1950}, Jacobi-Davidson \cite{G.Sleijpen1996} or LOBPCG \citep{Knyazev2001} can be applied for large-scale problems for some points in the parameterspace but such an approach becomes expensive when considering a large number of sample points.
This paper focuses on large scale eigenvalue problems. One class of examples is the estimation of the coercivity constant of parametrized elliptic partial differential equations \cite{Rozza2008}. Another is the computation of the inf-sup constant of PDEs from the minimal singular value of $\VM{A}(\VM{\omega})$, which is computed from the minimal eigenvalue of $\VM{A}(\VM{\omega})^T \VM{A}(\VM{\omega})$. The minimal eigenvalue problem also plays a role in the characterization of pseudospectra \cite{Karow2010}, \cite{Sirkovic2018}. Related problems are
the determination of statistical moments of the minimal eigenvalue when $\omega$ is the realization of random variables and the minimization or maximization of the $j$th largest eigenvalue of a parameter dependent Hermitian matrix; see \cite{KangalF2018Asmf} and the references therein.
There are several methods for this kind of problem. The first category concerns polynomial-based methods. In cite{Andreev2012} the authors approximate $\lambda_1(\VM{\omega})$ by sparse tensor products of Legendre polynomials and in \cite{Ghanem2007} polynomial chaos is advocated, a technique which is frequently used when the parameters are stochastic. These techniques have difficulties with the possible lack of smoothness of the minimal eigenvalue. Another approach is the so-called Successive Constraint Method (SCM), see \cite{Sirkovic2016} and the references therein. In paper \cite{Sirkovic2016}, lower and upper bounds are calculated for each sample in a finite subset of the parameterspace. The main disadvantage of this method is that it does not provide a way to calculate the minimal eigenvalue of a parameter sample other than the samples in the initial subset.
The idea of this paper is to approximate the minimal eigenvalue of a large scale matrix pencil by the minimal eigenvalue of a smaller matrix, which we call the reduced problem. The reduced problem is obtained by projection of the large scale matrix on a well-chosen subspace. The reduced problem adopts the same smoothness properties or lack thereof as the original large scale problem, which allows us to use less samples than methods based on smooth approximations such as polynomials.
The novelty of this paper is in the construction of the subspace to build the reduced eigenvalue problem. The subspace is constructed from samples of the associated eigenvector of the large scale matrix in the parameterspace. The selection of sample points is based on a greedy method \cite{Quarteroni2016}, with the aim to minimize the residual norm of the large scale eigenvalue problem with a minimum amount of samples. For each sampling point, a large scale eigenvalue problem is solved. Eigenvalue solvers such as Lanczos' method or Jacobi-Davidson usually compute more information than just one eigenvector approximation. The goal is to explore whether other (freely available) information can be used with the aim to further reduce the number of sample points and, consequently, the computational cost. In particular, we will study higher order Hermite interpolation sampling by inserting partial derivatives of the eigenvector towards the parameters in the subspace as well, and the addition of more than one eigenvector approximation for each sample point. Calculating the maximal eigenvalue or an eigenvalue that meets another condition (second maximal/minimal eigenvalue) can be done in a similar way as the method proposed here.
\medskip
The plan of the paper is as follows. In section 2 we introduce the necessary notation and we give some motivating examples. We explain theoretically why we add the first eigenvectors as well as the partial derivatives of the first eigenvector in section 3. Section 4 gives insight how eigenvectors change over the parameters. Subsequently we explain how we build up our subspace and we present the algorithm. In section 6 we illustrate the theory and algorithms numerically. We finish this paper with some concluding remarks and research possibilities.
\section{Notation and motivating examples}
We denote vectors by small bold letters and matrices by large bold letters. A subspace of $\mathbb{R}^n$ is denoted by calligraphic letters. We already mentioned that we restrict ourselves to the case where both $\VM{A}$ and $\VM{B}$ are symmetric, $\VM{B}$ is positive definite; $\VM{A}$ and $\VM{B}$ depend analytically on the parameter for all $\VM{\omega} \in \Omega$. From this last requirement, it follows that the inner product defined by
$$\left( \VM{x}, \VM{y}\right)_\VM{B} := \VM{x}^T \VM{B} \VM{y}, \VM{x},\VM{y} \in \mathbb{R}^n$$ and the associated $\VM{B}$-norm
$$\norm{\VM{x}}_\VM{B} := \left( \VM{x}, \VM{x}\right)_\VM{B}, \VM{x} \in \mathbb{R}^n$$ are well-defined. We call $\left(\lambda_i, \VM{x}_i\right), i = 1, \hdots, n$ an \emph{eigenpair} of the generalised eigenvalue problem $\left( \VM{A}, \VM{B} \right)$ if $$\VM{A} \VM{x}_i = \lambda_i \VM{B} \VM{x}_i, \quad \VM{x}_i \neq \VM{0}$$
where $\lambda_i$ and $\VM{x}_i$ are respectively called an \emph{eigenvalue} and an associated \emph{eigenvector}. In this case it is proven in \cite{Parlett1998} that the eigenvalues are real and we assume further that
\begin{equation}\label{eqn:ordering}
\lambda_1 \leq \lambda_2 \leq \hdots \leq \lambda_n.
\end{equation}
Eigenvectors are always assumed to be $\VM{B}$-orthonormal, meaning that $\VM{x}_i \VM{B} \VM{x}_j = \delta_{i,j}, i,j = 1,\hdots, n$.
We further define \begin{equation}
\VM{\VM{\Lambda}} = \mbox{diag}\left( \lambda_1, \lambda_2, \hdots, \lambda_n \right), \quad \VM{X}= [\VM{x}_1, \VM{x}_2, \hdots, \VM{x}_n] \label{eqn:0}
\end{equation} with $\VM{X}^T \VM{B} \VM{X} = \VM{I}_n$ then we can prove that
\begin{equation}
\VM{X}^T \VM{A} \VM{X} = \VM{\VM{\Lambda}}. \label{eqn:SVD_A}
\end{equation}
It immediately follows that if $\VM{A}$ is also positive definite, the eigenvalues of the couple $\left( \VM{A}, \VM{B} \right)$ are strictly positive.
Furthermore, if the eigenvalue is simple for $\VM{\omega} \in \Omega \subset \mathbb{R}^d$, there is an environment around $\VM{\omega}$ where this eigenvalue is differentiable as we enumerate the eigenvalues for all parametervalues in increasing order. If the eigenvalue has a multiplicity $m > 1$ for a given $\VM{\omega}$ and is simple in an open set around $\VM{\omega}$, then it can be decomposed in this open set in each direction into $m$ differentiable curves and something similar can be done for $\VM{B}$-normalized eigenvectors, see e.g \cite{Lax2007} ch. 9. This does not mean that it can also be decomposed in $d$-dimensional surfaces, see Example \ref{vbd:1}.
Remark that in view of \eqref{eqn:ordering}, $\VM{X}$ is not continuous in the parameter since in an open set around a point where not all eigenvalues are simple, the order of the eigenvectors may change.
\begin{eexample} \label{vbd:1}
Let $\Omega = [-0.5, 0.5] \times [-0.5, 0.5]$ be the parameterspace and let the matrices be
$$\VM{A}(\VM{\omega}) = \VM{W}(\VM{\omega}) \begin{bmatrix}
\omega_1+1 & \omega_2 & & & \\
\omega_2 & -\omega_1+1 & & & \\
& & 3 & & \\
& & & \ddots & \\
& & & & n
\end{bmatrix} \VM{W}(\VM{\omega})^T, \VM{B}(\VM{\omega}) = \VM{I}_n$$
with $\VM{W}(\VM{\omega}) = [\VM{w}_1, \VM{w}_2, \hdots, \VM{w}_n]$ an orthonormal ($n \times n$)-matrix for all $\VM{\omega} \in \Omega$. The eigenvalues are $\lambda_1(\VM{\omega}) = -\sqrt{\omega_1^2 + \omega_2^2}+1$, $\lambda_2(\VM{\omega}) = \sqrt{\omega_1^2 + \omega_2^2} +1$ and the other eigenvalues are equal to $3, \hdots, n$. The two minimal eigenvalues are displayed in Figure \ref{fig:ex1}. We observe that the eigenvalues cannot be decomposed in 2 smooth surfaces.
\begin{figure}[h]
\setlength{\figW}{5cm}
\centerline{\input{images/example_kegel.tex}}
\caption{Surfaces that represent $\lambda_1(\VM{\omega})$ and $\lambda_2(\VM{\omega})$ in Example \ref{vbd:1}} \label{fig:ex1}
\end{figure}
\end{eexample}
\begin{eexample} \label{vbd:2}
This example is taken from \cite[Example 4.4]{Sirkovic2016}. The parameterspace is $\Omega = [0.02, 0.5] \times [2, 8]$ and $\VM{A}(\VM{\omega}) = \sum_{i=1}^3 \theta_i(\VM{\omega}) \VM{A}_i$
with $\theta_i(\VM{\omega}), i = 1,2,3$ analytic functions and $\VM{B}$ is constant. The dimension of the problem is $1311$. The matrices originate from a finite element discretisation of a PDE. In Figure \ref{fig:ex2}, we show the minimal eigenvalue as a function of the parameters. More details can be found in \cite[Example 4.4]{Sirkovic2016}.
\begin{figure}[h]
\begin{center}
\setlength{\figW}{5cm}
\centerline{\input{images/example_coerc1.tex}}
\caption{Surface of $\lambda_1(\VM{\omega})$ in Example \ref{vbd:2}} \label{fig:ex2}
\end{center}
\end{figure}
\end{eexample}
It is immediately clear from the previous examples that an approximation method (including polynomial approximations) that relies on the smoothness of the surface, is not appropriate.
We consider projection methods and we describe in the next section why they are well suited in this case.
We end this section by introducing the concept of reduced eigenvalue problems. Let $\subs{V} \subset \mathbb{R}^n$ be a subspace spanned by the columns of an orthonormal matrix $\VM{V} := [\VM{v}_1, \VM{v}_2, \hdots, \VM{v}_m] \in \mathbb{R}^{n\times m}$ with $m$ the dimension of the subspace. We call $\left( \VM{V}^T \VM{A}(\VM{\omega}) \VM{V}, \VM{V}^T \VM{B}(\VM{\omega}) \VM{V} \right)$ the \emph{reduced eigenvalue problem}
on $\subs{V}$. We denote by $\left( \lambda^\subs{V}_i(\VM{\omega}), \VM{x}^\subs{V}_i(\VM{\omega}) \right), i = 1, \hdots, m$ an eigenpair of this reduced eigenvalue problem.
\section{Hermite interpolation by projection} \label{sect:Herm_int}
In this section, we derive Hermite interpolation properties of the reduced eigenvalue problem when the subspace is built with the first eigenvector and its partial derivatives in some sample points. Concretely, we show that the eigenvalue itself is interpolated as well as its first and second partial derivatives. To prove this, we need a characterisation of the first and second derivative of an eigenvalue and of the first derivative of an eigenvector. Since eigenvectors depend on the normalisation, we first outline the adopted setting, in which they are uniquely defined.
Let $\lambda_i^*$ be a simple eigenvalue with associated eigenvector $\VM{x}_i^*$ of $\VM{\omega}^* \in \Omega$ then there is an environment $\Omega^*$ around $\VM{\omega}^*$ and smooth functions
\begin{align}
\begin{aligned}
\lambda_i:& \Omega^* \rightarrow \mathbb{R} \\
& \VM{\omega} \mapsto \lambda_i(\VM{\omega})
\end{aligned}
\quad \text{ and } \quad
\begin{aligned}
\VM{x}_i:& \Omega^* \rightarrow \mathbb{R}^n \\
&\VM{\omega} \mapsto \VM{x}_i(\VM{\omega})
\end{aligned} \label{eqn:setting_deriv}
\end{align}
such that
\begin{equation}
\left\{\begin{matrix*}[c]
\VM{A}(\VM{\omega}) \VM{x}_i(\VM{\omega}) - \lambda_i(\VM{\omega}) \VM{B}(\VM{\omega}) \VM{x}_i(\VM{\omega})& = & 0 &,\forall \VM{\omega} \in \Omega^* & \left( (\lambda_i(\VM{\omega}), \VM{x}_i(\VM{\omega})) \text{ is eigenpair} \right)\\
\VM{x}_i(\VM{\omega})^T \VM{B}(\VM{\omega}) \VM{x}_i(\VM{\omega}) & = &1 &,\forall \VM{\omega} \in \Omega^* & \left( \text{ normalisation condition} \right) \\
\lambda_i(\VM{\omega}^*) & = &\lambda_i^*& & \\
\VM{x}_i(\VM{\omega}^*) & = &\VM{x}_i^* & &
\end{matrix*}\right. \label{eqn:condit_deriv}
\end{equation}
In the remainder of this section and the next section, we characterise the derivatives of the functions in \eqref{eqn:setting_deriv}. The derivative of a simple eigenvalue $\lambda_i$ with associated eigenvector $\VM{x}_i$ is
\begin{equation}
\pderiv{\lambda_i}{\omega_j} = \VM{x}_i^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_i,\quad j = 1, \hdots, d. \label{eqn:dlambda}
\end{equation}
This is a generalisation of the more known result for standard eigenvalue problems, see e.g \cite[Ch. 9]{Lax2007} . For the second derivative, it is sufficient to differentiate equation \eqref{eqn:dlambda}. In this way, we get the following expression for the second derivative
\begin{equation}
\pderiv[1]{\lambda_i}{\omega_j,\omega_k} = 2 \VM{x}_i^T \left( \pderiv{ \VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \pderiv{\VM{x}_i}{\omega_k} + \VM{x}_i^T \left( \pderiv[1]{\VM{A}}{\omega_j, \omega_k} - \pderiv{\lambda_i}{\omega_k} \pderiv{\VM{B}}{\omega_j} - \lambda_i \pderiv[1]{\VM{B}}{\omega_j, \omega_k} \right) \VM{x}_i, \quad j,k = 1, \hdots, d. \label{eqn:ddlambda}
\nonumber
\end{equation}
We characterize the derivative of the eigenvector associated with a simple eigenvalue as the solution of a system of linear equations. In this proof we need following lemma.
\begin{lemma} \label{lem:afgel_lambda}
It holds that
\begin{equation} \label{eqn:help_system_deriv}
\VM{x}_i^T \VM{B} \pderiv{\VM{x}_i}{\omega_j} = -\dfrac{1}{2} \VM{x}_i^T \pderiv{\VM{B}}{\omega_j} \VM{x}_i, \quad j = 1, \hdots, d.
\end{equation}
\begin{proof}
This follows from taking the derivative of the normalisation condition in \eqref{eqn:condit_deriv}.
\end{proof}
\end{lemma}
\begin{theorem} \label{ste:system_deriv}
The derivative of the eigenvector $\VM{x}_i$ associated with the simple eigenvalue $\lambda_i$ is characterized as the solution of the following system:
\begin{equation} \begin{bmatrix}
\lambda_i \VM{B} - \VM{A} & \VM{B} \VM{x}_i \\
\VM{x}_i^T \VM{B} & 0
\end{bmatrix} \begin{bmatrix} \pderiv{\VM{x}_i}{\omega_j} \\ \pderiv{\lambda_i}{\omega_j} \end{bmatrix} = \begin{bmatrix}
\left( \pderiv{\VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right ) \VM{x}_i \\
- \dfrac{ \VM{x}_i^T \pderiv{\VM{B}}{\omega_j} \VM{x}_i}{2}
\end{bmatrix}, \quad j = 1,\hdots, d \label{eqn:system_dx}.
\end{equation}
\begin{proof}
We start by differentiating $\left( \lambda_i \VM{B} - \VM{A} \right)\VM{x}_i = 0 $ to get
$$\left( \pderiv{\lambda_i}{\omega_j} \VM{B} + \lambda_i \pderiv{\VM{B}}{\omega_j} - \pderiv{\VM{A}}{\omega_j} \right) \VM{x}_i + \left( \lambda_i \VM{B} - \VM{A} \right) \pderiv{\VM{x}_i}{\omega_i} = 0, \quad j = 1, \hdots, d.$$
As $\left( \lambda_i \VM{B} - \VM{A} \right)$ is a singular matrix, the equation
\begin{equation} \label{eqn:sys_part1} \left( \lambda_i \VM{B} - \VM{A} \right) \pderiv{\VM{x}_i}{\omega_j} = -\left( \pderiv{\lambda_i}{\omega_j} \VM{B} + \lambda_i \pderiv{\VM{B}}{\omega_j} - \pderiv{\VM{A}}{\omega_j} \right) \VM{x}_i, \quad j = 1, \hdots, d
\end{equation}
is not sufficient to characterise the partial derivative of an eigenvector. The missing information is the information in the direction of $\VM{x}_i$ which is given in \eqref{eqn:help_system_deriv}. The combination of \eqref{eqn:sys_part1} and \eqref{eqn:help_system_deriv} leads to the proof of the theorem.
\end{proof}
\end{theorem}
We prove in the next property that if both the eigenvector and its partial derivatives are present in the subspace $\subs{V}$ for $\VM{\omega}^* \in \Omega$, the minimal eigenvalue of the reduced eigenvalue problem on $\subs{V}$ is a Hermite interpolant of degree two of the minimal eigenvalue of the pencil $\left( \VM{A}(\VM{\omega}^*), \VM{B}(\VM{\omega}^*) \right)$. Furthermore we state also the well-known property that the minimal eigenvalue of a reduced eigenvalue problem is bounded from below by the original eigenvalue problem.
\begin{property} \label{prop:interpol}
\begin{enumerate}
\item (Hermite interpolation) If $\left( \lambda_i, \VM{x}_i \right)$ is a simple eigenpair and if $\VM{x}_i \in \subs{V}$, then
\begin{itemize}
\item $\left(\lambda_i^\subs{V}, \VM{x}_i^\subs{V} \right) = \left(\lambda_i, \VM{V}^T \VM{x}_i \right)$ is an eigenpair of $ \left( \VM{V}^T \VM{A}\VM{V}, \VM{V}^T \VM{B} \VM{V} \right)$.
\item $ \pderiv{\lambda_i^\subs{V}}{\omega_j} = \pderiv{\lambda_i}{\omega_j}, \quad j = 1, \hdots d$.
\end{itemize}
If $\VM{x}_i, \pderiv{\VM{x}_i}{\omega_j} \in \subs{V}$, then
$$\pderiv{\lambda_i^\subs{V}}{\omega_j, \omega_k} = \pderiv{\lambda_i}{\omega_j, \omega_k}, \quad k = 1,\hdots, d$$
\item If $\subs{V}_1 \subset \subs{V}_2$ then it holds that
$$\lambda_1 \leq \lambda_1^{\subs{V}_2} \leq \lambda_1^{\subs{V}_1}, \quad \forall \omega \in \Omega, \quad j = 1,\hdots, d.$$
This means that if we extend the subspace, the estimation will be at least as good.
\end{enumerate}
\begin{proof}
\underline{Assertion 1:} \\
The first two statements are well-known results in the case of a standard eigenvalue problem, see e.g. \citep{KangalF2018Asmf} and the references therein. The proofs for the generalized eigenvalue problem are stated for the sake of completeness.
For the first statement, it is sufficient to see that $\VM{x}_i = \VM{V} \VM{V}^T \VM{x}_i$ to obtain $\VM{V}^T \VM{A} \VM{V} \VM{V}^T \VM{x}_i = \lambda_i \VM{V}^T \VM{B} \VM{V} \VM{V}^T \VM{x}_i$. \\
For the second, the result follows from
\begin{align*}
\pderiv{\lambda_i}{\omega_j} & = \VM{x}_i^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_i \\
& = \left( \VM{V} \VM{x}_i^\subs{V} \right)^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \VM{V} \VM{x}_i^\subs{V} \\
& = \left( \VM{x}_i^\subs{V}\right)^T \left( \VM{V}^T \pderiv{\VM{A}}{\omega_j}\VM{V} - \lambda_i^\subs{V} \VM{V}^T \pderiv{\VM{B}}{\omega_j} \VM{V} \right) \VM{x}_i^\subs{V} \\
& = \pderiv{\lambda_i^\subs{V}}{\omega_j}.
\end{align*}
If also $\pderiv{\VM{x}_i}{\omega_j} \in \subs{V}$, then there exists a vector $\VM{z}$ such that $\pderiv{\VM{x}_i}{\omega_j} = \VM{V} \VM{z}$. We prove that $\VM{z} = \pderiv{\VM{x}_i^\subs{V}}{\omega_j}$, i.e. the projection of $\pderiv{\VM{x}_i}{\omega_j}$ on $\subs{V}$ equals the derivative of the first eigenvector of the reduced eigenvalueproblem.
This derivative is uniquely determined by the system in \eqref{eqn:system_dx}, from which it follows that
$$
\left \{\begin{matrix}
\left(\lambda_i\VM{B} - \VM{A}\right) \VM{V} \VM{z} + \pderiv{\lambda^\subs{V}_i}{\omega_j} \VM{B} \VM{V} \VM{x}_i^\subs{V}& = & \left( \pderiv{\VM{A}}{\omega_j} - \lambda^\subs{V}_i \pderiv{\VM{B}}{\omega_j} \right) \VM{V} \VM{x}_i^\subs{V} \\
\left(\VM{x}_i^\subs{V}\right)^T \VM{V}^T \VM{B} \VM{z} & = & - \dfrac{ \left( \VM{x}_i, \VM{V}^T \pderiv{\VM{B}}{\omega_j} \VM{V} \VM{x}_i\right)}{2}
\end{matrix} \right.
$$
which means that $[\VM{z}, \pderiv{\lambda^\subs{V}_i}{\omega_j}]^T$ is the vector such that
$$
\left \{\begin{matrix}
\VM{V}^T \left(\lambda_i \VM{B} - \VM{A} \right) \VM{V} \VM{z} + \pderiv{\lambda^\subs{V}_i}{\omega_j} \VM{V}^T \VM{B} \VM{V} \VM{x}_i^\subs{V}& = & \VM{V}^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda^\subs{V}_i \pderiv{\VM{B}}{\omega_j} \right) \VM{V} \VM{x}_i^\subs{V} \\
\left(\VM{x}_i^\subs{V}\right)^T \VM{V}^T \VM{B} \VM{V} \VM{z} & = & - \dfrac{ \left( \VM{x}_i, \VM{V}^T \pderiv{\VM{B}}{\omega_j} \VM{V} \VM{x}_i\right)}{2}
\end{matrix} \right. .
$$
This is the system that uniquely determines the derivative of the first eigenvector of the reduced eigenvalue problem, so $\VM{z} = \pderiv{\VM{x}_i^\subs{V}}{\omega_j}$.
From equation \eqref{eqn:ddlambda}, we obtain
\begin{align*}
\pderiv{\lambda_i}{\omega_j, \omega_k} & = 2 \left( \VM{x}_i^\subs{V}\right)^T \VM{V}^T \left( \pderiv{ \VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \VM{V} \pderiv{\VM{x}_i}{\omega_k}^\subs{V} + \hdots \\
& \quad \left( \VM{x}_i^\subs{V}\right)^T \VM{V}^T \left( \pderiv[1]{\VM{A}}{\omega_j, \omega_k} - \pderiv{\lambda_i}{\omega_k} \pderiv{\VM{B}}{\omega_j} - \lambda_i \pderiv[1]{\VM{B}}{\omega_j, \omega_k} \right) \VM{V} \VM{x}_i^\subs{V}. \\
& = 2 \left( \VM{x}_i^\subs{V}\right)^T \left( \VM{V}^T \pderiv{ \VM{A}}{\omega_j} \VM{V} - \lambda_i^\subs{V} \VM{V}^T \pderiv{\VM{B}}{\omega_j} \VM{V} \right) \pderiv{\VM{x}_i^\subs{V}}{\omega_k} + \hdots \\
& \quad \left( \VM{x}_i^\subs{V}\right)^T \left( \VM{V}^T \pderiv[1]{\VM{A}}{\omega_j, \omega_k} \VM{V} - \pderiv{\lambda_i^\subs{V}}{\omega_k} \VM{V}^T \pderiv{\VM{B}}{\omega_j}\VM{V} - \lambda_i^\subs{V} \VM{V}^T \pderiv[1]{\VM{B}}{\omega_j, \omega_k} \VM{V} \right) \VM{x}_i^\subs{V} \\
& = \pderiv{\lambda_i^\subs{V}}{\omega_j, \omega_k}.
\end{align*}
\underline{Assertion 2:} This results is well known, we refer to \cite{Parlett1998} for a proof.
\end{proof}
\end{property}
\section{Characterisation of the derivative of an eigenvector}
The previous section showed that adding the derivative of the eigenvector to the subspace leads to higher-order Hermite interpolation in the eigenvalue. The aim of this section is to get more insight in the behaviour of eigenvalues as a function of the parameters by deriving an analytic formula for the derivative of the eigenvector. The next result can also be found in \cite{seyranian2003multiparameter}, but we give here our proof, which is based on the diagonalisation of $\lambda_1 B-A$ in \eqref{eqn:system_dx}, to show the connection with Theorem \ref{ste:system_deriv}.
\begin{theorem} \label{ste:dXdw}
If $\VM{x}_i$ is an eigenvector associated with a simple eigenvalue $\lambda_i$, then we have
\begin{equation} \label{eqn:dxdomega}
\pderiv{\VM{x}_i}{\omega_j} = - \dfrac{\VM{x}_i^T \pderiv{\VM{B}}{\omega_j} \VM{x}_i }{2} \VM{x}_i + \sum_{k=1, k \neq i}^n \left( \VM{x}_k^T \dfrac{ \left( \pderiv{\VM{A}}{\omega_j} - \lambda_i \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_i}{\lambda_i - \lambda_k} \right) \VM{x}_k, \quad j = 1, \hdots, d.
\end{equation}
\begin{proof}
Without loss of generality, we prove the statement for $i = 1$.
From \eqref{eqn:SVD_A} it follows immediately that system \eqref{eqn:system_dx} can be written as
$$\begin{bmatrix}[c|c]
\lambda_1 \left(\VM{X}^{-1} \right)^T \VM{X}^{-1} - \left(\VM{X}^{-1} \right)^T \VM{\Lambda} \VM{X}^{-1} & \left(\VM{X}^{-1} \right)^T \VM{e}_1 \\ \hline
\VM{e}_1^T \VM{X}^{-1} & 0
\end{bmatrix} \begin{bmatrix} \pderiv{\VM{x}_1}{\omega_j} \\ \pderiv{\lambda_1}{\omega_j} \end{bmatrix} = \begin{bmatrix}
\left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right ) \VM{x}_1 \\ \hline
- \dfrac{ \VM{x}_1^T \pderiv{\VM{B}}{\omega_j} \VM{x}_1}{2}
\end{bmatrix}. $$
Using $\VM{X} \VM{X}^T \VM{B} = \VM{I} $, we obtain
$$\begin{bmatrix}[c|c]
\lambda_1 \left(\VM{X}^{-1} \right)^T - \left(\VM{X}^{-1} \right)^T \VM{\Lambda} & \left(\VM{X}^{-1} \right)^T \VM{e}_1 \\ \hline
\VM{e}_1^T & 0
\end{bmatrix} \begin{bmatrix} \VM{X} ^T \VM{B} \pderiv{\VM{x}_1}{\omega_j} \\ \pderiv{\lambda_1}{\omega_j} \end{bmatrix} = \begin{bmatrix}
\left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right ) \VM{x}_1 \\ \hline
- \dfrac{ \VM{x}_1^T \pderiv{\VM{B}}{\omega_j} \VM{x}_1}{2}
\end{bmatrix}. $$
By multiplying from the left with the non-singular matrix $\begin{bmatrix}
\VM{X}^T & \VM{0} \\ 0 & 1
\end{bmatrix}$ we get
$$\begin{bmatrix}[c|c]
\lambda_1 \VM{I} - \VM{\Lambda} & \VM{e}_1 \\ \hline \VM{e}_1^T & 0
\end{bmatrix} \begin{bmatrix} \VM{X} ^T \VM{B} \pderiv{\VM{x}_1}{\omega_i} \\ \pderiv{\lambda_1}{\omega_j} \end{bmatrix} = \begin{bmatrix}
\VM{X}^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_1 \\ \hline
- \dfrac{ \VM{x}_1^T \pderiv{\VM{B}}{\omega_j} \VM{x}_1}{2}\end{bmatrix}.
$$
From this expression it follows that
\begin{align*}
\pderiv{\lambda_1}{\omega_j} & = \VM{x}_1^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_1 \\
\VM{x}_k^T \VM{B} \pderiv{\VM{x}_1}{\omega_j} & = \left\{\begin{matrix}
- \dfrac{ \VM{x}_k^T \pderiv{\VM{B}}{\omega_j} \VM{x}_1}{2} & ,k = 1\\
\dfrac{\VM{x}_k^T \left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_1}{\lambda_1 - \lambda_k} & ,k \neq 1.
\end{matrix}\right.
\end{align*}
Finally, multiplying with $\VM{X}$ from the left proves the statement.
\end{proof}
\end{theorem}
We observe that the weight of $\VM{x}_k, k = 2, \hdots, n$ in the expression for $\pderiv{\VM{x}_1}{\omega_j}$ depends on how close $\lambda_k$ is to $\lambda_1$ and depends on the projection of $\left( \pderiv{\VM{A}}{\omega_j} - \lambda_1 \pderiv{\VM{B}}{\omega_j} \right) \VM{x}_1, j = 1, \hdots,d$ on $\VM{x}_k$. In general, the closer $\lambda_k$ is to $\lambda_1$ the higher is the impact of $\VM{x}_k$. We also deduce that if $\VM{B}$ does not depend on the variables then the partial derivative is $\VM{B}$-orthonormal to $\VM{x}_1$.
Example \ref{ex:deriv} shows that the norm of the partial derivative of the eigenvector cannot be bounded uniformly on the set $\Omega$, despite $\VM{A}$ and $\VM{B}$ being analytic and $\VM{B}$ positive definite for $\VM{\omega} \in \Omega$.
\begin{eexample} \label{ex:deriv}
We reconsider example \ref{vbd:1} with $n = 2$ and $\VM{W} = \VM{I}_2$. In Figure \ref{fig:example_3}, we plot the eigenvalues $\lambda_1(\VM{\omega})$ and $\lambda_2(\VM{\omega})$ in (a) and (b) and the norm of $\pderiv{\VM{x}_1}{\omega_2}$ in (c) and (d) as a function of $\omega_2$, for fixed values of $\omega_1$.
\begin{figure}[h]
\setlength{\figW}{6cm}
\begin{subfigure}[b]{0.5\textwidth}
\hspace{0.8cm}
\input{images/eigw_kegel_0.5.tex}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\hspace{0.7cm}
\input{images/eigw_kegel_0.05.tex}
\caption{}
\end{subfigure} \\
\begin{subfigure}[b]{0.5\textwidth}
\input{images/eigv_norm_kegel_0.5.tex}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\input{images/eigv_norm_kegel_0.05.tex}
\caption{}
\end{subfigure}
\caption{(a) and (b): $\lambda_1(\VM{\omega})$ and $\lambda_2(\VM{\omega})$ with $\omega_1 = 0.5$ (left) and $\omega_1 = 0.05$ (right), (c) and (d): norm of $\pderiv{\VM{x}_1(\VM{\omega})}{\omega_2}$ with $\omega_1= 0.5$ (left) and $\omega_1= 0.05$ (right)} \label{fig:example_3}
\end{figure}
One can observe that for $\omega_2 = 0$, the norm of the partial derivative to $\omega_2$ becomes larger if $\omega_1$ goes to 0. It can be shown
by Theorem \ref{ste:dXdw} that this norm is unbounded. For $\omega_2 = 0$, the eigenvectors are equal to $$\VM{x}_1(\omega_1, 0) = \begin{bmatrix}
0 \\ 1
\end{bmatrix} \mbox{ and } \VM{x}_2(\omega_1, 0) = \begin{bmatrix}
1 \\ 0
\end{bmatrix}$$
from which we can deduce that
\begin{align*}
\dfrac{ \partial \VM{x}_1(\VM{\omega})}{\partial \omega_2} & = \VM{x}_2(\omega_1, 0)\dfrac{ \left( \pderiv{\VM{A}(\omega_1,0) }{\omega_2} \VM{x}_1(\omega_1, 0), \VM{x}_2(\omega_1, 0)\right)}{\lambda_1(\omega_1,0) - \lambda_2(\omega_1,0)} \\
& = \VM{x}_2(\omega_1, 0)\dfrac{ \left( \begin{bmatrix}
0 & 1\\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix}
1 \\ 0 \end{bmatrix} \right)}{\lambda_1(\omega_1,0) - \lambda_2(\omega_1,0)} \\
& = \VM{x}_2(\omega_1, 0)\dfrac{ 1}{-2 \omega_1}. \\
\end{align*}
We conclude that if $\omega_1$ goes to $0$, the norm of the partial derivative goes to infinity.
\end{eexample}
Although the analytic formula for the partial derivative gives much insight how eigenvectors depend on the other eigenvectors and eigenvalues, it is less suitable to compute derivatives. The reason is that we need all eigenvectors and eigenvalues and that in practice it is far too costly to calculate these. In practice we calculate the partial derivatives by solving system \eqref{eqn:system_dx}. Note that the cost of solving \eqref{eqn:system_dx} for all parameters is not negligible, since it depends on the dimension and the sparsity pattern of the matrices. Therefore we will propose an alternative construction of the subspace that adds approximations to the derivatives in the next section.
\section{Computing a global approximation} \label{sect:glob_approx}
\subsection{Subspace expansion}
Property \ref{prop:interpol} presents the key theory for the construction of the reduced problem: adding eigenvectors and partial derivatives leads to Hermite interpolation of degree two for the eigenvalue. The eigenvalue and associated eigenvector are computed by a subspace projection method. The computation of the derivatives of the eigenvector from \eqref{eqn:system_dx} is usually computationally expensive. Therefore we look for cheaper alternatives. Eq. \eqref{eqn:dxdomega} suggests that the derivatives mainly depend on the eigenvectors associated with eigenvalues closest to $\lambda_1$.In particular, this is the case when the spectrum is well-separated with only a few eigenvalues near $\lambda_1$. This implies that adding a few eigenvectors to the subspace in an interpolation point may serve as a computationally cheap surrogate for adding one eigenvector and its derivatives. Approximations to the associated eigenvectors of the eigenvalues nearest $\lambda_1$ are usually available from subspace methods. Indeed, when we use the Lanczos' method, the subspace produces reasonably good approximations to these eigenvectors, at no additional cost. Moreover, it should be noted that there is another reason why adding the second and third eigenvector may be beneficial: the eigenvector of the 2nd minimal eigenvalue may be a good approximation to the eigenvector associated with the minimal eigenvalue at another parameter value $\VM{\omega}_1 \in \Omega$. This means we probably do not need to solve a costly eigenvalue problem in $\VM{\omega}_1$. Example \ref{ex:vb3} gives a good illustration of such a situation. In addition, we see in the next section that adding the second eigenvector helps making a sharp upper bound for the error.
\begin{eexample} \label{ex:vb3}
We reconsider example \ref{vbd:1} where we take $n$ arbitrarily large.
If $\omega_2 = 0$, then one can verify that
$$\VM{x}_1(\omega_1,0) = \left\{\begin{matrix}
\VM{w}_1(\omega_1,0) & ,\omega_1 \leq 0\\
\VM{w}_2(\omega_1,0) & ,\omega_1 > 0
\end{matrix}\right.,
\quad
\VM{x}_2(\omega_1,0) = \left\{\begin{matrix}
\VM{w}_2(\omega_1,0) & ,\omega_1 \leq 0\\
\VM{w}_1(\omega_1,0) & ,\omega_1 > 0
\end{matrix}\right. .$$ Take $\omega^1 < 0$ and $\omega^2 > 0$ two points in the first dimension of $\Omega$. If both $\VM{w}_1$ and $\VM{w}_2$ do not strongly depend on $\VM{\omega}$, this means that $\VM{x}_1(\omega^1,0) \approx \VM{x}_2(\omega^2,0)$. Therefore we conclude that $\subs{V}_1 :=\text{span} \{ \VM{x}_1(\omega^i,0), i = 1,2 \} \approx \text{span} \{ \VM{x}_i(\omega^1,0), i = 1,2 \} =: \subs{V}_2$, but $\subs{V}_2$ is less costly to calculate than $\subs{V}_1$. This is illustrated in Figure \ref{fig:ex4}.
\begin{figure}[h]
\begin{center}
\input{images/eigw_kegel_0.tex}
\caption{Illustration corresponding to Example \ref{ex:vb3}. If both $\VM{w}_1$ and $\VM{w}_2$ do not depend much on $\omega_1$, then $\subs{V}_1 :=\text{span} \{ \VM{x}_1(\omega^i), i = 1,2 \} \approx \text{span} \{ \VM{x}_i(\omega^1), i = 1,2 \} =: \subs{V}_2$ but $\subs{V}_2$ is less costly to calculate than $\subs{V}_1$.} \label{fig:ex4}
\end{center}
\end{figure}
\end{eexample}
\subsection{Upper bound of the error} \label{upperb_error}
Before we can state the algorithm, we need a measure to qualify if the subspace is rich enough or not.
The subspace is rich enough if for every point in the parameterspace the difference between the minimal eigenvalue of the reduced problem and the minimal eigenvalue of the original large problem is smaller than a given tolerance. This means we need a way to measure the error for any given $\VM{\omega} \in \Omega$. An upper bound can be obtained only using the first eigenpair via an extension of the Bauer-Fike theorem (see \cite{Bauer1960} and more recently \cite{Saad2011}) to generalized eigenvalue problems. For this, we define the residual of an approximate eigenpair $\left( \hat{\lambda}, \VM{\hat{x}} \right)$ as $$\VM{r} = \VM{A} \VM{\hat{x}} - \hat{\lambda} \VM{B}\VM{\hat{x}}.$$
The proof of this extension is omitted as it is completely analogous to the original proof for the standard case.
\begin{theorem} \label{ste:Bauer_Fike} (Bauer-Fike for generalized eigenvalue problems) Let $\left( \tilde{\lambda}, \tilde{\VM{x}} \right)$ be an approximate eigenpair of $\left( \VM{A}, \VM{B}\right)$ where $\tilde{\VM{x}}$ is of $\VM{B}$-norm unity and $\tilde{\lambda} = \left( \VM{A} \tilde{\VM{x}}, \tilde{\VM{x}} \right)$. Then, there exists an eigenvalue $\lambda$ of $\left( \VM{A}, \VM{B} \right)$ such that
\begin{equation} \label{eqn:Ext_Bauer_Fike} \mid { \tilde{\lambda} - \lambda} \mid \leq \dfrac{ \left \| \VM{r} \right \|_2 }{ \sqrt{\lambda_1(\VM{B})}}.
\end{equation}
\end{theorem}
The residual norm is very cheap to calculate for a set of parameter samples if the dependency of $\VM{A}$ and $\VM{B}$ is affine. More precisely, if $\VM{A}(\VM{\omega})$ and $\VM{B}(\VM{\omega})$ can be written as $\VM{A}(\VM{\omega}) = \sum_{i=1}^{m_0} \theta_{A,i}(\VM{\omega}) \VM{A}_i$ resp. $\VM{B}(\VM{\omega}) = \sum_{i=1}^{m_1} \theta_{B,i}(\VM{\omega}) \VM{B}_i$ with $\theta_{A,i}, \theta_{B,j}, i= 1, \hdots m_0, j = 1, \hdots, m_1$ scalar functions then the residual is in this case
\begin{align*}
\VM{r}(\VM{\omega}) & = \VM{A}(\VM{\omega}) \tilde{\VM{x}}(\VM{\omega}) - \tilde{\lambda}(\VM{\omega}) \VM{B}(\VM{\omega}) \tilde{\VM{x}}(\VM{\omega}) \\
& = \sum_{i=1}^{m_0} \theta_{A,i}(\VM{\omega}) \left( \VM{A}_i \VM{V}\right) \VM{x}^\subs{V}(\VM{\omega}) - \tilde{\lambda}(\VM{\omega}) \sum_{i=1}^{m_1} \theta_{B,i}(\VM{\omega}) \left( \VM{B}_i \VM{V}\right) \VM{x}^\subs{V}(\VM{\omega}).
\end{align*}
This means that the precomputation of $\VM{A}_i, i = 1, \hdots, m_0$ and $\VM{B}_i, i = 1, \hdots, m_1$ reduces the cost of calculating the residual norm.
The drawback of \eqref{eqn:Ext_Bauer_Fike} is that this is usually a very crude bound. If we have also information about the second minimal eigenvalue, we can use the Kato-Temple theorem \cite{Kato1949}, \cite{Temple1952} for a sharper bound. We rephrase the theorem for the generalised eigenvalue problem.
\begin{theorem} \label{ste:KatoTemple}
(Kato-Temple for generalized eigenvalue problems) Let $\left( \tilde{\lambda}, \tilde{\VM{x}} \right)$ be an approximate eigenpair of $\left( \VM{A}, \VM{B}\right)$ where $\tilde{\VM{x}}$ is of $\VM{B}$-norm unity and $\tilde{\lambda} = \left( \VM{A} \tilde{\VM{x}}, \tilde{\VM{x}} \right)$. Assume that we know an interval $]\alpha, \beta[$ that contains $\tilde{\lambda}$ and one eigenvalue $\lambda$ of $\left( \VM{A}, \VM{B} \right)$. Then it holds
$$ - \dfrac{ \norm{\VM{r}}_2^2}{\lambda_1(\VM{B}) \left( \tilde{\lambda} - \alpha \right)} \leq \tilde{\lambda}-\lambda \leq \dfrac{ \norm{\VM{r}}_2^2}{\lambda_1(\VM{B})\left( \beta - \tilde{\lambda} \right)}.$$
\end{theorem}
If we now define $\delta$ as the distance to the second minimal eigenvalue, then we get
\begin{equation} |\tilde{\lambda} - \lambda_1 | \leq \dfrac{ \norm{\VM{r}}_2^2}{\lambda_1(\VM{B})\delta},\label{eqn:KatoTemple} \end{equation}
which is in general a sharper upper bound then the one from Theorem \ref{ste:Bauer_Fike}.
In the Bauer-Fike as well as in the Kato-Temple theorem the bound depends on the first eigenvalue of $\VM{B}$. As it is expensive to calculate this eigenvalue for every $\VM{\omega}$, we only calculate it for one parametervalue and use it for the whole domain. As already mentioned, we approximate the second, respectively third eigenvalue by the second, respectively third Ritz value from the Krylov subspace. Whether we use the Bauer-Fike theorem or the Temple-Kato theorem depends on how accurately we can determine the gap $\delta$ between the first two eigenvalues. If we add the first m eigenvectors $(m > 1)$ corresponding to every sample point to the subspace, we interpolate the first m eigenvalues in the sample points, and obtain in this way a global approximation of the first m eigenvalues over the parameter space. Hence we expect that the gap $\delta$ can be estimated well from the projected eigenvalue problem, and we make in this case use of the Kato-Temple theorem, where we replace $\delta$ by its estimate. If only one eigenvector is added in a sample point, an accurate estimate of the gap from the projected eigenvalue problem cannot be guaranteed. In the latter case the Bauer-Fike theorem is invoked instead.
\subsection{Algorithm} \label{sect:algo}
We now have all ingredients for an algorithm to compute the reduced problem. A remaining crucial element is the choice of sample points, which we now discuss. The proposed method is inspired by the reduced basis method for solving parametric
PDEs, see \cite{Quarteroni2016}.
We assume that the parameter space $\Omega$ is a Cartesian product of intervals in $\mathbb{R}$, so $$\Omega = \Omega_1 \times \Omega_2 \times \hdots \times \Omega_d = [a^1, b^1] \times [a^2, b^2] \times \hdots \times [a^d, b^d].$$ We first make an initial subspace $\subs{V}$ constructed with the first eigenvector(s) and the partial derivatives for all sample points in an initial set $\Omega_\text{init}$ (see line 1-4 in Algorithm \ref{algo:Redbasis}) and we check then if this constructed subspace is large enough by testing the upper bound on some training set $\Omega_\text{train}$. Whether we use the Kato-Temple theorem \ref{ste:KatoTemple} or the Bauer-Fike theorem \ref{ste:Bauer_Fike} depends on the number of eigenvectors we add per interpolation point, see previous section.
If the upper bound at all points in the training set $\Omega_\text{train}$ is not below a certain tolerance, we add the first eigenvectors and partial derivatives of the point in the training set where the upper bound is maximal (see line 13-22 in Algorithm \ref{algo:Redbasis}).
Once the upper bound is below the tolerance, these sample points are removed from the training set (see line 7-10 in Algorithm \ref{algo:Redbasis}). We are allowed to do this because this implies that the error on the eigenvalue is below the tolerance and furthermore the error cannot increase by the second assertion in Theorem \ref{prop:interpol} .
The initial set of sample points is decomposed as
\begin{equation} \label{eqn:algo_init2}
\Omega_\text{init} := \Omega^\text{init}_1 \times \Omega^\text{init}_2 \times \hdots \times \Omega^\text{init}_d
\end{equation}
where \begin{equation} \label{eqn:algo_init1}
\Omega_i^\text{init} = \{ a_i + (j-1) h_i | j = 1, 2, \hdots, n_i\}, h_i = \dfrac{b^i - a^i}{n_i-1}, i = 1, 2, \hdots, d.
\end{equation}
This is an initial set of $n_\text{init} = n_1 n_2 \hdots n_d$ points. The training set $\Omega_\text{train}$ is composed in a similar way but using a much finer grid. The algorithm is stated in pseudocode in Algorithm \ref{algo:Redbasis}. As we do experiments with adding partial derivatives to the subspace and with not adding them to the subspace, we state 'optionally' after the lines where we calculate the partial derivatives.
\begin{algorithm}[h]
\caption{Aim: Calculating a subspace$ \subs{V}$ such that $ | \lambda^\subs{V}_1(\VM{\omega}) - \lambda_1(\VM{\omega})| < \text{tol}, \forall \VM{\omega} \in \Omega$} \label{algo:Redbasis}
\hspace*{\algorithmicindent} \textbf{Input:} \begin{enumerate}
\item Matrix $\VM{A}(\VM{\omega})$ and $\VM{B}(\VM{\omega})$
\item All partial derivatives $\pderiv{\VM{A}(\VM{\omega})}{\omega_i}$ and $\pderiv{\VM{B}(\VM{\omega})}{\omega_i}, i = 1,2,\hdots,d$
\item Choose an initial set $\Omega_\text{init}$ and a training set $\Omega_\text{train}$ of sample points.
\item $n_\text{max}$ maximal number of iterations
\end{enumerate}
\hspace*{\algorithmicindent} \textbf{Output:} $\lambda^\subs{V}_1(\VM{\omega})$ such that $\max_{\VM{\omega} \in \Omega_\text{train}} | \lambda^\subs{V}_1(\VM{\omega}) - \lambda_1(\VM{\omega})| < \text{tol} $
\begin{algorithmic}[1]
\State Calculate $\left( \lambda_1(\VM{\omega}), \VM{x}_1(\VM{\omega}) \right), \VM{\omega} \in \Omega_\text{init}$
\State Estimate $\left( \tilde{\lambda}_j(\VM{\omega}), \tilde{\VM{x}}_j(\VM{\omega}) \right), j = 2, 3, \hdots, m, \VM{\omega} \in \Omega_\text{init}$
\State Calculate partial derivatives $\pderiv{\VM{x}_1(\VM{\omega})}{\omega_j}, j = 1, 2, \hdots, d, \VM{\omega} \in \Omega_\text{init}$ from system \eqref{eqn:system_dx} if $|\lambda_1 - \tilde{\lambda}_2|> 10^{-8}$ (If $\lambda_1$ is simple) (Optionally)
\State $ \subs{V}:= \text{span}\{ \VM{x}_1(\VM{\omega}), \tilde{\VM{x}}_2(\VM{\omega}) \hdots, \tilde{\VM{x}}_m(\VM{\omega}), \pderiv{\VM{x}_1(\VM{\omega})}{\omega_1}, \pderiv{\VM{x}_1(\VM{\omega})}{\omega_2}, \hdots, \pderiv{\VM{x}_1(\VM{\omega})}{\omega_d} | \VM{\omega} \in \Omega_\text{init} \} $
\State $u_\text{old}(\VM{\omega}) = 1, \VM{\omega} \in \Omega_\text{train}$
\For{$i=1, 2, \hdots, n_\text{max}$}
\State \underline{Update the training set}
\State Calculate new upper bound $u_\text{new}(\VM{\omega})$ for all $\VM{\omega} \in \Omega_\text{train} $ using Theorem \ref{ste:Bauer_Fike} if $m = 1$ or Theorem \ref{ste:KatoTemple} if $m > 1$
\State Upper bound $u(\VM{\omega}) = \min( u_\text{old}(\VM{\omega}), u_\text{new}(\VM{\omega}))$
\State $\Omega_\text{valid} = \{ \VM{\omega} | \VM{\omega} \in \Omega_\text{train} \text{ and } u(\VM{\omega}) < \text{tol} \}$
\State $\Omega_\text{train} = \Omega_\text{train} \setminus \Omega_\text{valid}$
\If{$\Omega_\text{train}$ empty}
\State break
\Else
\State \underline{Adding vectors to the subspace}
\State $\VM{\omega}^i = \text{arg} \max_{\VM{\omega} \in \Omega_\text{train}} u(\VM{\omega}) $
\State Calculate $\left( \lambda_1(\VM{\omega}^i), \VM{x}_1(\VM{\omega}^i) \right)$
\State Estimate $\left( \tilde{\lambda}_j(\VM{\omega}^i), \tilde{\VM{x}}_j(\VM{\omega}^i \right), j = 2,3, \hdots, m$
\If{$|\lambda_1 - \tilde{\lambda}_2|> 10^{-8}$} (Check if $\lambda_1$ is simple)
\State Calculate partial derivatives $\pderiv{\VM{x}_1(\VM{\omega}^i)}{\omega_j}, j = 1,2, \hdots, d$ from system \eqref{eqn:system_dx} (optionally) \\
\State $\subs{V} = \subs{V} \bigcup \{ \VM{x}_1(\VM{\omega}^i), \tilde{\VM{x}}_2(\VM{\omega}^i) \hdots, \tilde{\VM{x}}_m(\VM{\omega}^i), \} \left( \bigcup \{ \pderiv{\VM{x}_1(\VM{\omega}^i)}{\omega_1}, \pderiv{\VM{x}_1(\VM{\omega}^i)}{\omega_2}, \hdots, \pderiv{\VM{x}_1(\VM{\omega}^i)}{\omega_d} \} \text{(optionally)} \right)$
\Else
\State $\subs{V} = \subs{V} \bigcup \{ \VM{x}_1(\VM{\omega}^i), \tilde{\VM{x}}_2(\VM{\omega}^i) \hdots, \tilde{\VM{x}}_m(\VM{\omega}^i) \}$
\EndIf
\State Make basis for $\subs{V}$
\State $\Omega_\text{train} = \Omega_\text{train} \setminus \VM{\omega}^i$
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Saturation assumption}
Although the training set becomes smaller in every iteration, calculating the upper bound for all sample points in the training set can become expensive if many iterations are needed. Instead of updating all upper bounds, we select only those that potentially generate the highest upper bound for the error.
We therefore make use of the so-called saturation assumption, see \cite{Hesthaven2014}. Let $u^k$ be the upper bound at iteration $k$. The saturation assumption says that there exists a $C > 0$ such that
\begin{equation} \label{eqn:satur}
u^l(\VM{\omega}) < C u^k(\VM{\omega}), \forall l > k, \forall \VM{\omega} \in \Omega.
\end{equation}
In our method the saturation assumption is fulfilled for $C = 1$ as we only update the upper bound $u^k$ if the upper bound has decreased. We use this in the following way.
We first sort the parametervalues in $\Omega_\text{train}$ by the upper bound of the previous iteration in descending order. We recompute the upper bound and update after each computation the maximal upper bound $u_\text{max}$. If for a certain $\VM{\omega}$, it holds that $u^k(\VM{\omega}) < u_\text{max}$ we are allowed to skip all the next sample points in the sequence by \eqref{eqn:satur}. The next $\VM{\omega}$ for which we add the eigenvectors and all partial derivatives, is the $\VM{\omega}$ which has the upper bound $u_\text{max}$.
\section{Implementation details and numerical results}
All algorithms written in the previous sections are implemented in Matlab version R2017a. All experiments are performed on an Intel i5-6300U with 2.5 GHz and 8 GB RAM. In all examples, the linear systems in building the Krylov space and the systems of the form \eqref{eqn:system_dx} to compute the derivative of an eigenvector could still be solved efficiently using a direct method. Practically, because the used matrices are sparse, it is efficient to use the backslash-operator in Matlab for solving the systems of the form \eqref{eqn:system_dx}. We use the \texttt{eigs}-command in Matlab to calculate the minimal eigenvalue which uses the shift-and-invert Arnoldi's method with shift 0. This method makes a Krylov subspace and we adapt the code such that it also returns this space. We use the other vectors in the Krylov subspace to approximate the second eigenvector. In all examples a tolerance for the error of $10^{-5}$ is used.
We applied our algorithm to examples where the minimal eigenvalue corresponds to calculating the coercivity constant when solving PDEs \cite{Sirkovic2016} and to one example from structural mechanics. We compare the results from four cases: adding only the first eigenvector, adding the first two eigenvectors, adding the first eigenvector and all partial derivatives and finally the first two eigenvectors and all partial derivatives. In the next tables, we put the dimension of the found subspace, the number of sample points, the total computing time and the average time we needed to compute the partial derivatives and eigenvectors. \\
For Example \ref{vbd:2}, the initial set is built by discretising both dimensions of the parameterspace into $4$ to get an initial subset of $16$ points. For the training set, we discretise the first respectively the second dimension of the parameterspace into $25$ respectively $40$ points, to have a total of $1000$ points in our training set. The results can be found in Table \ref{tbl:vbd2}. We see that adding other information than only the first eigenvector is beneficial. The number of points where we need to calculate the second eigenvector and the partial derivative is much lower which results in a smaller computational time. The main benefit from adding the second eigenvector is that we can use the more strict upper bound of Kato-Temple. We also noticed that the time needed to compute the derivatives is much lower than calculating the eigenvalues. We see we get the best results when we add both the eigenvectors and the partial derivatives into the subspace.
\begin{table}[h]
\input{tables/results_ex4.tex}
\end{table}
The next example is also taken from \cite[Example 4.3]{Sirkovic2016}. The parameterspace is $\Omega = [-0.1, 0.1] \times [0.2, 0.3]$ and the matrix $\VM{A}$ is of the form $\VM{A}(\VM{\omega}) = \sum_{i=1}^{16} \theta_i(\VM{\omega}) \VM{A}_i$ with $\theta_i(\VM{\omega})$ analytic functions and matrix $\VM{B}$ is constant. The dimension of the problem is $2183$. As training set we choose a grid where we discretise the interval for $\omega_1$ into $40$ and $\omega_2$ into $25$ points. As initial set we discretise both intervals into 3.
The results are stated in Table \ref{tbl:coerc_1}. We see that in this case the needed subspace is low-dimensional which results in a small computational time. Similar conclusions can be drawn for this example.
\begin{table}[h]
\input{tables/results_ex3.tex}
\end{table}
In the previous examples matrix $\VM{B}$ was constant, since the article from which we took the examples considered only eigenvalue problems with constant $\VM{B}$. The next example originates from an application in structural mechanics and here matrix $\VM{B}$ depends on $\VM{\omega}$. \\
The matrices in this example are the stiffness and mass matrices arising from a finite element approximation of a vibrating beam that can deflect in both directions perpendicular to its own axis. The first parameter concerns the size of the beam along one of the axis (in interval $[0.1,1]$) and the second parameter deals with the stiffness and density of the material (in interval $[100, 1000]$). For the initial set, we discretise both dimensions into three. The matrices have size $1404$ and they both depend on at least one of the parameters. In this problem ,the stiffness and mass matrix both dependent on the two parameters. One is usually interested in the minimal eigenfrequencies which boils down to calculating the minimal eigenvalue. In Figure \ref{vb:bouwkunde} we depict the minimal eigenvalue over the parameterspace and Table \ref{tbl:bouwkunde} summarizes the results. We observe that only a small subspace was needed and that we benefit from including partial derivatives in the subspace since the computation of the eigenvectors takes much more time than solving the system to compute the partial derivatives.
\begin{figure}[h]
\begin{center}
\setlength{\figW}{5cm}
\input{images/example_bouwk1.tex}
\caption{The minimal eigenvalue $\lambda_1(\VM{\omega})$ over the parameter for the example from structural mechanics. We see the minimal eigenvalue does not smoothly depend on the variable. } \label{vb:bouwkunde}
\end{center}
\end{figure}
\begin{table}[h]
\input{tables/results_bouwkunde.tex}
\end{table}
\section{Conclusion}
The main contribution of this paper is that we showed that for calculating extreme eigenvalues of eigenpairs $(\VM{A}, \VM{B})$ where both matrices are symmetric and $\VM{B}$ is positive definite, it is beneficial to add, besides the first eigenvector, also its partial derivatives, or alternatively, the Ritz vector associated with the second and third Ritz values. We have proved that in the case of the addition of partial derivatives, the eigenvalues satisfy a Hermite interpolation property of order 2. Numerical examples confirm this statement and showed that this property can also be used in practice.
The question arises what we do if the system of linear equations for calculating the partial derivatives cannot be efficiently solved by direct solvers. A possibility is to first estimate the partial derivative from the Krylov subspace $\subs{V}_K$ obtained when calculating the first eigenvector as this space contains approximations of the vectors from which the Ritz-values are the closest to the minimal eigenvalue. Let $\VM{V}_K$ be a basis of this subspace then we first project the system in \eqref{eqn:system_dx} on $\subs{V}_K$ to obtain
\begin{equation} \label{eqn:init_sol_GMRES}
\begin{bmatrix}
\VM{V}_K^T \left( \lambda_1 \VM{B}-\VM{A} \right) \VM{V}_K & \VM{V}_K^T \VM{B} \VM{x}_1 \\
\VM{x}_1^T \VM{B} \VM{V}_K & 0
\end{bmatrix} \begin{bmatrix} \pderiv{\VM{x}_1}{\omega_i}^{\subs{V}_K} \\ \pderiv{\lambda^{\subs{V}_K}_1}{\omega_i} \end{bmatrix} = \begin{bmatrix}
\VM{V}_K^T \left(\pderiv{\VM{A}}{\omega_i} - \lambda_1 \pderiv{\VM{B}}{\omega_i} \right ) \VM{x}_1 \\
- \dfrac{ \VM{x}_1^T \pderiv{\VM{B}}{\omega_i} \VM{x}_1}{2}
\end{bmatrix}
\end{equation}
which has the same dimension as $\subs{V}_K$.
We can use this approximation as a starting vector in an iterative method like GMRES to compute the derivative.
\section*{Acknowledgements}
The authors thank the referees for their helpful remarks.
This work was supported by the project KU Leuven Research Council grant OT/14/074 and C14/17/072 and by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen).
|
2,877,628,089,524 | arxiv | \section{Introduction}
The study of galaxies at high redshifts opens an important window on
the process of galaxy formation and conditions in the early universe.
The detection of populations of galaxies at high redshifts is one of
the great challenges in observational cosmology. Currently three main
observational techniques are used to discover high redshift,
star-forming galaxies: (i) The Lyman-break drop-out technique, in
which a galaxy is imaged in a combination of three or more optical or
near-IR bands. The longer wavelength filters detect emission in the
rest-frame ultraviolet from ongoing star formation, whereas the
shorter wavelength filters sample the Lyman-break feature. Hence, a
Lyman-break galaxy appears blue in one colour and red in the other
\citep{steidel96, steidel99}. By shifting the whole filter set to
longer wavelengths, the Lyman-break feature can be isolated at higher
redshifts; (ii) Sub-millimetre emission, due to dust being heated when
it absorbs starlight \citep{smail97, hughes98}. The bulk of the energy
absorbed by the dust comes from the rest-frame ultra-violet and so the
dust emission is sensitive to the instantaneous star formation rate;
(iii) Ly-$\alpha$ line emission from star forming galaxies, typically
identified using either narrowband imaging \citep{hu98, kudritzki00,
gawiser07, ouchi07} or long-slit spectroscopy of gravitationally
lensed regions \citep{ellis01, santos04, stark07}. The Ly-$\alpha$
emission is driven by the production of Lyman-continuum photons and so
is dependent on the current star formation rate.
The Lyman-break drop-out and sub-millimetre detection methods are more
established than Ly-$\alpha$ emission as a means of identifying
substantial populations of high redshift galaxies. Nevertheless, in
the last few years there have been a number of \mbox{$\rmn{Ly}\alpha$}\ surveys which
have successfully found high redshift galaxies e.g. \citep{hu98,
kudritzki00}. The observational samples have grown in size such that
statistical studies of the properties of \mbox{$\rmn{Ly}\alpha$}\ emitters have now become
possible: for example, the SXDS Survey \citep{ouchi05,ouchi07} has
allowed estimates of the luminosity function (LFs) and clustering of
\mbox{$\rmn{Ly}\alpha$}\ emitters in the redshift range $3<z<6$, and the MUSYC survey
\citep{gronwall07,gawiser07} has also produced clustering measurements
at $z \sim 3$. Furthermore, the highest redshift galaxy ($z=6.96$)
robustly detected to date was found using the \mbox{$\rmn{Ly}\alpha$}\ technique
\citep{iye}. Taking advantage of the magnification of faint sources
by gravitational lensing, \citet{stark07} reported 6 candidates for
\mbox{$\rmn{Ly}\alpha$}\ emitters in the redshift range $8.7<z<10.2$, but these have yet
to be confirmed. The DAzLE Project \citep{horton} is designed to find
\mbox{$\rmn{Ly}\alpha$}\ emitters at $z=7.73$ and $z=8.78$. However, the small field of
view of the instrument ($6.83 \arcmin \times 6.83 \arcmin$) makes it
difficult to use to study large scale structure (LSS) at such
redshifts. On the other hand, the ELVIS Survey \citep{kim07a, kim07b}
would appear to offer a more promising route to study the LSS of very
high redshift galaxies ($z=8.8$).
Despite these observational breakthroughs, predictions of the
properties of star-forming \mbox{$\rmn{Ly}\alpha$}\ emitting galaxies are still in the
relatively early stages of development. Often these calculations
employ crude assumptions about the galaxy formation process to derive
a star formation rate and hence a \mbox{$\rmn{Ly}\alpha$}\ luminosity, or use
hydrodynamical simulations, which, due to the high computational
overhead, study relatively small cosmological volumes. \citet{hai}
made predictions for the escape fraction of \mbox{$\rmn{Ly}\alpha$}\ emission and the
abundance of \mbox{$\rmn{Ly}\alpha$}\ emitters using the Press-Schechter formalism and a
prescription for the dust distribution in galaxies. Radiative transfer
calculations of the escape fraction have been made by \citet{zheng02},
\citet{ahn04} and \citet{verhamme06} for idealized geometries, while
\citet{tasitsiomi06} and \citet{laursen07} applied these calculations
to galaxies taken from cosmological hydrodynamical
simulations. \citet{barton04} and \citet{furlanetto05} calculated the
number density of Ly-$\alpha$ emitters using hydrodynamical
simulations of galaxy formation. Nagamine et~al. (2006, 2008) used
hydrodynamical simulations to predict the abundance and clustering of
\mbox{$\rmn{Ly}\alpha$}\ emitters. The typical computational boxes used in these
calculations are very small ($\sim 10-30 h^{-1}$Mpc), which makes it
impossible to evolve the simulation accurately to $z=0$. Hence, it is
difficult to test if the galaxy formation model adopted produces a
reasonable description of present day galaxies. Furthermore, the small
box size means that reliable clustering predictions can only be
obtained on scales smaller than the typical correlation length of the
galaxy sample. As we will show in this paper, small volumes are
subject to significant fluctuations in clustering amplitude.
The semi-analytical approach to modelling galaxy formation allows us
to make substantial improvements over previous calculations of the
properties of \mbox{$\rmn{Ly}\alpha$}\ emitters. The speed of this technique means that
large populations of galaxies can be followed. The range of
predictions which can be made using semi-analytical models is, in
general, broader than that produced from most hydrodynamical
simulations, so that the model predictions can be compared more
directly with observational results. A key advantage is that the
models can be readily evolved to the present-day, giving us more faith
in the ingredients used; i.e. we can be reassured that the physics
underpinning the predictions presented for a high-redshift population
of galaxies would not result in too many bright/massive galaxies at
the present day.
The first semi-analytical calculation of the properties of \mbox{$\rmn{Ly}\alpha$}\
emitters based on a hierarchical model of galaxy formation was carried
out by \citet{dell1}. This is the model used throughout this work,
which has been shown to be successful in predicting the properties of
\mbox{$\rmn{Ly}\alpha$}\ emitters over a wide range of redshifts. The semi-analytical
model allows us to connect \mbox{$\rmn{Ly}\alpha$}\ emission to other galaxy
properties. \citet{dell2} showed that this model succesfully predicts
the observed \mbox{$\rmn{Ly}\alpha$}\ LFs and equivalent widths (EWs), along with some
fundamental physical properties, such as star formation rates (SFRs),
gas metallicities, and stellar and halo masses. In \citet{kim07a}, we
used the model to make further predictions for the LF of very high
redshift \mbox{$\rmn{Ly}\alpha$}\ emitters and to study the feasibility of current and
forthcoming surveys which aim to detect such high redshift
galaxies. \citet{kobayashi07} developed an independent semi-analytical
model to derive the luminosity functions of \mbox{$\rmn{Ly}\alpha$}\ emitters.
The focus of this paper is to use the model introduced by
\citet{dell1} to study the clustering of high-redshift \mbox{$\rmn{Ly}\alpha$}\ emitting
galaxies and to extend the comparison of model predictions with
current observational data. \citet{dell2} already gave an indirect
prediction of the clustering of \mbox{$\rmn{Ly}\alpha$}\ emitters by studying galaxy bias
as a function of \mbox{$\rmn{Ly}\alpha$}\ luminosity. However, these results depend on an
analytical model for the halo bias \citep{sheth01}, and furthermore
the linear bias assumption breaks down on small scales. Here we will
present an explicit calculation of the clustering of galaxies by
implementing the semi-analytical model on top of a large N-body
simulation of the hierarchical clustering of the dark matter
distribution. This allows us to predict the spatial distribution of
\mbox{$\rmn{Ly}\alpha$}\ emitting galaxies, and to create realistic maps of \mbox{$\rmn{Ly}\alpha$}\
emitters at different redshifts. These maps can be analysed with
simple statistical tools to quantify the spatial distribution and
clustering of galaxies at high redshifts. The N-body simulation used
in this work is the \textit{Millennium Simulation}, carried out by the
Virgo Consortium \citep{mill}. The simulation of the spatial
distribution of \mbox{$\rmn{Ly}\alpha$}\ emitters is tested by creating mock catalogues
for different surveys of \mbox{$\rmn{Ly}\alpha$}\ emitting galaxies in the range $3<z<9$.
The clustering of \mbox{$\rmn{Ly}\alpha$}\ emitters in our model is analysed with
correlation functions and halo occupation distributions. Taking
advantage of the large volume of the Millennium simulation, we also
compute the errors expected on correlation function measurements from
various surveys due to cosmic variance.
The outline of this paper is as follows: Section 2 gives a brief
description of the semi-analytical galaxy formation model and
describes how it is combined with the N-body simulation. In Section 3
we establish the range of validity of our simulated galaxy samples by
studying the completeness fractions in the model \mbox{$\rmn{Ly}\alpha$}\ luminosity
functions. Section 4 gives our predictions for the clustering of
\mbox{$\rmn{Ly}\alpha$}\ emitters in the range $0<z<9$. In Section 5 we compare our
simulation with recent observational data and we also make predictions
for future measurements (clustering and number counts) expected from
the ELVIS Survey. Finally, Section 6 gives our conclusions.
\section{The Model}
We use the semi-analytical model of galaxy formation, \texttt{GALFORM}, to
predict the properties of the \mbox{$\rmn{Ly}\alpha$}\ emission of galaxies and their
abundance as a function of redshift. The \texttt{GALFORM}\ model is fully
described in \citet{cole00} \citep[see also the review by][]{baugh06}
and the variant used here was introduced by \citet{baugh05} \citep[see
also][for a more detailed description]{lacey08}. The model computes
star formation histories for the whole galaxy population, following
the hierarchical evolution of the host dark matter haloes.
The merger histories of dark matter haloes are calculated using a
Monte Carlo method, following the formalism of the extended Press \&
Schechter theory \citep{press74, lacey93}. When using Monte Carlo
merger trees, the mass resolution of dark matter haloes can be
arbitrarily high, since the whole of the computer memory can be
devoted to one tree rather than a population of trees. In contrast,
N-body merger trees are constrained by a finite mass resolution due to
the particle mass, which is usually poorer than that typically adopted
for Monte Carlo merger trees. Discrepancies between the model
predictions obtained with Monte Carlo trees and those extracted
from a simulation only become evident fainter than some luminosity
which is set by the mass resolution of the N-body trees, as we will
see in the next section \citep[see also ][]{helly03}.
\begin{figure*}
\centering
\includegraphics[width=8.5cm]{plots/z0_lores.eps}
\includegraphics[width=8.5cm]{plots/z3.3_lores.eps}
\includegraphics[width=8.5cm]{plots/z5.7_lores.eps}
\includegraphics[width=8.5cm]{plots/z8.5_lores.eps}
\caption{ The spatial distribution of \mbox{$\rmn{Ly}\alpha$}\ emitting galaxies (coloured
circles) in a slice from the Millennium simulation, with the dark
matter distribution in green. The four panels are for redshifts in the
range $0<z<8.5$, as indicated in each panel. The colour of the
circles changes with the \mbox{$\rmn{Ly}\alpha$}\ luminosity of the galaxies, as shown
in the key in the upper-right corner of the first panel. Only galaxies
brighter than $\log(\llya\lunits) = 42.2$ are plotted. Each image
covers a square region $100\times 100 \mbox{$ h^{-1} \rmn{Mpc}$}^2$ across and having a
depth of 10\mbox{$ h^{-1} \rmn{Mpc}$}, which is less than one thousandth the volume of the
full simulation box.}
\label{fig.sample}
\end{figure*}
A critical assumption of the Baugh et al. model is that stars formed
in starbursts have a top-heavy initial mass function (IMF), where the
IMF is given by $\rm{d}N/\rm{d} \ln (m) \propto m^{-x}$ and
$x=0$. Stars formed quiescently in discs have a solar neighbourhood
IMF, with the form proposed by \citet{ken83}: $x=0.4$ for $m < 1
M_\odot$ and $x = 1.5$ for $m> 1 M_\odot$. Both IMFs cover the mass
range $0.15 M_\odot < m < 125 M_\odot$. Within the framework of
$\Lambda$CDM, Baugh et~al. argued that the top-heavy IMF is essential
to match the counts and redshift distribution of galaxies detected
through their sub-millimetre emission, whilst retaining the match to
galaxy properties in the local Universe, such as the optical and
far-IR luminosity functions and galaxy gas fractions and
metallicities. \citet{nagashima05a,nagashima05b} showed that such a
top-heavy IMF also results in predictions for the metal abundances in
the intra-cluster medium and in elliptical galaxies in much better
agreement with observations. \citet{lacey08} showed that the same
model predicts galaxy evolution in the IR in good agreement with
observations from {\em Spitzer}, and also discussed independent
observational evidence for a top-heavy IMF.
Reionization is assumed in our model to occur at $\zreion=10$
\citep{kogut03,dunkley08}. The photoionization of the intergalactic
medium (IGM) is assumed to suppress the collapse and cooling of gas in
haloes with circular velocities $V_c < 60 \rm{km/s}$ at redshifts
$z<\zreion$ \citep{benson02}. Recent calculations
\citep{hoeft06,okamoto08} imply that the above parameter values
overstate the impact of photoionization on gas cooling, and suggest
that photoionization only affects smaller haloes with $V_c \la 30 {\rm
km/s}$. Our model predictions, for the range of \mbox{$\rmn{Ly}\alpha$}\ luminosities we
consider in this paper, are not significantly affected by adopting the
lower $V_c$ cut. We neglect any attenuation of the \mbox{$\rmn{Ly}\alpha$}\ flux by
propagation through the IGM. This effect would suppress the observed
\mbox{$\rmn{Ly}\alpha$}\ flux mainly for $z \ga \zreion$, so this should only affect our
results for very high redshifts ($z \ga 10$).
The model used to predict the luminosities and equivalent widths of
the \mbox{$\rmn{Ly}\alpha$}\ galaxies is identical to that described in \citet{dell1,
dell2}. The \mbox{$\rmn{Ly}\alpha$}\ emission is computed by the following procedure: (i)
The integrated stellar spectrum of the galaxy is calculated, based on
its star formation history, including the effects of the distribution
of stellar metallicities, and taking into account the IMFs adopted for
different modes of star formation. (ii) The rate of production of
Lyman continuum (Lyc) photons is computed by integrating over the
stellar spectrum, and assuming that all of these ionizing photons are
absorbed by neutral hydrogen within the galaxy.
We calculate the fraction of \mbox{$\rmn{Ly}\alpha$}\ photons produced by these
Lyc photons, assuming Case B recombination \citep{osterbrock89}.
(iii) The observed \mbox{$\rmn{Ly}\alpha$}\ flux depends on the fraction of \mbox{$\rmn{Ly}\alpha$}\ photons which escape from
the galaxy ($f_{\rmn{esc}}$), which is assumed to be constant and
independent of galaxy properties.
Calculating the \mbox{$\rmn{Ly}\alpha$}\ escape fraction from first principles by
following the radiative transfer of the \mbox{$\rmn{Ly}\alpha$}\ photons is very
demanding computationally. A more complete calculation of the escape
fraction would have into account the structure and kinematic properties
of the intestellar medium (ISM) \citep{zheng02, ahn04, verhamme06}.
In our model, we adopt the simplest possible approach, which is to
fix the escape fraction, $f_{\rmn{esc}}$, to be the same for each
galaxy, without taking into account its dust properties. This results
in a surprisingly good agreement between the predicted number counts
and luminosity functions of emitters and the available observations at
$3 \la z \la 7$ \citep{dell1,dell2}. \citet{dell1} chose
$f_{\rmn{esc}} = 0.02$ to match the number counts at $z\approx 3$ at a
flux $f \approx 2 \times 10^{-17} {\rm erg cm^{-2} s^{-1}}$. The same
value is used in this work. This value for the \mbox{$\rmn{Ly}\alpha$}\ escape fraction
seems very small, but is consistent with direct observational
estimates for low redshift galaxies: \citet{Atek08} derive escape
fractions for a sample of nearby star-forming galaxies by combining
measurements of Ly$\alpha$, H$\alpha$ and H$\beta$, and find that most
have escape fractions of 3\% or less. \citet{dell2} also showed that
if a standard solar neighbourhhod IMF is adopted for all modes of star
formation, then a substantially larger escape fraction would be
required to match the observed counts of \mbox{$\rmn{Ly}\alpha$}\ emitters, and even then
the overall match would not be as quite good as it is when the
top-heavy IMF is used in bursts.
Once we obtain the galaxy properties from the semi-analytical model,
we plant these galaxies into a N-body simulation, in order to add
information about their positions and velocities. The simulation used
here is the \textit{Millennium Simulation} \citep{mill}. This
simulation adopts concordance values for the parameters of a flat
$\Lambda$CDM model, $\Omega_{\rm{m}} = 0.25$ and $\Omega_{\rm{b}} =
0.045$ for the densities of matter and baryons at $z=0$, $\rmn{h} =
0.73$ for the present-day value of the dimensionless Hubble constant,
$\sigma_8 = 0.9$ for the \textit{rms} linear mass fluctuations in a
sphere of radius $8 \rmn{h}^{-1}$Mpc at $z=0$ and $n=1$ for the slope
of the primordial fluctuation spectrum. The simulation follows
$2160^3$ dark matter particles from $z=127$ to $z=0$ within a cubic
region of comoving length $500\rmn{h}^{-1}\rmn{Mpc}$. The individual
particle mass is $8.6 \times 10^8\rmn{h}^{-1}\rmn{M_{\odot}}$, so the
smallest dark halo which can be resolved has a mass of $2 \times
10^{10}\rmn{h}^{-1}\rmn{M_{\odot}}$.
Dark matter haloes are identified using a \textit{Friends-Of-Friends}
(FOF) algorithm. To populate the simulation with galaxies from the
semi-analytical model, we use the same approach as in
\citet{benson00}. First, the position and velocity of the centre of
mass of each halo is recorded, along with the positions and velocities
of a set of randomly selected dark matter particles from each
halo. Second, the list of halo masses is fed into the semi-analytical
model in order to produce a population of galaxies associated with
each halo. Each galaxy is assigned a position and velocity within the
halo. Since the semi-analytical model distinguishes between central
and satellite galaxies, the central galaxy is placed at the centre of
mass of the halo, and any satellite galaxy is placed on one of the
randomly selected halo particles. Once galaxies have been generated,
and positions and velocities have been assigned, it is a simple
process to produce catalogues of galaxies with spatial information and
any desired selection criteria.
The combination of the semi-analytical model with the N-body
simulation is essential to study the detailed clustering of a desired
galaxy population, although the clustering amplitude on large scales
can also be estimated analytically \citep{dell2}. An example of the
output of the simulation is shown in the four images of
Fig.~\ref{fig.sample} which show redshifts $z=0$, $z=3.3$, $z=5.7$ and
$z=8.5$. The dark matter distribution (shown in green) becomes
smoother as we go to higher redshifts, due to the gravitational growth
of structures. As shown in Fig.~\ref{fig.sample}, for this particular
luminosity cut, the number density of \mbox{$\rmn{Ly}\alpha$}\ emitters varies at
different redshifts. As we will show in the next section, these
catalogues at high redshift are not complete at faint luminosities, so
we have to restrict our predictions to brighter luminosities as we go
to higher redshifts.
\section{Luminosity Functions}
The model presented by \citet{dell1, dell2} differs in two main ways
from the one presented in this paper: (i) there is a slight difference
in the values of the cosmological parameters used, and (ii) the
earlier work used a grid of halo masses together with an analytical
halo mass function, rather than the set of haloes from an N-body
simulation. In \S3.1, we investigate the impact of the different
choice of cosmological parameters on the luminosity function of \mbox{$\rmn{Ly}\alpha$}\
emitters, to see if the very good agreement with observational data
obtained by \cite{dell2} is retained on adopting the Millennium
cosmology. In \S3.2, we assess the completeness of our samples of
\mbox{$\rmn{Ly}\alpha$}\ emitters due to the finite mass resolution of the Millennium
simulation.
\subsection{Comparison of model predictions with observed luminosity functions}
\begin{figure}
\centering
\includegraphics[width=7cm,angle=90]{plots/clf_3.31.eps}
\includegraphics[width=7cm,angle=90]{plots/clf_5.72.eps}
\includegraphics[width=7cm,angle=90]{plots/clf_6.7.eps}
\caption{The cumulative luminosity functions of \mbox{$\rmn{Ly}\alpha$}\ emitters at
redshifts $z=3.3$ (Top), $z=5.7$ (Center) and $z=6.7$ (Bottom).
The blue points correspond to observational data (as indicated by the
key with full references in the text). The black and red curves correspond,
respectively, to the \texttt{GALFORM}\ predictions using the cosmological parameters of
the Millennium Simulation and those adopted in Le Delliou et al.}
\label{fig.clf}
\end{figure}
In this section, we investigate the impact on the model predictions of
the choice of cosmological parameters by re-running the model of
\citet{dell1, dell2}, keeping the galaxy formation parameters the same
but changing the cosmological parameters to match those used in the
Millennium simulation. To recap, the original \citet{dell2} model used
$\Omega_{\rm{m}} = 0.3$, $\Omega_{\Lambda} = 0.7$, $\Omega_{\rmn{b}} =
0.04$ , $\sigma_8 = 0.93$ and $h = 0.7$. In Fig.~\ref{fig.clf}, we
compare the cumulative luminosity functions obtained with \texttt{GALFORM}\
for the two sets of cosmological parameters with current observational
data in the redshift range $3<z<7$. The observational data is taken
from: \citet{kudritzki00} (crosses), \citet{cowie98} (asterisks),
\citet{gawiser07} (diamonds), \citet{ouchi07} (triangles and squares)
in the $z=3.3$ panel; \citet{ajiki03} (pluses), \citet{maier03}
(asterisks), \citet{hu04} (diamonds), \citet{rhoads03} (triangles),
\citet{shimasaku06} (squares) and \citet{ouchi07} (crosses) in the
$z=5.7$ panel; and \citet{taniguchi05} (crosses) and
\citet{kashikawa06} (asterisks and diamonds) in the $z=6.7$ panel. At
$z=3.3$, the two model curves agree very well, and are consistent with
the observational data shown. At $z=5.72$, the two models do not match
as well as in the previous case, but both are still consistent with
the observational data. Finally, at $z=6.7$ the differences are small
and both curves are consistent with observational data. The
conclusion from Fig.~\ref{fig.clf} is that there is not a significant
change in the model predictions on using these slightly different
values of the cosmological parameters. Furthermore, the observational
data is not yet sufficiently accurate to distinguish between the two
models or to motivate the introduction of further modifications to
improve the level of agreement, such as using a different \mbox{$\rmn{Ly}\alpha$}\ escape
fraction.
\subsection{The completeness of the Millennium galaxy catalogues}
The Millennium simulation has a halo mass resolution limit of $1.72
\times 10^{10} h^{-1}M_{\odot}$. In a standard \texttt{GALFORM}\ run, a grid
of haloes which extends to lower mass haloes than the Millennium
resolution is typically used, with $M_{\rm res} = 5 \times 10^{9}
h^{-1} M_{\odot}$ at $z=0$. A fixed dynamic range in halo mass is
adopted in these runs, but with the mass resolution shifting to
smaller masses with increasing redshift: for our standard setup, we
have $M_{\rm res} = 7.8\times 10^7 h^{-1} M_{\odot}$ and $1.4\times
10^7 h^{-1} M_{\odot}$ at $z=3$ and $6$ respectively. Therefore, when
putting \texttt{GALFORM}\ galaxies into the Millennium, our sample does not
contain galaxies which formed in haloes with masses below the
resolution limit of the Millennium. This introduces an incompleteness
into our catalogues when compared to the original \texttt{GALFORM}\
prediction. The incompleteness of the galaxy catalogues is more severe
for low luminosity galaxies because they tend to be hosted by low mass
haloes, as will be shown in the next section. Hereafter, we will use
\textit{N-body sample} to refer to the \texttt{GALFORM}\ galaxies planted in
the Millennium haloes, to distinguish them from the \textit{pure}
\texttt{GALFORM}\ catalogues generated using a grid of halos masses.
\begin{figure}
\centering
\includegraphics[width=7cm]{plots/completeness.eps}
\caption{Completeness of the Millennium galaxy catalogues with respect
to \mbox{$\rmn{Ly}\alpha$}\ luminosity or flux. (Top): The minimum luminosity down to
which the catalogues are 85$\%$ complete. (Bottom): The completeness
fraction as a function of redshift for a range of fluxes $-19 <
log(\flya \funits) <-17$, as indicated by the key.}
\label{fig.completeness}
\end{figure}
In order to quantify the incompleteness of the N-body sample as a
function of luminosity, we define the completeness fraction as the
ratio of the cumulative luminosity function for the N-body sample to
that obtained for a pure \texttt{GALFORM}\ calculation, and look for the
luminosity at which the completeness fraction deviates from unity.
The panels of Fig.~\ref{fig.completeness} give different views of the
completeness of the N-body samples. The top panel shows the luminosity
above which a catalogue can be considered as complete: we define the
completeness limit as the luminosity at which the completeness
fraction first drops to $0.85$. The figure clearly shows how the
luminosity corresponding to this completeness limit becomes
progressively brighter as we move to higher redshifts. For $z>9$ the
N-body sample is incomplete at all luminosities plotted.
The bottom panel of Fig.~\ref{fig.completeness} shows how the sample
becomes more incomplete at any redshift as we consider fainter
fluxes. A sample with galaxies brighter than $\log(\flya \funits) =
-19)$ is less than $70\%$ complete at all redshifts $z>5$, while a sample
with galaxies brighter than $\log(\flya \funits) = -17$ is always over
$90\%$ complete for $z<9$. The completeness fraction monotonically
decreases with increasing redshift until $z \sim 6$ for very faint
fluxes. For $z>6$ the completeness rises again: the shape of the bright
end of the luminosity function at this redshift is sensitive to the choice
of the redshift of reionization.
In summary, the requirement that our samples be at least $80\%$
complete restricts the range of validity of the predictions from the
Millennium simulation to redshifts below $9$, and fluxes brighter than
$\log(\flya \funits)>-17.5$.
\section{Clustering Predictions}
In this section we present clustering predictions using \mbox{$\rmn{Ly}\alpha$}\ emitters in the full
Millennium volume. To study the clustering of galaxies we calculate the two-point
correlation function, $\xi(r)$, of the galaxy distribution. In order to quantify the
evolution of the clustering of galaxies, we measure the correlation function over
the redshift interval $0<z<9$.
To calculate $\xi(r)$ in the simulaation, we use the standard
estimator (e.g. Peebles 1980):
\begin{equation}
1+\xi(r) = \frac{\langle DD \rangle }{\frac{1}{2} N_{\rm gal} n \Delta V(r)},
\label{eq.xi1}
\end{equation}
where $\langle DD \rangle$ stands for the number of distinct data
pairs with separations in the range $r$ to $r + \Delta r$, $n$ is the
mean number density of galaxies, $N_{\rm gal}$ is the total number of
galaxies in the simulation volume and $\Delta V(r)$ is the volume of a
spherical shell of radius $r$ and thickness $\Delta r$. This estimator
is applicable in the case of periodic boundary conditions. In the
correlation function analysis, we consider two parameters which help
us to understand the clustering behaviour of \mbox{$\rmn{Ly}\alpha$}\ galaxies: the
correlation length, $r_0$, and the galaxy bias, $b$, both of which are
discussed below.
\subsection{Correlation Length evolution}
A common way to characterize the clustering of galaxies is to fit a power-law
to the correlation function:
\begin{equation}
\xi(r) = \left( \frac{r}{r_0} \right)^{-\gamma},
\label{eq.xifit}
\end{equation}
where $r_0$ is the correlation length and $\gamma=1.8$ gives a good
fit to the slope of the observed correlation function over a
restricted range of pair separations around $r_0$ at $z=0$
(e.g. \citet{DavisPeebles83}). The correlation length can also be
defined as the scale where $\xi = 1$, and quantifies the amplitude of
the correlation function when the slope $\gamma$ is fixed.
\begin{figure}
\centering
\includegraphics[width=11cm,angle=90]{plots/xi_gal_dm.eps}
\caption{The correlation function predicted for \mbox{$\rmn{Ly}\alpha$}\ emitters (black
solid curve) for a range of redshifts, as indicated in each
panel. \mbox{$\rmn{Ly}\alpha$}\ emitters are included down to the completeness limit at
each redshift shown in Fig~\ref{fig.completeness}. The solid red
curve shows the correlation function of the dark matter at the same
epochs. The blue dashed line shows the power law fit of
Eq.~\eqref{eq.xifit}, evaluated in the range $1<r[\rmn{Mpc/h}]<10$, as
delineated by the vertical dashed lines.}
\label{fig.fullxi}
\end{figure}
Fig.~\ref{fig.fullxi} shows the correlation function of \mbox{$\rmn{Ly}\alpha$}\ emitting
galaxies, \mbox{$\xi_{\rmn{gal}}$}\ (solid black curves) of the full catalogues down to
the completeness limits at each redshift, calculated using
Eq.~\eqref{eq.xi1}. The red curve shows \mbox{$\xi_{\rmn{dm}}$}, the correlation
function of the dark matter. At $z=0$, \mbox{$\xi_{\rmn{dm}}$}\ is larger than \mbox{$\xi_{\rmn{gal}}$},
but for $z>0$ \mbox{$\xi_{\rmn{dm}}$}\ is increasingly below \mbox{$\xi_{\rmn{gal}}$}. We will study in
detail the comparison of the dark matter and \mbox{$\rmn{Ly}\alpha$}\ galaxy correlation
functions in \S4.2.
Another notable feature of Fig.~\ref{fig.fullxi} is that $\mbox{$\xi_{\rmn{gal}}$}(r)$
differs considerably from a power law, particularly on scales greater
than 10 \mbox{$ h^{-1} \rmn{Mpc}$}. When fitting Eq.~\eqref{eq.xifit} to the correlation
functions plotted in Fig.~\ref{fig.fullxi}, we use only the
measurements in the range [1,10] \mbox{$ h^{-1} \rmn{Mpc}$}, where $\mbox{$\xi_{\rmn{gal}}$}(r)$ behaves most
like a power law. We fix the slope $\gamma = 1.8$ for all $\mbox{$\xi_{\rmn{gal}}$}(r)$
to allow a comparison between different redshifts, although we note
that for $z<5$, the slope of $\mbox{$\xi_{\rmn{gal}}$}(r)$ is closer to $\gamma = 1.6$.
By using the power law fit we can compare the clustering amplitudes of
different galaxy samples. To determine the clustering evolution of
\mbox{$\rmn{Ly}\alpha$}\ emitters, we split the catalogues of \mbox{$\rmn{Ly}\alpha$}\ emitters into
luminosity bins. For each of these sub-samples, we calculate the
correlation function and then we obtain $r_0$ by fitting
Eq.~\eqref{eq.xifit} as described. Fig.~\ref{fig.r0_mhalo} (top) shows
the dependence of $r_0$ on luminosity for different redshifts in the
range $0<z<9$. The errors are shown by the area enclosed by the thin
solid lines for each set of points, and are calculated as the $90\%$
confidence interval of the $\chi^2$ fit of the correlation functions
to Eq.~\eqref{eq.xifit} (ignoring any covariance between pair
separation bins). The range of luminosities plotted is set by the
completeness limit of the simulation described in the previous
section. We also discard galaxy samples with fewer than 500 galaxies,
as in such cases, the errors are extremely large and the correlation
functions are poorly defined. The clustering in high redshift surveys
of \mbox{$\rmn{Ly}\alpha$}\ emitters is sensitive to the flux limit that they are able to
reach, as shown by Fig.~\ref{fig.r0_mhalo}.
The model predictions show modest evolution of $r_0$ with redshift for
most of the luminosity range studied. Over this redshift interval, on
the other hand, the correlation length of the dark matter changes
dramatically, as shown by Fig.~\ref{fig.fullxi}. Typically, at a
given redshift, we find that $r_0$ shows little dependence on
luminosity until a luminosity of \llya $\sim 10^{42} \lunits$ is
reached, brightwards of which there is a strong increase in clustering
strength with luminosity. This trend is even more pronounced at higher
redshifts. Galaxies at $z=0$ are less clustered than galaxies in the
range $3<z<7$, except at luminosities close to \llya $\sim 10^{40}
\lunits$. At $z=8.5$, $r_0$ increases from $r_0 \sim 5$ \mbox{$ h^{-1} \rmn{Mpc}$}\ at \llya
$\sim 10^{42} \lunits$ to $r_0 \sim 12$ \mbox{$ h^{-1} \rmn{Mpc}$}\ at \llya $> 10^{42.5}
\lunits$.
The growth of $r_0$ with limiting luminosity is related to the masses
of the haloes which host \mbox{$\rmn{Ly}\alpha$}\ galaxies. As shown in the bottom panel
of Fig.~\ref{fig.r0_mhalo}, there is not a simple relation between the
median mass of the host halo and the luminosity of \mbox{$\rmn{Ly}\alpha$}\ emitters.
For a given luminosity, \mbox{$\rmn{Ly}\alpha$}\ galaxies tend to be hosted by haloes of
smaller masses as we go to higher redshifts. In addition, for
redshifts $z>0$, there is a trend of more luminous \mbox{$\rmn{Ly}\alpha$}\ emitters
being found in more massive haloes. The key to explaining the trends
in clustering strength is to compare how the effective mass of the
haloes which host \mbox{$\rmn{Ly}\alpha$}\ emitting galaxies is evolving compared to the
typical or characteristic mass in the halo distribution ($M_{*}$)
\citep{mo96}; if \mbox{$\rmn{Ly}\alpha$}\ emitters tend to be found in haloes more
massive than $M_*$, then they will be more strongly clustered than the
dark matter. This difference between the clustering amplitude of
galaxies and mass is explored more in the next section. In a
hierarchical model for the growth of structures, haloes more massive
than $M_*$ are more clustered, and thus we expect a strong connection
between the evolution of $r_0$ and the masses of the
halos. Fig.~\ref{fig.r0_mhalo} shows that the dependence of $r_0$ (and
host halo mass) on luminosity becomes stronger at higher redshifts.
\subsection{The bias factor of \mbox{$\rmn{Ly}\alpha$}\ emitters}
\label{sec:bias}
The galaxy bias, $b$, quantifies the strength of the clustering of
galaxies compared to the clustering of the dark matter. One way to
calculate the bias is by taking the ratio of \mbox{$\xi_{\rmn{gal}}$}\ and \mbox{$\xi_{\rmn{dm}}$},
$\mbox{$\xi_{\rmn{gal}}$} = b^2\mbox{$\xi_{\rmn{dm}}$}$. Both correlation functions are estimated using
Eq.~\eqref{eq.xi1}. Since the simulation contains ten billion dark
matter particles, a direct pair-count calculation of \mbox{$\xi_{\rmn{dm}}$}\ would
demand a prohibitively large amount of computer time, so we extract
dilute samples of the dark matter particles, selecting randomly $\sim
10^7$ particles. In this way we only enlarge the pair-count errors on
\mbox{$\xi_{\rmn{dm}}$}\ (which nevertheless are still much smaller than for \mbox{$\xi_{\rmn{gal}}$}) but
obtain the correct amplitude of correlation function itself.
\begin{figure}
\centering
\includegraphics[width=12cm,angle=90]{plots/r0_mhalo.eps}
\caption{\textit{(Top):} The evolution of the correlation length $r_0$
as a function of \mbox{$\rmn{Ly}\alpha$}\ luminosity for several redshifts in the
range $0<z<9$, as indicated by the key. The thin solid coloured lines
shows the errors on the correlation length. \textit{(Bottom):} The
evolution of the median mass of halos which host \mbox{$\rmn{Ly}\alpha$}\ emitting
galaxies as a function of \mbox{$\rmn{Ly}\alpha$}\ luminosity, for the same range
of redshifts as above. }
\label{fig.r0_mhalo}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10cm, angle=90]{plots/bias_gal.eps}
\caption{The galaxy bias as a function of \mbox{$\rmn{Ly}\alpha$}\ luminosity at
different redshifts, as indicated by the key. The solid lines show the
results from the simulation and the dashed lines show the analytical
expression of SMT. The area enclosed by the thin solid lines shows the
error on the bias estimation for each redshift.}
\label{fig.gal_bias}
\end{figure}
To obtain the bias parameter of \mbox{$\rmn{Ly}\alpha$}\ emitters as a function of
luminosity, we split the full catalogue of galaxies at each redshift
into luminosity bins. For each of these bins we calculate
\mbox{$\xi_{\rmn{gal}}$}\ and divide by \mbox{$\xi_{\rmn{dm}}$}\ to get the sqaure of the bias. Due to
non-linearities, the ratio of \mbox{$\xi_{\rmn{gal}}$}\ and \mbox{$\xi_{\rmn{dm}}$}\ is not constant on
all scales. As a reasonable estimation of the bias we chose the mean
value over the range 6 \mbox{$ h^{-1} \rmn{Mpc}$}\ $<r<$ 30 \mbox{$ h^{-1} \rmn{Mpc}$}. Over these scales the bias
does seem to be constant and independent of scale. This range is quite
similar to the one used by \citet{gao05} to measure the bias parameter
of dark matter haloes in the Millennium Simulation.
The bias parameter can also be calculated approximately using various
analytical formalisms \citep{mo96,sheth01,mandelbaum05}. These
procedures relate the halo bias to $\sigma(m,z)$, the \textit{rms}
linear mass fluctuation within a sphere which on average contains mass
$m$. The bias factor for galaxies of a given luminosity is then
obtained by averaging the halo bias over the halos hosting these
galaxies. \citet{dell2} used the analytical expression of
\citet{sheth01} (hereafter SMT) to calculate the bias parameter for
the semi-analytical galaxies. This gives a reasonable approximation to
the large-scale halo bias measured in N-body simulations (e.g
\citet{angulo08}).
Fig.~\ref{fig.gal_bias} shows the bias parameter as a function of
luminosity for redshifts in the range $0<z<9$, and compares the direct
calculation using the N-body simulation (solid lines) with the
analytical estimation (dashed lines). In order to calculate the
uncertainty in our value of the bias, we assume an error on
$\mbox{$\xi_{\rmn{gal}}$}(r)$ of the form $\Delta \xi_{\rmn{gal}} =
2\sqrt{(1+\xi_{\rmn{gal}})/DD}$ \citep{hewitt82, baugh96}, and
assuming a negligible error in \mbox{$\xi_{\rmn{dm}}$}\ we get
\begin{equation}
\Delta b = \frac{1}{b\xi_{\rmn{dm}}}\sqrt{\frac{1+b^2\xi_{\rmn{dm}}}{DD}},
\end{equation}
for the error in the bias estimation. This error is shown in
Fig. \ref{fig.gal_bias} as the range defined by the thin solid lines
surrounding the bias measurement shown by the points.
The first noticeable feature of Fig.~\ref{fig.gal_bias} is the strong
evolution of bias with increasing redshift: From $z=0$ to $z=8.5$ the
bias factor increases from $b(z=0)\sim 0.8$ to $b(z=8.5)\sim 12$,
which means that the clustering amplitude of \mbox{$\rmn{Ly}\alpha$}\ emitters at $z=8.5$
is over 140 times the clustering amplitude of the dark matter at this
redshift. Another interesting prediction is the dependence of bias on
\mbox{$\rmn{Ly}\alpha$}\ luminosity. For $z > 3$ there seems to be a strong increase of
the bias with luminosity for bins where \llya\ $ > 10^{42}
\lunits$. The agreement between the analytic calculation of the bias
and the simulation result is reasonable over the range $0<z<5$, but
becomes less impressive as higher biases are reached. A similar
discrepancy was also noticed by \citet{gao05}, where they compared the
halo bias extracted from the simulation with different analytic
formulae (see also \citet{angulo08}).
Another way to describe galaxy clustering is through the halo
occupation distribution (HOD; \citet{benson00}, \citet{berlind03},
\citet{cooray02}). The HOD gives the mean number of galaxies which
meet a particular observational selection as a function of halo
mass. For flux-limited samples, the HOD can be broken down into the
contribution from central galaxies and satellite galaxies. In a simple
picture, the mean number of central galaxies is zero below some
threshold halo mass, $M_{min}$, and unity for higher halo masses. With
increasing halo mass, a second threshold is reached, $M_{crit}$, above
which a halo can also host a satellite galaxy. The number of
satellites is usually described by a power-law of slope $\beta$. In
the simplest case, three parameters are needed to describe the HOD
\citep{berlind02,hamana04}; more detailed models have been proposed to
describe the transition from 0 to 1 galaxy \citep{berlind03}.
\begin{figure}
\centering
\includegraphics[width=12cm,angle=90]{plots/hods.eps}
\caption{The HOD of \mbox{$\rmn{Ly}\alpha$}\ emitters at $z= 3.3$ ({\it top}) and $z =
4.9$ ({\it bottom}). Each set of points represents a model sample
with a different luminosity limit, as given by the key in the upper
panel. The dashed line in each panel correspond to a ``best'' fit
using the Berlind et al. (2003) parametrization. }
\label{fig.hod}
\end{figure}
We can compute the HOD directly from our model. The results are shown
in Fig.~\ref{fig.hod}, where we plot the HOD at two different
redshifts for different luminosity limits. For comparison, we plot the
HOD parametrization of \citet{berlind03} against our model
predictions. In general, this HOD does a reasonable job of describing
the model output, and is certainly preferred over a simple three
parameter model. However, for the $z=3.3$ case (top panel of
Fig.~\ref{fig.hod}), the shape of the model HOD for $\log
(M/M_{\odot}) > 13$ is still more complicated than can be accommodated
by the Berlind et al. parametrization, showing a flattening in the
number of satellites as a function of increasing halo mass. There is
less disagreement in the $z=4.9$ case (bottom panel), but our model
HOD becomes very noisy for large halo masses.
\section{Mock Catalogues}
\begin{table*}
\caption{Summary of survey properties and simulation results.}
\label{table.mock}
\begin{tabular}{@{}lcccccccccc}
\hline
\hline
(1) & (2) & (3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11)\\
\hline
Survey & $z_{\rmn{survey}}$ & $z_{\rmn{simulation}}$ & $\Delta z$ & Area $\arcm^2$ &
\ewobs [\AA{}] & \flya \funits & $N_{\rmn{obs}}$ & $N^{\rmn{median}}_{\rm{mock}} $ & 10-90\% & $C_v$\\
\hline
MUSYC & 3.1 & 3.06 & 0.04 & 961 & 80 & $1.5 \times 10^{-17}$ & 162 & 142 & 89-207
& 0.41\\
SXDS & 3.1 & 3.06 & 0.06 & 3538 & 328 & $1.1 \times 10^{-17} $ & 356 & 316 &
256-379 & 0.19\\
& 3.7 & 3.58 & 0.06 & 3474 & 282 & $2.7 \times 10^{-17}$ & 101 & 80 &
60-110 & 0.31\\
& 5.7 & 5.72 & 0.10 & 3722 & 335 & $7.4 \times 10^{-18}$ & 401 & 329 &
255-407 & 0.23\\
ELVIS & 8.8 & 8.54 & 0.10 & $\sim 3160$ &100& $3.7 \times 10^{-18} $ & -- & 20 &
14-29 & 0.37\\
\hline
\hline
\end{tabular}
\\
\begin{flushleft}
\indent Column (1) gives the name of the survey; (2) and (3) show the
redshift of the observations and nearest output from the simulations,
respectively; (4) shows the redshift width of the survey, based on the
FWHM filter width; (5) shows the area covered by each survey; (6) and
(7) show the equivalent width and \mbox{$\rmn{Ly}\alpha$}\ flux limits of the samples,
respectively; (8) shows the number of galaxies detected in each
survey; (9) and (10) show the median of the number of galaxies and the
10-90 percentile range found in the mock catalogues for each
survey. Finally, column (11) gives the fractional variation of the
number of galaxies, defined in Eq.~\eqref{eq.cv}.
\end{flushleft}
\end{table*}
In this section we make mock catalogues of \mbox{$\rmn{Ly}\alpha$}\ emitters for a
selection of surveys. In the previous section, we used the full
simulation box to make clustering predictions, exploiting the periodic
boundary conditions of the computational volume. The simulation is so
large that it can accommodate many volumes equivalent to those sampled
by current \mbox{$\rmn{Ly}\alpha$}\ surveys, allowing us to examine the fluctuations in
the number of emitters and their clustering. The characteristics of
the surveys we replicate are listed in Table~\ref{table.mock}.
The procedure to build the mock catalogues is the following:
\begin{enumerate}
\item We extract a catalogue of galaxies from an output of the
Millennium Simulation that matches (as closely as possible) the
redshift of a given survey. The simulation output contains 64
snapshots spaced roughly logarithmically in the redshift range
$[127,0]$.
\item We choose one of the axes (say, the z-axis) as the
line-of-sight, and we convert it to \textit{redshift space}, to match
what is observed in real surveys. To do this we replace $r_z$ (the
comoving space coordinate) with
\begin{equation}
s_z = r_z + \frac{v_z}{a\rm{H(z)}} \quad [\mbox{$ h^{-1} \rmn{Mpc}$}],
\end{equation}
where $v_z$ is the peculiar velocity along the z-axis, $a = 1/(1+z)$
and H($z$) is the Hubble parameter at redshift $z$.
\item We then apply the flux limit of the particular survey, to mimic the selection of
galaxies. Table \ref{table.mock} shows the flux limits of the surveys considered.
\item Then we extract many mock catalogues using the same geometry as the real survey.
We extract slices of a particular depth $\Delta z$ (different for each survey), and
within each slice we extract as many mock catalogues as possible using the same angular
geometry as the real sample. $\Delta z$ is determined using the transmission curves
of the narrow-band filters used in each survey. To derive the angular sizes we use:
\begin{equation}
\label{eq.tran_size}
D_t(\theta,z) = d_c(z) \Delta \theta,
\end{equation}
\begin{equation}
\label{eq.dc}
d_c(z) = \frac{c}{H_0} \int_0^z \frac{{\rm d}z'}{\sqrt{\Omega_m (1+z')^3 + \Omega_{\Lambda}}},
\end{equation}
where $D_t$ is the transverse comoving size in $h^{-1}$Mpc, $d_c$ is
the comoving radial distance, $c$ and $H_0$ are the speed of light and
the Hubble constant respectively, $\Omega_{\rm m}$ and
$\Omega_{\Lambda}$ are the density parameters of matter and the
cosmological constant respectively. Eq.~\eqref{eq.tran_size} is valid
for $\Delta \theta \ll 1 [\rm{radians}]$, which is the case for the
surveys we analyse in this work. We assume a flat cosmology.
\item From the line-of-sight axis we invert Eq. \eqref{eq.dc} to
obtain the redshift distribution of \mbox{$\rmn{Ly}\alpha$}\ galaxies within each mock
catalogue, converting galaxy position to redshift. This information is
then used to take into account the shape of the filter transmission
curve for each survey, which controls the minimum flux and equivalent
width as a function of redshift. The value given in
Table~\ref{table.mock} corresponds to the minimum flux and \ewobs\ at
the peak of the filter transmission curve. For redshifts at which the
transmission is smaller (the tails of the curve) the minimum flux and
\ewobs\ required for a \mbox{$\rmn{Ly}\alpha$}\ emitter to be included are proportionally
bigger.
\item Finally, we allow for incompleteness in the detection of \mbox{$\rmn{Ly}\alpha$}\
emitters at a given flux due to noise in the observed images (where
this information is available). To do this, we randomly select a
fraction of galaxies in a given \mbox{$\rmn{Ly}\alpha$}\ flux bin to match the
completeness fraction reported for the survey at that flux.
\end{enumerate}
Real surveys of \mbox{$\rmn{Ly}\alpha$}\ emitters usually lack detailed information about
the position of galaxies along the line-of-sight. Hence, instead of
measuring the spatial correlation function defined in
Eq.~\eqref{eq.xi1}, it is only possible to estimate the angular
correlation function, \wtheta, which is the projection on the sky of
$\xi(r)$.
We estimate \wtheta\ from mock catalogues using the following
procedure, which closely matches that used in real surveys. To compute
the angular correlation function we use the estimator \citep{landy93}:
\begin{equation}
w_{LS}(\theta) = \frac{\langle DD(\theta) \rangle - 2 \langle DR(\theta) \rangle + \langle RR(\theta) \rangle}{\langle RR(\theta) \rangle},
\label{eq.ls}
\end{equation}
where $\langle DR \rangle$ stands for data-random pairs, $\langle RR
\rangle$ indicates the number of random-random pairs and all of the
pair counts have been appropriately normalized. In the case of a
finite volume survey, this estimator is more robust than the one
defined in Eq.~\eqref{eq.xi1} because it is less sensitive to errors
in the mean density of galaxies, such as could arise from boundary
effects. In practice, the measured angular correlation function can be
approximated by a power law:
\begin{equation}
w(\theta) = A_w\left(\frac{\theta}{1^{\circ}}\right)^{-\delta},
\label{eq.powerlaw}
\end{equation}
where $A_w$ is the dimensionless amplitude of the correlation
function, and $\delta$ is related to slope of the spatial correlation
function, $\gamma$, from Eq.~\eqref{eq.xifit} by $\delta = \gamma -
1$. A relation between $r_0$ and $A_w$ can be obtained using a
generalization of Limber's equation \citep{simon07}.
Surveys of \mbox{$\rmn{Ly}\alpha$}\ emitters typically cover relatively small areas of
sky and can display significant clustering even on the scale of the
survey. As a result, the mean galaxy number density within the
survey area will typically differ from the cosmic mean value. If the
number of galaxies within the survey is used to estimate the mean
density, used in Eq.~\eqref{eq.ls}, rather than the unknown true
underlying density, this leads to a bias in the estimated correlation
function. This effect is known as the integral constraint (IC)
bias. \citet{landy93} show that when their estimator is used, the
expected value of the estimated correlation function $w_{LS}(\theta)$
is related to the true correlation function $w(\theta)$ by
\begin{equation}
\langle w_{LS}(\theta) \rangle = \frac{w(\theta)-w_{\Omega}}{1 +
w_{\Omega}} ,
\label{eq.xi_ic}
\end{equation}
where the integral constraint term $w_{\Omega}$ is defined as
\begin{equation}
w_{\Omega} \equiv \frac{1}{\Omega^2}\int d\Omega_1 d\Omega_2 w(\theta_{12}),
\label{eq.ic}
\end{equation}
integrating over the survey area, and is equal to the fractional
variance in number density over that area.
When the clustering is weak Eq. \eqref{eq.xi_ic} simplifies to $\langle
w_{LS}(\theta) \rangle \simeq w(\theta) - w_{\Omega}$. This motivates
the additive IC correction which is customarily used in practice:
\begin{equation}
w_{corr}(\theta) = w_{LS}(\theta) + w_{\Omega}.
\label{eq.wcorr}
\end{equation}
We use this to correct the angular correlation functions from our mock
catalogues. In order to estimate the term $w_{\Omega}$, we
approximate the true correlation function as a power law, as in
Eq. \eqref{eq.powerlaw}, and use
\begin{equation}
w_{\Omega} \simeq A_w \frac{\sum_i \langle RR_i \rangle \theta_i^{-\delta}}
{\sum \langle RR_i \rangle},
\label{eq.sigma}
\end{equation}
\citep{daddi00}, where $\langle RR \rangle$ are the same random pairs
as used in the estimate of $w_{LS}(\theta)$.
To quantify the sample variance expected for a particular survey, we
use the mock catalogues to calculate a \textit{coefficient of
variance} ($C_v$), which is a measure of the fractional variation in
the number of galaxies found in the mocks
\begin{equation}
C_v = \frac{N_{90} - N_{10}}{2N_{\rm{med}}},
\label{eq.cv}
\end{equation}
where $N_{10}$ and $N_{90}$ are the 10 and 90 percentiles of the
distribution of the number of galaxies in the mocks, respectively, and
$N_{\rm{med}}$ is the median. The value of $C_v$ allows us to compare
the sampling variance between different surveys in a quantitative way.
To analyse the clustering in the mock catalogues, we measured the
angular correlation function of each mock catalogue using the
procedure explained above. Then we fit Eq.~\eqref{eq.powerlaw} to
each of the mock \wtheta\ and we choose the median value of $A_w$ as
the representative power law fit. We fix the slope of \wtheta\ to
$\delta = 0.8$ for all surveys, except for ELVIS, where we found that
a steeper slope, $\delta = 1.2$, agreed much better with the simulated
data. To express the variation in the correlation function amplitude
found in the mocks, we calculate the 10 and 90 percentiles of the
distribution of $A_w$ for each set of mock surveys. We also calculate
\wtheta\ using the full transverse extent of the simulation, with the
same selection of galaxies as for the real survey. This estimate of
\wtheta, which we call the \textit{Model} \wtheta, represents an ideal
measurement of the correlation function without boundary effects (so
there is no need for the integral constraint correction).
The surveys we mimic are the following: the MUSYC Survey
\citep{gronwall07, gawiser07}, which is a large sample of \mbox{$\rmn{Ly}\alpha$}\
emitting galaxies at $z=3.1$; the SXDS Survey \citep{ouchi05,ouchi07},
which covers three redshifts: $z=3.1$, $z=3.7$ and $z=5.7$, and
finally, we make predictions for the forthcoming ELVIS survey
\citep{kim07a,kim07b}, which is designed to find \mbox{$\rmn{Ly}\alpha$}\ emitting
galaxies at $z=8.8$. We now describe the properties of the mock
catalogues for each of these surveys in turn.
\subsection{The MUSYC Survey}
\begin{figure}
\centering
\includegraphics[width=7cm]{plots/ew_comp.eps}
\caption{The observed $EW_{obs}$ distribution of the MUSYC survey at
$z=3.1$ (solid black line) and the simulation (solid green line).}
\label{fig.ew_musyc}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{plots/musyc_lores.eps}
\caption{An image of a mock catalogue of the MUSYC Survey of \mbox{$\rmn{Ly}\alpha$}\
emitters at $z=3.1$. The colour format and legend are the same as
used in Fig.~\ref{fig.sample}. The angular size of the image is
$31\arcmin \times 31 \arcmin$.}
\label{im.musyc}
\end{figure}
The Multi-wavelength Survey by Yale-Chile (MUSYC) \citep{quadri07,
gawiser06,gawiser07,gronwall07} is composed of four fields covering a total solid
angle of one square degree, each one imaged from the ground in the
optical and near-infrared. Here we use data from a single MUSYC field
consisting of narrow-band observations of \mbox{$\rmn{Ly}\alpha$}\ emitters made with the
CTIO 4-m telescope in the Extended Chandra Deep Field South (ECDFS)
\citep{gronwall07}. The MUSYC field, centred on redshift $z=3.1$,
contains 162 \mbox{$\rmn{Ly}\alpha$}\ emitters in a redshift range of $\Delta z \sim
0.04$ over a rectangular area of $31 \arcmin \times 31 \arcmin$ with
flux and \ewobs\ limits described in Table~\ref{table.mock}.
\begin{figure}
\centering
\includegraphics[width=7cm]{plots/ngal_musyc.eps}
\caption{Histogram of the number of \mbox{$\rmn{Ly}\alpha$}\ emitters found in the mock
MUSYC catalogues. The red line shows the median, the dashed blue
lines show the 10-90 percentile range, and the green line shows the
number of galaxies detected in the real survey.}
\label{fig.ngal_musyc}
\end{figure}
To test how well the model reproduces the \mbox{$\rmn{Ly}\alpha$}\ emitters seen in the
MUSYC survey, we first compare the predicted (green) and measured
(black) distributions of \mbox{$\rmn{Ly}\alpha$}\ equivalent widths in
Fig.~\ref{fig.ew_musyc}. Here the predicted distribution comes from
the full simulation volume. Overall, the simulation shows remarkably
good agreement with the real data, with a slight underestimation in
the range $200 < \ewobs [$\AA{}$] < 400$. For $\ewobs[$\AA{}$] > 400$
both distributions seem to agree well, although the number of detected
\mbox{$\rmn{Ly}\alpha$}\ emitters in the tail of the distribution is small.
For the MUSYC survey we built $252$ mock catalogues from the
Millennium simulation volume using the procedure outlined above.
Fig.~\ref{im.musyc} shows an example of one of these mock
catalogues. Many of the \mbox{$\rmn{Ly}\alpha$}\ emitters are found in high dark matter
density regions, and thus they are biased tracers of the dark
matter. Fig.~\ref{fig.ngal_musyc} shows the distribution of the number
of galaxies in the ensemble of mocks. The green line shows the number
detected in the real survey (162), which falls within the 10-90
percentile range of the mock distribution and is close to the median
(142). The 10-90 percentile range spans an interval of
$89<N_{\rm{gal}}<207$, indicating a large cosmic variance for this
survey configuration, with $C_v = 0.41$.
\begin{figure}
\centering
\includegraphics[width=8cm]{plots/w_musyc.eps}
\caption{ Angular clustering for the MUSYC Survey. Green circles show
\wtheta\ calculated from the observed catalogue \citep{gawiser07}. The
blue circles show the median \wtheta\ from all mock catalogues,
corrected for the integral constraint effect. The dark and light grey
shaded regions respectively show the 68\% and 95\% ranges of the
distribution of \wtheta\ measured in the mock catalogues. The red open
circles show the \textit{Model} correlation function, obtained using
the width of the entire simulation box (and the same EW, flux and
redshift limits). The dashed lines show the power-law fit to the
observed \wtheta\ (green) and the median fit to \wtheta\ from the mock
catalogues (blue). The amplitudes $A_w$ of these fits are also given
in the figure.}
\label{fig.w_musyc}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=7.2cm]{plots/sxds3.1_lores.eps}
\includegraphics[width=7.2cm]{plots/sxds3.6_lores.eps}
\includegraphics[width=7.2cm]{plots/sxds5.7_lores.eps}
\caption{Mock catalogues of the SXDS survey for redshifts $3.1$
(\textit{top}), $3.6$ (\textit{centre}) and $5.7$
(\textit{bottom}). The colour scheme and legend are the same as used
previously. The angular size of the image is $1.4^\circ \times
1.4^\circ$. }
\label{im.sxds}
\end{figure}
The next step is to compare the clustering in the simulations with the
real data. Fig.~\ref{fig.w_musyc} plots the correlation functions
from the mock catalogues alongside that measured in the real survey
\citep{gawiser07}. There is reasonable agreement between the mock
catalogue results and the observed data \footnotemark.
\footnotetext{After this paper was accepted for publication, we learned that the
MUSYC data shown in Fig.\ref{fig.w_musyc} are not corrected for the
IC. Including this correction would improve the agreement with the model.}
The median \wtheta\ from the
mocks is slightly higher than the observed values, but the observed
\wtheta\ is within the range containing 95\% of the mock \wtheta\
values (i.e. between the 2.5\% and 97.5\% percentiles, shown by the
light grey shaded region). We quantified this difference by fitting
the power law of Eq.~\eqref{eq.powerlaw} to both real and mock
data. The power-law fits were made over the angular range 1-10
arcmins. We find the value of $A_w$ (Eq.~\ref{eq.powerlaw}) for each
of the mock catalogue correlation functions by $\chi^2$-fitting (using
the same expression as in \S\ref{sec:bias} for the error on each model
datapoint) and then we plot the power law corresponding to the median
value of $A_w$. We find $A_w = 0.53^{+1.01}_{-0.33}$ for the mocks,
where the central value is the median, and the range between the error
bars contains 95\% of the values from the mocks. For the real data,
we find the best-fit $A_w$ and the 95\% confidence interval around it
by $\chi^2$-fitting, using the error bars on the individual datapoints
reported by \citeauthor{gawiser07}. This gives $A_w = 0.29 \pm 0.17$
for the real data. We again see that the observed value is within the
95\% range of the mocks, and is thus statistically consistent with the
model prediction. We also see that the 95\% confidence error bar on
the observed $A_w$ is much smaller than the error bar we find from our
mocks. This latter discrepancy arises from the small errors quoted on
\wtheta\ by \citet{gawiser07}, which are based on { modified} Poisson pair
count errors, but neglect variations between different sample volumes
(i.e. cosmic variance). On the contrary, using our mocks, we are
able to take cosmic variance fully into account. This underlines the
importance of including the cosmic variance in the error bars on
observational data, to avoid rejecting models by mistake.
The red open circles in Fig.~\ref{fig.w_musyc} show the correlation
function obtained using the full angular size attainable with the
Millennium simulation but keeping the same flux, EW and redshift
limits as in the MUSYC survey (averaging 7 different slices), and so
this measurement has a smaller sample variance. The area used here is
$\sim 120$ times bigger than the MUSYC area, so IC effects are
negligible on the scales studied here. We refer to this as the
\textit{Model} prediction for \wtheta.
The median of the mock correlation functions (including the IC
correction, blue circles) is seen to agree reasonably well with the
\textit{Model} correlation function (red open circles) for $\theta <
20 \arcm$. This shows that for this survey it is possible to obtain an
observational estimate of the correlation function which is unbiased
over a range of scales, by applying the integral constraint
correction. However, on large scales the median \wtheta\ of the mocks
(with IC correction included) lies above the Model \wtheta, which
shows that the IC correction is not perfect, even on
average. Presumably this failure is due (at least in part) to the fact
that the IC correction procedure assumes that \wtheta is a power law,
while the true \wtheta\ departs from a power law on large scales. It
is also important to note that these statements only apply to the
median \wtheta\ derived from the mock samples - the individual mocks
show a large scatter around the true \wtheta\ (as shown by the grey
shading), and the IC correction does not remove this. This scatter
rapidly increases at both small and large angular scales, so the best
constraints on \wtheta\ from this survey are for intermediate scales,
$1 \la \theta \la 5 \arcm$.
\subsection{The SXDS Surveys}
The Subaru/\textit{XMM-Newton} Deep Survey (SXDS)
\citep{ouchi05,ouchi07,kashikawa06} is a multi-wavelength survey
covering $\sim 1.3$ square degrees of the sky. The survey is a
combination of deep, wide area imaging in the X-ray with
\textit{XMM-Newton} and in the optical with the Subaru
Suprime-Cam. Here we are interested in the narrow-band observations at
three different redshifts: $3.1, 3.6$ and $5.7$ \citep{ouchi07}.
\begin{figure*}
\centering
\includegraphics[width=6.3cm,angle=90]{plots/ngal_sxds.eps}
\caption{The distribution of the number of galaxies in mock SXDS
catalogues, for $z=3.1$ \textit{(left)}, $z=3.6$ \textit{(centre)} and
$z=5.7$ \textit{(right)}. The red line shows the median of the number of
galaxies inside the mock catalogues, the blue lines show the 10-90
percentiles of the distribution, and the green line shows the
number observed in the SXDS.}
\label{fig.ng_sxds}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=6.5cm,angle=90]{plots/w_sxds.eps}
\caption{Angular correlation functions for the mock SXDS catalogues at
$z=3.1$ \textit{(Left)}, $z=3.6$ \textit{(Center)} and $z=5.7$
\textit{(Right)}. The blue circles show the median \wtheta\ from the
mock catalogues (after applying the IC correction). The dark and light
grey shaded regions respectively show the 68\% and 95\% ranges of the
distribution of \wtheta\ measured in the mock catalogues. The red open
circles are the Model \wtheta\ calculated using the full simulation
width, averaged over many slices. The green circles show the
observational data from Ouchi et al. The dashed lines show the
power-law fit to the observed \wtheta\ (green) and the median fit to
\wtheta\ from the mock catalogues (blue). The amplitudes $A_w$ of
these fits are also given in the figure. }
\label{fig.w_sxds}
\end{figure*}
We build mock SXDS catalogues following the same procedure as outlined
above. Fig.~\ref{im.sxds} shows examples of our mock catalogues for
each redshift. As in the previous case, we see that \mbox{$\rmn{Ly}\alpha$}\ emitters on
average trace the higher density regions of the dark matter
distribution. The real surveys have a well defined angular
size. However, the area sampled is slightly different at each
redshift. In order to keep the cross-like shape in our mock catalogues
and be consistent with the exact area surveyed, we scaled the
cross-like shape to cover the same angular area as the real survey at
each redshift.
Fig.~\ref{fig.ng_sxds} shows the distribution of the number of
galaxies in the mock catalogues for the three redshifts surveyed. The
median number of galaxies in the mocks at $z=3.1$ is 316, which
is remarkably similar to the observed number, 356. The 10-90
percentile range of the mocks covers 256--379 galaxies. The
coefficient of variation is $C_v = 0.19$, less than half the value
found for the MUSYC mock catalogues, $C_v = 0.41$. This reduction is
due mainly to the larger area sampled by the SXDS survey. In the
second slice ($z=3.6$), the redshift is only slightly higher than in
the previous case, but the number of galaxies is much lower. Looking
at the top panel of Fig.~\ref{fig.clf} we see that the observed LFs
are basically the same for these two redshifts. The difference between
the two samples is explained mostly by the different \mbox{$\rmn{Ly}\alpha$}\ flux limits
($1.2 \times 10^{-17} \funits$ for $z=3.1$ and $2.6 \times 10^{-17}
\funits$ for $z=3.6$). For the $z=3.6$ mocks, we find a median number
of 80 and 10-90\% range 60--110, in reasonable agreement
with the observed number of galaxies, 101. The fractional variation in
the number of galaxies, quantified by $C_v = 0.31$, is larger than in
the previous case, due to the smaller number of galaxies. The $z=5.7$
case is similar to the lower redshifts. The median number in the mocks
is 329, with 10-90\% range 255--407, again consistent with
the observed number, 401. The coefficient of variation for this survey
is $C_v = 0.23$, so the sampling variance is intermediate between that
for the $z=3.1$ and $z=3.6$ surveys.
The angular correlation functions of the mock catalogues are compared
with the real data in Fig.~\ref{fig.w_sxds}. The observational data
shown are preliminary angular correlation function measurements in the
three SXDS fields, with errorbars based on bootstrap resampling
(M. Ouchi, private communication). As in our comparison with the
MUSYC survey, we plot the median correlation function measured from
the mocks, after applying the IC correction, as a representative
\wtheta\ . As before, we also perform a $\chi^2$ fit of a power law to
the \wtheta\ measured in each mock, and to the observed values, to
determine the amplitude $A_w$. The fit is performed over the range $1
< \theta < 10 \arcm$ as before.
The left panel of Fig.~\ref{fig.w_sxds} shows the correlation
functions at $z=3.1$. According to both the error bars on the
observational data, and the scatter in \wtheta\ in the mocks (shown by
the grey shading), this survey provides useful constraints on the
clustering for $1 \la \theta \la 10 \arcm$, but not for smaller or
larger angles, where the scatter becomes very large. The fitted
amplitude $A_w$ for the observed correlation function is $A_w
=(0.32\pm 0.22)$ (95\% confidence, using the error bars reported by
Ouchi et al.), somewhat below the median value found in the mocks, $A_w
= 0.60$, but within the 95\% range for the mocks ($A_w =
0.23-1.35$). Based on the mocks, the model correlation function is
consistent with the SXDS data at this redshift.
Comparing these results with those we found for the MUSYC survey
(which has a very similar redshift and flux limit to SXDS at $z=3.1$),
we see that the results seem very consistent. The MUSYC survey has a
larger sample variance than SXDS, but the measured clustering
amplitude is very similar in the two surveys.
The middle panel of Fig.~\ref{fig.w_sxds} shows the correlation
function for the $z=3.6$ survey. In this case, the error bars on the
observational data and the scatter in the mocks are both larger, due
to the lower surface density of galaxies in this sample. For the
observed correlation amplitude, we obtain $A_w = 0.75 \pm 0.72$, while
for the mocks we find a median $A_w = 0.99$, with 95\% range
0.06--2.01, entirely consistent with the observational data.
The right panel of Fig.~\ref{fig.w_sxds} shows the correlation
function predictions for $z=5.7$. According to the spread of mock
catalogue results, the \wtheta\ measured here is the most accurate of
the three surveys, due to the large number of galaxies. For the mocks,
we find a median correlation amplitude $A_w = 0.82$, with 95\% range
0.42--1.49. For the observations, we find $A_w= 1.56 \pm 0.27$, if we
assume a slope $\delta=0.8$. The average correlation function in the
mocks agrees well with this slope over the range fitted, but the
observational data for \wtheta\ at this redshift prefer a flatter
slope. The model is however still marginally consistent with the
observational data at 95\% confidence. Similarly flat shapes were also
found in some previous surveys \citep{shimazaku04, hayashino04} in the
same field, but at redshifts $3.1$ and $4.9$ respectively. However,
these surveys are much smaller in terms of area surveyed and number of
galaxies (this is particularly so in \citet{shimazaku04}). This
behaviour in \wtheta\ might be produced by the high density regions
associated with protoclusters in the SXDS fields (M. Ouchi, private
communication), but still this behaviour of \wtheta\ must be confirmed
to prove that it is a real feature of the correlation function.
\subsection{ELVIS Survey}
\begin{figure}
\centering
\includegraphics[width=9cm]{plots/elvis_lores.eps}
\caption{An example of a mock catalogue for the ELVIS Survey. The
image shows the observed field of view (four strips). The legend and
colour format are the same as in Figs.\ref{im.musyc} and
\ref{im.sxds}.}
\label{im.elvis}
\end{figure}
One of the main goals of the public surveys planned for the Visible
and Infrared Survey Telescope for Astronomy (VISTA) is to find a
significant sample of very high redshift $(z \sim 8.8)$ \mbox{$\rmn{Ly}\alpha$}\
emitters. This program is called the Emission-Line galaxies with VISTA
survey (ELVIS) \citep{kim07a,kim07b}. The plan for ELVIS is to image
four strips of $11.6 \arcmin \times 68.27 \arcmin$, covering a total area
of 0.878$\rmn{deg}^2$, as shown in
Fig.~\ref{im.elvis}. This configuration is dictated by the layout of
the VISTA IR camera array. The only current detections of \mbox{$\rmn{Ly}\alpha$}\
emitters at $z>8$ are those of \citet{stark07}, which have not yet
been independently confirmed. \mbox{$\rmn{Ly}\alpha$}\ emitting galaxies at such
redshifts will provide us with valuable insights into the reionization
epoch of the Universe, as well as galaxy formation and evolution.
\begin{figure}
\centering
\includegraphics[width=7cm]{plots/ngal_elvis.eps}
\caption{Histogram of the number of galaxies in mock catalogues
expected for the ELVIS Survey. The red line shows the median of the
distribution, and the blue dashed lines the 10-90 percentiles of the
distribution.}
\label{fig.ngal_elvis}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{plots/w_elvis.eps}
\caption{Angular Correlation Functions in the mock catalogues of the
ELVIS Survey. The blue circles shows the median \wtheta\ from the mock
catalogues, while the orange circles shows the mean. The dark and
light grey shaded regions respectively show the 68\% and 95\% ranges
of the distribution of \wtheta\ in the mocks. The red open circles
show the Model \wtheta\, obtained using the full width of the
simulation box. The amplitude and slope of the median power-law fit to
the mocks are also given. }
\label{fig.w_elvis}
\end{figure}
For our mock ELVIS catalogues, we select galaxies with a minimum flux
of $\flya = 3.7 \times 10^{-18} \funits$ and $\ewobs > 100$ \AA{}, as
listed in Table~\ref{table.mock}. (The \ewobs\ limit is just a rough
estimate, although our predictions should not be sensitive to the
exact value chosen.) Fig.~\ref{im.elvis} shows the expected spatial
distribution of $z=8.5$ galaxies in one of the ELVIS mock
catalogues. Each mock catalogue has four strips, matching the
configuration planned for the real survey. The median number of
galaxies within the mock catalogues is 20, with a 10-90 percentile
spread of 14 to 29 galaxies, as shown in
Fig.~\ref{fig.ngal_elvis}. The fractional variation in number between
different mocks is $C_v=0.37$, which is quite large, but no worse than
for the MUSYC survey at $z=3.1$, even though that survey has 10 times
as many galaxies.
The angular correlation functions of the mock ELVIS catalogues were
calculated in the same way as before (including the integral
constraint correction). Fig.~\ref{fig.w_elvis} shows the median of the
\wtheta\ values measured from each mock catalogue (blue circles), and
also the mean (orange circles). In this case, the distribution of
\wtheta\ values within each angle bin is very skewed, due to the small
number of galaxies in the mock catalogues, and so the mean and median
can differ significantly. The dark and light grey shaded regions show
the ranges containing 68\% and 95\% of the \wtheta\ values from the
mocks, from which it can be seen that the cosmic variance for this
survey is very large. We also show the Model \wtheta\ (red circles),
which provides our best estimate of the true correlation function
based on the Millennium simulation, and was calculated by averaging
over 10 slices of the simulation, using the full width of the
simulation box. Even here, the error bars on \wtheta\ are quite large,
due to the very low number density of galaxies predicted. We see that
the mean and median \wtheta\ in the mocks lie close to the Model
values for $2\arcm < \theta < 20\arcm$, so in this sense they provide
an unbiased estimate.
The most noticeable feature of Fig.~\ref{fig.w_elvis} is the large
area covered by both the 68\% and 95\% ranges of the distribution of
\wtheta\ in the mocks, which extend down to $w(\theta)=0$. This
indicates that the ELVIS survey will only be able to put a weak upper
limit on the clustering amplitude of $z \sim 8.8$ \mbox{$\rmn{Ly}\alpha$}\ emitters, if
our model is correct. As before, we can quantify this by fitting a
power law to \wtheta\ in our mocks. We notice that the Model \wtheta\
for this sample has a significantly steeper slope, $\delta = 1.2$,
than the canonical value $\delta = 0.8$, and so we do our fits to the
mocks using $\delta = 1.2$. We find a median amplitude in the mocks
$A_w = 3.57^{+12.0}_{-33.5}$, where the error bars give the 95\%
range.
\section{Summary and conclusions}
We have combined a semi-analytical model of galaxy formation with a
high resolution, large volume N-body simulation to make predictions
for the spatial distribution of \mbox{$\rmn{Ly}\alpha$}\ emitters in a $\Lambda$CDM
universe.
Our model for \mbox{$\rmn{Ly}\alpha$}\ emitters is appealingly simple. Using the star
formation history predicted for each galaxy from the semi-analytical
model to compute the production of Lyman continuum photons, we find
that on adopting a constant escape fraction of \mbox{$\rmn{Ly}\alpha$}\ photons the
observed number of \mbox{$\rmn{Ly}\alpha$}\ emitters can be reproduced amazingly well
over a range of redshifts \citep{dell2}. Our modelling of \mbox{$\rmn{Ly}\alpha$}\
emission may appear overly simplistic on first comparison to other
calculations in the literature. For example, \citet{nagamine08}
predicted the clustering of \mbox{$\rmn{Ly}\alpha$}\ emitters in a gas-dynamic
simulation, modelling the \mbox{$\rmn{Ly}\alpha$}\ emission through a \mbox{$\rmn{Ly}\alpha$}\ escape
fraction or a duty cycle scenario. However, the fraction of active
emitters in the duty cycle scenario needs to be tuned at each
redshift, for which there is no physical justification. Since our
predictions for \mbox{$\rmn{Ly}\alpha$}\ emission are derived from a full model of galaxy
formation, it is straightforward to extract other properties of the
emitters, such as their stellar mass or the mass of their host dark
matter halo (Le Delliou et~al. 2006). In this paper we have extended
this work to include explicit predictions for the spatial and angular
clustering of \mbox{$\rmn{Ly}\alpha$}\ emitters.
We have studied how the clustering strength of \mbox{$\rmn{Ly}\alpha$}\ emitters depends
upon their luminosity as a function of redshift. Generally, we find
that \mbox{$\rmn{Ly}\alpha$}\ emitters show a weak dependence of clustering strength on
luminosity, until the brightest luminosities we consider are reached.
At the present day, \mbox{$\rmn{Ly}\alpha$}\ emitters display weaker clustering than the
dark matter. This changes dramatically at higher redshifts ($z > 3$),
with currently observable \mbox{$\rmn{Ly}\alpha$}\ emitters predicted to be much more
strongly clustered than the dark matter, with the size of the bias
increasing with redshift. We compared the simulation results with
analytical estimates of the bias. Whilst the analytical results show
the same trends as the simulation results, they do not match well in
detail, and this supports the use of an N-body simulation to study
the clustering.
A key advantage of using semi-analytical modelling is that the
evolution of the galaxy population can be readily traced to the
present day. This gives us some confidence in the star formation
histories predicted by the model. The semi-analytical model passes
tests on the predicted distribution of star formation rates at high
redshift (the number counts and redshifts of galaxies detected by
their sub-millimetre emission and the luminosity function of
Lyman-break galaxies), whilst also giving a reasonable match to the
present day galaxy luminosity function (Baugh et~al. 2005), and also
matching the observed evolution of galaxies in the infrared
\citep{lacey08}. Gas dynamic simulations as a whole struggle to
reproduce the present-day galaxy population, due to a combination of a
limited simulation volume (set by the need to attain a particular mass
resolution) and a tendency to overproduce massive galaxies. The small
box size typically employed in gas dynamic simulations means that
fluctuations on the scale of the box become nonlinear at low
redshifts, and their evolution can no longer be accurately modelled. A
further consequence of the small box size is that predictions for
clustering are limited to small pair separations
(e.g. \citet{nagamine08} use a box of side 33 \mbox{$ h^{-1} \rmn{Mpc}$}, limiting their
clustering predictions to scales $r \la 3 \mbox{$ h^{-1} \rmn{Mpc}$}$). By using a
simulation with a much larger volume than that of any existing \mbox{$\rmn{Ly}\alpha$}\
survey, we can subdivide the simulation box to make many mock
catalogues to assess the impact of sampling fluctuations (including
cosmic variance) on current and future measurements of the clustering
of \mbox{$\rmn{Ly}\alpha$}\ emitters.
We made mock catalogues of \mbox{$\rmn{Ly}\alpha$}\ emitters to compare with the MUSYC
($z=3$) and SXDS ($z=3-6$) surveys, and to make predictions for the
forthcoming ELVIS survey at $z \sim 9$. In the case of MUSYC and SXDS,
we found that the observed number of galaxies lies within the 10-90
percentile interval of the number of \mbox{$\rmn{Ly}\alpha$}\ emitters found in the
mocks.
We find that high-redshift clustering surveys underestimate their
uncertainties significantly if
they fail to account for cosmic variance in their error budget.
Overall, the measured angular correlation functions are
consistent with the model predictions. The clustering results in our
mock catalogues span a wide range of amplitudes due to the small
volumes sampled by the surveys, which results in a large cosmic
variance. ELVIS will survey \mbox{$\rmn{Ly}\alpha$}\ emitters at very high redshift ($z =
8.8$). Our predictions show that a single pointing will be strongly
affected by sample variance, due to the small volume surveyed and the
strong intrinsic clustering of the \mbox{$\rmn{Ly}\alpha$}\ emitters which will be
detected at this redshift. Many ELVIS pointings will be required to
get a robust clustering measurement.
We have shown that surveys of \mbox{$\rmn{Ly}\alpha$}\ emitters can open up a new window
on the high redshift universe, tracing sites of active star
formation. With increasing redshift, the environments where \mbox{$\rmn{Ly}\alpha$}\
emitters are found in current and planned surveys become increasingly
unusual, sampling the galaxy formation process in regions that are
likely to be proto-clusters and the progenitors of the largest dark
matter structures today. Our calculations show that with such strong
clustering, surveys of \mbox{$\rmn{Ly}\alpha$}\ emitters covering much larger
cosmological volumes are needed in order to minimize cosmic variance
effects.
\section*{Acknowledgements}
AO gratefully acknowledges a STFC Gemini studentship and support of
the European Commission's ALFA-II programme through its funding of the
Latin-American European Network for Astrophysics and Cosmology
(LENAC). CGL acknowledges support from the STFC rolling grant for
extragalactic astronomy and cosmology at Durham. CMB is supported by
a University Research Fellowship from the Royal Society. { We are very grateful
to Masami Ouchi and collaborators for kindly providing us with their
clustering measurements in advance of publication.}
We acknowledge a helpful report from the referee.
|
2,877,628,089,525 | arxiv | \section{Introduction}
In spite of the steadily increasing precision of the N-body simulations of astronomical systems, spherical analytic models are still ubiquitous in astrophysics and cosmology, ranging from globular clusters to galaxies, and from regular galaxy clusters to dark matter halos. Examples are many. While the Plummer sphere model~\citep{plummer} provides a good description of stars in globular clusters, the Jaffe~\citep{jaffe} and Hernquist~\citep{hernquist} ones are routinely used to represent the stellar distribution in spheroidal elliptical galaxies and bulges. Further, the pseudo-isothermal profile (see e.g.~\citealt{pseudoisothermal}) correctly describes the intermediate part of dark matter halos of spiral galaxies while the Navarro-Frank-White (NFW) model~\citep{nfw} represents the galaxy distribution in regular clusters and
provides a good analytic fit to the N-body simulations of pure dark matter halos within the standard $\Lambda$CDM context.
The aim of the present paper is twofold. First, we consider a {\it generic spherically symmetric} galactic model whose integrated mass profile $M (r)$ -- defined by Eq. (\ref{a2}) below -- is such that $M (r) \to 0$ as $r \to 0$. And we stress that -- because of a certain statement -- the considered model is {\it physically consistent} near the centre {\it only} provided that two conditions are satisfied: the circular velocity $v_c (r) \to 0$ and the gravitational field $g (r) \to 0$ as $r \to 0$ (they are defined by Eqs. (\ref{a5}) and (\ref{a6}) below). Second, we apply such a statement to a broad class of five-parameter spherical galactic models, which includes most of those used in astrophysics and cosmology, in particular the three-parameter family of Dehnen profiles~\citep{dehnen,dehnengerhard,tremaine1994}, the Jaffe, the Hernquist and the pseudo-isothermal spherical models, as well as the NFW, the Plummer sphere, the modified Hubble~\citep{binneytremaine} and the perfect sphere~\citep{perfectsphere} profiles~\footnote{Observe that the \citet{dehnen} and \citet{tremaine1994} models coincide up to a simple rescaling of the radius.}.
Surprisingly, nobody seems to have realized that the considered statement leads to new important results.
The paper is organized as follows. In Sect. 2 we emphasize the above statement, while in Sect. 3 we describe our broad class of five-parameter spherical galactic models. Sect. 4 is devoted to the application of the statement in question to a few considered models from a purely mathematical point of view. We see that for some of them -- e.g. the Jaffe, Herquist and NFW profiles -- as the central distance decreases the gravitational field either monotonically decreases becoming nonvanishing in the centre (Hernquist, NFW) or infinite there (Jaffe). In Sect. 5 we analyze the Jaffe and Hernquist mass models for {\it real} spheroidal elliptical galaxies and bulges. We find that, when the Jaffe and Hernquist models are used to describe the stellar population of spheroidal elliptical galaxies and bulges, the presence of the central supermassive black hole avoids the pathological behaviour in the neighbourhood of the centre. But we show that nonetheless both models can {\it only be trusted for} $r \gtrsim 0.2 \, R_e$ ($R_e$ being the effective radius). In Sect. 6 we discuss the NFW model for the distribution of galaxies in regular clusters. Sect. 7 addresses the application of the
NFW model to pure dark matter halos. In either case, the NFW model loses its validity towards the centre, thereby failing to predict a central cusp both in the distribution of galaxies in regular clusters and in pure dark matter halos. Moreover, we discuss the observed gamma-ray excess from the Galactic Centre (GC), its relationship with the NFW cusp and some alternative explanations. Finally, in Sect. 8 we draw our conclusions.
\section{Discussion of the statement}
Before committing ourselves with any specific model described by a spherically symmetric density profile $\rho (r)$, we stress a statement which ensures that {\it a given galactic model} makes sense close to the centre.
\
\noindent {\bf STATEMENT:} Suppose that an arbitrary spherically symmetric galactic model defined by the density profile $\rho (r)$ has integrated mass profile
\begin{equation}
M (r) \equiv 4 \pi \int_0^r d r^{\prime} ~ r^{\prime 2} \, \rho (r^{\prime})
\label{a2}
\end{equation}
such that
\begin{equation}
\label{a1a}
\lim_{r \rightarrow 0} M (r) = 0~.
\end{equation}
Then, in order for the mass model in question to be {\it physically consistent} near the centre the following two conditions should be met
\begin{equation}
\label{a1b}
\lim_{r \rightarrow 0} v_c^2 (r) = 0~,
\end{equation}
\begin{equation}
\label{a1c}
\lim_{r \rightarrow 0} g (r) = 0~,
\end{equation}
where $v_c (r)$ denotes the circular velocity while $g (r)$ stands for the gravitational field.
Indeed, given the fact that in the centre the mass is zero, a vanishing mass must give rise to a vanishing circular velocity and a vanishing gravitational field in the centre. So, whenever condition (\ref{a1a}) is met but one of the conditions (\ref{a1b}) or (\ref{a1c}) is not, then the considered mass model loses its physical meaning in the neighbourhood of the centre.
\section{A five-dimensional class of spherically symmetric galactic models}
Our models are defined by the following mass density profile
\begin{equation}
\rho ( r ) = \rho_0 \left(\frac{r}{a} \right)^{- \gamma} \left[ 1 + \left(\frac{r}{a} \right)^{\alpha} \right]^{(\gamma - \beta)/\alpha}~,
\label{a1}
\end{equation}
where $\rho_0$, $a$ are arbitrary positive constants, and $\alpha$, $\beta$, $\gamma$ are arbitrary parameters. Models of this sort are mentioned but not thoroughly discussed
by Mo, van den Bosch and White (\citealt{whitebook}, see also \citealt{binneytremaine}).
We shall see that for some of them $M (r)$ behaves as $M (r) \to \infty$ for $r \to \infty$, but this fact does not bother us, since realistic astronomical systems are spatially bounded with radius ${\cal R}$, hence the considered mass models should be cut at $r = {\cal R}$. Of course, such a truncation can affect other properties of the models, like for instance isothermality in the case of the regular isothermal sphere~\citep{binneytremaine}. In addition, we shall encounter models which exhibit a central density profile $\rho = {\rm constant}$ which is called a {\it central core}, whereas other models display a {\it central cusp}, namely they have $\rho (r) \to \infty$ as $r \to 0$. {\it A priori}, nobody worries about a central cusp since an infinite central density is not against any physical principle: indeed, the density is merely a derived quantity which cannot be directly measured, and what matters are the integrated mass profile, the circular velocity and the gravitational field: only $v_c (r)$ and $g (r)$ are directly measurable quantities.
Actually, the main point behind the present analysis is that -- given a certain density profile $\rho (r)$ -- it cannot absolutely be taken for granted that the observable quantities $v_c (r)$ and $g (r)$ possess a physically sensible behaviour towards the centre. Surprisingly, even though several properties of some models included in the considered family have been carefully analyzed, close to the centre so far insufficient attention has been paid to the circular velocity and no attention whatsoever to the gravitational field (with the exception of the regular isothermal sphere,~\citealt{binneytremaine}).
We should mention that after this Paper was nearly finished we have become aware of the exhaustive analysis of the same class of models described by Eq. (\ref{a1}) carried out in 1996 by Zhao~\citep{zhao}. Nevertheless, the overlap between the two Papers is nearly vanishing, since also Zhao does not consider the behaviour of the gravitational {\it field} $g (r)$. As far as the notations are concerned, the reader can recover Zhao's counterpart of our Eq. (\ref{a1}) by the replacements $\rho_0 \to C$, $r/a \to r$ and $\alpha \to 1/\alpha$.
\section{Mathematical discussion}
Starting from Eq. (\ref{a1}), the integra\-ted mass profile
reads
\begin{equation}
M (r) = 4 \pi \rho_0 \, a^3 \int_0^{r/a} d t ~ t^{(2 - \gamma)} \,
\bigl(1 + t^{\alpha} \bigr)^{(\gamma - \beta)/\alpha}~,
\label{a3}
\end{equation}
whose explicit form is
\begin{eqnarray}
\label{a4}
&\displaystyle M (r) = \frac{4 a^3 \pi \rho_0}{3 - \gamma} \left(\frac{r}{a} \right)^{3 - \gamma} \times \nonumber \\
&\displaystyle \times \, {_2 F}_1 \left[\frac{3 - \gamma}{\alpha}, \frac{\beta - \gamma}{\alpha}; 1+ \frac{3 - \gamma}{\alpha}; - \left(\frac{r}{a}\right)^{\alpha} \right]~,
\end{eqnarray}
where ${_2 F}_1\left(\cdot, \cdot; \cdot; \cdot \right)$ is the confluent hypergeometric function of the second kind. Correspondingly, the circular velocity and the gravitational field are defined as
\begin{equation}
v_c^2 ( r ) \equiv \left(\frac{G}{r} \right) M ( r )~,
\label{a5}
\end{equation}
\begin{equation}
g(r) \equiv - \, \left(\frac{G}{r^2} \right) M( r )~,
\label{a6}
\end{equation}
respectively~\citep{binneytremaine}. So, all we need to know is $M ( r )$.
Specifically, our task is to explicitly investigate the behaviour of $M ( r )$, $v_c^2 (r)$ and
$g (r)$ as $r \to 0$ for the above-mentioned models, even though our strategy can straightforwardly be extended to {\it arbitrary values} of $\alpha$, $\beta$ and $\gamma$.
In view of the forthcoming analysis it is therefore instrumental to evaluate $M(r)$, $v_c^2(r)$ and $g(r)$ as $r \to 0$ for $\alpha$, $\beta$ and $\gamma$ in specific ranges. We start with the case $\alpha = 1$,
$3 \leq \beta \leq 4$ and $0 \leq \gamma < 3$. Correspondingly, we find
\begin{equation}
\lim_{r \rightarrow 0} M(r) = 0~,
\label{Mgen1}
\end{equation}
while
\begin{equation}
\lim_{r \rightarrow 0} v_c^2(r) = \begin{cases}
0~, & 0 \leq \gamma < 2~, \\[8pt]
4 \pi G \rho_0 \, a^2~, & \gamma = 2~, \\[8pt]
\infty~, & 2 < \gamma <3~,
\end{cases}
\label{v2gen1}
\end{equation}
and
\begin{equation}
\lim_{r \rightarrow 0} g(r) = \begin{cases}
0~, & 0 \leq \gamma <1~, \\[8pt]
- 2 \pi G \rho_0 \, a~, & \gamma = 1~, \\[8pt]
- \infty~, & 1 < \gamma <3~,
\end{cases}
\label{Ggen1}
\end{equation}
for any value of $\beta$ in the above range. Next, we address the case $\alpha =2$, $2 \leq \beta \leq 5$ and $\gamma = 0$. Accordingly, we obtain
\begin{equation}
\label{M1a}
\lim_{r \rightarrow 0} M (r) = 0~,
\end{equation}
\begin{equation}
\label{M1b}
\lim_{r \rightarrow 0} v_c^2 (r) = 0~,
\end{equation}
\begin{equation}
\label{M1c}
\lim_{r \rightarrow 0} g (r) = 0~.
\end{equation}
regardless of the values of $\beta$ in the specified range. As a consequence, in the present case conditions (\ref{a1a}), (\ref{a1b}) and (\ref{a1c}) happen to be automatically satisfied.
Finally, we proceed to apply these results to the previously considered models.
\begin{enumerate}
\item {\bf NFW model} -- It corresponds to $\alpha=1$, $\beta=3$, $\gamma=1$. The integrated mass profile is
\begin{equation}
M(r) = 4 \pi \rho_0 \, a^3 \left[ {\rm ln} \left(1 + \frac{r}{a} \right)-\frac{r}{r+a}\right]~.
\label{NFW1}
\end{equation}
According to Eqs. (\ref{Mgen1}), (\ref{v2gen1}) and (\ref{Ggen1}), conditions (\ref{a1a}) and (\ref{a1b}) are met but condition (\ref{a1c}) is not, and we have $g (r) \to - 2 \pi G \rho_0 \, a$ as $r \to 0$. Thus, the NFW model exhibits a central {\it unphysical} cusp.
\item {\bf Dehnen models} -- They correspond to $\alpha=1$, $\beta=4$, $\gamma < 3$. The integrated mass profile is
\begin{equation}
M(r) = \frac{4 \pi \rho_0 a^3}{3 - \gamma} \left(\frac{r}{r + a} \right)^{3-\gamma}~.
\label{den1}
\end{equation}
Thanks to Eqs. (\ref{Mgen1}), (\ref{v2gen1}) and (\ref{Ggen1}), we see that conditions (\ref{a1a}) and (\ref{a1b}) are obeyed for $0 \leq \gamma < 2$ but conditions (\ref{a1c}) is satisfied for
$0 \leq \gamma < 1$. So, only for $0 \leq \gamma < 1$ are the Dehnen models physically consistent near the centre.
\item {\bf Hernquist model} -- It is the particular case of the Dehnen models with
$\gamma=1$. Hence, conditions (\ref{a1a}) and (\ref{a1b}) are met but condition (\ref{a1c}) is not, and we have $g (r) \to - 2 \pi G \rho_0 \, a$ as $r \to 0$. As a consequence, the Hernquist model is physically inconsistent in the neighbourhood of the centre.
\item {\bf Jaffe model} -- It is a particular case of the Dehnen models with $\gamma=2$. Thus, only condition (\ref{a1a}) is obeyed but conditions (\ref{a1b}) and (\ref{a1c}) are not, and we have $v_c^2(r) \to 4 \pi G \rho_0 \, a^2$ as $r \to 0$ and $g (r) \to - \, \infty$ as $r \to 0$. Therefore, the Jaffe model is physically inconsistent towards the centre.
\item {\bf Pseudo-isothermal sphere} -- It corresponds to $\alpha=2$, $\beta=2$, $\gamma=0$. The integrated mass profile is
\begin{equation}
M(r) = 4 \pi \rho_0 \, a^2 \left[ r - \, a \, {\rm tan}^{- 1} \left(\frac{r}{a}\right)\right]~.
\label{iso1}
\end{equation}
Owing to Eqs. (\ref{M1a}), (\ref{M1b}) and (\ref{M1c}), conditions (\ref{a1a}), (\ref{a1b}) and (\ref{a1c}) are met. Consequently, the Pseudo-isothermal sphere is physically consistent close to the centre.
\item {\bf Modified Hubble profile} -- It corresponds to $\alpha=2$, $\beta=3$, $\gamma=0$. The integrated mass profile is
\begin{eqnarray}
&\displaystyle M(r) = \frac{4 \pi \rho_0 \, a^3}{r^2+a^2} - \\ \label{Hub1}
&\displaystyle - \Bigg[ (r^2 + a^2) \, {\rm sinh}^{- 1} \left(\frac{r}{a} \right) - a \, r \left(1+ \frac{r^2}{a^2} \right)^{1/2} \Bigg]~. \nonumber
\end{eqnarray}
Due to Eqs. (\ref{M1a}), (\ref{M1b}) and (\ref{M1c}), conditions (\ref{a1a}), (\ref{a1b}) and (\ref{a1c}) are satisfied. Hence, the modified Hubble profile is physically consistent near the centre.
\item {\bf Perfect sphere model} -- It corresponds to $\alpha=2$, $\beta=4$, $\gamma=0$. The integrated mass profile is
\begin{eqnarray}
&\displaystyle M(r) = \frac{2 \pi \rho_0 \, a^3}{r^2+a^2} \times \\ \label{Perf1}
&\displaystyle \times \left[ (r^2+a^2) \, {\rm tan}^{- 1} \left(\frac{r}{a}\right) \nonumber
- a \, r \right]~.
\end{eqnarray}
Thanks to Eqs. (\ref{M1a}), (\ref{M1b}) and (\ref{M1c}), conditions (\ref{a1a}), (\ref{a1b}) and (\ref{a1c}) are met. So, the perfect sphere model is physically consistent in the neighbourhood of the centre.
\item {\bf Plummer sphere model} -- It corresponds to $\alpha=2$, $\beta=5$, $\gamma=0$. The integrated mass profile is
\begin{equation}
M(r) = \frac{4}{3}\pi \rho_0 r^3 \left(1+ \frac{r^2}{a^2} \right)^{-3/2}~.
\label{Plummer1}
\end{equation}
On account of Eqs. (\ref{M1a}), (\ref{M1b}) and (\ref{M1c}), conditions (\ref{a1a}), (\ref{a1b}) and (\ref{a1c}) are obeyed. So, the Plummer sphere model is physically consistent towards the centre.
\end{enumerate}
\section{Real spheroidal ellipticals and bulges}
The analysis carried out so far is formal in nature, since it merely refers to specific abstract models. For instance, models describing the stellar distribution inside spheroidal elliptical galaxies and bulges are invalid close to the centre because of the presence of a supermassive black hole (SMBH). Nonetheless, our previous results are important because they are alarm bells that some models can be pathological also {\it beyond} the SMBH. Below, we will carefully analyze the behaviour of such models in their realistic context.
\
The Dehnen models -- and in particular the Jaffe and Hernquist ones -- have routinely been used to represent the stellar distribution within spheroidal elliptical galaxies and bulges (just to quote a few papers, see~\citealt{hui1995,rix1997,gerhard1998,saglia2000,cappellari2006}).
Let us therefore discuss the effect of the central SMBH on a generic Dehnen model. Here, the relevant quantity is the {\it dynamical radius} ${\cal R}_g$, where the gravitational field of the SMBH and of the host galaxy are equal~\citep{binneytremaine}. We neglect the dark matter, because the central region of ellipticals and bulges is believed to be baryon dominated. It is then trivial to find that
${\cal R}_g$ is given by
\begin{equation}
{\cal R}_g = a \left\{\left[\frac{4 \pi \rho_0 \, a^3}{(3 - \gamma)M_{\rm SMBH}}\right]^{\frac{1}{3-\gamma}} - \,1 \right\}^{-1}~,
\label{rg}
\end{equation}
but since the term inside the square brackets is obviously much larger than 1 -- defining $M_e \equiv \bigl(4 \pi \rho_0 \, a^3 \bigr)/3$ -- Eq. (\ref{rg}) boils down to the following approximate expression
\begin{equation}
{\cal R}_g \simeq a \left(1-\frac{\gamma}{3} \right)^{\frac{1}{3-\gamma}} \left( \frac{M_{\rm SMBH}}{M_e}\right)^{\frac{1}{3 - \gamma}}~.
\label{rgapprox}
\end{equation}
Several relationships exist in the literature between $M_{\rm SMBH}$ and the central
one-dimensional velocity dispersion of the host galaxy $\sigma (0)$ as evaluated within a given aperture (this point has been carefully discussed in~\citealt{tremaine2002}). In order to be specific -- choosing the aperture $R_e/8$ -- $M_{\rm SMBH}$ reads~\citep{merrittferrarese2001}
\begin{equation}
M_{\rm SMBH} \simeq 1.30 \cdot 10^8 \left(\frac{\sigma (0)}{200 \, {\rm km} \, {\rm s}^{- 1}} \right)^{4.72} \, M_{\odot}~.
\label{24022020a}
\end{equation}
Thus, we conclude that the Dehnen models can make sense for a galactocentric distance {\it larger} than ${\cal R}_g$ as provided by Eqs. (\ref{rg}) or (\ref{rgapprox}) (more about this, later).
\bigskip
\centerline{ \ \ \ * \ \ \ * \ \ \ * \ \ \ }
\medskip
As a next step, we focus our attention on the Jaffe and Hernquist models. Since $a = 0.55 \, R_e$ for the Hernquist model, and $a = 1.31 \, R_e$ for the Jaffe one, by specializing Eq. (\ref{rgapprox}) to these cases, we get
\begin{equation}
{\cal R}_{g, J} \simeq 0.44 \, R_e \left(\frac{M_{\rm SMBH}}{M_e}\right)~,
\label{rgapproxJ}
\end{equation}
and
\begin{equation}
{\cal R}_{g, H} \simeq 0.45 \, R_e \left(\frac{M_{\rm SMBH}}{M_e}\right)^{1/2}~.
\label{rgapproxH}
\end{equation}
So, only for galactocentric distances larger than either ${\cal R}_{g, J}$ or ${\cal R}_{g, H}$
can the Jaffe or the Hernquist models be regarded as a realistic description of the stellar population of spheroidal ellipticals and bulges.
Incidentally, a slightly different discussion of the Hernquist model is contained in (\citealt{binneytremaine}, see Fig. 4.20), where -- denoting by $M_g$ the mass of the galaxy -- for $M_{\rm SMBH} = 0.002 \, M_g$ and $M_{\rm SMBH} = 0.004 \, M_g$ it is found
${\cal R}_{g, H} \simeq 0.026 \, R_e$ and ${\cal R}_{g, H} \simeq 0.037 \, R_e$, respectively.
\bigskip
\centerline{ \ \ \ * \ \ \ * \ \ \ * \ \ \ }
\medskip
We prefer to work henceforth with the dimensionless quantities defined as follows.
\begin{enumerate}
\item Radial distance: $r/R_e$.
\item Mass density: $\rho/\rho_0$.
\item Integrated mass profile: $M (r)/ \bigl(4 \pi \, \rho_0 \, R_e^3 \bigr)$.
\item Circular velocity square: $v_c^2 (r)/\bigl(4 \pi G \, \rho_0 \, R_e^2 \bigr)$.
\item Gravitational field: $g (r)/\bigl(2 \pi G \, \rho_0 \, R_e \bigr)$.
\end{enumerate}
We will replace $R_e$ by $a_{\rm NFW}$ for the NFW model.
\
We are now in a position to assess the validity of the Jaffe and Hernquist models. Because we are interested to investigate in great detail what happens around the centre, we plot
$\rho/\rho_0$, $M (r)/ \bigl(4 \pi \, \rho_0 \, R_e^3 \bigr)$, $v_c^2 (r)/\bigl(4 \pi G \, \rho_0 \, R_e^2 \bigr)$ and $g (r)/\bigl(2 \pi G \, \rho_0 \, R_e \bigr)$ versus $r/R_e$ in logarithmic scales in Figs.~\ref{figC1},~\ref{figC2},~\ref{figC3} and~\ref{figC4}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{rhoHJloglog.pdf}
\caption{\label{figC1} We report on the vertical axis $\rho/\rho_0$ and on the horizontal axis $r/R_e$, both in logarithmic scale.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{mHJloglog.pdf}
\caption{\label{figC2} We show on the vertical axis $M (r)/\bigl(4 \pi \, \rho_0 \, R_e^3 \bigr)$ and $r/R_e$ on the horizontal axis, both in logarithmic scale.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{vHJloglog.pdf}
\caption{\label{figC3} We exhibit on the vertical axis $v_c^2 (r)/\bigl(4 \pi G \, \rho_0 \, R_e^2 \bigr)$ and $r/R_e$ on the horizontal axis, both in logarithmic scale.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{gHJloglog.pdf}
\caption{\label{figC4} We report on the vertical axis $g (r)/\bigl(2 \pi G \, \rho_0 \, R_e \bigr)$ and $r/R_e$ on the horizontal axis, both in logarithmic scale.}
\end{figure}
The departure from similarity of the two models takes place around $r \simeq 0.2 \, R_e$, where it starts to become larger and larger as the galactocentric distance gets smaller and smaller. Moreover, the circular velocity curve for the Hernquist model is physically very well behaved while for the Jaffe model it is not. In addition, the gravitational field does not show any turn towards 0 for both models. Fortunately, we can make sense out of such a behaviour by recalling that historically both models have been devised in order to reproduce the De Vaucouleurs surface brightness profile upon projection, assuming a constant mass-to-light ratio. Accordingly, their shape should nearly coincide at, say, $r = 2 \, R_e$, as indeed it takes place in the considered figures. We are thus led to the guess that both models fail to fit the De Vaucouleurs law in projection for $r \lesssim 0.2 \, R_e$. A check of our guess can be obtained by projecting these models onto the sky. The results are shown in Fig.~\ref{fig2306a}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{proj.pdf}
\caption{\label{fig2306a} We show the projected Jaffe and Hernquist models as well as the De Vaucouleurs law, both in logarithmic scale. In all cases ${\rm I}/{\rm I}_{\rm ref}$ is the dimensionless surface brightness.}
\end{figure}
We see that Fig.~\ref{fig2306a} beautifully shows that indeed both the Jaffe and Hernquist model {\it can only be trusted} for $r \gtrsim 0.2 \, R_e$, if we want to stick to a {\it constant} luminous mass-to-light ratio $\Upsilon_{\rm lum}$ for ${\cal R}_g < r \lesssim 0.2 \, R_e$ (our conversion from surface mass density to surface brightness has been performed by assuming $\Upsilon_{\rm lum} = {\rm constant}$). Taking these model seriously in the latter range an unphysical gradient in $\Upsilon_{\rm lum}$ would necessarily show up, which could be confused with a colour gradient or a gradient of the total mass-to-light ratio $\Upsilon_{\rm tot}$, which might erroneously be interpreted as evidence for dark matter.
\section{NFW model and regular galaxy clusters}
Let us come back to the NFW profile, whose explicit form is
\begin{equation}
\rho ( r ) = \rho_0 \left(\frac{r}{a_{\rm NFW}} \right)^{- 1} \left[ 1 + \left(\frac{r}{a_{\rm NFW}} \right) \right]^{- 2}~,
\label{24022010c}
\end{equation}
which we plot in a log-log diagram in Fig.~\ref{fig7a2}. Moreover, we get the dimensionless square circular velocity $v^2_c (r)/\bigl(4 \pi G \, \rho_0 \, a^2_{\rm NFW} \bigr)$ and the dimensionless gravitational field $g (r)/\bigl(2 \pi G \, \rho_0 \, a_{\rm NFW} \bigr)$ upon the replacement $a \to a_{\rm NFW}$ in Eq. (\ref{NFW1}) and by employing Eqs. (\ref{a5}) and (\ref{a6}), respectively. These quantities are plotted versus $r/a_{\rm NFW}$ in Figs. (\ref{fig3}) and (\ref{fig4}).
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{rhoNFWloglog.pdf}
\caption{\label{fig7a2} We exhibit on the vertical axis $\rho (r)/\rho_0$ and on the horizontal axis $r/a_{\rm NFW}$, both in logarithmic scale.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{vNFWloglog.pdf}
\caption{\label{fig3} We report on the vertical axis $v_c^2 (r)/\bigl(4 \pi G \, \rho_0 \, a^2_{\rm NFW} \bigr)$ and on the horizontal axis $r/a_{\rm NFW}$, both in logarithmic scale.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{gNFWloglog.pdf}
\caption{\label{fig4} We show on the vertical axis $g (r)/\bigl(2 \pi G \, \rho_0 \, a_{\rm NFW} \bigr)$ and on the horizontal axis $r/a_{\rm NFW}$, both in logarithmic scale.}
\end{figure}
Nowadays, the overall distribution of galaxies in regular clusters is believed to be well described by an NFW model with $a_{\rm NFW} = R_{\rm vir}/c_{\rm gal}$, where $R_{\rm vir}$ denotes the virial radius and the galaxy concentration $c_{\rm gal}$ ranges from $c_{\rm gal} = 3.7$~\citep{carlsberg1997} to $c_{\rm gal} = 4.2$~\citep{vandermarel2000}. Unfortunately, the galaxy distribution in the central region is more uncertain. According to Adami et al., the luminosity profile of the brightest galaxies is significantly cusped in the center of the clusters (regardless of the redshift), but the luminosity profile of the fainter galaxies is significantly better fitted by a cored model~\citep{adami2001}. But Lin et al. claim that {\it all} galaxies are distributed according to a model (\ref{a1}) with $\alpha = 1.07$, $\beta = 3$, $\gamma = 1$ and $c_{\rm gal} = 2.71$, which is almost undistinguishable from an NFW profile~\citep{lin2004}.
Manifestly, the behaviour exhibited in Fig.~\ref{fig4} -- anticipated in item (i) of Sect. 4 -- shows that the galaxy distribution {\it cannot be} represented by an NFW profile all the way down to the centre. We stress that the situation is presently worse as compared to the one discussed in Sect. 5, since we have no handle to tell at which distance from the centre the NFW model breaks down.
\section{NFW model and dark matter halos}
It is well known that the NFW profile provides the classic analytic fit to the N-body simulations of collisionless cold dark matter particles. Within this context we have $a_{\rm NFW} \equiv r_{200} /c_h$, where $r_{200}$ is the radius where the overdensity is $200$ times larger than the mean cosmic density -- currently considered as the virial radius -- while $c_h$ is the halo concentration parameter, which depends on both the halo mass and its redshift~(\citealt{whitebook}). But for the present analysis we do not need to commit ourselves with any specific value.
Thus, to the extent that dark matter halos are correctly described by the NFW model, no central cusp is predicted because the model {\it does not} make physical sense near the centre, as already anticipated in item (i) of Sect. 4. From an observational standpoint, the situation is identical to the one encountered in Sect. 6: while at large enough distances from the centre there is no doubt that the NFW model is physically consistent -- and observationally correct -- we are totally unable to tell at which central distance it breaks down. We nevertheless note that even though this looks disappointing from a conceptual point of view, it makes the NFW model {\it in agreement with observations}. Indeed, the cusp has not been found even where it should, namely in bulgeless galaxies like the low surface brightness and dwarf ones (see e.g.~\citealt{adams2014,oh2015} and references therein).
As far as the dark matter halos are concerned -- even if embedded in real galaxies -- the NFW cusp with mass density profile $\rho (r)$ has been used by many researchers for a very specific purpose. But to appreciate this point some preliminaries are compelling. Let us assume that the non-baryonic cold dark matter is made of standard weakly interacting massive particles (WIMPs), which couple to ordinary ones with the weak interaction strength. Then, their mass $M_{\rm WIMP}$ must obey the lower bound $M_{\rm WIMP} \gtrsim 45 \, {\rm GeV}$, since otherwise they would have been detected at CERN as decay products of the ${\rm Z}^0$ (but some unconventional WIMPs can avoid such a bound, see e.g.~\citealt{escudero2017}). Even if WIMPs must be almost stable in order to survive until the present -- otherwise they could not be the dark matter today -- there is a small but nonvanishing probability that two of them annihilate into standard model particles. This possibility has been recognized long ago by many authors (for a review and a list of references, see~\citealt{jungman1996,bertone2005,berstrom2012}). Let us consider the centre of the Milky Way as an example. The crucial point is that -- since we are dealing with a two-body annihilation process -- the photon flux is proportional to the WIMP square density, namely $\rho^2 (r)$ (see Eq. (\ref{25112020a}) below). As a consequence, the presence of the NFW cusp would {\it greatly enhance the resulting photon flux}: this is the key-point. Either the fragmentation or the decay of the WIMP-produced particles would then give rise to photons which are manifestly expected to be in the gamma-ray band. The shape of the signal can be either a continuum or a single photon line. Denoting by $\psi$ the angle between the direction of the line of sight to a generic point P of the cusp and that to the Milky Way centre, the observed photon flux is
\begin{equation}
\label{25112020a}
\Phi (\psi) = \frac{\sigma v}{8 \pi} \sum_i B_i \, \frac{d N_i}{d E} \int_{\rm los} d l (\psi) ~
\left(\frac{\rho (l)}{M_{\rm WIMP}} \right)^2~,
\end{equation}
where the integration is performed along the line of sight (los) to P, $\sigma$ is the cross-section, $v$ is the WIMP relative velocity, $B_i$ is the branching ratio into the $i$-th final state, and $d N_i/d E$ is the corresponding differential photon yield (see e.g.~\citealt{wimp}).
What about observations towards the Galactic Centre? One should distinguish between lines and continuum.
Let us first address lines. Because WIMPs are supposed to be {\it cold} dark matter, their velocity dispersion is small. Therefore, in the case of a line arising from direct WIMP annihilation into two photons the energy of each of them should be very close to that of $M_{\rm WIMP}$. Alternatively, a line can arise from direct WIMP annihilation into a photon and a ${\rm Z}^0$ boson. From time to time, evidence for these lines has been claimed, but subsequent observations have never confirmed such a presumed detection. An example can be found
in~\citep{finkbeiner2013}.
Let us next turn our attention to the continuum. It lies below the WIMP mass and arises mainly from the decay of $\pi^0$ mesons produced in the final state fragmentation or annihilation. In 1998 the EGRET telescope aboard the Compton Gamma-Ray Observatory (CGRO) has reported a {\it gamma-ray excess from the GC} in the range $30 \, {\rm MeV} \lesssim E \lesssim 10 \, {\rm GeV}$~\citep{mayerH1998}. In a preliminary attempt, Cesarini et al. succeeded in explaining this excess in terms of the neutralino dark matter present in the minimal supergravity framework~\citep{wimp}. Subsequently, many authors expected a diffuse gamma-ray excess from the Milky Way centre. In 2017 the {\it Fermi}/LAT detector aboard the {\it Fermi} satellite reported {\it solid evidence for a gamma-ray excess from the GC}, as compared to naive expectations~\citep{ackermann2017}. The spectral energy distribution is not reported by the {\it Fermi collaboration}, which merely states that `{\it the excess at the GC around a few GeV is statistically significant}'. While nobody casts doubts about this result, its interpretation is a matter of strong debate. The spectrum of the excess is in good agreement with the assumption that a population of WIMPs with mass in the range $20 \, {\rm GeV} \lesssim M_{\rm WIMP} \lesssim 65 \, {\rm GeV}$ annihilate into standard model particles, which ultimately produce gamma-ray photons. Moreover, the shape of the excess spectrum is uniform across the inner Galaxy and exhibits approximate spherical symmetry within at least $10^{\circ}$ from the GC. In addition, according to the WIMP interpretation, it is required for the WIMP mass density $\rho (r) \propto r^{- 1.2}$, {\it consistent with a NFW cusp}, presumably made slightly steeper by the baryonic compression. Finally, it should be needed $\sigma v \sim 10^{- 26} \, {\rm cm}^{3} \,
{\rm s}^{- 1}$, which is indeed the typical value required for standard WIMPs to be produced through thermal freeze-out in the early Universe in order to be the right amount of non-baryonic cold dark matter today~\citep{calore2015,daylan2016}.
Beautiful as it is, this scenario is actually very hard to believe. In the first place, the {\it Fermi} collaboration remarks that `{\it however, control regions along the Galactic plane, where a DM signal is not expected, show excesses of similar amplitude relative to the local background}'~\citep{ackermann2017}. As already pointed out, observations have failed to detect a NFW-like cusp also in bulgeless galaxies, where no source confusion prevents its observability. Moreover, our results points to the same conclusion, namely no NFW cusp exists. Finally, the above sort of WIMPs find their natural motivation only within supersymmetric extensions of the standard model, where supersymmetry is broken close enough to the Fermi scale $\sim \, 250 \, {\rm GeV}$, but the LHC at CERN -- whose centre-of-mass energy is 14 TeV -- has not detected any supersymmetric particle at all.
For completeness, we mention two alternative scenarios which could explain the diffuse gamma-ray excess surrounding the GC. One is a population of about 500 centrally located millisecond pulsars, as strongly suggested by the HAWC observatory~\citep{carlson2014,petrovic2014,hooper2017}. The other is a recent outburst of cosmic rays~\citep{cholis2015}.
Thus, to date the nature of the gamma-ray excess around the GC remains a mystery.
\section{Conclusions}
We have first stressed the statement according to which any spherically symmetric galactic model whose integrated mass profile $M (r) \to 0$ as $r \to 0$ is physically consistent in the neighborhood of the centre only provided that the circular velocity $v_c (r) \to 0$ as $r \to 0$, and the gravitational field $g (r) \to 0$ as $r \to 0$. We have next applied the considered statement to some most used models from a class of five-parameter self-gravitating spherical galactic models, which are most frequently used in astrophysics and cosmology, like the Hernquist, Jaffe and NFW ones.
As is well known, the stellar population of spheroidal elliptical galaxies and of bulges are often described by either the Jaffe model or by the Hernquist one. We have shown that in both cases -- even taking the central SMBH into account -- they can be trusted only for galactocentric distances {\it larger than about 0.2 effective radii}.
We have next addressed the distribution of galaxies in regular clusters and the density profiles of pure dark matter halos, which are both believed to be well represented by an NFW model, even if with different values of the parameters. We have demonstrated that in either case such a description must break down towards the centre without a central cusp as that instead predicted by the NFW model. As far as regular clusters are concerned, the central galactic density profile can be determined by improving photometry, while for pure dark matter halos it is unclear whether N-body simulations with resolution better than about $1 \, {\rm kpc}$ solve the problem {\it before} the baryonic infall. Moreover -- at present -- in both cases we have no idea where such a failure starts to takes place.
In addition, we have schematically discussed the observed gamma-ray excess around the Milky Way centre. Our conclusion is that such a gamma-ray excess cannot be explained in terms of a cusp-enhanced WIMP annihilation into gamma-ray photons, contrary to the claim of several authors.
Finally, recently the important possibility that baryons alter the halo density profile close to the galactic center after their infall -- in such a way to wipe out the dark matter cusp, replacing it with a core -- has started to be systematically investigated by employing models of the form (\ref{a1}) for suitable values of the parameters not considered here (see e. g.~\citealt{dekel2017,freundlich2020}). We plan to extend the present analysis to that very interesting case in a future publication.
\section*{Acknowledgments}
We warmly thank Patrizia Caraveo, Stefano Ettori, Ortwin Gerhard and Andrea Macci\`o for discussions and criticism, and Giancarlo Setti for a careful reading of the manuscript. The work of MR is supported by an INFN TAsP grant and GG acknowledges contribution from the grant ASI-INAF 2015-023-R.1.
|
2,877,628,089,526 | arxiv | \section{Introduction}
Non-equilibrium phenomena in strongly interacting many-body systems often
provide complex interactions between fluctuation and dissipation processes,
which constitutes an important field of ongoing research~\cite{benA00,Henk08}.
Fluctuations in one dimension are so large that mean field methods are
irrelevant, instead exact results are necessary but not always available. One
simple
model possessing strong fluctuations and critical behavior is represented by the
diffusion-coagulation process of indistinguishable particles on a discrete
and infinite chain where each site of elementary size $a$ contains at most one
particle.
The dynamics is defined by particles $A$ that can hop between neighboring sites
$A+\O\rightarrow\O+A$ or $\O+A\rightarrow A+\O$ with a rate $\Gamma$ and
eventually coagulate $A+A\rightarrow A$ when two particles meet on the same site
with probability unity. This model is exactly solvable and the density of
particles in the continuum limit, when the product $\Gamma a^2=:{\mathscr D}$
(diffusion coefficient) is kept constant when $a$ goes to zero, is known to
decrease with time like $t^{-1/2}$ (see~\cite{benA90} for a detailed review)
in the long-time limit (scaling regime), instead of $t^{-1}$ in the mean-field
approximation, implying strong fluctuation effects. Such effects
have been observed experimentally, in the kinetics of quasi-particles called
excitons on long chains of polymers TMMC=(CH$_3$)$_4$N(MnCl$_3$) \cite{Kroo93},
and in other types of almost one-dimensional polymers \cite{Pras89,Kope90}.
Interesting quantities such as two-point correlation functions and response
functions ~\cite{benA98,Maye07,Dura11} can be explicitly evaluated in the
continuum limit, invalidating the direct applicability of
the fluctuation-dissipation theorem. Introducing external sources is a usual
tool to probe the dynamics and influence of time scales in the different
transient regimes. Influence of sources was studied
in the case of uniform particle deposition with a given constant
rate~\cite{Racz85,Doer89,Rey97} or charge deposition~\cite{Takayasu91} on
random chosen sites in one dimensional chains, or even in
membranes~\cite{Frisch92}. In the coagulation-diffusion
model, the equation of diffusion for the probability of finding an empty
interval of size $x$ is modified by a source term proportional to the size $x$.
This
equation admits solutions in terms of the Airy function, with eigenvalues
proportional to the zeros of this function. It shows interestingly that no
first-order rate equation can be written explicitly, except in the asymptotic
regime near the stationary state. Relaxation behavior was also studied
in the one-dimensional charge aggregation model~\cite{Takayasu89,Takayasu91},
where particles can coagulate by addition of their charge, and time power law
or stretched exponential dependence was found by looking how an excitation
charge (or pair of opposite charges) is dissipated into the system using
the Green's function behavior in the long-time regime.
Here we consider the dynamics of a coagulation-diffusion process
on a finite and semi-infinite chain
with a source of particles at the origin and eventually an asymmetric hopping
rate. The aim is to probe the different time scale regimes and steady states,
by varying the input current and particle drift, or biased
diffusion. Finite size scaling was previously studied in the case
of no source term, with
open and periodic boundary conditions~\cite{Alca94,Krebs95a,Krebs95b,Hinr97}.
Scaling law for the particle concentration $\rho_L(t)\simeq L^{-1}F_0(8{\mathscr D}
t/L^2)$ in a
finite chain of size $L$ and diffusion constant ${\mathscr D}$ was derived and
expressed in particular with Jacobi theta functions, reflecting the
Gaussian or diffusive character of the Green's function. In
the following, we consider the possibility of having different crossover
regimes in
the case of an input current at the origin, which introduces another time scale
in the system, or coherent length, after the characteristic time of diffusion
through the system $L^2/8{\mathscr D}$ is reached, and from an initial state where
every site is occupied by a single particle.
Such a model was already studied in details with particle inputs and
asymmetric diffusion/coagulation rates in reference~\cite{Hinr97}. The
authors were able to extract different asymptotic regimes for the particle
density as function of input rate of particles and biased rates in the
stationary state. The case with infinite input rate at both ends
was also studied previously~\cite{Der95} in relation with the Potts
model in one dimension (see also \cite{cheng89}).
The analytical treatment presented in this paper is reminiscent of the
empty-interval
method conveniently used for deriving the exact two-point correlation and
response functions~\cite{Krebs95a,Maye07,Dura11}, in the transient and critical
regimes. We can express the average density in the non-stationary
regime with a scaling function as
$\rho_L(t)=L^{-1}F_0(8{\mathscr D} t/L^2,k_{in}^2L^2)$, where $k_{in}$ is the typical
momentum of the input current $I_{{in}}$ in the continuum limit, expressed as
$I_{{in}}=k_{in}^2{\mathscr D}$. This scaling behavior can be exactly derived from the
linear equation of motions for the empty-interval probability. Solving these
equations is based on a different method than~\cite{Hinr97} and is structured
as follow: first we write the boundary conditions
at both ends of the chain, dependent on the input current, and combine
continuity/differentiability relations that include these
boundary terms into a general Green's formalism. Then the continuum limit is
derived in part 3, as well as the different
transport quantities by using the expression of the empty-interval probability.
In parts 5 and 6, we solve the local density in the semi-infinite and finite
cases and study the existence of different regimes by identifying the
crossover between the scaling regime and the finite size effects at
later times, and compute the coagulation rate.
\section{Empty interval probability method}
\begin{figure
\begin{center}
\includegraphics[scale=0.4,clip]{fig_process.eps}
\caption{\label{fig1}Example of a chain of length $N=7$ filled
partially with particles (disks). One time processes occur when
particles diffuse to the left or right with rate $\Gamma$ and
$\alpha\Gamma$ respectively. Particles can exit the last site on the right with
rate $\alpha\Gamma$ and enter from the left with a different rate $\beta\Gamma$ (input
current).}
\end{center}
\end{figure}
We consider a finite one-dimensional chain of $N$ sites filled with particles
($\bullet$) or empty ($\circ$). Particles can diffuse inside the chain
with asymmetric rate $\alpha\Gamma$ to the right, with $\alpha\ge 1$, and with rate
$\Gamma$ to the left, see Fig. 1.
They can also merge (coagulation) on the same site with probability unity. A
flux of new particles is introduced from the left hand side of the chain
with rate $\beta\Gamma$. Comparing these notations with
reference~\cite{Hinr97}, we have the corresponding rates: $a_L=c_L=\Gamma$,
$a_R=c_R=\alpha\Gamma$ for the biased diffusion $a_{L,R}$ and coagulation $c_{L,R}$
rates, $p_L=\beta\Gamma$, and $p_R=0$ for the particle inputs on the left
and right ends of the chain respectively. The authors also introduced a
parameter $q=\sqrt{\alpha}$ which represents the asymmetric diffusion and an input
of particles at the origin $p=p_L$. We follow the
same conditions here and take an initial configuration where
the system is full of particles. They were able to compute exact asymptotic
regimes for the density:
\begin{itemize}
\item In the semi-infinite and discrete case:
Exact asymptotic expansions far from the origin as function of finite input
rate $p$ and drift ($q$ non equal to 1).
\item In the continuum case and semi-infinite system:
Exact density expansions far from the origin as function of finite input
rate and drift.
\item In the continuum case, for infinite input rate $p$, they expressed the density as an exact scaling function of finite ratio $x/L$ in the limit where the system size $L$ and $x$ large (equation 2.62 in their paper). Expansions for $p$ finite (equation 2.65) are also given.
\end{itemize}
In this paper we use a different
scaling regime (time is kept finite, eventually large) and develop a different
approach to solve the set of equations of motion by finding
appropriate solutions combining the characteristic lengths and time
variables into a scaling form. We would like in particular to study in the
non-equilibrium state and for any initial condition the transition between
massless (for time smaller than the diffusion characteristic time) and massive
regimes with the conditions discussed just above.
\subsection{Definition of the model and equations of motion in the discrete
case}
A convenient way to describe in general coagulation-diffusion processes is to
introduce the empty-interval probability $E_{n_1,n_2}(t)={\rm Pr}(n_1\,\fbox{ d
}\,n_2)$ for $0\le n_1\le n_2\le N$, $d=n_2-n_1$ \cite{Hinr97}, which physically
represents
the probability to have empty sites at
least inside the interval $[n_1,n_2]$ of size $d$. The boundary
condition of zero size interval is given by $E_{n_1,n_1}(t)=1$,
which is the probability to find no particle. Inside the chain, we can write the
following equation of evolution
\begin{eqnarray}\nonumber
\frac{\partial E_{n_1,n_2}(t)}{\partial t}=
{\rm Pr}(\stackrel{\curvearrowleft}{ } n_1\,\fbox{$\bullet$ d-1 }\,n_2)+{\rm Pr}(n_1\,\fbox{ d-1 $\bullet$}\,n_2\stackrel{\curvearrowright}{ })
\\
-{\rm Pr}(\bullet\stackrel{\curvearrowright}{ } n_1\,\fbox{ d }\,n_2)-{\rm Pr}(n_1\,\fbox{ d }\,n_2\stackrel{\curvearrowleft}{ }\bullet).
\end{eqnarray}
In this equation, the transition rate ${\rm Pr}(\stackrel{\curvearrowleft}{ } n_1\,\fbox{$\bullet$ d-1
}\,n_2)$ on the right hand side is the rate at which a particle
located in box $[n_1,n_1+1]$ and near an empty interval of size $d-1$ jumps on the
left site. It is equal to the product of the rate $\Gamma$ (or $\alpha\Gamma$
if it jumps on the right site) and the probability ${\rm Pr}(n_1\,\fbox{$\bullet$d-1
}\,n_2)$ that such initial configuration exists. The latter probability
can be computed using conservative relations and empty interval probabilities
as shown below
\begin{eqnarray}\nonumber
{\rm Pr}(\stackrel{\curvearrowleft}{ } n_1\,\fbox{$\bullet$ d-1 }\,n_2)=\Gamma\times{\rm Pr}(n_1\,\fbox{$\bullet$
d-1 }\,n_2), \\
\nonumber
{\rm Pr}(n_1\,\fbox{$\bullet$ d-1 }\,n_2)+{\rm Pr}(n_1\,\fbox{$\circ$ d-1
}\,n_2)={\rm Pr}(n_1+1\,\fbox{ d-1 }\,n_2),
\\
{\rm Pr}(n_1\,\fbox{$\bullet$ d-1 }\,n_2)=E_{n_1+1,n_2}-E_{n_1,n_2},
\end{eqnarray}
One then obtains, for the dynamics inside the bulk the following equation
\begin{eqnarray} \label{disc}\fl
\frac{\partial E_{n_1,n_2}(t)}{\partial t}=\Gamma \Big [
E_{n_1-1,n_2}(t)+E_{n_1+1,n_2}(t)+E_{n_1,n_2-1}(t)+E_{n_1,n_2+1}(t)
\\ \nonumber
-4E_{n_1,n_2}(t)\Big ]
+(\alpha-1)\Gamma\Big [E_{n_1,n_2-1}(t)+E_{n_1-1,n_2}(t)-2E_{n_1,n_2}(t)\Big ].
\end{eqnarray}
The last term in brackets corresponds to the drift contribution
$(\alpha-1)\ne 0$ which vanishes when the dynamics is symmetric (no drift term).
The
first term in brackets is the classic diffusion process in the bulk.
\subsection{Boundary conditions}
Boundary conditions at locations $n_1=0$ and $n_2=N$ are found by writing the
equations of motion around these points. The treatment of these conditions is
done by imposing the continuity and differentiability of the interval
probability across the boundaries. We therefore need to determine
uniquely the probability functions for all index $n_1$ and $n_2$ inside and
outside the physical domain by extrapolation of the equations of motion. The
main advantage is then to use a general Fourier transform which depends only on
the initial conditions without introducing Dirichlet conditions. Contrary
to reference \cite{Hinr97}, we do not separate the time from the
space dependence, but look at a global solution that combines both time and space variables
inside a scaling parameter (see below). This method is similar in some sense to
the mirror symmetry method, albeit different, since we are able to construct
uniquely the solution for $E$ everywhere by continuity of this
function and its derivatives.
New particles can enter the left hand side of the chain with rate $\beta\Gamma$ and
diffuse through the system with rates $\Gamma$ (left) or $\alpha\Gamma$
(right) before eventually
exit the chain on the right with probability $\alpha\Gamma$. We can write (see
for example section 2.1 of reference \cite{Hinr97})
\begin{eqnarray}\nonumber
\frac{\partial E_{0,n_2}(t)}{\partial t}=
{\rm Pr}(0\,\fbox{ d-1 $\bullet$}\,n_2\stackrel{\curvearrowright}{ })-{\rm Pr}(0\,\fbox{ d }\,n_2\stackrel{\curvearrowleft}{ }\bullet)
-{\rm Pr}(\bullet\stackrel{\curvearrowright}{ } 0\,\fbox{ d }\,n_2).
\end{eqnarray}
The last probability is equal to ${\rm Pr}(\bullet\stackrel{\curvearrowright}{ } 0\,\fbox{ d
}\,n_2)=\beta\Gamma\times{\rm Pr}(0\,\fbox{ d }\,n_2)=\beta\Gamma E_{0,n_2}(t)$ and one obtains
\begin{eqnarray}\label{eq0}
\frac{\partial E_{0,n_2}(t)}{\partial t}=\Gamma\Big [
\alpha E_{0,n_2-1}(t)+E_{0,n_2+1}(t)-(1+\alpha+\beta)E_{0,n_2}(t)\Big ].
\end{eqnarray}
Comparing \eref{eq0} with \eref{disc}, we can formally extend the first index
$n_1$ to negative values, by imposing the relation
$\alpha E_{-1,n_2}(t)+E_{1,n_2}(t)=(1+\alpha-\beta)E_{0,n_2}(t)$ and which gives a
condition of continuity between probabilities with negative index $n_1=-1$ and
positive one $n_1=1$. By differentiating this relation with respect to time,
i.e.
\begin{eqnarray}\nonumber
\alpha \frac{\partial E_{-1,n_2}(t)}{\partial
t}+\frac{\partial E_{1,n_2}(t)}{\partial
t}=(1+\alpha-\beta)\frac{\partial E_{0,n_2}(t)}{\partial t},
\end{eqnarray}
and performing some algebra and simplifications involving the two
previous identities~\eref{disc} and~\eref{eq0}, one obtains formally
another
relation between quantities $E_{-2,n_2}$ and $E_{2,n_2}$, assuming that
~\eref{disc} holds for all negative locations $-n_1$
\begin{eqnarray}
\alpha^2 E_{-2,n_2}(t)+E_{2,n_2}(t)=\left [(1+\alpha-\beta)^2-2\alpha\right ]E_{0,n_2}(t).
\end{eqnarray}
These two relations obtained for $n_1=-1$ and $n_1=-2$ are simple enough
to suggest a general solution of type
\begin{eqnarray}\label{cond0}
\alpha^{n_1}E_{-n_1,n_2}(t)+E_{n_1,n_2}(t)={\mathscr A}(n_1)E_{0,n_2}(t).
\end{eqnarray}
The factor $\alpha^{n_1}$ is due to the fact that each time we take the time
derivative of~\eref{cond0}, the term $\alpha^{n_1}\partial_tE_{-n_1,n_2}(t)$
contains the unique contribution from the lowest index $-n_1-1$: $\alpha^{n_1}\alpha
E_{-n_1-1,n_2}=\alpha^{n_1+1} E_{-n_1-1,n_2}$, and coming from \eref{eq0}, hence
a general factor $\alpha^{n_1+1}$ appears. Many terms cancel each other in the
further simplifications by taking the time derivative of \eref{cond0} for
$E_{-n_1,n_2}(t)$ and by assuming recursively that \eref{cond0} holds for
all $E_{n,n_2}$ with $n\le n_1-1$. The initial conditions are
given by ${\mathscr A}(1)=(1+\alpha-\beta)$ and ${\mathscr A}(0)=2$ as found just above for these
particular cases. After some algebra, we find that ${\mathscr A}(n)$ satisfies the
discrete equation ${\mathscr A}(n+1)+\alpha{\mathscr A}(n-1)={\mathscr A}(1){\mathscr A}(n)$. The unique solution of
this equation is given by
\begin{eqnarray}\label{an}
{\mathscr A}(n)=r_1^{n}+r_2^{n},
\end{eqnarray}
with $r_1r_2=\alpha$ and $r_1+r_2=1+\alpha-\beta$.
On the right (open) boundary of
the chain, we have instead, by counting the different possibilities for
particles to create or destroy the empty interval $[n_1,N]$
\begin{eqnarray}\nonumber
\frac{\partial E_{n_1,N}(t)}{\partial t}=
{\rm Pr}(\stackrel{\curvearrowleft}{ } n_1\,\fbox{$\bullet$ d-1 }\,N)+{\rm Pr}(n_1\,\fbox{ d-1 $\bullet$}\,N\stackrel{\curvearrowright}{ })
-{\rm Pr}(\bullet\stackrel{\curvearrowright}{ } n_1\,\fbox{ d }\,N),
\end{eqnarray}
or, after using the probability relations,
\begin{eqnarray}\label{eqL}\fl
\frac{\partial E_{n_1,N}(t)}{\partial t}=\Gamma \Big [
\alpha E_{n_1-1,N}(t)+E_{n_1+1,N}(t)+\alpha E_{n_1,N-1}(t)-(1+2\alpha)E_{n_1,N}(t)
\Big ].
\end{eqnarray}
Comparing \eref{eqL} with \eref{disc}, we can see that the contribution
$E_{n_1,N+1}-E_{n_1,N}$ is missing, which corresponds to the fact
that no particle can enter from the right boundary. Assuming as before that
~\eref{disc} is true for $n_2\ge N$ by continuity, one obtains
the condition $E_{n_1,N+1}(t)=E_{n_1,N}(t)$, valid at all time, which gives a
first relation
for $n_2=N+1$. Taking the time derivative of this identity,
$\partial_tE_{n_1,N}(t)=\partial_tE_{n_1,N+1}(t)$, using~\eref{disc} and
the previous relation found for $n_2=N+1$, the next relation yields
$E_{n_1,N+2}(t)=(1-\alpha)E_{n_1,N}(t)+\alpha E_{n_1,N-1}(t)$.
One obtains a relation between $E_{n_1,N+2}$ and the physical
quantities $E_{n_1,N}$, $E_{n_1,N-1}$. More generally, by induction, we can try
to find a set of coefficients ${\mathscr B}(k,l)$ such that $E_{n_1,N+k}$ depends only
on physical quantities $E_{n_1,N-l}$ for $0\le l\le N-k+1$
\begin{eqnarray}\label{bn}
E_{n_1,N+k}(t)=\sum_{l=0}^{k-1}{\mathscr B}(k,l)E_{n_1,N-l}(t).
\end{eqnarray}
A closed form between coefficients ${\mathscr B}$ can be found as before
by considering the time derivative of \eref{bn} and assuming that \eref{bn}
holds from $E_{n_1,N+1}$ until $E_{n_1,N+k}$ for a given $k$. We also assume
that \eref{disc} holds for all $n_2>N$ by continuity. The term
$\partial_tE_{n_1,N+k}(t)$ contains the contribution $E_{n_1,N+k+1}(t)$
which can be expressed as function of $E_{n_1,N+k}(t)$, $E_{n_1,N+k-1}(t)$,
$\cdots$, $E_{n_1,N+1}(t)$ and other physical probabilities. Then coefficients
${\mathscr B}(k+1,l)$ are function of previous coefficients ${\mathscr B}(k'\le k,l')$. One
obtains after some algebra the discrete recursive equations
\begin{eqnarray}\nonumber
{\mathscr B}(k+1,0)={\mathscr B}(k,0)+{\mathscr B}(k,1)-\alpha{\mathscr B}(k-1,0),
\\ \nonumber
{\mathscr B}(k+1,l)={\mathscr B}(k,l+1)+\alpha{\mathscr B}(k,l-1)-\alpha{\mathscr B}(k-1,l),\textrm{ for $1\le
l\le k-2$},\\ \label{cond1}
{\mathscr B}(k+1,k-1)=\alpha{\mathscr B}(k,k-2),\textrm{ and }{\mathscr B}(k+1,k)=\alpha{\mathscr B}(k,k-1).
\end{eqnarray}
By inspection, the boundary conditions are ${\mathscr B}(1,0)=1$, ${\mathscr B}(k\ge
2,0)=1-\alpha$, and generally ${\mathscr B}(k,l)=(1-\alpha)\alpha^l$ for other values of $l$,
except for the last term ${\mathscr B}(k+1,k)=\alpha^k$. One obtains the general expression
\begin{eqnarray}\label{bn2}
E_{n_1,N+k}(t)=(1-\alpha)\sum_{l=0}^{k-2}\alpha^lE_{n_1,N-l}(t)+\alpha^{k-1}E_{n_1,N-k+1}
(t).
\end{eqnarray}
These continuity equations can be generalized to other boundary
conditions, for example when two sources are present at both ends of the chain.
In principle one obtains non-local kernel equations relating
positive and negative coordinates, such as~\eref{bn2}. The method developed
in this paper is quite straightforward, based on the discrete case.
However, there is no
guarantee that a simple solution can be found in the form of~\eref{bn}.
Moreover, for finite size systems, imposing two sources and
an asymmetric diffusion coefficient leads to work with two non-local kernels,
which renders the general expression for the interval probability hard to work
with, or even to write explicitly as function of the initial conditions.
\section{Continuum limit and symmetry equations}
In this section we consider the continuum limit of \eref{disc} satisfying the
different boundary conditions previously obtained.
If $a$ is the elementary lattice step, we introduce coordinates $x_1=n_1a$
and $x_2=n_2a$, while $L=Na$ is finite when both $a\rightarrow 0$ and
$N\rightarrow\infty$. In this case
$E_{n_1,n_2}(t)\rightarrow E(x_1,x_2;t)$ and \eref{disc} becomes the equation of
diffusion
\begin{eqnarray}\label{diff}
\frac{\partial E(x_1,x_2;t)}{\partial t}
={\mathscr D}
\left (
\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}
\right )E(x_1,x_2;t)-v
\left (
\frac{\partial}{\partial x_1}+\frac{\partial}{\partial x_2}
\right )E(x_1,x_2;t),
\end{eqnarray}
where ${\mathscr D}=\Gamma a^2$ is the diffusion coefficient in the limit $a\rightarrow
0$ and $\Gamma\rightarrow\infty$, and $v=2k_b{\mathscr D}$ is the drift velocity,
$k_b$ being the characteristic momentum from the scaling $\alpha=1+2k_b a$ (see
Table \ref{table1}). We can notice that $\alpha\rightarrow 1$ in the continuum limit.
\begin{table}[ht]
\caption{\label{table1}Notation and continuum limit for the physical quantities}
\small\addtolength{\tabcolsep}{-4pt}
\renewcommand{\arraystretch}{0.5}
\begin{tabular}{llll}
\\
\hline
\\
\hline
\\
a & lattice step & L & system size \\
$\Gamma$ & diffusion rate to the left &${\mathscr D} =\Gamma a^2$ & diffusion constant
\\
$\alpha\Gamma$ & diffusion rate to the right & $\beta\Gamma$ & input rate
of particles\\
$\alpha=1+2k_b a$ & scaling limit for $\alpha$ &$k_{in}=\sqrt{\beta}/a$ &
input momentum\\
$v=2k_b {\mathscr D}$ & drift velocity & ${\mathscr L}_0=\sqrt{8{\mathscr D} t}$ &
diffusion length \\
$I_{in}={\mathscr D} k_{in}^2$ & input current &
$I_{out}(t)=-\frac{{\mathscr D}}{2}\partial_1^2E(L,L;t)$ & output current\\
$\eta_L(t)=I_{out}(t)/I_{in}$ & current ratio
& $\rho(x;t)=\frac{1}{2}(\partial_1-\partial_2)E(x,x;t)$ & local particle
concentration
\\ $\rho_L(t)=L^{-1}\int_0^L\rho(x;t)\,dx$ & averaged
concentration & $\Lambda=\sqrt{k_b^2-k_{in}^2}$ & effective wavenumber
\\
$t_L=L^2/8{\mathscr D}$ & characteristic time & $\epsilon=L^2/{\mathscr L}_0^2=t_L/t$ & inverse time parameter
\\
$R(x;t)$ & local coagulation rate & $R(t)$ & global coagulation rate
\\
${\cal N}_L=\int_0^L\rho(x;t)d x$ & number of particles & &
\\
[1ex]
\hline
\end{tabular}
\label{table:notation}
\end{table}
Equation (\ref{diff}) is solved using a double Fourier transform
$E(x_1,x_2;t)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{d k_1d
k_2}{4\pi^2} \exp(\II k_1x_1+\II k_2x_2)\wit{E}(k_1,k_2;t)$, and the
evolution of the empty-interval probability as function of
initial conditions is given by
\begin{eqnarray}\nonumber\fl
E(x_1,x_2;t)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}
\!\frac{d x_1'd x_2'}{4\pi {\mathscr D} t\,}\:
\exp\Big [-\frac{1}{4{\mathscr D} t}(x_1-x_1'-v t)^2-\frac{1}{4{\mathscr D}
t}(x_2-x_2'-v t)^2\Big ]E(x_1',x_2';0) \\ \nonumber
=
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\!d x_1'd
x_2'\: {\cal W}{{\mathscr L}_0}{x_1-x_1'}{\cal W}{{\mathscr L}_0}{x_2-x_2'}E(x_1',x_2';0),
\\ \label{intE1}
{\cal W}{{\mathscr L}_0}{x}:=
\!\sqrt{\frac{2}{\pi{\mathscr L}_0^2}}\: \exp\Big
\{-2(x-v t)^2/{\mathscr L}_0^2\Big \}.
\end{eqnarray}
The integrals over the real axis in the previous expression are unrestricted.
We also have introduced the classical diffusion length
${\mathscr L}_0 :=\sqrt{8{\mathscr D} t\,}$, which acts as the typical scaling length in the
problem~\footnote{In the context of the coagulation-diffusion
problem in an infinite chain and without drift, the probability is invariant
by translation and $E(x_1,x_2;t)$
can be written as $E(x_2-x_1;t)$. In \eref{intE1}, the change of variable $y_1=x_2'-x_1'$
and $y_2=x_2'-x_2$, such that
$(x_1-x_1')^2+(x_2-x_2')^2=(x_2-x_1-x_2'+x_1')^2+2(x_2-x_2')(x_1-x_1')=(x_2-x_1-
y_1)^2+2y_2(y_2-y_1+x_2-x_1)$, and the Gaussian integration on $y_2$ lead to the
well-known one-interval solution
$E(x_2-x_1;t)=\int_{-\infty}^{\infty}\!\frac{d y_1}{\sqrt{\pi}\,\,{\mathscr L}_0}\:
\exp\Big
[-\frac{1}{{\mathscr L}_0^2}(x_2-x_1-y_1)^2\Big ]E(y_1;0)$.}. The different
physical parameters and
their continuous versions are given in Table \ref{table1}.
We now treat the boundary conditions in the continuous limit. On the left hand
side of the chain, around the origin, the symmetry
\eref{cond0} has a continuous solution given by
\begin{eqnarray}\label{sym0}
{\rm e}^{2k_b x_1}E(-x_1,x_2;t)+E(x_1,x_2;t)={\mathscr A}(x_1)E(0,x_2;t),
\end{eqnarray}
where ${\mathscr A}(n)\rightarrow{\mathscr A}(x=nL)$ satisfies the differential equation
\begin{eqnarray}\label{eqdiff0}
{\mathscr A}''(x)-2k_b{\mathscr A}'(x)+k_{in}^2{\mathscr A}(x)=0,
\end{eqnarray}
with initial conditions ${\mathscr A}(0)=2$ and ${\mathscr A}'(0)=2k_b$. This equation is
deduced from the discrete recursion for ${\mathscr A}(n)$, and from
the natural scaling $\beta=a^2k_{in}^2$ where $k_{in}$ is the input momentum. Indeed,
the input current is given by $I_{{in}}=\Gamma\beta$ (see next section) which
has the finite value $I_{{in}}={\mathscr D} k_{in}^2$, by replacing $\beta$ with the
corresponding scaling. In particular, the continuous limit of
\eref{cond0} for intervals incorporating the origin, $\alpha
E_{-1,n_2}(t)+E_{1,n_2}(t)=(1+\alpha-\beta)E_{0,n_2}(t)$, is
$\partial^2_{x_1}E(0,x_2;t)-2k_b\partial_{x_1}E(0,x_2;t)=-k_{in}^2 E(0,x_2;t)$. We
may then identify $k_{in}$ with the
inverse
of a coherent length inside the chain, in the sense that
empty intervals are suppressed by large input currents.
Then the solution for \eref{eqdiff0} is
given by ${\mathscr A}(x)=2\exp(k_b x)\cosh(x\sqrt{k_b^2-k_{in}^2})$. The $\cosh$ function
is transformed into a cosine function when $k_{in}>k_b$, or ${\mathscr A}(x)=2\exp(k_b
x)\cos(x\sqrt{k_{in}^2-k_b^2})$.
We also have a symmetry equation by exchanging the position variables
of the interval~\cite{Takayasu91,dura10}
\begin{eqnarray}\label{sym}
E(x_1,x_2;t)=2-E(x_2,x_1;t),
\end{eqnarray}
which holds even in presence of a drift term $v$.
The continuum version of the second boundary condition~\eref{bn2} can be found
by
noticing that the sum of terms proportional to $(1-\alpha)=-2k_b a$
becomes an integral, and coefficients $\alpha^l$ with $l\ge 0$,
in~\eref{bn}, have a finite limit ${\mathscr B}(x):=\exp(2k_b x)$ with $x=la$. Then one
obtains
\begin{eqnarray}\label{sym1}\fl
E(x_1,L+x_2;t)=\exp(2k_b x_2)E(x_1,L-x_2;t)-2k_b \int_0^{x_2}d y \exp(2k_b
y)E(x_1,L-y;t).
\end{eqnarray}
It is useful to define the modified function
$\widehat{E}(x_1,x_2;t):=\exp[-k_b(x_1+x_2)]E(x_1,x_2;t)$ in order to simplify the
different symmetry relations given by the set of three equations
\begin{eqnarray}\nonumber
\widehat{E}(x_1,x_2;t)+\widehat{E}(-x_1,x_2;t)={\widehat{\mathscr A}}(x_1)\widehat{E}(0,x_2;t),
\;{\widehat{\mathscr A}}(x_1):=2\cosh\Big (x_1\sqrt{k_b^2-k_{in}^2}\Big ),\textrm{ (a)}
\\ \nonumber
\widehat{E}(x_1,x_2+L;t)=\widehat{E}(x_1,L-x_2;t)-2k_b \int_0^{x_2}d y \exp[2k_b
(y-x_2)]\widehat{E}(x_1,L-y;t),\textrm{ (b)}
\\ \label{symtot}
\widehat{E}(x_1,x_2;t)+\widehat{E}(x_2,x_1;t)=2\exp[-k_b(x_1+x_2)].\textrm{ (c)}
\end{eqnarray}
In the following, we will consider two cases, as in \cite{Hinr97}. The
semi-infinite system
with $L=\infty$, where symmetries \eref{symtot} reduce
to (a) and (c), and the finite system with no drift term $k_b=0$. In both cases,
the interval probability function can be computed explicitly and for any
initial configuration of particles. In the next
section, we define the important transport quantities in the continuum limit
that are used in the next parts of the paper, such as the particle density as
function of space and time.
\subsection{Physical quantities}
We define the average density and coagulation rate inside
the chain. The different notations throughout the text for the
physical quantities can be found in Table \ref{table1}.
The local density is noted $\rho_{n}(t)$ (or $\rho(x;t)$ in the continuum
limit), and is defined by $a^{-1}{\rm Pr}(n\,\fbox{$\bullet$}\,n+1)$ which is equal
to $a^{-1}(1-E_{n,n+1}(t))\simeq -\partial_{2} E(x,x;t)$.
Short notation $\partial_{i}$, with $i=1,2$, is meant for partial derivation
with respect to component $x_i$.
Similarly, we can write
$\rho_{n}(t)=a^{-1}{\rm Pr}(n-1\,\fbox{$\bullet$}\,n)\simeq
\partial_{1} E(x,x;t)$, and therefore, by symmetrization,
\begin{eqnarray}
\rho(x;t)=\frac{1}{2}\Big (\partial_{1}-\partial_{2}\Big )E(x,x;t).
\end{eqnarray}
For systems with translational symmetry,
$E(x_1,x_2;t)=E(x_2-x_1;t)$, then $\partial_1=-\partial_2$. In this case,
the density is simply equal to $\rho(x;t)=-\partial_xE(x=0;t)$ and is
site-independent.
The current entering the system by unit of time at the origin can be defined as
the rate $\Gamma\beta$ times the probability that a particle is not present in
the interval $[0,1]$ (if a particle is already present, coagulation will occur)
\begin{eqnarray}\nonumber
I_{in}=\Gamma\beta\times
{\rm Pr}(0\,\fbox{$\circ$}\,1)=\Gamma\beta E_{0,1}(t)\simeq \Gamma\beta={\mathscr D}
k_{in}^2.
\end{eqnarray}
We also consider the local coagulation rate
$R_n(t)$ as the number of pairs of particles that coagulate in the box
$[n,n+1]$ per unit of time. In terms of probabilities, we can write
\begin{eqnarray}\nonumber
R_n(t)&=&\Gamma\left [
\alpha{\rm Pr}(n-1\fbox{$\bullet\stackrel{\curvearrowright}{ }\,\bullet$}\,n+1)+
{\rm Pr}(n\fbox{$\bullet\stackrel{\curvearrowleft}{ }\,\bullet$}\,n+2)
\right ]
\\ \nonumber
&=&\alpha \Gamma\left [1-E_{n-1,n}(t)-E_{n,n+1}(t)+E_{n-1,n+1}(t) \right ]
\\
&+&\Gamma\left [1-E_{n,n+1}(t)-E_{n+1,n+2}(t)+E_{n,n+2}\right ].
\end{eqnarray}
Indeed, we need at least two particles in two consecutive sites for
coagulation to occur. In the continuum limit $R_n(t)\rightarrow R(x;t)$, one
obtains
\begin{eqnarray}
R(x;t)=-\frac{1}{2}{\mathscr D}\left (\partial_{11}+\partial_{22}+6\partial_{12}\right
)E(x,x;t).
\end{eqnarray}
We have used the relation
$(\partial_1+\partial_2)E(x,x;t)=\partial_xE(x,x;t)=0$, deduced from the
symmetry property \eref{sym} or constraint $E(x,x;t)=1$. The
coagulation rate reduces to $R(x;t)=2{\mathscr D}\partial_{11}E(x,x;t)$ in the case of
translational symmetry, which corresponds to the curvature of the empty
interval probability. In a system of size $L$, we can also define a global
coagulation rate $R(t)$, which will be studied in the last section, by
considering the terms contributing to the loss and gain of particles, and
function of the
averaged density. First, we define a dimensionless integrated density
${\cal N}_L(t):=L\rho_L(t):=\int_0^L\rho(x;t)\,dx$, incorporating the evolution of
the number of particles as function of time. Using the input and output currents
$I_{in}$ and $I_{out}(t)$ respectively, one obtains the conservation equation
\begin{eqnarray}\label{coag}
\frac{d{\cal N}_L(t)}{dt}=I_{in}-I_{out}(t)-R(t),
\end{eqnarray}
where the output current $I_{out}$ at the end of the chain is given by the
product of the probability that a particle is present in the box $[N-1,N]$, and
the rate $\Gamma$, or explicitly
\begin{eqnarray}\nonumber
I_{out}(t)=\Gamma\times
{\rm Pr}(N-1\,\fbox{$\bullet$}\,N)=\Gamma\times \left [1-E_{N-1,N}(t)\right ]\simeq
-\frac{{\mathscr D}}{2} \partial_{1}^2E(L,L;t).
\end{eqnarray}
We considered in particular the fact that $\partial_{1}E(L,L;t)=0$, resulting
from the symmetry $E(L+x_1,L)=E(L-x_1,L)$. $R(t)$ can
therefore be deduced from~\eref{coag} if we know the density.
\section{Semi-infinite system}
When size $L$ is infinite, symmetries \eref{symtot} reduce to (a) and
(c) only. \eref{intE1} can be decomposed relatively to the origin
in four sectors, depending on the sign of the two coordinates
\begin{eqnarray}\nonumber\fl
E(x_1,x_2;t)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\!d x_1'd
x_2'\: \twe{{\mathscr L}_0}{x_1,x_1'}\twe{{\mathscr L}_0}{x_2,x_2'}\widehat{E}(x_1',x_2';0)
\\ \nonumber\fl
=\int_0^{\infty}\int_0^{\infty}\!d x_1'd
x_2'\:\left \{ \twe{{\mathscr L}_0}{x_1,x_1'}\twe{{\mathscr L}_0}{x_2,x_2'}\widehat{E}(x_1',x_2';0)
+\twe{{\mathscr L}_0}{x_1,-x_1'}\twe{{\mathscr L}_0}{x_2,x_2'}\widehat{E}(-x_1',x_2';0) \right .
\\ \label{Egen}\fl
+\left .\twe{{\mathscr L}_0}{x_1,x_1'}\twe{{\mathscr L}_0}{x_2,-x_2'}\widehat{E}(x_1',-x_2';0)
+\twe{{\mathscr L}_0}{x_1,-x_1'}\twe{{\mathscr L}_0}{x_2,-x_2'}\widehat{E}(-x_1',-x_2';0)\right \},
\end{eqnarray}
with the notation $\twe{{\mathscr L}_0}{x,y}:={\cal W}{{\mathscr L}_0}{x-y}\exp(k_b y)$.
From \eref{symtot}, we can deduce the corresponding values
in each of the sectors containing negative coordinates (after dropping the time
argument for simplification)
\begin{eqnarray}\nonumber\fl
\widehat{E}(-x_1',x_2')={\widehat{\mathscr A}}(x_1')\widehat{E}(0,x_2')-\widehat{E}(x_1',x_2'),
\\ \nonumber\fl
\widehat{E}(x_1',-x_2')=-{\widehat{\mathscr A}}(x_2')\widehat{E}(0,x_1')+4{\rm e}^{-k_b
x_1'}\cosh(k_b x_2')-\widehat{E}(x_1',x_2'),
\\ \nonumber\fl
\widehat{E}(-x_1',-x_2')={\widehat{\mathscr A}}(x_1')\Big [2\cosh(k_b x_2')-\widehat{E}(0,x_2')\Big ]
-{\widehat{\mathscr A}}(x_2')\Big [2\cosh(k_b x_1')-\widehat{E}(0,x_1')\Big ]
\\ \label{symLinf}
+4\sinh(k_b x_1')\cosh(k_b x_2')+\widehat{E}(x_1',x_2').
\end{eqnarray}
The last equation can be deduced from the first two by performing symmetry
operation on the first negative coordinate $-x_1'\rightarrow x_1'$, then on
the second $-x_2'\rightarrow x_2'$. Another relation is also possible, by
performing the same symmetry
operation on the second coordinate $-x_2'$ first, then on $-x_1'$. A
prescription is necessary in this case to obtain the correct answer: the final
result will be given by taking half the sum of these two operations,
yielding the third identity \eref{symLinf}.
We then insert these expressions in \eref{Egen}, and rearrange all terms
such that the time dependent interval probability is expressed as a sum of
different contributions, function of initial condition $E(x_1,x_2;0)$ with
$x_1<x_2$, in
addition to those which do not depend on it. One obtains, after some algebra,
and using symmetry properties in exchanging the variables of integration
$x_1'\leftrightarrow x_2'$, the following general expression
\begin{eqnarray}\nonumber\fl
E(x_1,x_2;t)=
\int_{0}^{\infty}\int_{0}^{\infty}d x_1'd x_2'\left
[K(x_1,x_1')K(x_2,x_2')-K(x_2,x_1')K(x_1,x_2')\right ]
E(x_1',x_2';0)\theta(x_2'-x_1')
\\ \nonumber\fl
+\int_{0}^{\infty}\int_{0}^{\infty}d x_1'd x_2'\:{\widehat{\mathscr A}}(x_1')\left [
\twe{{\mathscr L}_0}{x_1,-x_1'}K(x_2,x_2')-\twe{{\mathscr L}_0}{x_2,-x_1'}K(x_1,x_2') \right
]E(0,x_2';0)
\\ \nonumber\fl
+1+\int_{0}^{\infty}\int_{0}^{\infty}d x_1'd x_2'\:
\left [{\cal W}{{\mathscr L}_0}{x_1-x_1'}{\cal W}{{\mathscr L}_0}{x_2+x_2'}
-{\cal W}{{\mathscr L}_0}{x_2-x_1'}{\cal W}{{\mathscr L}_0}{x_1+x_2'}
\right ]\left [1+{\rm e}^{-2k x_2'}\right ]
\\ \nonumber\fl
+\left [
{\cal W}{{\mathscr L}_0}{x_1+x_1'}{\cal W}{{\mathscr L}_0}{x_2+x_2'}
-{\cal W}{{\mathscr L}_0}{x_2+x_1'}{\cal W}{{\mathscr L}_0}{x_1+x_2'}\right ]{\rm e}^{-2k_b x_2'}
\\ \label{ELinfty}\fl
+\int_{0}^{\infty}\int_{0}^{\infty}d x_1'd x_2'\:
\Big [
K(x_2,x'_1)K(x_1,x'_2)-K(x_1,x'_1)K(x_2,x'_2)
\Big ]
\theta(x'_2-x'_1)
\\ \nonumber\fl
+\int_{0}^{\infty}\int_{0}^{\infty}d x_1'd x_2'\:
{\widehat{\mathscr A}}(x_1')\left
[\twe{{\mathscr L}_0}{x_1,-x_1'}\twe{{\mathscr L}_0}{x_2,-x_2'}-
\twe{{\mathscr L}_0}{x_2,-x_1'}\twe{{\mathscr L}_0}{x_1,-x_2'}
\right ]2\cosh(k_b x_2'),
\end{eqnarray}
where $\theta(x)$ is the usual Heaviside function, equal to unity if $x>0$,
$\theta(0)=1/2$, and
zero otherwise. The kernel $K$ is given by
\begin{eqnarray}
K(x_1,x_1'):=
\left [\twe{{\mathscr L}_0}{x_1,x_1'}-\twe{{\mathscr L}_0}{x_1,-x_1'}\right ]{\rm e}^{-k_b x_1'}.
\end{eqnarray}
\eref{ELinfty} is consistent with the fact that $E$ can be expressed generally
as $E(x_1,x_2)=1+A(x_1,x_2)$, where $A$ is an antisymmetric function:
$A(x_1,x_2)=-A(x_2,x_1)$. Equation (\ref{ELinfty}) is also general in the sense
that any kind of initial conditions can be implemented.
As an application, we consider an initial system
entirely filled with particles $E(x_1,x_2;0)=0$, which
simplifies \eref{ELinfty} since the first two integrals
vanish.
After some algebra, we find that the concentration is the sum of different
contributions
\begin{eqnarray}\nonumber
\rho(x;t)=\rho_0(x,k_b;t)+\frac{1}{4}\exp\left [ (k_b-\Lambda)x-k_{in}^2 {\mathscr D} t\right
]\Big \{
(k_b-\Lambda)n_0(x,k_b;t)n_0(x,\Lambda;t)
\\ \label{conc}
+2\rho_0(x,k_b;t)n_0(x,\Lambda;t)-2\rho_0(x,\Lambda;t)n_0(x,k_b;t)
\Big \}+\rho_1(x,k_b;t),
\end{eqnarray}
where we introduced the momentum $\Lambda:=\sqrt{k_b^2-k_{in}^2}$. Functions
$\rho_0$, $n_0$, and $\rho_1$ are defined by the expressions
\begin{eqnarray}\nonumber
\rho_0(x,k;t):=\frac{1}{\sqrt{\pi {\mathscr D} t}}\exp\Big [-\frac{(x-2k{\mathscr D}
t)^2}{4{\mathscr D} t}\Big ]
-k{\rm e}^{2kx}{\rm erfc}\left (\frac{x+2k{\mathscr D} t}{2\sqrt{{\mathscr D} t}} \right )
\\ \nonumber
n_0(x,k;t):={\rm erfc}\left(\frac{x-2k{\mathscr D} t}{2\sqrt{{\mathscr D} t}} \right )+
{\rm e}^{2kx}{\rm erfc}\left
(\frac{x+2k{\mathscr D} t}{2\sqrt{{\mathscr D} t}} \right ),
\\ \nonumber
\rho_1(x,k;t):=\frac{1}{2\sqrt{2\pi {\mathscr D} t}}\left [
{\rm erfc}\left(\frac{-x+2k{\mathscr D} t}{\sqrt{2{\mathscr D} t}} \right )
-{\rm e}^{4kx}{\rm erfc}\left
(\frac{x+2k{\mathscr D} t}{\sqrt{2{\mathscr D} t}} \right ) \right ]
\\ \nonumber
+\frac{k{\rm e}^{4kx}}{2}{\rm erfc}^2\left
(\frac{x+2k{\mathscr D} t}{2\sqrt{{\mathscr D} t}} \right )
-\frac{1}{4}\partial_x{\cal G}_k(x,0){\cal G}_k(x,0)
\\ \label{rhoLinf}
-\frac{k{\rm e}^{2kx}}{\sqrt{\pi{\mathscr D} t}}\int_0^{\infty}d y\:
\exp\left [-\frac{(x-y-2k{\mathscr D} t)^2}{4{\mathscr D} t}\right ]
{\rm erfc}\left
(\frac{x+y+2k{\mathscr D} t}{2\sqrt{{\mathscr D} t}} \right ).
\end{eqnarray}
\begin{figure
\subfigure[]{
\includegraphics[scale=0.35,clip]{fig_densityTime_k1.eps}
\label{fig_density_time}
}
\subfigure[]{
\includegraphics[scale=0.35,clip]{fig_densityKin_k1.eps}
\label{fig_density_kin}
}
\caption{(a) Density profile $\rho(x;t)$ as function of
time $t$ in presence of drift $k_b=1$ (${\mathscr D} =1$). Far from the origin, the
density is uniform and equal
to $1/\sqrt{2\pi{\mathscr D} t}$. (b) Density profile in absence of
drift $k_b=0$ at time
$t=1$ (${\mathscr D}=1$)
for several $k_{in}$. The density at the origin (inset) is maximum
for a finite value of $k_{in}$, function of $k_b$.
When $k_{in}$ is large, the asymptotic behavior of
$\rho(0;t)$ is given by the same value as $k_{in}=0$.}
\end{figure}
In the last equation, function ${\cal G}_k(x,y)$ is given in \ref{app1}, see
\eref{defGk}. When no source and drift are present
$k_b=k_{in}=0$, the density profile is given by the simple expression
\begin{eqnarray}\label{conc0}
\rho(x;t)=\frac{1}{\sqrt{\pi{\mathscr D} t}}
\exp\left (-\frac{x^2}{4{\mathscr D} t}\right )
{\rm erfc}\left (\frac{x}{2\sqrt{{\mathscr D} t}} \right )
+\frac{1}{\sqrt{2\pi{\mathscr D} t}}
{\rm erf}\left (\frac{x}{\sqrt{2{\mathscr D} t}} \right ),
\end{eqnarray}
from which we recover the bulk density $\rho(x\gg
1;t)=(2\pi{\mathscr D} t)^{-1/2}$ far from the origin. At the origin the
density is larger by a factor $\sqrt{2}$: $\rho(x=0;t)=(\pi{\mathscr D} t)^{-1/2}$,
see Fig. 2(b). In Fig. 2(a) is represented the density profile for a drift
momentum $k_b=1$ (we set ${\mathscr D} =1$), and for several values of time and input
momentum $k_{in}$.
The density at the origin grows with $k_{in}$ as expected.
In Fig. 2(b) we plotted the density profile for several input
momenta $k_{in}$ at fixed time and in absence of drift.
At the origin, the concentration is given by
\begin{eqnarray}
\rho(0;t)=\rho_0(0,k_b;t)+\exp(-k_{in}^2 {\mathscr D} t)\left [
k_b-\Lambda+\rho_0(0,k_b;t)-\rho_0(0,\Lambda;t) \right ].
\end{eqnarray}
The density $\rho(0;t)$ is increasing with $k_{in}$ up to a finite value, then
decreases as the input current becomes large, see inset of Fig. 2(b). The
asymptotic value is equal to the value in absence of current. This feature is
characteristic of coagulation-diffusion processes, since coagulation prevents
the system to become
overpopulated and limits the amount of particles that can be injected into the
system. At the same time, diffusion tends to disperse the
incoming particles, with the existence of an optimal current $k_{in}\simeq 1$. In
the next section, we focus our
analysis on the finite size system.
\section{Finite system with $k_b=0$}
Here we consider the case of a system of size $L$, in absence of
drift, $k_b=0$. In \eref{intE1}, the integration
covers the entire plane ${\mathbb{R}}^2$ inside which only the region ${\cal D}_0:=\{0\le x_1\le
x_2\le L\}$ is of physical meaning. Symmetries \eref{symtot} are used to fold
the plane ${\mathbb{R}}^2$ into ${\cal D}_0$, so that integrations are
made only on the physical region given by initial condition $E(x_1,x_2;0)$.
Equation (\ref{intE1}) can be decomposed into sectors of area $L\times L$
\begin{eqnarray}\nonumber
E(x_1,x_2;t)=\int_{0}^{L}\int_{0}^{L}
d x_1'd
x_2'\sum_{m,n=-\infty}^{+\infty}{\cal W}{{\mathscr L}_0}{x_1-x_1'-mL}{\cal W}{{\mathscr L}_0}{x_2-x_2'-nL}
\\ \label{intE2}
\times E(x_1'+mL,x_2'+nL;0).
\end{eqnarray}
For example, for $m\ge 0$, we can show recursively the following identities
(after dropping the time argument for simplification)
\begin{eqnarray}\fl\nonumber
E(x_1+mL,x_2) =
\left\{ \begin{array}{rl}
(-1)^pE(x_1,x_2)+\sum_{k=1}^{p}(-1)^{k+1}{\mathscr A}([m-2k]L+x_1)E(0,x_2)
\mbox{, $m=2p$} \\
(-1)^pE(L-x_1,x_2)+\sum_{k=1}^{p}(-1)^{k+1}{\mathscr A}([m-2k]L+x_1)E(0,x_2)
\mbox{, $m=2p+1$.}
\end{array} \right.
\end{eqnarray}
We can define the geometric sum
$\varphi_p(x):=\sum_{k=1}^{p}(-1)^{k+1}{\mathscr A}(2[p-k]L+x)$, with the condition
$\varphi_0(x)=0$, and which can be expressed as
\begin{eqnarray}\label{vp}
\varphi_p(x)=\frac{\cos k_{in}[(2p-1)L+x]-(-1)^p\cos k_{in}(L-x)}{\cos k_{in} L}.
\end{eqnarray}
The previous equation then becomes
\begin{eqnarray}\label{cond_pos}
E(x_1+mL,x_2) =
\left\{ \begin{array}{cl}
(-1)^pE(x_1,x_2)+\varphi_p(x_1)E(0,x_2)
\mbox{, $m=2p$} \\
(-1)^pE(L-x_1,x_2)+\varphi_p(x_1+L)E(0,x_2)
\mbox{, $m=2p+1$.}
\end{array} \right.
\end{eqnarray}
Also, when $m\le 0$, one obtains after some algebra
\begin{eqnarray}\label{cond_neg}
E(x_1+mL,x_2) =
\left\{ \begin{array}{cl}
(-1)^pE(x_1,x_2)+\varphi_p(2L-x_1)E(0,x_2)
\mbox{, $m=-2p$} \\
(-1)^{p}E(L-x_1,x_2)+\varphi_{p}(L-x_1)E(0,x_2)
\mbox{, $m=-2p+1$.}
\end{array} \right.
\end{eqnarray}
These expressions can be put into a more compact form such as
\eref{cond_pos} with $p$ running from negative
to positive values using the symmetry property
$\varphi_{-p}(x)=\varphi_p(2L-x)$, in which case
\eref{cond_neg} is equivalent to \eref{cond_pos} by extrapolation.
Equivalently, we also have two different sets of relations for $E(x_1,x_2+nL)$,
with $n$ either positive or negative, even or odd. However, we can use the
identity $E(x_1,x_2+nL)=2-E(x_2+nL,x_1)$ and
Eqs. (\ref{cond_pos})-(\ref{cond_neg}) to deduce them. It is then
sufficient to express $E(x_1+2pL,x_2+2qL)$ in terms of
$E(x_1,x_2)$ inside the physical domain ${\cal D}_0$. This is done by applying
the symmetries on the first argument $x_1+2pL\rightarrow x_1$, then on the
second $x_2+2qL\rightarrow x_2$, and inversely:
\begin{eqnarray}\nonumber\fl
E(x_1+2pL,x_2+2qL)=(-1)^{p+q}E(x_1,x_2)-(-1)^p\varphi_q(x_2)E(0,
x_1)+(-1)^q\varphi_p(x_1)E(0,x_2)
\\ \fl
+2(-1)^p[1-(-1)^q]+2\varphi_p(x_1)[1-(-1)^q]-\varphi_p(x_1)\varphi_q(x_2)
\\ \nonumber\fl
E(x_1+2pL,x_2+2qL)=(-1)^{p+q}E(x_1,x_2)-(-1)^p\varphi_q(x_2)E(0,
x_1)+(-1)^q\varphi_p(x_1)E(0,x_2)
\\ \fl
+2[1-(-1)^q]-2\varphi_q(x_2)[1-(-1)^p]+\varphi_p(x_1)\varphi_q(x_2).
\end{eqnarray}
The result depends therefore on the paths chosen in the plane
to map the point $(x_1+2pL,x_2+2qL)$ onto the physical domain ${\cal D}_0$.
The correct prescription is to take half the sum of the two previous
identities
\begin{eqnarray}\nonumber\fl
E(x_1+2pL,x_2+2qL)=(-1)^{p+q}E(x_1,x_2)-(-1)^p\varphi_q(x_2)E(0,
x_1)+(-1)^q\varphi_p(x_1)E(0,x_2)
\\ \label{symE}
+[1+(-1)^p][1-(-1)^q]+\varphi_p(x_1)[1-(-1)^q]-\varphi_q(x_2)[1-(-1)^p].
\end{eqnarray}
\begin{figure
\begin{center}
\includegraphics[scale=0.5,angle=270,clip]{Fig3D.eps}
\caption{\label{figEsym} Example of surface $E(x_1,x_2)$ after symmetrization
for the particular initial
conditions $E(x_1,x_2>x_1;0)=\exp[-\alpha(x_2-x_1)^2]$,
$\alpha=10$ and $k_{in}=\pi$.}
\end{center}
\end{figure}
The resulting expression has correct symmetries and continuity. The
continuity between domains of size $(L\times L)$ is satisfied using at the
boundaries $\varphi_{p+1}(0)=\varphi_p(2L)+2(-1)^p$.
In particular, $E(x_1,x_2)$ can be formally written as before as
$E(x_1,x_2)=1+A(x_1,x_2)$
where $A$ is antisymmetric. In Fig. 3 is plotted the resulting surface
after symmetrization for a Gaussian distribution $E(x_1,x_2>x_1;0)$, with an
input current, and which satisfies continuous conditions in the entire plane.
To obtain the solution for the interval probability at any time, the double sum
in \eref{intE2} can be further reduced using \eref{symE} for odd and even
integers $m$ and $n$,
\begin{eqnarray}\nonumber
\int_0^L\int_0^Ldx_1'\,dx_2'
\sum_{m,n=-\infty}^{+\infty}{\cal W}{{\mathscr L}_0}{x_1-x_1'-mL}{\cal W}{{\mathscr L}_0}{x_2-x_2'-nL
}
E(x_1'+mL,x_2'+nL)
\\ \nonumber
=\int_0^L\int_0^Ldx_1'\,dx_2'
\sum_{p,q=-\infty}^{+\infty}\sum_{\epsilon,\epsilon'=0,1}
{\cal W}{{\mathscr L}_0}{x_1-x_1'-\epsilon L-2pL}{\cal W}{{\mathscr L}_0}{x_2-x_2'-\epsilon' L-2qL}
\\
\times E(x_1'+\epsilon L+2pL,x_2'+\epsilon' L+2qL).
\end{eqnarray}
We then replace $E(x_1'+\epsilon L+2pL,x_2'+\epsilon' L+2qL)$ by its
value \eref{symE} in ${\cal D}_0$, and the double sum over $(p,q)$ depends explicitly
on the
two Gaussian series
\begin{eqnarray}\label{def_psichi}\fl
\Psi(x,y):=\sum_{p=-\infty}^{+\infty}{\cal W}{{\mathscr L}_0}{x-y-2pL}(-1)^p,
\;
\chi(x,y):=\sum_{p=-\infty}^{+\infty}{\cal W}{{\mathscr L}_0}{x-y-2pL}\varphi_p(y),
\end{eqnarray}
where function $\Psi(x,y)$ is anti-periodic:
$\Psi(x,y+2L)=\Psi(x+2L,y)=-\Psi(x,y)$. For example, the first term on the
right hand side of \eref{symE} gives
\begin{eqnarray}\nonumber\fl
\sum_{p,q=-\infty}^{+\infty}{\cal W}{{\mathscr L}_0}{x_1-x_1'-2pL}{\cal W}{{\mathscr L}_0}{x_2-x_2'-2qL}
(-1)^{p+q}E(x_1',x_2')=
\Psi(x_1,x_1')\Psi(x_2,x_2')E(x_1',x_2').
\end{eqnarray}
The other sums over $(p,q)$ are performed using additional functions
$\Psi_s(x,y):=\Psi(x,y)-\Psi(x,-y)$ and $\chi_s(x,y):=\chi(x,y)+\chi(x,y+L)$,
and symmetries $E(L+x_1',x_2')=E(L-x_1',x_2')$, $E(x_1',L+x_2')=E(x_1',L-x_2')$.
After rearranging the different terms and performing a variable change in the
integration over $(x_1',x_2')$, one finally obtains
\begin{eqnarray}\nonumber\fl
E(x_1,x_2;t)&=&1+G(x_1)-G(x_2)-G(x_1)F(x_2)+G(x_2)F(x_1)+F(x_1)-F(x_2)
\\ \nonumber\fl
&-&F(x_1)F(x_2)+H(x_1,x_2)
\\ \nonumber\fl
&+&G(x_1)\int_0^Ld x_2'\Psi_s(x_2,x_2')E(0,x_2';0)
-G(x_2)\int_0^Ld x_1'\Psi_s(x_1,x_1')E(0,x_1';0)
\\ \label{res_tot}
&+&\int_0^Ld x_2'\,\int_0^{x_2'}d x_1'\left \{
\Psi_s(x_1,x_1')\Psi_s(x_2,x_2')-\Psi_s(x_2,x_1')\Psi_s(x_1,x_2')
\right \}E(x_1',x_2';0),
\end{eqnarray}
where we defined the functions
\begin{eqnarray}
F(x):=\int_0^L\Psi_s(x,x')\,d x',\;
G(x):=\int_0^L\chi_s(x,x')\,d x',
\end{eqnarray}
and the contribution coming from the double integral over the
two ordered space variables
\begin{eqnarray}
H(x_1,x_2):=2\int_0^Ld x_1'\Psi_s(x_1,x_1')\,\int_0^{x_1'}d
x_2'\Psi_s(x_2,x_2').
\end{eqnarray}
In formula (\ref{res_tot}), the terms independent of the initial conditions
$E(x_1',x_2';0)$ in the first two lines contribute to the long time regime.
It can be checked again that $E(x_1,x_2)=1+A(x_1,x_2)$
where $A$ is antisymmetric, and in particular $E(x_1,x_1)=1$. In the following
we take an initial configuration where particles occupy every site.
\subsection{Expression of the density in terms of Elliptic functions}
Previous functions $\Psi_s$ and $\chi_s$ appearing in \eref{res_tot} can
be expressed in terms of Jacobi elliptic functions $\theta_3$ and $\theta_4$,
after performing the sum over the integers in \eref{def_psichi}. Similar
expressions were found before for the coagulation model with periodic boundary
conditions \cite{Krebs95a}. The details of the computation are given in
\ref{app2} and we find
\begin{eqnarray}\nonumber
\fl\Psi_s(x,y)
=
\sqrt{\frac{2}{\pi {\mathscr L}_0^2}}
\left \{
{\rm e}^{-\frac{2}{{\mathscr L}_0^2}(x-y)^2}
\theta_4\left (\frac{4iL^2}{{\mathscr L}_0^2}\frac{x-y}{L},{\rm e}^{-\frac{8L^2}{{\mathscr L}_0^2}}
\right )
-{\rm e}^{-\frac{2}{{\mathscr L}_0^2}(x+y)^2}
\theta_4\left (\frac{4iL^2}{{\mathscr L}_0^2}\frac{x+y}{L},{\rm e}^{-\frac{8L^2}{{\mathscr L}_0^2}}
\right )
\right \},
\\ \nonumber
\fl
\chi_s(x,y)
=
\sqrt{\frac{2}{\pi{\mathscr L}_0^2}}\frac{{\rm e}^{-\frac{2}{{\mathscr L}_0^2}(x-y)^2}}{\cos(k_{in} L)}
\left \{
\Re\,\left [ {\rm e}^{ik_{in}(y-L)}
\theta_3\left (
\frac{4iL^2}{{\mathscr L}_0^2}\frac{x-y}{L}-k_{in} L,{\rm e}^{-\frac{8L^2}{{\mathscr L}_0^2}}
\right )
\right ]
\right .
\\ \label{def_psischis}
-\left .\cos[k_{in} (y-L)]\theta_4\left
(\frac{4iL^2}{{\mathscr L}_0^2}\frac{x-y}{L},{\rm e}^{-\frac{8L^2}{{\mathscr L}_0^2}}\right )
\right \}+(y\rightarrow y+L).
\end{eqnarray}
\subsection{Small time behavior}
For times $t$ small compare to the characteristic time $t_L:=\frac{L^2}{8{\mathscr D}}$
which is the time for the particles to diffuse through the chain, the ratio
$L^2/{\mathscr L}_0^2$ is large, and we can replace $\theta_3$ and $\theta_4$
in \eref{def_psischis} by unity, since the modulus $\exp(-8L^2/{\mathscr L}_0^2)$
is exponentially small. In this case, one simply obtains
\begin{eqnarray}\label{def_psischisSmall}
\Psi_s(x,y)
=
\sqrt{\frac{2}{\pi {\mathscr L}_0^2}}
\left \{
{\rm e}^{-\frac{2}{{\mathscr L}_0^2}(x-y)^2}
-{\rm e}^{-\frac{2}{{\mathscr L}_0^2}(x+y)^2}
\right \},\;\chi_s(x,y)=0.
\end{eqnarray}
It is then straightforward to evaluate $F(x)={\rm erf}\left (\sqrt{2}x/{\mathscr L}_0
\right )$ and $G(x)=0$. The local density can be generally expressed in terms of
functions $F$, $G$ and $H$ as
\begin{eqnarray}\label{eq_rho}
\rho(x;t)=(1-F(x))G'(x)+(1-F(x)+G(x))F'(x)+\partial_1H(x,x).
\end{eqnarray}
Using $\partial_1H(x,x)=2\,(\pi{\mathscr L}_0^2)^{-1/2}{\rm erf}\left (2x/{\mathscr L}_0 \right
)$, we recover \eref{conc0} and the system behaves like a system of
semi-infinite size without input current. In particular, the integrated density
${\cal N}_L(t)$ can be expanded in terms of large parameter $L/{\mathscr L}_0\gg 1$
\begin{eqnarray}\label{rho_LargeL}
{\cal N}_L(t)\simeq\frac{2L}{\sqrt{\pi}{\mathscr L}_0}+\frac{1}{2}-\frac{1}{\pi}+\left\{\frac{
{\mathscr L}_0 } { \sqrt{ 2\pi }L}-\frac{
{\mathscr L}_0^3 } { 4\sqrt{ 2\pi }L^3}\right \}\exp(-2L^2/{\mathscr L}_0^2),
\end{eqnarray}
where the first term is the $t^{-1/2}$ law and the corrections are
exponentially small in $L^2/{\mathscr L}_0^2$.
\subsection{Large time expansion}
In this section, we analyze the long-time limit of \eref{def_psischis}, when
${\mathscr L}_0\gg L$. In this limit, it is
sufficient to study the behavior of the elliptic functions
\begin{eqnarray} \nonumber
\theta_3(z,\exp(-\alpha\epsilon))=1+2\sum_{n=1}^{\infty}\exp(-\alpha\epsilon
n^2)
\cos(2n z),
\\
\theta_4(z,\exp(-\alpha\epsilon))=1+2\sum_{n=1}^{\infty}\exp(-\alpha\epsilon
n^2)(-1)^n
\cos(2n z),
\end{eqnarray}
where $\alpha>0$, $z$ complex, and $\epsilon:=L^2/{\mathscr L}_0^2$
is the small parameter of the expansion. We can use
the Dirac comb identity
$\sum_{n=-\infty}^{\infty}\delta(x-n)=\sum_{n=-\infty}^{\infty}
\exp(2i\pi nx)$ to rewrite $\theta_3(z,\exp(-\alpha\epsilon))$ as
\begin{eqnarray}\nonumber
\theta_3(z,\exp(-\alpha\epsilon))=\int_{-\infty}^{+\infty}d
x\sum_{n=-\infty}^{\infty}\delta(x-n)\exp(-\alpha\epsilon
x^2)\cos(2x z)
\\ \nonumber
=
\int_{-\infty}^{+\infty}d
x\sum_{n=-\infty}^{\infty}\exp(-\alpha\epsilon
x^2+2i\pi nx)\cos(2x z)
=\sqrt{\frac{\pi}{\alpha \epsilon}}\sum_{n=-\infty}^{\infty}\exp\left
[-\frac{1}{\alpha
\epsilon}\Big (z+n\pi\Big )^2\right ].
\end{eqnarray}
For $\theta_4(z,\exp(-\alpha\epsilon))$, the expression is identical,
with instead a shift of $\pm \pi/2$ in the $z$ argument
\begin{eqnarray}\nonumber
\theta_4(z,\exp(-\alpha\epsilon))=\sqrt{\frac{\pi}{\alpha
\epsilon}}\sum_{n=-\infty}^{\infty}\exp\left [-\frac{1}{\alpha
\epsilon}\Big (z+n\pi-\frac{\pi}{2}\Big )^2\right ].
\end{eqnarray}
Setting $\alpha=8$, $z=4i\epsilon (x-y)/L=:4i\epsilon (u-v)$, with $u:=x/L$ and
$v:=y/L$, one obtains the asymptotic limit for the $\theta_4$ function in
\eref{def_psischis}
\begin{eqnarray}\label{theta4}
\theta_4(4i\epsilon(u-v),\exp(-8\epsilon))\simeq
\sqrt{\frac{\pi}{2\epsilon}}{\rm e}^{-\pi^2/32\epsilon+2\epsilon
(u-v)^2}\cos\left [\frac{\pi}{2}(u-v)\right ].
\end{eqnarray}
In the case of the $\theta_3$ function present in \eref{def_psischis}, we take
instead
$z=4i\epsilon (u-v)-k_{in} L$. The resulting $\theta_3$ function is
then complex and
\begin{eqnarray}\nonumber
\theta_3(4i\epsilon(u-v)-k_{in} L,\exp(-8\epsilon))\simeq
\sqrt{\frac{\pi}{8\epsilon}}\sum_{n=-\infty}^{\infty}
\exp\Big [-\frac{(n\pi-k_{in} L)^2}{8\epsilon}\Big ]
\\ \label{theta3}
\times\exp\Big [2\epsilon(u-v)^2+i(k_{in} L-n\pi)(u-v) \Big ].
\end{eqnarray}
Most of the terms in the sum are exponentially small unless $k_{in} L$ is close
to $n\pi$ with $n$ integer. Using \eref{theta4} and
\eref{theta3}, we can evaluate directly
scaling functions $\Psi_s(x,y)=:L^{-1}\widetilde\Psi_s(u,v)$
and $\chi_s(x,y)=:L^{-1}\widetilde\chi_s(u,v)$. In
particular, one obtains asymptotically
\begin{eqnarray}\nonumber
\widetilde\Psi_s(u,v)\simeq2{\rm e}^{-\pi^2/32\epsilon
}\sin\Big (\frac{\pi u}{2} \Big )\sin\Big (\frac{\pi v}{2} \Big ),
\\ \label{asympt}
\widetilde\chi_s(u,v)\simeq\frac{1}{\cos(k_{in} L)}\left \{
-{\rm e}^{-\pi^2/32\epsilon}\cos\Big [\frac{\pi (u-v)}{2} \Big ]\cos[k_{in} L(v-1)]
\right .
\\ \nonumber
+\left .\frac{1}{2}\sum_{n=-\infty}^{\infty}{\rm e}^{-(n\pi-k_{in} L)^2/8\epsilon}
\cos\Big [k_{in} L(v-1)+(k_{in} L-n\pi)(u-v) \Big ]
\right \}+(v\rightarrow v+1).
\end{eqnarray}
The integration over $v$ can be performed in the previous expansion, $F$
and $G$ are asymptotically given by
\begin{eqnarray}\nonumber
F(x)\simeq \frac{4}{\pi}{\rm e}^{-\pi^2/32\epsilon}\sin\Big (\frac{\pi x}{2L}\Big
)
\\ \label{FG}
G(x)\simeq
{\rm e}^{-\pi^2/32\epsilon}
\frac{\pi}{k_{in}^2L^2-\pi^2/4}\sin\Big (\frac{\pi x}{2L}\Big )
+{\rm e}^{-k_{in}^2L^2/8\epsilon}
\frac{\cos[k_{in}(x-L)]}{\cos(k_{in} L)}.
\end{eqnarray}
In the last sum of \eref{asympt}, only the term $n=0$ does not
vanish after integration over variable $v$.
Taking into account the dominant terms \eref{FG}, and using the fact that $H$
can be approximated by
\begin{eqnarray}\label{H}
H(x_1,x_2)\simeq
\frac{16}{\pi^2}{\rm e}^{-\pi^2/16\epsilon}
\sin\Big (\frac{\pi x_1}{2L}\Big )\sin\Big (\frac{\pi x_2}{2L}\Big
)=F(x_1)F(x_2),
\end{eqnarray}
one obtains for the expression for integrated density ${\cal N}_L(t)$, using
\eref{eq_rho}
\begin{eqnarray}\nonumber\fl
{\cal N}_L(t)&=&\frac{16k_{in}^2L^2}{\pi(4k_{in}^2L^2-\pi^2)}{\rm e}^{-\pi^2t/32t_L}+
\frac{16\pik_{in}
L\sin(k_{in}
L)-16k_{in}^2L^2-4\pi^2}{\pi\cos(k_{in}
L)(4k_{in}^2L^2-\pi^2)}{\rm e}^{-\left (\pi^2+4k_{in}^2L^2\right )t/32t_L}
\\ \label{intc}
\fl&+&\frac{1-\cos(k_{in} L)}{\cos(k_{in} L)}{\rm e}^{-k_{in}^2L^2t/8t_L},
\end{eqnarray}
where $t_L:=L^2/8{\mathscr D}$ is the diffusion time across the system. In the
absence of input current, or $k_{in}=0$, the integrated density simply
decreases like ${\cal N}_L(t)\simeq 4\pi^{-1}{\rm e}^{-\pi^2 t/32t_L}$.
\begin{figure}[ht
\begin{center}
\subfigure[ ]
{
\includegraphics[scale=0.32,clip]{fig_densityTimeLarge.eps}
\label{fig_density_Large}
}
\subfigure[ ]
{
\includegraphics[scale=0.32,clip]{fig_density.eps}
\label{fig_density}
}
\caption{
(a) Averaged number of particle
${\cal N}_L(t)=L\rho_L(t)=\int_0^L\rho(x;t)\,dx$ as
function of time in units of $t_L=L^2/8{\mathscr D}$ (logarithmic scale), for
different values of $k_{in}
L$. The chain is initially filled with particles. The
curves with symbols are the numerical resolution of exact density function
using expressions \eref{def_psischis}. The red dashed curve for $k_{in} L=0.2$ is
the long time behavior \eref{intc}, which fits the exact solution for $t>t_L$.
Black line is the density decay for the scaling regime, $L\gg 1$, given by
\eref{conc0}.
(b) Asymptotic regime ${\mathscr L}_0> L$ or $t>t_L$. Magenta
dashed line shows the exponential decay $4\pi^{-1}\exp\{-\pi^2/32\epsilon\}$
in the limit $k_{in} L=0$, and the black dashed line is the asymptotic
fit, \eref{fit}, for $k_{in} L=0.5$.}
\end{center}
\end{figure}
In Fig. 4(a) is represented the evolution of the number of
particles ${\cal N}_L(t)$ as function of time. For time values less than the
diffusion
time $t_L$, the number decreases like $t^{-1/2}$, and follows closely the
result for the bulk \eref{conc0}. After reaching the diffusion time $t_L$, the
number decreases exponentially like $\exp(-\pi^2t/32t_L)$, independent of the
input current. Then, after a crossover time $t_c$, the long-time regime is
characterized by the exponential decay ${\cal N}_L\sim\exp(-k_{in}^2L^2t/8t_L)$
which depends on $k_{in}$.
This behavior can be seen explicitly in Fig. 4(b), where the crossover
is clearly visible on the averaged number ${\cal N}_L(t)$ as
function of time (here in units of $t_L$) and for different values of $k_{in} L$.
After a sharp decreasing behavior dominated
mainly by the second term of \eref{intc}, the asymptotic regime is
accurately given by
\begin{eqnarray}\label{fit}
{\cal N}_L\simeq\frac{1-\cos(k_{in} L)}{\cos(k_{in} L)}\exp(-k_{in}^2L^2t/8t_L),
\end{eqnarray}
which is represented by the black dashed curve for $k_{in} L=0.5$ in Fig. 4(b).
The characteristic or relaxation time for this process is
actually independent of the system size $L$, and is equal to $8t_L/k_{in}^2
L^2=({\mathscr D} k_{in}^2)^{-1}$.
The different curves appear to decrease more slowly as
$k_{in} L$ is small. The crossover time $t_c$ is determined by comparing the
second and third terms in \eref{intc}, in the limit of small $k_{in} L$ relatively
to $\pi/2$:
\begin{eqnarray}
t_c=\frac{32t_L}{\pi^2}\log\left \{ \frac{4}{\pi[1-\cos(k_{in} L)]}\right \}.
\end{eqnarray}
For example, one obtains $t_c/t_L\simeq 13$ for $k_{in} L=0.2$, $t_c/t_L\simeq 18$
for $k_{in} L=0.1$, and $t_c/t_L\simeq 33$ for $k_{in} L=0.01$, in accordance with
data displayed in Fig. 4(a) and (b).
We can define a transfer ratio through the finite system as
\begin{eqnarray}\label{def_eta}
\eta_L(t):=I_{out}(t)/I_{in}=-\frac{1}{2} k_{in}^{-2}\partial_1^2E(L,L;t),
\end{eqnarray}
which measures the loss of particles through the system in presence of an input
current. From the general expression (\ref{res_tot}), the current
$I_{out}$ and coefficient $\eta_L$ can be evaluated with initial
conditions where the chain is entirely filled with particles $E(x_1,x_2;0)=0$:
\begin{eqnarray}\label{eq_eta}\fl
\eta_L(t)=-\frac{1}{2}k_{in}^{-2}\Big [
\{1-F(L)\}G''(L)+\{1+G(L)\}F''(L)-F(L)F''(L) +\partial_1^2H(L,L)\Big ].
\end{eqnarray}
\begin{figure
\subfigure[ ]
{
\includegraphics[scale=0.35,clip]{fig_eta_time.eps}
\label{fig_eta_time}
}
\subfigure[ ]
{
\includegraphics[scale=0.35,clip]{fig_eta.eps}
\label{fig_eta}
}
\caption{
(a) Current ratio $\eta_L(t)=I_{out}(t)/I_{in}$ as
function of time $t$, in units of $t_L=L^2/8{\mathscr D}$, for different values of
parameter $k_{in} L$ in the asymptotic regime $t\gg t_L$, or when
$\epsilon=t_L/t$ is small. The current
ratio tends to a constant value, around $1/2$ when $k_{in} L$ is small, before
decreasing at later times.
(b) Current ratio as function of $k_{in} L$ for
different time values.
}
\end{figure}
Using approximations (\ref{FG}) and \ref{H}), one obtains
\begin{eqnarray}\nonumber
\eta_L(t)=\frac{2\pi}{4k_{in}^2L^2-\pi^2}{\rm e}^{-\pi^2t/32t_L}-
\frac{4k_{in}^2L^2-\pi^2}{2\pik_{in}^2L^2\cos(k_{in} L)}{\rm e}^{-\left (\pi^2+4k_{in}^2L^2\right
)t/32t_L}
\\ \label{eta}
+\frac{1}{2\cos(k_{in} L)}{\rm e}^{-k_{in}^2L^2t/8t_L}.
\end{eqnarray}
Figure 5(a) represents $\eta_L(t)$ as function of time, in units
of $t_L$, and for several values of input current $k_{in} L$. We
notice first a sharp decreasing of the output current,
then a crossover towards a regime with a less pronounced variation.
In particular, in the limit of small $k_{in} L\ll 1$, or very low current,
$\eta_L(t)$ is close to $1/2$. In this limit, one obtains the following
expansion
\begin{eqnarray}
\eta_L(t)\simeq \frac{1}{2}+\left [
\frac{\pi}{2k_{in}^2L^2}+\frac{\pi}{4}-\frac{4}{\pi}-\frac{\pi t}{16t_L}
\right ]{\rm e}^{-\pi^2t/32t_L },
\end{eqnarray}
for which the value $1/2$ is reached after an interval of time $t_L$.
Oppositely, figure 5(b) represents the ratio $\eta_L(t)$ as function
of $k_{in} L$ for different time values. As the size $L$ of the system increases,
the ratio goes to zero monotonically as expected. Finally, we can use
expressions \eref{intc} and \eref{eta} to compute the coagulation rate
$R(t)$, defined by \eref{coag}, as function of time. In the long
time limit,
and for small input current $k_{in} L\ll 1$, the following expansion is obtained
\begin{eqnarray}
R(t)\simeq \frac{1}{2} I_{in}+\frac{\pi}{16
t_L}{\rm e}^{-\pi^2t/32t_L}+8I_{in}{\rm e}^{-\pi^2t/32t_L}\left
[\frac{3}{2\pi}-\frac{1}{2}+\frac{\pi}{32}-\frac{\pi t}{128 t_L} \right ],
\end{eqnarray}
which shows that half of the input particles coagulate, plus corrective terms
which are exponentially small, the last term being negative in this limit.
These corrections arise from the finiteness of the system and depend on the
time for the input particles to reach the opposite border.
\section{Conclusion}
In this paper, we presented an application of the empty interval method to
the dynamic properties in a reaction-diffusion process, with semi-infinite
and finite geometries, as in \cite{Hinr97}. The method developed here is
well adapted in computing the particle density using only a two-space
variable interval probability which
satisfies a classical linear equation of diffusion, and which measures
specifically
the probability of having an empty space between two given sites. The essential
point here was to find a different method from~\cite{Hinr97}, to treat
the boundary conditions, since there
is no possibility to use translation invariance, by incorporating the boundary
terms into general symmetries
of the probability function. This can be done by extending the problem outside
the physical domain, and by introducing a mirror-image like method that takes
exactly into account the continuity and differentiability relating negative
(unphysical) and positive (physical) interval sizes in the discrete form of the
master equation. The effect of a current at
the origin, which probes the dynamics for
a finite or semi-infinite system, is to induce different time scales, one
short time scaling regime, where the density scales like $t^{-1/2}$, and two
exponential decays, once the time reaches the typical diffusion time scale
through the chain, whose relaxation constant depends on the current value. We
were also able to compute the coagulation rate in the asymptotic regime by
studying the balance between the different reaction rates. The semi-infinite
chain with asymmetry diffusion rate shows also the existence of an optimal
current which maximizes the particle density near the origin. This method can
also be implemented to treat other boundary conditions and/or initial particle
configurations.
\ack{We would like to acknowledge M. Henkel and J. Richert for informal
discussions on this topic.}
|
2,877,628,089,527 | arxiv | \section{Introduction}
Precise stellar parameters, such as effective temperature, surface gravity, metallicity, stellar mass, and stellar radius, are crucial for several reasons in astronomy. Amongst these, there are the precise characterization of planetary systems \citep[e.g.][]{Tor12,ME13c}, discovery of the possible link between the properties of stars and the existence of a planet \citep[e.g.][]{Adi13,Bea13,ME13}, and the complete and accurate picture of Galactic evolution \citep[e.g.][]{Edv93,McW08,Min13}.
In the ever-growing exoplanetary field\footnote{More than 1700 discovered exoplanets, see \url{www.exoplanet.eu}}, accurate and precise stellar parameters are necessary for the precise characterization of exoplanets. The main bulk of the discovered exoplanets has been found using radial velocities and/or the photometric transit technique. Separately, these techniques only partly characterize the planet. With radial velocities, a constraint is put on the planetary mass ($M_p\sin i$), while the transit technique is used to determine the planetary radius ($R_p$). Good knowledge of both these properties is essential for understanding the different kinds of planets and their distributions in the Galaxy \citep[e.g.][]{Buch14,Dum14,Marcy14}.
However, these planetary characteristics (mass, radius, and thus mean density) are highly dependent on the knowledge of the stellar characteristics ($M_p \propto M_{\ast}^{2/3}$ and $R_p \propto R_{\ast}$) \citep[e.g. ][]{Tor12,ME13c}. The stellar mass and radius, in turn, depend on the effective temperature, surface gravity, and the metallicity of the star, therefore it is extremely important to obtain precise atmospheric stellar properties.
Furthermore, to minimize the errors and to obtain comparable results, a uniform analysis is required \citep{Tor08,Tor12,San13} to guarantee the best possible homogeneity in the results. By homogeneously deriving precise stellar parameters we also gain more than just improving planetary parameters. Observational and theoretical works have shown that the processes of planet formation and evolution seem to depend on several stellar properties, such as stellar metallicity and mass \citep[e.g. ][]{But06,Udry07,Bow10,John10,Mayor11,Sou11b,Mor12,ME13,Adi13}. With large samples of planet hosts with homogeneously derived stellar and planetary parameters, we can look for correlations between the various parameters and statistically evaluate them. These correlations will allow us to narrow down the theories of planet formation.
Not just exoplanetary science benefits from having precise, accurate, and homogeneous stellar properties. These can also be useful to explain the formation and evolution of stars and thus of our Galaxy, which consists of different structures all with different properties. It has been shown for example that there is a difference in metallicity (iron and other heavy elements) between the thin disk and the thick disk \citep[e.g.][]{Edv93,Bens05,Hay08,Adi13b}. To properly understand the different stellar populations and their origins in the Milky Way, we need precise and homogeneous stellar parameters.
To derive a set of precise stellar properties (effective temperature $T_{\mathrm{eff}}$, surface gravity $\log g$, metallicity [Fe/H], and microturbulence $\xi$), high-resolution spectroscopy is usually the best approach. Commonly, two methods are used to analyse these spectra: spectral synthesis and spectral line analysis. The first method compares observed spectra with synthetic ones, for example with the code SME \citep{Val96} or MATISSE \citep{Rec06}. Spectral line analysis, as used in this work, makes use of the equivalent width (EW) of absorption lines (usually the \ion{Fe}{i} and \ion{Fe}{ii} lines) to demand excitation and ionization equilibrium.
\begin{figure*}[th!]
\begin{center}
\includegraphics[width=5.8cm]{DeltaTeff_specLC_ART.ps}
\includegraphics[width=5.8cm]{DeltaFeh_specLC_ART.ps}
\includegraphics[width=5.8cm]{DeltaVT_specLC_ART.ps}
\caption{Differences of the spectroscopic results (left to right: effective temperature, metallicity, and microturbulence) as a function of the difference in $\log$g (defined as `constrained with transit $\log g$ - unconstrained').}
\label{FigFL1}
\end{center}
\end{figure*}
Both methods have been shown to provide surface gravities that are not well constrained and do not compare well with surface gravities as obtained from other non-spectroscopic methods, such as asteroseismology or stellar models \citep[e.g. ][]{Tor12,Hub13,ME13c}. This surface gravity is important for the determination of the stellar mass and especially the stellar radius as shown in \citet{ME13c}.
In this work, we take a closer look at the surface gravity and its effect on the determination of the other atmospheric parameters. In Section \ref{Meth}, we present the uniform spectroscopic method we use. Section \ref{Tra} handles the effect of fixing the surface gravity to a value obtained by transit photometry and a possible correction formula. The same study is then done for the more accurate surface gravities as obtained by asteroseismolgy (Section \ref{Seis}). Finally, we discuss in Section \ref{Dis}.
\section{Spectroscopic method}\label{Meth}
Over the years, we have developed a homogeneous method to derive stellar parameters \citep[e.g.][]{San04,Sou08,Sou11b,Tsa13}. This method is based on the analysis of iron lines from high-resolution spectra. Details of this method can be found in \citet{San13} and references therein. Here we only give an overview of the method.
EWs of iron lines (\ion{Fe}{I} and \ion{Fe}{II}) are automatically calculated with the code ARES \citep[Automatic Routine for line Equivalent widths in stellar Spectra -][]{Sou07} for which the large lists with stable lines of \citet{Sou08} and \citet{Tsa13} are used for stars hotter and cooler than 5200\,K, respectively. These EWs are then used together with a grid of ATLAS plane-parallel model atmospheres \citep{Kur93} to determine the atmospheric stellar parameters, $T_{\mathrm{eff}}$, $\log g$, [Fe/H], and $\xi$. Therefore, we use the MOOG code\footnote{\url{http://www.as.utexas.edu/~chris/moog.html}} \citep{Sne73} in which we assume Local Thermodynamic Equilibrium (LTE).
By imposing excitation and ionization equilibrium, the atmospheric parameters are determined using an iterative minimization code based on the Downhill Simplex Method \citep{Pre92}.
The same method can be used whilst fixing the surface gravity to a predetermined value (see next Sections). In this case however, ionization equilibrium will not be imposed as this is the main condition for determining the surface gravity. As a direct result, we do not use the \ion{Fe}{II} lines anymore. The value for the metallicity is thus determined by only using the \ion{Fe}{I} lines.
\begin{longtab}
\begin{longtable}{cccccccc}
\caption{\label{TabPar} Stellar (unconstrained) spectroscopic parameters used in this work. The last two columns contain the surface gravities as calculated with Equation \ref{EqFit}, resp. Equation \ref{EqFit2}}\\
\hline\hline
Name & T$_{eff,spec}$ & $\log g_{spec}$ & [Fe/H]$_{spec}$ & $\xi_{spec}$ & Ref. & $\log g_{corr,1}$ & $\log g_{corr,2}$ \\
& (K) & (dex) & (dex) & (km s$^{-1}$) & & (dex) & (dex) \\
\hline
\endfirsthead
\caption{continued.}\\
\hline\hline
Name & T$_{eff,spec}$ & $\log g_{spec}$ & [Fe/H]$_{spec}$ & $\xi_{spec}$ & Ref. & $\log g_{corr,1}$ & $\log g_{corr,2}$ \\
& (K) & (dex) & (dex) & (km s$^{-1}$) & & (dex) & (dex) \\
\hline
\endhead
\hline
\endfoot
\input{TablePar.tex}
\end{longtable}
\input{TableParRef.tex}
\end{longtab}
\begin{longtab}
\begin{longtable}{cccccc}
\caption{\label{TabParFix} Stellar spectroscopic parameters where the surface gravity was fixed to either the value from the photometric transit light curve or a value obtained through asteroseismology. }\\
\hline\hline
Name & T$_{eff,fix}$ & $\log g_{fix}$ & [Fe/H]$_{fix}$ & $\xi_{fix}$ & Method\\
& (K) & (dex) & (dex) & (km s$^{-1}$) & \\
\hline
\endfirsthead
\caption{continued.}\\
\hline\hline
Name & T$_{eff,fix}$ & $\log g_{fix}$ & [Fe/H]$_{fix}$ & $\xi_{fix}$ & Method\\
& (K) & (dex) & (dex) & (km s$^{-1}$) & \\
\hline
\endhead
\hline
\endfoot
\input{TableParFix.tex}
\end{longtable}
\end{longtab}
\section{Surface gravity from transits}\label{Tra}
For stars with transiting planets, an independent measurement of the surface gravity can be obtained using the effective temperature and metallicity from the spectroscopic analysis, and the stellar density which is obtained directly from the transit light curve through the formula
\begin{equation}
\rho_{\ast} + k^3\rho_p = \frac{3\pi}{\mathrm{G}P^2}\left(\frac{a}{R_{\ast}}\right)^3,
\end{equation}
\noindent where $\rho_{\ast}$ and $\rho_p$ are the stellar and planetary density, $P$ the period of the planet, $a$ the orbital separation, G the gravitational constant, and $R_{\ast}$ the stellar radius \citep{Winn11}. Since the constant coefficient $k$ is usually small, the second term on the left is negligible. All parameters on the right come directly from analysing the transit light curve.
The surface gravities can then be obtained through isochrone fitting using the PARSEC isochrones \citep{Bre12} and a $\chi^2$ minimization process \citep[for details, see ][]{ME13c}. They showed that the spectroscopic and photometric surface gravities do not compare well with each other. The $\log g$ values obtained through the photometric transit light curve compare best with literature values (but note that most literate values also come from photometric methods).
In this work, we used the sample of 87 stars from \citet{ME13c}. All these stars are of spectral type F, G or K and are known to be orbited by a transiting planet (according to the online catalog \url{www.exoplanet.eu}). They were observed with different high-resolution spectrographs and analysed in \citet{ME13c} with our method (see Table \ref{TabPar}).
In order to test the effect the surface gravity has on the determination of the other three atmospheric parameters, we redid the same spectroscopic analysis as performed in \citet{ME13c}, but we fixed the surface gravity to the value obtained through the photometric transit light curve. The results can be found in Table \ref{TabParFix}. The errors of the effective temperature, metallicity and microturbulence were set to the errors of the unconstrained values. Not all spectra were suitable to derive atmospheric parameters whilst fixing one parameter due to their lower signal-to-noise ratio (S/N). For these lower S/N stars we did not always reach the rigorous convergence we apply in the analysis and we preferred not to lighten it. In the end, we got results for 76 out of the 87 stars. This subsample is representable for the complete sample.
For 12 of the cooler stars, where the shorter linelist of \citet{Tsa13} was used, we did not always converge to a good microturbulence determination because of the small EW interval of the measured \ion{Fe}{i} lines. Following \citet{ME13b}, the microturbulence was derived with the empirical formula \citep[taken from ][]{Ram13}
\begin{multline}\label{EqVt}
\xi_t = 1.163 + 7.808\cdot10^{-4}\cdot (T_{\mathrm{eff}}-5800) \\- 0.494 \cdot (\log g - 4.30) - 0.05\cdot [Fe/H].
\end{multline}
This formula is comparable to what \citet{Tsa13} found, using 451 FGK dwarfs with parameters derived following our method. In this work, however, we gave preference to the formula of \citet{Ram13}, since they include the metallicity of the star in the relation.
We compared the stellar parameters obtained from fixing the surface gravity to the photometric light curve value with the parameters obtained with no constraints on the surface gravity \citep[taken from ][]{ME13c}. All three parameters compare well, with mean differences of $19$\,K, $0.02$\,dex and $0.0$\,km/s for the effective temperature, metallicity, and microturbulence, respectively.
In Figure \ref{FigFL1}, the differences in the spectroscopic parameters (defined as `constrained with transit $\log g$ - unconstrained') are plotted against the difference in surface gravity (defined as 'photometric - spectroscopic'). All three parameters are anticorrelated with the difference in surface gravity.
Because of these trends, we calculated the median absolute deviations (MAD) as well, which is an easy way to quantify variation. We find that the MADs are $66.5$\,K, $0.03$\,dex and $0.13$\,km/s for the effective temperature, metallicity, and microturbulence, respectively. Since these values are within the errorbars of the parameters, these trends are thus small enough so that we are confident that the surface gravity does not have a large effect on the determination of other atmospheric parameters using our method of spectral line analysis with the linelists of \citet{Sou08} and \citet{Tsa13}.
The differences in the spectroscopic parameters become constant for higher absolute differences of the surface gravity. This is in contrast with the results from \citet{Tor12} where the differences were linearly correlated with the surface gravity difference, also for the larger differences. In their work, they used two spectral synthesis methods, SPC \citep[Stellar Parameter
Classification - ][]{Buch12} and SME \citep[Spectroscopy Made Easy - ][]{Val96}. They also tested for a spectral line analysis method, but the sample was too small for any firm conclusions.
\subsection{Correction with temperature}\label{CorrTr}
The differences in photometric and spectroscopic surface gravity seem to depend on the (unconstrained) effective temperature as can be seen in Figure \ref{Figlogg}, where a decreasing linear trend is noticeable. The same trend is found for the microturbulence, which is closely related to the effective temperature (as seen from Equation \ref{EqVt}). Comparing the $\log$g differences with metallicities reveals no additional trends.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=7cm]{DeltaLogg_Teff_fit_ART.ps}
\caption{Surface gravity difference (`photometric - spectroscopic') versus the (unconstrained) effective temperature. A linear fit is shown with the solid black curve.}
\label{Figlogg}
\end{center}
\end{figure}
We fitted the trend with temperature with a linear function, taking into account the errors on both datasets (see Figure \ref{Figlogg}). We used the complete sample of 87 stars and followed the procedure as described in Numerical Recipes in C \citep{Pre92C} to obtain 1-sigma errors on the coefficients. We found the following relation:
\begin{equation}\label{EqFit}
\log g_{LC} - \log g_{spec} = -4.57\pm0.25\cdot 10^{-4} \cdot T_{\mathrm{eff}} + 2.59 \pm 0.15
\end{equation}
This formula is valid for stars with an effective temperature between $4500$\,K and $7050$\,K. It can be used to correct for the spectroscopic surface gravity when no transit light curve is available (and thus even for stars without planets). Using this formula assumes that the $\log g$ value coming from the transit is the more accurate one. As we will show later (see Section \ref{Seis}), these values can also suffer from inaccuracies. By applying this formula, we corrected our spectroscopic surface gravities for the sample of 87 stars (see Table \ref{TabPar}). The resulting values compare, as expected, better with the photometric surface gravities.
As an additional test, we selected a subsample of our sample of stars, the ones with the highest S/N spectra (38 out of 87 stars). The coolest stars were hereby left out of the sample. We then again redid the spectroscopic analysis, but this time we fixed the surface gravity to the value corrected using Equation \ref{EqFit}..
We compare the spectroscopic parameters obtained from fixing the surface gravity to the formula corrected value ('corr') with the unconstrained spectroscopic parameters ('spec') and the ones obtained from fixing the surface gravity to the photometric light curve value ('LC'). All parameters compare really well (see Figure \ref{FigFixCorr}), with mean differences of $13$\,K, $0.02$\,dex and $0.02$\,km/s for the effective temperature, metallicity, and microturbulence, respectively for the difference between the corrected values and the spectroscopic values. For the differences between the corrected and the photometric results, we find mean differences of $-21$\,K, $-0.01$\,dex and $-0.06$\,km/s for the effective temperature, metallicity, and microturbulence, respectively. No obvious trends are present. For completeness we calculated the MADs again. For the difference between the corrected values and the spectroscopic values, we find a MAD of $38$\,K, $0.02$\,dex and $0.08$\,km/s for the effective temperature, metallicity, and microturbulence, respectively. The difference between the corrected values and the photometric values gives a MAD of $42$\,K, $0.02$\,dex and $0.08$\,km/s for the effective temperature, metallicity, and microturbulence, respectively. These are thus well within the error bars.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=5.8cm]{Teff_specCorr_ART.ps}
\includegraphics[width=5.8cm]{Feh_specCorr_ART.ps}
\includegraphics[width=5.8cm]{Vt_specCorr_ART.ps}\\
\includegraphics[width=5.8cm]{Teff_LCCorr_ART.ps}
\includegraphics[width=5.8cm]{Feh_LCCorr_ART.ps}
\includegraphics[width=5.8cm]{Vt_LCCorr_ART.ps}
\caption{Comparisons of the spectroscopic effective temperatures, metallicities, and microturbulences (left to right). In the top panels we compare the unconstrained results (`spec') with the results using a fixed surface gravity from the correction formula (`corr'). In the bottom panels the comparison is shown between the two fixed results ($\log g$ from transit (`LC') and $\log g$ from the formula (`corr')).}
\label{FigFixCorr}
\end{center}
\end{figure*}
\section{Surface gravity from asteroseismology}\label{Seis}
As \citet{Hub13} showed, the surface gravities obtained through the stellar density from the transit light curve may also be less accurate when the eccentricity or the impact parameter of the transiting planet are under- or overestimated or fixed whilst fitting the light curve. Asteroseismic $\log g$'s on the other hand are more accurate. Although most of the planets from our sample in the previous Section have almost circular orbits, it is still worth, especially since \citet{Hub13} show clear trends, to check if a similar relation can be found to correct spectroscopic surface gravities if one would use asteroseismic surface gravities.
We used a sample compiled from the literature for which the asteroseismic parameters, the maximum frequency $\nu_{max}$ and the large separation $\Delta\nu$, are precisely determined and we have access to high-resolution spectra with moderate to high signal-to-noise. In the end, we have a sample of 86 stars, subsamples of the samples in \citet{Chap14} and \citet{Bru10}. The first work contains asteroseismic data obtained with the Kepler space telescope \citep{Bor09}. The latter compiles a sample of stars analysed with HARPS \citep{Mayor03}.
Spectroscopic parameters for the sample of \citet{Chap14} are gathered from \citet{Mol13}. Their work contains spectroscopic parameters for Kepler targets derived by several methods, one of which is our method with the linelist of \citet{Sou08} as described in Section \ref{Meth}. There are 74 stars in common. The 12 stars from \citet{Bru10} have been spectroscopically analysed with our method either in previous works \citep{San05,Sou08,Tsa13,San13} or in this work.
For stars that were previously not yet analysed by our team, we gathered per star 40 spectra from the HARPS archives (taken from the long asteroseismology series). We shifted them to the reference frame and added them together. Given that these stars are bright, this gives for a high S/N spectrum in the end. We then analysed them following the method described in Section \ref{Meth}. The results are in Table \ref{TabPar}.
The surface gravities of the final sample of 86 stars are then obtained through isochrone fitting using the PARSEC isochrones \citep{Bre12} in the web interface for the Bayesian estimation of stellar parameters\footnote{\url{http://stev.oapd.inaf.it/cgi-bin/param}} \citep[for details, see ][]{Das06}. As input parameters we needed the large separation $\Delta\nu$, the maximum frequency $\nu_{max}$, the effective temperature T$_{\mathrm{eff}}$, and the metallicity [Fe/H]. As Bayesian priors we assumed the lognormal initial mass function from \citet{Chab03} and a constant star formation rate.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=7.0cm]{LoggLogg_seism_ART.ps}
\caption{Asteroseismic versus spectroscopic surface gravity.}
\label{FigloggSA}
\end{center}
\end{figure}
As expected, the spectroscopic and the asteroseismic surface gravities do not compare well (see Figure \ref{FigloggSA}). As before, we redid, for most of the sample, the same spectroscopic analysis as performed in \citet{ME13c}, but this time we fixed the surface gravity to the asteroseismic value. The results can be found in Table \ref{TabParFix}.
\begin{figure*}[th!]
\begin{center}
\includegraphics[width=5.8cm]{DeltaTeffvsDeltaLogg.ps}
\includegraphics[width=5.8cm]{DeltaFehvsDeltaLogg.ps}
\includegraphics[width=5.8cm]{DeltaVtvsDeltaLogg.ps}
\caption{Differences of the spectroscopic results (left to right: effective temperature, metallicity, and microturbulence) as a function of the difference in $\log$g (defined as `constrained with asteroseismic $\log g$ - unconstrained').}
\label{FigFA1}
\end{center}
\end{figure*}
We compared the parameters obtained from fixing the surface gravity to the asteroseismic value with the parameters obtained with no constraints on the surface gravity. All parameters compare well, with mean differences of $68$\,K, $0.04$\,dex, and $0.15$\,km/s for the effective temperature, metallicity, and microturbulence, respectively.
In Figure \ref{FigFA1}, the differences in the spectroscopic parameters (defined as `constrained with asteroseismic $\log g$s - unconstrained') are plotted against the difference in surface gravity (defined as `asteroseismic - spectroscopic'). All parameters are slightly anticorrelated with the difference in surface gravity, although most values stay within errorbars. Furthermore, we see the same converging trends as before.
Because of these trends, we again calculated the median absolute deviations (MAD) to quantify the variation. We find that the MADs are $28.5$\,K, $0.02$\,dex, and $0.06$\,km/s for the effective temperature, metallicity, and microturbulence, respectively. Since these values are definitely within the errorbars of the parameters, these trends are thus small enough so that we are again confident that the surface gravity does not have a large effect on the determination of other atmospheric parameters using our method of spectral line analysis and the mentioned linelists. This result confirms the results from Section \ref{Tra}.
\subsection{Correction with temperature}\label{CorrS}
The differences in asteroseismic and spectroscopic surface gravity also seem to depend on the (unconstrained) effective temperature as can be seen in Figure \ref{Figlogg2}, where a decreasing linear trend is again noticeable. The same trend is found for the microturbulence while comparing the $\log$g differences with metallicities reveals no additional trends.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=6.8cm]{DeltaLogg_Teff_fit_seism_ART.ps}
\caption{Surface gravity difference (`asteroseismic - spectroscopic') versus the (unconstrained) effective temperature. A linear fit is shown with the solid black curve.}
\label{Figlogg2}
\end{center}
\end{figure}
We applied the same procedure as in Section \ref{CorrTr} on the complete sample of 86 stars.
We found the following relation:
\begin{equation}\label{EqFit2}
\log g_{seis} - \log g_{spec} = -3.89\pm0.23\cdot 10^{-4} \cdot T_{\mathrm{eff}} + 2.10 \pm 0.14
\end{equation}
This formula is comparable to the fit presented in Section \ref{CorrTr} for the overlapping temperature range ($5200$\,K till $7000$\,K). This may be somehow surprising since the transit $\log g$ may be less accurate than the asteroseismic one, as showed by \citet{Hub13}. However, we note that in our sample of transiting hosts, most planets have nearly circular orbits which strengthens the accuracy for the derived surface gravity through the transit light curve.
Given the better accuracy of asteroseismic surface gravities as compared to photometric surface gravities, we prefer Equation \ref{EqFit2} to correct for the spectroscopic surface gravity. Since we barely have asteroseismic data for stars cooler than 5200\,K, we cannot guarantee the accuracy of this formula for that temperature range and Equation \ref{EqFit} may thus be preferred for cooler stars.
\section{Summary and discussion}\label{Dis}
In this work we derived spectroscopic parameters (effective temperature, metallicity, surface gravity and microturbulence) for a sample of FGK dwarfs in several ways. First we left the surface gravity free in the spectroscopic analysis as described in Section \ref{Meth} \citep[for the values, see ][and this work]{ME13c,Mol13}. Afterwards, we reran the same analysis whilst fixing the surface gravity to different values:
\begin{itemize}
\item A value obtained through the photometric transit light curve.
\item A value obtained through the large separation and maximum frequency from asteroseismology.
\item A value obtained through an empirical formula, using the effective temperature and the unconstrained surface gravity.
\end{itemize}
We find that, in almost all cases, the resulting stellar atmospheric parameters ($T_{\mathrm{eff}}$, [Fe/H], $\xi_t$) compare well within errorbars although there are slight trends noticable which correlate with the difference in surface gravity. The trends quickly converge and the differences in atmospheric parameters stay stable even for very large differences in surface gravity.
Differences between the constrained and the unconstrained atmospheric parameters can lead to differences in the values for the stellar mass and radius, and thus the planetary mass and radius. On average, the difference for the efffective temperature is about $70$\,K and for the metallicity about $0.04$\,dex. Using these numbers and the calibration formulae from \citet{Tor10}, we find that the resulting stellar mass and stellar radius will, on average, only differ by about $2-3\%$ and $1-1.5\%$, respectively, for FGK dwarfs. This will lead to an average difference of $1.3-2\%$ and $1-1.5\%$ for the planetary mass and radius. These differences are well within the precision that can currently be achieved \citep[e.g.][]{Hub13}.
It seems that the difference between the spectroscopic surface gravities and the photometric or asteroseismic ones is dependent on the effective temperature. By fitting a linear relation to the data, we obtained a correction formula for the surface gravity obtained with our spectroscopic method. Since asteroseismic surface gravities are the most accurate, we recommend to use Equation \ref{EqFit2} rather than Equation \ref{EqFit}. For stars cooler than 5200\,K, we have little asteroseismic data and as such we cannot guarantee the accuracy of the formula for cooler stars. However, since Equation \ref{EqFit} from the photometric $\log g$s is comparable to the one coming from asteroseismic $\log g$s, the former may be used with caution for the cooler stars.
We note that although the surface gravity as calculated through these formulas may be more accurate, it cannot be more precise than the original unconstrained surface gravity, since the error bars of the spectroscopic $\log g$ are factored in when calculating the corrected value for $\log g$. Regardless, the value will definitely be more accurate than the one from the unconstrained MOOG analysis using our proposed linelists. As such the corrected value is better used for calculating other stellar parameters like the stellar mass and radius in case no additional methods can be used to derive the surface gravity, such as a transit light curve or asteroseismology.
For the other spectroscopic parameters the question remains whether the original spectroscopic parameters are accurate and can thus be used without performing the spectroscopic analysis again. In the case of the effective temperature, we compared our values with values obtained with the accurate and trusted InfraRed Flux Method (IRFM). We have 19 values from the transit sample from \citet{Max11b} and 21 from \citet{Cas11} of which 2 from the transit sample and 19 from the asteroseismic sample. The comparisons can be seen in Figure \ref{FigIRFM}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=9cm]{Teff_specIRFM_all_ART2.ps}
\caption{Comparisons of the unconstrained (left) and constrained (right) spectroscopic temperatures with literature temperatures obtained through the IRFM method (taken from \citet{Max11b}, represented by green triangles, and \citet{Cas11}, represented by blue circles).}
\label{FigIRFM}
\end{center}
\end{figure}
For stars cooler than 6300\,K, the results compare well. We find mean differences of $-75 \pm 100$\,K and $-66\pm74$\,K, for our unconstrained and constrained temperatures, respectively. For the total sample, we find mean differences of $-106\pm122$\,K and $-122\pm138$\,K, respectively. For both the unconstrained and the constrained values, the hotter stars show larger differences, where the spectroscopic temperatures are larger than the ones from the IRFM. This may be an effect of the linelist used for the spectral line analysis. This linelist was calibrated for solar-like stars and the resulting effective temperatures may be overestimated for stars that are much hotter than our Sun \citep[see also][]{Sou11b}.
Given on one hand the marginal difference between comparing the IRFM temperatures with the constrained or the unconstrained temperatures and on the other hand the fact that fixing the surface gravity barely affects the other atmospheric parameters, we can be confident about the results of our unconstrained spectroscopic analysis for the derivation of the effective temperature, metallicity, and microturbulence of FGK dwarfs.
\citet{Tor12} did a similar analysis, but they used an analysis based on synthetic spectra. As already mentioned in Section \ref{Tra}, our results are better constrained than those from an analysis with synthetic spectra and the linelist of \citet{Val05}. They found a linear relation between the temperature and metallicity differences with the surface gravity difference. For surface gravity differences $\Delta \log g \sim 0.5$\,dex, they found differences in temperature of about 350\,K and in metallicity of about 0.20\,dex. With our spectral line analysis method and the carefully selected linelist, we have differences of only 120\,K and 0.05\,dex for temperature and metallicity, respectively.
To conclude, when atmospheric stellar parameters of FGK dwarfs are derived with high-resolution spectroscopy using our ARES+MOOG method, as described in \citet[][and references therein]{San13}, and the linelist of \citet{Sou08} or \citet{Tsa13}, we are confident that the resulting effective temperature, metallicity, and microturbulence are accurate and precise\footnote{We note that the effective temperatures are slightly overestimated for the hotter stars, as mentioned in \citet{Sou11b}.}. The less accurate surface gravity can then easily be corrected using Equation \ref{EqFit2} (or Equation \ref{EqFit} for the coolest stars). This method will always work with high-resolution spectra, even when no other means are available for the determination of surface gravity, like a transit light curve or asteroseismology.
\begin{acknowledgements}
We like to thank the anonymous referee for the fruitful discussion on our paper.
This work made use of the ESO archive and the Simbad Database. This work was supported by the European Research Council/European Community under the FP7 through Starting Grant agreement number 239953. N.C.S. was supported by FCT through the Investigador FCT contract reference IF/00169/2012 and POPH/FSE (EC) by FEDER funding through the program "Programa Operacional de Factores de Competitividade - COMPETE. V.Zh.A., S.G.S., and I.M.B. acknowledge the support of the Funda\c c\~ao para a Ci\^encia e a Tecnologia (FCT) in the form of grant references SFRH/BPD/70574/2010, SFRH/BPD/47611/2008, and SFRH/BPD/87857/2012. IMB acknowledges support from the EC Project SPACEINN (FP7-SPACE-2012-312844).
\end{acknowledgements}
\bibliographystyle{aa}
|
2,877,628,089,528 | arxiv | \section{Background and results}
Let $X$ be a topological space consisting of functions which are analytic in the unit disk $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1\}$ and which satisfies some customary desirable properties, such as that the evaluation $f \mapsto f(\lambda)$ is a continuous functional on $X$ for each $\lambda \in \mathbb{D}$, and that the function $z \mapsto zf(z)$ is contained in the space $X$ whenever $f \in X$. A function $g \in X$ is said to be \textit{cyclic} if there exists a sequence of analytic polynomials $\{p_n\}_n$ for which the polynomial multiples $\{gp_n\}_n$ converge to the constant function $1$ in the topology of the space.
The well-known Hardy classes $H^p$ are among the very few examples of analytic function spaces in which the cyclicity phenomenon is completely understood. The cyclic functions $g$ are of the form \begin{equation} \label{outerdef}
g(z) = \exp\Big( \int_\mathbb{T} \frac{\zeta + z}{\zeta - z} \log(|g(\zeta)|) \, d\textit{m}(\zeta)\Big), \quad z \in \mathbb{D},
\end{equation} where $d\textit{m}$ is the (normalized) Lebesgue measure of the unit circle $\mathbb{T} = \{ z\in \mathbb{C} : |z| = 1\}$. Functions as in \eqref{outerdef} are called \textit{outer functions}. The \textit{inner functions} are of the form \begin{equation} \label{innerdef}
\theta(z) = \prod_{n} \frac{\conj{\alpha_n}}{|\alpha_n|}\frac{z - \alpha_n}{1-\conj{\alpha_n}z} \cdot \exp\Big( -\int_\mathbb{T} \frac{\zeta + z}{\zeta - z} d\nu(\zeta)\Big), \quad z \in \mathbb{D},
\end{equation} where $\nu$ is a positive finite Borel measure on $\mathbb{T}$ and $\{\alpha_n\}_n$ is a Blaschke sequence. It is clear that if the Blaschke product on the left is non-trivial, then $\theta$ vanishes at points in $\mathbb{D}$ and therefore cannot be cyclic in any reasonable space of analytic functions $X$. The right factor is a \textit{singular inner function}, and it is well-known that if a function $g \in H^p$ has a singular inner function as a factor, then $g$ is not cyclic in $H^p$. As a consequence, if $\{p_n\}_n$ is a sequence of polynomials for which we have $$\lim_{n \to \infty} \theta(z)p_n(z) = 1, \quad z \in \mathbb{D},$$ then necessarily the Hardy class norms of the sequence must explode: $$\lim_{n \to \infty} \, \|\theta p_n\|^p_{H^p} := \lim_{n \to \infty} \int_\mathbb{T} |\theta p_n|^p d\textit{m} = \infty$$ for finite $p \geq 1$, or in case $p = \infty$, $$ \lim_{n \to \infty} \|\theta p_n\|_\infty := \lim_{n \to \infty}\sup_{z \in \mathbb{D}} |\theta(z)p_n(z)| = \infty.$$
When other norms are considered, cyclic singular inner functions might exist, and here the Bergman spaces $L^p_a(\mathbb{D})$ provide a famous set of examples. The Bergman norms are of the form $$\|g\|^p_{L^p(\mathbb{D})} := \int_\mathbb{D} |g(z)|^p dA(z),$$ where $dA$ is the normalized area measure of $\mathbb{D}$. After a sequence of partial results by multiple authors, Boris Korenblum in \cite{korenblum1975extension} and James Roberts in \cite{roberts1985cyclic} independently characterized the cyclic singular inner functions in the Bergman spaces in terms of the vanishing on certain subsets of $\mathbb{T}$ of the corresponding singular measure $\nu$ appearing in \eqref{innerdef}. A construction of a singular inner function which is cyclic in the classical Bloch space appears in \cite{anderson1991inner}.
The purpose of this note is to discuss and apply a recent theorem of Thomas Ransford from \cite{ransford2021decay} which deals with singular inner functions which decay slowly near the boundary of the disk. Here is the statement of Ransford's theorem.
\begin{thm} \thlabel{pre-ransfordtheorem}
Let $w:[0,1) \to (0,1)$ be any function satisfying $\lim_{r \to 1^-} w(r) = 0$. Then there exists a singular inner function $\theta$ for which we have \begin{equation} \label{loweresttheta}
\min_{|z| < r} |\theta(z)| \geq w(r), \quad r \in (0,1).
\end{equation}
\end{thm}
We refer the reader to the article \cite{ransford2021decay} for the clever proof of the statement, which relies on an application of Frostman's classical construction of a measure with prescribed smoothness properties, and estimates of the Poisson kernel. We will apply Ransford's theorem in a slightly different form than it is stated in \cite{ransford2021decay}. The following consequence of his theorem is the pivotal point of our discussion.
\begin{cor} \thlabel{ransfordtheorem} Let $w:[0,1] \to (0,1)$ be any decreasing function satisfying $\lim_{t \to 1} w(t) = 0$. There exists a singular inner function $\theta = S_\nu$ and a sequence of analytic polynomials $\{p_n\}_n$ such that
\begin{enumerate}[(i)]
\item $\lim_{n \to \infty} \theta(z) p_n(z) = 1, \quad z \in \mathbb{D},$
\item $\sup_{z \in \mathbb{D}} |\theta(z) p_n(z)| w(|z|) \leq 2$.
\end{enumerate}
\end{cor}
\begin{proof}
Apply Ransford's \thref{pre-ransfordtheorem} to the function $w$ to produce a singular inner function $\theta$ satisfying \eqref{loweresttheta}. For integers $n \geq 2$ we set $r_n := 1- 1/n$ and $Q_n(z) := 1/\theta(r_n z)$. Then $Q_n$ is holomorphic in a neighbourhood of the closed disk $\conj{\mathbb{D}}$, and because we are assuming that $w$ is decreasing, we have the estimate $$\sup_{z \in \mathbb{D}} |Q_n(z)|w(|z|) \leq \sup_{z \in \mathbb{D}} \frac{w(|z|)}{w(r_n|z|)} \leq 1.$$ We can approximate $Q_n$ by an analytic polynomial $p_n$ so that $$\sup_{z \in \mathbb{D}} |Q_n(z) - p_n(z)| \leq 1/n.$$
Then \begin{gather*}
\sup_{z \in \mathbb{D}} |\theta(z) p_n(z)| w(|z|) \leq \sup_{z \in \mathbb{D}} \Big(|\theta(z) Q_n(z)| + 1/n \Big) w(|z|) \leq 2
\end{gather*}
It is clear from the construction that $\theta(z)p_n(z) \to 1$ as $n \to \infty$, for any $z \in \mathbb{D}$.
\end{proof}
\thref{ransfordtheorem} says that there exists cyclic singular inner functions in essentially any space of analytic functions defined in terms of a growth condition, or in any space in which such a \textit{growth space} is continuously embedded.
In the proofs of our main results, which will be stated shortly, we will concern ourselves with the following \textit{weak} type of cyclicity of singular inner functions. Let $Y$ be some linear space of analytic functions which is contained in $H^1$. We want to investigate if there exists a singular inner function $\theta$ and a sequence of polynomials $\{p_n\}_n$ such that \begin{equation}
\label{weakcyclicity}
f(0) = \int_\mathbb{T} f \, d\textit{m} = \lim_{n \to \infty} \int f \conj{\theta p_n} \,d\textit{m}
\end{equation} holds for all $f \in Y$. The above situation means that the sequence $\{\theta p_n\}_n$ converges to the constant $1$, weakly over the space $Y$. Now, clearly if $Y$ is too large of a space (say, $Y = H^2$), then \eqref{weakcyclicity} can never hold for all $f \in Y$. However, if $Y$ is sufficiently small, then the situation in \eqref{weakcyclicity} might occur. For instance, in the extreme case when $Y$ is a set of analytic polynomials, then any singular inner function $\theta$ and any sequence of polynomials $\{p_n\}_n$ which satisfies $\lim_{n \to \infty} p_n(z) = 1/\theta(z)$ for $z \in \mathbb{D}$, is sufficient to make \eqref{weakcyclicity} hold. Philosophically speaking, it is the uniform smoothness of the functions in the class $Y$ that allows the existence of singular inner functions $\theta$ for which the above situation occurs.
Such weak cyclicity of singular inner functions plays a role in the theory of approximations in model spaces $K_\theta$ and the broader class of de Branges-Rovnyak spaces $\mathcal{H}(b)$. Recall that the space $K_\theta$ is constructed from an inner function $\theta$ by taking the orthogonal complement of the subspace $$\theta H^2 := \{ \theta h : h \in H^2\}$$ in the Hardy space $H^2$:
\begin{equation*}
K_\theta = H^2 \ominus \theta H^2.
\end{equation*}
For background on the spaces $K_\theta$ one can consult the books \cite{cauchytransform} and \cite{garcia2016introduction}.
Note that the integrals on the right-hand side in \eqref{weakcyclicity} represents the inner product in $H^2$ between a function $f$ and a polynomial multiple of $\theta$. Under insignificat assumptions on $Y$, a straightforward argument will show that if $\eqref{weakcyclicity}$ occurs, then the intersection between $Y$ and $K_\theta$ is trivial: $Y \cap K_\theta = \{0\}$. As a consequence of these observations, Ransford's theorem, and some further analysis, we will be able to construct model spaces $K_\theta$ which have trivial intersections with many examples of spaces $Y$, even ones defined by very mild smoothness or continuity conditions. Consequently, we will be able to construct spaces $K_\theta$ in which every non-zero function behaves \textit{badly} in some precise way.
In one of our main results, we will show that the famous approximation theorem of Aleksandrov from \cite{aleksandrovinv} on density in $K_\theta$ of functions which extend continuously to the boundary, is in fact essentially sharp, as it cannot be extended to any class of functions satisfying an estimate on their modulus of continuity. By a \textit{modulus of continuity} $\omega$ we mean a function $\omega: [0,\infty) \to [0, \infty)$ which is continuous, increasing, satisfies $\omega(0) = 0$, and for which $\omega(t)/t$ is a decreasing function with $$\lim_{t \to 0^+} \omega(t)/t = \infty.$$ For such a function $\omega$ we define $\Lambda^\omega_a$ to be the space of functions $f$ which are analytic in $\mathbb{D}$, extend continuously to $\conj{\mathbb{D}}$, and satisfy \begin{equation}\label{mocspacedef}
\sup_{z,w \in \conj{\mathbb{D}}, z \neq w} \frac{|f(z)-f(w)|}{\omega(|z-w|)} < \infty.
\end{equation} Then $\Lambda^\omega_a$ is the space of analytic functions on $\mathbb{D}$ which have a modulus of continuity dominated by $\omega$. We make $\Lambda^\omega_a$ into a normed space by introducing the quantity $$\|f\|_\omega := \|f\|_\infty + \sup_{z,w \in \conj{\mathbb{D}}, z \neq w} \frac{|f(z)-f(w)|}{\omega(|z-w|)}.$$
By a theorem of Tamrazov from \cite{tamrazov1973contour}, we could have replaced the supremum over $\conj{\mathbb{D}}$ by a supremum over $\mathbb{T}$, and obtain the same space of functions (we remark that a nice proof of this result is contained in \cite[Appendix A]{bouya2008closed}). One of our two main results is the following optimality statement regarding Aleksandrov's density theorem.
\begin{thm} \thlabel{mocnondensitymodelspaces}
Let $\omega$ be a modulus of continuity. There exists a singular inner function $\theta$ such that $$\Lambda^\omega_a \cap K_\theta = \{0\}.$$
\end{thm}
This statement will be proved in Section \ref{aleksandrovoptimality}. In fact, we will see that \thref{mocnondensitymodelspaces} is a consequence of a variant, and in some directions a strengthening, of a theorem of Dyakonov and Khavinson from \cite{starinvsmooth}. For a sequence of positive numbers $\bm{\lambda} = \{\lambda_n\}_{n=0}^\infty $ we define the class \begin{equation}
\label{H2wdef}
H^2_{\bm{\lambda}} = \Big\{ f = \sum_{n=0}^\infty f_n z^n \in Hol(\D) : \sum_{n=0}^\infty \lambda_n|f_n|^2 < \infty \Big\}.
\end{equation}
The other of our main results, proved in Section \ref{badmodelspacesection}, reads as follows.
\begin{thm} \thlabel{badmodelspace}
Let $\bm{\lambda} = \{\lambda_n\}_{n=0}^\infty$ be any increasing sequence of positive numbers with $\lim_{n \to \infty} \lambda_n = \infty$. Then there exists a singular inner function $\theta$ such that $$K_\theta \cap H^2_{\bm{\lambda}} = \{0\}.$$
\end{thm}
The result can be compared to the mentioned result of Dyakonov and Khavinson in \cite{starinvsmooth}, from which the above result can be deduced in the special case $\bm{\lambda} = \{ (k+1)^{\alpha}\}_{k=0}^\infty$ with any $\alpha > 0$.
The theory of de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a well-known generalization of the theory of model spaces $K_\theta$. The symbol of the space $b$ is now any analytic self-map of the unit disk, and we have $\mathcal{H}(b) = K_b$ whenever $b$ is inner. For background on $\mathcal{H}(b)$ spaces, see \cite{sarasonbook}, or \cite{hbspaces1fricainmashreghi} and \cite{hbspaces2fricainmashreghi}. A consequence of the authors work in collaboration with Alexandru Aleman in \cite{dbrcont} is that the above mentioned density theorem of Aleksandrov generalizes to the broader class of $\mathcal{H}(b)$ spaces: any such space admits a dense subset of functions which extend continuously to the boundary. Since \thref{mocnondensitymodelspaces} proves optimality of Aleksandrov's theorem for inner functions $\theta$, one could ask if at least for outer symbols $b$ any improvement of the density result in $\mathcal{H}(b)$ from \cite{dbrcont} can be obtained. In Section \ref{hboptimalitysec} we remark that this is not the case, and the result in \cite{dbrcont} is also essentially optimal, even for outer symbols $b$. This is shown to be equivalent to a theorem of Khrushchev from \cite{khrushchev1978problem}.
In the last Section \ref{questionsec} we list a few questions we have not found an answer for, and some ideas for further research.
\section{Proof of \thref{badmodelspace}}
\label{badmodelspacesection}
In the proof of the theorem we will need to use the following crude construction of an integrable weight with large moments.
\begin{lemma} \thlabel{momentlemma}
Let $\{\lambda_n\}_{n=0}^\infty$ be a decreasing sequence of positive numbers with $\lim_{n \to \infty} \lambda_n = 0$. There exists a non-negative function $\Lambda \in L^1([0,1])$ which satisfies $$\lambda_n \leq \int_{0}^1 x^{2n+1} \Lambda(x) \,dx, \quad n \geq 0.$$
\end{lemma}
\begin{proof}
Recall that the sequence $(1-1/n)^n = \exp(n\log(1-1/n))$ is decreasing and satisfies $$\lim_{n \to \infty} (1-1/n)^n = e^{-1}.$$ It follows that $$\inf_{x \in (1-1/n, 1)} x^{2n+1} \geq \alpha$$ for some constant $\alpha > 0$ which is independent of $n$. For $n \geq 1$, we define the intervals $I_n = \big( 1 - 1/n, 1 - 1/(n+1) \big)$. Our function $\Lambda$ will be chosen to be of the form $$\Lambda(x) = \sum_{n=0}^\infty 1_{I_n} c_n,$$ where $1_{I_n}$ is the indicator function of the interval $I_n$ and the $c_n$ are positive constants to be chosen shortly. Note that \begin{gather} \label{firstest} \int_0^1 x^{2N+1} \Lambda(x) \, dx \geq \int_{1-1/N}^1 x^{2N+1} \Lambda(x) \, dx \geq \alpha \sum_{n = N}^\infty |I_n| c_n. \end{gather} We choose $$c_n = \alpha^{-1}|I_n|^{-1}(\lambda_n - \lambda_{n+1}).$$ This choice of coefficients $c_n$ makes $\Lambda$ integrable over $[0,1]:$ \begin{gather*}
\int_{0}^1 \Lambda(x) dx = \sum_{n=1}^\infty |I_n|c_n = \alpha^{-1} \sum_{n=1}^\infty \lambda_n - \lambda_{n+1} \\ =
\lim_{M \to \infty} \alpha^{-1} \sum_{n=1}^M \lambda_n - \lambda_{n+1} = \lim_{M \to \infty} \alpha^{-1} (\lambda_1 - \lambda_{M+1}) \\ = \alpha^{-1} \lambda_1 \end{gather*} In the last step we used the assumption that the sequence $\{\lambda_n\}_n$ converges to zero. Moreover, by \eqref{firstest} and the choice of $c_n$ we can estimate
\begin{gather*}\int_0^1 x^{2N+1} \Lambda(x) \, dx \geq \alpha \sum_{n=N}^\infty |I_n|c_n \\ = \lim_{M \to \infty} \alpha \sum_{n=N}^M |I_n| c_n = \lim_{M \to \infty} \sum_{n=N}^\mathcal{M} \lambda_n -\lambda_{n+1} \\ = \lim_{M \to \infty} \lambda_N - \lambda_{M+1} = \lambda_N.\end{gather*} The proof is complete.
\end{proof}
The significance of the above lemma is the estimate \begin{equation} \label{Lambdanorm}
\sum_{k=0}^\infty \lambda_k|f_k|^2 \leq c \int_\mathbb{D} |f(z)|^2 \Lambda(|z|) dA(z)
\end{equation} for some numerical constant $c > 0$ and any function $f$ which is holomorphic in a neighbourhood of the closed disk $\conj{\mathbb{D}}$. The estimate can be vereified by direct computation of the integral on the right-hand side, using polar coordinates.
We will also use the following well-known construction.
\begin{lemma} \thlabel{multiplier}
For any function $g \in L^1([0,1])$ there exists a positive and increasing function $w: [0,1) \to \mathbb{R}$ which satisfies $$\lim_{t \to 1^-} w(t) = \infty$$ and $$wg \in L^1([0,1]).$$
\end{lemma}
\begin{proof} The integrability condition on $g$ implies that $$\lim_{t \to 1^-} \int_{t}^1 |g(x)| dx = 0.$$
Thus there exists a sequence of intervals $\{I_n\}_{n=1}^\infty$ which have $1$ as the right end-point and which satisfy $I_{n+1} \subset I_n$ for all $n \geq 1$, and $$\int_{I_n} |g(x)| dx \leq 4^{-n}.$$ If we set $$w(t) = 1_{[0,1) \setminus I_1} + \sum_{n=1}^\infty 2^n 1_{I_n \setminus I_{n+1}},$$ where $1_{I_n \setminus I_{n+1}}$ is the indicator function of the set difference $I_n \setminus I_{n+1}$, then $w$ is increasing, satisfies $\lim_{t \to 1^-} w(t) = \infty$, and $$\int_{I_n \setminus I_{n+1}} w(x)|g(x)| dx \leq 2^{-n}$$ for all $n \geq 1$. Consequently $$\int_{0}^1 w(x)|g(x)|dx \leq \int_{0}^1 |g(x)|dx + \sum_{n} \int_{I_n \setminus I_{n+1}} w(x)|g(x)| dx < \infty.$$
\end{proof}
\begin{proof}[Proof of \thref{badmodelspace}]
Let $\Lambda$ be the function in \thref{momentlemma} which corresponds to the sequence $\{1/\lambda_n\}_{n=0}^\infty$. That is, $\Lambda$ satisfies $$\frac{1}{\lambda_n} \leq \int_0^1 x^{2n+1} \Lambda(x) \, dx, \quad n \geq 0,$$ and $\Lambda \in L^1[0,1]$. Now, let $w$ be a positive decreasing function which satisfies $\lim_{x \to 1^-} w(x) = 0$ and $$\int_0^1 \frac{\Lambda(x)}{w^2(x)} \,dx < \infty.$$ Existence of such a function follows readily from \thref{multiplier}. Apply \thref{ransfordtheorem} to $w$ and obtain a corresponding inner function $\theta$ and a sequence of polynomials $\{p_n\}_n$ for which the conclusions $(i)$ and $(ii)$ of \thref{ransfordtheorem} hold. We will show that for this $\theta$ we have $K_\theta \cap H^2_{\bm{\lambda}} = \{0\}.$
Indeed, assume $f \in K_\theta \cap H^2_{\bm{\lambda}} = \{0\},$ but that in fact $f$ is non-zero. Since both $K_\theta$ and $H^2_{\bm{\lambda}}$ are invariant for the backward shift operator, we may without loss of generality assume that $f(0) \neq 0$. Fix an integer $n$ and let \begin{equation}
\label{gdef} g(z) = \theta(z)p_n(z) - 1, z \in \mathbb{D}.
\end{equation} Let $\{f_k\}_k, \{g_k\}_k$ be the sequences of Taylor coefficients of $f$ and $g$, respectively. Since $f \in K_\theta$, we have
\begin{gather*}
|f(0)| = \Big\vert \int_\mathbb{T} f \, d\textit{m} \Big\vert = \Big\vert \int_\mathbb{T} f\conj{\theta p_n - 1} \, d\textit{m}\Big\vert = \lim_{r \to 1^-} \Big\vert \sum_{k=0}^\infty r^{2k} f_k \conj{g_k} \Big\vert \\ \leq \limsup_{r \to 1^-}\Big( \sum_{k=0}^\infty \lambda_k r^{2k} |f_k|^2 \Big)^{1/2} \Big( \sum_{k=0}^\infty \frac{1}{\lambda_n} |r^k g_k|^2 \Big)^{1/2} \\
\end{gather*}
Using inequality \eqref{Lambdanorm} on the term on the right-hand side in the last expression (with $\lambda_n$ replaced by $1/\lambda_n$), we obtain
\begin{gather*}
|f(0)| \leq C \limsup_{r \to 1^-} \Big( \sum_{k=0}^\infty \lambda_k |f_k|^2 \Big)^{1/2} \Big(\int_\mathbb{D} |g(rz)|^2 \Lambda(|z|)d A(z) \Big)^{1/2} \\ = C \Big( \sum_{k=0}^\infty \lambda_k |f_k|^2 \Big)^{1/2} \Big(\int_\mathbb{D} |g(z)|^2 \Lambda(|z|)d A(z) \Big)^{1/2}.
\end{gather*}
By assertion in part $(ii)$ of \thref{ransfordtheorem}, the function $|g(z)|^2\Lambda(|z|)$ is dominated pointwise in $\mathbb{D}$ by the integrable function $$\frac{4\Lambda(|z|)}{w^2(|z|)}, \quad z \in \mathbb{D}$$ independently of which polynomial $p_n$ is used to defined $g$ in \eqref{gdef}. But if we let $n \to \infty$ in \eqref{gdef}, then $|g(z)|^2\Lambda(|z|) \to 0$, and so we infer from the computation above and the dominated convergence theorem that $f(0) = 0$, which is a contradiction. The conclusion is that $K_\theta \cap H^2_{\bm{\lambda}} = \{ 0\}$, and the proof of the theorem is complete.
\end{proof}
\section{Proof of \thref{mocnondensitymodelspaces}}
\label{aleksandrovoptimality}
\thref{mocnondensitymodelspaces} will follow immediately from \thref{badmodelspace} together with the following embedding result for the spaces $\Lambda^\omega_a$.
\begin{lemma} \thlabel{mocspecestimatelemma}
Let $\omega$ be a modulus of continuity. There exists an increasing sequence of positive numbers $\bm{\alpha} = \{ \alpha_n\}_{n=0}^\infty$ satisfying $\lim_{n \to \infty} \alpha_n = \infty$ and such that for any $f \in \Lambda^\omega_a$ we have the estimate \begin{equation} \label{specembedding}
\sum_{n=0} \alpha_n|f_n|^2 \leq C \|f\|^2_\omega
\end{equation} where $C > 0$ is a numerical constant and $\{f_n\}_n$ is the sequence of Taylor coefficients of $f$.
\end{lemma}
\begin{proof}
For each $r \in (0,1)$ we have the estimate \begin{equation} \label{valuestospectrum}
\sum_{n=0} (1-r^{2n})|f_n|^2 = \int_{\mathbb{T}} |f(\zeta) - f(r\zeta)|^2 d\textit{m}(\zeta) \leq \omega(1-r)^2 \|f\|^2_\omega.
\end{equation} Since $\lim_{t \to 0} \omega(t) = 0$, for each positive integer $N$ there exists a number $r_N \in (0,1)$ such that $\omega(1-r_N) \leq \frac{1}{2^N}$. Since $\lim_{n \to \infty} r_N^{2n} = 0$, there exists an integer $K(N)$ such that $r_N^{2n} < 1/2$ for $n \geq K(N)$. Then \begin{equation*}
\sum_{n = K(N)}^\infty \frac{|f_n|^2}{2} \leq \sum_{n=K(N)}^\infty (1-r_N^{2n})|f_n|^2 \leq \frac{1}{4^N}\|f\|^2_\omega.
\end{equation*} Consequently \begin{equation} \label{specpartialestimate}
\sum_{n=K(N)} 2^N |f_n|^2 \leq \frac{1}{2^{N-1}} \|f\|^2_\omega.
\end{equation} We can clearly choose the sequence of integers $K(N)$ to be increasing with $N$. If we define the sequence $\bm{\alpha}$ by the equation $\alpha_n = 1$ for $n \leq K(1)$, and $\alpha_n = 2^n$ for $K(N) \leq n < K(N+1)$, then \eqref{specembedding} follows readily from \eqref{specpartialestimate} by summing over all $N \geq 1$.
\end{proof}
\begin{proof}[Proof of \thref{mocnondensitymodelspaces}]
\thref{mocspecestimatelemma} implies that $\Lambda^\omega_a$ is contained in some space of the form $H^2_{\bm{\alpha}}$ as defined in \eqref{H2wdef}. If $\theta$ is a singular inner function given by \thref{badmodelspace} such that $H^2_{\bm{\alpha}} \cap K_\theta = \{0\}$, then obviously we also have that $\Lambda^\omega_a \cap K_\theta = \{0\}$, and so the claim follows.
\end{proof}
\section{The case of $\mathcal{H}(b)$ spaces}
\label{hboptimalitysec}
Here we prove the optimality of the continuous approximation theorem for the larger class of $\mathcal{H}(b)$ spaces.
\begin{prop}
Let $\omega$ be a modulus of continuity. There exists an outer function $b:\mathbb{D} \to \mathbb{D}$ such that $$\Lambda^\omega_a \cap \mathcal{H}(b) = \{ 0 \}.$$
\end{prop}
\begin{proof}
By a result of Khrushchev noted in \cite[Theorem 2.4]{khrushchev1978problem}, there exists a closed subset $E$ of the circle $\mathbb{T}$ with the property that for no non-zero integrable function $h$ supported on $E$ is the Cauchy integral $$C_h(z) = \int_{\mathbb{T}} \frac{h(\zeta)}{1-z\conj{\zeta}} d\textit{m}(\zeta)$$ a member of the space $\Lambda^\omega_a$. It suffices thus to construct an $\mathcal{H}(b)$ space for which every function can be expressed as such a Cauchy integral. The simplest choice for the space symbol $b$ is the outer function with modulus $1$ on $\mathbb{T} \setminus E$ and $1/2$ on $E$. Then $b$ is invertible in the algebra $H^\infty$, and a consequence of the general theory (see \cite[Theorem 20.1 and Theorem 28.1] {hbspaces2fricainmashreghi}) is that every function in the space $\mathcal{H}(b)$ is a Cauchy integral of a function $h$ which is square-integrable on $\mathbb{T}$ and supported only on $E$. Thus $\Lambda^\omega_a \cap \mathcal{H}(b) = \{ 0 \}$.
\end{proof}
\section{Some ending questions and remarks}
\label{questionsec}
Since Ransford's theorem seems to be such a powerful tool in establishing results of the kind mentioned here, we are wondering whether it can be further applied. In particular, the following questions come to mind.
\begin{enumerate}
\item Are our methods strong enough to prove that there exists model spaces $K_\theta$ which admit no non-zero functions in the Wiener algebra of absolutely convergent Fourier series? The result is known, and has been noted in \cite{limani2021abstract}. However, it was proved as a consequence of a complicated construction of a cyclic singular inner function in the Bloch space. Is it so that Ransford's construction is sufficient to prove the non-density result for the Wiener algebra in the fashion presented here?
\item For $p > 2$, the Banach spaces $\ell^p_a$ consisting of functions $f\in Hol(\D)$ with Taylor series $\{f_n\}_{n=0}^\infty$ satisfying $$\|f\|^p_{\ell^p_a} := \sum_{n=0}^\infty |f_n|^p < \infty$$ are of course larger than the space $H^2 = \ell^2_a$. Do there exist cyclic singular inner functions in these spaces?
\end{enumerate}
\subsubsection*{Acknowledgements} The author would like to thank Christopher Felder for his reading of the manuscript and for his helpful suggestions for improvements. He would also like to thank Adem Limani for very useful discussions and important comments.
\bibliographystyle{siam}
|
2,877,628,089,529 | arxiv | \section{Introduction}
The classical Perron-Frobenius theorem guarantees that an irreducible nonnegative matrix always has a unique (up to scaling) entrywise positive eigenvector with eigenvalue equal to the spectral radius of the matrix. There is a long history of nonlinear extensions of the Perron-Frobenius theorem to functions defined on a cone that are homogeneous (of degree one) and preserve the partial order induced by the cone. Functions defined on the positive orthant $\mathbb{R}^n_{> 0}$ are of particular interest in applications. Here we use $\mathbb{R}^n_{>0}$ to denote the set of vectors in $\mathbb{R}^n$ with strictly positive entries and $\mathbb{R}^n_{\ge 0}$ denotes the vectors with nonnegative entries. For $x, y \in \mathbb{R}^n$, we say that $x \le y$ when $y-x \in \mathbb{R}^n_{\ge 0}$. By \emph{order-preserving}, we mean that if $x,y$ are in the domain of $f$ and $x \ge y$, then $f(x) \ge f(y)$; and \emph{homogeneous} means that $f(t x) = t f(x)$ for all $t > 0$.
General results for establishing the existence and uniqueness of eigenvectors of order-preserving homogeneous functions $f:\mathbb{R}^n_{>0}\rightarrow \mathbb{R}^n_{>0}$ have been proved by Morishima \cite{Morishima64}, Oshime \cite{Oshime83}, and Nussbaum \cite{Nussbaum86,Nussbaum88,Nussbaum89}. These results require strong assumptions on $f$ in order to guarantee both existence and uniqueness of the positive eigenvectors of $f$. More general sufficient conditions for the existence of eigenvectors in $\mathbb{R}^n_{>0}$ have been established by Schneider and Turner \cite{ScTu72}, Nussbaum \cite{Nussbaum88,Nussbaum89} and Gaubert and Gunawardena \cite{GaGu04}. All of these conditions prove the existence of a positive eigenvector by either directly or indirectly showing that $f$ has an invariant subset in $\mathbb{R}^n_{>0}$ that is bounded in Hilbert's projective metric. We will describe Hilbert's projective metric (denoted $d_H$) in more detail in the next section, for now we just note that any order-preserving, homogeneous map $f$ on $\mathbb{R}^n_{>0}$ is nonexpansive with respect to $d_H$, that is $d_H(f(x),f(y)) \le d_H(x,y)$ for all $x, y \in \mathbb{R}^n_{>0}$. Two recent papers \cite{AkGaHo20,LLN16} have studied the eigenvector existence problem by explicitly seeking necessary and sufficient conditions for certain invariant subsets to be bounded in $(\mathbb{R}^n_{>0},d_H)$. We give a brief summary of some of the main results of these papers here.
In \cite{GaGu04}, the authors describe three special classes of invariant subsets of $\mathbb{R}^n_{>0}$. For any $\alpha, \beta > 0$, the \emph{sub-eigenspace} corresponding to $\alpha$ is the set
$$S_\alpha(f) := \{x \in \mathbb{R}^n_{>0} : \alpha x \le f(x)\}$$
and the \emph{super-eigenspace} corresponding to $\beta$ is
$$S^\beta(f) := \{x \in \mathbb{R}^n_{>0} : f(x) \le \beta x\}.$$
The intersection $S_\alpha^\beta(f) := S_\alpha(f) \cap S^\beta(f)$ is called a \emph{slice space}.
Proving that any one of these sets is nonempty and bounded in Hilbert's projective metric is sufficient to guarantee that $f$ has a positive eigenvector. One of the main results in \cite{GaGu04} is a necessary and sufficient condition for all super-eigenspaces of $f$ to be $d_H$-bounded. A similar condition is also be given for sub-eigenspaces. Later, \cite{AkGaHo20} gave a necessary and sufficient condition for all slice spaces for $f$ to be $d_H$-bounded. All three of these conditions can be described in terms of two directed hypergraphs $\mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(f)$ that were introduced in \cite{AkGaHo15}.
Recall that a \emph{directed hypergraph} is a set of nodes $N$ together with a collection of directed hyperarcs. A \emph{directed hyperarc} is an ordered pair $(\mathbf{t},\mathbf{h})$ where $\mathbf{t}, \mathbf{h} \subseteq N$. We refer to $\mathbf{t}$ as the \emph{tail} of the hyperarc and $\mathbf{h}$ as the \emph{head}. The two hypergraphs $\mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(f)$ both have nodes $N = \{1, \ldots, n\}$, and a set of directed hyperarcs $(I,\{j\})$ where $I \subset N$ and $j \in N$. The hyperarcs of $\mathcal{H}^-_0(f)$ are the pairs $(I,\{j\})$ such that $j \notin I$ and
$$\lim_{t \rightarrow \infty} f(\exp(-te_I))_j = 0.$$
Here $\exp$ is the entrywise natural exponential function and $e_I \in \mathbb{R}^n$ is the vector with entries
$$(e_I)_i := \begin{cases}
1 & \text{ if } i \in I \\
0 & \text{ if } i \in I^c.
\end{cases}$$
As we will see in the next section, if $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving and homogeneous, then $f$ extends continuously to $\mathbb{R}^n_{\ge 0}$, so the condition defining the hyperarcs of $\mathcal{H}^-_0(f)$ can be replaced by $f(e_{N \backslash I})_j = 0$. We will use $N \backslash I$ and $I^c$ interchangeably for subsets of $N$. The hyperarcs of $\mathcal{H}^+_\infty(f)$ are $(I, \{j\})$ such that $j \notin I$ and
$$\lim_{t \rightarrow \infty} f(\exp(te_I))_j = \infty.$$
We say that a subset $I \subseteq N$ is \emph{invariant} in a hypergraph if there are no hyperarcs from a subset of $I$ to a subset of $I^c$. Since $f$ is order-preserving, if there is a hyperarc in $\mathcal{H}^-_0(f)$ (or $\mathcal{H}^+_\infty(f))$ from a subset of $I$ to $j \in I^c$, then $(I,\{j\})$ is also a hyperarc of $\mathcal{H}^-_0(f)$ (or $\mathcal{H}^+_\infty(f)$). Therefore $I$ is invariant in one of the graphs $\mathcal{H}^-_0(f)$ or $\mathcal{H}^+_\infty(f)$ if and only if there are no hyperarcs $(I,\{j\})$ in that hypergraph.
Although not originally described in terms of hypergraphs, \cite[Theorem 5]{GaGu04} says the following.
\begin{theorem}[Gaubert-Gunawardena] \label{thm:GG}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Then all super-eigenspaces $S^\beta(f)$ are bounded in $(\mathbb{R}^n_{>0},d_H)$ if and only if $\mathcal{H}^+_\infty(f)$ has no nonempty invariant sets $J \subsetneq N$.
\end{theorem}
A corresponding condition involving the hypergraph $\mathcal{H}^-_0(f)$ is equivalent to all sub-eigenspaces of $f$ being $d_H$-bounded. An even more general result was proved as part of \cite[Theorem 1.2]{AkGaHo20}.
\begin{theorem}[Akian-Gaubert-Hochart] \label{thm:AGH}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous.
All slices spaces $S_\alpha^\beta(f)$ are bounded in $(\mathbb{R}^n_{>0}, d_H)$ if and only if for every pair of nonempty disjoint sets $I, J \subset N$, either $I^c$ is not invariant in $\mathcal{H}^-_0(f)$ or $J^c$ is not invariant in $\mathcal{H}^+_\infty(f)$.
\end{theorem}
Both of these theorems are sufficient to prove that there exists an eigenvector of $f$ in $\mathbb{R}^n_{>0}$. In fact, they both guarantee that the \emph{eigenspace}
$$E(f) := \{x \in \mathbb{R}^n_{>0} : x \text{ is an eigenvector of } f\}$$
is nonempty and bounded in Hilbert's projective metric. The condition of Theorem \ref{thm:GG} is more restrictive but its conclusion is stronger since if all super-eigenspaces are $d_H$-bounded, then so are all slice spaces. Even the conclusion that all slice spaces are bounded in $(\mathbb{R}^n_{>0},d_H)$ is quite strong. In \cite{AkGaHo20}, it is shown to be equivalent to several other conditions on the function $f$, including that for any $n$-by-$n$ diagonal matrix $D$ with positive diagonal entries, the map $D f$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^n_{>0}$.
Another recent result in nonlinear Perron-Frobenius theory is \cite[Theorem 5.1]{LLN16} which directly addresses when the eigenspace $E(f)$ is nonempty and $d_H$-bounded:
\begin{theorem}[Lemmens-Lins-Nussbaum] \label{thm:LLN}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. The eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$ if and only if for every nonempty proper subset $J \subset N$, there exists $x \in \mathbb{R}^n$ such that
\begin{equation} \label{illum}
\max_{j \in J} \frac{f(x)_j}{x_j} < \min_{i \in J^c} \frac{f(x)_i}{x_i}.
\end{equation}
\end{theorem}
Theorem \ref{thm:LLN} actually implies slightly more, since it shows that small perturbations of $f$ will also have a $d_H$-bounded set of eigenvectors, see \cite[Corollary 3.5]{LLN16}.
In general the conditions of these three theorems are progressively more difficult to check. In \cite{LLN16}, the authors suggest a non-deterministic algorithm to check the conditions of Theorem \ref{thm:LLN}. The algorithm involves testing random points in $\mathbb{R}^n_{>0}$ until \eqref{illum} has been verified for every nonempty $J \subsetneq N$. If the algorithm halts, then $f$ has a positive eigenvector. But this approach is less satisfactory than a straightforward combinatorial test, particularly when the function $f$ has symbolic parameters with unspecified values.
In Section \ref{sec:exist} of this paper, we use Theorem \ref{thm:LLN} to prove a new necessary and sufficient condition for the eigenspace $E(f)$ of an order-preserving homogeneous function $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ to be nonempty and bounded in Hilbert's projective metric. This new condition (Theorem \ref{thm:super}) is easier to check than the condition of Theorem \ref{thm:LLN}. It also complements the hypergraph conditions of Theorems \ref{thm:GG} and \ref{thm:AGH}, allowing one to check the hypergraph conditions first, and then check the new condition only for subsets of $N$ where the hypergraph condition fails. Section \ref{sec:conv} shows how the results of Section \ref{sec:exist} can be improved for multiplicatively convex functions. We give simple necessary and sufficient conditions for the eigenspace $E(f)$ to be nonempty and bounded in $(\mathbb{R}^n_{>0}, d_H)$ when $f$ is multiplicatively convex. We also generalize a theorem of Hu and Qi \cite[Theorem 5]{HuQi16} which gives necessary and sufficient conditions for the existence of entrywise positive eigenvectors for strongly nonnegative tensors.
In Section \ref{sec:unique} we prove that if an order-preserving and homogeneous function $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is also real analytic and the eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$, then $f$ has a unique positive eigenvector up to scaling. This is precisely analogous to the situation for nonnegative matrices. By \emph{real analytic}, we mean that each entry of $f$ is a real analytic function on all of $\mathbb{R}^n_{>0}$. To prove this result, we show that a real analytic nonexpansive map on a Banach space with the fixed point property cannot have a bounded set of fixed points with more than one element.
When an order-preserving and homogeneous map $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ has a unique positive eigenvector $u \in \mathbb{R}^n_{>0}$ with $\|u\|=1$, it is sometimes important to know whether the normalized iterates $f^k(x)/\|f^k(x)\|$ converge to $u$ for all $x \in \mathbb{R}^n_{>0}$ as $k \rightarrow \infty$. In Section \ref{sec:iterates} we give a new condition on the derivative of $f$ at $u$ that is sufficient to guarantee that the iterates converge. We also point out that if $f$ has an eigenvector in $\mathbb{R}^n_{>0}$, then the normalized iterates of $f+\mathrm{id}$ always converge to an eigenvector of $f$, even if the normalized iterates of $f$ do not.
We conclude in Section \ref{sec:examples} with detailed examples of how to use these new results in applications. These include a population biology model and a Shapley operator associated with a stochastic game. In Theorem \ref{thm:Mplus}, we give complete necessary and sufficient conditions for existence and uniqueness of entrywise positive eigenvectors for functions in the class $\mathcal{M}_+$ which includes the order-preserving homogeneous functions associated with the H-eigenvector problem for nonnegative tensors. We also translate the results of Section \ref{sec:exist} to the setting of topical functions, which are functions $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ that are order-preserving and additively homogeneous.
\section{Preliminaries}
A \emph{closed cone} in $\mathbb{R}^n$ is a closed convex set $K$ such that (i) $t K \subseteq K$ for all $t > 0$ and (ii) $K \cap (-K) = \{0\}$. A closed cone $K \subset \mathbb{R}^n$ induces a partial ordering $x \le_K y$ when $y-x \in K$. We write $x \ll_K y$ when $y-x$ is contained in the interior of $K$, denoted $\operatorname{int} K$. The \emph{standard cone} in $\mathbb{R}^n$ is the closed cone $\mathbb{R}^n_{\ge 0}$ and we will write $x \le y$ for the partial ordering induced by the standard cone, and likewise, $x \ll y$ means that $x_i < y_i$ for all $i \in N$.
If $K$ is a closed cone and $x,y \in K$, then we say that $x$ is \emph{comparable} to $y$ and write $x \sim_K y$ if there exist constants $\alpha, \beta > 0$ such that $\alpha x \le_K y \le_K \beta x$. It is easy to see that comparability is an equivalence relation on $K$. The equivalence classes under this relation are called the \emph{parts} of $K$. If $K$ has nonempty interior, then $\operatorname{int} K$ is a part of $K$. For $x \in \mathbb{R}^n_{\ge 0}$, we define the \emph{support} of $x$ to be
$$\operatorname{supp}(x) := \{j \in N : x_j > 0\}.$$
The parts of $\mathbb{R}^n_{\ge 0}$ are the subsets $\mathbb{R}^J_{>0} = \{ x \in \mathbb{R}^n_{\ge 0} : \operatorname{supp}(x)=J \}$.
Let $K \subset \mathbb{R}^n$ and suppose that $x, y \in K$ with $x \sim_K y$ and $y \ne 0$. \emph{Hilbert's projective metric} is
$$d_H(x,y) := \log \left( \frac{M(x/y)}{m(x/y)} \right) $$
where
$$M(x/y) := \inf \{ \beta > 0 : x \le_K \beta y \}$$
and
$$m(x/y) := \sup \{ \alpha > 0 : \alpha y \le_K x \}.$$
We adopt the convention that $d_H(0,0) = 0$ and $d_H(x,y) = \infty$ if $x$ and $y$ are not comparable. When $K = \mathbb{R}^n_{\ge 0}$ and $x, y\in \mathbb{R}^n_{>0}$,
$$m(x/y) = \min_{i \in N} x_i/y_i,\hspace*{0.5cm} M(x/y) = \max_{i \in N} x_i/y_i,$$
and
$$d_H(x,y) = \log \max_{i, j \in N} \left( \frac{y_i \, x_j}{x_i \, y_j } \right).$$
Hilbert's projective metric has the following properties \cite[Proposition 2.1.1]{LemmensNussbaum}.
\begin{proposition} \label{prop:dH}
Let $K$ be a closed cone in $\mathbb{R}^n$. Then
\begin{enumerate}
\item $d_H(x,y) \ge 0$ and $d_H(x,y) = d_H(y,x)$ for all $x, y \in K$.
\item $d_H(x,z) \le d_H(x,y) + d_H(y,z)$ for any $x, y, z$ in a part of $K$.
\item $d_H(\alpha x, \beta y) = d_H(x,y)$ for all $\alpha, \beta >0$ and $x, y \in K$.
\end{enumerate}
\end{proposition}
The use of Hilbert's projective metric to prove results in linear Perron-Frobenius theory can be traced back independently to Birkhoff \cite{Birkhoff57} and Samelson \cite{Samelson57}. Their methods also apply to nonlinear Perron-Frobenius theory because if $f: K \rightarrow K$ is order-preserving with respect to $\le_K$ and homogeneous, then $f$ is nonexpansive with respect to $d_H$ \cite[Proposition 2.1.3]{LemmensNussbaum}.
Let $K$ be a closed cone in $\mathbb{R}^n$.
Let $f:K \rightarrow K$ be an order-preserving homogeneous function. The \emph{cone spectral radius} of $f$ is
$$r_K(f) := \lim_{k \rightarrow \infty} \|f^k\|_K^{1/k}$$
where $\|f\|_K := \sup_{x \in K} \|f(x)\|/\|x\|$. If $K$ has nonempty interior, then an equivalent formula for the cone spectral radius is
\begin{equation} \label{rlim}
r_K(f) = \lim_{k \rightarrow \infty} \|f^k(x)\|^{1/k}
\end{equation}
which does not depend on the choice of $x \in \operatorname{int} K$ \cite[Proposition 5.3.6]{LemmensNussbaum}. Note that neither formula for $r_K(f)$ depends on which norm is used for $\mathbb{R}^n$.
Although it can be difficult to know whether or not an order-preserving homogeneous function has an eigenvector in the interior of $K$, the following well-known result guarantees the existence of an eigenvector in $K$ \cite[Corollary 5.4.2]{LemmensNussbaum}.
\begin{theorem} \label{thm:KR}
Let $K$ be a closed cone in $\mathbb{R}^n$ and let $f:K \rightarrow K$ be order-preserving and homogeneous. Then there exists a nonzero $x \in K$ such that $f(x) = r_K(f) x$.
\end{theorem}
An order-preserving homogeneous function $f: K \rightarrow K$ can have several eigenvectors and more than one eigenvalue. However, if $f$ has two eigenvectors in the same part of $K$, then those eigenvectors must have the same eigenvalue \cite[Corollary 5.2.2]{LemmensNussbaum}. Furthermore, $r_K(f)$ is the maximum of all of the eigenvalues of $f$ \cite[Proposition 5.3.6]{LemmensNussbaum}. Although Theorem \ref{thm:KR} does not guarantee that $f$ has an eigenvector in the interior of $K$, if $x \in \operatorname{int} K$ is an eigenvector of $f$, then the eigenvalue corresponding to $x$ must be $r_K(f)$.
Suppose $K \subset \mathbb{R}^n$ is a closed cone with nonempty interior and $f: K \rightarrow K$ is order-preserving and homogeneous. The \emph{upper Collatz-Wielandt number} for $f$ is
\begin{equation} \label{cw}
r(f) := \inf_{x \in \operatorname{int} K} M(f(x)/x),
\end{equation}
and the \emph{lower Collatz-Wielandt number} for $f$ is
\begin{equation} \label{lcw}
\lambda(f) := \sup_{x \in \operatorname{int} K} m(f(x)/x).
\end{equation}
From the definition, it is clear that $\lambda(f) \le r(f)$ always. It is well known that the upper Collatz-Wielandt number $r(f)$ is always equal to the \emph{cone spectral radius} $r_K(f)$ \cite[Theorem 5.6.1]{LemmensNussbaum}.
If an order-preserving and homogeneous function $f:\mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ has a positive eigenvector $x \in \mathbb{R}^n_{>0}$ such that $f(x) = \mu x$, then the eigenvalue $\mu$ must equal both the upper and lower Collatz-Wielandt numbers, i.e., $r(f) = \lambda(f)=\mu$. In fact, we will prove a stronger result in Lemma \ref{lem:mindisp}. Another quick observation about Collatz-Wielandt numbers is the following.
\begin{lemma} \label{lem:cworder}
Let $f, g: \mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ be order-preserving and homogeneous. If $f(x) \le g(x)$ for all $x \in \mathbb{R}^n_{\ge 0}$, then $r(f) \le r(g)$ and $\lambda(f) \le \lambda(g)$.
\end{lemma}
\begin{proof}
This follows immediately from the definitions \eqref{cw} and \eqref{lcw}.
\end{proof}
Every order-preserving homogeneous function $f: \mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n_{> 0}$ extends continuously to the closed cone $\mathbb{R}^n_{\ge 0}$. In fact, the following result \cite[Corollary 2]{BuSp00} (see also \cite[Corollary 4.6]{BuSpNu03} and \cite[Theorem 5.1.5]{LemmensNussbaum}) actually says a little more. Note that when working with the extended real line $\overline{\mathbb{R}} = [-\infty, \infty]$, we use the usual order topology induced by the natural linear ordering on $\overline{\mathbb{R}}$.
\begin{theorem}[Burbanks-Sparrow] \label{thm:BS}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Then $f$ extends continuously to a map $\underline{f}:\mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$. The map $f$ also extends continuously to a map $\overline{f}:(0,\infty]^n \rightarrow (0,\infty]^n$. Both extensions are order-preserving and homogeneous.
\end{theorem}
We will write $f$ instead of $\overline{f}$ or $\underline{f}$ when working with these extensions, as long as the meaning is clear. Note that when we say that $\overline{f}$ is order-preserving and homogeneous, we are using the usual conventions that $c < \infty$ for all $c \in \mathbb{R}$ and $c \cdot \infty = \infty$ for all $c > 0$.
We can make the extension of $f$ to $(0,\infty]^n$ somewhat more familiar by observing that $\overline{f} = L \underline{g} L$ where $g = L f L$ and $L:[0,\infty]^n\rightarrow [0,\infty]^n$ is the entrywise reciprocal function
\begin{equation} \label{L}
L(x)_j := \begin{cases}
x_j^{-1} & \text{ if } 0 < x_j < \infty \\
\infty & \text{ if } x_j = 0 \\
0 & \text{ if } x_j = \infty.
\end{cases}
\end{equation}
For any order-preserving homogeneous function $f:(0,\infty]^n \rightarrow (0,\infty]^n$, we can define the upper and lower Collatz-Wielandt numbers $r(f)$ and $\lambda(f)$ using \eqref{cw} and \eqref{lcw}, and we allow the possibility that they might be infinite. We define the \emph{support} of $x \in (0,\infty]^n$ to be $\operatorname{supp}(x) = \{j \in N : x_j < \infty \}$. We can also define the \emph{parts} of $(0,\infty]^n$ to be the sets
$$(0,\infty]^J := \{x \in (0,\infty]^n : \operatorname{supp}(x) = J \}.$$
Note that Hilbert's projective metric is defined on each part and is finite for any pair of vectors in the same part.
Using the reciprocal function $L$, we see that $\lambda(f) = r(LfL)^{-1}$ and $r(f) = \lambda(LfL)^{-1}$. Because $f$ extends continuously to $(0,\infty]^n$, the hypergraph $\mathcal{H}^+_\infty(f)$
has a hyperarc $(I,\{j\})$ with $j \notin I$ if and only if $f(\omega_I)_j = \infty$ where $\omega_I \in (0,\infty]^n$ is given by
$$(\omega_I)_i := \begin{cases}
\infty & \text{ if } i \in I \\
1 & \text{ if } i \in I^c.
\end{cases}$$
Since $\omega_I = L(e_{N \backslash I})$, it follows that $\mathcal{H}^+_\infty(f) = \mathcal{H}^-_0(LfL)$.
The following iterative formula for the lower Collatz-Wielandt number is similar to \eqref{rlim}.
\begin{lemma} \label{lem:llim}
Let $f: \mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ (or $f: (0,\infty]^n \rightarrow (0,\infty]^n$) be order-preserving and homogeneous. For every $x \in \mathbb{R}^n_{>0}$,
$$\lambda(f) = \lim_{k \rightarrow \infty} \left(\min_{i \in N} f^k(x)_i\right)^{1/k}.$$
\end{lemma}
\begin{proof}
If $f(x)_i = 0$ for some $i \in N$ and $x \in \mathbb{R}^n_{>0}$, then $\lambda(f) = 0$ by definition. At the same time, $f^k(x)_i = 0$ for all $k \in \mathbb{N}$ since $f$ is order-preserving and there exists some $\beta > 0$ such that $f(x) \le \beta x$. Also, $f^k(y)_i = 0$ for all $y \in \mathbb{R}^n_{>0}$ since $x$ and $y$ are comparable. Therefore $\lim_{k \rightarrow \infty} \left( \min_{i \in N} f^k(y)_i \right)^{1/k} = 0$ for all $y \in \mathbb{R}^n_{>0}$.
Suppose now that $f(x)_i > 0$ for all $i \in N$. Let $g = L f L$. Observe that
\begin{align*}
r(g) = \inf_{x \in \mathbb{R}^n_{>0}} \max_{i \in N} \frac{g(x)_i}{x_i} &= \inf_{y = Lx \in \mathbb{R}^n_{>0}} \max_{i \in N} \frac{y_i}{f(y)_i} \\
&= \left(\sup_{y = Lx \in \mathbb{R}^n_{>0}} \min_{i \in N} \frac{f(y)_i}{y_i}\right)^{-1} = \lambda(f)^{-1}.
\end{align*}
For $x \in \mathbb{R}^n_{>0}$ and $k \in \mathbb{N}$, $\min_{i \in N} f^k(x)_i = (\max_{i \in N} g^k(y)_i)^{-1}$ , where $y = Lx$ and $k \in \mathbb{N}$. Working with the supremum norm $\|x\|_\infty = \max_{i \in N} |x_i|$, we have by \eqref{rlim}:
$$\lim_{k \rightarrow \infty} \left(\min_{i \in N} f^k(x)_i \right)^{1/k}= \lim_{k \rightarrow \infty} \left(\max_{i \in N} g^k(y)_i \right)^{-1/k} = r(g)^{-1} = \lambda(f).$$
\end{proof}
For an order-preserving homogeneous function $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$, we define the \emph{minimum displacement} of $f$ to be $\delta(f) := \inf \{ d_H(x,f(x)) : x \in \mathbb{R}^n_{>0} \}$.
\begin{lemma} \label{lem:mindisp}
Let $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Then the minimum displacement of $f$ satisfies
$$\delta(f) = \log r(f) - \log \lambda(f).$$
\end{lemma}
\begin{proof}
By \eqref{rlim} and Lemma \ref{lem:llim}, the following is true for any $x \in \mathbb{R}^n_{>0}$.
\begin{align*}
\log r(f) - \log \lambda(f) &= \lim_{k \rightarrow \infty} \frac{1}{k} \log \left(\max_{i \in N} f^k(x)_i \right) - \lim_{k \rightarrow \infty} \frac{1}{k} \log \left(\min_{j \in N} f^k(x)_j\right) \\
&= \lim_{k \rightarrow \infty} \frac{1}{k} \log \left(\max_{i, j \in N} \frac{f^k(x)_i}{f^k(x)_j} \right) \\
&= \lim_{k \rightarrow \infty} \frac{1}{k} \log \left(\max_{i, j \in N} \frac{f^k(x)_i \, x_j}{f^k(x)_j \, x_i} \right) \\
&= \lim_{k \rightarrow \infty} \frac{1}{k} d_H(x,f^k(x)).
\end{align*}
It is known that $\lim_{k \rightarrow \infty} \frac{1}{k} d_H(x,f^k(x)) = \delta(f)$ for all $x \in \mathbb{R}^n_{>0}$ \cite[Theorem 1]{GaVi12}, so we conclude that $\log r(f) - \log \lambda(f) = \delta(f)$.
\end{proof}
If $f$ has an eigenvector $x \in \mathbb{R}^n_{>0}$, then $d_H(x,f(x)) = 0$, so $\delta(f) = 0$. In that case Lemma \ref{lem:mindisp} proves that $r(f) = \lambda(f)$. However, there are simple examples where $\delta(f) = 0$, but $f$ has no eigenvectors in $\mathbb{R}^n_{>0}$.
Another interesting corollary of Theorems \ref{thm:KR} and \ref{thm:BS} is the following. A similar slightly weaker observation was made in \cite[Theorem 2]{ChFrLi13}.
\begin{corollary} \label{cor:formaleig}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Then $f$ has an eigenvector $y \in (0,\infty]^n$ with $y_i < \infty$ for some $i \in N$ such that $f(y) = \lambda(f) y$.
\end{corollary}
\begin{proof}
Note that $LfL$ is order-preserving and homogeneous on $\mathbb{R}^n_{>0}$ and it has a continuous extension to $\mathbb{R}^n_{\ge 0}$ by Theorem \ref{thm:BS}. Furthermore, the continuous extension is order-preserving and homogeneous and $r(LfL) = \lambda(f)^{-1}$. Then, by Theorem \ref{thm:KR}, there is a nonzero $x \in \mathbb{R}^n_{\ge 0}$ such that $LfL(x) = \lambda(f)^{-1}x$. So $y = Lx$ is an eigenvector of the continuous extension of $f$ to $(0,\infty]^n$ with eigenvalue $\lambda(f)$, and since $x \ne 0$, there is at least one entry of $y$ that is not infinite.
\end{proof}
\section{Existence of eigenvectors} \label{sec:exist}
Before stating the main result of this section, we need to introduce a class of auxiliary functions defined for any order-preserving homogeneous $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$. For $\alpha \in [-\infty, \infty]$ and $J \subseteq N$, let $P^J_\alpha: [-\infty,\infty]^n \rightarrow [-\infty,\infty]^n$ be the projection
\begin{equation} \label{PJ}
P^J_\alpha(x)_j := \begin{cases}
x_j & \text{ if } j \in J \\
\alpha & \text{ otherwise.}
\end{cases}
\end{equation}
Observe that for all $x \in [0,\infty]^n$, $P^J_0 x \le x \le P^J_\infty x$. For any order-preserving homogeneous function $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_0$, we define
$$f^J_0 := P^J_0 f P^J_0 \hspace*{0.5cm} \text{ and } \hspace*{0.5cm} f^J_{\infty} := P^J_\infty f P^J_\infty.$$
By Theorem \ref{thm:BS}, both $f^J_0: \mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ and $f^J_\infty: (0,\infty]^n \rightarrow (0,\infty]^n$ are well-defined order-preserving and homogeneous functions. Note also that
\begin{equation} \label{monotonicity}
f^J_0(x) \le f(x) \le f^J_\infty(x)
\end{equation}
for all $x \in \mathbb{R}^n_{>0}$.
Now the main result of this section is:
\begin{theorem} \label{thm:super}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. The eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0}, d_H)$ if and only if for all nonempty proper subsets $J \subset N$,
\begin{equation} \label{superIllum}
r(f^J_0) < \lambda(f^{N \backslash J}_\infty).
\end{equation}
\end{theorem}
Before proving Theorem \ref{thm:super} we need the following lemmas.
\begin{lemma} \label{lem:near}
Let $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. For any $v \in \mathbb{R}^J_{>0}$ and $w \in \mathbb{R}^{N \backslash J}_{>0}$, let $x = v+tw$ where $t > 0$ is a small constant. Then
$$\lim_{t \rightarrow 0} \frac{f(x)_j}{x_j} = \frac{f^J_0(v)_j}{v_j} \text{ and } \lim_{t \rightarrow 0} \frac{f(x)_i}{x_i} =\frac{f^{N \backslash J}_\infty(w)_i}{w_i}$$
for all $j \in J$ and $i \in J^c$.
\end{lemma}
\begin{proof}
The first limit follows immediately from Theorem \ref{thm:BS}. The second follows from Theorem \ref{thm:BS} with the additional observation that
$$\lim_{t \rightarrow 0} \frac{f(x)_i}{x_i} = \lim_{t \rightarrow 0} \frac{f(t^{-1} x)_i}{t^{-1} x_i} = \frac{f^{N \backslash J}_\infty(w)_i}{w_i}$$
for all $i \in J^c$.
\end{proof}
\begin{lemma} \label{lem:super}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Let $J$ be a nonempty proper subset of $N$. Then there exists $x \in \mathbb{R}^n_{>0}$ such that
$$\max_{j \in J} \frac{f(x)_j}{x_j} < \min_{i \in J^c} \frac{f(x)_i}{x_i}$$
if and only if $r(f^J_0) < \lambda(f^{N \backslash J}_\infty).$
\end{lemma}
\begin{proof}
($\Rightarrow$) Suppose that $\max_{j \in J} {f(x)_j}/{x_j} < \min_{i \in J^c} {f(x)_i}/{x_i}$ for some $x \in \mathbb{R}^n_{>0}$. By the definitions of the upper and lower Collatz-Wielandt numbers and \eqref{monotonicity},
\begin{align*}r(f^J_0) \le \max_{j \in J} \frac{f^J_0(x)_j}{x_j} &\le \max_{j \in J} \frac{f(x)_j}{x_j} \\
&< \min_{i \in J^c} \frac{f(x)_i}{x_i} \le \min_{i \in J^c} \frac{f^{N\backslash J}_\infty(x)_i}{x_i} \le \lambda(f^{N\backslash J}_\infty).
\end{align*}
\noindent
($\Leftarrow$) Suppose that $r(f^J_0) < \lambda(f^{N \backslash J}_\infty)$.
By the definitions of the upper and lower Collatz-Wielandt numbers, we may choose $v \in \mathbb{R}^J_{>0}$ and $w \in \mathbb{R}^{N \backslash J}_{>0}$ such that $\max_{j \in J} f^J_0(v)_j/v_j$ is close enough to $r(f^J_0)$ and $\min_{i \in J^c} f^{N \backslash J}_\infty(w)_i/w_i$ is close enough to $\lambda(f^{N \backslash J}_\infty)$ so that
$$\max_{j \in J} \frac{f^J_0(v)_j}{v_j}< \min_{i \in J^c} \frac{f^{N \backslash J}_\infty(w)_i}{w_i}.$$
Then, by Lemma \ref{lem:near}, there exists $x \in \mathbb{R}^n_{>0}$ such that
$$\max_{j \in J} \frac{f(x)_j}{x_j} < \min_{i \in J^c} \frac{f(x)_i}{x_i}.$$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:super}]
Theorem \ref{thm:super} follows immediately from Theorem \ref{thm:LLN} and Lemma \ref{lem:super}.
\end{proof}
The following result shows that it is not always necessary to check
the conditions of Theorem \ref{thm:super}
for every $J \subset N$.
\begin{theorem} \label{thm:quick}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Suppose that \eqref{superIllum} holds for some proper nonempty subset $J \subset N$. If $f^J_0$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^J_{>0}$, then \eqref{superIllum} also holds for all nonempty $I \subseteq J$. If $f^{N \backslash J}_\infty$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^{N \backslash J}_{>0}$, then \eqref{superIllum} holds for all $I \supseteq J$ with $I \ne N$.
\end{theorem}
\begin{proof}
Suppose that $f^J_0$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^J_{>0}$. Consider any nonempty proper subset $I \subset J$. Let $D: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be the linear transformation given by
$$D(x)_j := \begin{cases}
x_j & \text{ if } j \in I \\
(1-\epsilon)x_j & \text{ if } j \in J \backslash I \\
0 & \text{ otherwise}
\end{cases}$$
where $\epsilon>0$ is a small constant.
Note that $D f^J_0 (x) \le f^J_0 (x)$ for all $x \in \mathbb{R}^n_{\ge 0}$, so $r(D f^J_0) \le r(f^J_0)$ by Lemma \ref{lem:cworder}.
As long as $\epsilon$ is sufficiently small, Theorem \ref{thm:LLN} will imply that $D f^J_0$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^J_{>0}$ since all of the inequalities \eqref{illum} which hold for the map $f^J_0$ will also hold for $Df^J_0$. Let $v \in \mathbb{R}^J_{>0}$ be one such eigenvector. Then $D f^J_0(v) = r(D f^J_0) v$. This means that
$$\frac{f^J_0(v)_j}{v_j} = \begin{cases}
r(Df^J_0) & \text{ if } j \in I \\
(1-\epsilon)^{-1} r(D f^J_0) & \text{ if } j \in J \backslash I.\\
\end{cases}$$
If $\epsilon > 0$ is small enough, then by \eqref{superIllum},
$$(1-\epsilon)^{-1} r(Df^J_0) < \lambda(f^{N \backslash J}_\infty).$$
By the definition of the lower Collatz-Wielandt number, we can choose $w \in \mathbb{R}^{N \backslash J}_{>0}$ such that
$$(1-\epsilon)^{-1} r(Df^J_0) < \min_{j \in J^c} \frac{f^{N \backslash J}_\infty(w)_j}{w_j}.$$
Then
$$\max_{i \in I} \frac{f^J_0(v)_i}{v_i} = r(Df^J_0) < (1-\epsilon)^{-1} r(Df^J_0) = \min_{i \in J\backslash I} \frac{f^J(v)_i}{v_i} < \min_{j \in J^c} \frac{f^{N \backslash J}_\infty(w)_j}{w_j}.$$
By Lemma \ref{lem:near}, we can choose $x \in \mathbb{R}^n_{>0}$ such that
$$\max_{i \in I}\frac{f(x)_i}{x_i} < \min_{j \in I^c} \frac{f(x)_j}{x_j}.$$
From this and \eqref{monotonicity}, we conclude that
\begin{align*}r(f^I_0) \le \max_{j \in I} \frac{f^J_0(x)_j}{x_j} &\le \max_{j \in I} \frac{f(x)_j}{x_j} \\
&< \min_{i \in I^c} \frac{f(x)_i}{x_i} \le \min_{i \in I^c} \frac{f^{N\backslash I}_\infty(x)_i}{x_i} \le \lambda(f^{N\backslash I}_\infty).
\end{align*}
The proof that \eqref{superIllum} holds for all $I \supseteq J$ when $f^{N \backslash J}_\infty$ has a nonempty and $d_H$-bounded set of eigenvectors in $\mathbb{R}^{N \backslash J}_{>0}$ is essentially the same.
\end{proof}
In order to check the conditions of Theorem \ref{thm:super}, the following lemma can also be helpful.
\begin{lemma} \label{lem:AsubB}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Suppose that $A \subseteq B \subseteq N$. Then
\begin{enumerate}[(a)]
\item $r(f^A_0) \le r(f^B_0)$, \text{ and }
\item $\lambda(f^A_\infty) \ge \lambda(f^B_\infty)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that $f^A_0(x) \le f^B_0(x)$ for all $x \in \mathbb{R}^n_{\ge 0}$. Therefore $r(f^A_0) \le r(f^B_0)$ by Lemma \ref{lem:cworder}.
Since $f^A_\infty(x) \ge f^B_\infty(x)$ for all $x \in (0, \infty]^n$, a similar argument shows that $\lambda(f^A_\infty) \ge \lambda(f^B_\infty)$.
\end{proof}
The following condition can show that an order-preserving homogeneous function has no eigenvectors in $\mathbb{R}^n_{>0}$.
\begin{theorem} \label{thm:none}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. If
$$r(f^J_0) > \lambda (f^{N \backslash J}_\infty)$$
for some nonempty proper subset $J \subset N$, then $f$ has no eigenvectors in $\mathbb{R}^n_{>0}$.
\end{theorem}
\begin{proof}
By Lemma \ref{lem:AsubB}, we have
$$r(f) \ge r(f^J_0) > \lambda (f^{N \backslash J}_\infty) \ge \lambda(f).$$
Then the minimum displacement $\delta(f) = r(f) - \lambda(f) > 0$ by Lemma \ref{lem:mindisp}. This means that $f$ cannot have an eigenvector in $\mathbb{R}^n_{>0}$.
\end{proof}
Notice that there is a middle ground between the conditions of Theorems \ref{thm:super} and \ref{thm:none} where $r(f^J_0) \le \lambda(f^{N \backslash J}_\infty)$ for all nonempty proper $J \subset N$, but $r(f^J_0) = \lambda(f^{N \backslash J}_\infty)$ for at least one $J$. In that case, it is possible that $f$ has no eigenvectors in $\mathbb{R}^n_{>0}$, or that the eigenspace $E(f)$ is unbounded in $(\mathbb{R}^n_{>0}, d_H)$. For example consider the linear maps corresponding to the nonnegative matrices
$$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \text{ and } \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.$$
The first is the identity matrix and every element of $\mathbb{R}^2_{>0}$ is an eigenvector, while the second matrix has no eigenvectors in $\mathbb{R}^2_{>0}$. The interested reader can easily check that $r(f^{\{1\}}_0) = \lambda(f^{\{2\}}_\infty)=1$ and $r(f^{\{2\}}_0) \le \lambda(f^{\{1\}}_\infty)$ for each matrix.
\begin{remark} \label{rem:irred}
If the function $f$ in Theorem \ref{thm:super} is differentiable and the Jacobian matrix $f'(x)$ is irreducible for all $x \in \mathbb{R}^n_{>0}$, then any eigenvector of $f$ in $\mathbb{R}^n_{>0}$ will be unique (up to scaling). This is also true, even if $f'(x)$ is not irreducible, as long as $f'(x)$ has a unique positive eigenvector for every $x \in \mathbb{R}^n_{>0}$ \cite[Corollary 6.4.8]{LemmensNussbaum}. In that case, the condition of Theorem \ref{thm:super} is both necessary and sufficient for $f$ to have an eigenvector in $\mathbb{R}^n_{>0}$.
\end{remark}
\subsection{Using hypergraphs to detect eigenvectors} \label{sec:hypergraph}
The results in this section can be combined with the hypergraph conditions from Theorems \ref{thm:GG} and \ref{thm:AGH} to give a very general method for checking whether an order-preserving homogeneous $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ has any eigenvectors in $\mathbb{R}^n_{>0}$. The following two lemmas show how the upper and lower Collatz-Wielandt numbers of the auxiliary functions $f^J_0$ and $f^{N \backslash J}_\infty$ relate to the hypergraphs $\mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(f)$. For a hypergraph $\mathcal{H}$ and a subset of nodes $J$, the \emph{reach} of $J$ in $\mathcal{H}$, denoted $\operatorname{reach}(J,\mathcal{H})$, is the smallest invariant subset of $\mathcal{H}$ containing $J$.
\begin{lemma} \label{lem:hyperorder}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous and let $J \subset N$. If $\operatorname{reach}(J,\mathcal{H}^+_\infty(f)) = I$, then $\lambda(f^{N \backslash J}_\infty) = \lambda(f^{N \backslash I}_\infty)$. If $\operatorname{reach}(J^c, \mathcal{H}^-_0(f)) = I^c$, then $r(f^J_0) = r(f^I_0)$.
\end{lemma}
\begin{proof}
Fix any $x \in \mathbb{R}^n_{>0}$. Let $x^k = (f^J_0)^k(x)$ for $k \in \mathbb{N}$. Since there exists $\beta > 0$ such that $f^J_0(x) \le \beta x$ and $f$ is order-preserving, it follows that $x^{k+1} \le \beta x^k$ for all $k \in \mathbb{N}$. In particular, $\operatorname{supp}(x^{k+1}) \subseteq \operatorname{supp}(x^k)$ for each $k$. Let $I_k = \operatorname{supp}(x^k)$. Note that $I_{k+1} = I_k$ if and only if $I^c_k$ is invariant in $\mathcal{H}^-_0(f)$. So the sets $I_k$ eventually stabilize at some $k = m$ where $I_m = \operatorname{reach}(J^c,\mathcal{H}^-_0(f))^c$. Let $I = I_m$. We can choose $y \in \mathbb{R}^n_{>0}$ such that $P^I_0y = x^m$, and then $x^k = (f^I_0)^{k-m}(y)$ for all $k \ge m$. Therefore $r(f^J_0) = r(f^I_0)$ by \eqref{rlim}.
The proof that $\lambda(f^{N \backslash J}_\infty) = \lambda(f^{N \backslash I}_\infty)$ when $I = \operatorname{reach}(J,\mathcal{H}^+_\infty(f))$ is essentially the same, but uses Lemma \ref{lem:llim} in place of \eqref{rlim}.
\end{proof}
\begin{lemma} \label{lem:connect}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Consider any $J \subset N$. The following are equivalent.
\begin{enumerate}[(a)]
\item \label{item:a} There is no nonempty $A \subseteq J$ such that $A^c$ is invariant in $\mathcal{H}^-_0(f)$.
\item \label{item:b} $\operatorname{reach}(J^c,\mathcal{H}^-_0(f)) = N$.
\item \label{item:c} $r(f^J_0) = 0$.
\end{enumerate}
Similarly, the following are also equivalent.
\begin{enumerate}[(a),resume]
\item \label{item:d} There is no nonempty $B \subseteq J^c$ such that $B^c$ is invariant in $\mathcal{H}^+_\infty(f)$.
\item \label{item:e} $\operatorname{reach}(J,\mathcal{H}^+_\infty(f)) = N$.
\item \label{item:f} $\lambda(f^{N\backslash J}_\infty) = \infty$.
\end{enumerate}
\end{lemma}
\begin{proof}
It is obvious from the definition of $\operatorname{reach}(J^c,\mathcal{H}^-_0(f))$ that conditions \ref{item:a} and \ref{item:b} are equivalent.
\ref{item:b} $\Rightarrow$ \ref{item:c}. If $\operatorname{reach}(J^c,\mathcal{H}^-_0(f)) = N$, then $r(f^J_0) = r(f^\varnothing_0) = 0$.
\ref{item:c} $\Rightarrow$ \ref{item:a}. Suppose that $r(f^J_0) = 0$. If $A \subseteq J$ and $f^J_0(e_A)_j > 0$ for all $j \in A$, then there exists $\alpha > 0$ such that $f^J_0(e_A) \ge \alpha e_A$. But then $(f^J_0)^k(e_A) \ge \alpha^k e_A$ and so $r(f^J_0) \ge \alpha$ by \eqref{rlim}. This is a contradiction, so we conclude that $f^J_0(e_A)_j = 0$ for some $j \in A$. Therefore there is a hyperarc from $A^c$ into $A$ in $\mathcal{H}^-_0(f)$, so $A^c$ is not invariant in $\mathcal{H}^-_0(f)$.
To prove that conditions \ref{item:d}, \ref{item:e}, and \ref{item:f} are equivalent, let $g = L f L$, where $L$ is the entrywise reciprocal function from \eqref{L}. Observe that $\mathcal{H}^+_\infty(f) = \mathcal{H}^-_0(g)$ and $f^{N \backslash J}_\infty = L g^{N \backslash J}_0 L$, so $\lambda(f^{N \backslash J}_\infty) = \infty$ if and only if $r(g^{N \backslash J}_0) = 0$. Then \ref{item:d}, \ref{item:e}, \ref{item:f} are equivalent to each other because \ref{item:a}, \ref{item:b}, \ref{item:c} are equivalent.
\end{proof}
The following corollary is a restatement of Theorem \ref{thm:AGH} using Lemma \ref{lem:connect}.
\begin{corollary} \label{cor:AGH2}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Then the following are equivalent.
\begin{enumerate}
\item All slices spaces $S_\alpha^\beta(f)$ are $d_H$-bounded.
\item For every $J \subset N$, either $r(f^J_0) = 0$ or $\lambda(f^{N\backslash J}_\infty) = \infty$.
\item For every $J \subset N$, either $\operatorname{reach}(J^c,\mathcal{H}^-_0(f)) = N$ or $\operatorname{reach}(J,\mathcal{H}^+_\infty(f)) = N$.
\end{enumerate}
\end{corollary}
Corollary \ref{cor:AGH2} suggests the following strategy for checking whether the eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$. First, check whether $\mathcal{H}^+_\infty(f)$ has any invariant subsets $J$. If not, then $E(f)$ is nonempty and $d_H$-bounded by Theorem \ref{thm:GG}. If there are invariant sets $J$ in $\mathcal{H}^+_\infty(f)$, then for each one, check whether $\operatorname{reach}(J^c,\mathcal{H}^-_0(f))=N$. If there are any sets $J \subset N$ such that $\operatorname{reach}(J, \mathcal{H}^+_\infty(f)) \ne N$ and $\operatorname{reach}(J^c, \mathcal{H}^-_0(f)) \ne N$, then those are the only sets where we need to check \eqref{superIllum} in Theorem \ref{thm:super}. We give examples in Section \ref{sec:examples} to demonstrate this process and show how we can further reduce the number of sets $J$ to check by using Theorem \ref{thm:quick} and Lemma \ref{lem:AsubB}.
For some order-preserving homogeneous functions $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ there are faster ways to confirm that $f$ has a positive eigenvector. The \emph{directed graph associated with} $f$ is the digraph $\mathcal{G}(f)$ with vertices $N$ and an arc from $i$ to $j$ when
$$\lim_{t \rightarrow \infty} f(\exp(te_{\{j\}}))_i = \infty.$$
Note that $(i,j)$ with $i \ne j$ is a arc of $\mathcal{G}(f)$ if and only if $(\{j\},\{i\})$ is a hyperarc of $\mathcal{H}^+_\infty(f)$. Also observe that the direction of the hyperarc $(\{j\},\{i\}) \in \mathcal{H}^+_\infty(f)$ is the reverse of $(i,j) \in \mathcal{G}(f)$.
For a nonnegative matrix $A= \begin{bmatrix} a_{ij} \end{bmatrix}_{i, j \in N}$, $\mathcal{G}(A)$ is equivalent to the usual directed graph associated with a matrix which has an arc from $i$ to $j$ precisely when $a_{ij} \ne 0$.
In \cite[Theorem 2]{GaGu04}, the authors point out that if $\mathcal{G}(f)$ is strongly connected, then $f$ has an eigenvector in $\mathbb{R}^n_{>0}$. This directly generalizes the classical Perron-Frobenius theorem for irreducible matrices, since $A \in \mathbb{R}^{n \times n}_{\ge 0}$ is irreducible if and only if $\mathcal{G}(A)$ is strongly connected. This nonlinear version of the Perron-Frobenius theorem follows immediately from Theorem \ref{thm:GG}. If $\mathcal{G}(f)$ is strongly connected, then $\mathcal{H}^+_\infty(f)$ has no invariant subsets, so Theorem \ref{thm:GG} implies all super-eigenspaces of $f$ are bounded in $(\mathbb{R}^n_{>0}, d_H)$. As described in the introduction, this guarantees that $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$. If $\mathcal{G}(f)$ is not strongly connected, then its nodes can be partitioned into strongly connected components. A strongly connected component is called a \emph{final class} if no arcs leave the component. The following sufficient condition for existence of a positive eigenvector generalizes \cite[Theorem 2]{GaGu04} and is generally easier to check than Theorem \ref{thm:super} since the graph $\mathcal{G}(f)$ and its connected components can be computed relatively quickly even when the dimension $n$ is large.
\begin{theorem} \label{thm:graphCond}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. If $\mathcal{G}(f)$ has a unique final class $C$ and $r(f^{N \backslash C}_0) < r(f)$, then the eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0}, d_H)$.
\end{theorem}
\begin{proof}
Suppose temporarily that $\lambda(f) < r(f)$. We may assume without loss of generality that $r(f) = 1$ by replacing $f$ with $r(f)^{-1}f$. By Theorem \ref{thm:KR}, there is an eigenvector $v \in \mathbb{R}^n_{\ge 0}$ such that $f(v) = v$. By Corollary \ref{cor:formaleig}, there is also an eigenvector $w \in (0,\infty]^n$ such that $f(w) = \lambda(f) w$. Let $J = \operatorname{supp}(v)$ and $I = \operatorname{supp}(w)$. For any $x \in \mathbb{R}^n_{>0}$ we can find $\alpha, \beta > 0$ such that $\alpha v \le x \le \beta w$. Then $\alpha v \le f^k(x) \le \beta \lambda(f)^k w$. This means that $\lim_{k \rightarrow \infty} f^k(x)_i = 0$ for all $i \in I$, while $f^k(x)_j$ is bounded below for all $j \in J$. Therefore $I$ and $J$ are disjoint. Note that $r(f^J_0) = 1$. Since $r(f^{N \backslash C}_0) < r(f) = 1$, Lemma \ref{lem:AsubB} implies that $J$ cannot be a subset of $N \backslash C$. So there exists some $j \in J \cap C$. Since $C$ is the unique final class of $\mathcal{G}(f)$, there must be a path in $\mathcal{G}(f)$ from every $i \in N$ to $j$. Choose $i \in I$ and let $m$ be the length of a path from $i$ to $j$ in $\mathcal{G}(f)$. Then by the definition of $\mathcal{G}(f)$, we must have
$$\lim_{t \rightarrow \infty} f^m(\exp(te_{\{j\}}))_i = \infty.$$
At the same time, since $j \notin I$, there is a $\beta > 0$ such that $\exp(te_{\{j\}}) \le \beta w$ for all $t>0$. Then
$$f^m(\exp(te_{\{j\}}))_i \le \beta \lambda(f)^m w_i < \beta w_i < \infty$$
for all $t > 0$. This is a contradiction, so we conclude that $\lambda(f) = r(f) = 1$.
Let $A$ be a nonempty proper subset of $N$. If $A \subseteq N \backslash C$, then
$$r(f^A_0) \le r(f^{N \backslash C}_0) < r(f) = \lambda(f) \le \lambda(f^{N \backslash A}_\infty)$$
by Lemma \ref{lem:AsubB}. Therefore \eqref{superIllum} holds for all $A \subseteq N \backslash C$. On the other hand, if $A \cap C \ne \varnothing$, then there is a $j \in A \cap C$ and a path in $\mathcal{H}^+_\infty(f)$ from $j$ to every $i \in N$. This means that $\operatorname{reach}(A,\mathcal{H}^+_\infty(f)) = N$, so $\lambda(f^{N \backslash A}_\infty) = \infty$ by Lemma \ref{lem:connect}. Then \eqref{superIllum} also holds for $A$. Since \eqref{superIllum} holds for all nonempty proper $A \subset N$, we conclude by Theorem \ref{thm:super} that $E(f)$ is nonempty and bounded in $(R^n_{>0},d_H)$.
\end{proof}
\section{Multiplicatively convex functions.} \label{sec:conv}
Many order-preserving homogeneous functions $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ have the additional property that $\log \circ f \circ \exp$ is convex. Recall that a function $g:D \rightarrow \mathbb{R}$ is \emph{convex} when its \emph{epigraph}
$$\operatorname{epi}(f) := \{ (x,y) \in D \times \mathbb{R} : y \ge f(x) \}$$
is convex. A vector-valued function $f:D \rightarrow \mathbb{R}^n$ is \emph{convex} if each entry of $f$ is convex. When $\log \circ f \circ \exp$ is convex, we will say that $f$ is \emph{multiplicatively convex}. Note that $f:\mathbb{R}^m_{>0} \rightarrow \mathbb{R}^n_{>0}$ is multiplicatively convex if and only if for all $x, y \in \mathbb{R}^m_{>0}$ and $0 \le \theta \le 1$,
$$f(x^\theta y^{1-\theta}) \le f(x)^{\theta} f(y)^{1-\theta} $$
where this notation is understood to indicate entrywise products and powers. We will continue to use this notation throughout the rest of this paper.
The following result is a special case of \cite[Proposition 6.1]{Nussbaum86}. The proof is short, so we include it here.
\begin{lemma} \label{lem:prop61}
Let $f, g:\mathbb{R}^m_{>0} \rightarrow \mathbb{R}^n_{>0}$ be multiplicatively convex. Then $f+g$ is multiplicatively convex, and if $f$ is also order-preserving, then $f \circ g$ is multiplicatively convex.
\end{lemma}
\begin{proof}
By H{\"o}lder's inequality,
\begin{align*}
f(x^\theta y^{1-\theta})_i + g(x^\theta y^{1-\theta})_i &\le f(x)_i^\theta f(y)_i^{1-\theta} + g(x)_i^\theta g(y)_i^{1-\theta} \\
&\le [f(x)_i + g(x)_i]^\theta [f(y)_i + g(y)_i]^{1-\theta}
\end{align*}
for each $i \in N$ and $0 < \theta < 1$. Therefore $f+g$ is multiplicatively convex. If $f$ is also order-preserving, then
\begin{align*}
f(g(x^\theta y^{1-\theta})) &\le f(g(x)^\theta g(y)^{1-\theta})\\
&\le f(g(x))^{\theta} f(g(y))^{1-\theta},
\end{align*}
which proves that $f \circ g$ is multiplicatively convex.
\end{proof}
An immediate consequence of Lemma \ref{lem:prop61} is that any order-preserving, homogeneous, and linear map $f:\mathbb{R}^m_{>0} \rightarrow \mathbb{R}^n_{>0}$ is multiplicatively convex.
Other important classes of order-preserving, homogeneous, multiplicatively convex functions include matrices over the max-times algebra (see e.g., \cite{MuPe15}), and the class $\mathcal{M}_+$ introduced in \cite{Nussbaum89} (see also \cite[Proposition 3.1 and Equation 3.15]{Nussbaum86} and \cite[Section 6.6]{LemmensNussbaum}). We will discuss the class $\mathcal{M}_+$ in more detail in Section \ref{sec:examples}.
When $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving, homogeneous, and multiplicatively convex, it is possible to give very general conditions for existence of eigenvectors in $\mathbb{R}^n_{>0}$, even when the eigenspace $E(f)$ might not be bounded in Hilbert's projective metric.
Let $C_1, \ldots, C_m$ denote the strongly connected components of $\mathcal{G}(f)$. Borrowing some terminology from the theory of nonnegative matrices (see e.g., \cite[Definition 2.3.8]{BermanPlemmons}), we will say that $C_j$ is a \emph{basic class} of $f$ if $r(f^{C_j}_0) = r(f)$.
In \cite{HuQi16}, Hu and Qi define what they call strongly nonnegative tensors. These tensors correspond to a special class of order-preserving, homogeneous, and multiplicatively convex functions on $\mathbb{R}^n_{>0}$. The following definition extends their notion of strong nonnegativity to all order-preserving, homogeneous, and multiplicatively convex functions on $\mathbb{R}^n_{>0}$.
\begin{definition}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex. Let $\mathcal{C}$ denote the set of strongly connected components of $\mathcal{G}(f)$. We will say that $f$ is \emph{strongly nonnegative} if $r(f^{C}_0) = r(f)$ for every final class $C \in \mathcal{C}$ and $r(f^{C}_0) < r(f)$ when $C \in \mathcal{C}$ is not a final class. That is, $f$ is strongly nonnegative if and only if its basic and final classes coincide.
\end{definition}
The following theorems are the main results of this section. The first theorem shows the relationship between the upper and lower Collatz-Wielandt numbers and the strongly connected components of $\mathcal{G}(f)$.
\begin{theorem} \label{thm:basicClass}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex. Let $\mathcal{C}$ denote the set of strongly connected components of $\mathcal{G}(f)$ and let $\mathcal{F}$ denote the set of final classes of $\mathcal{G}(f)$. Then
$$r(f) = \max_{C \in \mathcal{C}} r(f^{C}_0) ~~~~ \text{ and } ~~~~ \lambda(f) = \min_{C \in \mathcal{F}} r(f^{C}_0).$$
\end{theorem}
The second theorem gives very general conditions for the existence of eigenvectors when $f$ is multiplicatively convex, even when the eigenspace $E(f)$ might not be bounded in Hilbert's projective metric. Note that conditions \ref{item:strongNonneg} and \ref{item:analyticConverse} generalize \cite[Theorem 5]{HuQi16} and corresponding results for reducible nonnegative matrices \cite[Theorem 2.3.10]{BermanPlemmons}.
\begin{theorem} \label{thm:convMain}
Let $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex.
\begin{enumerate}[(a)]
\item \label{item:convex} $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$ if and only if $f$ is strongly nonnegative and $\mathcal{G}(f)$ has only one final class.
\item \label{item:strongNonneg} If $f$ is strongly nonnegative, then $E(f)$ is nonempty.
\item \label{item:analyticConverse} If $f$ is real analytic and $E(f)$ is nonempty, then $f$ is strongly nonnegative.
\end{enumerate}
\end{theorem}
More is known for max-times functions: \cite[Theorem 3.4]{BaStVa95} and \cite[Theoreme 2.2.4]{Gaubert92} independently proved that a matrix in the max-times algebra has an entrywise positive eigenvector if and only if all of its final classes are basic. So it is possible for a multiplicatively convex function that is not strongly nonnegative to have an entrywise positive eigenvector if it is not real analytic.
In general, it is much easier to check the conditions in Theorem \ref{thm:convMain} than the conditions of Theorem \ref{thm:super}. The strongly connected components of $\mathcal{G}(f)$ can be computed efficiently, and there are at most $n$ Collatz-Wielandt numbers to compute. For more details about the complexity aspects of this problem, see \cite[Section 4.2.3]{AkGaHo20}. See also \cite[Corollary 4.4]{AkGaHo20} which gives a sufficient condition for all slice spaces to be $d_H$-bounded when $f$ is multiplicatively convex.
To prove Theorems \ref{thm:basicClass} and \ref{thm:convMain}, we need some preliminary results.
\begin{lemma} \label{lem:convConst}
Let $g: (0,\infty) \rightarrow (0,\infty)$ be order-preserving and multiplicatively convex. Then $\lim_{x \rightarrow \infty} g(x) < \infty$ if and only if $g$ is constant. If $g$ is also real analytic, then $\lim_{x \rightarrow \infty} g(x) < \infty$ if and only if $g'(x) = 0$ for some $x > 0$.
\end{lemma}
\begin{proof}
If $g$ is constant, then $\lim_{x \rightarrow \infty} g(x) < \infty$. Conversely, if $g$ is not constant, there exists $x < y$ such that $g(x) < g(y)$. Then the epigraph of $\log \circ g \circ \exp$ has a support line with a positive slope. It follows that $\lim_{x \rightarrow \infty} g(x) = \infty$.
If $g$ is also real analytic and $g'(x) = 0$ for some $x \in \mathbb{R}$, then $g'(y) = 0$ for all $y < x$, by convexity. But then $g'$ is identically zero on all of $(0,\infty)$ which means that $g$ is constant.
\end{proof}
\begin{lemma} \label{lem:convEq}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex. Suppose that there are no arcs leaving $J$ in $\mathcal{G}(f)$ for some nonempty $J \subset N$. Then
$$f(x)_j = f^J_0(x)_j = f^J_\infty(x)_j$$
for all $x \in \mathbb{R}^n_{>0}$ and $j \in J$.
\end{lemma}
\begin{proof}
No arcs leaving $J$ in $\mathcal{G}(f)$ means that $\lim_{t \rightarrow \infty} f(\exp(te_{\{i\}}))_j < \infty$ for all $j \in J$ and $i \in N \backslash J$. Since $f$ is order-preserving and homogeneous,
$$\lim_{t \rightarrow \infty} f(x \exp(te_{\{i\}}))_j \le (\max_{k \in N} x_k) \lim_{t \rightarrow \infty} f(\exp(t e_{\{i\}}))_j < \infty$$
for all $x \in \mathbb{R}^n$. By Lemma \ref{lem:convConst} this means that $t \mapsto f(x\exp(t e_{\{i \}}))_j$ is constant. Therefore the value of $f(x)_j$ does not change if we change the entries $x_i$ with $i \in N \backslash J$.
By letting $x_i \rightarrow 0$ (respectively, $x_i \rightarrow \infty$) for all $i \in N \backslash J$, we get $f(x)_j = f^J_0(x)_j$ (and $f(x)_j = f^J_\infty(x)_j$) for all $j \in J$.
\end{proof}
\begin{lemma} \label{lem:bdryEig}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex. If $C$ is a final class of $\mathcal{G}(f)$, then $f^C_0$ and $f^C_\infty$ both have eigenvectors with support equal to $C$ and eigenvalue equal to $r(f^C_0) = \lambda(f^C_\infty)$.
\end{lemma}
\begin{proof}
The directed graph $\mathcal{G}(f^C_\infty)$ associated with $f^C_\infty$ on $\mathbb{R}^C_{>0}$ has nodes $C$ and contains all of the arcs between pairs $i, j \in C$ that are in $\mathcal{G}(f)$. Since $C$ is a strongly connected component of $\mathcal{G}(f)$, we see that $\mathcal{G}(f^C_\infty)$ is strongly connected. Therefore $f^C_\infty$ has an eigenvector $v \in (0,\infty]^n$ with $\operatorname{supp}(v) = C$ and eigenvalue $\lambda(f^C_\infty)$ by Theorem \ref{thm:graphCond}. By Lemma \ref{lem:convEq}, $f^C_0(x)_j = f^C_\infty(x)_j$ for all $x \in \mathbb{R}^n_{>0}$ and $j \in J$. If we choose $x \in \mathbb{R}^n_{>0}$ such that $v = P^C_\infty x$, then $P^C_0 x$ is an eigenvector of $f^C_0$ with the same eigenvalue as $v$. This means that $r(f^C_0) =\lambda(f^C_\infty)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:basicClass}]
Let $C_1, \ldots, C_m$ denote the strongly connected components of $\mathcal{G}(f)$.
These components are the vertices of a directed acyclic graph with an arc from $C_i$ to $C_j$ whenever there is an arc in $\mathcal{G}(f)$ from a vertex in $C_i$ to a vertex in $C_j$. Since the vertices of a directed acyclic graph have a topological ordering, we can assume that the strongly connected components of $\mathcal{G}(f)$ are ordered so that there are no paths from $C_i$ to $C_j$ in $\mathcal{G}(f)$ when $i < j$. We may also assume that the final classes are $C_1, \ldots, C_k$ where $k$ is the number of final classes in $\mathcal{G}(f)$.
By Theorem \ref{thm:KR}, $f$ has an eigenvector $v \in \mathbb{R}^n_{\ge 0}$ with eigenvalue $r(f)$. Let $j$ be the minimal index such that $C_j \cap \operatorname{supp}(v) \ne \varnothing$. Let $J = C_1 \cup \ldots \cup C_j$.
By Lemma \ref{lem:convEq}, $f^J_0(v)_i = f(v)_i$ for all $i \in J$. Note that this equation is true even if some entries of $v$ are zero because both $f$ and $f^J_0$ extend continuously to the boundary of $\mathbb{R}^n_{\ge 0}$ by Theorem \ref{thm:BS}.
Therefore $P^J_0 v$ is an eigenvector of $f^J_0$ with eigenvalue $r(f)$. Since $\operatorname{supp}(v) \subseteq C_j \cup \ldots \cup C_m$, $P^J_0v = P^{C_j}_0 v$ so $P^{C_j}_0 v$ is an eigenvector of $f^{C_j}_0$ with eigenvalue $r(f)$. This implies that $r(f^{C_j}_0) \ge r(f)$. Since $r(f^{C_i}_0) \le r(f)$ by Lemma \ref{lem:AsubB} for each $1 \le i \le m$, we see that $r(f) = \max_{1 \le i \le m} r(f^{C_i}_0)$.
To prove the corresponding equation for $\lambda(f)$, note that $f$ also has an eigenvector $w \in (0,\infty]^n$ with eigenvalue $\lambda(f)$ by Corollary \ref{cor:formaleig}. If there is a path from $j$ to $i$ in $\mathcal{G}(f)$, then $\lim_{t \rightarrow \infty} f^p(\exp(te_{\{i\}}))_j = \infty$ where $p$ is the length of the path. If $w_i = \infty$, then there is a constant $\beta > 0$ such that $\exp(te_{\{i\}}) \le \beta w$ for all $t > 0$. Therefore
$$\beta \lambda(f)^p w_j = \beta f^p(w)_j \ge \lim_{t \rightarrow \infty} f^p(\exp(t e_{\{i\}}))_j = \infty $$
so $w_j = \infty$. This means that if $j \in \operatorname{supp}(w)$ (so that $w_j < \infty$) and $i \notin \operatorname{supp}(w)$, then there cannot be a path from $j$ to $i$ in $\mathcal{G}(f)$. It also means that if $j \in \operatorname{supp}(w)$, then any other vertex in the same strongly connected component with $j$ must also be in the support of $w$.
Let $j$ be the smallest index such that $C_j \cap \operatorname{supp}(w) \ne \varnothing$. By the above argument, there cannot be any arcs in $\mathcal{G}(f)$ from a vertex in $C_j$ to a vertex in $C_i$ when $i < j$. Therefore $C_j$ is a final class of $\mathcal{G}(f)$ and $C_j \subseteq \operatorname{supp}(w)$.
By Lemma \ref{lem:convEq}, $f_\infty^{C_j}(P^{C_j}_\infty w)_i = f(w)_i = \lambda(f) w_i$ for all $i \in C_j$. Therefore $P^{C_j}_\infty w$ is an eigenvector of $f_\infty^{C_j}$ with eigenvalue $\lambda(f)$. By Lemma \ref{lem:bdryEig}, $r(f^{C_j}_0) = \lambda(f^{C_j}_\infty) = \lambda(f)$. Since $r(f^{C_i}_0) = \lambda(f^{C_i}_\infty) \ge \lambda(f)$ for all other final classes $C_i$ by Lemmas \ref{lem:AsubB} and \ref{lem:bdryEig}, it follows that $\lambda(f) = \min_{1 \le i \le k} r(f^{C_i}_0)$.
\end{proof}
\begin{lemma} \label{lem:classCap}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and multiplicatively convex.
If $\mathcal{G}(f)$ has a unique final class $C$, then $f$ is strongly nonnegative if and only if $r(f^{N \backslash C}_0) < r(f^C_0)$.
\end{lemma}
\begin{proof}
Suppose that $r(f^{N \backslash C}) < r(f^C_0)$. If $B$ is any non-final strongly connected component of $\mathcal{G}(f)$, then $B \subseteq N \backslash C$. Therefore $r(f^B_0) \le r(f^{N \backslash C}_0) < r(f^C_0)$ by Lemma \ref{lem:AsubB} and $r(f^C_0) = r(f)$ by Theorem \ref{thm:basicClass}. Thus $f$ is strongly nonnegative.
Conversely, if $f$ is strongly nonnegative, then $r(f^B_0) < r(f) = r(f^C_0)$ for every strongly connected component $B$ of $\mathcal{G}(f)$ other than $C$. The directed graph
$\mathcal{G}(f^{N \backslash C}_0)$ has nodes $N \backslash C$ and an arc from $i$ to $j$ precisely when
$$\lim_{t \rightarrow \infty} f^{N \backslash C}_0(\exp(te_{\{j\}}))_i = \infty.$$
If there is an arc from $i$ to $j$ in $\mathcal{G}(f^{N \backslash C}_0)$, then there is an arc from $i$ to $j$ in $\mathcal{G}(f)$ by \eqref{monotonicity}. So for any $i \in N \backslash C$, the strongly connected component $A[i]$ of $\mathcal{G}(f^{N \backslash C}_0)$ which contains $i$ is a subset of the corresponding strongly connected component $B[i]$ of $\mathcal{G}(f)$ that contains $i$. Thus $r(f^{A[i]}_0) \le r(f^{B[i]}_0)$ for all $i \in N \backslash C$ by Lemma \ref{lem:AsubB}. Then, by Theorem \ref{thm:basicClass},
$$r(f^{N \backslash C}_0) = \max_{i \in N \backslash C} r(f^{A[i]}_0) \le \max_{i \in N \backslash C} r(f^{B[i]}_0) < r(f^C_0).$$
\end{proof}
\begin{lemma} \label{lem:samederiv}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, real analytic, and multiplicatively convex. Then $\mathcal{G}(f) = \mathcal{G}(f'(x))$ for every $x \in \mathbb{R}^n_{>0}$.
\end{lemma}
\begin{proof}
By the chain rule,
$$\left. \frac{\partial}{\partial t} f(x \exp( t e_{\{j\}}))_i \right|_{t = 0} = x_j f'(x)_{ij}$$
for any $x \in \mathbb{R}^n_{>0}$.
Therefore, $\lim_{t \rightarrow \infty} f(x \exp(te_{\{j\}})_i = \infty$ if and only if $f'(x)_{ij} > 0$ by Lemma \ref{lem:convConst}. Since $f$ is $d_H$-nonexpansive, $\lim_{t \rightarrow \infty} f(x \exp(te_{\{j\}}))_i = \infty$ if and only if $\lim_{t \rightarrow \infty} f(\exp(te_{\{j\}}))_i = \infty$ which is equivalent to $(i,j) \in \mathcal{G}(f)$. Therefore $\mathcal{G}(f) = \mathcal{G}(f'(x))$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:convMain} \ref{item:convex}] Suppose that $f$ is strongly nonnegative and $\mathcal{G}(f)$ has a unique final class $C$. By Lemma \ref{lem:classCap}, $r(f^{N \backslash C}_0) < r(f^C_0) = r(f)$. Then $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0},d_H)$ by Theorem \ref{thm:graphCond}.
Conversely, suppose that $E(f)$ is nonempty and $d_H$-bounded. Let $C$ be a final class of $\mathcal{G}(f)$. Then $r(f^C_0) = \lambda(f^C_\infty)$ by Lemma \ref{lem:bdryEig}. Since $E(f)$ is nonempty, we also know that $\lambda(f) = r(f)$ by Lemma \ref{lem:mindisp}. Using Lemma \ref{lem:AsubB}, we have
$$\lambda(f) \le \lambda(f^C_\infty) = r(f^C_0) \le r(f)$$
which implies that $r(f^C_0) = r(f)$ so $C$ is a basic class. If $\mathcal{G}(f)$ had another final class $B \subset N$ different from $C$, then the same argument would show that $r(f^B_0) = r(f) = r(f^C_0)$. But $B \subseteq N \backslash C$, so
$$r(f^B_0) \le r(f^{N \backslash C}_0) < \lambda(f^C_\infty) = r(f^C_0)$$
by Theorem \ref{thm:super} and Lemma \ref{lem:AsubB}. So we conclude that $C$ must be the only final class of $\mathcal{G}(f)$. Since $C$ is the unique final class and $r(f^{N \backslash C}_0) < r(f^C_0)$, Lemma \ref{lem:classCap} implies that $f$ is strongly nonnegative.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:convMain} \ref{item:strongNonneg}]
Let $J$ be the union of the final classes of $\mathcal{G}(f)$. Since $f$ is strongly nonnegative, each final class $C$ is basic. By Lemma \ref{lem:bdryEig}, there is an eigenvector in $\mathbb{R}^C_{>0}$ with eigenvalue $r(f)$. Let $u \in \mathbb{R}^n_{\ge 0}$ be the sum of one such eigenvector from each final class. Observe that $\operatorname{supp}(u) = J$ and $u$ is an eigenvector of $f$ with eigenvalue $r(f)$ by Lemma \ref{lem:convEq}.
We can assume without loss of generality that $N \backslash J = \{1, \ldots, m\}$ and $J = \{m+1, \ldots, n\}$ for some $m < n$. Let $K = \{x+tu : x \in \mathbb{R}^{N \backslash J}_{\ge 0} \text{ and } t\ge 0\}$. Then $K$ is a closed cone contained in $\mathbb{R}^n_{\ge 0}$ and $f$ leaves $K$ invariant by Lemma \ref{lem:convEq}. We can identify $K$ with $\mathbb{R}^{m+1}_{\ge 0}$ via the invertible map $V: \mathbb{R}^{m+1}_{\ge 0} \rightarrow K$ defined by
$$V(x)_i := \begin{cases}
x_i & \text{ when } i \le m \\
x_{m+1} u_i & \text{ when } i > m.
\end{cases}$$
Note that $V$ is linear and order-preserving. Let $g = V^{-1} \circ f \circ V$. Then $g:\mathbb{R}^{m+1}_{>0} \rightarrow \mathbb{R}^{m+1}_{>0}$ is order-preserving, homogeneous, and multiplicatively convex. The digraph $\mathcal{G}(g)$ has a unique final class $B = \{m+1\}$. The other strongly connected components of $\mathcal{G}(g)$ have vertices in $N \backslash J$ and are exactly the same as the corresponding strongly connected components of $\mathcal{G}(f)$. Since $r(g^B_0) = r(f^J_0) = r(f)$ and $r(g^A_0) = r(f^A_0) < r(f)$ for any other strongly connected component $A$ of $\mathcal{G}(g)$, it follows that $g$ is strongly nonnegative. Therefore, Theorem \ref{thm:convMain} \ref{item:convex} implies that $g$ has an eigenvector $w \in \mathbb{R}^{m+1}_{>0}$. Then $V(w) \in \mathbb{R}^{n}_{>0}$ is an eigenvector of $f$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:convMain} \ref{item:analyticConverse}] Suppose that $f$ is real analytic and $u \in \mathbb{R}^n_{>0}$ is an eigenvector of $f$. We may also assume that $r(f) = 1$ by scaling $f$. Thus, $f(u) = u$. Since $f$ is order-preserving, $A := f'(u)$ is a nonnegative matrix; and since $f$ is homogeneous, $u$ is an eigenvector of $A$ with eigenvalue $r(f) = 1$. By Lemma \ref{lem:samederiv}, $\mathcal{G}(A) = \mathcal{G}(f)$. So $\mathcal{G}(f)$ and $\mathcal{G}(A)$ have the same strongly connected components and the same final classes.
Let $J$ be the union of the final classes of $\mathcal{G}(f)$ and let $I = N \backslash J$. We can assume without loss of generality that $I = \{1, \ldots, m\}$ and $J = \{m+1, \ldots, n\}$ for some $m < n$. Then we can write $A$ and $u$ in block form:
$$A = \left[ \begin{array}{c|c}
A_{I} & * \\ \hline
0 & A_{J}
\end{array} \right] ~~~~ \text{ and } ~~~~ u = \left[ \begin{array}{c}
u_{I} \\
u_{J}
\end{array} \right].$$
Observe that $A_J u_J = u_J$. Let $x = \begin{bmatrix} 0 \\ u_J \end{bmatrix}$. Then $Ax \ge x$.
For every $i \in I$, there is a path in $\mathcal{G}(A)$ from $i$ to some $j \in J$. Therefore, there is some $p \in \mathbb{N}$ such that $A^p x \gg 0$. Then, since $x_j > 0$ for all $j \in J$, it follows that $A^p e_J \gg 0$ as well. This means that $A^p (u-\epsilon e_J) \ll u$ for all $\epsilon > 0$. Note that $A^p$ is the derivative of $f^p$ at $u$ by the chain rule for Fr\'{e}chet derivatives. So we have $f^p(u-\epsilon e_J) \ll u$ when $\epsilon > 0$ is sufficiently small. Then
$$(f^{N \backslash J}_0)^p(u) = (f^{N \backslash J}_0)^p( u - \epsilon e_J ) \le f^p( u - \epsilon e_J ) \ll u.$$
Therefore there is a positive constant $\beta < 1$ such that $(f^{N \backslash J}_0)^p(u) \le \beta u$. By \eqref{rlim}, this means that $r(f^{N \backslash J}_0) < 1$.
Let $C$ be a strongly connected component of $\mathcal{G}(f)$. If $C$ is not final, then $C \subseteq N \backslash J$, so Lemma \ref{lem:AsubB} shows that $r(f^{C}_0) \le r(f^{N \backslash J}_0) < 1 = r(f)$. If $C$ is final, then $P^{C}_0 u$ is an eigenvector of $f^{C}_0$ with eigenvalue 1 by Lemma \ref{lem:convEq}. Therefore the final classes are exactly the basic classes of $\mathcal{G}(f)$.
\end{proof}
\color{black}
\section{Uniqueness of eigenvectors} \label{sec:unique}
Suppose that $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving and linear. If $f$ has two linearly independent eigenvectors $x, y \in \mathbb{R}^n_{>0}$, then every vector in the linear span of $\{x,y\}$ is also an eigenvector corresponding to the spectral radius. Since $\operatorname{span} \{x,y\} \cap \mathbb{R}^n_{>0}$ is not bounded in Hilbert's projective metric, we see that the set of eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ is nonempty and $d_H$-bounded if and only if $f$ has a unique eigenvector in $\mathbb{R}^n_{>0}$ up to scaling. Of course, this is a well known part of linear Perron-Frobenius theory.
Strikingly, it turns out that the exact same statement is true if $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving, homogeneous, and real analytic.
\begin{theorem} \label{thm:uniqueEig}
Let $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving, homogeneous, and real analytic. The eigenspace $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0}, d_H)$ if and only if $f$ has a unique eigenvector in $\mathbb{R}^n_{>0}$ up to scaling.
\end{theorem}
In order to prove Theorem \ref{thm:uniqueEig}, we will prove a general result about the uniqueness of fixed points of nonexpansive maps on a real Banach space $X$. To do this, we need the following simple observation about Banach space norms.
In what follows $X^*$ denotes the dual space of $X$ and $\|\cdot\|_*$ is the dual norm. For $x^* \in X^*$, we write $\inner{x,x^*}$ to represent $x^*(x)$. The set valued \emph{duality map} $J: X \rightrightarrows X^*$ is given by
$$J(x) := \{x^* \in X^* : \|x^*\|_* = \|x \| \text{ and } \inner{x,x^*} = \|x\| \|x^*\|_* \}.$$
The following is \cite[Lemma 3.3]{Lins21}. This observation is certainly not original to \cite{Lins21}, but we don't know of another reference.
\begin{lemma} \label{lem:subgradient}
Let $X$ be a real Banach space and let $x, y \in X$ with $\|x\|=1$. Then there exists $x^* \in J(x)$ such that
\begin{equation} \label{eq:subgradient}
\lim_{t \rightarrow \infty} \|t x - y\| - \inner{t x - y, x^*} = 0.
\end{equation}
\end{lemma}
A real Banach space $X$ has the \emph{fixed point property} if for every closed, bounded, convex subset $C \subset X$ and any nonexpansive map $f:C \rightarrow C$, $f$ has a fixed point in $C$. All finite dimensional Banach spaces have the fixed point property, as do many infinite dimensional spaces.
For a real Banach space $X$, we will say that a map $f: X \rightarrow X$ is \emph{real analytic} if $t \mapsto \inner{f(x+ty),z^*}$ is a real analytic function for all $x,y \in X$, $z^* \in X^*$, and $t \in \mathbb{R}$.
\begin{theorem} \label{thm:uniqueFixed}
Let $X$ be a real Banach space with the fixed point property. Let $f:X \rightarrow X$ be nonexpansive and real analytic. If $f$ has more than one fixed point, then the set of fixed points of $f$ is unbounded.\end{theorem}
\begin{proof}
We can assume without loss of generality that $f(0) = 0$. Otherwise, if $x_0$ is any fixed point of $f$, then replace $f$ by the function $x \mapsto f(x+x_0)-x_0$. Now, suppose that $f$ also has a nonzero fixed point $w$. Let $v = w/\|w\|$.
For any $v^* \in J(v)$ and $0 < t < \|w\|$, observe that
$$\inner{f(tv),v^*} \le \|f(tv) - 0\| \le \|tv - 0\| = t$$
by nonexpansiveness. Also
\begin{align*}
\|w\|-\inner{f(tv),v^*} &= \inner{w-f(tv),v^*} \\
&\le \|w - f(tv)\| & (\text{since } \|v^*\|_* =1) \\
&\le \|w - tv\| & (\text{nonexpansiveness}) \\
&= \|w\| - t.
\end{align*}
Combining the two inequalities for $\inner{f(tv),v^*}$, we see that $\inner{f(tv),v^*} = t$ for all $0 < t < \|w\|$. Since $t \mapsto \inner{f(tv),v^*}-t$ is a real analytic function that is identically zero on the interval $0 \le t \le \|w\|$, it follows that
\begin{equation} \label{tv}
\inner{f(tv),v^*} = t
\end{equation}
for all $t \in \mathbb{R}$ and $v^* \in J(v)$.
Let $h(x) = \inf_{v^* \in J(v)} \inner{x,v^*}$ and
let $H_\alpha = \{x \in X : h(x) \ge \alpha\}$ for all $\alpha \in \mathbb{R}$. Each $H_\alpha$ is closed and convex since it is the intersection of the closed half-spaces $\{x \in X : \inner{x,v^*} \ge \alpha\}$ where $v^* \in J(v)$. The sets $H_\alpha$ are also nonempty since they contain $tv$ for all $t \ge \alpha$.
Choose any $x \in X$. Then
\begin{align*}
\sup_{v^* \in J(v)} \inner{tv- f(x),v^*} &= \sup_{v^* \in J(v)} \inner{f(tv)- f(x),v^*} & (\text{by } \eqref{tv}) \\
&\le \|f(tv) - f(x) \| & (\text{since } \|v^*\|_* = 1) \\
& \le \|tv-x\|. & (\text{nonexpansiveness})
\end{align*}
By Lemma \ref{lem:subgradient}, there is an $x^* \in J(v)$ such that for every $\epsilon > 0$,
$$\|tv-x\| \le \inner{tv-x,x^*} + \epsilon$$
for all $t >0$ sufficiently large. Therefore
$$\sup_{v^* \in J(v)} \inner{tv- f(x),v^*} \le \inner{tv - x,x^*} + \epsilon \le \sup_{v^* \in J(v)} \inner{tv-x,v^*} + \epsilon$$
when $t$ is large. Equivalently,
$$\inf_{v^* \in J(v)} \inner{x,v^*} \le \inf_{v^* \in J(v)} \inner{f(x),v^*} + \epsilon.$$
Since $\epsilon$ was arbitrary, we see that
$$h(x) = \inf_{v^* \in J(v)} \inner{x,v^*} \le \inf_{v^* \in J(v)} \inner{f(x),v^*} = h(f(x)).$$
It follows that each $H_\alpha$ is invariant under $f$.
Since $f$ is nonexpansive and $f(0)=0$, the closed ball $B_R := \{x \in X : \|x\| \le R \}$ is also invariant under $f$ for all $R > 0$. Therefore, the sets $H_\alpha \cap B_R$ are closed, convex, bounded, and invariant under $f$. They are also nonempty as long as $R \ge \alpha$. Since $f$ has the fixed point property, we conclude that $H_\alpha \cap B_R$ must contain a fixed point of $f$ for every $\alpha > 0$ and every $R$ large enough. Note that if $x \in H_\alpha$, then $\|x\| \ge \alpha$, so we see that $f$ has an unbounded set of fixed points.
\end{proof}
\begin{remark}
The function $h$ in the proof of Theorem \ref{thm:uniqueFixed} is an example of a horofunction, and the sets $H_\alpha$ are horoballs on $X$. Horofunctions have proven to be extremely useful tools for analyzing nonexpansive maps. See e.g., \cite{Beardon90, GaVi12, Karlsson01, LLN16, Lins07, Lins21}.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:uniqueEig}]
Suppose that $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving, homogeneous, and real analytic. If $f$ has a unique eigenvector in $\mathbb{R}^n_{>0}$ up to scaling, then $E(f)$ is nonempty and bounded in $(\mathbb{R}^n_{>0}, d_H)$. To prove the converse, note that $d_H$ is a metric on the set $\Sigma = \{x \in \mathbb{R}^n_{>0} : x_n = 1\}$ by Proposition \ref{prop:dH}. The function $\hat{f}(x) = f(x)/(f(x)_n)$ maps $\Sigma$ into $\Sigma$ and is $d_H$-nonexpansive. Every eigenvector $x \in \mathbb{R}^n_{>0}$ of $f$ corresponds to a fixed point $\hat{x} = x/x_n$ of $\hat{f}$ in $\Sigma$, and every fixed point of $\hat{f}$ is an eigenvector of $f$. This means that $\hat{f}$ has a bounded set of fixed points in $(\Sigma,d_H)$.
The entrywise natural logarithm $\log:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n$ is an isometry from $(\Sigma,d_H)$ onto the subspace $V = \{x \in \mathbb{R}^n : x_n = 0\}$ with the variation norm
\begin{equation} \label{varnorm}
\|x\|_\text{var} := \max_{i \in N} x_i - \min_{j \in N} x_j.
\end{equation}
The inverse of $\log$ is the entrywise natural exponential function $\exp:\mathbb{R}^n \rightarrow \mathbb{R}^n_{>0}$. The translated map $\exp \circ \hat{f} \circ \log$ is nonexpansive on $(V,\|\cdot\|_\text{var})$ and it is also a real analytic function with a bounded set of fixed points, so it must only have one fixed point $x \in V$ by Theorem \ref{thm:uniqueFixed}. Then $\exp(x)$ is the unique fixed point of $\hat{f}$ in $\Sigma$ and therefore all eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ are multiples of $\exp(x)$.
\end{proof}
\section{Convergence of iterates to a unique eigenvector} \label{sec:iterates}
In some applications, it is important to know whether the iterates $f^k(x)$ will converge (after normalizing) to a single vector for all $x \in \mathbb{R}^n_{>0}$. Of course, this can only happen if $f$ has a unique eigenvector in $\mathbb{R}^n_{>0}$ up to scaling. One known sufficient condition for the iterates to converge is if $f$ is differentiable at an eigenvector $u \in \mathbb{R}^n_{>0}$ and the derivative $f'(u)$ is primitive. In that case, $f^k(x)/\|f^k(x)\| \rightarrow u$ as $k \rightarrow \infty$ for all $x \in \mathbb{R}^n_{>0}$ \cite[Corollary 6.5.8]{LemmensNussbaum}. Here we suggest a somewhat more general sufficient condition. Recall that a strongly connected component of a directed graph is \emph{primitive} if there is an $m \in \mathbb{N}$ such that every ordered pair of vertices in the component can be connected by a path of length exactly $m$.
\begin{theorem} \label{thm:convergence}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be order-preserving and homogeneous. Suppose that $f$ has an eigenvector $u \in \mathbb{R}^n_{>0}$ with $\|u\|= 1$ and $f$ is differentiable at $u$. If $\mathcal{G}(f'(u))$ has a unique final class, then $u$ is unique, that is, all other eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ are scalar multiples of $u$. If, in addition, the final class is primitive, then $\lim_{k \rightarrow \infty} f^k(x)/\|f^k(x)\| = u$ for all $x \in \mathbb{R}^n_{>0}$.
\end{theorem}
The condition that $\mathcal{G}(f'(u))$ has a unique final class that is primitive is sufficient but not necessary for the normalized iterates of $f$ to converge to the unique positive unit eigenvalue $u$. We shall see in the proof, however, that this condition is necessary for $r(f)$ to be a simple eigenvalue of the derivative $f'(u)$ and for all other eigenvalues of $f'(u)$ to have absolute value strictly smaller than $r(f)$. This is an important consideration since it guarantees a linear rate of convergence for the iterates $f^k(x)/\|f^k(x)\|$.
To prove Theorem \ref{thm:convergence} we use the following linear algebra proposition. Here $A = \begin{bmatrix} a_{ij} \end{bmatrix}_{i,j \in N}$ will be an $n$-by-$n$ nonnegative matrix. We use $\rho(A)$ to denote the \emph{spectral radius} of $A$ which is the maximum of the absolute values of the eigenvalues of $A$. For a nonnegative $n$-by-$n$ matrix, the spectral radius $\rho(A)$ and the cone spectral radius $r(A)$ are the same (see e.g., \cite[Proposition 5.3.6 and Corollary 5.4.2]{LemmensNussbaum}). For any subset $J \subset N$, we let $A_J$ denote the principal submatrix $\begin{bmatrix}a_{ij} \end{bmatrix}_{i,j \in J}$. We don't claim that this proposition is original, but since we are unaware of a reference, we include a proof.
\begin{proposition} \label{prop:uniquefinal}
Let $A$ be an $n$-by-$n$ nonnegative matrix. Then $A$ has a unique (up to scaling) positive eigenvector $u \in \mathbb{R}^n_{>0}$ if and only if $\mathcal{G}(A)$ has a unique final class $C$ and $\rho(A_C) > \rho(A_{N \backslash C})$. If, in addition, the unique final class $C$ is primitive, then all other eigenvalues of $A$ have absolute value strictly less than $\rho(A)$.
\end{proposition}
\begin{proof}
In order for $A$ to have a positive eigenvector, or for $\mathcal{G}(A)$ to have a unique final class, $A$ must have at least one positive entry in each row. Then $A$ maps $\mathbb{R}^n_{>0}$ into itself and $A$ is multiplicatively convex by Lemma \ref{lem:prop61}. For any $J \subseteq N$, $r(A^J_0) = \rho(A_J)$ and $r(A^{N \backslash J}_0) = \rho(A_{N \backslash J})$. Therefore Theorem \ref{thm:convMain}\ref{item:convex} and Theorem \ref{thm:uniqueEig} imply that $A$ has a unique eigenvector in $\mathbb{R}^n_{>0}$ up to scaling if and only if $\mathcal{G}(A)$ has a unique final class $C$ and $\rho(A_{N \backslash C}) < \rho(A_C)$.
If the unique final class $C$ of $\mathcal{G}(A)$ is primitive, then $A_C$ is a primitive matrix, so $\rho(A) = \rho(A_C)$ is the only eigenvalue of $A_C$ with maximum modulus \cite[Theorem 8.5.3]{HornJohnson}. Since $\rho(A_{N \backslash C}) < \rho(A_C)$, it follows that $\rho(A)$ is also the only eigenvalue of $A$ with maximum modulus.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:convergence}]
We can assume without loss of generality that $r(f) =1$ since we can replace $f$ by $r(f)^{-1}f$. Then all multiples of $u$ are fixed points of $f$. Since $f$ is order-preserving, $A := f'(u)$ is a nonnegative matrix; and since $f$ is homogeneous, $u$ is an eigenvector of $A$ with eigenvalue $r(f) = 1$. Since $u$ has all positive entries, this means that $\rho(A) = 1$.
Let $C$ denote the unique final class of $\mathcal{G}(A)$. Note that $\rho(A_C) = r(A^C_0)$ and $\rho(A_{N \backslash C}) = r(A^{N \backslash C}_0)$ in the notation of Section \ref{sec:exist}. Since $A$ has an eigenvector in $\mathbb{R}^n_{>0}$, $A$ must be strongly nonnegative by Theorem \ref{thm:convMain} \ref{item:analyticConverse}. Therefore $\rho(A_C) > \rho(A_{N \backslash C})$. Then Proposition \ref{prop:uniquefinal} implies that $u$ is the unique eigenvector of $A$ corresponding to $\rho(A)$ up to scaling. By \cite[Corollary 6.4.7]{LemmensNussbaum}, $u$ is also the unique positive eigenvector of $f$ in $\mathbb{R}^n_{>0}$ with $\|u\| = 1$.
If $C$ is primitive, then Proposition \ref{prop:uniquefinal} also implies that all other eigenvalues of $A$ have absolute value strictly less than $\rho(A)$.
By the chain rule for Fr\'echet derivatives, $(f^k)'(u) = A^k$. Therefore $u$ is the unique positive eigenvector of $(f^k)'(u)$ with $\|u \|=1$ for all $k \in \mathbb{N}$. By \cite[Corollary 6.4.7]{LemmensNussbaum}, $u$ is also the unique positive eigenvector of $f^k$ in $\mathbb{R}^n_{>0}$ with $\|u\| = 1$ for all $k \in \mathbb{N}$.
For any $x \in \mathbb{R}^n_{>0}$, there are constants $\alpha, \beta > 0$ such that $\alpha u \le x \le \beta u$. Since $f$ is order-preserving, $\alpha u \le f^k(x) \le \beta u$ for all $k \in \mathbb{N}$. By \cite[Theorem 8.1.7]{LemmensNussbaum} it follows that there is a period $p \in \mathbb{N}$ and a point $y \in \mathbb{R}^n_{\ge 0}$ such that $\lim_{k \rightarrow \infty} f^{kp}(x) = y$ and $y$ is periodic with period $p$ under iteration by $f$. Furthermore, $\alpha u \le y \le \beta u$, so $y \in \mathbb{R}^n_{>0}$. But then $f^p(y) = y$, so $y$ must be a multiple of $u$ since $u$ is the only eigenvector of $f^p$ in $\mathbb{R}^n_{>0}$ up to scaling. From this we see that $f^k(x)/\|f^k(x)\|$ converges to $u$ as $k \rightarrow \infty$.
\end{proof}
Theorem \ref{thm:convergence} requires $f$ to be differentiable at the eigenvector $u \in \mathbb{R}^n_{>0}$. There are other conditions which can show that the normalized iterates of $f$ converge to an eigenvector when $f$ is not differentiable. See \cite[Section 6.5]{LemmensNussbaum} for details. Our next result shows that even when the iterates of $f$ do not converge to an eigenvector, the normalized iterates of $f+\mathrm{id}$ always do, as long as $f$ has an eigenvector in $\mathbb{R}^n_{>0}$. A similar result, \cite[Theorem 11]{GaSt20}, proves that the normalized iterates of the function $x \mapsto x^{1/2} f(x)^{1/2}$ converge to an eigenvector of $f$ when $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ is order-preserving, homogeneous, and has an eigenvector in $\mathbb{R}^n_{>0}$. The result in \cite{GaSt20} is a multiplicative version of Krasnoselskii-Mann iteration. The observation about $f+\mathrm{id}$ appears to be new and applies to any polyhedral cone.
\begin{theorem} \label{thm:fplusI}
Let $K$ be a polyhedral cone with nonempty interior in a finite dimensional real Banach space $X$ and let $\mathrm{id}$ denote the identity operator on $X$. Let $f: \operatorname{int} K \rightarrow \operatorname{int} K$ be order-preserving and homogeneous and let $g = f+\mathrm{id}$. If $f$ has an eigenvector in $\operatorname{int} K$, then $g^k(x)/\|g^k(x)\|$ converges to an eigenvector of $f$ as $k \rightarrow \infty$ for every $x \in \mathbb{R}^n_{>0}$.
\end{theorem}
Before proving Theorem \ref{thm:fplusI}, we briefly recall some facts about Hilbert's projective metric on cones other than the standard cone $\mathbb{R}^n_{\ge 0}$. Let $K$ be a closed cone with nonempty interior in a finite dimensional real Banach space $X$ and let $K^*$ denote the dual cone $\{ x^* \in X^* : x^*(x) \ge 0 \text{ for all } x \in K \}$. Then for any $x, y \in K \backslash \{0\}$,
$$d_H(x,y) = \sup_{\varphi, \psi \in K^* \backslash \{ 0 \}} \log \left( \frac{ \varphi(x) }{ \psi(x) } \, \frac{ \psi(y) }{ \varphi(y) } \right).$$
Note that if $x \in \operatorname{int} K$, then $\varphi(x) > 0$ for all $\varphi \in K^* \backslash \{0 \}$. See \cite[Section 2.1]{LemmensNussbaum} for more details.
\begin{proof}[Proof of Theorem \ref{thm:fplusI}]
Let $\hat{g}(x) = g(x)/\|g(x)\|$ for all $x \in \operatorname{int} K$. We will prove that $\hat{g}$ has no periodic orbits in $\operatorname{int} K$ with period $p > 1$. Since $f$ and therefore $g$ both have an eigenvector in $\operatorname{int} K$, it follows that the orbit $\hat{g}^k(x) = g^k(x)/\|g^k(x)\|$ is $d_H$-bounded. Then since $K$ is polyhedral, the iterates $\hat{g}^k(x)$ converge to a periodic orbit in $\operatorname{int} K$ as $k \rightarrow \infty$ by \cite[Corollary 4.2.5 and Proposition 2.2.3]{LemmensNussbaum}. So proving that there are no nontrivial periodic orbits will guarantee that $\hat{g}^k(x)$ converges to an eigenvector of $g$. And any eigenvector of $g$ is also an eigenvector of $f$.
Suppose by way of contradiction that $\hat{g}$ has a periodic orbit in $\operatorname{int} K$ with period $p > 1$. Chose some $y$ in this orbit. Let $y^0 = y$ and $y^{k+1} = g(y^k)$ for each $k \in \mathbb{N}$. Then $y^p$ is a scalar multiple of $y$. Also, $y^1 = g(y) = y+f(y)$. Then recursively,
\begin{equation} \label{recursive}
y^{k+1} = y^k+ f(y^k) = y + f(y^1) + \ldots + f(y^k)
\end{equation}
for all $k \in \mathbb{N}$.
Choose $m \in \{1, \ldots, p-1\}$ such that $d_H(y,y^m)$ is maximal. Let $M = d_H(y,y^m)$. Since $g$ is nonexpansive, this implies that $d_H(y^k,y^j) \le M$ for all $k, j \in \mathbb{N}$.
Since $d_H(y,y^m) = M$, there are linear functionals $\varphi, \psi$ in the dual cone $K^*$ such that
$$d_H(y,y^m) = \log \left( \frac{ \varphi(y^m) }{ \psi(y^m) } \, \frac{ \psi(y) }{ \varphi(y) } \right) = M.$$
Then
$$\frac{ \varphi(y^m) }{ \psi(y^m) } = e^M \frac{ \varphi(y) }{ \psi(y) }.$$
Let $a = \varphi(y)/\psi(y)$ so $\varphi(y^m)/\psi(y^m) = a e^M$.
For any $k \in \mathbb{N}$,
$$\log \left( \frac{\varphi(y^k)\psi(y)}{\psi(y^k) \varphi(y)} \right) \le d_H(y,y^k) \le M = \log \left( \frac{\varphi(y^m)\psi(y)}{\psi(y^m) \varphi(y)} \right)$$
which implies that
$$\frac{\varphi(y^k)}{\psi(y^k)} \le \frac{\varphi(y^m)}{\psi(y^m)} = a e^M.$$
Similarly,
$$\log \left( \frac{\varphi(y^m)\psi(y^k)}{\psi(y^m) \varphi(y^k)} \right) \le d_H(y^k,y^m) \le M = \log \left( \frac{\varphi(y^m)\psi(y)}{\psi(y^m) \varphi(y)} \right)$$
so
$$a = \frac{\varphi(y)}{\psi(y)} \le \frac{\varphi(y^k)}{\psi(y^k)}.$$
Therefore
$$a \le \frac{\varphi(y^k)}{\psi(y^k)} \le a e^M$$
for all $k \in \mathbb{N}$.
\color{black}
Let $b = \min \{ \varphi(f(y^k))/\psi(f(y^k)) : 0 \le k \le p-1\}$. Since $f$ is $d_H$-nonexpansive, $d_H(f(y^k),f(y^j)) \le M$ for all $k, j \in \mathbb{N}$. Therefore
$$b \le \frac{\varphi(f(y^k))}{\psi(f(y^k))} \le b e^M$$
for all $k \in \mathbb{N}$. Suppose that $b \le a$. Then
$$\varphi(f(y^k)) - a e^M \psi(f(y^k)) \le \varphi(f(y^k)) - b e^M \psi(f(y^k)) \le 0$$
for all $k \in \mathbb{N}$. At the same time, $\varphi(y) - a \psi(y) = 0$ so $\varphi(y) - a e^M \psi(y) < 0$. Combined with \eqref{recursive}, we must have
$\varphi(y^m) - a e^M \psi(y^m) < 0$, but that contradicts the fact that $\varphi(y^m)/\psi(y^m) = a e^M$.
Now suppose that $b > a$. Then
$$\varphi(f(y^k)) - a \psi(f(y^k)) > \varphi(f(y^k)) - b \psi(f(y^k)) \ge 0$$
for all $k \in \mathbb{N}$. Using \eqref{recursive} we have $\varphi(y^p) - a\psi(y^p) > 0$. But $y^p$ is a scalar multiple of $y$. So $\varphi(y^p)/\psi(y^p) = \varphi(y)/\psi(y) = a$ which is a contradiction. We conclude that $\hat{g}$ cannot have a nontrivial periodic orbit, so $\hat{g}^k(x)$ converges to an eigenvector of $f$ for every $x \in \operatorname{int} K$.
\end{proof}
\section{Applications and examples} \label{sec:examples}
\subsection{Functions in the class $\mathcal{M}$}
A large class of order-preserving, homogeneous, and real analytic functions $f:\mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ was introduced by Nussbaum in \cite{Nussbaum89}. The entries of these functions are built from various means of the entries of $x \in \mathbb{R}^n_{\ge 0}$. Let $r \in \mathbb{R} \backslash \{0\}$ and let $\sigma \in \mathbb{R}^n_{\ge 0}$ be a probability vector, that is, $\sum_{i \in N} \sigma_i = 1$. The \emph{$(r,\sigma)$-mean} of $x \in \mathbb{R}^n_{\ge 0}$ is
\begin{equation} \label{Mrsigma}
M_{r\sigma}(x) := \left( \sum_{i \in N} \sigma_i x_i^r \right)^{1/r}.
\end{equation}
When $r = 0$, we define
\begin{equation} \label{Mrsigma0}
M_{r\sigma}(x) := \prod_{i \in \operatorname{supp}(\sigma)} x_i^{\sigma_i}.
\end{equation}
Note that if $r = 1, 0$, or $-1$, then $M_{r \sigma}(x)$ corresponds to a weighted arithmetic, geometric, or harmonic mean, respectively. Let $x,y \in \mathbb{R}^n_{>0}$ and $0 < \theta < 1$. When $r > 0$, H\"older's inequality implies that
\begin{align*}
M_{r \sigma}(x^\theta y^{1-\theta}) &\le \left( \sum_{i \in N} (\sigma_i x_i^r)^\theta (\sigma_i y_i^r)^{1-\theta} \right)^{1/r} \\
&\le M_{r \sigma}(x)^{\theta} M_{r \sigma}(y)^{1-\theta}
\end{align*}
so $M_{r \sigma}$ is multiplicatively convex. When $r=0$, $ \log \circ M_{r \sigma} \circ \exp$ is linear so $M_{0 \sigma}$ is multiplicatively convex.
\color{black}
Following \cite{Nussbaum89} (see also \cite[Section 6.6]{LemmensNussbaum}), we say that a function $f:\mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{>0}$ is in \emph{class} $M$ if each entry of $f$ is a positive linear combination of $(r,\sigma)$-means. We also say that $f$ is in \emph{class} $M_+$ (alternatively, $M_-$) if $f$ is in class $M$ and each $(r, \sigma)$-mean in the formula for $f$ has $r \ge 0$ ($r < 0$). Then \emph{class} $\mathcal{M}$ is the smallest class of functions $f: \mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ that contains class $M$ and is closed under addition and composition. Likewise, \emph{class} $\mathcal{M}_+$ and $\mathcal{M}_-$ are the smallest classes containing $M_+$ and $M_-$, respectively, that are closed under addition and composition. Note that all functions in class $\mathcal{M}$ are real analytic on $\mathbb{R}^n_{>0}$.
By Lemma \ref{lem:prop61}, every $f \in \mathcal{M}_+$ is multiplicatively convex. Therefore we have the following result which combines Theorems \ref{thm:convMain} and \ref{thm:uniqueEig}.
\begin{theorem} \label{thm:Mplus}
Let $f: \mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ be in $\mathcal{M}_+$. Then $f$ has an eigenvector in $\mathbb{R}^n_{>0}$ if and only if $f$ is strongly nonnegative. If $f$ does have an eigenvector in $\mathbb{R}^n_{>0}$, then it is unique up to scaling if and only if $\mathcal{G}(f)$ has a unique final class $C$.
\end{theorem}
Functions in class $\mathcal{M}_-$ are not multiplicatively convex so we do not have such simple conditions for existence and uniqueness of eigenvectors there. However, the results of Section \ref{sec:exist} and Section \ref{sec:unique} still let us find necessary and sufficient conditions on the symbolic parameters of any function in $\mathcal{M}$ to have a unique entrywise positive eigenvector up to scaling. Our first detailed example is a function in $\mathcal{M}_-$ that comes from a population biology model.
\begin{example} \label{ex:Schoen}
The following order-preserving homogeneous function $f:\mathbb{R}^4_{>0} \rightarrow \mathbb{R}^4_{>0}$ was introduced by Schoen \cite{Schoen86} in a population model. The function was analyzed in detail in \cite[Section 3]{Nussbaum89}, and necessary and sufficient conditions were given there for the existence and uniqueness of an eigenvector in $\mathbb{R}^4_{>0}$. In what follows we will show how the methods described in this paper make the analysis easier and lead to the same conclusions. Let
$$f\left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \right) :=
\begin{bmatrix}
a_1 x_1 + b_1 \theta(x_1,x_2) + c_1 \theta(x_1,x_4) + d_1 \theta(x_2,x_3) \\
a_2 x_2 + b_2 \theta(x_1,x_2) + c_2 \theta(x_1,x_4) + d_2 \theta(x_2,x_3) \\
a_3 x_3 + b_3 \theta(x_3,x_4) + c_3 \theta(x_1,x_4) + d_3 \theta(x_2,x_3) \\
a_4 x_4 + b_4 \theta(x_3,x_4) + c_4 \theta(x_1,x_4) + d_4 \theta(x_2,x_3)
\end{bmatrix}$$
where $a_i, b_i, c_i, d_i$ are nonnegative constants, and $\theta(s,t) := (s^{-1}+t^{-1})^{-1}$. In the original model, the parameters $a_i$ were negative, but they can be made positive by adding a multiple of the identity to $f$ without affecting the existence of eigenvectors. Observe that $\theta(0,t) = 0$ and $\theta(\infty,t) = t$. We make the following additional assumptions about $f$:
\begin{itemize}
\item $a_i > 0$ for all $i$.
\item $d_1, c_2, c_3$ and $d_4$ are all positive.
\item $a_1 < a_2+b_2$, $a_2<a_1+b_1$, $a_3<a_4+b_4$, and $a_4<a_3+b_3$.
\end{itemize}
With these assumptions, we can find the hypergraphs $\mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(f)$ associated with $f$. Since each $a_i >0$, the hypergraph $\mathcal{H}^-_0(f)$ has no hyperarcs. Figure \ref{fig:Schoen} shows the minimal hyperarcs of $\mathcal{H}^+_\infty(f)$. The only invariant sets in $\mathcal{H}^+_\infty(f)$ are the singleton sets $\{1\}, \{2\}, \{3\}, \{4\}$, and the sets $\{1,2\}$, $\{1,3\}$, $\{2,4\}$, and $\{3,4\}$.
All other nonempty $J \subsetneq N$ have $\operatorname{reach}(J,\mathcal{H}^+_\infty(f)) = N$, so we do not need to check \eqref{superIllum} for those. It turns out that we won't need to check \eqref{superIllum} for all eight of the invariant sets either. If we can verify \eqref{superIllum} for each of the four invariant doubleton sets, then we will see that Theorem \ref{thm:quick} shows that \eqref{superIllum} also holds for $J = \{1\}, \{2\}, \{3\},$ and $\{4\}$, and thus $f$ has an eigenvector in $\mathbb{R}^4_{>0}$.
\begin{figure}[ht] \label{fig:Schoen}
\begin{center}
\begin{tikzpicture}
\path (1,0) node[draw,shape=circle,scale=0.75] (A1) {1};
\path (2,0) node[draw,shape=circle,scale=0.75] (A4) {4};
\path (-2,0) node[draw,shape=circle,scale=0.75] (A2) {2};
\path (-1,0) node[draw,shape=circle,scale=0.75] (A3) {3};
\draw[thick,->] (A1) to [bend right=42] (A2);
\draw[thick,->] (A4) to [bend right=45] (A2);
\draw[thick,->] (A1) to [bend right=42] (A3);
\draw[thick,->] (A4) to [bend right=45] (A3);
\draw[thick,->] (A2) to [bend right=45] (A1);
\draw[thick,->] (A3) to [bend right=42] (A1);
\draw[thick,->] (A2) to [bend right=45] (A4);
\draw[thick,->] (A3) to [bend right=42] (A4);
\fill[left color=gray, right color=white,opacity=0.2] (A1) to [bend right=42] (A2) to [bend left=45] (A4) -- cycle;
\fill[left color=gray, right color=white,opacity=0.2] (A1) to [bend right=42] (A3) to [bend left=45] (A4) -- cycle;
\fill[left color=white, right color=gray,opacity=0.2] (A2) to [bend right=45] (A1) to [bend left=42] (A3) -- cycle;
\fill[left color=white, right color=gray,opacity=0.2] (A2) to [bend right=45] (A4) to [bend left=42] (A3) -- cycle;
\end{tikzpicture}
\end{center}
\caption{The hypergraph $\mathcal{H}^+_\infty(f)$ for the maps in Example \ref{ex:Schoen}. }
\end{figure}
Below we write out \eqref{superIllum} with explicit formulas for $f^J_0$ and $f^{N\backslash J}_\infty$ for each of the four invariant doubleton sets. We will see that \eqref{superIllum} is always satisfied for two of them, and the remaining two amount to a necessary and sufficient condition on the parameters $a_i, b_i, c_i,$ and $d_i$ for the eigenspace $E(f)$ to be nonempty and bounded in $(\mathbb{R}^4_{>0},d_H)$. Since $f$ is differentiable on $\mathbb{R}^4_{>0}$ and the derivative is always irreducible, it follows from Remark \ref{rem:irred} that any eigenvector of $f$ in $\mathbb{R}^4_{>0}$ must be unique up to scaling. This means that the conditions below are necessary and sufficient for $f$ to have \emph{any} eigenvectors in $\mathbb{R}^4_{>0}$.
\begin{description}
\item[Case 1] $J = \{1,2\}$:
$$
r \left(
\begin{bmatrix}
a_1 x_1 + b_1 \theta(x_1,x_2)\\
a_2 x_2 + b_2 \theta(x_1,x_2)\\
0 \\
0
\end{bmatrix} \right) <
\lambda \left(
\begin{bmatrix}
\infty \\
\infty \\
a_3 x_3 + b_3 \theta(x_3,x_4)+c_3 x_4 + d_3 x_3 \\
a_4 x_4 + b_4 \theta(x_3,x_4)+c_4 x_4 + d_4 x_3
\end{bmatrix} \right).
$$
By applying Theorem \ref{thm:super} to the map
$$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \mapsto
\begin{bmatrix}
a_1 x_1 + b_1 \theta(x_1,x_2)\\
a_2 x_2 + b_2 \theta(x_1,x_2)\\
\end{bmatrix}$$
we see that is has a nonempty and bounded eigenspace in $(\mathbb{R}^2_{>0},d_H)$ if and only if $a_1 < a_2+b_2$ and $a_2 < a_1 + b_1$.
Since we have assumed that those inequalities are true, we can use Theorem \ref{thm:quick} to conclude that \eqref{superIllum} holds for $J = \{1\}$ and $J = \{2\}$ if it holds for $J = \{1,2\}$. Furthermore Theorem \ref{thm:super} can also be used to show that $f^{N \backslash J}_\infty$ has an eigenvector in $\mathbb{R}^{\{3,4\}}_{>0}$. This implies that $\lambda(f^{N \backslash J}_\infty)$ is equal to the cone spectral radius of the function
$$\begin{bmatrix} x_3 \\ x_4 \end{bmatrix} \mapsto
\begin{bmatrix}
a_3 x_3 + b_3 \theta(x_1,x_2)+c_3 x_4 + d_3 x_3\\
a_4 x_4 + b_4 \theta(x_1,x_2)+c_4 x_4 + d_4 x_3\\
\end{bmatrix}.$$
It is not hard to calculate the spectral radii of these two functions. Since they are both defined on $\mathbb{R}^2_{>0}$, one can explicitly solve for an eigenvector of the form $\begin{bmatrix} \tau \\ 1-\tau \end{bmatrix}$ where $0 < \tau <1$. Once the eigenvector is known, then the corresponding eigenvalue can be computed directly. This is done in \cite[Lemma 3.9]{Nussbaum89}.
\item[Case 2] $J = \{1,3\}$:
$$
r \left(
\begin{bmatrix}
a_1 x_1 \\
0 \\
a_3 x_3 \\
0 \\
\end{bmatrix} \right) < \lambda \left(
\begin{bmatrix}
\infty \\
a_2 x_2 + b_2 x_2 + c_2 x_4 + d_2 x_2 \\
\infty \\
a_4 x_4 + b_4 x_4 + c_4 x_4 + d_4 x_2
\end{bmatrix} \right).
$$
In this case, both $r(f^J_0)$ and $\lambda(f^{N \backslash J}_\infty)$ are eigenvalues of linear maps which correspond to the 2-by-2 nonnegative matrices
$$\begin{bmatrix} a_1 & 0 \\ 0 & a_3 \end{bmatrix} \text{ and } \begin{bmatrix} a_2+b_2+d_2 & c_2 \\ d_4 & a_4+b_4+c_4 \end{bmatrix}$$
respectively. Since we have assumed that $c_2, d_4 > 0$, the later matrix is irreducible and so has a unique positive eigenvector (up to scaling) with eigenvalue equal to $\lambda(f^{N \backslash J}_\infty)$. It is easy to see that $r(f^J_0) = \max \{a_1,a_3\}$. Since we assumed that $a_1 < a_2+b_2$ and $a_3 < a_4+b_4$, we see that $r(f^J_0) < \lambda(f^{N \backslash J}_\infty)$ is always true in this case.
\item[Case 3] $J = \{2,4\}$:
$$r\left(
\begin{bmatrix}
0 \\
a_2 x_2 \\
0 \\
a_4 x_4 \\
\end{bmatrix} \right) <
\lambda \left(
\begin{bmatrix}
a_1 x_1 + b_1 x_1 + c_1 x_1 + d_1 x_3 \\
\infty \\
a_3 x_3 + b_3 x_3 + c_3 x_1 + d_3 x_3 \\
\infty \\
\end{bmatrix} \right).$$
Similar to Case 2, this condition is automatically true since $a_2 < a_1+b_1$ and $a_4 < a_3+b_3$.
\item[Case 4] $J = \{3,4\}$:
$$r\left(
\begin{bmatrix}
0 \\
0 \\
a_3 x_3 + b_3 \theta(x_3,x_4) \\
a_4 x_4 + b_4 \theta(x_3,x_4)
\end{bmatrix} \right) <
\lambda \left(
\begin{bmatrix}
a_1 x_1 + b_1 \theta(x_1,x_2) + c_1 x_1 + d_1 x_2 \\
a_2 x_2 + b_2 \theta(x_1,x_2) + c_2 x_1 + d_2 x_2 \\
\infty \\
\infty \\
\end{bmatrix} \right).$$
Similar to Case 1, the function $f^J_0$ has a nonempty and $d_H$-bounded eigenspace in $\mathbb{R}^J_{>0}$ since $a_3 < a_4+b_4$ and $a_4 < a_3+b_3$. By Theorem \ref{thm:quick}, if we verify the inequality above for $J = \{3,4\}$, then we will also verify \eqref{superIllum} for $\{3\}$ and $\{4\}$ as well. The function $f^{N\backslash J}_\infty$ also has an eigenvector in $\mathbb{R}^{N \backslash J}_{>0}$. So it is possible to explicitly compute both $r(f^{\{3,4\}}_0)$ and $\lambda(f^{\{1,2\}}_\infty)$ in order to check whether or not $r(f^{\{3,4\}}_0) < \lambda(f^{\{1,2\}}_\infty)$.
\end{description}
Together the conditions $r(f^{\{1,2\}}_0) < \lambda(f^{\{3,4\}}_\infty)$ and $r(f^{\{3,4\}}_0) < \lambda(f^{\{1,2\}}_\infty)$ from Case 1 and Case 4 are necessary and sufficient for $f$ to have an eigenvector in $\mathbb{R}^4_{>0}$. Note that these two inequalities are equivalent to the conditions given in \cite[Theorem 3.9]{Nussbaum89}. The method used to find these conditions in \cite{Nussbaum89} involves finding all eigenvectors of $f$ corresponding to $r(f)$ on the boundary of $\mathbb{R}^4_{\ge 0}$ and then checking a condition on a Gateaux derivative at each one. Our method is simpler and always results in a set of necessary and sufficient conditions for any order-preserving homogeneous function $f:\mathbb{R}^n_{>0} \rightarrow \mathbb{R}^n_{>0}$ to have a nonempty and $d_H$-bounded eigenspace.
\end{example}
\subsection{Nonnegative tensors} \label{sec:nonnegtensor}
An \emph{order-$d$ tensor} is an array of real numbers $\mathcal{A} = [\![ a_{j_1\cdots j_d} ]\!] \in \mathbb{R}^{n_1 \times \cdots \times n_d}$. Here we focus tensors with constant dimension $n_1 = \ldots = n_d = n$. The nonnegative eigenvalue problem for tensors seeks to find an eigenvalue $\lambda \in \mathbb{R}$ and an eigenvector $x \in \mathbb{R}^n$ such that
$$\mathcal{A}x^{(d-1)} = \lambda x^{[d-1]}$$
where $x^{[d-1]}:= (x_1^{d-1}, \ldots, x_n^{d-1})$ and
$$(\mathcal{A}x^{(d-1)})_i := \sum_{1 \le j_2, \ldots, j_d \le n} a_{i j_2 \cdots j_d} x_{j_2} \cdots x_{j_n}.$$
This problem was independently introduced by Lim \cite{Lim05} and Qi \cite{Qi05}. It is referred to by Qi as the H-eigenproblem to distinguish it from the Z-eigenproblem which seeks eigenpairs $(\lambda, x) \in \mathbb{R} \times \mathbb{R}^n$ such that
$$\mathcal{A}x^{(d-1)} = \lambda x.$$
If the entries of $\mathcal{A}$ are all nonnegative, then we can rephrase the H-eigenproblem by letting
$$f(x)_i = ((\mathcal{A}x^{(d-1)})_i)^{1/(d-1)} \text{ for } i \in N.$$
Then $f: \mathbb{R}^n_{\ge 0} \rightarrow \mathbb{R}^n_{\ge 0}$ is order-preserving and homogeneous. As long as there is a positive entry $a_{i j_2 \cdots j_d}$ in $\mathcal{A}$ for every $i \in N$, then $f$ maps $\mathbb{R}^n_{>0}$ into $\mathbb{R}^n_{>0}$ and we can ask whether or not $f$ has an eigenvector $x \in \mathbb{R}^n_{>0}$. If there is, then the eigenpair $(r(f)^{d-1},x)$ is a solution to the H-eigenproblem. This problem and related generalizations have been studied in the context of general nonlinear Perron-Frobenius theory before, see e.g., \cite{AkGaHo20,FrGaHa13,GaTuHe19}.
In \cite[Theorem 5]{HuQi16}, Hu and Qi give necessary and sufficient conditions for an order-$d$ nonnegative tensor $\mathcal{A} \in \mathbb{R}^{n \times \cdots \times n}$ to have an H-eigenvector in $\mathbb{R}^n_{>0}$. Their condition does not require or imply that the eigenspace is bounded in Hilbert's projective metric. Here we observe that the order-preserving homogeneous function $f$ associated with a nonnegative tensor $\mathcal{A}$ is in the class $\mathcal{M}_+$. Therefore Theorem \ref{thm:Mplus} gives necessary and sufficient conditions for $f$ to have a unique eigenvector in $\mathbb{R}^n_{>0}$, up to scaling.
\begin{example} \label{ex:tensor}
The following example corresponds to an order-3 nonnegative tensor in $\mathbb{R}^{4 \times 4 \times 4}$. It is a slight variation of \cite[Example 5.4]{AkGaHo20}. Unlike that example, the hypergraph condition of Theorem \ref{thm:AGH} fails for this map. It turns out that the existence of an eigenvector in $\mathbb{R}^n_{>0}$ will depend on the values of the parameters of the function $f$, and Theorem \ref{thm:Mplus} lets us give precise necessary and sufficient conditions for this map to have a unique (up to scaling) eigenvector in $\mathbb{R}^n_{>0}$.
$$f\left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \right) :=
\begin{bmatrix}
\sqrt{a_1 x_1 x_2 + a_2 x_2^2} \\
\sqrt{b_1 x_1^2 + b_2 x_1 x_2 + b_3 x_2^2} \\
\sqrt{c_1 x_1^2 + c_2 x_1 x_2 + c_3 x_2 x_3} \\
\sqrt{d_1 x_1 x_4 + d_2 x_3^2 + d_3 x_4^2} \\
\end{bmatrix}.$$
\begin{figure}[ht] \label{fig:Hpm2}
\begin{center}
\begin{tikzpicture}
\begin{scope}
\draw (-1.75,0.5) node {$\mathcal{H}^-_0(f)$};
\path (45:1) node[draw,shape=circle,scale=0.75] (A1) {1};
\path (135:1) node[draw,shape=circle,scale=0.75] (A2) {2};
\path (225:1) node[draw,shape=circle,scale=0.75] (A3) {3};
\path (315:1) node[draw,shape=circle,scale=0.75] (A4) {4};
\draw[thick,<-] (A1) to [bend right=30] (A2);
\draw[thick,->] (A1) to [bend right=20] (A3);
\draw[thick,->] (A2) to [bend left=30] (A3);
\fill[top color=white, bottom color=black,opacity=0.5] (A1) to [bend right=20] (A3) to [bend right=30] (A2);
\end{scope}
\begin{scope}[xshift=5cm]
\draw (-1.75,0.5) node {$\mathcal{H}^+_\infty(f)$};
\path (45:1) node[draw,shape=circle,scale=0.75] (B1) {1};
\path (135:1) node[draw,shape=circle,scale=0.75] (B2) {2};
\path (225:1) node[draw,shape=circle,scale=0.75] (B3) {3};
\path (315:1) node[draw,shape=circle,scale=0.75] (B4) {4};
\draw[thick,->] (B1) to [bend right=20] (B2);
\draw[thick,->] (B1) to [bend left=0] (B3);
\draw[thick,->] (B1) to [bend left=0] (B4);
\draw[thick,->] (B2) to [bend right=20] (B1);
\draw[thick,->] (B2) to [bend right=0] (B3);
\draw[thick,->] (B3) to [bend right=0] (B4);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The hypergraphs $\mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(f)$ for the maps in Example \ref{ex:tensor}. }
\end{figure}
The minimal hyperarcs of $\mathcal{H}^+_\infty(f)$ and $\mathcal{H}^-_0(f)$ are shown in Figure \ref{fig:Hpm2}.
The only invariant subsets of $\mathcal{H}^+_\infty(f)$ are $J = \{3,4\}$ and $J = \{4\}$. Observe that $\operatorname{reach}(\{4\}^c,\mathcal{H}^-_0(f)) = \{1,2,3\}$ and $\operatorname{reach}(\{3,4\}^c,\mathcal{H}^-_0(f)) = \{1,2,3\}$, so both possible choices of $J$ fail the conditions in Corollary \ref{cor:AGH2}.
\begin{figure}[ht] \label{fig:G}
\begin{center}
\begin{tikzpicture}
\draw (-1.85,0.5) node {$\mathcal{G}(f)$};
\path (45:1) node[draw,shape=circle,scale=0.75] (B1) {1};
\path (135:1) node[draw,shape=circle,scale=0.75] (B2) {2};
\path (225:1) node[draw,shape=circle,scale=0.75] (B3) {3};
\path (315:1) node[draw,shape=circle,scale=0.75] (B4) {4};
\draw[thick,->] (B1) to [bend right=20] (B2);
\draw[thick,->] (B3) to [bend left=0] (B1);
\draw[thick,->] (B4) to [bend left=0] (B1);
\draw[thick,->] (B2) to [bend right=20] (B1);
\draw[thick,->] (B2.135) arc(45:315:2.5mm);
\draw[thick,->] (B3.135) arc(45:315:2.5mm);
\draw[thick,->] (B3) to [bend right=0] (B2);
\draw[thick,->] (B4) to [bend right=0] (B3);
\draw[thick,->] (B4.45) arc(135:-135:2.5mm);
\draw[thick,->] (B1.45) arc(135:-135:2.5mm);
\end{tikzpicture}
\end{center}
\caption{The directed graph $\mathcal{G}(f)$ in Example \ref{ex:tensor}. }
\end{figure}
Since $f \in \mathcal{M}_+$, Theorem \ref{thm:Mplus} applies to $f$. The directed graph $\mathcal{G}(f)$ is shown in Figure \ref{fig:G}. Observe that $\mathcal{G}(f)$ has a unique final class $C = \{1,2\}$. Therefore $f$ has an eigenvector in $\mathbb{R}^n_{>0}$ (which is necessarily unique up to scaling) if and only if $f$ is strongly nonnegative, which by Lemma \ref{lem:classCap} is equivalent to $r(f^{\{3,4\}}_0) < r(f^{\{1,2\}}_0)$.
Writing this condition down, we have
$$r \left(
\begin{bmatrix}
0 \\
0 \\
0 \\
\sqrt{d_2 x_3^2 + d_3 x_4^2} \\
\end{bmatrix} \right) < r \left(
\begin{bmatrix}
\sqrt{a_1 x_1 x_2 + a_2 x_2^2} \\
\sqrt{b_1 x_1^2 + b_2 x_1 x_2 + b_3 x_2^2} \\
0 \\
0 \\
\end{bmatrix}\right).$$
It is easy to see that $r(f^{\{3,4\}}_0) = \sqrt{d_3}$, so the necessary and sufficient condition for $f$ to have an (unique) eigenvector in $\mathbb{R}^4_{>0}$ can be stated as follows
$$\sqrt{d_3}< r \left(
\begin{bmatrix}
\sqrt{a_1 x_1 x_2 + a_2 x_2^2} \\
\sqrt{b_1 x_1^2 + b_2 x_1 x_2 + b_3 x_2^2} \\
\end{bmatrix}\right).$$
\end{example}
\subsection{Topical functions and stochastic games}
The methods of this paper also apply to \emph{topical functions} which are order-preserving and additively homogeneous functions $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$. A function is \emph{additively homogeneous} if $T(x+c e_N} %\newcommand{\one}{\mathbf{1}) = T(x) + c e_N} %\newcommand{\one}{\mathbf{1}$ for all $x \in \mathbb{R}^n$ and $c \in \mathbb{R}$. For a topical function $T$, we say that $x \in \mathbb{R}^n$ is an \emph{additive eigenvector} with \emph{eigenvalue} $\lambda \in \mathbb{R}$ if $T(x) = x+\lambda e_N} %\newcommand{\one}{\mathbf{1}$. If $T$ is topical, then the function $f = \exp \circ T \circ \log$ is order-preserving and homogeneous on $\mathbb{R}^n_{>0}$. In fact, the functions $\exp$ and $\log$ are order-preserving isometries between $(\mathbb{R}^n_{>0},d_H)$ and $(\mathbb{R}^n,\|\cdot\|_\text{var})$, were $\|\cdot\|_\text{var}$ is the variation norm defined in \eqref{varnorm}. Therefore $T$ is nonexpansive with respect to the variation norm.
Furthermore, $T$ has an additive eigenvector $x \in \mathbb{R}^n$ if and only if $\exp(x)$ is an eigenvector of $f$ in $\mathbb{R}^n_{>0}$. Thus all of the existence and uniqueness results for eigenvectors of order-preserving homogeneous functions on $\mathbb{R}^n_{>0}$ can be translated to corresponding results about additive eigenvectors for topical functions on $\mathbb{R}^n$. In some applications, it is more natural to work in the additive setting.
Working with topical functions requires some adjustments to the notation and terminology of the previous sections.
Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be topical. We define the upper and lower Collatz-Wielandt numbers of a topical function to be
$$r(T) := \inf_{x \in \mathbb{R}^n} \max_{i \in N} T(x)_i - x_i$$
and
$$\lambda(T) := \sup_{x \in \mathbb{R}^n} \min_{i \in N} T(x)_i - x_i$$
respectively. There are iterative formulas analogous to \eqref{rlim} for both
$$r(T) = \lim_{k \rightarrow \infty} \max_{i \in N} \frac{T^k(x)_i}{k}$$
and
$$\lambda(T) = \lim_{k \rightarrow \infty} \min_{i \in N} \frac{T^k(x)_i}{k},$$
and both formulas hold for all $x \in \mathbb{R}^n$.
As with order-preserving homogeneous functions on $\mathbb{R}^n_{>0}$, topical functions extend continuously to $(-\infty,\infty]^n$ and to $[-\infty, \infty)^n$ \cite[Theorem 1]{BuSp00}, and the extensions are order-preserving and additively homogeneous \cite[Lemmas 6 \& 7]{BuSp00}. We can extend the definition of the upper Collatz-Wielandt number $r(T)$ to order-preserving, additively homogeneous $T:[-\infty, \infty)^n \rightarrow [-\infty,\infty)^n$ and likewise, we can extend the definition of $\lambda(T)$ to order-preserving additively homogeneous $T:(-\infty, \infty]^n \rightarrow (-\infty, \infty]^n$. For any subset $J \subset N$, we define $T^J_{-\infty} = P^J_{-\infty} T P^J_{-\infty}$ and $T^J_{\infty} = P^J_{\infty} T P^J_{\infty}$. With this notation, we have the following restatement of Theorem \ref{thm:super}.
\begin{theorem} \label{thm:topical}
Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be topical. The set of additive eigenvectors of $T$ is nonempty and bounded in the variation norm if and only if
\begin{equation} \label{topicalCond}
r(T^J_{-\infty}) < \lambda(T^{N \backslash J}_{\infty})
\end{equation}
for every nonempty proper subset $J \subset N$.
\end{theorem}
Following \cite{AkGaHo20}, we define two hypergraphs $\mathcal{H}^-_\infty(T)$ and $\mathcal{H}^+_\infty(T)$ for any topical function $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ as follows. The nodes of both hypergraphs are the set $N$ and
the hyperarcs of $\mathcal{H}^-_\infty(T)$ (respectively, $\mathcal{H}^+_\infty(T)$) are pairs $(I,\{j\})$ such that $I \subset N$, $j \in N \backslash I$, and
$$\lim_{t \rightarrow -\infty} T(te_I)_j = -\infty \hspace*{1cm} (\text{resp.}, \lim_{t \rightarrow \infty} T(te_I)_j = \infty).$$
Note that $\mathcal{H}^-_\infty(T) = \mathcal{H}^-_0(f)$ and $\mathcal{H}^+_\infty(T) = \mathcal{H}^+_\infty(f)$ when $f = \exp \circ T \circ \log$. Therefore each of the results in Subsection \ref{sec:hypergraph} can be translated directly to the additive setting. In particular, we have the following corollary of Lemma \ref{lem:connect}.
\begin{lemma} \label{lem:connect2}
Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be topical and let $J$ be a proper nonempty subset of $N$. If $\operatorname{reach}(J^c,\mathcal{H}^-_\infty(T)) = N$, then $r(T^J_{-\infty}) = -\infty$. If $\operatorname{reach}(J,\mathcal{H}^+_\infty(T)) = N$, then $r(T^{N \backslash J}_\infty) = \infty$. In either case, \eqref{topicalCond} holds for $J$.
\end{lemma}
Our next application is one where it is more natural to work in an additive setting.
A (finite) zero-sum \emph{stochastic game} consists of the following:
\begin{enumerate}
\item A finite state space, which we will assume is $N$,
\item two finite sets of actions $A$, $B$ (one for each player),
\item a payoff function $r:N \times A \times B \rightarrow \mathbb{R}$, and
\item a transition function $\rho$ which assigns a probability vector in $\mathbb{R}^n$ to each $(i,a,b) \in N \times A \times B$.
\end{enumerate}
At each stage of the game, Player A pays Player B a payoff $r(i,a,b)$ which depends on the state $i$ and the choices $a \in A$ and $b \in B$ of the two players during that stage. Then the state of the game transitions from $i$ to $j$ with probability equal to $\rho(i,a,b)_j$ for the next stage of the game. Naturally, player A wishes the payoffs at each stage to be as small as possible, while player B seeks the opposite.
The \emph{Shapley operator} for a stochastic game is a topical function $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by
$$T(x)_i = \inf_{a \in A} \sup_{b \in B} \left( r(i,a,b) + \sum_{j = 1}^n x_j \rho(i,a,b)_j \right).$$
These functions were introduced in \cite{Shapley53} to find the expected average payoff of a repeated stochastic game over several stages.
Given a stochastic game in state $i$, the Shapley operator lets us recursively compute the expected total payoff from player A to player B over $k$ stages if both players are following their optimal strategies. The expected total payoff in that case is $T^k(0)_i$. If $T$ has an additive eigenvector, then the long run average payoff
$$\lim_{k \rightarrow \infty} \frac{T^k(0)_i}{k}$$
does not depend on the initial state $i$, and is equal to $r(T)$.
\begin{example} \label{ex:game}
Consider a simple two-player game with three states. Suppose that player B controls what happens in state 1, while player A controls states 2 and 3. In state 1, player B can choose to stay in state 1, in which case the payoff (from player A to B) is $r_1$. Or player B can choose a different payoff $r_2$, but then the state will either remain in state 1 with probability $p_1$, or transition to state 2 with probability $1-p_1$ for the next stage. In state 2, player A chooses between payoffs $r_3$ and $r_4$. Choosing $r_3$ keeps the game in state 2, while $r_4$ sends the game to state 1 with probability $p_2$ or to state 3 with probability $1-p_2$. Finally, in state 3, player A can choose between staying in state 3 for a payoff of $r_5$, or transitioning the game to state 1 for a payoff of $r_6$. Figure \ref{fig:game} shows the states and transition choices facing the players.
\begin{figure}[ht] \label{fig:game}
\begin{center}
\begin{tikzpicture}
\path (-4,2) node[draw,shape=rectangle,scale=1] (r1) {$r_1$};
\path (-2,2) node[draw,shape=circle,scale=1] (s1) {1};
\path (0,2) node[draw,shape=rectangle,scale=1] (r2) {$r_2$};
\path (2,2) node[draw,shape=circle,scale=1] (s2) {2};
\path (4,2) node[draw,shape=rectangle,scale=1] (r3) {$r_3$};
\path (-2,0) node[draw,shape=rectangle,scale=1] (r6) {$r_6$};
\path (0,-2) node[draw,shape=rectangle,scale=1] (r5) {$r_5$};
\path (2,0) node[draw,shape=rectangle,scale=1] (r4) {$r_4$};
\path (0,0) node[draw,shape=circle,scale=1] (s3) {3};
\draw[thick,->] (s1) to (r1);
\draw[thick,->] (s1) to (r2);
\draw[thick,->] (s2) to (r3);
\draw[thick,->] (s2) to (r4);
\draw[thick,->] (s3) to (r5);
\draw[thick,->] (s3) to (r6);
\draw[thick,black,->] (r1) to [bend right=40] (s1);
\draw[thick,black,->] (r2) to [bend right=40] (s1);
\draw[thick,black,->] (r2) to [bend left=40] (s2);
\draw[thick,black,->] (r3) to [bend left=40] (s2);
\draw[thick,black,->] (r4) to (s3);
\draw[thick,black,->] (r4) to (s1);
\draw[thick,black,->] (r5) to [bend left=40] (s3);
\draw[thick,black,->] (r6) to (s1);
\draw (-1,2.5) node[above] {$p_1$};
\draw (1,2.5) node[above] {$1-p_1$};
\draw (0,1) node[above right] {$p_2$};
\draw (1,0) node[below] {$1-p_2$};
\end{tikzpicture}
\end{center}
\caption{The game board for Example \ref{ex:game}.}
\end{figure}
The Shapley operator for this game is
$$T(x) := \begin{bmatrix} (r_1 + x_1) \vee (r_2 + p_1 x_1 + (1-p_1) x_2) \\
(r_3 + x_2) \wedge (r_4+p_2 x_1 + (1-p_2) x_3) \\
(r_5+ x_3) \wedge (r_6 + x_1)
\end{bmatrix}$$
where $\vee$ is the max operation and $\wedge$ denotes min. The minimal hyperarcs of the hypergraphs $\mathcal{H}^+_\infty(T)$ and $\mathcal{H}^-_\infty(T)$ are shown in Figure \ref{fig:HT}.
\begin{figure}[ht] \label{fig:HT}
\begin{center}
\begin{tikzpicture}
\begin{scope}
\draw (-1.85,0.5) node {$\mathcal{H}^-_\infty(T)$};
\path (150:1) node[draw,shape=circle,scale=0.75] (A1) {1};
\path (30:1) node[draw,shape=circle,scale=0.75] (A2) {2};
\path (-90:1) node[draw,shape=circle,scale=0.75] (A3) {3};
\draw[thick,->] (A1) to (A2);
\draw[thick,->] (A1) to (A3);
\draw[thick,->] (A3) to (A2);
\end{scope}
\begin{scope}[xshift=5cm]
\draw (-1.85,0.5) node {$\mathcal{H}^+_\infty(T)$};
\path (150:1) node[draw,shape=circle,scale=0.75] (B1) {1};
\path (30:1) node[draw,shape=circle,scale=0.75] (B2) {2};
\path (-90:1) node[draw,shape=circle,scale=0.75] (B3) {3};
\draw[thick,->] (B2) to (B1);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The hypergraphs $\mathcal{H}^-_\infty(T)$ and $\mathcal{H}^+_\infty(T)$ for the Shapley operator in Example \ref{ex:game}. }
\end{figure}
Every proper subset $J \subset N = \{1,2,3\}$ either has $\operatorname{reach}(J^c,\mathcal{H}^-_\infty(T)) = N$ or $\operatorname{reach}(J,\mathcal{H}^+_\infty(T)) = N$, except $J = \{1,3\}$ and $J = \{1\}$. Therefore, Theorem \ref{thm:topical} and Lemma \ref{lem:connect2} imply that $T$ has a nonempty and bounded set of additive eigenvectors in $(\mathbb{R}^n,\|\cdot\|_\text{var})$ if and only if the following two inequalities hold.
\begin{description}
\item[Case 1] $J = \{1\}$
$$r_1 = r\left( \begin{bmatrix}
r_1 + x_1 \\
-\infty \\
-\infty
\end{bmatrix} \right) < \lambda \left( \begin{bmatrix}
\infty \\
r_3 + x_2 \\
r_5 + x_3
\end{bmatrix} \right) = \min \{r_3, r_5\}.$$
\item[Case 2] $J = \{1,3\}$
$$r_1 = r\left( \begin{bmatrix}
r_1 + x_1 \\
-\infty \\
(r_5+x_3) \wedge (r_6+x_1)
\end{bmatrix} \right) < \lambda \left( \begin{bmatrix}
\infty \\
r_3 + x_2 \\
\infty
\end{bmatrix} \right) = r_3.$$
\end{description}
From these two cases, we see that $r_1 < \min \{r_3, r_5\}$ is necessary and sufficient for the set of additive eigenvectors of $T$ to be nonempty and bounded in the variation norm.
\end{example}
\subsection*{Acknowledgement} The author wishes to thank Roger Nussbaum and St\'ephane Gaubert for their careful reading and helpful comments. Thanks also to Bas Lemmens for his encouragement and suggestions.
|
2,877,628,089,530 | arxiv | \section{Introduction}
\label{sec:intro}
Transportation networks
\cite{albert2002review,newman2003review,newman_book2006,dorogovtsev_book2006,cohen_book2010,barrat_book2012},
like computer networks, railway systems, water pipelines
or energy grids, are ubi\-quitous in highly technological
societies. Since the well functioning of these societies depends heavily
on transportation networks,
large-scale (cascading) failure are in particular threatening.
Previous work on cascading failures in networks have often
analyzed the occurrence of past failures \cite{chen2005,bao2009b} or
studied phase transitions,
as a function of some network parameter, from a resilient
to a failure phase \cite{deArcangelis1985,crucitti2004,dobson2007,bao2009}.
In this work we are concerned always with the
case of a design in such a way that a failure is prevented, i.e.,
resilient networks,
the typical task of an engineer. In particular, we are interested
on how to make a network fail-safe against the failure of one link
by including enough, but
as small as possible, \emph{backup capacity}, such that a cascading failure
is prevented (called ``$N-1$ criterion'' for power transmission).
To gain a fundamental understanding of the
problem, no real-world networks are studied here. Instead,
this study is performed for three different network ensembles, two of
them are highly relevant for transport processes in spatial settings,
another simple model is included for comparison.
Here, the behavior over the range of (almost) each \emph{complete}
ensemble is
addressed, this means in particular the properties of typical as well as
extremely resilient and extremely weak networks are investigated. Namely, the
distribution of backup capacities is obtained for almost the complete
support, for backup capacities which appear about $10^{-80}$ less likely than
in typical networks. This requires to apply
special but simple numerical \emph{importance-sampling techniques},
as explained in section \ref{sec:method}, to obtain the
\emph{large-deviation} properties of the networks.
For many problems in science and in statistics, the large-deviation
properties play an important role \cite{denHollander2000,dembo2010}.
Only for few cases analytical results can be obtained.
Thus, most problems have to be studied by numerical simulations
\cite{practical_guide2009}, in particular by Monte Carlo (MC) techniques
\cite{newman1999,landau2000}. Classically, MC simulations have
been applied to ensembles of random systems in the following way:
For a finite number of independently drawn instances from the ensemble,
arbitrary properties of these instances have been
calculated using MC simulations.
Later, it has been noticed that MC simulations can be
used via introducing an artificial
``temperature'' to
sample the random ensemble such that the
large-deviation properties of the (almost) full ensemble
can be obtained \cite{align2002}. The results are
re-weighted in a way that the results for
the original quenched ensemble are obtained. In this way, the
large-deviation properties of the distribution of alignment
scores for protein comparison were studied
\cite{align2002,align_long2007,newberg2008},
which is of importance to calculate
the significance of results of protein-data-base queries \cite{durbin2006}.
Motivated by these results, similar approaches have been applied to other
problems like the distribution of the number of components of
Erd\H{o}s-R\'enyi (ER) random networks \cite{rare-graphs2004},
the size of the largest components of ER random networks and of
two-dimensional grids \cite{largest-2011},
the partition function of Potts models \cite{partition2005},
the distribution of ground-state energies of spin glasses
\cite{pe_sk2006} and of directed polymers in random media \cite{monthus2006},
the distribution of Lee–-Yang zeros for spin glasses \cite{matsuda2008},
the distribution of success probabilities of error-correcting codes
\cite{iba2008}, the distribution of free energies of RNA secondary
structures \cite{rnaFreeDistr2010}, and some large-deviation properties
of random matrices \cite{driscoll2007,saito2010}.
To the knowledge of the author, no corresponding study has been
performed to obtain the large-deviation properties of transport networks,
in particular of failure-resilient networks.
Here, the large-deviation approach is applied to a simple yet often used
transport model on three standard random network ensembles.
Thus, this work serves in particular as a proof of principle that
measuring large-deviation properties of transport networks
is possible and allows one to obtain useful insight. This
shows that similar approaches
should be applicable to more complex transport networks, e.g., dynamic networks
of oscillators as used to study energy grids \cite{filatrella2008,rohden2012}.
The remainder
of the paper is organized as follows. First, in Sec.\ \ref{sec:ensembles},
the different network ensembles
under scrutiny are presented. Then, the backup capacity is introduced,
which is used to describe how resilient a network is. In Sec.\
\ref{sec:tests} a couple of test simulations are presented,
which explore the concept of the backup capacity. In the
following section, the large-deviation approach is presented.
Within the main Sec.\ \ref{sec:results}, all results are given. The paper
is closed by a summary and a discussion.
\section{Network Ensembles}
\label{sec:ensembles}
This work is about the resilience of network ensembles, since such
ensembles are used often in theoretical studies about various network
properties. This type of approach is different from the question
how, e.g., the most resilient network for a given set of nodes and real-space
positions
looks like. For such a setup the notion of an ensemble makes less sense
and is thus not covered in this work. Also no existing networks are studied
empirically here since that would be beyond the scope of the work.
Nevertheless, the present approach allows to gain insights
about the behavior of typical and atypical network instances, leading to
general design principles for resilient networks.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.3\textwidth]{sample_erdoes_renyi.eps}
\includegraphics[clip,width=0.3\textwidth]{sample_small_world.eps}
\includegraphics[clip,width=0.3\textwidth]{sample_spatial.eps}
\caption{
\label{fig:samples}
Three different ensembles are treated. For each ensemble,
a sample network with $N=100$ nodes is shown:
Erd\H{o}s-R\'enyi random networks (top, with randomly placed
nodes), small-world networks (middle)
and two-dimensional spatial networks (bottom).
}
\end{figure}
The most simple type of random networks
is the Erd\H{os}-R\'enyi (ER)
network ensemble \cite{erdoes1960}. It makes no assumption about the
topology of the network, i.e., in particular it exhibits no spacial structure,
see Fig.\ \ref{fig:samples} (top). Thus, it
is ideally suited for comparison with more
complex network ensembles as well as single network instances
to see what effect the structures of these more complex
networks have regarding their behavior.
To create an
ER random network, one starts with an empty network of $N$ nodes.
For each pair $i,j$ of nodes, with some given probability $p_{ij}^{\rm ER}$
the link $\{i,j\}$ is added to the network. Here,
\begin{equation}
p_{ij}^{\rm ER}=c/(N-1)
\end{equation}
is chosen such that, on average, each node
has $c$ connections. Nevertheless, all possible networks have a nonzero
(although sometimes quite small) probability, even the complete (fully connected)
as well as the empty network.
Since ER random networks have a minimum
amount of structure,
they often serve as a suitable null model
when comparing to other ensembles of random networks.
Next, a widely studied model of networks is considered
here, the \emph{small-world} (SW) ensemble
\cite{watts1998,amaral2000,barrat2001}. This ensemble was found, e.g., to represent
the U.S. power grid well \cite{watts1998,amaral2000,motter2002}
and was used for modelling other transport networks as well \cite{bao2009}.
For this model, in a first step
$N$ nodes are distributed on a ring and connected
with their direct and second-nearest neighbors.
Thus, each node has four links.
Next, for each existing
link, with probability $p$ (here $p=0.1$) it is disconnected at one terminal
node and reconnected with a completely randomly chosen node, hence
keeping the average number of links per node.
Thus, for $p\to 1$ the SW networks
become more similar to a modified ER
ensemble where the average number of neighbors is $c=4$ and where the
actual number of links does not fluctuate. For $p$ small, this results
in a mixture of many local short-range and few long-range links,
see Fig.\ \ref{fig:samples} (middle), which is a key characteristics of
many existing real-world
networks. Note that for easy comparison between different network models,
$c=4$ was chosen here for the ER ensemble, such that all network ensembles
have the same average number of neighbors.
Since many existing transport networks are embedded on a two-dimensional
(earth) surface, also a \emph{spatial} (two-dimensional)
model \cite{barthelemy2011}
is included in the present study, which exhibits even more
spatial structure than the SW ensemble. Here, $N$ nodes are distributed
randomly with uniform weight in a $[0,1]^2$-plane. Afterward,
for each pair $i,j$ of nodes, with probability
\begin{equation}
p_{ij}^{\rm sp}=f\cdot (1+\sqrt{N\pi}d_{ij}/\alpha)^{-\alpha}
\end{equation}
the link $\{i,j\}$ is added to the network, where $d_{ij}$ is the
Euclidean distance between nodes $i,j$.
A sample network is shown in Fig.\ \ref{fig:samples} (bottom).
Here, values $f=0.95$ and $\alpha=3$
are chosen, which results also in an average number of neighbors
close to $4$. Note that several variants
of spatial networks exist in the literature.
Although the model appears to be in particular useful
for surface-embedded transportation networks,
it is so far less established than the SW model,
so the many results for the SW
model are included here as well.
\section{Resilience}
\label{sec:resilience}
The quantity to describe the resilience of a network against
a failure leading to cascading failures is based on a rather simple
(i.e., not time-dependent) yet
established quantity to measure loads in transport networks
\cite{motter2002,motter2004,bao2009}.
The loads are given by the assumption that for each pair $i,j$
of nodes, a unit one of some quantity has to be transported
between $i,j$. This requires the network to be connected, i.e., to consist of a
single connected component. For the above random ensembles it means that they
are restricted to the subset of connected network instances.
For the SW model and the spatial model
basically all network instances are connected using the given parameters,
while for the ER model typically only a small fraction
of networks is connected (37\% for $N=50$, 15\% for $N=100$, 2.5\% for $N=200$
and 0.005\% for $N=400$). Note that the sampling used here, see
Sec.\ \ref{sec:method}, ensures that only connected networks are sampled.
For the transport between any pair $i,j$ of nodes the shortest path is chosen
(if several shortest paths exist, the transportation
is divided equally among the different paths). This is performed
for all pairs of nodes, which are connected in the network. The actual
load
$l_{i,j}$ for an link $e=\{i,j\}$ is the total number of (possibly sums
of fractional) shortest
paths which run through the link. Hence, the load is equal to the
well known \emph{edge-betweenness}, which can be calculated
conveniently using a fast algorithm \cite{newman2001shortest-path}.
Now, the \emph{backup capacity} $c_{\rm b}$ is defined. For this purpose,
the link
$ e_{\max}={\rm argmax}_{\{i,j\}}\, l_{i,j}$
exhibiting the highest load is removed from
the network. Next, all loads are recalculated, resulting in load
values $\{\tilde l_{i,j}\}$
of the modified network. The \emph{backup capacity} is defined as
the highest increment in the load, i.e.,
\begin{equation}
c_{\rm b} = \max_{\{i,j\}} (\tilde l_{i,j} - l_{i,j})\,.
\end{equation}
If the network is disconnected by the removal of $ e_{\max}$,
$c_{\rm b}=\infty$ is chosen, i.e., such networks are actually
disregarded as well.
The (or one) link which exhibits the maxi\-mum increase in load is
denoted here by $e_{\rm incr-max}$.
Note that in some link the load will actually decrease, but that
is no problem for the definition.
Hence, the backup capacity represents a rough and rather safe
estimate of how much the capacities have to be chosen above
the actual load values to make the transportation network resilient against
the failure of one link.
\section{Test simulations}
\label{sec:tests}
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{resilience_full.eps}
\caption{Scatter plot of the full backup capacity
$c_{\rm b}^{\rm full}$, which is optimized over the removal
of every link, one at a time, as a function of
actually used backup capacity $c_{\rm b}$ (where only the link with the
largest load is removed), for small-world networks with $N=100$ and $p=0.1$.
\label{fig:resilience_full}
}
\end{figure}
To verify whether concentrating on the removal of the link with the
largest load (leading to backup capacity $c_{\rm b}$)
is justified, simulations were performed such that
for each network the necessary backup capacity was also maximized over
the single-link removal of \emph{all links},
resulting in a maxi\-mum backup
capacity $c_{\rm b}^{\rm full}$.
Clearly, $c_{\rm b}^{\rm full}\ge c_{\rm b}$ holds for any network.
In Fig.\ \ref{fig:resilience_full},
results for small-world networks with $N=100$ nodes
and rewiring probability $p=0.1$ are displayed,
the results for other network ensembles look
simi\-lar. As visible, there
is a strong correlation between the full backup capacity
$c_{\rm b}^{\rm full}$ and the actually used backup capacity $c_{\rm b}$.
A linear correlation coefficient $0.85$ was found.
In fact, for about 1/3 of all networks, both are exactly equal
and for more than 90\% $c_{\rm b}^{\rm full}\le 2 c_{\rm b}$ holds.
Hence, due to the strong correlation, when optimizing the network topology
with respect $c_{\rm b}$ one will also obtain very efficient networks
with respect to $c_{\rm b}^{\rm full}$.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{cascade.eps}
\caption{Probability $P_{\rm cascade}$ of a cascade which disconnects
the network as a function of the fraction $f_{\rm b}$ of the backup
capacity added to the link capacities.
For small-world networks with different sizes $N=20$, $N=50$,
$N=100$ and $p=0.1$.
\label{fig:cascade}
}
\end{figure}
Thus, for best efficiency, for the main simulations
the above quantity $c_{\rm b}$ was evaluated, i.e.
only one link was removed and the
load recalculated. Nevertheless, in principle one is interested
in global events, i.e., in \emph{cascading failures} \cite{crucitti2004}.
The basic assumption is
that when the backup capa\-city is not sufficient, a cascading failure
will be triggered frequently. This assumption was also checked explicitly
in this work for the sample setup of SW networks ($p=0.1$): After
the calculation of the backup capacity $c_{\rm b}$ the capacity of every
link was increased by the amount $f_b c_{\rm b}$, where $f_b$ is
a factor in the range $[0,2.5]$, except for the link $e_{\rm incr-max}$
which exhibits the
largest load increase after the initial removal. For this link
the capacity is increased only by $c_{\rm p}-\epsilon$, where $\epsilon>0$.
Thus, this link will for sure fail
next with the recalculated load distribution since the
capacity is not sufficient. Thus it has to be
removed as well and the load distribution has to be recalculated again. Now
even more links may fail. This process is repeated, until no more links fail,
i.e., the network is able to redistribute the load, or until
a cascade of failures result in the
network breaking apart, i.e. a complete breakdown.
In Fig.\ \ref{fig:cascade}, the proba\-bili\-ty
$P_{\rm cascade}$ of a cascading failure leading to a network breakdown
is shown as a function of $f_b$. Clearly, for $f_b<1$ almost
all failures trigger a network breakdown.
On the other hand, when increasing $f_b$ cascading failures
become less likely. Hence, it is justified to calculate just the
backup capacity $c_{\rm b}$, involving only two calculations
of the load distribution, to learn about the resilience of the
network against an event leading to a cascading failure.
\section{Simulation and reweighting method}
\label{sec:method}
To determine the distribution $P(c_{\rm b})$ for any measurable quantity $c_{\rm b}$,
here denoting the backup capacity of a network, \emph{simple sampling} is
straightforward: One generates a certain number $K$ of network samples
and obtains $c_{\rm b}(G)$ for each sample $G$.
This means each network $G$ will appear with its natural ensemble
probability $Q(G)$.
The probabili\-ty to measure a value of $c_{\rm b}$ is given by
\begin{equation}
P(c_{\rm b}) = \sum_{G} Q(G)\delta_{c_{\rm b}(G),c_{\rm b}} \label{eq:PS}
\end{equation}
Therefore, by calculating a histogram of the values for $c_{\rm b}$, a good
estimation for $P(c_{\rm b})$ is obtained.
Nevertheless, $P(c_{\rm b})$ can only be measured in a regime where $P(c_{\rm b})$
is relatively large, about $P(c_{\rm b})>1/K$. Unfortunately, the distribution
decreases for many systems exponentially fast in the system size $N$ when moving
away from its typical (peak) value.
This means, even for moderate system
sizes $N$, the distribution will be unknown on almost its complete support.
To estimate $P(c_{\rm b})$ for a much larger range, even possibly on the
full support of $P(c_{\rm b})$, where probabilities smaller
than $10^{-10}$ may appear,
a different approach is used \cite{align2002}.
For self-containedness, the method is outlined subsequently.
The basic idea is to use an additional
Boltzmann factor $\exp(-c_{\rm b}(G)/T)$, $T$ being a ``temperature''
parameter, in the following manner:
A Markov-chain MC simulation \cite{newman1999,landau2000}
is performed, where in each step
$t$ from the current network $G(t)$ a candidate network $G^*$ is created:
A node $i$ of the current network is selected randomly,
with uniform weight $1/N$,
and all adjacent links are deleted. For all feasible links $\{i,j\}$, the link
is added with a weight corresponding to the natural weight $Q(G)$, e.g.,
with probability $c/(N-1)$ for ER random networks. For SW and spatial
networks it is done correspondingly, see Sec.\ \ref{sec:ensembles}.
Next, it is checked whether network $G^*$ is connected, i.e., consists of one single
connected component, because only for a connected network the backup capa\-city
is defined. If $G^*$ is not connected, it is rejected, i.e. $G(t+1)=G(t)$.
Note that it has to be made sure that the initial network is connected. This
was achieved by gene\-rating candidates for the initial network until a connected
instance was found.
If the candidate network is connected,
the backup capacity $c_{\rm b}(G^*)$ is calculated. Finally,
the candidate network is \emph{accepted}, ($G(t+1)=G^*$)
with the Metropolis proba\-bility
\begin{equation}
p_{\rm Met} = \min\left\{1,e^{-[c_{\rm b}(G^*)-c_{\rm b}(G(t))]/T}\right\}\,.
\end{equation}
Otherwise the candidate network is also rejected ($G(t+1)=G(t)$). By construction,
the algorithm fulfills detailed balance. Clearly the algorithm is also
ergodic, since within $N$ steps, each possible network may be constructed, in
principle. Thus,
in the limit of infinite long Markov chains,
the distribution of networks will follow the probability
\begin{equation}
q_T(G) = \frac{1}{Z(T)} Q(G)e^{-c_{\rm b}(G)/T}\,, \label{eq:qT}
\end{equation}
where $Z(T)$ is the a priori unknown normalization factor.
The distribution for $c_{\rm b}$ at temperature $T$ is given by
\begin{eqnarray}
P_T(c_{\rm b}) & = &\sum_{G} q_T(G) \delta_{c_{\rm b}(G),c_{\rm b}} \nonumber\\
& \stackrel{(\ref{eq:qT})}{=} &
\frac{1}{Z(T)}\sum_{G} Q(G)e^{-c_{\rm b}(G)/T} \delta_{c_{\rm b}(G),c_{\rm b}} \nonumber \\
& = & \frac{e^{-c_{\rm b}/T}}{Z(T)} \sum_{G} Q(G) \delta_{c_{\rm b}(G),c_{\rm b}}
\nonumber \\
& \stackrel{(\ref{eq:PS})}{=} &
\frac{e^{-c_{\rm b}/T}}{Z(T)} P(c_{\rm b}) \nonumber\\
\Rightarrow \quad P(c_{\rm b}) & = & e^{c_{\rm b}/T} Z(T) P_T(c_{\rm b}) \label{eq:rescaling}
\end{eqnarray}
Hence, the target distribution $P(c_{\rm b})$ can be estimated, up to a normalization
constant $Z(T)$, from sampling at finite temperature $T$. For each
temperature, a specific range of the distribution $P(c_{\rm b})$ will be sampled:
Using a positive temperature allows to sample the region of
a distribution left to its peak (values
smaller than the typical value),
while negative temperatures are used to access the right tail.
Temperatures of large absolute value will cause a sampling of the
distribution close to its typical value, while temperatures of small
absolute value
are used to access the tails of the distribution. Hence, by choosing a
suitable set of temperatures, $P(c_{\rm b})$ can be measured over a large
range, possibly on its full support.
The normalization constants $Z(T)$ can easily be obtained by including
a histogram obtained from simple sampling, which corresponds
to temperature $T=\pm\infty$, which means $Z\approx 1$
(within numerical accuracy). Using suitably chosen
temperatures $T_{+1}$, $T_{-1}$, one measures histograms which overlap
with the simple sampling histogram on its left and right border,
respectively. Then the corresponding normalization
constants $Z(T_{\pm 1})$ can be obtained by the requirement that
after rescaling the histograms according to
(\ref{eq:rescaling}), they must agree
in the overlapping regions with the simple sampling histogram within
error bars. This means, the histograms are ``glued'' together,
similar to the multi-histogram approach of Ferrenberg and Swendsen
\cite{ferrenberg1989}. In the same
manner, the range of covered
$c_{\rm b}$ values can be extended iteratively to the left and to
the right by choosing additional suitable temperatures
$T_{\pm 2}, T_{\pm 3}, \ldots$ and gluing the
resulting histograms one to the other. A pedagogical explanation
and examples of
this procedure can be found in Ref.\ \cite{align_book}.
In order to obtain the correct result, the MC simulations must be
equilibrated. The equilibration of the simulation can be monitored by starting
with two different initial networks, respectively:
\begin{itemize}
\item First an unbiased
random network is taken, which means that the measure of interest is
close to its typical value.
\item Second, one uses a very atypical network, e.g., a fully connected
network.
\end{itemize}
In any case,
for the two different initial conditions,
the evolution of $c_{\rm b}(G(t))$ will approach from two different
extremes, which allows for a simple equilibration test:
equilibration is achieved if
the measured values of $c_{\rm b}$ agree within the range of fluctuations.
For the simulations performed in this work,
equilibration was achieved always within 200 Monte Carlo sweeps (i.e., $200*N$
Monte Carlo steps).
\section{Results}
\label{sec:results}
Simulations where performed for ER, SW and spatial networks. For
each type, several number of nodes were considered,
to study finite-size effects. The evaluation of the backup
capacity is rather involved, compared, e.g., to past large-deviation
studies of the largest component of networks \cite{largest-2011}.
Thus, the largest networks
under scrutiny here exhibit $N=400$ nodes. Nevertheless, many
existing transportation networks are of similar size.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{distr_sw.eps}
\caption{Probability distribution $P(c_{\rm b})$ for the backup capacity
$c_{\rm b}$,
for small-world networks ($p=0.1$) with different sizes $N=50$,
$100$, $200$ and $400$. Standard error bars are at most of
order of symbol size. The line shows a fit of the tail ($c_{\rm b}\ge 20000$)
of the $N=400$ data
to an exponential ($a\exp({-c_{\rm b}/\tilde c})$, $a=2.4(1)\times 10^{-5}$,
$\tilde c=1022(1)$). The inset enlarges the region near
$c_{\rm b}=0$, the line being a guide to the eyes only.
\label{fig:distr}
}
\end{figure}
Figure \ref{fig:distr} shows the distribution of the backup capacity for
almost the full support for the ensemble of SW networks. Note that
probabilities as small as $10^{-22}$ are easily obtained which are
clearly out of reach using conventional simulation techniques.
Typical, very reliable and very unreliable networks are accessible
using the large-deviation approach.
Typical networks, near the peak of the distributions, exhibit a rather
small backup capacity. Very unreliable networks, where the backup capacity has
to be large to prevent cascading failures, are very rare and located
in the tails of the distributions to the right. The tails of the distribution
follow exponentials very well, as a fit to $a\exp({-c_{\rm b}/\tilde c})$
to the tail of the data for the largest networks revealed,
resulting in $a=2.4(1)\times 10^{-5}$,
$\tilde c=1022(1)$. Below a more detailed analysis via an extrapolation of
the large-deviation rate function is given, supporting that the limiting
distribution is exponential.
An inspection of the networks in the far tail of the distribution
showed that the most unreliable networks have a quite
special structure. They consist of a small core of connected nodes,
e.g., a triangle of nodes in the simplest case,
see Fig.\ \ref{fig:worst_case}. All remaining ($N-3$)
nodes are connected, directly or indirectly, to one of two of these
core nodes, roughly partitioned equally into two sets, i.e. about $N/2$
per set of nodes.
This means, a large number of $(N/2)^2$ shortest-path connections
runs from one set through a single link of the core to
the other set. This single link exhibits the highest load,
while the other core links are not used much. By removing this
extreme-load link,
the load is redistributed completely in the core, hence, increased
by an amount of $\sim N^2/4$. Clearly, such networks are very
rare by chance and rather special.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.39\textwidth]{worst_case.eps}
\caption{Network with highest backup capacity. Two large $O(N)$
subnetworks are connected trough a small core network (a triangle, here).
The links
with the highest loads are shown in bold. Removing the highest load
from the core part results in an increase of the load through the
other core links by an amount $N^2/4$.
\label{fig:worst_case}
}
\end{figure}
On the other hand, there are also networks which have even a
much more resilient structure than typical
networks, since they require only a rather small backup capacity. They
are located near the origin of the distribution and are also rather
rare (probability $< 10^{-9}$ for the largest case considered here). Below
we will identify some structural network properties that make
it very resilient.
The inset of Fig.\ \ref{fig:distr} shows also that
with increasing network size, the typical backup capacity, i.e.,
the location of the peak, grows.
A more detailed study, also invol\-ving larger sizes which were studied by
simple sampling, exhibits that the growth is linear (not shown here).
For comparison with the most simple network model, Erd\H{os}-R\'enyi (ER)
random networks \cite{erdoes1960}, the corresponding results are shown
in Fig.\ \ref{fig:distr:er}. The peak of the distribution again
moves (linearly) to the right but is located relatively much
further left compared to the SW case.
Again an exponential fits the data well, now for almost over the full support.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{distr_er.eps}
\caption{Probability distribution $P(c_{\rm b})$ for the backup capacity
$c_{\rm b}$,
for Erd\H{os}-R\'enyi networks ($c=4$) with different sizes $N=50$,
$100$, $200$ and $400$. Standard error bars are at most of
order of symbol size. The line shows a fit over almost the full support
($c_{\rm b}\ge 5000$) of the $N=400$ data
to an exponential ($a\exp({-c_{\rm b}/\tilde c})$, $a=0.015(2)$,
$\tilde c=223(1)$). The inset enlarges the region near
$c_{\rm b}=0$, lines being guides to the eyes only.
\label{fig:distr:er}
}
\end{figure}
For comparison with the realistic spatial network model,
the corresponding results are shown
in Fig.\ \ref{fig:distr:spatial}. Typically, these networks are more resilient
(smaller value of $c_{\rm b}$) than the SW networks, but less resilient
than the ER model. Again,
most of the distribution can be well fitted by an exponential.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{distr_spatial.eps}
\caption{Probability distribution $P(c_{\rm b})$ for the
backup capacity, for spatial random networks
($\alpha=3$, $f=0.95$) with different sizes $N=50$, $N=100$, $N=200$
and $N=400$. Standard error bars are at most of order of symbol size.
The line shows a fit over most of the support
($c_{\rm b}\ge 10000$) of the
$N=400$ data
to an exponential ($a\exp({-c_{\rm b}/\tilde c})$, $a=2.7(4)\times 10^{-5}$,
$\tilde c=392(1)$). The inset enlarges the region near $c_{\rm b}=0$,
lines being guides to the eyes only.
\label{fig:distr:spatial}
}
\end{figure}
Comparing the insets of Fig.\ \ref{fig:distr}, \ref{fig:distr:er} and
\ref{fig:distr:spatial}, one observes that both ER random networks as well
spatial random networks typically require a lower backup capacity
compared to the SW model.
Typical values of the backup capacity $c_{\rm b}$ for the ER model are
located at small values ($c_{\rm b}\approx 40$ for $N=400$). The
spatial networks
are typically almost as resilient ($c_{\rm b}\approx 100$ for $N=400$).
For the SW ensemble, typical networks need much larger backup capacity
($c_{\rm b}\approx 500$ for $N=400$).
Correspondingly, large values of $c_{\rm b}$ are very unlikely for the
ER ensemble (a density of
$10^{-80}$ at $c_{\rm b}=40000$ for $N=400$) and quite unlikely for
the spatial networks (a density of $10^{-50}$).
For the SW model the density of $10^{-22}$ at rightmost tail is relatively
larger. These quantitative differences in the large-deviation behavior
are also reflected by the constants obtained by fitting an exponential
to the tail of the distribution. The drop of the tail is strongest for the
ER model, followed by the spatial ensemble and finally by the SW networks.
Thus, the SW ensemble relatively favors less resilient networks compared
to the other two ensembles.
For the ER model this behavior is no
big surprise
because the ER model does not exhibit any spatial structure,
allowing for arbitrary network topologies in particular many
long-range links (equivalent to
a high dimension of the system) which lead to a strong resilience.
In particular the network where all pairs of nodes are directly
connected (complete network) is contained in the ensemble, which
is not possible for the SW ensemble because the average number of neighbours
is fixed to 4.
On the other hand, the SW and the spatial model are both embedded
in a low-dimensional structure, which might indicate that both should need
larger backup capacities than the ER ensemble. Nevertheless, the
spatial ensemble seems to be more similar to the ER networks with respect
to the resilience. First, one should note that for the spatial networks
in principle a complete network is possible (but even more unlikely
than for ER random networks)
in contrast to the SW ensemble. Second, the SW model exhibits indeed
some long-range links, which can be used to decrease to overall load, hence
the backup capacity. Nevertheless, by accident few of the long-range links
will be very suitable for many of the shortest paths, acquiring
much of the load, while many other long-range links carry only a small load.
This ``channeling effect'' leads to a rather large backup capacity,
thus a large load has to be rerouted after the failure.
Opposite to this, for the spatial model, due to the
distribution of the nodes in a two-dimensional plane, the total
all-to-all traffic is distributed more over different paths,
leading to a more uniform distribution
of the load in the plane, in turn requiring less backup capacity.
Thus,
using a true two-dimensional model, like the one applied here, appears from
the present results the most meaningful approach within this field.
This type of networks exhibit on the one hand spatial structure, as needed
for most real-world applications, one the other hand, the resilience is typically,
and also optimally, rather large. This is opposed to the SW model, which
is often used to model power grids
\cite{watts1998,amaral2000,motter2002}, but exhibits rather low resilience
and is not embedded in a two-dimensional plane.
Clearly, by increasing the
fraction $p$ of randomized links, the SW model can be made more resilient,
but that will render it much more similar to the ER model (without fluctuating
number of links), but less
finite-dimensional, i.e., less realistic.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{rate_fct_sw.eps}
\caption{Rate function $\phi$ for the rescaled backup capacity
$r=c_{\rm b}/(N^2/4)$,
for small-world networks ($p=0.1$) with different sizes $N=50$,
$N=100$, $N=200$, $N=400$ and for the extrapolation $N\to\infty$.
The line represents a power law fit to the extrapolated values.
The
inset shows a sample extrapolation (using a power law plus
constant) for $r=0.4$.
\label{fig:rate_fct_sw}
}
\end{figure}
Next, some information about the limiting distribution for large network
sizes is obtained using the so-called \emph{rate function}
\begin{equation}
\phi = - \frac 1 N \log P(c_{\rm b})
\end{equation}
which is a standard quantity in large-deviation theory
\cite{denHollander2000,touchette2009}. It displays the leading behavior under
the assumption that away from the typical instances, the distribution
decays exponentially fast in the system size, i.e., $P(c_{\rm b})\sim e^{-N\phi}$.
In Fig.\ \ref{fig:rate_fct_sw} the rate function
as a function of the rescaled backup capacity
$r=c_{\rm b}/(N^2/4)$ is shown for SW networks. This rescaling to
$r$, motivated by the above finding about the most unresilient networks,
ensures that the
maximum of the support for the rate function is close to $r=1$.
This allows for a comparison
and extrapolation of the results for different sizes $N$.
Just from looking at the data, the rate function seems to approach a limiting
shape for larger networks.
To make this statement quantitatively precise, an extrapolation
to $N\to\infty$ was performed in the following way: For selected
fixed values of $r$, the rate function $\phi$ was considered
as a function of the system size $N$ and fitted to a power
law $\phi_r(N)=\phi_r^{\infty}+b_r N^{-c_r}$, which is a typical
finite-size behavior found in statistical mechanics models. The inset
of Fig.\ \ref{fig:rate_fct_sw} shows the SW data and the resulting
fit for the case $r=0.4$ (with $\phi_{0.4}^{\infty}=0.041(7)$,
$b_{0.4}=1.9(6)$ and $c_{0.4}=0.72(9)$). The resulting
extrapolated values $\phi_r^{\infty}$ are also displayed in
Fig.\ \ref{fig:rate_fct_sw} together with a fit to a power
law $\phi_r^{\infty}=\alpha r^\beta$ ($\alpha=0.102(5)$,$\beta=0.97(10)$,
i.e., close to a linear behavior), which is compatible with
an exponential distribution ($\beta=1$) for the backup capacity in the thermodynamic
limit. For the two other network types, the exponential nature of
the tails of the distributions is even more obvious from the data
shown in Figs.\ \ref{fig:distr:er} and \ref{fig:distr:spatial} directly,
hence corresponding analyses of the rate function are omitted here.
The fact that the data can be so well described by the rate function in the
thermodynamic limit indicates that the problem studied here may be well
accessible using analytical large-deviation approaches, which often are based
on obtaining the rate function.
Finally, we want to understand the source of resilience in principle.
Trivially, the higher the load in the most-loaded link, the more load has to be
redistributed when this link is removed, i.e., the larger the needed backup
capacity. More interesting it is to ask
which network structures lead to resilient networks, without looking
at the actual load values. Here, selected results are shown for
the connection between the resilience and the number of links and,
respectively, the diameter of a network.
First, the relationship of the resilience to the number of links is
investigated. For this purpose, the $N=400$ networks obtained during the
simulations at different temperatures
$T$ were binned according to the number $n_{\rm e}$ of links. For the
networks in each bin, the average backup capacity $c_{\rm b}$ was
evaluated. The result is shown in Fig.\
\ref{fig:correlation_edges} for the ER and the spatial random networks (for the
SW ensemble, the number of edges does not vary). One sees that if only
few links are available, the backup capacity is very large, which
is meaningful,
because having more links allows to distribute the load making a network more
resilient. Interestingly, a sharp drop as a function of $n_{\rm e}$ is visible,
looking like a phase transition. This drop appears for ER random networks at a
smaller number of edges, which is meaningful, because ER networks exhibit no
constraints, thus one has more ``freedom'' to arrange the links such that
a high resilience can be obtained. Note that
typical networks exhibit a
very small backup capacity compared the the ``high-backup capacity phase''
in the left part of Fig.\ \ref{fig:correlation_edges}. Thus, this
transition is not investigated more thoroughly here.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.49\textwidth]{correlation_edges.eps}
\caption{The average resilience $c_{\rm b}$ as a function of the
number $n_{\rm e}$ of links for ER and spatial networks of size $N=400$.
\label{fig:correlation_edges}
}
\end{figure}
Thus, including more edges leads, not surprisingly, to a higher resilience.
Note that examples exist, where adding more links sometimes also
decreases the stability of a network \cite{witthaut2012}.
Anyway,
for real networks, including more links leads almost always to larger
costs (see also remarks below). Thus,
it would be interesting to see how the resilience correlates with other
topological measures of the network. For this purpose, a scatter plot of the
backup capacity versus the diameter $d$ of a network (i.e., the
longest among all
shortest $i\leftrightarrow j$ paths) was recorded, see inset of Fig.\
\ref{fig:correlation_diameter} for the SW model. One can observe that large
backup capacities go along with large diameters. Note that for the SW model,
the differences in backup capacity can not originate from fluctuations of the
number of edges.
The positive relation between diameter and backup capacity can be
seen even better in the main part of Fig.\ \ref{fig:correlation_diameter},
where a binning of the networks with respect to $c_{\rm b}$ was
performed (shown here just for typical and very resilient networks,
i.e., small values of $c_{\rm b}$)
and within each bin the average
diameter was evaluated. Again, the positive correlation between diameter and
backup capacity is visible, for all three network ensembles. In particular
very small backup capacities, i.e, the most resilient graphs (which are
not accessible using standard simple-sampling simulation approaches) are related
to extremely small diameters. This shows that the diameter is a key quantity when
considering and optimizing resilience of transport networks.
Note that again
ER ensemble and spatial networks are very similar behaving, although very
different in definition. For the ER ensemble, for the most resilient instances
obtained, an average diameter of $d\approx 2$ was measured, close to the value of $d=1$
of the complete network. Note that a complete network minus one edge has already
diameter two. This shows that actually the most resilient networks were sampled
during the simulations.
On the other hand, the SW ensemble exhibits for
the same backup capacity a larger average diameter. This may occur on the first sight
interesting since it allows for longer paths leading to the same resilience. On the
other hand, this effect is only visible for slightly higher backup capacities.
Therefore, in the region of extremely resilient networks, which are most interesting, the
SW ensemble does not contribute at all, because such resilient networks do simple
not exist there. Furthermore, extrapolating the SW data of $d$ by eye to small
values of $c_b$ leads to small values of the diameter, which simply cannot be
obtained in this ensemble.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=0.45\textwidth]{correlation_diameter.eps}
\caption{The average diameter $d$ as a function of the
resilience $c_{\rm b}$ for SW, ER and spatial networks of size $N=400$,
in the range of small value of $c_{\rm b}$.
The inset shows a scatter plot of the data for the entire range of
backup capacities for the SW case.
\label{fig:correlation_diameter}
}
\end{figure}
Note finally that for real networks costs are an important issue.
In order to observe how the resilience scales with the costs
of a network, one would have to take into account the spatial length
of the links and built upon that the costs which are a function of the number
of links and their lengths. This is certainly beyond the
scope of the current work which has its focus on abstract but standard and widespread
network ensembles. Nevertheless, the general results as
shown here will likely persist, in particular the strong correlation
with the diameter of a network, which should be minimized for given costs.
\section{Summary and outlook}
\label{sec:summary}
The resilience of simple models of transportation networks against
failures of highly-loaded links were studied here.
For the \emph{random networks}, three different ensembles were considered:
The Erd\H{o}s-R\'enyi ensemble is the
most simple model for random networks, exhibiting no spatial structure
at all, but serves well as a null model for comparison.
Small-world networks are also very simple but are used often to
model real-world transportation
networks still rather well, like, e.g., energy grids. Finally, spatial
networks are considered here, which are more sophisticated,
but not well established.
They might serve in the future as standard models for
surface-based transportation.
To model the resilience against single-link failures (leading to
cascading failures) the \emph{backup capacity} is defined, which
describes the amount of additional capacity, which one has to
be included in the links to prevent a failure. The lower the backup
capacity is, the more resilient, i.e., the better, is the structure of the network.
Here, a \emph{large-deviation} approach was used to study
the distribution of the backup capacity.
Since the method allows to access a distribution (almost) on its complete
support, one can study the scaling behavior not only of the typical and
average but also of the best and the worst network instances.
Networks leading to very small probability
densities of the backup capacity such as $10^{-80}$
could be generated and studied with the correct weight via introducing a
bias and reweighting the results for the analysis.
The main results are as follows: Trivially, by including more links,
a network can be made more resilient. More interestingly, for all types of networks,
even for the SW ensemble with fixed number of links,
the most-resilient
networks can be obtained by minimizing the diameter of the network.
The typical backup capacity, on the other hand, grows linearly
with the number of nodes in the network. In particular, spatial (two-dimensional)
networks appear most promising for future studies of resilience of models of
real-world transportation networks.
Furthermore, using the rate function
approach, the shape of the distribution could be extracted in the
thermodynamic limit, which is exponential. In particular, the
large-deviation property is fulfilled, which means that it appears
promising to use standard mathematical large-deviation techniques, e.g.,
generating functions, to study the distribution of the backup capacity
more rigorously.
Hence, this study shows that the full range of transportation
networks ranging from the rare very resilient, over typical to the
exponential rare very susceptible networks can be studied numerically
using large-deviation techniques. Here, a rather simple and unspecific
transportation model, yet widely used in the literature,
was used. Hence, it appears to be very promising
to apply similar approaches to more realistic and specific models
of transportation networks, e.g., time-dependent ac currents
based on Kuramoto oscillators to model energy grids
\cite{filatrella2008,rohden2012,witthaut2012}.
\section*{Acknowledgements}
The author thanks Frank den Hollander for interesting discussions. The
author is grateful to Timo Dewenter for critically reading the manuscript.
Financial support was obtained via the
Lower Saxony research network ``Smart Nord'' which
acknowledges the support of the Lower Saxony Ministry of Science
and Culture through the ``Nieder\-s\"achsi\-sches Vorab'' grant program
(grant ZN 2764/ ZN 2896).
The simulations were performed at the HERO cluster of the University
of Oldenburg funded by the DFG (INST 184/108-1 FUGG) and the
ministry of Science and Culture (MWK) of the Lower Saxony State.
\bibliographystyle{epj}
|
2,877,628,089,531 | arxiv | \section{Introduction}
In 1967 Winfree proposed a model for coupled oscillators that spontaneously synchronize \cite{winfree2001geometry}. Beyond a critical coupling strength, the oscillators overcome the disordering effect of the dissimilarities in their natural frequencies and spontaneously lock their cycles. Kuramoto later simplified Winfree's model and solved it exactly \cite{kuramoto1975international}. Since then, the study of sync has matured into a vibrant field \cite{strogatz2004sync,pikovsky2003synchronization,acebron2005kuramoto}. On the theoretical side, theorists have twisted Kuramoto's model in various ways resulting in rich phenomena like glassy behavior \cite{daido1992quasientrainment, bonilla1993glassy,iatsenko2014glassy, ottino2018volcano} and chimeras states \cite{kuramoto2002coexistence,abrams2004chimera,abrams2008solvable}. On the applied side, coupled oscillators have found use in neurobiology \cite{montbrio2015macroscopic, pazo2014low, o2016dynamics,luke2013complete}, cardiac dynamics \cite{peskin1975mathematical, grudzinski2004modeling,gois2009analysis}, and the bunching of school buses \cite{saw2018bus}. An interesting application of sync is the design of `bio-inspired' algorithms. Here, by aping the simplicity of coupled oscillator models, researchers have devised resource-efficient algorithms to procure synchrony in the lab, useful for networked computing and robotics~\cite{werner-allen05,hong05,babaoglu2007firefly,tyrrell10}.
A story with many parallels to the sync story has evolved in the study of swarms. In 1995 Vicsek proposed a simple model of swarming agents \cite{vicsek1995novel}, which -- like the sync transition of coupled oscillators -- showed a transition from disorder to order: beyond a critical coupling strength, the agents switched from a gas-like, incoherent state to one in which the agents moved as a coherent flock. The novelty of the out-of-equilibrium nature of the flocking transition -- the system is out of equilibrium since agents constantly consume energy to propel themselves -- piqued the minds of physicists and other theorists, which in turn helped give rise to the field of active matter \cite{ramaswamy2010mechanics,marchetti2013hydrodynamics,sanchez2012spontaneous}, a field perhaps more vibrant than the field of synchronization. In a final mirroring of the sync story, the bio-inspired community has also mimicked the minimalism of Vicsek's and other models of swarming to design novel algorithms for optimization \cite{dorigo2010ant,meng2014new,meng2016new,mirjalili2017salp,yang2013swarm,shi2001particle} and robotic swarms \cite{hamann2018swarm,schmickl2011beeclust,savkin2004coordinated,merkle2007swarm,yang2013swarm}.
These stories demonstrate swarming and synchronization are intimately related. In a sense, the two effects are `spatiotemporal opposites': in synchronization the units self-organize in time but not in space; in swarming the units self-organize in space but not in time. Given this conceptual twin\-ship, it is natural to wonder about the possibility of units which can self-organize in both space and time -- that is, to wonder how swarming and synchronization might interact. And more pertinent to this work, to wonder how a mix of swarming and synchronization could be useful in technology.
Several researchers have started to address these questions by analyzing systems that both sync and swarm. Von Brecht and Unimsky have generalized swarming particles by endowing them with an internal polarization vector \cite{von2016anisotropic}, equivalent to an oscillator's phase. Others have attacked the problem from the other direction, by considering synchronizing oscillators able to move around in space~\cite{uriu2013dynamics,sevilla2014synchronization,stilwell2006sufficient,frasca2008synchronization}. In these studies, however, the oscillators' movements affect their phase dynamics, but not the other way around; thus, the interaction between swarming and synchronization is only one-way. Two-way interaction between swarming and synchronization has also been considered: The pioneering work is the Iwasa-Tanaka model of chemotactic oscillators \cite{tanaka2007general,iwasa2010hierarchical}, i.e., oscillators which interact through a background diffusing chemical. More recent works were carried out by Starnini et al \cite{starnini2016emergence}, Belovs et al \cite{belovs2017synchronized}, and O'Keeffe et al \cite{o2017oscillators} who proposed `bottom-up' toy models without reference to a background medium. O'Keeffe et al called the elements of their systems `swarmalators', to capture their twin identities as swarming oscillators, and to distinguish them from the mobile oscillators mentioned above for which the coupling between swarming and synchronization is~unidirectional.
\begin{figure*}[t]
\includegraphics[width = 2 \columnwidth]{swarmalator_states_allonerow.png}
\caption{\textbf{Swarmalator states}. Scatter plots in the $(x,y)$ plane, where the swarmalators are colored according to their phase. (a) Static sync for $(J,K) = (0.1, 1)$. (b) Static async $(J,K) = (0.1, -1)$. (c) Static phase wave $(J, K) = (1, 0)$. (d) Splintered phase wave $ (J,K) = (1, -0.1) $. (e) Active phase wave $(J,K) = (1,-0.75)$.}
\label{stationary_states_2d}
\end{figure*}
This paper reviews research on swarmalators and other systems that mix swarming and synchronization. Our main motivation is to identify if the interplay of sync and swarming can be useful for bio-inspired computing and related engineering domains. We outline the theoretical work on swarmalators, experimental realizations, and finally conjecture on their technological utility.
\section{Models combining swarming and synchronization}
To combine synchronization with swarming, we first define what we mean by swarming. While to our knowledge there is no universally agreed on definition, swarming systems typically have at least one of two key features: ($i$) aggregation, arising from a balance between the units' mutual attraction and repulsion and ($ii$) alignment, referring to the units' tendency to align their orientation in space and move in a flock. Synchronization can thus be combined with either aggregation alone, alignment alone, or with both aggregation and alignment. In what follows, we present classes of model based on these three ways to combine sync and swarming.
\begin{figure} [h!]
\includegraphics[width= 0.99 \linewidth]{phase_diagram.pdf}
\caption{\textbf{Phase diagram}. Locations of states of the model defined by equations~\eqref{x_eom_model} and \eqref{theta_eom_model} in the $(J,K)$ plane. The straight line separating the static async and active phase wave states is a semi-analytic approximation given by $J \approx 1.2 K$ -- see Eq.~(18) in \cite{o2017oscillators}. Black and red dots show simulation data. The red dashed line simply connects the red dots and was included to make the boundary visually clearer. Note, this figure has been reproduced from \cite{o2017oscillators}.}
\label{phase_diagram_model1}
\end{figure}
\subsection{Aggregation and Synchronization}
The paradigmatic model of biological aggregation has the form
\begin{align}
&\dot{\mathbf{x}}_i = \frac{1}{N} \sum_{j \neq i}^N \mathbf{I}_{\rm att}( \mathbf{x_j} - \mathbf{x_i}) - \mathbf{I}_{\rm rep}( \mathbf{x_j} - \mathbf{x_i} ), \label{agg}
\end{align}
\noindent
where $\mathbf{x}_i \in \mathbb{R}^d$ (usually with $d \leq 3$) is the $i$-th particle's position, $\mathbf{I}_{\rm att}$ captures the attraction between particles, and $\mathbf{I}_{\rm rep}$ captures the repulsion between them. The competition between $\mathbf{I}_{\rm att}$ and $\mathbf{I}_{\rm rep}$ gives rise to congregations of particles with sharp boundaries, in accordance with many biological systems (see \cite{topaz2004swarming,mogilner1999non} for a~review).
The paradigmatic model in synchronization is the Kuramoto model \cite{kuramoto91}:
\begin{align}
\dot{\theta_i} = \omega_i + \frac{K}{N} \sum_{j \neq i}^N \sin(\theta_j - \theta_i). \label{kuramoto}
\end{align}
\noindent
Here $\theta_i \in S^1$ and $\omega_i$ are the phase and natural frequency of the $i$-th oscillator. The sine term captures the oscillators' coupling, where $K > 0$ means oscillators tend to synchronize, and $K < 0$ means oscillators tend to desynchronize. As mentioned, beyond a critical coupling strength $K_c$, a fraction of oscillators overcome the disordering effects imposed by their distributed natural frequencies $\omega_i$ and spontaneously synchronize.
To study the co-action of synchronization and aggregation, a natural strategy would be to stitch the aggregation model \eqref{agg} and the Kuramoto model \eqref{kuramoto} together. This was the approach taken by O'Keeffe et al \cite{o2017oscillators} who proposed the following swarmalator model:
\begin{align}
&\dot{\mathbf{x}}_i = \frac{1}{N} \sum_{ j \neq i}^N \Bigg[ \frac{\mathbf{x}_j - \mathbf{x}_i}{|\mathbf{x}_j - \mathbf{x}_i|} \Big( 1 + J \cos(\theta_j - \theta_i) \Big) - \frac{\mathbf{x}_j - \mathbf{x}_i}{ | \mathbf{x}_j - \mathbf{x}_i|^2}\Bigg]\label{x_eom_model} \\
& \dot{\theta_i} = \omega_i + \frac{K}{N} \sum_{j \neq i}^N \frac{ \sin(\theta_j - \theta_i)}{ |\mathbf{x}_j - \mathbf{x}_i| } . \label{theta_eom_model}
\end{align}
\noindent
Equation~\eqref{x_eom_model} models phase-dependent aggregation and Equation~\eqref{theta_eom_model} models position-dependent synchronization. The interaction between the space and phase dynamics is captured by the term $1 + J \cos(\theta_j - \theta_i) $. If $J > 0$, ``like attracts like'': swarmalators are preferentially attracted to other swarmalators with the same phase, while $J < 0$ indicates the opposite. For simplicity the authors considered identical swarmalators $\omega_i = \omega$, and by a change of reference they set $\omega = 0$.
The swarmalator model exhibits five long-term collective states. Figure~\ref{stationary_states_2d} showcases these states as scatter plots in the $(x,y)$ plane, where swarmalators are represented by dots and the color of each dot represents the swarmalator's phase $\theta$ (color, recall, can be mapped to $S^1$ and so can be used to represent swarmalators' phases). The parameter dependence of these states are encapsulated in the phase diagram shown in Figure~\ref{stationary_states_2d}. The first three states, named the \textit{static sync}, \textit{static async}, and \textit{static phase wave}, are -- as their names suggest -- static in the sense that the individual swarmalators are stationary in both position and phase. This stationarity allows the density of these swarmalators $\rho(\mathbf{x}, \theta)$ in these states to be constructed explicitly (the density $\rho(\mathbf{x}, \theta)$ is interpretted in the Eulerian sense, so that $\rho(\mathbf{x} + \mathbf{dx}, \theta + d \theta)$ gives the fraction of swarmalators with positions between $\mathbf{x}$ and $\mathbf{x} + d \mathbf{x}$ and phases between $\theta$ and $\theta + d \theta$). In the remaining \textit{splintered phase wave} and \textit{active phase waves} states swarmalators are no longer stationary. In the splintered phase wave state swarmalators execute small oscillation in both space and phase within each cluster. In the active phase wave the swarmalators split into counter-rotating groups -- in both space and phase -- so that $ \langle \dot{\theta_i} \rangle = \langle \dot{\phi_i} \rangle = 0$, where $\phi_i = \arctan(y_i / x_i)$ is the spatial angle of the $i$-th oscillator and angle brackets denote population averages. The conservation of these two quantities follows from the governing equations; the pairwise terms are odd and thus cancel under summation. Three-dimensional analogues of the five collective states were also reported.
The stability properties of the static async state are unusual. Via a linear stability analysis in density space, an integral equation for the eigenvalues $\lambda$ was derived. Yet numerical solutions to this integral equation, in the parameter regime where the static async state should be stable, produced a leading eigenvalue so small in magnitude that its sign could not be determined reliably. Thus, the stability of the state could not be analytically confirmed. There is however a parameter value at which the magnitude of $\lambda$ increases sharply, which was used to derive a pseudo critical parameter value $K_c \approx - 1.2 J$ marking the apparent (i.e.~as observed in simulations) destabilization of the state. Nevertheless, the true stability of the static async state remains a puzzle.
\begin{figure}[h]
\includegraphics[width= 0.60 \columnwidth]{ring_state.pdf}
\caption{\textbf{Ring phase state.} Swarmalators are represented by colored dots in the $(x,y)$ plane where the color of each dot represents its phase. The state is found by numerically integrating the governing equations in \cite{o2018ring} with $(J_1, J_2, K, N) = (0,0.8,0,100)$.}
\label{ring}
\end{figure}
Some extensions of the work in \cite{o2017oscillators} have been carried out. One addresses the conspicuously empty upper right quadrant in the $(J,K)$ parameter plane in Figure~\ref{phase_diagram_model1}, where only the trivial static sync state appears. By adding phase noise to \eqref{theta_eom_model}, Hong discovered \cite{hong2018active} the active phase wave exists for $K > 0$. Unexpectedly, the splintered phase wave was not observed. O'Keeffe, Evers, and Kolokolnikov \cite{o2018ring} have extended \eqref{x_eom_model} by allowing phase similarity to affect the spatial repulsion term, as well as the spatial attraction term (i.e.~they multiply the second term in \eqref{x_eom_model} by a term $1+J_2 \cos(\theta_j - \theta_i)$). This led to the emergence of ring states, an example of which is depicted in Figure~\ref{ring}. They constructed and analyzed the stability of these ring states explicitly for a population of given size $N$. Analytic results for arbitrary $N$ are potentially useful for robotic swarms, which presumably are realized in the small $N$ limit. Another offshoot of this analysis was a heuristic method to predict the number of clusters which form in the splintered phase wave state; they viewed each cluster of synchronized swarmalators as one giant swarmalator which let them re-imagine the splintered phase wave state as a ring, allowing them to leverage their analysis. A precise description of the number of clusters formed is an open problem.
\textbf{Iwasa-Tanaka model}. Iwasa and Tanaka proposed and studied a `swarm-oscillator' model in a series of papers \cite{tanaka2007general, iwasa2011juggling, iwasa2017mechanism, iwasa2010dimensionality, iwasa2012various}. The inspiration for their work comes from chemotactic oscillators, i.e., oscillators moving around in a diffusing chemical which mediates their interactions. They began with the general~model
\begin{align}
\dot{\mathbf{X}}_i(t) &= f(\mathbf{X}_i) + k g(S(\mathbf{r}_i,t)) \\
m \ddot{r_i}(t) &= - \gamma \dot{r}_i - \sigma(\mathbf{X}_i) \nabla S \\
\tau \partial_{\tau} S(\mathbf{r}, t) &= -S + d \nabla^2 S + \sum_i h(\mathbf{X}) \delta(\mathbf{r} - \mathbf{r}_i),
\end{align}
\noindent
where $\mathbf{X}_i$ represents the internal state (which will later be identified as a phase), $\mathbf{r}$ represents the position of the $i$-th oscillator, and $S$ represents the concentration of the background chemical. By means of a center manifold calculation and a phase reduction technique they derived the simpler equations
\begin{align}
\dot{\psi}_i(t) &= \sum_{j \neq i} e^{- |\mathbf{R}_{ji}|} \sin( \Psi_{ji} - \alpha |\mathbf{R}_{ji}| - c_1 ) \label{Tanaka_x} \\
\dot{\mathbf{r}}_i(t) &= c_3 \sum_{j \neq i} \hat{\mathbf{R}}_{ji} e^{- |R_{ji}|} \sin( \Psi_{ji} - \alpha |\mathbf{R}_{ji}| - c_2 ) \label{Tanaka_theta}
\end{align}
\noindent
where $\mathbf{R}_{ji} = \mathbf{R}_j - \mathbf{R}_i$, $\Psi_{ji} = \psi_j - \psi_i$, and $\psi_i$ is the $i$-th oscillator's phase. We call this Iwasa-Tanaka model.
Notice the space-phase coupling in this model is somewhat peculiar; in contrast to the swarmalator model given by \eqref{x_eom_model} and \eqref{theta_eom_model} the relative position $\mathbf{R}_{ji}$ and relative phase $\Phi_{ji}$ appear \textit{inside} the sine terms in both the $\dot{r}_i$ and $\dot{\psi}_i$ equations. Another difference between the two models is that $\dot{\mathbf{r}}_i$ in \eqref{Tanaka_x} has no hardshell repulsion term, which means the oscillators can occupy the same position in~space.
The Iwasa-Tanaka model has rich collective states. An exhaustive catalogue of these states with respect to the model's four parameters is an ongoing effort \cite{iwasa2012various}. Highlights include a family of clustered states \cite{tanaka2007general,iwasa2010hierarchical,iwasa2010dimensionality} in which swarmalators collect in synchronous groups. The spatial distributions of these groups depend on their phase, similar to the splintered phase wave (Figure~\ref{stationary_states_2d}). The authors speculate this phase clustering is reminiscent of the chemotactic cell sorting during biological development \cite{tanaka2007general}. The Iwasa-Tanaka model also produces ring states \cite{iwasa2010hierarchical}, as well as an interesting `juggling' state \cite{iwasa2011juggling} in which the population forms a ``rotating triangular structure whose corers appear to `catch' and `throw' individual elements'' -- in other words, the population juggles the elements around the corners of a triangle (see Figure 1 in \cite{iwasa2011juggling}). Aside from theoretical novelty, this juggling could conceivably be exploited in robotic swarms, potentially allowing some form of relay between the elements.
\subsection{Alignment and synchronization}
In our proposed taxonomy, systems that combine alignment and synchronization are characterized by an internal phase $\theta$ and an orientation $\beta$ without a dynamic spatial degree of freedom. In other words, the particles' position $\mathbf{x}$ might affect their $\theta$ and $\beta$ dynamics, but $\mathbf{x}$ itself does not evolve in time. Although units characterized by just a phase $\theta$ and orientation $\beta$ might seem odd, Leon and Liverpool studied a system with units which meet these criteria: a class of soft active fluids which constitute a `new type of nonequilibrium soft matter -- a space-time liquid crystal' \cite{leoni2014synchronization}. They developed a phenomenological theory of these soft fluids which allowed them to derive dynamical equations for order parameters quantifying the orientation order, phase order, and orientation-phase order. These revealed collective states which maximize each of these order parameters: aligned states with orientational order but no phase order, sync'd states with phase order but no orientational order, and states with both phase and orientational order. They were able to partially analyze these states, and conjectured the states could be realized in protein-filaments mixtures, such as cell cytoskeletons \cite{toner1995long,ahmadi2006hydrodynamics} or tissue-forming cells \cite{sachs2005pattern}.
\subsection{Alignment, aggregation, and synchronization}
Systems with units that align, aggregate, and synchronize -- and therefore have dynamic state variables $\mathbf{x}, \theta, \beta$ -- are the least well studied. A small study was carried out in \cite{o2017oscillators}, with the aim of checking if the swarmalator states reported are generic, i.e., robust to the inclusion of alignment dynamics (it was found they were). Beyond this preliminary study, the space of possible behaviors arising from alignment, aggregation, and synchronization is largely unexplored.
\subsection{Alignment and aggregation}
While not strictly within our proposed taxonomy of swarmalator systems, we give a brief review of studies on aggregation and alignment. We do this because alignment can be viewed as a type of synchronization, where instead of units adjusting to a common phase $\theta$, units instead adjust a common orientation $\beta$. Or put another way, because an internal phase $\theta$ and orientation $\beta$ are formally equivalent -- both being circular variables -- particles aligning their orientations is analogous to oscillators aligning their phases.
Vicsek set the paradigm of this class of models with a beautiful and simple model:
\begin{align}
& \mathbf{r}_i(t + \delta t)= \mathbf{r}_i(t) + (\nu \delta t) \hat{n} \\
& \beta_i(t + \delta t) = \langle \beta_j \rangle_{|r_j - r_i|<r} + \eta_i(t) \label{vicsek},
\end{align}
\noindent
where $\hat{n} = (\cos(\beta), \sin(\beta))$, $\eta_i(t)$ is a white noise, $\nu$ is the particles' speed, and $r$ is the coupling range. For sufficiently strong noise this model exhibits a flocking transition, where particles switch from erratic incoherent movements to moving in a unidirectional flock.
Leibchen and Levis \cite{liebchen2017collective, levis2018activity} modified the Vicsek model as follows:
\begin{align}
&\dot{\mathbf{r}}_i= \nu \vec{n}_i \\
& \dot{\beta_i} = \omega_i + \frac{K}{\pi R^2} \sum_{j \in \partial_i}^N \sin(\theta_j - \theta_i) + \sqrt{2D} \eta_i,
\end{align}
\noindent
where $\partial_i$ is the set of neighbours of the $i$-th particle, $K$ is the phase coupling, and $D$ is the noise strength. The key new feature is the $\omega_i$ term, signifying the particles have an intrinsic rotation. They find three collective states: a disordered state in which neither spatial order nor phase order exists, a chiral state in which macroscopic `droplets' of synchronized oscillators form in a sea of otherwise desynchronized oscillators, and finally a \textit{mutual flocking} state in which oscillators with `opposite chirality cooperate to move coherently at a relative angle, forming non-rotating flocks'. The theoretical novelty of the latter state is that -- in contrast to regular, non-moving, locally-coupled oscillators coupled in two or three dimensions -- long-range synchronous clusters are observed, despite the fact that the oscillators are only locally coupled.
\section{Swarmalators in the real-world}
To our knowledge, there are just two works which unequivocally realize swarmalators in the real world -- by this we mean, precisely, a real-world system in which a bidirectional space-phase coupling is unambiguously exhibited. These works are:
\textbf{Swarmalatorbots}. Bettstetter et al first realized the collective states of the swarmalator model (Figure~\ref{stationary_states_2d}) in the lab using small robots \cite{gniewek19}. They programmed `swarmalatorbots' whose governing equations were derived from the swarmalator model \eqref{x_eom_model}, \eqref{theta_eom_model}.
\textbf{Magnetic domain walls}. To our knowledge, the first realization of a natural swarmalator system was found by Hrabec et al when studying the magnetic domain walls \cite{hrabec2018velocity}. Ordinarily, domain walls are described by a single spatial degree of freedom $\mathbf{x}$. But as the authors note, the dynamics on the walls also depend on their internal structure. The authors minimally describe this internal state by a one dimensional polarization angle of the internal magnetic field $\theta$. This allows the wall to be viewed as a point particle with position $\mathbf{x}$ and phase $\theta$ -- to be viewed as a swarmalator \cite{hrabec2018velocity}. An experimental study of coupling between two domain walls -- usually just one wall is studied -- revealed the walls can synchronize, which in turn affects the walls' velocity. Richer space-phase dynamics, such as families of Lissajous curves, are also reported.
\begin{center}
***
\end{center}
Beyond swarmalatorbots and magnetic domain walls, there are many systems in which a bidirectional coupling between swarming and synchronization might exist. We list these candidate swarmalator systems below:
\textbf{Myxobacteria}. Myxobacteria are bacteria commonly found in soil, and can produce interesting collective effects \cite{igoshin2001pattern}. The phase variable of myxobacteria characterizes the internal growth cycle of the cell. Groups of cells interact through cell-to-cell contact, theorized to provide a channel through which swarming and synchronization can couple. Populations of myxobacteria can exhibit a `ripple phase', in which complex patterns of waves propagate through population \cite{igoshin2001pattern}. In contrast to other wave phenomena in biological systems, such as the well-studied Dictyostelium discoideum, these rippling waves do not annihilate on collision. In \cite{igoshin2001pattern} a Fokker-Planck type equation was used to analyze rippling waves. Realizing them in a microscopic swarmalator system is an open problem.
\textbf{Biological microswimmers}. `Microswimmers' is an umbrella term for self-propelled micro\-organisms confined to fluids, such as celia, bacteria, and sperm \cite{elgeti2015physics}. Groups of micro\-swimmers show swarming behavior as a result of cooperative goal seeking, such as searching for food or light. They can also synchronize: Here the phase variable is associated with the rhythmic beating of swimmers' tails which -- through hydrodynamic interactions -- can synchronize with the beatings of others swimmers' tails. Whether or not this hydrodynamics provides a bidirectional coupling between sync and swarming -- as required of swarmalators -- is unclear, but to us seems plausible. Researchers have developed models of sperm under this assumption \cite{yang2008cooperation} and found clusters of synchronized sperm consistent with real data~\cite{hayashi1996insemination}. Vortex arrays of sperm have also been reported \cite{riedel2005self}. Here, sperm self-organize into subgroups arranged in a lattice; within each subgroup, the sperm move in a vortex, wherein a correlation between the their angular velocity and their phase velocity is realized, reminiscent of the splintered phase wave state (Figure~\ref{stationary_states_2d}). Simulations of realistic models have been carried out \cite{belovs2017synchronized} which show other interesting~phenomena.
\textbf{Japanese tree frogs}. During mating season, male Japanese tree frogs attract females by croaking rhythmically. The croaking of neighbouring frogs tend to anti-synchronize due to a precedence effect: croaking shortly after a rival makes a frog look less dominant. Researchers have theorized that this competition leads to mutual interaction between frogs' space and phase dynamics, coupling sync and swarming. Models based on these assumption produce ring-like states where frogs arrange themselves on the borders of fields with interesting space-phase patterns, some of which are consistent with data on real tree-frogs' behavior collected in the wild.
\textbf{Magnetic colloids}. Similar to magnetic domain walls, a colloid's constituent particles can synchronize the orientation of their magnetic dipole vector when sufficiently close. When in solution, they are free to move around in space, creating a feedback loop between their space and phase dynamics. In ferromagnetic colloids confined to liquid-liquid interfaces, Snezhko and Aranson \cite{snezhko2011magnetic} found this interaction leads to the formation of `asters' -- star-like arrangements of particles whose spatial angles correlate with the orientations of their magnetic dipole vectors, equivalent to the static phase wave (Figure~\ref{stationary_states_2d}). Yan et al explored how synchronization is useful in colloids of Janus particles \cite{yan2012linking}. Janus particles are micrometer-sized spheres with one hemisphere covered with nickle which gives them non-standard magnetic properties. In particular when subject to a precessing magnetic field, they oscillate about their centers of mass. This oscillation creates a coupling between particles, giving rise to `synchronization-selected' self-assembly. For example, zig-zag chains of particles and microtubes of synchronized particles can be realized.
\section{Swarmalators in bio-inspired computing}
The computer engineering community has been inspired by coupled oscillators to develop new techniques for synchronization in communication and sensor networks~\cite{mathar96,hong05,werner-allen05,simeone08,babaoglu2007firefly,tyrrell10}. Here, instead of Kuramoto-type oscillators \eqref{kuramoto}, models of pulse-coupled oscillators have been borrowed. As the name suggests, pulse-coupled oscillators communicate by exchanging short signals; this time-discrete variant of coupling is a more natural fit for application in technology, where smooth, continuous coupling -- as exemplified by Kuramoto oscillators -- is costly to achieve.
The canonical model of pulse-coupled oscillators is the Peskin model \cite{peskin1975mathematical}, defined by \begin{equation}
\dot{x_i} = S_0 - \gamma x_i
\label{peskin}
\end{equation}
\noindent
with $S_0, \gamma > 0$ and $x_i$ is a voltage like state variable for the $i$-th oscillator. When the oscillators' voltage reach a threshold value they ($i$)~fire a pulse which instantaneously raises the voltage of all the other oscillators ($ii$)~reset their voltage to zero, along with any other oscillator whose voltage was raised above the threshold on account of receiving a pulse. The synchronization properties of the Peskin model are well-studied \cite{bottani1995pulse,mirollo1990synchronization}. More recently, estimates for the convergence speed have been derived~\cite{o2015synchronization,o2016transient}.
The simplicity, distributed nature, adaptability, and scalability of pulse-coupled oscillators make them attractive from an engineering point of view, where temporal coordination is often a goal. For example, synchronization is required for many tasks in different layers of computing and communications systems~\cite{bregni02} such as in the alignment of transmission slots for efficient medium access~\cite{Roberts:1975:APS}, scheduling of sleep cycles for energy efficiency~\cite{ye04}, and coordination of sensor readings to capture a scene from different perspectives~\cite{aghajan:book}. It is the relative synchrony of the networked units, not necessarily the absolute time, that is relevant here. The synchronization precision required is determined by the specific task. The achievable precision depends on the environment and hardware, and is influenced by deviations in delays and phase rates. Experiments in laboratory environments show that the precision is in the order of hundred microseconds with low-cost programmable sensor platforms \cite{werner-allen05,pagliari11} and a few microseconds with field-programmable gate array (FPGA)-based radio boards~\cite{brandner16:cn}. Further use cases can be found in acoustics (synchronizing multiple loudspeakers) and energy systems (synchronizing decentralized grids \cite{rohden12,skardal15}), to give two examples.
Synchronization is also important in robotics in order to perform coordinated movements -- and this is where temporal and spatial coordination unite. As mentioned, Bettstetter and colleagues~\cite{gniewek19} extended the swarmalator model \cite{o2017oscillators} so that it could be applied to mobile robots. They implemented the extended model in the Robot Operating System~2 (ROS\,2) and experimentally demonstrated that the space-time patterns achieved in theory (Figure\,\ref{stationary_states_2d}) can be reproduced in the real world. Beyond realizing these specific states, we conjecture that swarmalator-type models will enable novel self-assembly procedures in other robotics groups, in turn enabling collaborative actions in monitoring, exploration, and manipulation. As envisioned in \cite{gniewek19}, underwater robots could be designed, which -- in imitation of biological microswimmers -- could both swim in formation and synchronize their fin movements, thereby enhancing their functional capabilities.
More speculatively, the swarmalator concept could be useful in the design of autonomous transport systems; for example, when multiple vehicles driving in a convoy have to avoid collisions with other convoys due to crossings and for the purpose of overtaking. The model could also enable self-configuring distributed antenna arrays (or loudspeakers) in which multiple antenna elements (or sound sources) automatically arrange their positions and orientations to create specific radiation patterns used to send radio (audio) signals in a synchronous way. Another promising application field is the planning and replanning of processes in factories, where products and machines must follow a certain space-time order and where standard optimization methods reach their limits due to the high system complexity. A final, more playful, application is in art: Artistic aerial light shows created by drone swarms running the swarmalator model produce charming visual displays.
\section{Discussion}
The implicit strategy in the theoretical works we reviewed is: given a model, which spatiotemporal patterns emerge? Future work could consider the inverse strategy: given a desired spatiotemporal pattern, which form should the model take on? -- a question often considered in engineering contexts. From a more general perspective, one has to design the local rules and interactions that guide a self-organizing system toward a desired global state \cite{prehofer05:commag}. Von Brecht et al have studied this inverse question for swarming systems~\cite{von2012soccer}; perhaps their tools could be extended to swarmalators.
Future work could also extend the reviewed models, which -- recall -- were designed to be as minimal as possible. A wealth of synchronization phenomena have been found by adorning the Kuramoto model with new features like mixed-sign coupling \cite{hong2011kuramoto,maistrenko2014solitary,hong2016phase, hong2016correlated}, non-local coupling \cite{abrams2004chimera, abrams2008solvable}, and delayed interactions \cite{yeung1999time}. Equipping swarmalators with these features would likely cure the poverty of phenomena in the first, third, and fourth quadrants of the $(J,K)$ plane (Figure~\ref{phase_diagram_model1}) and produce new dynamics in Iswasa-tanaka and Vicesek type models, too. Phase models more sophisticated than the Kuramoto model could also be explored; the Winfree model \cite{winfree67} or the newly introduced Janus oscillators \cite{nicolaou2019multifaceted} would be exciting to experiment with. Swarmalators with discrete and spatially local coupling also merit study and are useful in computer and communication systems. Stitching the Peskin model \eqref{peskin} to the aggregation equation \eqref{agg} or the Vicsek model \eqref{vicsek} seems the natural way to do this. Continuing to study finite systems of swarmalators \cite{o2018ring} is also pertinent, since many robotic systems lie in the low $N$ regime. And finally, beyond richer phase models, the spatial dynamics of swarmalators could also be generalized. Even within the framework of the aggregation equation~\eqref{agg}, a menagerie of spatial patterns have been catalogued \cite{kolokolnikov2011stability}. Swarmalator counterparts to these states would be exciting to explore.
We hope to have outlined that swarmalators have great potential in computing and other fields of technology. Here we see swarmalators only as a case study in a broader class of systems with large technological utility: systems whose units have both spatial and internal degrees of freedom. The one-dimensional phase $\theta$ of a swarmalator is perhaps the simplest instance of an internal degree of freedom, which can more generally be represented by a feature vector~$\mathbf{f}$. This vector could describe a particle's (three dimensional) magnetic or electric field; a person's political affiliation, mood, or health state; a bacteria's phase in a non-circular developmental cycle; and the mode of a robot, machine, or vehicle. The collective states arising when particles' features $\mathbf{f}$ and positions $\mathbf{x}$ defines a wide, scarcely investigated area. We anticipate an exploration of this area will bear rich fruit in both Nature and technology. \\
\section*{Acknowledgements}
The work of C.\,Bettstetter is supported by the Austrian Science Fund (FWF) [grant P30012] and the Popper Kolleg {\it Networked Autonomous Aerial Vehicles}.
\bibliographystyle{apsrev}
|
2,877,628,089,532 | arxiv | \section{Introduction}
Multipole moments for static asymptotic flat spacetimes have been first defined by Geroch \cite{GerochI,GerochII}. Later on, Hansen \cite{Hansen} extended this result to the stationary case. Other authors have also interpreted and proposed novel approaches to compute the multipole moments \cite{Thorne,Simon-Beig,Quevedo,Simon,FHP,HP,Sotiriou,fodor2020calculation}.
It is well-known that a spherically symmetric black hole solution like the Reissner-Nordstr\"{o}m one \cite{Reissner,Nordstrom} can be entirely characterized by the mass and electric charge, being the two monopole moments of the gravitational and electric fields, respectively. The same statement can be announced for Kerr's black hole \cite{Kerr} where now the first two multipole moments; the mass and angular momentum, define the source perfectly in stationary spacetimes. In both scenarios, the black hole solutions fulfill the no-hair conjecture \cite{Israel,Carter,Robinson}.
On the other hand, the exterior gravitational field of a neutron star cannot be explicitly comprehended unless multipole moments of higher orders are considered, such as the moment of inertia and quadrupole moment, in addition to its mass and radius, since, for instance, they might have some deformation in the mass distribution \cite{PhysRevD.91.103003}. Here, the multipole moments can even play a whole in trying to understand the equation of state for a star \cite{Laarakkers_1999}. In this context, the study of multipole moments is really useful for a better understanding of the physical measurable quantities of any type of sources in General Relativity with an astrophysical meaning; for instance, black holes or neutron stars \cite{Berti15,BertiStergioulas}, and other interesting physical features related to them like geodesics, shadows, and lensing effects or quasinormal modes, among others.
As a matter of fact, Fintan D. Ryan \cite{Ryan95,Ryan97} provided in 1995 a practical application for the multipole moments by extracting physical information from the gravitational radiation emitted by an object orbiting around a central body (see \cite{Pappas15b} for example in scalar-tensor theories). Actually, the recent detection of gravitational waves produced by the coalescence of binary black hole mergers \cite{LIGO}, suggest that the measure of the spin-induced quadrupole moment in compact binary mergers permits a clear distinction between binary black hole systems and binary systems containing another type of exotic compact objects \cite{Krishnendu}. In this regard, it is quite clear that the multipole structure of astrophysical objects is per se of great physical and mathematical relevance, and it deserves to be taken into account.
Several years ago, in 1996, a complete description for multipole moments in stationary vacuum systems had been accomplished by Hernández-Pastora, J.L. \cite{pastora}. He was able to work out a mapping among the $4N$-parameter exact stationary solution with the $4N$ arbitrary Geroch-Hansen multipole moments \cite{GerochI,Hansen}. This $4N$ exact solution is the vacuum specialization of the $N$-soliton electrovac solution derived in \cite{RMJ} via the Sibgatullin method \cite{Sibgatullin,Manko1993}; recalling that Sibgatullin's method provides an exact stationary solution in the entire spacetime once it is established any specific form of the Ernst potentials \cite{Ernst} on the symmetry axis (the axis data). In 1998, Manko and Ruiz published this result again in a section of a relatively famous paper \cite{MankoRuiz}, but now lacking of proof and containing fewer details. The main idea of the thesis \cite{pastora} and the paper \cite{MankoRuiz} is quite clear; the parameters composing the axis data can be related to arbitrary multipole moments and vice-versa; therefore, the whole spacetime can be completely represented with a whole physical meaning. Naturally, one might expect to extend this result to the electrovacuum scenario, which to our knowledge, has not been investigated yet (or just ignored due to its complexity).
Moreover, a natural demand due to the increase of numerical models fitting astrophysical observations is to construct analytical solutions \cite{1995ApJ...444..306S,PhysRevLett.108.231104} in order to bring more qualitative and quantitative understanding in what has been measured—making more evident the need to directly relate the physical quantities to the mathematical parameters that appear in the generating techniques.
The present paper aims to introduce a concise general formulation for electrovac spacetimes in terms of a multipole moment structure, extending the earlier results provided in \cite{pastora}. Ernst and Hauser have shown that the Ernst potentials fall into the class of elliptic differential equations, implying that they satisfy a quite interesting and useful property: the behavior of the Ernst potentials, $\mathcal{E}$ and $\Phi$, on the symmetry axis is sufficient to perform an analytic continuation of them
to the whole space. Meaning that, in order for us to fully connect a solution of the Einstein-Maxwell equations in axistationary spacetimes with the multipole moments, we only need to consider the relationship between the Ernst potentials and the multipole moments on the symmetry axis. The outline of our paper is the following. In Sec. \ref{Nsol} the N-soliton solution for electrovac spacetimes \cite{RMJ} as well as some basic backgrounds are first explained. Afterward, the path provided by Hernández-Pastora, J.L. \cite{pastora} is revisited in Sec. \ref{multipolevacuum} with the main purpose of entering the reader to the electrovacuum case that is described in Sec. \ref{multipoleelectrovacuum}. In Sec. \ref{examples} the multipole structure is developed for the particular case 2-soliton solution, where two results previously considered in the literature \cite{TomiSato} are taken into account to test the validity of our result and we also discuss how to use the $N$-soliton to construct approximated solutions possessing arbitrary multipole moments and use the monopole-dipole source as an example.
\section{The Extended N-soliton solution}\label{Nsol}
The use of solitonics techniques for constructing exact solutions of the Einstein equations was first introduced by Belinski and Zakharov in 1978 \cite{Belinsky:1971nt,Belinsky:1979mh} by means of their formulation of the inverse scattering method (ISM). In particular, the study of stationary axisymmetric spacetimes possesses an enormous physical interest because they can describe, in an idealized way, for instance, the exterior region of black holes, neutron stars, and accretion flow.
The stationary and axisymmetric spacetimes, which admits $G_2$ as isometry group, possess a set of completely integrable equations \cite{stephani_kramer_maccallum_hoenselaers_herlt_2003,martini1985geometric}. Several authors have introduced generating techniques based on these symmetries, and most of them, constructed employing the two Ernst potentials \cite{osti_6803745,Neugebauer_1983}. In fact, they have construct $N$-soliton solutions in terms of determinants for rational Ernst potentials on the symmetry axis \cite{PhysRevD.50.6179,PhysRevD.50.4993,Neugebauer_1980,etde_5927960}. However, some of these generating techniques have problems in constructing extreme, sub-extreme and hyper-extreme objects at the same time or even a problem in interpreting the independents' mathematical parameters contained in the solution. One advantage of using the Sibgatullin integral method is the direct relation of the solution parameters with the physical properties of the objects to be described, and, as we shall prove, all parameters that appear to construct the solution have a map one-to-one with the physical parameters, the multipole moments.
Considering axisymmetric stationary spacetimes, the idea of adding solitons to the background is the following. Consider the background as being the Minkowski space. The Ernst potentials for such spacetime is $\mathcal{E}=1$ and $\Phi=0$. By adding rational functions on the symmetry axis, you are adding solitons into the background. That is, the solution
\begin{equation}
\mathcal{E}(z,0)=1+\dfrac{e_1}{z-\beta_1},\qquad\Phi(z,0)=\dfrac{f_1}{z-\beta_1}
\end{equation}
is equivalent to a ``1-soliton solution''. Notice that by adding an arbitrary rational function, the solutions are asymptotically flat (at least on the symmetry axis). The majority of interesting cases fall into the $N$-soliton solutions possessing first order poles. Hence, let us consider the general N-soliton electrovac solution characterized by the Ernst potentials \cite{Ernst} on the symmetry axis given in terms of a polynomial quotient \cite{RMJ}:
\vspace{-0.1cm}
\begin{align}
\mathcal{E}(\rho=0,z)=e(z)=\dfrac{z^N+\sum_{l=1}^{N}{a_lz^{N-l}}}{z^N+\sum_{l=1}^{N}{b_lz^{N-l}}}=\dfrac{P(z)}{R(z)}\label{ePR},\\
\Phi(\rho=0,z)=f(z)=\dfrac{\sum_{l=1}^{N}{c_lz^{N-l}}}{z^N+\sum_{l=1}^{N}{b_lz^{N-l}}}=\dfrac{Q(z)}{R(z)}\label{fPR},
\end{align}
\noindent where $a_l,b_l,c_l$, $k=1,\cdots N$ are $3N$ arbitrary complex constants. It is worth noting that the higher order coefficients have been chosen to give an appropriate asymptotic behavior to the potentials at spatial infinity. Moreover, it is assumed that the previous quotients are irreducible and thus $R$ posses only roots of multiplicity one. Or in an equivalent form
\vspace{-0.1cm}
\begin{equation} e(z)=1+\sum_{l=0}^{N}\dfrac{e_l}{z-\beta_l},\qquad
f(z)=\sum_{l=1}^{N}\dfrac{f_l}{z-\beta_l},\end{equation}
\noindent while the coefficients $a_l,b_l,c_l$ are related to the ones $e_l,f_l,\beta_l$ through the relations
\begin{equation}
\begin{aligned}
e_l=\dfrac{P(\beta_l)}{\prod\limits_{k\ne l}^{N}(\beta_l-\beta_k)};\qquad\qquad & f_l=\dfrac{Q(\beta_l)}{\prod\limits_{k\ne l}^{N}(\beta_l-\beta_k)}; \\
R(\beta_l)=0.
\end{aligned}
\end{equation}
Papapetrou has shown that the most general line element for a spacetime with the prescribed symmetries, in the absence of cosmological constant, can be written in the following form \cite{Papapetrou1964zz}:
\begin{equation}\label{PapapetrouLE}
ds^2=F(dt^2-\omega d\varphi)^2-F^{-1}[e^{2\gamma}(d\rho^2+dz^2)+\rho^2d\varphi^2]
\end{equation}
Here the coordinate system utilized is composed of what is called the Weyl canonical coordinates $(t,\varphi,\rho,z)$. The metric functions depend upon the spatial coordinates $z\in(-\infty,\infty)$ and $\rho\in[0,\infty)$. The function $\omega$ is related to the angular momentum of the sources, which can represent rotations around the axis at $\rho=0$.
In 1995, E. Ruiz \emph{et al.} \cite{RMJ} found, via the Sibgatullin integral method \cite{Sibgatullin,Manko1993}, a very concise form to write the general solution of this $N$-soliton problem containing $3N$ complex arbitrary parameters. In other words, the Ernst potentials and their corresponding metric function are written in a very simple way. As a matter of fact, this general solution simplifies the study of some particular metrics with a real physically meaningful. The expressions for the metric functions and for the Ernst potentials are given by
\begin{equation}\label{ernstr}
\mathcal{E}(\rho,z)=\dfrac{E_+}{E_{-}};\qquad\qquad \Phi(\rho,z)=\dfrac{W}{E_{-}}
\end{equation}
\begin{equation} E_\pm =
\begin{vmatrix}
1 & 1 & \dots & 1\\
\pm 1 & \frac{r_1}{\alpha_1-\beta_1} & \dots & \frac{r_{2N}}{\alpha_{2N}-\beta_{1}} \\
\vdots & \vdots & \ddots & \vdots\\
\pm 1 & \frac{r_1}{\alpha_1-\beta_N} & \dots & \frac{r_{2N}}{\alpha_{2N}-\beta_{N}} \\
0 & \frac{h_1(\alpha_1)}{\alpha_1-\beta_1^*} & \cdots & \frac{h_1(\alpha_{2N})}{\alpha_{2N}-\beta_1^*}\\
\vdots & \vdots & \ddots & \vdots\\
0 & \frac{h_N(\alpha_1)}{\alpha_1-\beta_N^*} & \cdots & \frac{h_N(\alpha_{2N})}{\alpha_{2N}-\beta_N^*}\\
\end{vmatrix}\qquad\qquad
W=
\begin{vmatrix}
0 & f(\alpha_1) & \dots & f(\alpha_{2N})\\
-1 & \frac{r_1}{\alpha_1-\beta_1} & \dots & \frac{r_{2N}}{\alpha_{2N}-\beta_{1}} \\
\vdots & \vdots & \ddots & \vdots\\
-1 & \frac{r_1}{\alpha_1-\beta_N} & \dots & \frac{r_{2N}}{\alpha_{2N}-\beta_{N}} \\
0 & \frac{h_1(\alpha_1)}{\alpha_1-\beta_1^*} & \cdots & \frac{h_1(\alpha_{2N})}{\alpha_{2N}-\beta_1^*}\\
\vdots & \vdots & \ddots & \vdots\\
0 & \frac{h_N(\alpha_1)}{\alpha_1-\beta_N^*} & \cdots & \frac{h_N(\alpha_{2N})}{\alpha_{2N}-\beta_N^*}\\
\end{vmatrix}
\end{equation}
\begin{equation}
F=\frac{D}{2E_{+}E_{-}},
\qquad e^{2\gamma}=\frac{D}{2K_{0}K_{0}^{*}\prod\limits_{n=1}^{2N}r_{n}},
\qquad \omega=\frac{2 \operatorname{Im}\left(E_{-} H^{*}-E^{*}_{-} G-W I^{*}\right)}{D},
\end{equation}
\vspace{-0.1cm}
\begin{align}
D&=E_{+} E^{*}_{-}+E^{*}_{+} E_{-}+2 W W^{*}, \nonumber\\\nonumber\\
H&=
\begin{vmatrix}
z & 1 & \cdots & 1 \\
-\beta_{1} & \frac{r_{1}}{\alpha_{1}-\beta_{1}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{1}} \\
\vdots & \vdots & \ddots & \vdots \\
-\beta_{N} & \frac{r_{1}}{\alpha_{1}-\beta_{N}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{N}} \\
e^{*}_{1} & \frac{h_{1}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{1}} & \cdots & \frac{h_{1}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{1}} \\
\vdots & \vdots & \ddots & \vdots \\
e^{*}_{N} & \frac{h_{N}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{N}} & \cdots & \frac{h_{N}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{N}}
\end{vmatrix},\qquad
G=\begin{vmatrix}
0 & g_{1} & \dots & g_{2N}\\
-1 & \frac{r_{1}}{\alpha_{1}-\beta_{1}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{1}} \\
\vdots & \vdots & \ddots & \vdots \\
-1 & \frac{r_{1}}{\alpha_{1}-\beta_{N}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{N}} \\
0 & \frac{h_{1}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{1}} & \cdots & \frac{h_{1}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{1}} \\
\vdots & \vdots & \ddots & \vdots \\
0 & \frac{h_{N}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{N}} & \cdots & \frac{h_{N}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{N}}
\end{vmatrix},\qquad\\
I&=\begin{vmatrix}
\sum\limits_{l=1}^{N} f_{l} & 0 & f\left(\alpha_{1}\right) & \dots & f\left(\alpha_{2 N}\right) \\
z & 1 & 1 & \dots & 1 \\
-\beta_{1} & -1 & \frac{r_{1}}{\alpha_{1}-\beta_{1}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{1}} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-\beta_{N} & -1 & \frac{r_{1}}{\alpha_{1}-\beta_{N}} & \cdots & \frac{r_{2 N}}{\alpha_{2 N}-\beta_{N}} \\
e^{*}_{1} & 0 & \frac{h_{1}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{1}} & \cdots & \frac{h_{1}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{1}} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
e^{*}_{N} & 0 & \frac{h_{N}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{N}} & \cdots & \frac{h_{N}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{N}}
\end{vmatrix}, \qquad
K_{0}=
\begin{vmatrix}
1 & \cdots & 1 \\
\frac{1}{\alpha_{1}-\beta_{1}} & \cdots & \frac{1}{\alpha_{2 N}-\beta_{1}} \\
\vdots & \vdots & \ddots \\
\frac{1}{\alpha_{1}-\beta_{N}} & \cdots & \frac{1}{\alpha_{2 N}-\beta_{N}} \\
\frac{h_{1}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{1}} & \cdots & \frac{h_{1}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{1}} \\
\vdots & \vdots & \ddots \\
\frac{h_{N}\left(\alpha_{1}\right)}{\alpha_{1}-\beta^{*}_{N}} & \cdots & \frac{h_{N}\left(\alpha_{2 N}\right)}{\alpha_{2 N}-\beta^{*}_{N}}
\end{vmatrix}, \\\nonumber\\
g_{n}&=r_{n}+\alpha_{n}-z, \qquad h_{l}(\alpha_{n})=e_{l}^{*}+2f_{l}^{*}f(\alpha_{n}),
\end{align}
\noindent where $r_n=\sqrt{\rho^2+(z-\alpha_n)^2}$ are the distances from the value $\alpha_{n}$ defining the location of the sources to any arbitrary point $(\rho,z)$ off the symmetry axis. In this case, $\alpha_n$ are the $2N$ roots satisfying the following characteristic equation:
\begin{equation} P(z)R^*(z)+P^*(z)R(z)+2Q(z)Q^*(z)=0. \end{equation}
An important point to be underlined from this solution is the fact that at least a priori, these $3N$ complex parameters do not necessarily have a physical meaning unless we link them first to the Geroch-Hansen multipole moments \cite{GerochII,HP,Simon,Sotiriou}. A first development that helped us to contour this problem was a previous work provided by by Hernández-Pastora \cite{pastora} and by Manko \emph{et al.} \cite{MankoRuiz}, in which has been analyzed vacuum solutions ($\Phi=0$) relating the $2N$ parameters $a_l$ and $b_l$ with the corresponding $2N$ multipole moments.
\vspace{-0.1cm}
\section{Relations between the Ernst potentials and multipole moments in vacuum case}\label{multipolevacuum}
\vspace{-0.1cm}
As has been shown in \cite{Simon,HP,fodor2020calculation}, the Geroch-Hansen multipole moments \cite{Hansen,GerochII}, $P_n$ and $Q_n$ , for a given stationary axisymmetric exact solution can be obtained from their corresponding coefficients expansion of the Ernst potentials $\xi$ and $q$ evaluated on the symmetry axis. That is when $z\rightarrow\infty$, namely,
\vspace{-0.1cm}
\begin{equation}
\xi=\sum_{k=0}^{\infty}m_{k}z^{-k-1}, \qquad q=\sum_{k=0}^{\infty}q_{k}z^{-k-1}\label{mqmultipole},
\end{equation}
\noindent being $\xi$ and $q$ related to $\mathcal{E}$ and $\Phi$ in the form
\vspace{-0.1cm}
\begin{equation} \mathcal{E}=\dfrac{1-\xi}{1+\xi}, \qquad \Phi=\dfrac{q}{1+\xi}. \label{ernstxiq} \end{equation}
the coefficients $m_k$ and $q_k$ are related to the multipole moments $P_n$ and $Q_n$. We are interested in showing that indeed it is possible to associate the Geroch-Hansen coefficients, $m_k$ and $q_k$, with the constants $a_l$, $b_l$ and $c_l$, and hence, to characterize physically the Ernst equations. As it is shown in \cite{Hoenselaers86,Simon}, the multipole moments $P_k$ and $Q_k$ can be written in terms of the the power series expansion coefficients $m_k$ and $q_k$, of the Ernst potentials on the symmetry axis\footnote{Once $m_k$ and $q_k$ are known, the multipole moments $P_k$ and $Q_k$ can be constructed or vice-versa.}. The real and imaginary parts of the multipole $P_k$ are associated with the mass and angular multipole moments, respectively \cite{Hansen}. In the same way, the real and imaginary parts of the multipole $Q_k$ are related to the electric and magnetic field multipole moments \cite{Simon}. As pointed out in reference \cite{HP}, it is enough to know the behavior of the Ernst potentials $\mathcal{E}$ and $\Phi$ on the symmetry axis to perform an analytic continuation on them into the whole space \cite{HauserErnst,stephani_kramer_maccallum_hoenselaers_herlt_2003}. Hence, the multipole coefficients $m_k$ and $q_k$ seem to have a significant hole in the present development. Moreover, it is possible to show that the multipole moments uniquely characterize the geometry of the spacetime \cite{Simon-Beig,Simon}. The arbitrariness of these coefficients brings us the question of which might be the condition they should fulfill to satisfy the relation between them and $a_l$, $b_l$ and $c_l$. The substitution of Eqs.\ \eqref{ePR}-\eqref{fPR} into \eqref{mqmultipole} allows us to find
\vspace{-0.1cm}
\begin{eqnarray} \dfrac{R(z)-P(z)}{R(z)+P(z)}=\sum_{k=0}^{\infty}m_{k}z^{-k-1},\qquad
\dfrac{2\,Q(z)}{R(z)+P(z)}=\sum_{k=0}^{\infty}q_{k}z^{-k-1}\label{qmutipolerelations}.\end{eqnarray}
Before continuing, first we are going to revisit the analysis employed in references \cite{pastora,MankoRuiz} regarding the vacuum case. So, taking into account only the first equality Eq.\ (\ref{qmutipolerelations}), in the absence of electromagnetic field, after equating the coefficients with the same powers of $z$, we obtain
\vspace{-0.1cm}
\begin{align} \label{mank}
&\frac{1}{2}(b_1-a_1) = m_0, \nonumber\\
&\frac{1}{2}(b_2-a_2)=m_1+\frac{1}{2}(b_1+a_1)m_0, \nonumber\\
&\vdots\nonumber\\
&\frac{1}{2}(b_N-a_N)=m_{N-1}+\frac{1}{2}(b_1+a_1)m_{N-2}+\cdots+\frac{1}{2}(b_{N-1}+a_{N-1})m_0, \nonumber\\
&0=m_n+\frac{1}{2}(b_1+a_1)m_{n-1}+\cdots+\frac{1}{2}(b_N+a_N)m_{n-N}\text{, for } n\ge N.
\end{align}
It follows that the simple redefinitions $A_l=\frac{1}{2}(b_l-a_l)$, $B_l=\frac{1}{2}(b_l+a_l)$ with $l=1,2\cdots N$, $A_0=B_0=1$, permits us to get straightforwardly the following system of algebraic equations
\vspace{-0.1cm}
\begin{align}
&A_{n+1}=\sum_{l=0}^{n}B_l\,m_{n-l},\qquad n=0,1,\dots,N-1, \label{Aequantion}\\
&0=\sum_{l=0}^{N}B_l\,m_{n-l},\qquad n\ge N. \label{Bequation}
\end{align}
The above set of equations constitute an algebraic system with infinite equations for a finite number of variables, which may give us the opportunity to write $a_l$ and $b_l$ in terms of $m_n$. In order to describe the $N$-soliton problem, the authors in \cite{RMJ} used a set of $2N$ arbitrary parameters, $a_l$ and $b_l$. Thus, in principle, it would be possible to use a set of 2$N$ coefficients $m_k$ to describe such a problem. In what follows, it is outlined a generalization of the compatibility condition which ensures a similar system.
In order to find the $N$ variables $B_l$, $N$ equations are needed. Then, inside the infinity set of parameters $m_n$, take $N$ elements $\{m_{n_1},m_{n_2},\dots,m_{n_N}\}$ in such way that $n_i\ge N$ with $i=1,2,\dots,N$. Consider, now, the equation \eqref{Bequation} for this set of $n_i$.
\vspace{-0.1cm}
\begin{equation}\sum_{l=1}^{N}B_l\,m_{n_{i}-l}=-m_{n_{i}}\label{Bequation2},\end{equation}
\noindent or equivalently
\vspace{-0.1cm}
\begin{equation}
\begin{pmatrix}
m_{n_{1}-1} & m_{n_{1}-2} & \cdots & m_{n_{1}-N+1} & m_{n_{1}-N}\\
m_{n_{2}-1} & m_{n_{2}-2} & \cdots & m_{n_{2}-N+1} & m_{n_{2}-N}\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
m_{n_{N}-1} & m_{n_{N}-2} & \cdots & m_{n_{N}-N+1} & m_{n_{N}-N}\\
\end{pmatrix}
\begin{pmatrix}
B_1\\
B_2\\
\vdots\\
B_N
\end{pmatrix}
=-
\begin{pmatrix}
m_{n_{1}}\\
m_{n_{2}}\\
\vdots\\
m_{n_{N}}\\
\end{pmatrix}\label{Bequation3},
\end{equation}
Defining a new object, $L_i$, as a $i\times i$ matrix:
\vspace{-0.1cm}
\begin{equation}
L_i=\begin{pmatrix}
m_{n_{1}-N+i-1} & m_{n_{2}-N+i-1} & \cdots & m_{n_{i-1}-N+i-1} & m_{n_{i}-N+i-1}\\
m_{n_{1}-N+i-2} & m_{n_{2}-N+i-2} & \cdots & m_{n_{i-1}-N+i-2} & m_{n_{i}-N+i-2}\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
m_{n_{1}-N} & m_{n_{2}-N} & \cdots & m_{n_{i-1}-N} & m_{n_{i}-N}\\
\end{pmatrix},
\end{equation}
Where it is straightforward to observe that the system \eqref{Bequation3} only have solution when $\det L_N\ne 0$. In order to shortening the notation on the following equations, let us use $|-|$ to correspond the determinant of a matrix. Using the Cramer's rule to find the coefficients $B_l$, we obtain:
\vspace{-0.1cm}
\begin{equation}
B_l=(-1)^l |L_N|^{-1}
\begin{vmatrix}
m_{n_{1}} & m_{n_{1}-1} & \cdots & m_{n_{1}-(l-1)} & m_{n_{1}-(l+1)} & \cdots & m_{n_{1}-N}\\
m_{n_{2}} & m_{n_{2}-1} & \cdots & m_{n_{2}-(l-1)} & m_{n_{2}-(l+1)} & \cdots & m_{n_{2}-N}\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\
m_{n_{N}} & m_{n_{N}-1} & \cdots & m_{n_{N}-(l-1)} & m_{n_{N}-(l+1)} & \cdots & m_{n_{N}-N}\\
\end{vmatrix}.
\end{equation}
Due to this high symmetric structure, it is possible to rewrite $B_l$ as:
\vspace{-0.1cm}
\begin{equation}
B_l=|L_N|^{-1}
\begin{vmatrix}
0 & m_{n_{1}} & m_{n_{2}} & \cdots & m_{n_{N-1}} & m_{n_{N}}\\
0 & & & & &\\
\vdots & & & & &\\
1 & & & L_N & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix},
\end{equation}
\noindent where the row corresponding to the ``1'' in the first column is the ($l+1$)-th row. By using this result for $B_l$ the coefficients $A_l$ can be found by means of Eq.\ \eqref{Aequantion} and , in a similar way, can be written as:
\begin{equation}
A_{l+1}=|L_N|^{-1}
\begin{vmatrix}
m_l & m_{n_{1}} & m_{n_{2}} & \cdots & m_{n_{N-1}} & m_{n_{N}}\\
m_{l-1} & & & & &\\
\vdots & & & & &\\
m_0 & & & L_N & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix}.
\end{equation}
Nonetheless, $A_l$ and $B_l$ are described in terms of, at most, $N^2+2N$ independent coefficients $m_k$ and by hypothesis it asserts that they must be written in terms of $2N$ coefficients $m_k$. Therefore, it is necessary to restrict the set $\{m_{n_{i}}\}$, which is solution of the equation \eqref{Bequation2}, and its condition is only respected in case when the set $\{m_{n_{i}}\}$ is chosen with $n_1=N$, $n_2=N+1$, $\dots$, $n_N=2N-1$. Then, $|L_i|$ can be written as:
\vspace{-0.1cm}
\begin{equation}
|L_i|=\begin{vmatrix}
m_{i-1} & m_{i} & \cdots & m_{2i-3} & m_{2i-2}\\
m_{i-2} & m_{i-1} & \cdots & m_{2i-4} & m_{2i-3}\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
m_{0} & m_{1} & \cdots & m_{i-2} & m_{i-1}\\
\end{vmatrix},
\end{equation}
And then, $A_l$ and $B_l$, written in terms of $2N$ $m_k$'s, take the form:
\vspace{-0.1cm}
\begin{equation}
A_{l+1}=|L_N|^{-1}
\begin{vmatrix}
m_l & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
m_{l-1} & & & & &\\
\vdots & & & & &\\
m_0 & & & L_N & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix},
\end{equation}
\begin{equation}
B_l=|L_N|^{-1}
\begin{vmatrix}
0 & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
0 & & & & &\\
\vdots & & & & &\\
1 & & & L_N & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix}.
\end{equation}
Notice that in $B_l$ the only elements that do not repeat are $m_0$ and $m_{2N-1}$, since the diagonals on the principal direction are constituted by equal elements, with the exception of the elements in the first column. Because of that, $m_1$ appears twice, $m_2$ appears three times, until $m_{N-1}$ and $m_N$ which appear $N$ times, and then, $m_{N+1}$ appears $N-1$ times and so on.
For completeness, we are going to deduce now the condition that coefficients must satisfy when calculating $L_i$ for $i>N$. In order to complete this statement, let us now consider
\vspace{-0.1cm}
\begin{equation}
|L_{N+1}|=\begin{vmatrix}
m_{N} & m_{N+1} & \cdots & m_{2N-1} & m_{2N}\\
m_{N-1} & m_{N} & \cdots & m_{2N-2} & m_{2N-1}\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
m_{0} & m_{1} & \cdots & m_{N-1} & m_{N}\\
\end{vmatrix},
\end{equation}
\noindent and after making a cofactor expansion, we find:
\vspace{-0.1cm}
\begin{equation}
|L_{N+1}|=(-1)^N| L_N| \sum_{k=0}^{N}B_k m_{2N-k}.
\end{equation}
However, by hypothesis, the sum $\sum\limits_{k=0}^{N}B_k m_{n-k}$ is equal to zero for all $n\ge N$. Thus, $m_k$ must be in such way that $|L_{N+1}|=0$. By induction, it is straightforward to see that all determinants $|L_n|=0$ for all $n> N$, proving the following lemma, which was first stated in reference \cite{pastora} and revisited in \cite{MankoRuiz} reads:
\begin{lemma} \label{lemma1}
Given all multipole coefficients $m_i$, once fixed the set of coefficients $\{m_{i}\}$ with $i=0,1,\dots, 2N-1$, the necessary and sufficient condition for this set to describe the behaviour of the Ernst potentials on the symmetry axis as a polynomial quotient \eqref{ePR} is that the determinant $|L_n|$ be nonzero for $n=N$ and zero for all $n>N$.
\end{lemma}
\vspace{-0.2cm}
\section{Relations between the Ernst potentials and multipole moments in electrovacuum case}\label{multipoleelectrovacuum}
\vspace{-0.2cm}
As mentioned before, Lemma \ref{lemma1} is not a new result, however, its general proof might remained unnoticed in literature since it was first presented in the thesis \cite{pastora} but it became well-known in \cite{MankoRuiz} without proof. As a matter of fact, one may bear in mind to consider a similar analysis outlined in the vacuum case but now applied to the electrovacuum case, which to our knowledge has been ignored due to its complexity, in order to extend the result already given in \cite{pastora}. In order to try to generalize these results for the cases where the electromagnetic field is present, a similar analysis for the equation \eqref{qmutipolerelations} will be done, so that in the end the $3N$ variables $a_l$, $b_l$ and $c_l$ can be written in terms of $3N$ coefficients $m_k$ and $q_k$ related with the multipole moments. Equating the coefficients with the same powers of $z$, we find:
\vspace{-0.1cm}
\begin{align}
&c_1 = q_0,\nonumber\\
&c_2=q_1+\frac{1}{2}(b_1+a_1)q_0\nonumber\\
&\vdots\nonumber\\
&c_N=q_{N-1}+\frac{1}{2}(b_1+a_1)q_{N-2}+\cdots+\frac{1}{2}(b_{N-1}+a_{N-1})q_0\nonumber\\
&0=q_n+\frac{1}{2}(b_1+a_1)q_{n-1}+\cdots+\frac{1}{2}(b_N+a_N)q_{n-N}, \quad \text{, for } n\ge N.
\end{align}
Such system can be summarized into:
\vspace{-0.1cm}
\begin{align}
&c_{n+1}=\sum_{l=0}^{n}B_l\,q_{n-l},\qquad n=0,1,\dots,N-1,\label{cequantion}\\
&0=\sum_{l=0}^{N}B_l\,q_{n-l},\qquad n\ge N. \label{Bqequation}
\end{align}
Notice that the above Eqs.\ (\ref{cequantion})-(\ref{Bqequation}) have the same structure as the Eqs.\ \eqref{Aequantion}-\eqref{Bequation}. In addition, the same function $B_l$, which was already evaluated in terms of the coefficients $m_k$, will be evaluated now in terms of the coefficients $q_k$. Since it is the same function $B_l$, when it is written in terms of $m_k$ or $q_k$ it must be equivalent.
Due to the fact that equation's structure contains the same aspect as in the vacuum case, we will introduce a new index here to the determinant $L_i$ in order to differentiate whether it is written in terms of $m_k$ or $q_k$, that is, $L_{i,m}$ and $L_{i,q}$. That is:
\begin{equation}
L_{i,q}=\begin{pmatrix}
q_{i-1} & q_{i} & \cdots & q_{2i-3} & q_{2i-2}\\
q_{i-2} & q_{i-1} & \cdots & q_{2i-4} & q_{2i-3}\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
q_{0} & q_{1} & \cdots & q_{i-2} & q_{i-1}\\
\end{pmatrix}.
\end{equation}
Therefore, the equation for $B_l$ in terms of $q_k$ is given by:
\vspace{-0.1cm}
\begin{equation}
B_l=|L_{N,q}|^{-1}
\begin{vmatrix}
0 & q_{N} & q_{N+1} & \cdots & q_{2N-2} & q_{2N-1}\\
0 & & & & &\\
\vdots & & & & &\\
1 & & & L_{N,q} & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix}.
\end{equation}
Since the variable $B_l$ must be the same independently of whether it is written in terms of the $m_k$ or $q_k$, the following relation is obtained:
\begin{align}\label{Bcondition}
&B_l=\\&|L_{N,m}|^{-1}
\begin{vmatrix}
0 & m_{N} & m_{N+1} & \cdots & m_{2N-1}\\
0 & & & &\\
\vdots & & & &\\
1 & & & L_{N,m} & \\
\vdots & & & &\\
0 & & & & \\
\end{vmatrix}
=|L_{N,q}|^{-1}
\begin{vmatrix}
0 & q_{N} & q_{N+1} & \cdots & q_{2N-1}\\
0 & & & &\\
\vdots & & & &\\
1 & & & L_{N,q} & \\
\vdots & & & &\\
0 & & & & \\
\end{vmatrix}\nonumber
\end{align}
Given that the value of $B_l$ is defined the for $m_k$ coefficients, from the equality above, we conclude that, for a fixed $l$, the set of variables $q_k$, loose one degree of freedom. Furthermore, knowing that $l$ ranges from 1 to $N$, one notices that $N$ of the $|q_k|$ variables, for $k=0,1\cdots 2N-1$, are not free, i.e., $N$ variables from the set $q_k$ can be described as a function of $2N$ variables from $m_k$ and $N$ variables of $q_k$. Consequently, one can generalize lemma \ref{lemma1}.
\begin{lemma}\label{lemma2}
Given all multipole coefficients $m_i$ and $q_i$, once fixed a set of coefficients $\{m_{i}\}$ with $i=0,1,\dots, 2N-1$ and a subset of $N$ coefficients $q_i$ contained in $\{q_i\}$ with $i=0,1,\dots, 2N-1$, the necessary and sufficient conditions for those $3N$ variables to describe the behaviour of the Ernst potentials on the symmetry axis as polynomial quotient \eqref{ePR} and \eqref{fPR} is that the determinant $|L_n,_{m}^{q}|$ be nonzero for $n=N$ and zero for all $n>N$ and that equation \eqref{Bcondition} is valid.
\end{lemma}
The final representation for the $3N$ variables $A_l$, $B_l$ e $c_l$ written in terms of $2N$ coefficients $m_k$ and $N$ coefficients $q_k$ is given by
\vspace{-0.1cm}
\begin{equation}
A_{l+1}=|L_{N,m}|^{-1}
\begin{vmatrix}\label{Al}
m_l & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
m_{l-1} & & & & &\\
\vdots & & & & &\\
m_0 & & & L_{N,m} & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix},
\end{equation}
\vspace{-0.1cm}
\begin{equation}\label{Bl}
B_l=|L_{N,m}|^{-1}
\begin{vmatrix}
0 & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
0 & & & & &\\
\vdots & & & & &\\
1 & & & L_{N,m} & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix},
\end{equation}
\vspace{-0.1cm}
\begin{equation}\label{cl}
c_l=|L_{N,m}|^{-1}
\begin{vmatrix}
q_l & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
q_{l-1} & & & & &\\
\vdots & & & & &\\
q_0 & & & L_{N,m} & & \\
\vdots & & & & &\\
0 & & & & & \\
\end{vmatrix}.
\end{equation}
\noindent where the relation between the variables $A_l$ e $B_l$ with the variables $a_l$ e $b_l$ defined in \eqref{ePR} and \eqref{fPR} is given by
\begin{equation}\label{ab}
a_l=B_l-A_l, \qquad b_l=B_l+A_l.
\end{equation}
Thereby, we conclude the Section showing that the electrovacuum $N$-soliton solution on the z-axis can be written in terms of the multipole moments in the following compact way:
\begin{equation}
P(z)=z^{N}+\sum_{l=1}^{N}a_{l}z^{N-l} = z^{N}+\sum_{l=1}^{N}(B_{l}-A_{l})z^{N-l}=
\end{equation}
\begin{equation}
=\sum_{l=0}^{N}B_{l}z^{N-l}-\sum_{l=1}^{N}\sum_{k=0}^{l-1}B_{k}m_{l-1-k}z^{N-l} = \sum_{l=0}^{N}B_{l}z^{N-l}-\sum_{l=0}^{N-1}\sum_{k=0}^{l}B_{k}m_{l-k}z^{N-l-1}
\end{equation}
\begin{equation}
=\sum_{l=0}^{N}B_{l}z^{N-l}-\sum_{l=0}^{N-1}B_l\sum_{k=0}^{N-1-l}m_{k}z^{N-l-k-1},
\end{equation}
where the first and second terms from the above equality can be written as
\begin{equation}
\sum_{l=0}^{N}B_{l}z^{N-l}=|L_N|^{-1}
\begin{vmatrix}
z^N & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\
z^{N-1} & & & & &\\
\vdots & & & L_N & &\\
z & & & & & \\
1 & & & & & \\
\end{vmatrix},
\end{equation}
\begin{equation}
\sum_{l=0}^{N-1}B_l\sum_{k=0}^{N-1-l}m_{k}z^{N-l-k-1} =|L_{N,m}|^{-1}
\begin{vmatrix}
\sum\limits_{k=0}^{N-1}m_{k}z^{N-k-1} & m_{N} & m_{N+1} & \cdots & m_{2N-1}\\[10pt]
\sum\limits_{k=0}^{N-2}m_{k}z^{N-k-2} & & & &\\
\vdots & & & L_{N,m} &\\
m_0 & & & & \\
0 & & & & \\
\end{vmatrix},
\end{equation}
Hence, we can write $P(z)$, $R(z)$ and $Q(z)$ in the very simple form
\begin{equation}
P(z)=|L_{N,m}|^{-1}
\begin{vmatrix}
z^{N}-\sum\limits_{k=0}^{N-1}m_{k}z^{N-k-1} & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\[10pt]
z^{N-1}-\sum\limits_{k=0}^{N-2}m_{k}z^{N-k-2} & & & & &\\
\vdots & & & L_{N,m} & &\\
z-m_0 & & & & & \\
1 & & & & & \\
\end{vmatrix},
\end{equation}
\begin{equation}
R(z)=|L_{N,m}|^{-1}
\begin{vmatrix}
z^{N}+\sum\limits_{k=0}^{N-1}m_{k}z^{N-k-1} & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\[10pt]
z^{N-1}+\sum\limits_{k=0}^{N-2}m_{k}z^{N-k-2} & & & & &\\
\vdots & & & L_{N,m} & &\\
z+m_0 & & & & & \\
1 & & & & & \\
\end{vmatrix},
\end{equation}
\begin{equation}
Q(z) =|L_{N,m}|^{-1}
\begin{vmatrix}
\sum\limits_{k=0}^{N-1}q_{k}z^{N-k-1} & m_{N} & m_{N+1} & \cdots & m_{2N-2} & m_{2N-1}\\[10pt]
\sum\limits_{k=0}^{N-2}q_{k}z^{N-k-2} & & & & &\\
\vdots & & & L_{N,m} & &\\
q_0 & & & & & \\
0 & & & & & \\
\end{vmatrix}.
\end{equation}
Possessing in hands Lemma \ref{lemma1} and \ref{lemma2}, in order to ensure that the multipole coefficients will describe a solution of the Ernst potentials which are rational on the symmetry axis, we need all coefficients $m_i$ and $q_i$, since we need to ensure that $|L_N,_{m}^{q}|$ be nonzero for $n=N$ and zero for all $n>N$. However, the solution will only be described in terms of $3N$ variables. This lead us to the next part of this work that is to prove that, given a set of $3N$ multipole coefficients, we they will describe a behaviour of a Ernst potentials which are rational on the symmetry axis.
\subsection{Multipole moments of the N-Soliton solution}
So far, we have given the relations and conditions for writing the $3N$ parameters of the N-soliton solution, $a_l$, $b_l$ and $c_l$, in terms of the multipole coefficients $m_l$ and $q_l$. Now, a stronger result can be achieved by studying the inverse relation of these coefficients. That is, we will show that , in fact, $L_{N+1},_{m}^{q}$ is always zero for the N-soliton solution, and the conditions in Lemmas \ref{lemma1} and \ref{lemma2} are always satisfied for such solution. For this purpose, consider the series below:
\begin{equation}
\dfrac{\sum\limits_{l=1}^{N}e_l z^{N-l}}{z^N+\sum\limits_{k=1}^{N}d_k z^{N-k}}
\end{equation}
This series has the same shape as in equation \eqref{qmutipolerelations} (they are the same apart from a factor). Therefore, in order to write $m_l$ and $q_l$ in terms of $a_l$, $b_l$ and $c_l$, it is necessary to see how to expand the above series in terms of negative powers of $z$. By canceling the term $z^N$ and focusing on the denominator, we notice that it is possible to expand it in the following power series
\begin{equation}
\dfrac{1}{1+\sum\limits_{k=1}^{N}d_k z^{-k}}=\sum_{j=0}^{\infty}(-1)^j\left(\sum_{k=1}^{N}d_k z^{-k}\right)^j.
\end{equation}
However
\begin{equation}
(d_1 z^{-1}+d_2 z^{-2}+\cdots+ d_N z^{-N})^j=\sum_{k_1+k_2+\cdots+k_N=j} \dfrac{j!}{k_1!k_2!\cdots k_N!}\prod_{t=1}^{N}(d_t z^{-t})^{k_t},
\end{equation}
therefore
\begin{equation}
\dfrac{\sum\limits_{l=1}^{N}e_l z^{N-l}}{z^N+\sum\limits_{k=1}^{N}d_k z^{N-k}}=\sum\limits_{l=1}^{N}e_l z^{-l}\sum_{j=0}^{\infty}\sum_{k_1+k_2+\cdots+k_N=j} \dfrac{j!}{k_1!k_2!\cdots k_N!}\prod_{t=1}^{N}(d_t z^{-t})^{k_t}.
\end{equation}
Now, we need to find the general coefficient for this power series. That is, we must write
\begin{equation}
\dfrac{\sum\limits_{l=1}^{N}e_l z^{N-l}}{z^z+\sum\limits_{k=1}^{N}d_k z^{N-k}}\equiv \sum_{\alpha=0}^{\infty}h_\alpha z^{-\alpha-1},
\end{equation}
and find the coefficients $h_\alpha$. After some simple calculations, we find
\begin{equation}
h_{\alpha}=\sum_{l=1}^{N}e_l \theta_{l,\alpha},
\end{equation}
where
\begin{equation}
\theta_{l,\alpha}=\left\{\begin{matrix}
&0, \mbox{ if } \alpha < l,\\
&\sum\limits_{k_1+2 k_2+\cdots+N k_N=\alpha-l}(-1)^{k_1+k_2+\cdots+k_N}\dfrac{(k_1+k_2+\cdots+k_N)!}{k_1!k_2!\cdots k_N!}\prod\limits_{t=1}^{N}(d_t)^{k_t}, \mbox{ if } \alpha\ge l.
\end{matrix}\right.
\end{equation}
From the above equation, it is possible to find a relation between the $h_\alpha$
\begin{equation}
h_{\alpha+N}=-\sum_{l=1}^{N}d_l h_{\alpha+N-l}.
\end{equation}
This shows that $h_{\alpha+N}$ is a linear combination of the set $\{h_\alpha,h_{\alpha+1},\cdots, h_{\alpha+N-1}\}$ with fixed coefficients $d_l$. This implies that the last column of the matrix whose determinant is $L_{N+1},_{m}^{q}$ is a linear combination of the first $N$ columns. Finally, we can write:
\begin{equation}\label{multipolecoe}
m_{\alpha}=\frac{1}{2}\sum_{l=1}^{N}(b_l-a_l) \theta_{l,\alpha}, \qquad\qquad q_{\alpha}=\sum_{l=1}^{N}c_l \theta_{l,\alpha},
\end{equation}
\begin{equation}
\theta_{l,\alpha}=\left\{\begin{matrix}
&0, \mbox{ if } \alpha < l,\\
&\sum\limits_{k_1+2 k_2+\cdots+N k_N=\alpha-l}(-1)^{k_1+k_2+\cdots+k_N}\dfrac{(k_1+k_2+\cdots+k_N)!}{k_1!k_2!\cdots k_N!}\prod\limits_{t=1}^{N}\left(\dfrac{b_l+a_l}{2}\right)^{k_t}, \mbox{ if } \alpha\ge l.
\end{matrix}\right.
\end{equation}
For this reason, not only $|L_{N+1},_{m}^{q}|$ but $L_{N+k},_{m}^{q}\ k\geq 1,$ is zero for all $N$-soliton solutions. Moreover, this implies that all $m_n$ are determined for $n\ge 2N$, and $q_n$ are determined for $n\ge N$. Finally, using this, we can improve Lemma \ref{lemma2} and state the best form of our result.
\begin{theorem}\label{theo1}
As in Lemma \ref{lemma2} fix a set of coefficients $\{m_{i}\}_{i=1}^{2N}$ and a subset $\{q_{i_1},\ q_{i_2},\ \dots , q_{i_N}\}\subset\{q_i\}_{i=1}^{2N}$. Then, these $3N$ variables describe the behavior of Ernst potentials on the symmetry axis as polynomial quotient \eqref{ePR} and \eqref{fPR} if and only if the determinant $|L_N,_{m}^{q}|\neq0$.
\end{theorem}
Another interesting outcome evolves from these previous results. First, notice that Lemma \ref{lemma2} imposes two conditions in the multipoles in order to describe a $N$-soliton solution, the equation \eqref{Bcondition} be valid and $|L_N,_{m}^{q}|\neq0$. However, we proved that equation \eqref{Bcondition} is always valid in the case of a $N$-soliton solution. Arriving in the Theorem \ref{theo1}, which states that $|L_N,_{m}^{q}|\neq0$ is the only condition to establish the relation between the multipoles and the solution. But notice that, if we have $|L_N,_{m}^{q}|=0$ and $|L_{N-1},_{m}^{q}|\neq0$, we are describing a system of $N-1$ soliton, and the equations are still valid. However, due to the results in the present Section, if a given stationary axisymmetric spacetime, solution of the Einstein-Maxwell equations, such that its multipole moments satisfy \eqref{Bcondition} and $|L_N,_{m}^{q}|\neq0$, then this solution can be approximated as a $N$-soliton solution. Moreover, it worth mentioned that it is simple to construct exact spacetimes possessing finite coefficients $m_k$ and $q_k$. When $a_l=-b_l$ for $l=1,2,\cdots, N$, then the value of $\theta_{l,\alpha}$ is $0$ for $l\neq\alpha$ and $1$ for $l=\alpha$.
In order to exemplify how the results of the present paper should be interpreted and used in practical terms, we give a series of examples below.
\section{Solutions from prescribed multipole moments}\label{examples}
\subsection{2-Soliton solution}
Based on the $2$-soliton solution, let us find the solution associated with the given multipoles moments. In order to clarify how the method derived in the present paper should be used, we will start with know multipole moments are written and then construct the metric solution associated with them. The $2$-soliton solution can describe several interesting cases, from the Tomimatsu-Sato with $\delta=2$ to two interacting Kerr-Newman-like black holes \cite{TomiSato,1981GReGr..13..195N,KRAMER1980259,Herdeiro_2008,CABRERAMUNGUIA2020135945}. However, the physical parameterization of these solutions is somehow obscure due to the fact that the direct relation between the multipole coefficients and the $N$-soliton solutions has not been made until now. Consider then, the Ernst potentials of the $2$-soliton solution in the symmetry axis:
\begin{align}
\mathcal{E}(\rho=0,z)=\dfrac{z^2+a_1 z+a_2}{z^2+b_1 z+b_2},\\
\Phi(\rho=0,z)=\dfrac{c_1 z+c_2}{z^2+b_1 z+b_2}
\end{align}
Notice here that the $2$-soliton solution is given in terms of $6$ arbitrary parameters $\{a_l, b_l, c_l\}$, $l=1,2$. Let us see how these parameters are connected to the multipole moments. Using the equation \eqref{multipolecoe}, the multipole coefficients are written as:
\begin{align*}
&m_0=\dfrac{b_1-a_1}{2}\,,\\
&m_1=\frac{a_1^2-b_1^2+2 b_2-2 a_2}{4}\,,\\
&m_2=\frac{b_1^3-a_1^3-a_1^2 b_1+a_1 \left(4 a_2+b_1^2\right)-4 b_1 b_2}{8}\,,\\
&m_3=\frac{ (a_1^2-b_1^2) \left((a_1+b_1)^2-4 (a_2+b_2)\right)-2 (a_2-b_2) \left((a_1+b_1)^2-2 (a_2+b_2)\right)}{16}\,,
\end{align*}
And
\begin{align*}
&q_0=c_1\,,\\
&q_1=c_2-\frac{c_1 (a_1+b_1)}{2}\,,\\
&q_2=\frac{c_1 (a_1+b_1)^2-2 c_2 (a_1+b_1)-2 c_1 (a_2+b_2)}{4}\,,\\
&q_3=\frac{4 c_1 (a_1+b_1) (a_2+b_2)-c_1 (a_1+b_1)^3+2 c_2 (a_1+b_1)^2-4 c_2 (a_2+b_2)}{8} \,.
\end{align*}
Theorem \ref{theo1} give us the necessary and sufficient conditions for the coefficients $m_k's$ and $q_k's$ describe a $2$-soliton solution. With their form in hands (written in terms of $\{a_l, b_l, c_l\}$), it is easy to show from the above discussion that it is needed only 6 parameters to describe the Ernst potentials. First of all, notice that:
\begin{equation}
L_{n,m}=L_{n,q}=0\,, \quad \forall\,n \ge 3.
\end{equation}
and
\begin{equation}
L_{2,m}\neq0,\qquad L_{2,q}\neq0.
\end{equation}
\begin{equation}
(L_{2,m})^{-1}
\begin{vmatrix}
0 & m_{2} & m_{3} \\
1 & m_{1} & m_{2} \\
0 & m_{0} & m_{1} \\
\end{vmatrix}
=(L_{2,q})^{-1}
\begin{vmatrix}
0 & q_{2} & q_{3} \\
1 & q_{1} & q_{2} \\
0 & q_{0} & q_{1} \\
\end{vmatrix}
\end{equation}
\begin{equation}
(L_{2,m})^{-1}
\begin{vmatrix}
0 & m_{2} & m_{3} \\
0 & m_{1} & m_{2} \\
1 & m_{0} & m_{1} \\
\end{vmatrix}
=(L_{2,q})^{-1}
\begin{vmatrix}
0 & q_{2} & q_{3} \\
0 & q_{1} & q_{2} \\
1 & q_{0} & q_{1} \\
\end{vmatrix}
\end{equation}
Hence, the above parameterization in fact satisfies the Theorem \ref{theo1}; and therefore describe a $2$-soliton solution. And trough equations \eqref{Al}, \eqref{Bl} and \eqref{cl}, we can find the inverse relation:
\begin{align*}
&a_1=\frac{-m_{2} m_{0}^2+m_{1}^2 m_{0}-m_{3} m_{0}+m_{1} m_{2}}{m_{0} m_{2}-m_{1}^2}\,,\qquad a_2=\frac{m_{1}^3+(m_{3}-2 m_{0} m_{2}) m_{1}-m_{2}^2+m_{0}^2 m_{3}}{m_{0} m_{2}-m_{1}^2}\,,\\
&b_1=\frac{m_{1}^3+(m_{3}-2 m_{0} m_{2}) m_{1}-m_{2}^2+m_{0}^2 m_{3}}{m_{0} m_{2}-m_{1}^2}\,,\qquad b_2=\frac{m_{1}^3 + m_{2}^2 + m_{0}^2 m_{3} -
m_{1} (2 m_{0} m_{2} + m_{3})}{m_{1}^2 - m_{0} m_{2}}\,.
\end{align*}
Finally, the coefficients $c_l$ can be found by \eqref{cl}:
\begin{equation}
c_1=q_0\,,\qquad c_2=\frac{m_{1}^2 q_{1}-m_{1} m_{2} q_{0}+m_{0} (m_{3} q_{0}-m_{2} q_{1})}{m_{1}^2-m_{0} m_{2}}\,.
\end{equation}
\subsubsection{Tomimatsu-Sato solution with \texorpdfstring{$\delta$=$2$}{d=2}}
Now, let us proceed in the same way for the particular case of multipoles moments associated with the Tomimatsu-Sato solution \cite{TomiSato}. We will find that the Ernst potentials on the symmetry axis are recovered from the corresponding multipole moments. In order to clarify how the method derived in the preceding sections should be used, consider the multipole moments below and then construct the metric solution associated with them. Due to the complexity of higher orders, we will consider a solutions with only the first 6 multipoles $P_n$:
\begin{align*}
&P_0=M\,,\\
&P_1=i a M\,,\\
&P_2=-\frac{1}{4} M \left(3 a^2+M^2\right)\,,\\
&P_3=-\frac{1}{2} i a M \left(a^2+M^2\right)\,,\\
&P_4=\frac{1}{112} M \left(35 J^4+66 J^2 M^2+11 M^4\right)\,,\\
&P_5=\frac{1}{112} M \left(35 J^4+66 J^2 M^2+11 M^4\right)\,.
\end{align*}
Therefore, the coefficients $m_k$ are:
\begin{align*}
&m_0=M\,,\\
&m_1=ia M\,,\\
&m_2=-\frac{1}{4} M \left(3 a^2+M^2\right)\,,\\
&m_3=-\frac{1}{2} i a M \left(a^2+M^2\right)\,,\\
&m_4=\frac{1}{16} \left(5 a^4 M+10 a^2 M^3+M^5\right)\,,\\
&m_5=\frac{1}{16} i \left(3 a^5 M+10 a^3 M^3+3 a M^5\right)\,.
\end{align*}
With the coefficients $m_k$ in hand, it is easy to show from the discussion above that only 4 parameters are needed to describe the Ernst potentials. First of all, notice that:
\begin{equation}
L_{n,m}=0\,, \quad \forall\,n \ge 3.
\end{equation}
and
\begin{equation}
L_{2,m}=\frac{1}{4} \left(M^4-a^2 M^2\right)\,.
\end{equation}
In order to relate the multipole coefficients $m_k$ and $q_k$ with the Ernst coeficients $a_l$ and $b_l$ the following relations must be true:
\begin{equation}
B_1=(L_{2,m})^{-1}
\begin{vmatrix}
0 & m_{2} & m_{3} \\
1 & m_{1} & m_{2} \\
0 & m_{0} & m_{1} \\
\end{vmatrix}=-i a
\end{equation}
\begin{equation}
B_2=(L_{2,m})^{-1}
\begin{vmatrix}
0 & m_{2} & m_{3} \\
0 & m_{1} & m_{2} \\
1 & m_{0} & m_{1} \\
\end{vmatrix}=\frac{1}{4} \left(M^2-a^2\right)
\end{equation}
which can be verified after a straightforward calculation. Consequently:
\begin{align*}
&B_1=-i a\,,\qquad B_2=\frac{1}{4} \left(M^2-a^2\right)\,,\\
&A_1=M\,,\qquad A_2=0\,.
\end{align*}
Using the equation \eqref{ab} yields:
\begin{align*}
&a_1=- (M+ia)\,,\qquad a_2=\frac{1}{4} \left(M^2-a^2\right)\,,\\
&b_1=M-i a\,,\qquad b_2=\frac{1}{4} \left(M^2-a^2\right)\,.
\end{align*}
Thus, the Ernst potentials $\mathcal{E}$ can now be evaluated with the relations \eqref{ab}:
\begin{equation}
\mathcal{E}=\dfrac{1-\xi}{1+\xi}=\dfrac{z^2-(M+i a)z+\frac{M^2-a^2}{4}}{z^2+(M-i a)z+\frac{M^2-a^2}{4}}
\end{equation}
recovering, then, the Ernst potential on the symmetry axis for the Tomimatsu-Sato solution with distorsion parameter $\delta=2$ \cite{TomiSato}.
\subsection{Approximated solutions}
We already saw how the multipole moments, under certain conditions, are exactly matched to the $N$-soliton solution. We will elucidate how to find approximated solutions describing physical objects possessing the required multipole moments by applying the methodology derived in the present paper can be utilized to construct physical objects with the required multipole moments or, at least, find approximated solutions. Hence, consider the first seven gravitational multipole moments as given in \cite{fodor2020calculation}:
\begin{align}
P_0&=m_0 \ , \label{eqp0m0} \\
P_1&=m_1 \ , \\
P_2&=m_2 \ , \\
P_3&=m_3+\frac{1}{5}q^{*}_0 S_{10} \ , \\
P_4&=m_4
-\frac{1}{7}m^{*}_0 M_{20}
+\frac{3}{35}q^{*}_1 S_{10}
+\frac{1}{7}q^{*}_0(3S_{20}-2H_{20}) \ , \\
P_5&=m_5
-\frac{1}{21}m^{*}_1 M_{20}
-\frac{1}{3}m^{*}_0 M_{30}
+\frac{1}{21}q^{*}_2 S_{10}
+\frac{1}{21}q^{*}_1(4S_{20}-3H_{20}) \notag\\
&+\frac{1}{21}q^{*}_0\left(
q^{*}_0 q_0 S_{10}
-m^{*}_0 m_0 S_{10}
+14S_{30}+13S_{21}-7H_{30}
\right) \ , \\
P_6&=m_6
-\frac{5}{231}m^{*}_2 M_{20}
-\frac{4}{33}m^{*}_1 M_{30}
+\frac{1}{33}m^{*2}_0 m_0 M_{20}
-\frac{1}{33}m^{*}_0(18M_{40}+8M_{31}) \notag\\
&+\frac{1}{33}q^{*}_3 S_{10}
+\frac{1}{231}q^{*}_2(25S_{20}-20H_{20})
+\frac{2}{231}q^{*}_1(35S_{30}+37S_{21}-21H_{30}) \notag\\
&-\frac{1}{1155}(37q^{*}_1m^{*}_0
+13q^{*}_0m^{*}_1)m_0 S_{10}
+\frac{1}{33}q^{*2}_0
\left(5q_0 S_{20}-4m_0 Q_{20}+3q_1 S_{10}\right) \notag\\
&+\frac{10}{231}q^{*}_1q^{*}_0 q_0 S_{10}
+\frac{2}{33}q^{*}_0m^{*}_0
\left(2m_0 H_{20}-3q_0 M_{20}-2m_1 S_{10}\right) \\
&+\frac{1}{33}q^{*}_0
\left(30S_{40}+32S_{31}-24H_{31}-12H_{40}\right) \ . \notag
\end{align}
And the electromagnetic moments:
\begin{align}
Q_0&=q_0 \ , \\
Q_1&=q_1 \ , \\
Q_2&=q_2 \ , \\
Q_3&=q_3-\frac{1}{5}m^{*}_0 H_{10} \ , \\
Q_4&=q_4
+\frac{1}{7}q^{*}_0 Q_{20}
-\frac{3}{35}m^{*}_1 H_{10}
-\frac{1}{7}m^{*}_0(3H_{20}-2S_{20}) \ , \\
Q_5&=q_5
+\frac{1}{21}q^{*}_1 Q_{20}
+\frac{1}{3}q^{*}_0 Q_{30}
-\frac{1}{21}m^{*}_2 H_{10}
-\frac{1}{21}m^{*}_1(4H_{20}-3S_{20}) \notag\\
&+\frac{1}{21}m^{*}_0\left(
m^{*}_0 m_0 H_{10}
-q^{*}_0 q_0 H_{10}
-14H_{30}-13H_{21}+7S_{30}
\right) \ , \\
Q_6&=q_6
+\frac{5}{231}q^{*}_2 Q_{20}
+\frac{4}{33}q^{*}_1 Q_{30}
+\frac{1}{33}q^{*2}_0 q_0 Q_{20}
+\frac{1}{33}q^{*}_0(18Q_{40}+8Q_{31}) \notag\\
&-\frac{1}{33}m^{*}_3 H_{10}
-\frac{1}{231}m^{*}_2(25H_{20}-20S_{20})
-\frac{2}{231}m^{*}_1(35H_{30}+37H_{21}-21S_{30})\notag\\
&-\frac{1}{1155}(37m^{*}_1q^{*}_0
+13m^{*}_0q^{*}_1)q_0 H_{10}
+\frac{1}{33}m^{*2}_0
\left(5m_0 H_{20}-4q_0 M_{20}+3m_1 H_{10}\right) \notag\\
&+\frac{10}{231}m^{*}_1m^{*}_0 m_0 H_{10}
+\frac{2}{33}m^{*}_0q^{*}_0
\left(2q_0 S_{20}-3m_0 Q_{20}-2q_1 H_{10}\right) \label{eqq6q6}\\
&-\frac{1}{33}m^{*}_0
\left(30H_{40}+32H_{31}-24S_{31}-12S_{40}\right) \ . \notag
\end{align}
Where
\begin{align}
M_{ij}&=m_i m_j-m_{i-1}m_{j+1} \ , &
S_{ij}&=m_i q_j-m_{i-1}q_{j+1} \ , \label{eqmijsij}\\
Q_{ij}&=q_i q_j-q_{i-1}q_{j+1} \ , &
H_{ij}&=q_i m_j-q_{i-1}m_{j+1} \ , \label{eqqijhij}
\end{align}
Notice here that the Multipole moment $P_{n}$ ($Q_{n}$) is linear on $m_{n}$ ($q_{n}$) and do not depend on the higher orders of the coefficients $m_{n}$ ($q_{n}$). Hence, we can chose $m_{n}$ ($q_{n}$) so that the multipole moment $P_{n}$ ($Q_{n}$) is the desired one. If, for instance, we want to describe a pole-dipole source, that is, a source only possessing a mass and electric
monopole moment and angular momentum and magnetic dipole moments, we can set all $m_k$ and $q_k$ such that \cite{HP}:
\begin{equation}
\begin{aligned}
&P_0=m,\qquad \qquad Q_0=e,\\
&P_1=i m a,\qquad\qquad Q_1=i e \mu,\\
&P_n=0,\qquad\qquad Q_n=0, \quad \text{for all }n\ge 2.
\end{aligned}
\end{equation}
By means of the general equations \eqref{eqp0m0}-\eqref{eqq6q6}, we can determine the first seven multipole coefficients $m_k$ and $q_k$ univocally:
\begin{equation}\label{mdipole}
\begin{aligned}
&m_0=m,\quad m_1=i m a,\quad m_2=0,\quad m_3=-\frac{1}{5} i e^2 m (a-\mu ),\\
&m_4=\dfrac{1}{7}a^2m^3-\dfrac{8}{35}a m e^2 \mu+\dfrac{3}{35} m e^2\mu^2,\\
&m_5=\frac{3}{35} i m e^2 (a-\mu ) (e^2-m^2)-\dfrac{1}{21}i a m(a^2 m^2-e^2\mu^2),\\
&m_6=\frac{1}{21} a^2 m^3 \left(m^2-2 e^2\right)-\frac{2}{35}e^2 \mu m^3 (a-\mu )+\frac{1}{105} e^4 \mu m (16 a-11 \mu ),
\end{aligned}
\end{equation}
and
\begin{equation}\label{qdipole}
\begin{aligned}
&q_0=e,\quad e_1=i e \mu,\quad q_2=0,\quad q_3=-\frac{1}{5} i e m^2 (a-\mu ),\\
&q_4=-\dfrac{1}{7}\mu^2e^3+\dfrac{8}{35}a m^2 e \mu-\dfrac{3}{35}e a^2 m^2,\\
&q_5=\frac{3}{35} i e m^2 (a-\mu ) (e^2-m^2)+\frac{1}{21} i e \mu (e^2 \mu^2 -a^2m^2),\\
&q_6=\frac{1}{21} e^3 \mu ^2 \left(e^2-2 m^2\right)+\frac{2}{35} a e^3 m^2 (a-\mu )+\frac{1}{105} a e m^4 (16 \mu -11 a).
\end{aligned}
\end{equation}
Hence, although the multipole moments of order higher than 1 are zero, the multipole coefficients are not. Notice that the coefficients presented here differ from those presented on the papers \cite{HP, Sotiriou} based on the recent paper of Fodor et al \cite{fodor2020calculation}. It is interesting to compare our solution with the previous one in the limit $\mu=a$:
\begin{equation}
\begin{aligned}
&m_0=m,\quad m_1=i m a,\quad m_2=0,\quad m_3=0,\\
&m_4=\frac{1}{7} a^2 m \left(m^2-e^2\right),\\
&m_5=-\frac{1}{21} i a^3 m \left(m^2-e^2\right),\\
&m_6=\frac{1}{21} a^2 m \left(e^2-m^2\right)^2,
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&q_0=e,\quad e_1=i e \mu,\quad q_2=0,\quad q_3=0,\\
&q_4=-\frac{1}{7} a^2 e \left(e^2-m^2\right),\\
&q_5=\frac{1}{21} i a^3 e \left(e^2-m^2\right),\\
&q_6=\frac{1}{21} a^2 e \left(e^2-m^2\right)^2.
\end{aligned}
\end{equation}
This agrees with the previous results presented in \cite{HP,Sotiriou} (although the authors only presented the coefficients until the fifth-order). But, if, instead of considering the limit $\mu=a$, we consider $e^2=m^2$, the coefficients do not agree. Under the special case of $\mu=a$ and $e^2=m^2$, all coefficients $m_k$ and $q_k$ vanish for $k\ge2$. For the special case, these multipole coefficients describe a $N$-soliton solution. In fact, the expansion
\begin{equation}
\xi(\rho=0,z)=\dfrac{m}{z}+i \dfrac{a m}{z^2},
\end{equation}
\begin{equation}
q(\rho=0,z)=\pm\dfrac{m}{z}\pm i \dfrac{a m}{z^2},
\end{equation}
implies
\begin{equation}
\mathcal{E}(\rho=0,z)=\dfrac{z^2-m z-i a m}{z^2+m z+ i a m},
\end{equation}
\begin{equation}
\Phi(\rho=0,z)=\dfrac{\pm m z\pm i a m}{z^2+m z+ i a m},
\end{equation}
Which describes a $2$-soliton solution, and the solution for the whole spacetime is given by equation \eqref{ernstr}.
However, for the general case where $e^2\ne m^2$ and $\mu\ne a$, it does not seem that the determinants $L_N,_{m}^{q}$ become zero for some order $N$. Therefore, the solution for the monopole-dipole source can not be exactly described as a $N$-soliton solution. Yet, we can use the $N$-soliton solution in order to approach the monopole-dipole solution. We can successfully describe the monopole-dipole solution as a $N$-soliton solution up to the order $N$. For instance, the corresponding $7$-soliton solution based on formulae \eqref{multipolecoe} is simply given by
\begin{equation}
\mathcal{E}(\rho=0,z)=\dfrac{z^7-m_0 z^6-m_1 z^5-m_2 z^4-m^3 z^3-m_4 z^2-m_5 z-m_6}{z^7+m_0 z^6+m_1 z^5+m_2 z^4+m^3 z^3+m_4 z^2+m_5 z+m_6}
\end{equation}
\begin{equation}
\Phi(\rho=0,z)=\dfrac{q_0 z^6+q_1 z^5+q_2 z^4+q^3 z^3+q_4 z^2+q_5 z+q_6}{z^7+m_0 z^6+m_1 z^5+m_2 z^4+m^3 z^3+m_4 z^2+m_5 z+m_6}
\end{equation}
where $m_k$ and $q_k$ are given by equations \eqref{mdipole} and \eqref{qdipole}. And again, we can construct the solution in the whole space by means of equation \eqref{ernstr}. This solution will have
\begin{equation}
\begin{aligned}
&P_0=m,\qquad \qquad Q_0=e,\\
&P_1=i m a,\qquad\qquad Q_1=i e \mu,\\
&P_n=0,\qquad\qquad Q_n=0, \quad \text{for }6\ge n\ge 2,\\
&P_n\neq0,\qquad\qquad Q_n\neq0, \quad \text{for } n\ge 6.
\end{aligned}
\end{equation}
That is, the above identification allows us to describe an approximate solution for the monopole-dipole source in charged spacetimes. For the vacuum case, see for instance reference \cite{HernandezPastora:1998mc}
\section{Conclusions}
Solutions of the Einstein-Maxwell equations for stationary axisymmetric spacetimes proved to be not only interesting from the theoretical point of view but also to have experimental applications. Keeping in mind that, we briefly revised the family of solutions introduced by Manko and Ruiz \cite{RMJ} named ``Extended N-soliton solution'' focusing on clarifying what is understood as soliton in the context of general Relativity.
The present paper was devoted to constructing a direct relation between the multipole moments and the mathematical parameters which appear. We also demonstrated how to construct exact solutions from their given multipole moments in the general $N$-soliton case. That is, this work extends the previous development of Manko and Ruiz to include electromagnetic fields. As mentioned by the authors in \cite{MankoRuiz}, ``It is remarkable that (4.4) is a linear system of algebraic equations when either one looks for the form of $m_n$ in terms of $a_l$ and $b_l$ , or vice versa, when one wants to see how the constants $a_l$ and $b_l$ depend on $m_n$'' (equation 4.4 in their paper is equivalent to equation \eqref{mank}). We could extend their sentence to include the parameters $q_n$ and $c_l$ and also say that it is remarkable that the physical parameterization of the $N$-soliton solution relies on solving a linear system.
In this way, the direct link was made between the coefficients of the multipole expansion in General Relativity and the $3N$ parameters of the $N$ -soliton solution. This result has been summarized in the Lemmas of Sections \ref{multipolevacuum} and \ref{multipoleelectrovacuum}, and are extensions of the Lemma already presented in the literature by the above-mentioned authors. Furthermore, the theorem presented in the same Section shows that any set of multipole moments satisfying a not so restrict condition can build an $N$-soliton solution.
Notice that for particular cases, we could have a solution having, for instance, $|L_n,_{m}|$ be nonzero for $n=N_1$ and zero for all $n>N_1$; and $|L_n,_{q}|$ be nonzero for $n=N_2$ and zero for all $n>N_2$ for $N_1\neq N_2$. All results remains valid and we would describe a $N_2$-soliton solution if, for instance, $N_2> N_1$.
Although compact, the results of the present paper can get even simple when symmetries are considered. Consider, for instance, spacetimes possessing equatorial symmetry/antisymmetry. Pachón-Contreras et al. \cite{Pach_n_2006} and Ernst et al. \cite{Ernst_2006,Ernst_2007} deduced how such symmetries are expressed in terms of the Ernst potentials on the symmetry axis. They found that, for the symmetric case, $m_k$ must be real for $k$ even and imaginary for $k$ odd. While $q_k$ has its behavior depending on a parameter $\epsilon$. When $\epsilon=1$, $q_k$ obeys the same rule: it must be real for $k$ even and imaginary for $k$ odd. On the other hand, for $\epsilon=-1$, $q_k$ must be imaginary for $k$ even and real for $k$ odd. For the antisymmetric case, $m_k$ vanishes for $k$ odd and $q_k$ vanishes for $k$ even. It is straightforward to see that equations \eqref{Al}-\eqref{cl}, where the mathematical parameters of the solution, $a_l$, $b_l$ and $c_l$, are written in terms of the determinants of the physical parameters, the multipole coefficients, $m_k$ and $q_k$, become even simpler! In fact, the proofs contained in reference \cite{Ernst_2006} become quite more direct if one makes use of equation \eqref{multipolecoe}, deduced in the present paper.
We also conclude that a generic solution of Einstein’s equation coupled with electromagnetism, whose multipole moments satisfy this condition, can be approximated as an $N$-soliton solution.
\ack
One of the authors, ESCF , would like to thanks Professors Betti Hartmann and Gyula Fodor for their support and helpful comments during the preparation of the present work. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)-Finance Code 001. ICM acknowledges the financial support of SNI-CONACyT, M\'exico, grant with CVU No. 173252.
\section*{References}
\bibliographystyle{hhieeetr}
|
2,877,628,089,533 | arxiv | \section{Historical Background}
Today we know that galactic globular clusters are old stellar systems and
people are therefore often surprised by the presence of hot stars in these
clusters. As the following paragraphs will show hot stars have been known to
exist in globular clusters for quite some time:
Barnard (1900) reports the detection of stars in globular clusters that were
much brighter on photographic plates than they appeared visually: {\it ``Of
course the simple explanation of this peculiarity is that these stars, so
bright photographically and so faint visually, are shining with a much bluer
light than the stars which make up the main body of the clusters''}.
In 1915 Harlow Shapley started a project to obtain colours and magnitudes of
individual stars in globular and open clusters (Shapley 1915a) hoping that
{\it ``considerable advance can be made in our understanding of the internal
arrangement and physical characteristics''} of these clusters. In the first
globular cluster studied (M~3, Shapley 1915b) he found a double peaked
distribution of colours, with a red maximum and a blue secondary peak. He
noticed that - in contrast to what was known for field dwarf stars - the stars
in M~3 became bluer as they became fainter. Ten Bruggencate (1927, p.130) used
Shapley's data on M~3 and other clusters to plot magnitude versus colour
(replacing luminosity and spectral type in the Hertzsprung-Russell diagram)
and thus produced the first colour-magnitude diagrams\footnote{Shapley (1930,
p.26, footnote) disliked the idea of plotting individual data points - he
thought that the small number of measurements might lead to spurious results.}
({\sc ``Farbenhelligkeitsdiagramme''}). From these colour-magnitude diagrams
(CMD's) ten Bruggencate noted the presence of a giant branch that became bluer
towards fainter magnitudes, in agreement with Shapley (1915b). In addition,
however, he saw a horizontal branch ({\sc ``horizontaler Ast''}) that parted
from the red giant branch and extended far to the blue at constant brightness.
Greenstein (1939) produced a colour-magnitude diagram for M~4 (again noting
the presence of a sequence of blue stars at constant brightness) and mentioned
that {\it ``the general appearance of the colour-magnitude diagram of M4 is
almost completely different from that of any galactic {\rm (i.e. open)}
cluster''}. He also noticed that - while main-sequence B and A type stars were
completely missing - there existed a group of bright stars above the
horizontal branch and on the blue side of the giant branch. Similar stars
appeared also in the CMD's presented by Arp (1955). As more CMD's of globular
clusters were obtained it became apparent that the horizontal branch
morphology varies quite considerably between individual clusters. The clusters
observed by Arp (1955) exhibited extensions of the blue horizontal branch
towards bluer colours and fainter visual magnitudes, i.e. towards hotter
stars\footnote{The change in slope of the horizontal branch is caused by the
decreasing sensitivity of B$-$V to temperature on one hand and by the
increasing bolometric correction for hotter stars on the other hand.} (see
Fig.~\ref{cmd}). In some of Arp's CMD's (e.g. M~15, M~2) these {\bf blue
tails} are separated from the horizontal part by gaps (see also
Fig.~\ref{cmds_obs}).
About 25 years after their discovery first ideas about the nature of the
horizontal branch stars began to emerge: Hoyle \& Schwarzschild (1955) were
the first to identify the horizontal branch with post-red giant branch (RGB)
stars that burn helium in their cores.
Sandage \& Wallerstein (1960) noted a correlation between the metal abundance
and the horizontal branch morphology seen in globular cluster CMD's: the
horizontal branch became bluer with decreasing metallicity. Faulkner (1966)
managed for the first time to compute zero age horizontal branch (HB) models
that qualitatively reproduced this trend of HB morphology with metallicity
(i.e. for a constant total mass stars become bluer with decreasing
metallicity) without taking any mass loss into account but assuming a rather
high helium abundance of Y = 0.35. Iben \& Rood (1970), however, found that
{\it ``In fact for the values of Y and Z most favored (Y $\ge$ 0.25
$\rightarrow$ 0.28, Z = $10^{-3} \rightarrow 10^{-4}$), individual tracks are
the stubbiest. We can account for the observed spread in color along the
horizontal branch by accepting that there is also a spread in stellar mass
along this branch, bluer stars being less massive (on the average) and less
luminous than redder stars. It is somewhat sobering to realize that this
conclusion comes near the end of an investigation that has for several years
relied heavily on aesthetic arguments against mass loss and has been guided by
the expectation of obtaining, as a final result, individual tracks whose color
amplitudes equal the entire spread in color along the observed horizontal
branches''}. In the same paper they found that {\it ``During most of the
double-shell-source phase, models evolve upwards and to the red along a
secondary giant branch\footnote{This secondary giant branch is called
asymptotic giant branch (AGB) later in the text and consists of stars with a
hydrogen and a helium burning shell.} that, for the models shown, approaches
the giant branch defined by models burning hydrogen in a shell.''}
Comparing HB models to observed globular cluster CMD's Rood (1973) found that
an HB that {\it ``\ldots is made up of stars with the same core mass and
slightly varying total mass, produces theoretical c-m diagrams very similar to
those observed. \ldots A mass loss of perhaps 0.2~M$_\odot$ with a random
dispersion of several hundredths of a solar mass is required somewhere along
the giant branch.''} The assumption of mass loss also diminished the need for
very high helium abundances.
Sweigart \& Gross (1974, 1976) computed HB tracks including semi-convection
and found that this inclusion considerably extends the temperature range
covered by the tracks. However, Sweigart (1987) noted that {\it ``For more
typical globular cluster compositions, however, the track lengths are clearly
too short to explain the observed effective temperature distributions along
many HB's, and thus semiconvection does not alleviate the need for a spread in
mass (or some other parameter), a point first emphasized by Rood (1973)''}.
\begin{figure}[ht]
\vspace*{11.cm}
\special{psfile=cmd.ps hscale=45 vscale=45 hoffset=40 voffset=-25 angle=0}
\caption{Colour-magnitude diagram of M~3 (Buonanno et al. 1994) with
the names of the principal sequences (some of which will be used in
this paper).}
\label{cmd}
\end{figure}
Caloi (1972) investigated zero age HB locations of stars with very low
envelope masses ($\le$ 0.02~\Msolar; extended or {\bf extreme HB} = EHB) and
found that they can be identified with subdwarf B stars in the field
(Greenstein 1971). Sweigart et al. (1974) and Gingold (1976) studied the
post-HB evolution and found that -- in contrast to the more massive blue HB
stars -- EHB models do not ascend the second (asymptotic) giant branch (AGB).
Thus our current understanding sees {\bf blue horizontal branch stars} as stars
that burn helium in a core of about 0.5~\Msolar\ and hydrogen in a shell. Their
hydrogen envelopes vary between $\ge$ 0.02~\Msolar\ (less massive envelopes
belong to EHB stars which do not have any hydrogen shell burning) and 0.1 --
0.15~\Msolar. Depending on the mass of their hydrogen envelopes they evolve to
the asymptotic giant branch (BHB stars) or directly to the white dwarf domain
(EHB stars, AGB manqu\'e stars). For a review see Sweigart (1994).
But blue horizontal branch stars are neither the brightest nor the bluest stars
in globular clusters: Already Shapley (1930, p.30) remarked that {\it
``Occasionally, there are abnormally bright blue stars, as in Messier~13, but
even these are faint absolutely, compared with some of the galactic B
stars''}. This statement refers to stars like those mentioned by Barnard
(1900) which in colour-magnitude diagrams lie above the horizontal branch and
blueward of the red giant branch (see Fig.~\ref{cmd}). This is also the region
where one would expect to find central stars of planetary nebulae, which are,
however, rare in globular clusters: Until recently Ps1 (Pease 1928), the
planetary nebula in M~15 with its central star K~648, remained the only such
object known in globular clusters (see also Jacoby et al. 1997).
The bright blue stars are clearly visible in the colour-magnitude diagrams of
Arp (1955). Apart from analyses of individual stars like vZ~1128 in M~3 (Strom
\& Strom 1970, and references therein) and Barnard~29 in M~13 (Traving 1962,
Stoeckley \& Greenstein 1968) the first systematic work was done by Strom et
al. (1970). All stars analysed there show close to solar helium content,
contrary to the blue horizontal branch stars, which in general are depleted in
helium (Heber 1987). Strom et al. identified the brightest and bluest stars
with models of post-AGB stars (confirming the ideas of Schwarzschild \& H\"arm
1970) and the remaining ones with stars evolving from the horizontal branch
towards the AGB. This means that all of the stars in this study are in the
double-shell burning stage. Zinn et al. (1972) performed a systematic search
for such stars using the fact that they are brighter in the U band than all
other cluster stars. This also resulted in the name {\bf UV Bright Stars} for
stars brighter than the horizontal branch and bluer than the red giant branch.
Zinn (1974) found from a spectroscopic analysis of UV bright stars in 8
globular clusters {\it ``a strong correlation between the presence of supra-HB
stars in a globular cluster and the presence of HB stars hotter than
log~${T_{eff}}$ = 4.1''}. Harris et al. (1983) extended the compilation of UV
bright stars in globular clusters and de Boer (1987) gave another list of UV
bright stars in globular clusters, together with estimates of effective
temperatures and luminosities.
De Boer (1985) found from analyses of IUE spectra of UV bright stars in 7
globular clusters that their contribution to the total cluster intensity ranges
{\it ``from, on average, over 50\% at 1200~\AA\ to a few percent at
3000~\AA.''} Most of the UV bright stars found in ground based searches are
cooler than 30,000~K, although theory predicts stars with temperatures up to
100,000~K (e.g. Sch\"onberner 1983) The ground based searches, however, are
biased towards these cooler stars due to the large bolometric corrections for
hotter stars. It is therefore not very surprising that space based searches in
the UV (Ultraviolet Imaging Telescope, Stecher et al. 1997) discovered a
considerable number of additional {\em hot} UV bright stars in a number of
globular clusters.
Space based observatories also contributed a lot of other information about
hot stars in globular clusters: UIT observations showed the unexpected presence
of blue HB stars in metal-rich globular clusters like NGC~362 (Dorman et al.
1997) and 47~Tuc (O'Connell et al. 1997). At about the same time Hubble Space
Telescope (HST) observations of the core regions of globular clusters showed
long blue tails in metal-rich bulge globular clusters (Rich et al. 1997).
Observations of the very dense core regions of globular clusters show that the
colour-magnitude diagrams seen there may differ considerably from those seen
in the outer regions of the same clusters (e.g. Sosin et al. 1997). The most
recent addition to the family of hot stars in globular clusters are the white
dwarfs seen in HST observations of M~4 (Richer et al. 1995, 1997), NGC~6752
(Renzini et al. 1996) and NGC~6397 (Cool et al. 1996), which unfortunately are
at the very limit for any spectroscopic observations even with 10m class
telescopes.
The interest in old hot stars like blue horizontal branch and UV bright stars
has been revived and extended by the discovery of the UV excess in elliptical
galaxies (Code \& Welch 1979; de Boer 1982) for which they are the most likely
sources (Greggio \& Renzini 1990, Brown et al. 1997).
\section{Spectroscopic Analysis Methods} Much of the discussion and findings
described above are based solely on the photometric properties of hot stars in
globular clusters. Much more physical information regarding their evolutionary
status can be gained from spectroscopic analyses: From spectra of various
resolutions in combination with photometric data we can determine their
atmospheric parameters (effective temperature, surface gravity, and helium
abundance) and compare those to the predictions of the stellar evolutionary
theory. The disadvantage of spectroscopic observations (compared to photometric
ones) is the fact that they require larger telescopes and/or more observing
time: For the observations of the blue HB stars in M~15 we used the 3.5m
telescope of the German-Spanish Calar Alto observatory in Spain and the
targets in NGC~6752 were mostly observed with the NTT at the ESO La Silla
observatory in Chile.
To determine effective temperatures and surface gravities we compare various
spectroscopic and photometric observations to their theoretical counterparts.
Depending on the available observational and theoretical data and the amount
of software sophistication a wide variety of analysis methods is currently
available. The following paragraphs attempt to give an overview that allows to
judge our results -- for detailed information we refer the
reader to the cited papers.
\subsection{Effective Temperature}
\begin{figure}[t]
\vspace*{9cm}
\special{psfile=plotfitlc.ps hscale=50 vscale=50 hoffset=-0 voffset=280
angle=270}
\caption{The fit of the flux distribution ({\bf left}) and line profiles ({\bf
right}) of B~491 in NGC~6752. Plotted are the observed data (grey lines)
and a model atmosphere for [M/H] = $-$1.5, \teff\ = 28,500~K and \logg\ = 5.3.
These parameters were determined from the flux distribution and the line
profiles of H$_\beta$ to H$_\delta$ only. It can be seen that the
corresponding model also fits the higher Balmer lines quite well.}
\label{plotfit}
\end{figure}
The ideal temperature indicator is insensitive to variations of surface
gravity because it then allows to derive the effective temperature
independently from the surface gravity. The UV flux distribution meets this
requirement for all blue HB stars and the Balmer jump fulfills it for stars
with effective temperatures between about 11,000~K and
30,000~K\footnote{Johnson UBV photometry becomes rather gravity independent
for temperatures above about 20,000~K - at the same time, however, it also
loses temperature sensitivity. Str\"omgren uvby photometry stays temperature
sensitive up to higher effective temperatures but is not available for most
globular clusters.}. As interstellar extinction changes (reddens) the flux
distribution of a star the observational data must be corrected for this
effect. We dereddened the observed spectra using the extinction law of Savage
\& Mathis (1979) and the appropriate reddening values for the respective
globular cluster. For the analysis published ATLAS9 model spectra (Kurucz
1992) for the metallicity closest to the globular cluster metallicity were
used. If not mentioned otherwise the quality of the fit was judged by eye. An
example is shown in Fig.~\ref{plotfit}. Temperature determinations that
include UV data (e.g. IUE spectrophotometry) are in general more reliable than
those relying solely on optical observations, as the UV flux distribution is
more sensitive to temperature variations than the optical continuum. However,
as only a very limited amount of UV spectrophotometry is available such data
were used only by Heber et al. (1986), de Boer et al. (1995), Cacciari et al.
(1995), and Moehler et al. (1997b).
If only {\bf optical spectrophotometry} is available (most stars in NGC~6752
and M~15) the overall continuum slope and the Balmer jump should be fitted
simultaneously, including - if possible - optical photometric data as well. It
turns out that straylight from red (i.e. cool) neighbours can cause problems
for optical spectrophotometry\footnote{This problem does not affect the IUE
data as the flux of cool stars decreases rapidly towards shorter
wavelengths.}: A model that fits the Balmer jump (and the BV photometry)
cannot fit the spectrophotometric continuum longward of 4000~\AA\ but instead
predicts too little flux there - the straylight from the cool star causes a
red excess. An attempt to quantify these effects is described in Moehler et
al. (1997b).
Crocker et al. (1988) fit the continuum ($\lambda\lambda$ 3450--3700~\AA\ and
4000--5100~\AA ) in their spectrophotometric data and employ a $\chi^2$ test
to find the best fit. In addition they use the star's position along the
observed HB to obtain another estimate of its temperature and finally average
both values for \teff .
\subsection{Surface Gravity}
Provided the effective temperature has been determined as described above the
surface gravity can be derived by fitting the shape of the Balmer line
profiles at a fixed temperature. For this purpose the spectra are normalized
and corrected for Doppler shifts introduced by the radial velocities of the
stars. The model spectra have to be convolved with a profile representing the
instrumental resolution, which was generally determined from the FWHM of the
calibration lines (for more details see Moehler et al. 1995). We computed (at
fixed temperature) for H$_\beta$ to H$_\delta$ the squared difference between
the observed spectrum and the theoretical line profile and used the sum of
these differences as estimator for the quality of the fit (Moehler et al.
1995, 1997b, see also Fig.~\ref{plotfit}). Crocker et al. (1988) used the same
lines and employed a $\chi^2$ test to determine \logg . In addition they
corrected their results for the subsolar helium abundance normally present in
blue HB stars.
\subsection{Simultaneous Determination of \teff\ and \logg}
For the cooler stars (below about 20,000~K) one can use a combination of {\bf
optical photometry} and {\bf Balmer line profile fits} to determine effective
temperature and surface gravity simultaneously: Reddening free indices (Q for
Johnson UBV photometry, Moehler et al. 1995; [c1], [u-b] for Str\"omgren uvby
photometry, de Boer et al. 1995) in comparison with theoretical values allow
to determine a relation between effective temperature and surface gravity.
Fits to the lower Balmer lines (H$_\beta$ to H$_\delta$) yield another relation
between \teff\ and \logg\ and from its intersection with the photometric
relation effective temperature and surface gravity can be derived (for
examples see de Boer et al. 1995 and Moehler \& Heber 1998).
For stars below about 8,500~K (Moehler et al. 1995, M~15) the Balmer lines
depend more on \teff\ than on \logg. In these cases the Balmer lines are used
to estimate the temperature and \logg\ is derived from the Q value.
Including also the higher Balmer lines (H$_\epsilon$ to H$_{10}$) allows to
derive effective temperature {\em and} surface gravity by fitting {\bf all
Balmer lines (H$_\beta$ to H$_{10}$)} simultaneously (Bergeron et al. 1992;
Saffer et al. 1994). This method has been used for the UV bright stars
(Moehler et al. 1998a) and the blue HB stars in metal-rich globular clusters.
We used the procedures developed by Bergeron et al. (1992) and Saffer et al.
(1994), which employ a $\chi^2$ test to establish the best fit. Using only the
lower Balmer lines (H$_\beta$ to H$_\delta$) yields rather shallow minima of
$\chi^2$, which allow a large range of values for \teff\ and \logg .
\subsection{Helium Abundances}
Helium abundances were either derived from the simultaneous fitting of the
Balmer and \ion{He}{i}/\ion{He}{ii} lines (Moehler et al. 1998a) or from
measured equivalent widths that are compared to theoretical curves-of-growth
for the appropriate values of effective temperature and surface gravity
(Moehler et al. 1997b).
\subsection{Model atmospheres}
Most of the stars discussed here are in a temperature--gravity range where LTE
(local thermal equilibrium) is a valid approximation for the calculation of
model atmospheres (Napiwotzki 1997). For the older data published ATLAS model
spectra were used: ATLAS6 (Kurucz 1979) by Crocker et al. (1988) resp. ATLAS9
(Kurucz 1992) by de Boer et al. (1995) and Moehler et al. (1995, 1997b). The
stars in NGC~6752 (Moehler et al. 1997b) required an extension of the model
atmosphere grid to higher surface gravities, for which we used an updated
version of the code of Heber (1983). The new fit procedures (Bergeron et al.
1992; Saffer et al. 1994) which we employed for the recent data (Moehler et
al. 1998a) required line profiles for the higher Balmer lines (shortward of
H$_\delta$) that are not available from Kurucz. We therefore calculated model
atmospheres using ATLAS9 (Kurucz 1991, priv. comm.) and used the LINFOR
program (developed originally by Holweger, Steffen, and Steenbock at Kiel
university) to compute a grid of theoretical spectra that contain the Balmer
lines H$_\alpha$ to H$_{22}$ and \ion{He}{i} lines. For those stars which show
\ion{He}{ii} lines in their spectra (and are thus considerably hotter than the
bulk of our programme stars) it is necessary to take non-LTE effects into
account (Napiwotzki 1997; Moehler et al. 1998a).
\section{Gaps and Blue Tails}
\begin{figure}[t]
\vspace*{10.5cm}
\special{psfile=cmds_obs.ps hscale=43 vscale=43 hoffset=40 voffset=-25 angle=0}
\caption{Colour-magnitude diagrams of NGC~6752 (Buonanno et al. 1986), NGC~288
(Buonanno et al. 1984), M~5 (Buonanno et al. 1981), NGC~6397 (Alcaino et al.
1987), M~92 (Buonanno et al. 1983b), M~15 (Buonanno et al. 1983a). The wide
gaps in the brighter and more horizontal part of some HB's are in reality
populated by (variable) RR~Lyrae stars, which are omitted from the plots. The
gaps discussed in this paper are located at fainter magnitudes in the (mostly
vertical) blue tails (most pronounced in NGC~6752, NGC~288, and M~15). }
\label{cmds_obs}
\end{figure}
As mentioned above the blue tails seen in many CMD's of globular clusters are
often separated from the more horizontal part of the HB by gaps at varying
brightness (examples are shown in Fig.~\ref{cmds_obs}; for a list of globular
clusters with blue tails see Fusi Pecci et al. [1993]; Catelan et al. [1998]
and Ferraro et al. [1998] give comprehensive lists of clusters that show gaps
and/or bimodal horizontal branches). Such gaps can be found already in Arp's
(1955) CMD's and have caused a lot of puzzlement, since they are not
predicted by any canonical HB evolution. One of the first ideas was that the
gaps are created by diverging evolutionary paths that turn a unimodal
distribution on the ZAHB into a bimodal one as the stars evolve away from the
ZAHB (Newell 1973; Lee et al. 1994). Evolutionary calculations, however, do not
show any such behaviour as horizontal branch stars spend most of their
lifetime close to the ZAHB (Dorman et al. 1991; Catelan et al. 1998). Rood \&
Crocker (1985) suggested that the gaps separate two groups of HB stars that
differ in, e.g., CNO abundance or core rotation. A more extreme version of
this idea was suggested by Iben (1990): blue tail stars are produced
differently from the blue HB stars, e.g. by merging of two helium white
dwarfs. So far, no precursor systems of such stars have been observed. Quite
recently, Caloi (1999) proposed a change in the stellar atmospheres from
convection to diffusion as an explanation for the gaps around (B$-$V)$_0$ = 0,
whereas Catelan et al. (1998) suggested that at least some of the gaps may be
due to statistical fluctuations. More detailed descriptions of possible
explanations for the gaps can be found in Crocker et al. (1988), Catelan et
al. (1998), and Ferraro et al. (1998).
The need for more information on the stars along the blue tails led to our
project to obtain atmospheric parameters for blue HB and blue tail stars in
several globular clusters: NGC~6397 (de Boer et al. 1995), NGC~6752 (Heber et
al. 1986; Moehler et al. 1997b), and M~15 (Moehler et al. 1995, 1997a). To
enlarge our sample we also used the data of NGC~288, M~5, and M~92 published
by Crocker et al. (1988). The CMD's of these clusters can be found in
Fig.~\ref{cmds_obs}.
\subsection*{Evolutionary status}
In Fig.~\ref{plottga1} the physical parameters of the HB stars analysed by
Crocker et al. (1988; M~5, M~92, NGC~288), de Boer et al. (1995; NGC~6397), and
Moehler et al. (1995, 1997a, M~15; 1997b, NGC6752) are compared to
evolutionary tracks. The zero-age HB (ZAHB) marks the position where the HB
stars have settled down and started to quietly burn helium in their cores. The
terminal-age HB (TAHB) is defined by helium exhaustion in the core of the HB
star. The distribution of stars belonging to an individual cluster is hard to
judge in this plot but it is obvious that the observed positions in the (\logg
, \teff)-diagram fall mostly above the ZAHB and in some cases even above the
TAHB\footnote{Preliminary results of Bragaglia et al. (1999) indicate
deviations from this trend}. An indication of a low-temperature gap can be seen
at \logt\ $\approx$ 4.1. The gaps seen in the CMD of NGC~6752 and in the M~15
data of Durrell \& Harris 1993 (from which the two hottest stars in M~15 were
selected) are visible in the (\logg , \teff) plane at about 24,000~K, where
they separate BHB from EHB stars. In all other clusters the stars above and
below the gaps are blue horizontal branch stars cooler than 20,000~K.
\begin{figure}
\vspace*{8.6cm}
\special{psfile=plottga1.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The results of Crocker et al. (1988; M~5, M~92, NGC~288), de Boer
et al. (1995; NGC~6397) and Moehler et al. (1995, 1997a, M~15; 1997b, NGC6752)
compared to evolutionary tracks from Dorman et al. (1993). ZAHB and TAHB
stand for zero-age and terminal-age HB (see text for details). Also given is
the position of the helium main-sequence (Paczynski 1971).}
\label{plottga1}
\end{figure}
Independent of the occurrence of any gaps stars with temperatures between
11,000 (\logt\ = 4.04) and 20,000~K (\logt\ = 4.30) show lower gravities than
expected from canonical scenarios, whereas stars outside this temperature
range are well described by canonical HB and EHB evolutionary tracks. The UIT
observations of M~13 (Parise et al. 1998) and the HUT spectra of M~79 (Dixon
et al. 1996) also suggest lower than expected gravities for blue HB stars.
Whitney et al. (1998) use UIT observations of the hot stars in $\omega$ Cen to
claim that the extreme HB stars -- which agree with theoretical expectations
in our results -- have lower than expected luminosities, which would mean
higher than expected gravities. These deviations in \logg\ could indicate that
some assumptions used for the calculations of model atmospheres may not be
appropriate for the analysis of the BHB stars (see also de Boer et al. 1995,
Moehler et al. 1995):
Diffusion might lead to peculiar abundance patterns, because radiative
levitation might push up some metals into the atmospheres whereas other
elements might be depleted due to gravitational settling. Line blanketing
effects of enhanced metals may change the atmospheric structure. We found,
however, that even an increase of 2 dex in [M/H] results in an increase of
only 0.1~dex in \logg\footnote{This is consistent with the findings of Leone \&
Manfr\`e (1997) that Balmer-line gravities can be underestimated by 0.25 dex
if a solar metal abundance is assumed for metal-rich helium weak stars.}.
Another effect of diffusion might be a stratification of the atmosphere, i.e.
an increase of helium abundance with depth, which has been predicted for white
dwarf atmospheres (Jordan \& Koester, 1986). In order to affect the Balmer
jump significantly the transition from low to high He abundance must take place
at an optical depth intermediate between the formation depths of the Paschen
and the Balmer continua. Such a fine tuning is unlikely to occur.
Rapid rotation rotation of the stars -- if neglected in the model atmospheres
-- would lower the determined gravities. This effect, however, becomes
significant only if the rotation velocity exceeds about half of the break-up
velocity. As measured rotation velocities for HB stars are small (Peterson et
al. 1995) this possibility can be ruled out as well.
As we did not find any systematic effects in our analysis that are large
enough to explain the observed offsets in surface gravity we assume for now
that the physical parameters are correct and look for a scenario that can
explain them\footnote{Scenarios like the merging of two helium white dwarfs
(Iben \& Tutukov 1984) or the stripping of red giant cores (Iben \& Tutukov
1993, Tuchman 1985) may produce stars that deviate from the ZAHB. Such stars,
however, are either too hot (merger) or too short-lived (stripped core) to
reproduce our results.}:
\subsubsection*{Deep mixing}
Abundance variations (C, N, O, Na, Al) in globular cluster red giant stars
(Kraft 1994, Kraft et al. 1995, Pilachowski et al. 1996) suggest that
nuclearly processed material from deeper regions is mixed to the surface of
the stars. Depending on the element considered this mixing has to reach down
into varying depths. The enhancement of aluminium, for instance, requires the
mixing to extend down into the hydrogen burning (= helium producing) shell
(e.g. Cavallo et al. 1998). This means that any mixing that dredges up
aluminium will also dredge up helium ({\it helium mixing} or {\it deep
mixing}). Table~\ref{tab-mix} lists the evidence for deep mixing for the
clusters shown in Fig.~\ref{cmds_obs}.
\begin{table}
\begin{tabular}{lll}
Cluster & Mixing & Reference\\
\hline
NGC 6752 & probable & Shetrone 1997, IAU Symp. 189(P), 158\\
NGC 6397 & probable & Bell et al. 1992, AJ 104, 1127\\
M 92 & probable & Shetrone 1996, AJ 112, 1517\\
M 15 & probable & Sneden et al. 1997, AJ 114 1964\\
M 5 & unlikely & Sneden et al. 1992, AJ 104, 2121\\
NGC 288 & unlikely & Dickens et al. 1991, Nature 351, 212\\
\hline
\end{tabular}
\caption[]{References concerning deep mixing in the clusters shown in
Fig.~\ref{cmds_obs}}
\label{tab-mix}
\end{table}
\suppressfloats
Such ``helium mixed'' red giant stars evolve to higher luminosities and
therefore lose more mass than their canonical counterparts. The resulting HB
stars then have less massive hydrogen envelopes and are thus hotter than in the
canonical case. In addition the higher helium abundance in the hydrogen
envelopes of the HB stars increases the efficiency of the hydrogen shell
burning and thereby leads to higher luminosities at a given effective
temperature. This increase in luminosity leads to lower gravities for ``deep
mixed'' HB stars than predicted by canonical evolution. For a more detailed
discussion of the effects of deep mixing on post-RGB evolution see Sweigart
(1997, 1999).
\begin{figure}[!]
\vspace*{8.6cm}
\special{psfile=plottga2.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The results from Fig.~\ref{plottga1} compared to evolutionary tracks
that take into account the effects of helium mixing (Sweigart 1999). $\Delta
X_{mix}$ gives the amount of He mixing with 0 indicating an unmixed track (for
details see Sweigart 1999). Also given is the position of the helium
main-sequence (Paczynski 1971).}
\label{plottga2}
\end{figure}
From Fig.~\ref{plottga2} it can be seen that most stars hotter than 11,000~K
are well fitted by tracks that assume deep mixing\footnote{The good fit of the
helium-mixed tracks to the stars in NGC~288 is problematic as there is no
evidence for deep mixing in this cluster (cf. Table~\ref{tab-mix}).}. The
cooler stars, however, are better explained by canonical evolution. As deep
mixing leads to hotter and brighter blue HB stars it is possible that cool
blue HB stars result from unmixed RGB stars. Unfortunately it is not possible
to determine the envelope helium abundance of a blue HB star because almost
all of these stars are helium-deficient due to diffusion (Heber 1987). As it
remains unclear what causes deep mixing (although rotation probably plays a
role, Sweigart \& Mengel 1979) we do also not know whether all RGB stars
within one cluster experience the same degree of mixing.
\subsection*{Masses}
Knowing effective temperatures and surface gravities of the stars allows to
determine the theoretical brightness at the stellar surface, which together
with the absolute brightness of the star yields its radius and thus its mass
(see de Boer et al. 1995, Moehler et al. 1995, 1997b). The distances to the
globular clusters (necessary to determine the absolute brightnesses of the
stars) were taken from the compilation of Djorgovski (1993). The results are
plotted in Fig.~\ref{plottlm1} and can be summarized as follows:
\begin{figure}[!]
\vspace*{8.6cm}
\special{psfile=plottlm1.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The masses of the stars derived with the distances listed by
Djorgovski (1993) compared to evolutionary calculations from Dorman et al.
(1993). The solid line gives the ZAHB for [Fe/H] = $-$1.48, the dashed line
marks the ZAHB for [Fe/H] = $-$2.26.}
\label{plottlm1}
\end{figure}
While the masses of the stars in M~5 and NGC~6752 scatter around the canonical
values the blue HB stars in all other clusters show masses that are
significantly lower than predicted by canonical HB evolution - even for
temperatures cooler than 11,000~K where we saw no deviation in surface gravity
from the canonical tracks. The fact that the stars in two of the clusters show
``normal'' mass values makes errors in the analyses an unlikely cause for the
problem (for a more detailed discussion see Moehler et al. 1995, 1997b, and de
Boer et al. 1995). Also the merger models of Iben (1990) cannot explain these
masses since the resulting stars are much hotter. However, if some of the {\bf
distance moduli} we used were too small this could cause such an effect --
larger distances would result in brighter absolute magnitudes, i.e. larger
radii and thus larger masses.
\subsubsection*{Distances to Globular Clusters }
Using {\sc Hipparcos} data for local subdwarfs several authors (Reid 1997,
1998; Gratton et al. 1997; Pont et al. 1998) determined distances to globular
clusters by main sequence fitting. The results are given in
Table~\ref{tab-hipp} and show that the new distance moduli are in general
larger than the old ones, in some cases by up to \magpt{0}{4} \ldots
\magpt{0}{6}\footnote{An increase of \magpt{0}{2} in (m-M)$_{\rm V}$
increases the mass of a cluster star by 20\%.}.
\begin{table}
\begin{tabular}{llllllll}
\hline
Cluster & \multicolumn{2}{c}{[Fe/H]} & \multicolumn{5}{c}{(m-M)$_0$}\\
& ZW84 & CG97 & D93 & R97 & R98 & G97 & P98 \\
\hline
47 Tuc & $-$0.71 & $-$0.67 & 13.31 & & 13.56 & 13.64 & \\
M 71 & $-$0.58 & $-$0.70 & 12.96 & & 13.19 & & \\
{\bf NGC 288} & $-$1.40 & $-$1.05 & 14.62 & & 15.00 & 14.96 & \\
{\bf M 5} & $-$1.40 & $-$1.10 & 14.40 & 14.45 & & 14.62 & \\
NGC 362 & $-$1.28 & $-$1.12 & 14.67 & & & 15.06 & \\
M 13 & $-$1.65 & $-$1.41 & 14.29 & 14.48 & 14.45 & 14.47 & \\
{\bf NGC 6752} & $-$1.54 & $-$1.43 & 13.12 & 13.20 & 13.16 & 13.34 & \\
{\bf NGC 6397} & $-$1.91 & $-$1.82 & 11.71 & & 12.24 & & \\
M 30 & $-$2.13 & $-$1.88 & 14.35 & 14.95 & & 14.96 & \\
M 68 & $-$2.09 & $-$1.95 & 14.84 & 15.29 & & 15.33 & \\
{\bf M 15} & $-$2.15 & $-$2.12 & 15.11 & 15.38 & & & \\
{\bf M 92} & $-$2.24 & $-$2.15 & 14.38 & 14.93 & & 14.82 & 14.70 \\
\hline
\end{tabular}
\caption[]{{\sc Hipparcos} based distance moduli for globular clusters
compared to older values. The data are taken from Zinn \& West (1984, ZW84),
Carretta \& Gratton (1997, CG97), Djorgovski (1993, D93), Reid (1997, R97;
1998, R98), Gratton et al. (1997, G97) and Pont et al. (1998, P98). The
clusters discussed here are marked in bold font.}
\label{tab-hipp}
\end{table}
\begin{table}[!]
\begin{tabular}{lllll}
\hline
Cluster & \multicolumn{2}{c}{[Fe/H]} & $<\frac{M}{M_{ZAHB}}>_{D93}$
& $<\frac{M}{M_{ZAHB}}>_{R}$ \\
\hline
NGC 288 & $-$1.40 & $-$1.05 & 0.52 $\pm$ 0.12 & 0.69 $\pm$ 0.16 \\
M 5 & $-$1.40 & $-$1.10 & 0.87 $\pm$ 0.21 & 0.95 $\pm$ 0.22 \\
NGC 6752 & $-$1.54 & $-$1.43 & 1.04 $\pm$ 0.52 & 1.10 $\pm$ 0.55 \\
NGC 6397 & $-$1.91 & $-$1.82 & 0.62 $\pm$ 0.16 & 0.78 $\pm$ 0.20 \\
M 15 & $-$2.15 & $-$2.12 & 0.81 $\pm$ 0.49 & 1.10 $\pm$ 0.67 \\
M 92 & $-$2.24 & $-$2.15 & 0.66 $\pm$ 0.25 & 1.10 $\pm$ 0.42 \\
\hline
\end{tabular}
\caption[]{The average ratio of calculated mass (as described in the text)
to the mass on the ZAHB (for the respective temperature) of HB stars in
globular clusters plotted in Figs.~\ref{plottlm1} and \ref{plottlm2}. The
distance moduli were taken from Djorgovski (1993, D93) and Reid
(1997, 1998, R).}
\label{tab-mass}
\end{table}
\begin{figure}[!]
\vspace*{8.6cm}
\special{psfile=plottlm2.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The masses of the stars derived with the new distances listed by Reid
(1997, 1998) compared to evolutionary tracks from Dorman et al. (1993). The
solid line gives the ZAHB for [Fe/H] = $-$1.48, the dashed line marks the ZAHB
for [Fe/H] = $-$2.26.}
\label{plottlm2}
\end{figure}
It is interesting to note that for M~5 and NGC~6752 (where the masses almost
agree with the canonical expectations) the new distances are close to the old
ones, whereas for the metal poor clusters M~15, M~92, and NGC~6397 the new
distance moduli are \magpt{0}{3} -- \magpt{0}{6} larger than the old ones,
thereby greatly reducing the mass discrepancies (see also Heber et al., 1997).
The resulting new masses are plotted in Fig.~\ref{plottlm2} and in
Table~\ref{tab-mass} we list the average ratio between the mass calculated for
an HB star (as described in the text) and the supposed ZAHB mass for its
temperature (from Dorman et al. 1993). It can be seen that in all cases
(except NGC~6752) the agreement between expected and calculated mass improves
with the new distance moduli, although the masses in NGC~288 remain
significantly too low. From our observations we therefore favour the longer
distance scale for globular clusters as suggested by most analyses of the {\sc
Hipparcos} data.
\section{Blue HB Stars in Metal-Rich Globular Clusters }
As mentioned in Section~1 metal-rich globular clusters tend to have red
horizontal branches. This is plausible as according to canonical stellar
evolutionary theory metal-rich HB stars have to have much smaller envelope
masses than metal-poor HB stars to achieve the same temperature. Therefore a
fine tuning of mass loss is required to produce blue HB stars in metal-rich
environments in the framework of classical stellar evolution. Deep mixing or
the merging of two helium white dwarfs offer other, more exotic, possibilities
to produce hot stars. Yi et al. (1997, 1998) discuss possible mechanisms to
produce blue HB stars in elliptical galaxies and d'Cruz et al. (1996) describe
mechanisms to create extreme HB stars in metal-rich open clusters like
NGC~6791, where Liebert et al. (1994) found subdwarf B stars. Despite this
recent theoretical work it came as a surprise when blue HB stars really showed
up in metal-rich globular clusters:
UIT images of 47~Tuc ([Fe/H] = $-$0.71; O'Connell et al. 1997) and of
NGC~362\footnote{\rm While not exactly metal-rich NGC~362 has been famous as
part of the second-parameter pair NGC~288/NGC~362: Both clusters have similar
metallicities, but NGC~288 shows a well populated blue HB, whereas NGC~362
shows almost only red HB stars.} ([Fe/H] = $-$1.28; Dorman et al. 1997) show
the presence of blue stars. Colour-magnitude diagrams of the central regions of
NGC~6388 ([Fe/H] = $-$0.60) and NGC~6441 ([Fe/H] = $-$0.53) obtained with the
Hubble Space Telescope (HST) show sloped blue HB's and long blue tails in
both clusters (Rich et al. 1997). The slope means that in these clusters bluer
HB stars are visually brighter than redder ones, in contrast to canonical
expectations. Their brighter luminosities require the blue HB stars to have
lower gravities, which can be caused by rotation, deep mixing and/or higher
primordial helium abundance (Sweigart \& Catelan 1998).
\begin{figure}[!]
\vspace*{8.6cm}
\special{psfile=plottg3.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The effective temperatures and surface gravities of the blue stars in
47~Tuc, NGC~362, NGC~6388, and NGC~6441 compared to an evolutionary track for
[Fe/H] = $-$0.47 from Dorman et al. (1993, solid line). Only probable cluster
members are plotted. The dashed lines mark the region of deep mixed tracks for
[Fe/H] = $-$0.5 of Sweigart (priv. comm.).}
\label{plottg3}
\end{figure}
To find out what really causes the unexpected presence of blue HB stars in
these metal-rich clusters we decided to perform a spectroscopic investigation.
Unfortunately we were not too lucky with weather and technical conditions and
the number of observed stars is small. In addition, some of the blue stars in
47~Tuc and NGC~362 turned out to be field HB stars or SMC main sequence stars.
Those stars, that are members of the clusters, are confirmed to be blue HB
stars with effective temperatures between 7,500~K and 15,000~K. Due to the
weather conditions we could not observe the fainter and therefore hotter stars
in these clusters, which would be especially interesting for the question of
deep mixing. More details can be found in the forthcoming papers Moehler,
Landsman, Dorman (47~Tuc, NGC~362) and Moehler, Catelan, Sweigart, Ortolani
(NGC~6388, NGC~6441). The results of our spectroscopic analyses are plotted in
Fig.~\ref{plottg3} and show that so far there is no evidence for deep mixing or
primordial helium enrichment in these stars\footnote{The helium white dwarf
merging model of Iben (1990) is unable to produce stars with so low
temperatures, because available hydrogen envelope masses are small ($<
10^{-4}$~\Msolar).}. More spectra, especially of fainter stars, are necessary
to verify this statement.
\section{Hot UV Bright Stars in Globular Clusters}
As mentioned in Section~1 optical searches for UV bright stars in globular
clusters yielded mainly stars cooler than 30,000~K (the majority of which was
even cooler than 15,000~K) due to the increasing bolometric corrections for
hotter stars. The vast majority of stars selected this way will evolve either
from the blue HB to the asymptotic giant branch or from there to the white
dwarf domain. It is rather unlikely to find post-EHB stars this way as they
spend only short time in such cool regions (if they reach them at all, see
Fig.~\ref{uvbs_tg}). In addition their overall fainter magnitudes work against
their detection. Searches in the ultraviolet regime, on the other hand, will
favour hotter stars and thereby increase the chance to detect post-EHB stars.
We therefore decided to spectroscopically analyse the many hot UV bright stars
that were found in globular clusters by the Ultraviolet Imaging Telescope.
Details of the observations, reduction, and analyses can be found in Moehler
et al. (1998a). The main goal was to find out how the physical parameters of
these stars compare to evolutionary tracks.
The derived effective temperatures and gravities of the target stars are
plotted in Fig.~\ref{uvbs_tg} and compared to various evolutionary tracks.
Three of the stars (in NGC~6121 and NGC~6723) appear to fit the post-early
AGB\footnote{Post-early AGB stars left the asymptotic giant branch before the
thermally pulsing stage} track, while the remaining targets (in NGC~2808 and
NGC~6752\footnote{including three stars analysed by Moehler et al. (1997b)})
are consistent with post-EHB evolutionary tracks. Like the extreme HB stars
themselves (Moehler et al. 1997b) the post-EHB stars show subsolar helium
abundances probably caused by diffusion. In contrast the three post-early AGB
stars - which are supposed to be successors to helium-deficient blue HB stars
- have approximately solar helium abundances. This agrees with the expectation
that already during the early AGB stages convection is strong enough to
eliminate any previous abundance patterns caused by diffusion.
\begin{figure}[ht]
\vspace*{8.6cm}
\special{psfile=uvbs_tg.ps hscale=45 vscale=45 hoffset=10 voffset=255
angle=270}
\caption{The atmospheric parameters of the UV bright stars compared to
evolutionary tracks. The solid lines mark the ZAHB and post-EHB evolutionary
tracks for [Fe/H] = $-$1.48 (Dorman et al. 1993). The dashed lines give
post-AGB (0.565~\Msolar) and post-early AGB (0.546~\Msolar) tracks from
Sch\"onberner (1983). All evolutionary tracks are labeled with the mass of the
respective model in units of 10$^{-3}$\Msolar. The filled symbols are from
Moehler et al. (1997b, 1998a), the open symbols are taken from Conlon et al.
(1994), Heber \& Kudritzki (1986) and Heber et al. (1993).}
\label{uvbs_tg}
\end{figure}
As expected, the two clusters with populous EHB's (NGC~2808 and NGC~6752) have
post-EHB stars but no post-AGB stars. The number ratio of post-EHB to EHB
stars in NGC~6752, however, is much lower than expected from stellar
evolutionary theory: 6\% instead of 15 -- 20\%. This discrepancy has first
been noted by Landsman et al. (1996) and has been confirmed by our studies,
which verified all four post-EHB candidates, but found no additional ones
(Moehler et al. 1997b, 1998a). The clusters NGC~6723 and M~4, on the other
hand, do not have an EHB population, although they do have stars blueward of
the RR Lyrae gap (which are potential progenitors of post-early AGB stars).
The lack of genuine post-AGB stars may be understood from the different
lifetimes: The lifetime of Sch\"onberner's (1983) post-early AGB track is
about 10 times longer than that of his lowest mass post-AGB track. Thus, even
if only a small fraction of stars follow post-early AGB tracks, those stars
may be more numerous than genuine post-AGB stars. Due to their relatively long
lifetime, post-early AGB stars are unlikely to be observed as central stars of
planetary nebulae since any nebulosity is probably dispersed before the
central star is hot enough to ionize it. These different life times in
combination with the fact that a considerable number of globular clusters
stars (all post-EHB stars and some post-BHB stars) do not reach the thermally
pulsing AGB stage could be an explanation for the lack of planetary nebulae in
globular clusters reported by Jacoby et al. (1997).
\section{Abundance Patterns of UV Bright Stars in Globular Clusters}
Up to now detailed abundance analyses have been performed mainly for post-AGB
stars in the field of the Milky Way (McCausland et al. 1992 and references
therein; Conlon 1994; Napiwotzki et al. 1994; Moehler \& Heber 1998), for
which the population membership is difficult to establish. The summarized
result of these analyses is that the abundances of N, O, and Si are roughly
1/10 of the solar values, while Fe and C are closer to 1/100 solar.
McCausland et al. (1992) and Conlon (1994) interpret the observed abundances
as the results of dredge-up processes on the AGB, i.e. the mixing of
nuclearly processed material from the stellar interior to the surface.
Standard stellar evolutionary theories (Renzini \& Voli 1981; Vassiliadis \&
Wood 1993) do not predict any dredge-up processes for the low-mass precursors
of these objects. Nevertheless the planetary nebula Ps~1 in M~15 as well as the
atmosphere of its central star K~648 are both strongly enriched in carbon when
compared to the cluster carbon abundance\footnote{A preliminary analysis of
ZNG1 in M~5 also shows evidence for a third dredge-up, but no trace of a nebula
(Heber \& Napiwotzki 1999).} (Adams et al. 1984; Heber et al. 1993),
pinpointing the dredge-up of triple $\alpha$ processed material to the stellar
surface and suggesting a possible connection between dredge-up and planetary
nebula ejection (Sweigart 1998). This discrepancy may be solved by newer
evolutionary calculations which are able to produce a third dredge-up also in
low-mass AGB stars (Herwig et al. 1997).
Napiwotzki et al. (1994) on the other hand suggest that the photospheric
abundances are caused by gas-dust separation towards the end of the AGB phase:
If the mass loss at the end of AGB ceases rapidly gas can fall back onto the
stellar surface while the dust particles are blown away by radiative pressure.
This process has been proposed by Bond (1991) to explain the extreme iron
deficiencies seen in some cooler post-AGB stars and is described in more
detail by Mathis \& Lamers (1992).
As iron is very sensitive to depletion by gas-dust separation the iron
abundance is the crucial key to the distinction between dredge-up and gas-dust
separation. To verify any elemental depletion, however, one has to know the
original abundance of the star, which is generally not the case for field
stars. Therefore UV bright stars in globular clusters with known metallicities
provide ideal test cases for this problem and we started a project to derive
iron abundances from high-resolution UV spectra obtained with HST.
\subsection*{Abundance analysis of Barnard~29 in M~13 and ROA~5701 in
$\omega$ Cen}
\begin{figure}[!ht]
\vspace{10cm}
\special{psfile=ghrs_uvbs.ps hscale=50 vscale=50 hoffset=-5 voffset=280
angle=270}
\caption{The abundances derived for ROA~5701 ({\bf left}) and Barnard~29 ({\bf
right}) compared to those of other stars in $\omega$ Cen resp. M~13. The solid
line connects the abundances derived for ROA~5701 resp. Barnard~29. In the
{\bf left} panel the short dashed lines mark the cool UV bright stars in
$\omega$ Cen (filled circles, Gonzalez \& Wallerstein 1994), the dotted lines
connect the abundances of the red giants (open circles, Paltoglou \& Norris
1989; Brown et al. 1991; Brown \& Wallerstein 1993; Smith et al. 1995). In
the {\bf right} panel the dotted and short-dashed lines mark the abundances of
the red giants in M~13 taken from Smith et al. (1996, circles) resp. Kraft et
al. (1997, squares).}
\label{ghrs_uvbs}
\end{figure}
For Barnard~29 a detailed abundance analysis from optical spectra has been
done by Conlon et al. (1994) and we use their results for \teff\ and \logg\
for our analysis. For ROA~5701 we determined these parameters from IUE
low-resolution spectra, optical photometry, and optical spectroscopy. For the
iron abundances of both stars we used GHRS spectra of 0.07~\AA\ resolution
that cover the range 1860 $-$ 1906 \AA. The abundances have been derived using
the classical curve-of-growth technique. We computed model atmospheres for the
appropriate values of effective temperature, surface gravity, and cluster
metallicity and used the LINFOR spectrum synthesis package (developed
originally by Holweger, Steffen, and Steenbock at Kiel university) for the
further analysis. A more detailed description of our analysis can be found in
Moehler et al. (1998b).
For ROA~5701 we find an iron abundance of $\log{\epsilon_{Fe}} = 4.89 \pm
0.12$ ([Fe/H] = $-$2.61) and for Barnard~29 we get $\log{\epsilon_{Fe}} = 5.38
\pm 0.14$ ([Fe/H] = $-$2.12). Both stars thus show iron abundances
significantly below the mean cluster abundances of [Fe/H] $\approx$ $-$1.5
\ldots $-$1.7. To look for any abundance trends in Barnard~29 and ROA~5701 in
comparison with other stars in these clusters we used the abundances of C, N,
O, Si in addition to iron. For ROA~5701 we determined these abundances from
optical high-resolution spectra. Abundance analyses of Barnard~29 have been
performed by Conlon et al. (1994, N, O, Si) and Dixon \& Hurwitz\footnote{They
also give an iron abundance of $\log{\epsilon_{Fe}} = 5.30^{+0.22}_{-0.26}$,
somewhat higher than ours.} (1998, C).
\begin{figure}[ht]
\vspace{10cm}
\special{psfile=ghrs_cno.ps hscale=50 vscale=50 hoffset=-5 voffset=280
angle=270}
\caption[]{The Fe abundances vs. the sum of CNO abundances for ROA~5701 ({\bf
left panel}) and Barnard~29 ({\bf right panel}) compared to those of red
giants and cool UVBS in $\omega$ Cen and M~13, respectively. For references
see Fig.~\ref{ghrs_uvbs}.}
\label{ghrs_cno}
\end{figure}
It can be seen in Fig.~\ref{ghrs_uvbs} that N, O, and Si in our two objects
show a behaviour similar to that in red giant stars. C seems to be depleted in
ROA~5701, but this abundance is based on an upper limit for one line only
(4267~\AA) which may be affected by non-LTE effects. One should note here,
however, that Gonzalez \& Wallerstein (1994) find strong enhancements of CNO
and s-process elements for the brightest of the cool UV bright stars in
$\omega$~Cen which they interpret as evidence for a third dredge-up. Taking
the sum of C+N+O as indicator for the original iron abundance (cf.
Fig.~\ref{ghrs_cno}) shows that ROA~5701 and Barnard~29 were not born iron
depleted and -- contrary to the brightest cool UV bright stars in $\omega$ Cen
-- also do not show any evidence for a third dredge up. The results of our
analysis thus favour the gas-dust separation scenario as explanation for the
abundance patterns of low-mass post-AGB and post-early AGB stars.
\section{Summary}
About nine years ago we began a spectroscopic study of blue horizontal branch
stars to find the reason for the gaps seen in the CMD's of many globular
clusters. While we haven't yet achieved this goal the study led to others
which altogether resulted in some interesting findings about the evolutionary
status of hot stars in globular clusters:
\subsection*{Blue Horizontal Branch Stars}
We studied stars above and below the gaps seen along the blue horizontal branch
in the CMD's of many globular clusters (cf. Fig.~\ref{cmds_obs}) and found that
most of the stars below the gaps are physically the same as the stars above
the gaps, i.e. blue horizontal branch stars with a helium burning core and a
hydrogen burning shell. So far extreme horizontal branch stars have been
verified spectroscopically only in two clusters (NGC~6752 and M~15).
The blue HB stars with temperatures between 11,000~K and 20,000~K show lower
gravities than expected from canonical stellar evolution, which can be
explained by deep mixing. The lower than expected masses that are found for
most stars cooler than 20,000~K can be understood if we assume that the
distance moduli to the globular clusters are larger than previously thought.
Analyses of BHB and EHB stars within the same cluster will provide a crucial
test of these two hypotheses.
We verified that most of the blue stars seen in the colour-magnitude diagrams
of several metal-rich globular clusters are indeed blue horizontal branch
stars in these clusters. We did so far not find any significant evidence for
deep mixing or a higher primordial helium abundance in these metal-rich
globular clusters but have currently too few data to draw any firm conclusions.
\subsection*{UV Bright Stars in Globular Clusters}
Analyses of hot UV bright stars in globular clusters uncovered a lack of
genuine post-AGB stars -- we found only post-early AGB and post-EHB stars. This
may be an explanation for the lack of planetary nebulae in globular clusters
seen by Jacoby et al. (1997). Abundance analyses of post-AGB stars in two
globular clusters suggest that gas and dust separate during the AGB phase.
\subparagraph{Acknowledgements}
I want to thank M. Catelan, K.S. de Boer, U. Heber, W.B. Landsman, M. Lemke,
R. Napiwotzki, S. Ortolani, and A.V. Sweigart for their collaboration on these
projects. Thanks go also to the staff of the La Silla, ESO, and Calar Alto,
DSAZ, and HST observatories for their support during and after observations. I
gratefully acknowledge support for this work by the DFG (grants Mo~602/1,5,6),
DARA (grant 50~OR~96029-ZA), Alexander von Humboldt-Foundation (Feodor Lynen
fellowship) and Dr. R. Williams as director of the Space Telescope Science
Institute (DDRF grant).
\subsection*{References}
\litanf
\item Adams S., Seaton M.J., Howarth I.D., Auri\'ere M., Walsh J.R., 1984,
MNRAS 207, 471
\item Alcaino G., Buonanno R., Caloi V., Castellani V., Corsi C.E.,
Iannicola G., Liller W., 1987, AJ 94, 917
\item Arp H.C., 1955, AJ 60, 317
\item Barnard E.E., 1900, ApJ 12, 176
\item Bell R.A., Briley M.M., Norris J.E., 1992, AJ 104, 1127
\item Bergeron, P., Saffer, R.A., Liebert, J., 1992, ApJ 394, 228
\item Bond H.E., 1991, in {\it Evolution of Stars: the Photospheric
Abundance Connection}, eds.\ G.~Michaud, A.~Tutukov, IAU Symp. 145
(Kluwer: Dordrecht) p.~341
\item Bragaglia, A., C. Cacciari, C., Carretta, E., Fusi Pecci, F., 1999, in
{\it The 3$^{rd}$ Conf. on Faint Blue Stars}, eds. A.G.D. Philip, J.
Liebert \& R.A. Saffer (Cambridge:CUP), p. 447
\item Brown J.A., Wallerstein G., Cunha K., Smith V.V., 1991, A\&A 249, L13
\item Brown J.A., Wallerstein G., 1993, AJ 106, 133
\item Brown T.M., Ferguson H.C., Davidsen A.F., Dorman B., 1997, ApJ 482,
685
\item Buonanno R., Corsi C.E., Fusi Pecci F., 1981, MNRAS, 196, 435
\item Buonanno R., Buscema G. Corsi C., Iannicola G., Fusi Pecci F., 1983a,
A\&AS 51, 83
\item Buonanno R., Buscema G. Corsi C., Iannicola G., Smriglio F., 1983b,
A\&AS 53, 1
\item Buonanno R., Corsi C.E., Fusi Pecci F., Alcaino G., Liller W., 1984,
A\&AS 57, 75
\item Buonanno R., Caloi V., Castellani V., Corsi C.E., Fusi Pecci F.,
Gratton R., 1986, A\&AS 66, 79
\item Buonanno R., Corsi C.E., Buzzoni A., Cacciari C., Ferraro F.R., Fusi
Pecci F., 1994, A\&A 290, 69
\item Cacciari C., Fusi Pecci F., Bragaglia A., Buzzoni A., 1995, A\&A 301, 684
\item Caloi V., 1972, A\&A 20, 357
\item Caloi V., 1999, A\&A in press
\item Carretta E., Gratton R.G. 1997, A\&AS 121, 95
\item Catelan M., Borissova J., Sweigart A.V., Spassova N., 1998, ApJ 494, 265
\item Cavallo R.M., Sweigart A.V., Bell R.A., 1998, ApJ 492, 575
\item Code A.D., Welch G.A., 1979, ApJ 228, 95
\item Conlon E.S., 1994, in {\it Hot Stars in the Galactic Halo}, eds. S.
Adelman, A. Upgren, C.J. Adelman, CUP, p. 309
\item Conlon E.S., Dufton, P.L., Keenan, F.P., 1994, A\&A 290, 897
\item Cool A.M., Piotto G., King I.R., 1996, ApJ 468, 655
\item Crocker D.A., Rood R.T., O'Connell R.W., 1988, ApJ 332, 236
\item d'Cruz N.L., Dorman B., Rood R.T., O'Connell R.W., 1996, ApJ 466, 359
\item de Boer K.S., 1982, A\&AS 50, 247
\item de Boer K.S., 1985, A\&A 142, 321
\item de Boer K.S., 1987, in {\it The 2$^{nd}$ Conference on Faint Blue Stars},
eds. A.G.D. Philip, D.S. Hayes, J. Liebert, Davis Press, p.~95
\item de Boer K.S., Schmidt J.H.K., Heber U., 1995, A\&A 303, 95
\item Dickens R.J., Croke B.F.W., Cannon R.D., Bell R.A., 1991, Nature 351,
212
\item Dixon W.V., Davidsen A.F., Dorman B., Ferguson H.C., 1996, AJ 111, 1936
\item Dixon W.V., Hurwitz M., 1998, ApJ 500, L29
\item Djorgovski S., 1993, in {\it Structure and Dynamics of Globular Clusters},
eds. S.G. Djorgovski \& G. Meylan, ASP Conf. Ser. 50, p. 373
\item Dorman, B., Lee Y.-W., VandenBerg D.A., 1991, ApJ 366, 115
\item Dorman, B., Rood, R.T., O'Connell, W.O., 1993, ApJ 419, 596
\item Dorman B., Shah R.Y., O'Connell R.W., Landsman W.B., Rood R.T., et al.,
1997, ApJ 480, L31
\item Durrell P.R., Harris W.E., 1993, AJ 105, 1420
\item Faulkner J., 1966, ApJ 144, 978
\item Ferraro F.R., Paltrinieri B., Fusi Pecci F., Dorman B., Rood R.T.,
1998, ApJ 500, 311
\item Fusi Pecci F., Ferraro F.R., Bellazzini M., Djorgovski S., Piotto G.,
Buonanno R., 1993, AJ 105, 1145
\item Gingold R.A., 1976, ApJ 204, 116
\item Gonzalez G., Wallerstein G., 1994, AJ 108, 1325
\item Gratton R.G., Fusi Pecci F., Carretta E., Clementini G., Corsi C.E.,
Lattanzi M. 1997, ApJ 491, 749
\item Greenstein J.L., 1939, ApJ 90, 387
\item Greenstein J.L., 1971, in {\it White Dwarfs}, ed. W.J. Luyten,
IAU Symp. 42, (Reidel), p. 46
\item Greggio L., Renzini A., 1990, ApJ 364, 35
\item Harris H.C., Nemec J.M., Hesser J.E., 1983, PASP 95, 256
\item Heber U., 1983, A\&A 118, 39
\item Heber U., 1987, Mitt. Astron. Ges. 70, 79
\item Heber U., Kudritzki R.P., Caloi V., Castellani V., Danziger J., Gilmozzi
R., 1986, A\&A 162, 171
\item Heber U., Kudritzki R.P., 1986, A\&A 169, 244
\item Heber U., Dreizler S., Werner, K., 1993, Acta Astron. 43, 337
\item Heber U., Moehler S., Reid I.N., 1997, in {\it HIPPARCOS Venice '97},
ed. B. Battrick, ESA-SP 402, p. 461
\item Heber U., Napiwotzki R., 1999, in {\it The 3$^{rd}$ Conf. on Faint
Blue Stars}, eds. A.G.D. Philip, J. Liebert \& R.A. Saffer
(Cambridge:CUP), p. 439 ({\tt astro-ph/9809129})
\item Herwig F., Bl\"ocker T., Sch\"onberber D., El Eid M., 1997, A\&A 324,
L81
\item Hoyle F., Schwarzschild M., 1955, ApJS 2, 1
\item Iben I. Jr., 1990, ApJ 353, 215
\item Iben I.Jr., Rood R.T., 1970, ApJ 161, 587
\item Iben I. Jr., Tutukov A.V., 1984, ApJS 54, 335
\item Iben I. Jr., Tutukov A.V., 1993, ApJ 418, 343
\item Jacoby G.H., Morse J. A., Fullton L.K., Kwitter K.B., Henry R.B.C,
1997, AJ 114, 2611
\item Jordan S., Koester D., A\&AS 65, 367
\item Kraft R.P., 1994, PASP 106, 553
\item Kraft R.P., Sneden C., Langer G.E., Shetrone M.D., Bolte M., 1995, AJ
109, 2586
\item Kraft R.P., Sneden C., Smith G.H., Shetrone M.D., Langer G.E.,
Pilachowski C.A., 1997, AJ 113, 279
\item Kurucz R.L., 1979, ApJS 40, 1
\item Kurucz R.L., 1992, in {\it The Stellar Populations of Galaxies},
eds. B. Barbuy \& A. Renzini, IAU Symp. 149 (Kluwer:Dordrecht), 225
\item Landsman W.B., Sweigart A.V., Bohlin R.C., Neff S.G., O'Connell R.W.,
et al., 1996, ApJ 472, L93
\item Lee Y.-W., Demarque P., Zinn R., 1994, ApJ 423, 248
\item Leone F., Manfr\`e M., 1997, A\&A 320, 257
\item Liebert J., Saffer R.A., Green E.M., 1994, AJ 107, 1408
\item Mathis J.S., Lamers, H.J.G.L.M., 1992 A\&A 259, L39
\item McCausland R.J.H., Conlon E.S., Dufton P.L., Keenan F.P., 1992, ApJ 394,
298
\item Moehler S., Heber U., de Boer K.S., 1995, A\&A 294, 65
\item Moehler S., Heber U., Durrell P., 1997a, A\&A 317, L83
\item Moehler S., Heber U., Rupprecht G., 1997b, A\&A 319, 109
\item Moehler S., Landsman W., Napiwotzki R., 1998a, A\&A 335, 510
\item Moehler S., Heber U., 1998, A\&A 335, 985
\item Moehler S., Heber U., Lemke M., Napiwotzki R., 1998b, A\&A 339, 537
\item Napiwotzki R., 1997 A\&A 322, 256
\item Napiwotzki R., Heber U., K\"oppen, J., 1994, A\&A 292, 239
\item Newell E.B., 1973, ApJS 26, 37
\item O'Connell R.W., Dorman B., Shah R.Y., Rood R.T., Landsman W.B., et al.,
1997, AJ 114, 1982
\item Paczynski B., 1971, Acta Astron. 21, 1
\item Paltoglou G., Norris J.E., 1989, ApJ 336, 185
\item Parise R.A., Bohlin R.C., Neff S.G., O'Connell R.W., Roberts M.S.,
et al., 1998, ApJ 501, L67
\item Pease F.G., 1928, PASP 40, 342
\item Peterson R.C., Rood R.T., Crocker D.A., 1995, ApJ 453, 214
\item Pilachowski C.A., Sneden C., Kraft R.P., Langer G.E., 1996, AJ 112, 545
\item Pont F., Mayor M., Turon C., VandenBerg D.A. 1998, A\&A 329, 87
\item Reid I.N. 1997, AJ 114, 161
\item Reid I.N. 1998, AJ 115, 204
\item Renzini A., Voli, M., 1981, A\&A 94, 175
\item Renzini A., Bragaglia A., Ferraro F.R., Gilmozzi R., Ortolani S.,
et al., 1996, ApJ 465, L23
\item Rich R.M., Sosin C., Djorgovski S.G., Piotto G., King I.R., et al.,
1997, ApJ 484, L25
\item Richer H.B., Fahlmann G.G., Ibata R.A., Stetson P.B., Bell R.A.,
et al., 1995, ApJ 451, L17
\item Richer H.B., Fahlmann G.G., Ibata R.A., Pryor C., Bell R.A.,
et al., 1997, ApJ 484, 741
\item Rood R.T., 1973, ApJ 184, 815
\item Rood R.T., Crocker D.A., 1985, in {\it Horizontal-Branch and
UV-Bright Stars''}, ed. A.G.D. Philip (Schenectady: L.Davis Press), p. 99
\item Saffer R.A., Bergeron P., Koester D., Liebert J., 1994, ApJ 432, 351
\item Sandage A.R., Wallerstein G., 1960, ApJ 131, 598
\item Savage B.D., Mathis F.S., 1979, ARAA 17,73
\item Sch\"onberner D., 1983, ApJ 272, 708
\item Schwarzschild M., H\"arm R., 1970, ApJ 160, 341
\item Shapley H., 1915a, Contr. Mt. Wilson 115
\item Shapley H., 1915b, Contr. Mt. Wilson 116
\item Shapley H., 1930, {\it Star Clusters}, The Maple Press Company, York,
PA, USA,
\item Shetrone M.D., 1996, AJ 112, 1517
\item Shetrone M.D., 1997, in {\it Fundamental Stellar Properties: The
Interaction between Observation and Theory}, IAU Symp. 189 (poster
proceedings) (Kluwer: Dordrecht), p. 158
\item Smith V.V., Cunha K., Lambert D.L., 1995, AJ 110, 2827
\item Smith G.H., Shetrone M.D., Bell R.A., Churchill C.W., Briley M.M., 1996,
AJ 112, 1511
\item Sneden C., Kraft R.P., Prosser C.F., Langer G.E., 1992, AJ 104, 2121
\item Sneden C., Kraft R.P., Shetrone M.D., Smith G.H., Langer G.E.,
Prosser C.F., 1997, AJ 114, 1964
\item Sosin C., Piotto G., Djorgovski S.G., King I.R., Rich R.M., Dorman B.,
Liebert J., Renzini A., 1997, in {\it Advances in Stellar Evolution},
eds. R.T. Rood \& A. Renzini, CUP, p. 92
\item Stecher T., Cornett R.H., Greason M.R., Landsman W.B., Hill J.K.,
et al., 1997, PASP 109, 584
\item Stoeckley R., Grennstein J.L., 1968, ApJ 154, 909
\item Strom S.E., Strom K.M., 1970, ApJ 159, 195
\item Strom S.E., Strom K.M., Rood R.T., Iben I.Jr., 1970, A\&A 8, 243
\item Sweigart A.V., 1987, ApJS 65, 95
\item Sweigart A.V., 1994, in {\it Hot Stars in the Galactic Halo}, eds. S.
Adelman, A. Upgren, C.J. Adelman, CUP, p. 17
\item Sweigart A.V., 1997, ApJ 474, L23
\item Sweigart A.V. 1998, to appear in {\it New Views on the Magellanic
Clouds}, eds. Y.-H. Chu, J. Hesser \& N. Suntzeff, IAU Symp. 190 (ASPC)
\item Sweigart A.V. 1999, in {\it The 3$^{rd}$ Conf. on Faint
Blue Stars}, ed. A.G.D. Philip, J. Liebert \& R.A. Saffer
(Cambridge:CUP), 3 ({\tt astro-ph/9708164})
\item Sweigart A.V., Mengel J.G., Demarque P., 1974, A\&A 30, 13
\item Sweigart A.V., Gross P.G., 1974, ApJ, 190, 101
\item Sweigart A.V., Gross P.G., 1976, ApJS 32, 367
\item Sweigart A.V., Mengel J.G., 1979, ApJ 229, 624
\item Sweigart A.V., Catelan M., 1998, ApJ 501, L63
\item ten Bruggencate P., 1927, {\it Sternhaufen}, Julius Springer Vlg.,
Berlin
\item Traving G., 1962, ApJ 135, 439
\item Tuchman Y., 1985, ApJ 288, 248
\item Vassiliadis E., Wood P.R., 1993, ApJ 413, 641
\item Whitney J.H., Rood R.T., O'Connell R.W., D'Cruz N.L., Dorman B.,
et al., 1998, ApJ 495, 284
\item Yi S., Demarque P., Kim Y.-C., 1997, ApJ 482, 677
\item Yi S., Demarque P., Oemler A. Jr., 1998, ApJ 492, 480
\item Zinn R., 1974, ApJ, 193, 593
\item Zinn R.J., Newell E.B., Gibson J.B., 1972, A\&A 18, 390
\item Zinn R., West M.J. 1984, ApJS 55, 45
\end{list}
\end{document}
\\
|
2,877,628,089,534 | arxiv | \section{Introduction}
Black-box optimization is the problem of optimizing an objective function $f : \mathcal{X} \mapsto \mathbb{R}$ within the constraints of a given evaluation budget $B$.
In other words, the objective is to obtain a point ${\bf x} \in \mathcal{X}$ with the lowest possible evaluation value $f({\bf x})$ within the $B$ function evaluations.
In black-box optimization, no algebraic representation of $f$ is given, and no gradient information is available.
Black-box optimization involves problems such as hyperparameter optimization of machine learning algorithms~\cite{NIPS2011_4443,EggFeuBerSnoHooHutLey13,ilievski2017efficient,Golovin:2017:GVS:3097983.3098043}, parameter tuning of agent-based simulations~\cite{Yang2009AgentBasedSO}, and aircraft design~\cite{10.1007/978-3-540-31880-4_43}.
In black-box optimization, it is generally assumed that the computational cost for evaluating the point is large; thus it is important to search efficiently with as low budgets as possible.
For example, it is reported that the experiment of hyperparameter optimization of Online LDA takes about 12 days for 50 evaluations~\cite{NIPS2012_4522}.
The performance of deep neural networks (DNN) is known to be very sensitive to hyperparameters, and it has been actively studied in recent years~\cite{Bergstra:2012:RSH:2188385.2188395,Domhan:2015:SUA:2832581.2832731,ilievski2017efficient,DBLP:journals/jmlr/LiJDRT17,DBLP:conf/aistats/KleinFBHH17,pmlr-v80-falkner18a}.
Because the experiment of DNN also takes a long time to learn corresponding to one hyperparameter, a lower evaluation budget can be used for hyperparameter optimization.
Bayesian optimization is an efficient method for black-box optimization.
Bayesian optimization is executed by repeating the following steps:
(1) Based on the data observed thus far, it constructs a surrogate model that considers the uncertainty of the objective function.
(2) It calculates the acquisition function to determine the point to be evaluated next by using the surrogate model constructed in step (1).
(3) By maximizing the acquisition function, it determines the point to be evaluated next.
(4) It then updates the surrogate model based on the newly obtained data, then returns to step (2).
However, Bayesian optimization, which constructs a surrogate model for the entire search space, can show bad performance in the low budget setting, because an optimization method cannot sample points sufficiently.
In the low budget setting, we believe that a search should be performed locally; however, in Bayesian optimization, to estimate uncertainty in the search space by surrogate model, the points are sampled for the search space globally.
Therefore, the lack of local search degrades the performance of Bayesian optimization.
If there is no prior knowledge of the problem, the search space tends to be widely defined.
When the search space is widely defined, the search will be performed more globally, degrading performance.
In this study, we develop a heuristic method that refines the search space for Bayesian optimization when evaluation budget is low.
The proposed method performs division to reduce the volume of the search space.
The proposed method makes it possible to perform Bayesian optimization within the local search space determined to be promising.
We confirm that Bayesian optimization with the proposed method outperforms Bayesian optimization alone (that is, Bayesian optimization without the proposed method) by the experiments on the six benchmark functions and the hyperparameter optimization of the three machine learning algorithms (multi-layer perceptron (MLP), convolutional neural network (CNN), LightGBM).
We also experiment with Simultaneous Optimistic Optimization (SOO)~\cite{NIPS2011_4304} and BaMSOO~\cite{pmlr-v33-wang14d}, which are search-space division algorithms, in order to confirm the validity of the refinement of the search space by the proposed method.
\section{Background}
\subsection{Bayesian Optimization}
\label{sec:bo}
Algorithm \ref{alg: bo} shows the algorithm of Bayesian optimization, which samples and evaluates the initial points (line 1),
constructs the surrogate model (line 3),
finds the next point to evaluate by optimizing the acquisition function (line 4),
evaluates the point selected and receives the evaluation value (line 5),
and updates the data (line 6).
The main components of Bayesian optimization are the surrogate model and the acquisition function.
In this section, we describe Bayesian optimization using a Gaussian process as the surrogate model and the expected improvement (EI) as the acquisition function.
\begin{algorithm}
\caption{Bayesian Optimization}
\label{alg: bo}
\begin{algorithmic}[1]
\REQUIRE objective function $f$, search space $\mathcal{X}$, initial sample size $n$, surrogate model $\mathcal{M}$, acquisition function $\alpha({\bf x} \mid \mathcal{M})$
\STATE sample and evaluate initial points: $\mathcal{D}_0 = \{({\bf x}_{0,1}, y_{0,1}), \cdots, ({\bf x}_{0, n}, y_{0, n})\}$
\FOR{$t = 1,2,\cdots$}
\STATE construct surrogate model $\mathcal{M}$ by $\mathcal{D}_{t-1}$
\STATE find ${\bf x}_t$ by optimizing the acqusition function $\alpha$ : ${\bf x}_t = \mathop{\rm arg~max}\limits_{{\bf x} \in \mathcal{X}} \alpha({\bf x} \mid \mathcal{M})$
\STATE evaluate ${\bf x}_t$ and receive: $y_t = f({\bf x}_t)$
\STATE update the data $\mathcal{D}_t = \mathcal{D}_{t-1} \cup \{({\bf x}_t, y_t)\}$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsubsection{Gaussian Process}
A Gaussian process~\cite{Rasmussen:2005:GPM:1162254} is the probability distribution over the function space characterized by the mean function $\mu: \mathcal{X} \to \mathbb{R}$ and the covariance function $\sigma^2: \mathcal{X} \to \mathbb{R}_{\geq 0}$.
We assume that data set $\mathcal{D}_t = \{({\bf x}_1, y_1) \cdots, ({\bf x}_{t}, y_{t})\}$ and observations ${\bf y} = ( y_{1}, \cdots , y_t )$ are obtained.
The mean $\mu({\bf x}_{t+1})$ and variance $\sigma^2({\bf x}_{t+1})$ of the predicted distribution $f({\bf x}_{t+1}) \sim \mathcal{N}(\mu({\bf x}_{t+1}), \sigma^2({\bf x}_{t+1}))$ of a Gaussian process with respect to ${\bf x}_{t+1}$ can be calculated using the kernel function $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}_{\geq 0}$ as follows:
\begin{align}
\mu({\bf x}_{t+1}) & = {\bf k}^{\top} {\bf K}^{-1} {\bf y}, \\
\sigma^2({\bf x}_{t+1}) & = k({\bf x}_{t+1}, {\bf x}_{t+1}) - {\bf k}^{\top} {\bf K}^{-1} {\bf k}.
\end{align}
Here,
\begin{align}
{\bf K} &=
\begin{bmatrix}
k({\bf x}_{1}, {\bf x}_{1}) & \cdots & k({\bf x}_{1}, {\bf x}_{t}) \\
\vdots & \ddots & \vdots\\
k({\bf x}_{t}, {\bf x}_{1}) & \cdots & k({\bf x}_{t}, {\bf x}_{t})
\end{bmatrix}, \\
\mathbf{k} &= \begin{bmatrix}k({\bf x}_{t+1}, {\bf x}_{1}) & \cdots & k({\bf x}_{t+1}, {\bf x}_{t})\end{bmatrix}^{\top}.
\end{align}
The squared exponential kernel (Equation (\ref{eq:se})) is one of the common kernel functions.
\begin{align}
\label{eq:se}
k_{\rm SE} ({\bf x}, {\bf x}^{\prime}) & = \sigma_f \exp \left(-\frac{r^2({\bf x}, {\bf x}^{\prime})}{2}\right), \\
\label{eq:kernel_r}
r^2({\bf x}, {\bf x}^{\prime}) & = \frac{\|{\bf x} - {\bf x}^{\prime}\|^2}{l}.
\end{align}
Here, $\sigma_f$ is a parameter that adjusts the scale of the whole kernel function, and $l$ is a parameter of sensitivity to the difference between the two inputs ${\bf x}, {\bf x}^{\prime}$.
\subsubsection{Expected Improvement}
The EI \cite{jones1998efficient} is a typical acquisition function in Bayesian optimization, and it represents the expectation value of the improvement amount for the best evaluation value of the candidate point.
Let the best evaluation value to be $f_{\min}$, EI for the point ${\bf x}$ is calculated as follow:
\begin{equation}
\label{eq: ei_origin}
\alpha_{\rm EI}({\bf x} \mid \mathcal{D}) = \mathbb{E}[\max\{f_{\min} - f({\bf x}), 0\} \mid {\bf x}, \mathcal{D}].
\end{equation}
When we assume that the objective function follows a Gaussian process, Equation (\ref{eq: ei_origin}) can be calculated analytically as follows:
\begin{equation}
\begin{split}
\alpha_{\rm EI}({\bf x} \mid \mathcal{D}) &=\begin{cases}
\sigma(x) \cdot \phi (Z) + (f_{\min} - \mu(x)) \cdot \Phi (Z) & (\sigma({\bf x}) > 0) \\
0 & (\sigma({\bf x}) = 0),
\end{cases} \\
Z &= \frac{f_{\min} - \mu({\bf x})}{\sigma({\bf x})}.
\end{split}
\end{equation}
Here, $\Phi$ and $\phi$ are the cumulative distribution function and probability density function of the standard normal distribution, respectively.
\subsection{Related Work}
\subsubsection{Bayesian optimization}
In Bayesian optimization, the design of surrogate models and acquisition functions are actively studied.
The tree-structured Parzen Estimator (TPE) algorithm \cite{NIPS2011_4443,pmlr-v28-bergstra13}, Sequential Model-based Algorithm Configuration (SMAC) \cite{hutter2011sequential} and Spearmint \cite{NIPS2012_4522} are known as powerful Bayesian optimization methods.
The TPE algorithm, SMAC, and Spearmint use a tree-structured parzen estimator, a random forest, and a Gaussian process as the surrogate model, respectively.
The popular acquisition functions in Bayesian optimization include the EI \cite{jones1998efficient}, probability of improvement \cite{article_pi}, upper confidence bound (UCB) \cite{Srinivas:2010:GPO:3104322.3104451}, mutual information (MI) \cite{pmlr-v32-contal14}, and knowledge gradient (KG) \cite{Frazier2009TheKP}.
However, there are few studies focusing on search spaces in Bayesian optimization.
A prominent problem in Bayesian optimization is the boundary problem~\cite{SwerskyDThesis} that points sampled concentrate near the boundary of the search space.
Oh et al. addressed this boundary problem by transforming the ball geometry of the search space using cylindrical transformation~\cite{pmlr-v80-oh18a}.
Wistuba et al. proposed using the previous experimental results to prune the search space of hyperparameters where there seems to be no good point \cite{10.1007/978-3-319-23525-7_7}.
In contrast to Wistuba's study, we propose a method to refine the search space without prior knowledge.
Nguyen et al. dynamically expanded the search space to cope with cases where the search space specified in advance does not contain a good point~\cite{inproceedings}.
In contrast to Nguyen's study, we focus on refining the search space rather than expanding.
\subsubsection{Search-Space Division Algorithm}
The proposed method is similar to methods such as Simultaneous Optimistic Optimization (SOO)~\cite{NIPS2011_4304} and BaMSOO~\cite{pmlr-v33-wang14d} in that it focuses on the division of the search space.
SOO is an algorithm that generalizes the DIRECT algorithm~\cite{Jones1993}, which is a Lipschitz optimization method, and the search space is expressed as a tree structure and the search is performed using hierarchical division.
BaMSOO is a method that makes auxiliary optimization of acquisition functions unnecessary by combining SOO with Gaussian process.
Wang et al. reported that BaMSOO shows better performance than SOO in experiments on some benchmark functions~\cite{pmlr-v33-wang14d}.
In the proposed method and search-division algorithms, SOO and BaMSOO, the motivation for optimization is different;
the proposed method divides the search space to identify a promising initial region for Bayesian optimization, while the search-division algorithms divide the search space to identify a good solution.
\section{Proposed Method}
In Bayesian optimization, there are many tasks with a low available evaluation budget.
For example, in hyperparameter optimization of machine learning algorithms, budget would be limited in terms of computing resources and time.
In this study, we focus on Bayesian optimization when there is not enough evaluation budget available.
Nguyen et al. state that Bayesian opitmization using a Gaussian process as the surrogate model and UCB as the acquisition function has the following relationships between the volume of a search space and the cumulative regret (the sum of differences between the optimum value and the evaluation value at each time)~\cite{inproceedings}.
(i) A larger space will have larger (worse) regret bound.
(ii) A low evaluation budget will make the difference in the regrets more significant.
Nguyen et al. give above description for cumulative regret~\cite{inproceedings}, but converting it to simple regret is straightforward, such as ~\cite{NIPS2016_6118}.
We therefore believe that in the low budget setting, making the search space smaller is also important in terms of the regret for Bayesian optimization in general.
In this study, we try to improve the performance of Bayesian optimization with the low budget setting by introducing a heuristic method that refines a given search space.
We assume that we have an arbitrary hypercube $\mathcal{X} \subseteq \mathbb{R}^d$ ($d$: the number of dimensions) as a search space.
Our method refines the search space by division, and outputs a region $\mathcal{X}_S (\subseteq \mathcal{X})$ considered to be promising.
As a result, Bayesian optimization can be executed with the refined search space $\mathcal{X}_S$ as the initial search space instead of the original search space $\mathcal{X}$.
\subsection{Integrating with Bayesian Optimization}
\label{sec:integrating_bo}
Algorithm \ref{alg:refine_frame} shows Bayesian optimization with the proposed method.
This method calculates the budget $B_{\rm ref}$ for refining the search space from the whole budget $B$ (line 1),
refines the promising search space (line 2),
performs optimization with the search space $\mathcal{X}_S$ refined in the line 2 as the initial search space (line 3).
We will describe ${\rm refine\_search\_space}(d, \mathcal{X}, B_{\rm ref})$ (line 2) in Section \ref{sec:proposal_algorithm}.
\begin{algorithm}
\caption{Bayesian optimization with the proposed method}
\label{alg:refine_frame}
\begin{algorithmic}[1]
\REQUIRE budget $B$, search space $\mathcal{X}$, the number of dimension $d$, ratio for refining $\gamma \ (0 \leq \gamma \leq 1)$
\STATE $B_{\rm ref} \gets \gamma \cdot B$
\STATE $\mathcal{X}_{S} \gets {\rm refine\_search\_space}(d, \mathcal{X}, B_{\rm ref})$
\STATE Bayesian optimization with the search space $\mathcal{X}_S$ until a budget reach $B$
\end{algorithmic}
\end{algorithm}
Figure \ref{fig:basic_idea} shows a conceptual design of the proposed method.
In the Figure \ref{fig:basic_idea} on the right, Bayesian optimization is executed on the search space which has been refined by the proposed method.
\begin{figure}[t]
\centering
\includegraphics[width=70mm]{figs/basic_idea.eps}
\caption{Conceptual design of Bayesian optimization with the proposed method. The figure on the left shows Bayesian optimization without the proposed method, and the figure on the right shows Bayesian optimization with the proposed method. The blue balls are the points sampled by Bayesian optimization and the red balls are the points sampled by the proposed method. The gray region on the right shows the discarded region by the proposed method.}
\label{fig:basic_idea}
\end{figure}
\subsection{Refining the Search Space}
\subsubsection{Calculation of the Budget}
\label{sec:budget_ratio}
Corresponding to the whole budget $B$, we set the budget $B_{\rm ref}$ used for the proposed method to $B_{\rm ref} = \gamma \cdot B$ (in Algorithm~\ref{alg:refine_frame}, line 1).
We calculate $\gamma$ by $\gamma = 0.59 \cdot \exp(- 0.033 B / d)$ with respect to the number of dimension $d \ (\in \mathbb{N})$.
If the evaluation budget increases to infinity (that is, $B \to \infty$), there is no need for refining the search space (that is, $\gamma \to 0$).
We note that $B_{\rm ref}$ is maximum budget for the proposed method, not used necessarily in fact; $B_{\rm ref}$ is used for determining the division number.
We show the details about how $B_{\rm ref}$ is used in Section \ref{sec:division_number}.
\subsubsection{Algorithm}
\label{sec:proposal_algorithm}
The proposed method refines the promising region by dividing the region at equal intervals for each dimension.
Figure \ref{fig:refine} shows that refining the search space by the proposed method when the number of dimensions is $d = 2$ and the division number is $K = 3$.
The proposed method randomly selects a dimension without replacement, divides the region corresponding to the dimension into $K$ pieces, leaves only the region where the evaluation value of the center point of the divided region is the best.
The proposed method repeats this operation until the division of the regions corresponding to all dimensions is completed.
\begin{figure}[t]
\centering
\includegraphics[width=90mm]{figs/refine.eps}
\caption{Refining the search space by the proposed method for the number of dimensions $d = 2$ and the division number $K = 3$. The proposed method divides the region corresponding to a certain dimension into $K$ pieces, and evaluates the center points. The proposed method leaves the region where the center point is the best, divides the region corresponding to another certain dimension again into $K$ pieces, and repeats this.}
\label{fig:refine}
\end{figure}
Algorithm \ref{alg:refine_search} shows the algorithm of refining the search space.
We denote the set of integers between $1$ and $n$ (including $1$ and $n$) by $[n]$ throughout the paper.
We describe how to set the division number $K$ (in Algorithm \ref{alg:refine_search}, line 1) in the next section.
\begin{algorithm}
\caption{${\rm refine\_search\_space}(d, \mathcal{X}, B_{\rm ref})$}
\label{alg:refine_search}
\begin{algorithmic}[1]
\REQUIRE the number of dimension $d$, search space $\mathcal{X}$, budget for refining $B_{\rm ref}$
\STATE $K = \mathop{\rm arg~max}\limits_{k \in \mathbb{N} \setminus 2 \mathbb{Z}} \{ B_k : B_k \leq B_{\rm ref} \}$
\IF{$K \leq 1$}
\STATE return $\mathcal{X}$ \COMMENT{need not divide in this case}
\ENDIF
\STATE initialization : $\mathcal{X}_S = \mathcal{X}$
\FOR {$i = 1$ to $d$}
\STATE randomly select an index $d_i$ without replacement from index set of dimensions
\STATE divide $\mathcal{X}_S$ into $\{ X_{k} \}_{k=1}^K$ with respect to dimension $d_i$
\STATE $k^{*} = \mathop{\rm arg~min}\limits_{k \in [K]} f(x_k)$, $x_k$ is a center point of $X_k$
\STATE update the search space : $\mathcal{X}_S \gets X_{k^{*}}$
\ENDFOR
\STATE return $\mathcal{X}_S$
\end{algorithmic}
\end{algorithm}
\subsubsection{Division Number}
\label{sec:division_number}
We need to set the division number $K$ to adjust how much the search space is refined.
If we set the division number $K$ to an even number, the evaluation budget for refining is calculated by $B_K^{\rm even} = d K$.
However, when $K$ is an odd number, the evaluation budget for refining is calculated by $B_K^{\rm odd} = K + \sum_{l=1}^{d-1} (K - 1)$ because the center point of the search space refined before can be reused for the next evaluation.
Therefore, we set the division number $K$ to an odd number $K \in \mathbb{N} \setminus 2 \mathbb{Z}$ so that the evaluation budget for refining $B_K$ approaches $B_{\rm ref}$ most closely according to Equation (\ref{eq:K}).
\begin{equation}
\label{eq:K}
K = \mathop{\rm arg~max}\limits_{k \in \mathbb{N} \setminus 2 \mathbb{Z}} \{ B_k : B_k \leq B_{\rm ref} \}.
\end{equation}
\section{Experiments}
In this section, we assess the performance of the proposed method through the benchmark functions and the hyperparameter optimization of machine learning algorithms to confirm the effectiveness of the proposed method in the low budget setting.
\subsection{Baseline Methods}
We use GP-EI (Bayesian optimization using Gaussian process as the surrogate model and the EI as the acquisition function), TPE \cite{NIPS2011_4443} and SMAC \cite{hutter2011sequential} as the baseline methods of Bayesian optimization.
In this experiment, we refer to GP-EI with the proposed method as {\it Ref+GP-EI}.
Likewise, we refer to TPE and SMAC with the proposed method as {\it Ref+TPE} and {\it Ref+SMAC}, respectively.
We use the GPyOpt\footnote{https://github.com/SheffieldML/GPyOpt}, Hyperopt\footnote{https://github.com/hyperopt/hyperopt} and SMAC3\footnote{https://github.com/automl/SMAC3} library to obtain the results for GP-EI, TPE and SMAC, respectively.
We set the parameters of GP-EI, TPE, and SMAC to the default values of each library and use the center point of the search space as the initial starting point for SMAC.
We also experiment with SOO~\cite{NIPS2011_4304} and BaMSOO~\cite{pmlr-v33-wang14d}, which are search-space division algorithms, in order to confirm the validity of the refinement of the search space by the proposed method.
\subsection{Benchmark Functions}
In the first experiment, we assess the performance of the proposed method on the benchmark functions that are often used in black-box optimization.
Table \ref{tab:bench} shows the six benchmark functions used in this experiment.
\begin{table*}[tb]
\caption{Name, definition formula, number of dimension, and search space of the benchmark functions. The coefficients appearing in the Branin, Shekel, and Harmann functions are shown in \cite{Benchmark2014}.}
\label{tab:bench}
\begin{center}
\begin{tabular}{l|c|c|c}
\hline
{\footnotesize Name} & {\footnotesize Definition} & {\footnotesize Dim $d$} & {\footnotesize Search Space $\mathcal{X}$} \\
\hline \hline
{\footnotesize Sphere} & {\footnotesize $f({\bf x}) = \sum_{i=1}^d x_i^2$} & {\footnotesize $5$} & {\footnotesize $[-5, 10]^{5}$} \\
\hline
{\footnotesize $k$-tablet $(k = \lfloor d/4 \rfloor)$} & {\footnotesize $f({\bf x}) = \sum_{i=1}^k x_i^2 + \sum_{i=k+1}^d (100 x_i)^2$} & {\footnotesize $5$} & {\footnotesize $[-5, 10]^{5}$} \\
\hline
{\footnotesize RosenbrockChain} & {\footnotesize $f({\bf x}) = \sum_{i=1}^{d-1}\left(100(x_{i+1} - x_i^2)^2 + (x_i - 1)^2\right)$} & {\footnotesize $5$} & {\footnotesize $[-5, 10]^{5}$} \\
\hline
{\footnotesize Branin} & {\footnotesize \begin{tabular}{c} $f({\bf x}) = a (x_2 - b x_1^2 + c x_1 - r)^2$\\ \hspace{10mm}$ + s (1 - t) {\rm cos}(x_1) + s$ \end{tabular} } & {\footnotesize $2$} & {\footnotesize $[-5, 10] \times [0, 15]$} \\
\hline
{\footnotesize Shekel $(m = 5)$} & {\footnotesize $f({\bf x}) = - \sum_{i=1}^5 \Bigl( \sum_{j=1}^4 (x_j - C_{ji})^2 + \beta_{i} \Bigr)^{-1}$} & {\footnotesize $4$} & {\footnotesize $[0, 10]^4$} \\
\hline
{\footnotesize Hartmann} & {\footnotesize $f({\bf x}) = - \sum_{i=1}^4 \alpha_i \exp \Bigl( - \sum_{j=1}^6 A_{ij} (x_j - P_{ij})^2 \Bigr)$} & {\footnotesize $6$} & {\footnotesize $[0, 1]^6$} \\
\hline
\end{tabular}
\end{center}
\end{table*}
\subsubsection{Experimental Setting}
We run 50 trials for each experiment, and we set the evaluation budget to $B = 10 \times d$ in each trial.
We assess the performance of each method using the mean and standard error of the best evaluation values in 50 trials.
In SOO and BaMSOO, we set the division numner $K = 3$, which is the same setting in ~\cite{pmlr-v28-valko13}.
For BaMSOO, we use the Mat$\acute{e}$rn $5/2$ kernel, which is one of the common kernel functions.
This equation is given by $k_{\rm M52} ({\bf x}, {\bf x}^{\prime}) = \sigma_f (1 + \sqrt{5 r^2({\bf x}, {\bf x}^{\prime})} + 5 r^2({\bf x}, {\bf x}^{\prime}) / 3) \exp (- \sqrt{5 r^2({\bf x}, {\bf x}^{\prime})}) ,$ where $r^2({\bf x}, {\bf x}^{\prime}) = \|{\bf x} - {\bf x}^{\prime}\|^2 / l$.
We set the initial hyperparameters to $\sigma_f = 1$ and $l = 0.25$ and update them by maximizing the data likelihood after each iteration.
\subsubsection{Results}
Figure \ref{fig:exp_bench} and Table \ref{tab:exp_bench} show the mean and standard error of the best evaluation values in 50 trials on the six benchmark functions.
Ref+GP-EI and BaMSOO show competitive performance in RosenbrockChain and Branin function, but Ref+GP-EI shows better performance than all the other methods in other benchmark functions.
Furthermore, Ref+GP-EI, Ref+TPE and Ref+SMAC outperform GP-EI, TPE and SMAC in all the benchmark functions, respectively.
Figure \ref{fig:discuss_typical_behavior} shows the typical behavior of each method on the Hartmann function.
Ref+GP-EI, Ref+TPE and Ref+SMAC sample many points with good evaluation values after refining the search space whereas other methods have not been able to sample points sufficiently with good evaluation values even at the end of the search.
\begin{figure*}[tb]
\centering
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/SphereFunction_result.eps}
\subcaption{Sphere}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/KTabletFunction_result.eps}
\subcaption{$k$-tablet $(k = d/4)$}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/RosenbrockChainFunction_result.eps}
\subcaption{RosenbrockChain}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/BraninFunction_result.eps}
\subcaption{Branin}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/ShekelFunction_result.eps}
\subcaption{Shekel}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/HartmannFunction_result.eps}
\subcaption{Hartmann}
\label{fig:mean1}
\end{minipage}
\caption{Sequences of the mean and standard error of the best evaluation values on the benchmark functions. The x-axis denotes the number of evaluations and the y-axis denotes the mean and standard error of the best evaluation values (averaged over 50 trials).}
\label{fig:exp_bench}
\end{figure*}
\begin{figure*}[tb]
\centering
\includegraphics[width=75mm]{figs/behavior/hartmann_behavior.eps}
\caption{Typical behavior of each method on the Hartmann function. The x-axis denotes the number of evaluations and the y-axis denotes the evaluation value. The black dotted line represents the moment that the proposed method refines the search space.}
\label{fig:discuss_typical_behavior}
\end{figure*}
\begin{table*}[tb]
\caption{Mean and standard error of the best evaluation values on the benchmark functions. The bold line shows the best mean in all the methods. The values in the $k$-tablet function and the RosenbrockChain function are multiplied by $10^3$ and $10^2$ with the original values, respectively. }
\label{tab:exp_bench}
\small
\begin{center}
{\tabcolsep = 1mm
\begin{tabular}{c||c|c|c|c|c|c|c|c}
\hline
Problem & GP-EI & Ref+GP-EI & TPE & Ref+TPE & SMAC & Ref+SMAC & SOO & BaMSOO \\
\hline \hline
Sphere & \begin{tabular}{c} $0.420$\\ $\pm 0.04$ \end{tabular} & \begin{tabular}{c} ${\bf 0.0145}$\\ $\pm 0.001$ \end{tabular} & \begin{tabular}{c} $12.5$\\ $\pm 0.8$ \end{tabular} & \begin{tabular}{c} $0.694$\\ $\pm 0.05$ \end{tabular} & \begin{tabular}{c} $17.4$\\ $\pm 1.0$ \end{tabular} & \begin{tabular}{c} $0.883$\\ $\pm 0.04$ \end{tabular} & \begin{tabular}{c} $21.4$\\ $\pm 0.3$ \end{tabular} & \begin{tabular}{c} $0.665$\\ $\pm 0.09$ \end{tabular} \\
\hline
$k$-tablet & \begin{tabular}{c} $1.24$\\ $\pm 1$ \end{tabular} & \begin{tabular}{c} ${\bf 0.0663}$\\ ${\pm 0.006}$ \end{tabular} & \begin{tabular}{c} $64$\\ $\pm 6$ \end{tabular} & \begin{tabular}{c} $3.95$\\ $\pm 0.4$ \end{tabular} & \begin{tabular}{c} $110$\\ $\pm 8$ \end{tabular} & \begin{tabular}{c} $5.77$\\ $\pm 0.5$ \end{tabular} & \begin{tabular}{c} $216$\\ $\pm 4$ \end{tabular} & \begin{tabular}{c} $1.72$\\ $\pm 0.4$ \end{tabular} \\
\hline
Rosen & \begin{tabular}{c} $7.47$\\ $\pm 1$ \end{tabular} & \begin{tabular}{c} ${\bf 1.53}$\\ ${\pm 0.01}$ \end{tabular} & \begin{tabular}{c} $32.6$\\ $\pm 4$ \end{tabular} & \begin{tabular}{c} $4.22$\\ $\pm 0.5$ \end{tabular} & \begin{tabular}{c} $38.7$\\ $\pm 3$ \end{tabular} & \begin{tabular}{c} $5.10$\\ $\pm 0.6$ \end{tabular} & \begin{tabular}{c} $46.4$\\ $\pm 1$ \end{tabular} & \begin{tabular}{c} $2.09$\\ $\pm 0.3$ \end{tabular} \\
\hline
Branin & \begin{tabular}{c} $1.08$\\ $\pm 0.09$ \end{tabular} & \begin{tabular}{c} ${\bf 0.42}$\\ ${\pm 0.003}$ \end{tabular} & \begin{tabular}{c} $3.08$\\ $\pm 0.4$ \end{tabular} & \begin{tabular}{c} $1.13$\\ $\pm 0.09$ \end{tabular} & \begin{tabular}{c} $2.81$\\ $\pm 0.4$ \end{tabular} & \begin{tabular}{c} $1.24$\\ $\pm 0.2$ \end{tabular} & \begin{tabular}{c} $2.31$\\ $\pm 0$ \end{tabular} & \begin{tabular}{c} $0.582$\\ $\pm 0.02$ \end{tabular} \\
\hline
Shekel & \begin{tabular}{c} $-5.28$\\ $\pm 0.4$ \end{tabular} & \begin{tabular}{c} ${\bf -6.79}$\\ ${\pm 0.5}$ \end{tabular} & \begin{tabular}{c} $-0.701$\\ $\pm 0.07$ \end{tabular} & \begin{tabular}{c} $-2.2$\\ $\pm 0.2$ \end{tabular} & \begin{tabular}{c} $-0.77$\\ $\pm 0.05$ \end{tabular} & \begin{tabular}{c} $-1.82$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} $-1.63$\\ $\pm 0$ \end{tabular} & \begin{tabular}{c} $-1.63$\\ $\pm 0$ \end{tabular} \\
\hline
Hartmann & \begin{tabular}{c} $-2.99$\\ $\pm 0.008$ \end{tabular} & \begin{tabular}{c} ${\bf -3.03}$\\ ${\pm 0.003}$ \end{tabular} & \begin{tabular}{c} $-2.45$\\ $\pm 0.03$ \end{tabular} & \begin{tabular}{c} $-2.97$\\ $\pm 0.006$ \end{tabular} & \begin{tabular}{c} $-2.54$\\ $\pm 0.04$ \end{tabular} & \begin{tabular}{c} $-2.97$\\ $\pm 0.005$ \end{tabular} & \begin{tabular}{c} $-2.41$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} $-2.42$\\ $\pm 0.02$ \end{tabular} \\
\hline
\end{tabular}
}
\end{center}
\end{table*}
\subsection{Hyperparameter Optimization}
In the second experiment, we assess the performance of the proposed method on the hyperparameter optimization of machine learning algorithms.
We experiment with the following three machine learning algorithms, that are with the low budget setting in many cases.
\begin{itemize}
\item MLP
\item CNN
\item LightGBM~\cite{ke2017lightgbm}
\end{itemize}
Table \ref{tab:mlp_cnn_hyperparameter} shows the four hyperparameters of MLP and their respective search spaces.
The MLP consists of two fully-connected layers and SoftMax at the end.
We set the maximum number of epochs during training to 20, and the mini-batch size to 128.
We use the MNIST dataset that has $28 \times 28$ pixel grey-scale images of digits, each belonging to one of ten classes.
The MNIST dataset consists of $60,000$ training images and $10,000$ testing images.
In this experiment, we split the training images into the training dataset of $50,000$ images and the validation dataset of $10,000$ images.
The CNN consists of two convolutional layers with batch normalization and SoftMax at the end.
Each convolutional layer is followed by a $3 \times 3$ max-pooling layer.
The two convolutional layers are followed by two fully-connected layers with ReLU activation.
We use the same hyperparameters and search spaces used by the MLP problem above (Table \ref{tab:mlp_cnn_hyperparameter}).
We set the maximum number of epochs during training to 10, and the mini-batch size to 128.
We use the MNIST dataset and split like the MLP problem.
\begin{table}[tb]
\begin{center}
\begin{tabular}{c}
\begin{minipage}{0.47\hsize}
\begin{center}
\caption{Details of four hyperparameters of MLP and CNN optimized on MNIST dataset.}
\label{tab:mlp_cnn_hyperparameter}
\begin{tabular}{|c|c|}
\hline
Hyperparameter & Search Space \\
\hline \hline
learning rate of SGD & $[0.001, 0.20]$ \\
\hline
momentum of SGD & $[0.80, 0.999]$ \\
\hline
num of hidden nodes & $[50, 500]$ \\
\hline
dropout rate & $[0.0, 0.8]$ \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}{0.47\hsize}
\begin{center}
\caption{Details of four hyperparameters of LightGBM optimized on Breast Cancer Wisconsin dataset.}
\label{tab:lightgbm_hyperparameter}
\begin{tabular}{|c|c|}
\hline
Hyperparameter & Search Space \\
\hline \hline
learning rate & $[0.001, 0.10]$ \\
\hline
colsample bytree & $[0.1, 1.0]$ \\
\hline
reg lambda & $[0.0, 100.0]$ \\
\hline
max depth & $[2, 7]$ \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{tabular}
\end{center}
\end{table}
Table \ref{tab:lightgbm_hyperparameter} shows the four hyperparameters of LightGBM and their respective search spaces.
We use the Breast Cancer Wisconsin dataset~\cite{Dua:2017} that consists of $569$ data instances.
In the experiment using this dataset, we use $80 \%$ of the data instances as the training dataset, and the evaluation value is calculated using $7$-fold cross validation.
\subsubsection{Experimental Setting}
We run 50 trials for each experiment, and we set the evaluation budget to $B = 20$ in each trial.
For all experiments, we use the misclassification rate on the validation dataset as the evaluation value.
For all the problems, we regard the integer-valued hyperparameters as continuous variables by using rounded integer values when evaluating.
\subsubsection{Results}
Figure \ref{fig:exp_hpo} and Table \ref{tab:hpo} show the mean and standard error of the best evaluation values for 50 trials on the hyperparameter optimization of the three machine learning algorithms.
Similar to the experiment of the benchmark functions, Ref+GP-EI, Ref+TPE and Ref+SMAC outperform GP-EI, TPE and SMAC in all the hyperparameter optimization of machine learning algorithms, respectively.
Likewise, Ref+GP-EI, Ref+TPE and Ref+SMAC show equal or better performance to SOO and BaMSOO in all problems.
\begin{figure*}[tb]
\centering
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/MnistMLP_result.eps}
\subcaption{MLP with MNIST dataset}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/MnistCNN_result.eps}
\subcaption{CNN with MNIST dataset}
\label{fig:mean1}
\end{minipage}
\begin{minipage}[t]{.47\textwidth}
\centering
\includegraphics[width=70mm]{figs/experiment_result/BCWLightGBM_result.eps}
\subcaption{LightGBM with Breast Cancer Wisconsin dataset}
\label{fig:mean1}
\end{minipage}
\caption{The sequences of the mean and standard error of the best evaluation values on the hyperparameter optimizations. The x-axis denotes the number of evaluations and the y-axis denotes the mean and standard error of the best evaluation values (averaged over 50 trials).}
\label{fig:exp_hpo}
\end{figure*}
\begin{table*}[tb]
\caption{Mean and standard error of the best evaluation values on the hyperparameter optimization of the machine learning algorithms. The bold line shows the best mean in all the methods. The values in each problem are multiplied by $10^2$ with the original values.}
\label{tab:hpo}
\small
\begin{center}
{\tabcolsep = 1mm
\begin{tabular}{c||c|c|c|c|c|c|c|c}
\hline
Problem & GP-EI & Ref+GP-EI & TPE & Ref+TPE & SMAC & Ref+SMAC & SOO & BaMSOO \\
\hline \hline
MLP & \begin{tabular}{c} $1.73$\\ $\pm 0.02$ \end{tabular} & \begin{tabular}{c} $1.71$\\ $\pm 0.008$ \end{tabular} & \begin{tabular}{c} $1.80$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} $1.71$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} $1.77$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} ${\bf 1.69}$\\ ${\pm 0.01}$ \end{tabular} & \begin{tabular}{c} $1.74$\\ $\pm 0.008$ \end{tabular} & \begin{tabular}{c} $1.75$\\ $\pm 0.01$ \end{tabular} \\
\hline
CNN & \begin{tabular}{c} $0.781$\\ $\pm 0.01$ \end{tabular} & \begin{tabular}{c} $0.722$\\ $\pm 0.006$ \end{tabular} & \begin{tabular}{c} $0.766$\\ $\pm 0.007$ \end{tabular} & \begin{tabular}{c} ${\bf 0.706}$\\ ${\pm 0.008}$ \end{tabular} & \begin{tabular}{c} $0.749$\\ $\pm 0.008$ \end{tabular} & \begin{tabular}{c} $0.723$\\ $\pm 0.006$ \end{tabular} & \begin{tabular}{c} $0.722$\\ $\pm 0.006$ \end{tabular} & \begin{tabular}{c} $0.741$\\ $\pm 0.007$ \end{tabular} \\
\hline
LightGBM & \begin{tabular}{c} $10.5$\\ $\pm 0.1$ \end{tabular} & \begin{tabular}{c} ${\bf 9.72}$\\ ${\pm 0.04}$ \end{tabular} & \begin{tabular}{c} $10.5$\\ $\pm 0.07$ \end{tabular} & \begin{tabular}{c} $9.82$\\ $\pm 0.05$ \end{tabular} & \begin{tabular}{c} $10.2$\\ $\pm 0.09$ \end{tabular} & \begin{tabular}{c} $9.89$\\ $\pm 0.04$ \end{tabular} & \begin{tabular}{c} $10.6$\\ $\pm 0.005$ \end{tabular} & \begin{tabular}{c} $10.6$\\ $\pm 0.01$ \end{tabular} \\
\hline
\end{tabular}
}
\end{center}
\end{table*}
\section{Conclusion}
In this study, we developed a simple heuristic method for Bayesian optimization with the low budget setting.
The proposed method refines the promising region by dividing the region at equal intervals for each dimension.
By refining the search space, Bayesian optimization can be executed with a promising region as the initial search space.
We experimented with the six benchmark functions and the hyperparameter optimization of the three machine learning algorithms (MLP, CNN, LightGBM).
We confirmed that Bayesian optimization with the proposed method outperforms Bayesian optimization alone in all the problems including the benchmark functions and the hyperparameter optimization.
Likewise, Bayesian optimization with the proposed method shows equal or better performance to two search-space division algorithms.
In future work, we plan to adapt the proposed method for noisy environments.
Real-world problems such as hyperparameter optimization are often noisy; thus, making the optimization method robust is important.
Furthermore, because we do not consider the variable dependency at present, we are planning to refine the search space taking the variable dependency into consideration.
{\small
\bibliographystyle{unsrt}
|
2,877,628,089,535 | arxiv | \section{Introduction} \label{sec:intro}
The knowledge of star's ages is important for many areas of modern astrophysics. For example, the studies of the structure and dynamic
of our Galaxy or formation and evolution of planetary systems require precise stellar ages.
There are several technics for age-dating of stars and ensembles of stars, but no direct method exists.
For stars other than the Sun, usually model-dependent methods are applied.
Till now, one of most accurate methods is based on the mass-radius relation applied to double-lined eclipsing binaries.
The most recent and extensive discussion on age determination and its importance was given by \cite{Soderblom2010} with a special emphasis on low-mass stars.
Mass is the most primary parameter of a star and all evolutionary computations start from fixing its value.
The direct determination of a mass is possible only for binary systems, so-called dynamical mass.
Spectroscopic data alone yield masses with an accuracy of a factor $\sin i$, where $i$ is the inclination angle of the orbital axis.
If the inclination of the orbit is unknown, only a mass ratio can be derived.
The absolute values of masses and radii can be obtained only for double-lined eclipsing binaries.
If eclipses do not occur then long-baseline interferometry may help but such observations are rather rarely available.
Then, for the determination of the absolute values of masses and radii we need the astrometric orbit, angular diameters and accurate parallaxes.
The most simple configurations of a two-star system are detached binaries where individual stellar component's evolution is not altered by the mass transfer.
Usually the coevality and similar chemical compositions for both components are assumed
because they should be formed almost simultaneously from the same cloud of interstellar gas.
The works by \cite{Torres2010} and \cite{Eker2014} constitute excellent compilations of detached double-lined eclipsing binaries (DDLEBs)
with the values of masses and radii determined with an accuracy of 3\% or better.
Because of precise estimates of stellar parameters, double-lined eclipsing binaries
are used as benchmarks for testing the theory of stellar evolution models at various ages and masses.
The method has been explored since many years. Most recently, \cite{Higl2017} selected 19 systems
with masses spanning from about 0.6 to 14 $M_{\sun}$ to constrain various
parameters of models and theory. They concluded that in the case of stars less massive than about 1.2 $M_{\sun}$,
there is a need for diffusion. For stars with convective core, they obtained some preference for
overshooting. An overview of the earlier studies on this subject can be found also in \cite{Higl2017}.
Independent constraints on stellar parameters can be obtained from asteroseismic modelling.
If pulsational modes can be well identified then, both, parameters of a model (e.g., mass, effective temperature, radius) and parameters of theory
(e.g., overshooting form convective regions, opacity data) can be constrained .
Thus, double-lined eclipsing binary with pulsating components suitable for seismic studies would be the best gauges,
in particular, for calibration of free parameters that cannot be derived from first principles, e.g.,
the amount of convective overshooting, mixing length parameter describing the efficiency of convective transport
or parameters of rotational-induced mixing.
Up to now there is no in-depth studies of such case for massive stars because there are not many double-lined eclipsing massive binaries
with pulsating components or there is a lack of reliable identification of pulsational modes. The promising case is a single-lined eclipsing
binary V381 Car with a pulsating primary \citep{Jerzykiewicz1992, Freyhammer2005}. Till now three frequencies were detected corresponding
to low order p modes \citep{Freyhammer2006}. However, the simultaneous binary and asteroseismic modelling of the V381 Car system awaits
more time series photometric and spectroscopic data in order to detect spectral lines of the secondary star and to get unambiguous identification of pulsational modes.
Recently, an attempt to such combined analysis was made by \cite{Tkachenko2016} for Spica which is a spectroscopic binary with pulsating component
but not the eclipsing one. Unfortunately, insufficient frequency resolution and detection of only three pulsational frequencies with no firm mode identification
prevented detailed seismic modelling.
Here, we present the determination of age for a homogenous sample of stars. We focus on all double-lined eclipsing binaries with two B-type components
on main sequence for which accurate values (i.e., usually below 3\%) of masses and radii were derived. Based on works by \cite{Torres2010} and \cite{Eker2014},
this sample consists of 38 binary systems. The more recent determination of masses and radii were also included in some cases.
We chose B-type stars because of their importance for evolution of the chemical composition and the structure of galaxies.
Moreover, stars with masses above 8$M_{\sun}$ produce a collapsing core and explode as a supernova.
Therefore, knowledge of properties of B-type main-sequence stars and their age is crucial for a better understanding of next evolutionary stages.
An important argument is also that in comparison to more or less massive stars, their structure and related phenomena are relatively simple. For example
we do not encounter such problems as strong mass loss or efficient convection in the outer layers. Thus, B-type eclipsing systems of the SB2 type
are also good indicators of age for open cluster and associations.
In this paper we apply the definition of the zero age as a time on the Zero Age Main Sequence (ZAMS).
In Sect.\,2, we introduce the selected sample of B-type binary stars. Sect.\,3 is devoted to the assessment of uncertainties in estimation of age
resulting from the adopted theoretical parameters describing various effects, i.e., metallicity, initial hydrogen abundance, rotation and overshooting from the convective core.
Results of the age determinations are given in Sect.\,4 and a few examples are described in details. Conclusions and future plans close the paper.
\section{The sample of binary stars}
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig1.eps}
\caption{The position of components of our sample of 38 binaries on the Hertzsprung-Russell diagram.
The evolutionary tracks were computed for the initial hydrogen abundance $X_0=0.70$, metallicity $Z=0.014$ and the AGSS09 solar mixture.
There is shown the effect of rotation, $V_{\rm rot}$, and overshooting from the convective core, $\alpha_{\rm ov}$.}
\label{fig1}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig2.eps}
\caption{The mass-radius diagram with the positions of the components of 38 selected binaries. There are shown also the lines of
ZAMS, TAMS} and isochrones from $\log (t/{\rm yr})=6$ to $\log (t/{\rm yr})=8$.
They were computed for the metallicity $Z=0.014$, initial hydrogen abundance $X_0=0.70$, without including rotation and convective core overshooting.
\label{fig2}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig3.eps}
\caption{The mass-luminosity diagram with the positions of components of 38 selected binaries. The lines of ZAMS and TAMS are for the same parameters as in Fig.\,2.}
\label{fig3}
\end{figure}
For age determination we selected uniform sample of binary stars.
All of them are double-lined eclipsing binaries with two components of B spectral type evolving on main sequence.
All of them are considered to be detached binary systems thus, in the first approximation, one can assume that there is no interaction between both components and their evolution
proceeds independently.
We use the data on dynamical masses and radii from \cite{Torres2010} supplemented with the more recent catalogue
by \cite{Eker2014} as well as other recent determinations. These parameters are known in most cases with an accuracy of 3\% or better
In total, there are 38 DDLEBs that meet the above criteria. In Table\,1, we list the basic parameters of these stars, i.e.,
the name, HD number, spectral types, the orbital period, $P$, the maximum brightness in the Johnson $V$ filter, $V_{\rm max}$,
mass, $M$, radius, $R$, effective temperature, $T_{\rm{eff}}$, luminosity, $\log{L/L_{\odot}}$, metallicity [m/H],
the projected value of the rotational velocity, $V_{\rm rot}\sin i$, the approximate value of the synchronous rotation.
In case of the systems AH Cep and V380 Cyg, two sets of stellar parameters were listed, from two different sources.
In Fig.\,1 we depicted their positions on the Hertzsprung-Russell diagram. The values of the effective temperature, $T_{\rm eff}$,
were taken from \cite{Torres2010} and \cite{Eker2014}. The values of luminosities, $L/L_{\sun}$, were computed from
the effective temperatures, $T_{\rm eff}$, and radii, $R$.
In the case of V380 Cyg, the parameters from two determinations were included, i.e., from Pavlovski \& Southworth (2009) and Tkachenko et al. (2014).
The evolutionary tracks for four values of masses (3, 5, 9, 15 $M_{\sun}$) were depicted.
They were computed with the Warsaw$-$New Jersey code \citep[e.g.,][]{Pamyatnykh1999} assuming the initial hydrogen abundance by mass $X_0=0.70$,
metallicity Z=0.014, OPAL opacities \citep{Iglesias1996} and the solar heavy element mixture by \cite{Asplund2009}, hereafter AGSS09.
The effects of rotation and overshooting from the convective core on the evolutionary tracks are shown as well.
In this paper, we adopted the abundance of metallicity by mass $Z=0.014$ as determined by \cite{Nieva2012} for galactic B-type stars.
From Figs.\,2 and 3 one can see that these stars obey quite well the mass-radius and mass-luminosity relations, respectively.
One star clearly stands out from these dependencies. This is the case of the more massive component of the binary V380 Cyg, whose
evolutionary stage is still uncertain. We show the two sets of parameters of V380 Cyg,
as determined by \cite{Pavlovski2009} and \cite{Tkachenko2014}, because they are quite different.
In the mass-radius diagram shown in Fig.\,2, we plotted the lines of ZAMS (Zero Age Main Sequence) and TAMS (Terminal Age Main Sequence) as well as
isochrones from the age $\log (t/{\rm yr})=6.0$ to $\log (t/{\rm yr})=8.0$, assuming the initial hydrogen abundance $X_0=0.70$, metallicity $Z=0.014$, no-rotation and no-overshooting.
Such diagrams are often used for the age determination from accurate masses and radii.
The values of the basic orbital parameters of the selected binary systems are depicted as histograms in Fig.\,4.
As one can see, most of the systems have the mass ratio, $M_B/M_A$, higher than 0.7 and only three systems have $M_B/M_A<0.6$; these are V380 Cyg, V379 Cep and V1331 Aql.
The majority of the systems (27) have short orbital periods with $P_{\rm orb}< 5$\,d and their orbits are not very elongated, i.e.,
$e<0.3$ for 34 systems. The system with the longest orbital period, $P\approx 100$ \,d, is V379 Cep which is a quadruple system with two binaries \citep{Harmanec2007}.
One of them is the eclipsing binary consisting of two B-type stars. The inclination angle of the orbital axis is larger than $i=75^\circ$ for 31 binaries.
Two systems have the value of $i$ below $65^\circ$, these are IM Mon \citep{Bakis2011} and V497 Cep \citep{Yakut2007}.
\startlongtable
\begin{deluxetable*}{lclcccccclll}
\tablecaption{Parameters of double-lined eclipsing binaries with two components of B spectral type.
Detailed description is given in the text. The last column contains the references.\label{chartable}}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{System} & \colhead{Star} &
\colhead{SpT} & \colhead{$P$} & \colhead{$M$} &
\colhead{$R$} & \colhead{$T_{\rm eff}$} &
\colhead{$\log L/L_{\odot}$ } & \colhead{[m/H]} &
\colhead{$V_{\rm rot} \sin i$ } & \colhead{$V_{\rm synch}$} & \colhead{Ref.} \\
\colhead{} & \colhead{} & \colhead{} & \colhead{$V_{\rm max}$} & \colhead{[$M_{\odot}$]} &
\colhead{[$R_{\odot}$]} & \colhead{[K]} & \colhead{} &
\colhead{} & \colhead{[km/s]} & \colhead{[km/s]} & \colhead{}
}
\startdata
AH Cep & A & B0.5Vn & 1.77 & 15.26$\pm$0.35 & 6.346$\pm$0.071 & 29900$\pm$1000 & 4.461$\pm$0.059 & -- & 185$\pm$30 & 181 & 1 \\
HD 216014 & B & B0.5Vn & 6.81 & 13.44$\pm$0.25 & 5.836$\pm$0.085 & 28600$\pm$1000 & 4.311$\pm$0.062 & & 185$\pm$30 & 167 & \vspace{5pt}\\
& A & B0.2V & 1.77 & 14.30$\pm$1.00 & 5.60$\pm$0.10 & 31000$\pm$3000 & 4.415$\pm$0.169 & -- & 200 & 160 & 13 \\
& B & B2V & 6.88 & 12.60$\pm$0.90 & 4.70$\pm$0.10 & 29000$\pm$4000 & 4.147$\pm$0.240 & & 170 & 134 & \vspace{5pt}\\
V578 Mon & A & B1V & 2.41 & 14.54$\pm$0.08 & 5.41$\pm$0.04 & 30000$\pm$500 & 4.330$\pm$0.030 & -0.30$\pm$0.13 & 117$\pm$5 & 109 & 1,14,15 \\
HD 259135 & B & B2V & 8.55 & 10.29$\pm$0.06 & 4.29$\pm$0.05 & 25750$\pm$435 & 3.860$\pm$0.031 & & 94$\pm$4 & 90 & \vspace{5pt}\\
HI Mon & A & B0V & 1.57 & 14.20$\pm$0.30 & 5.130$\pm$0.110 & 30000$\pm$500 & 4.282$\pm$0.034 & -- & 150$\pm$25 & 165 & 2 \\
HD 51076 & B & B0.5V & 9.45 & 12.20$\pm$0.20 & 4.990$\pm$0.070 & 29000$\pm$500 & 4.199$\pm$0.032 & & 150$\pm$25 & 160 & \vspace{5pt}\\
V453 Cyg & A & B0.4IV & 3.89 & 13.82$\pm$0.35 & 8.445$\pm$0.068 & 27800$\pm$400 & 4.583$\pm$0.026 & -0.25$\pm$0.05 & 109$\pm$3 & 111 & 1,16,17 \\
HD 227696 & B & B0.7IV & 8.28 & 10.64$\pm$0.22 & 5.420$\pm$0.068 & 26200$\pm$500 & 4.094$\pm$0.035 & & 98$\pm$5 & 71 & \vspace{5pt}\\
V380 Cyg & A & B1.5II-III & 12.43 & 13.13$\pm$0.24 & 16.22$\pm$0.26 & 21750$\pm$280 & 4.723$\pm$0.026 & 0.05$\pm$0.12 & 98$\pm$4 & 66 & 3, 7 \\
HD 187879 & B & B2V & 5.68 & 7.78$\pm$0.10 & 4.060$\pm$0.084 & 21600$\pm$550 & 3.508$\pm$0.048 & & 32$\pm$6 & 17 & \vspace{5pt}\\
& A & B1.5II-III & 12.43 & 11.43$\pm$0.19 & 15.71$\pm$0.13 & 21700(fixed) & 4.691$\pm$0.007 & & 98$\pm$4 & 64 & 4, 7 \\
& B & B2V & 5.68 & 7.00$\pm$0.14 & 3.819$\pm$0.048 & 23840$\pm$500 & 3.626$\pm$0.038 & & 32$\pm$6 & 16 & \vspace{5pt}\\
CW Cep & A & B0.5V & 2.73 & 13.05$\pm$0.20 & 5.640$\pm$0.120 & 28300$\pm$1000 & 4.263$\pm$0.064 & -- & 132 & 102 & 1, 6 \\
HD 218066 & B & B0.5V & 7.59 & 11.91$\pm$0.20 & 5.140$\pm$0.120 & 27700$\pm$1000 & 4.145$\pm$0.066 & & 138 & 93 & \vspace{5pt}\\
NY Cep & A & B0.5V & 15.28 & 13.0$\pm$1.0 & 6.800$\pm$0.700 & 28500$\pm$1000 & 4.438$\pm$0.108 & -- & 75$\pm$10 & 23 & 2 \\
HD 217312 & B & B2V & 7.43 & 9.0$\pm$1.0 & 5.400$\pm$0.500 & 23100$\pm$1000 & 3.872$\pm$0.110 & & 125$\pm$14 & 18 & \vspace{5pt}\\
V346 Cen & A & B1.5III & 6.32 & 11.8$\pm$1.4 & 8.2$\pm$0.3 & 26500$\pm$1000 & 4.474$\pm$0.073 & -- & 165$\pm$15 & 66 & 2, 9 \\
HD 101837 & B & B2V & 8.54 & 8.4$\pm$0.8 & 4.2$\pm$0.2 & 24000$\pm$1000 & 3.721$\pm$0.083 & & 140$\pm$15 & 34 & \vspace{5pt}\\
DW Car & A & B1V & 1.33 & 11.34$\pm$0.18 & 4.561$\pm$0.050 & 27900$\pm$1000 & 4.054$\pm$0.063 & -- & 182$\pm$3 & 174 & 1 \\
HD 305543 & B & B1V & 9.68 & 10.63$\pm$0.20 & 4.299$\pm$0.058 & 26500$\pm$1000 & 3.913$\pm$0.066 & & 177$\pm$3 & 164 & \vspace{5pt}\\
V379 Cep & A & B2IV & 99.76 & 10.56$\pm$0.23 & 7.909$\pm$0.120 & 22025$\pm$428 & 4.121$\pm$0.036 & -- & -- & 4 & 2 \\
HD 197770 & B & --- & 6.33 & 6.09$\pm$0.13 & 3.040$\pm$0.040 & 20206$\pm$374 & 3.141$\pm$0.034 & & -- & 2 & \vspace{5pt}\\
V1331 Aql & A & B1V & 1.36 & 10.10$\pm$0.11 & 4.240$\pm$0.030 & 25400$\pm$100 & 3.829$\pm$0.009 & -- & -- & 158 & 2 \\
HD 173198 & B & B1.5V & 7.80 & 5.29$\pm$0.10 & 4.040$\pm$0.030 & 20100$\pm$140 & 3.379$\pm$0.014 & & -- & 150 & \vspace{5pt}\\
QX Car & A & B2V & 4.48 & 9.25$\pm$0.12 & 4.291$\pm$0.091 & 23800$\pm$500 & 3.725$\pm$0.041 & -- & 120$\pm$10 & 48 & 1 \\
HD 86118 & B & B2V & 6.64 & 8.46$\pm$0.12 & 4.053$\pm$0.091 & 22600$\pm$500 & 3.585$\pm$0.043 & & 110$\pm$10 & 46 & \vspace{5pt}\\
V399 Vul & A & B3IV-V & 4.90 & 7.57$\pm$0.08 & 5.820$\pm$0.030 & 19000$\pm$320 & 3.598$\pm$0.030 & -- & 61$\pm$5 & 60 & 2 \\
HD 194495 & B & B4V & 7.06 & 5.46$\pm$0.03 & 3.140$\pm$0.080 & 18250$\pm$520 & 2.992$\pm$0.054 & & 39$\pm$7 & 32 & \vspace{5pt}\\
V1388 Ori & A & B2.5IV-V & 2.19 & 7.42$\pm$0.16 & 5.600$\pm$0.080 & 20500$\pm$500 & 3.697$\pm$0.044 & -- & 125$\pm$10 & 130 & 1 \\
HD 42401 & B & B3V & 7.40 & 5.16$\pm$0.06 & 3.760$\pm$0.060 & 18500$\pm$500 & 3.172$\pm$0.049 & & 75$\pm$15 & 87 & \vspace{5pt}\\
V497 Cep & A & B3V & 1.20 & 6.89$\pm$0.46 & 3.690$\pm$0.030 & 19500$\pm$950 & 3.245$\pm$0.087 & -- & -- & 155 & 5 \\
BD+61 2213& B & B4V & 8.95 & 5.39$\pm$0.40 & 2.920$\pm$0.030 & 17500$\pm$950 & 2.877$\pm$0.087 & & -- & 123 & \vspace{5pt}\\
V539 Ara & A & B3V & 3.17 & 6.240$\pm$0.066 & 4.516$\pm$0.084 & 18100$\pm$500 & 3.293$\pm$0.051 & -- & 75$\pm$8 & 71 & 1 \\
HD 161783 & B & B4V & 5.71 & 5.314$\pm$0.060 & 3.428$\pm$0.083 & 17100$\pm$500 & 2.955$\pm$0.055 & & 48$\pm$5 & 60 & \vspace{5pt}\\
CV Vel & A & B2.5V & 6.89 & 6.086$\pm$0.044 & 4.089$\pm$0.036 & 18100$\pm$500 & 3.207$\pm$0.049 & -- & 31$\pm$2 & 30 & 1 \\
HD 77464 & B & B2.5V & 6.69 & 5.982$\pm$0.035 & 3.950$\pm$0.036 & 17900$\pm$500 & 3.158$\pm$0.049 & & 19$\pm$1 & 29 & \vspace{5pt}\\
LT CMa & A & B4V & 1.76 & 5.59$\pm$0.20 & 3.590$\pm$0.070 & 17000$\pm$500 & 2.985$\pm$0.054 & -- & 109$\pm$10 & 103 & 2 \\
HD 53303 & B & B6.5V & 7.44 & 3.36$\pm$0.14 & 2.040$\pm$0.050 & 13140$\pm$800 & 2.047$\pm$0.108 & & 67$\pm$10 & 59 & \vspace{5pt}\\
IM Mon & A & B4V & 1.19 & 5.50$\pm$0.24 & 3.150$\pm$0.040 & 17500$\pm$350 & 2.922$\pm$0.036 & 0.2$\pm$0.15 & 147$\pm$15 & 134 & 2,12 \\
HD 44701 & B & B6.5V & 6.55 & 3.32$\pm$0.16 & 2.360$\pm$0.030 & 14500$\pm$550 & 2.344$\pm$0.067 & & 90$\pm$25 & 100 & \vspace{5pt}\\
AG Per & A & B3.4V & 2.03 & 5.359$\pm$0.160 & 2.995$\pm$0.071 & 18200$\pm$800 & 2.946$\pm$0.079 & -- & 94$\pm$23 & 75 & 1 \\
HD 25833 & B & B3.5V & 6.72 & 4.890$\pm$0.130 & 2.605$\pm$0.070 & 17400$\pm$800 & 2.747$\pm$0.083 & & 70$\pm$9 & 65 & \vspace{5pt}\\
U Oph & A & B5V & 1.68 & 5.273$\pm$0.091 & 3.484$\pm$0.021 & 16440$\pm$250 & 2.901$\pm$0.027 & -- & 125$\pm$5 & 100 & 1 \\
HD 156247 & B & B6V & 5.87 & 4.739$\pm$0.072 & 3.110$\pm$0.034 & 15590$\pm$250 & 2.710$\pm$0.029 & & 115$\pm$5 & 91 & \vspace{5pt}\\
DI Her & A & B5V & 10.55 & 5.170$\pm$0.110 & 2.681$\pm$0.046 & 17000$\pm$800 & 2.732$\pm$0.083 & -- & 108 & 13 & 1, 8 \\
HD 175227 & B & B5V & 8.42 & 4.524$\pm$0.066 & 2.478$\pm$0.046 & 15100$\pm$700 & 2.457$\pm$0.082 & & 116 & 12 & \vspace{5pt}\\
EP Cru & A & B5V & 11.08 & 5.020$\pm$0.130 & 3.590$\pm$0.035 & 15700$\pm$500 & 2.847$\pm$0.056 & -- & 141$\pm$5 & 16 & 2 \\
HD 109724 & B & B5V & 8.69 & 4.830$\pm$0.130 & 3.495$\pm$0.034 & 15400$\pm$500 & 2.790$\pm$0.057 & & 138$\pm$5 & 16 & \vspace{5pt}\\
V760 Sco & A & B4V & 1.73 & 4.969$\pm$0.090 & 3.015$\pm$0.066 & 16900$\pm$500 & 2.823$\pm$0.055 & -- & 95$\pm$10 & 88 & 1 \\
HD 147683 & B & B4V & 6.99 & 4.609$\pm$0.073 & 2.641$\pm$0.066 & 16300$\pm$500 & 2.645$\pm$0.058 & & 85$\pm$10 & 77 & \vspace{5pt}\\
MU Cas & A & B5V & 9.65 & 4.657$\pm$0.093 & 4.195$\pm$0.058 & 14750$\pm$800 & 2.874$\pm$0.095 & 0.22 & 21$\pm$2 & 22 & 1, 18 \\
BD+59 22 & B & B5V & 10.80 & 4.575$\pm$0.088 & 3.670$\pm$0.057 & 15100$\pm$800 & 2.798$\pm$0.093 & & 22$\pm$2 & 19 & \vspace{5pt}\\
GG Lup & A & B7V & 1.85 & 4.106$\pm$0.044 & 2.380$\pm$0.025 & 14750$\pm$450 & 2.382$\pm$0.054 & -- & 97$\pm$8 & 65 & 1 \\
HD 135876 & B & B9V & 5.59 & 2.504$\pm$0.023 & 1.726$\pm$0.019 & 11000$\pm$600 & 1.593$\pm$0.095 & & 61$\pm$5 & 47 & \vspace{5pt}\\
V615 Per & A & B7V & 13.72 & 4.075$\pm$0.055 & 2.291$\pm$0.141 & 15000$\pm$1100 & 2.370$\pm$0.128 & $-0.3$ & 28$\pm$5 & 8 & 5, 11 \\
& B & --- & 13.02 & 3.179$\pm$0.051 & 1.903$\pm$0.094 & 13000$\pm$1300 & 1.960$\pm$0.293 & & 8$\pm$5 & 7 & \vspace{5pt}\\
BD+03\_3821& A & B8V & 3.66 & 4.040$\pm$0.110 & 3.770$\pm$0.030 & 13140$\pm$1500 & 2.580$\pm$0.198 & -- & 109$\pm$2 & 52 & 2 \\
HD 174884 & B & --- & 7.98 & 2.720$\pm$0.110 & 2.040$\pm$0.020 & 12044$\pm$100 & 1.896$\pm$0.017 & & 60$\pm$3 & 28 & \vspace{5pt}\\
V1665 Aql & A & B9V & 3.88 & 3.970$\pm$0.400 & 4.130$\pm$0.100 & 12300$\pm$350 & 2.545$\pm$0.054 & -- & -- & 54 & 2 \\
HD 175677 & B & --- & 8.11 & 3.660$\pm$0.370 & 2.600$\pm$0.060 & 11650$\pm$310 & 2.048$\pm$0.050 & & -- & 34 & \vspace{5pt}\\
$\zeta$ Phe & A & B6V & 1.67 & 3.921$\pm$0.045 & 2.852$\pm$0.015 & 14400$\pm$800 & 2.497$\pm$0.097 & -- & 85$\pm$8 & 86 & 1 \\
HD 6882 & B & B8V & 3.95 & 2.545$\pm$0.026 & 1.854$\pm$0.011 & 12000$\pm$600 & 1.806$\pm$0.087 & & 75$\pm$8 & 56 & \vspace{5pt}\\
YY Sgr & A & B5V & 2.63 & 3.900$\pm$0.130 & 2.560$\pm$0.030 & 14800$\pm$700 & 2.451$\pm$0.083 & -- & 58$\pm$2 & 49 & 2, 10 \\
HD 173140 & B & B6V & 10.17 & 3.480$\pm$0.090 & 2.330$\pm$0.050 & 14125$\pm$670 & 2.288$\pm$0.084 & & 39$\pm$2 & 45 & \vspace{5pt}\\
V398 Lac & A & B9V & 5.41 & 3.830$\pm$0.350 & 4.890$\pm$0.180 & 11000$\pm$500 & 2.497$\pm$0.085 & -- & 79$\pm$2 & 46 & 2 \\
HD 210180 & B & --- & 8.75 & 3.290$\pm$0.320 & 2.450$\pm$0.110 & 10900$\pm$450 & 1.881$\pm$0.082 & & 19$\pm$2 & 23 & \vspace{5pt}\\
V413 Ser & A & B8V & 2.26 & 3.680$\pm$0.050 & 3.210$\pm$0.050 & 11100$\pm$300 & 2.147$\pm$0.049 & -- & 44$\pm$5 & 72 & 2,10 \\
HD 171491 & B & B9V & 7.99 & 3.360$\pm$0.040 & 2.930$\pm$0.050 & 10350$\pm$280 & 1.947$\pm$0.049 & & 38$\pm$7 & 66 & \vspace{5pt}\\
$\chi^2$ Hya& A & B8V & 2.27 & 3.605$\pm$0.078 & 4.390$\pm$0.039 & 11750$\pm$190 & 2.518$\pm$0.029 & -- & 112$\pm$10 & 98 & 1 \\
HD 96314 & B & B8V & 5.65 & 2.632$\pm$0.049 & 2.159$\pm$0.030 & 11100$\pm$230 & 1.803$\pm$0.038 & & 60$\pm$6 & 48 & \vspace{5pt}\\
V906 Sco & A & B9V & 2.79 & 3.378$\pm$0.071 & 4.521$\pm$0.035 & 10400$\pm$500 & 2.332$\pm$0.084 & 0.14$\pm$0.06 & 80$\pm$5 & 82 & 1,19,20 \\
HD 162724 & B & B9V & 5.96 & 3.253$\pm$0.069 & 3.515$\pm$0.039 & 10700$\pm$500 & 2.163$\pm$0.082 & 0.03$\pm$0.02 & 62$\pm$8 & 64 & \vspace{5pt}\\
$\eta$ Mus & A & B8V & 2.40 & 3.300$\pm$0.040 & 2.140$\pm$0.020 & 12700$\pm$100 & 2.029$\pm$0.016 & -- & 34$\pm$2 & 45 & 2 \\
HD 114911 & B & B8V & 4.78 & 3.290$\pm$0.040 & 2.130$\pm$0.040 & 12550$\pm$300 & 2.005$\pm$0.045 & & 44$\pm$2 & 45 & \vspace{5pt}\\
V799 Cas & A & B8V & 7.70 & 3.080$\pm$0.400 & 3.230$\pm$0.140 & 11550$\pm$14 & 2.222$\pm$0.038 & -- & -- & 21 & 2,10 \\
HD 18915 & B & B8.5V & 8.82 & 2.970$\pm$0.400 & 3.200$\pm$0.140 & 11210$\pm$14 & 2.162$\pm$0.038 & & -- & 21 & \vspace{5pt}\\
PV Cas & A & B9.5V & 1.75 & 2.816$\pm$0.050 & 2.301$\pm$0.020 & 10200$\pm$250 & 1.711$\pm$0.043 & -- & -- & 66 & 1 \\
HD 240208 & B & B9.5V & 9.72 & 2.757$\pm$0.054 & 2.257$\pm$0.019 & 10190$\pm$250 & 1.693$\pm$0.043 & & -- & 65 & \vspace{5pt}\\
\enddata
\tablecomments{1) \cite{Torres2010}, 2) \cite{Eker2014}, 3) \cite{Pavlovski2009},
4) \cite{Tkachenko2014}, 5) \cite{Yildiz2011}, 6) \cite{Nha1975}, 7) \cite{Prugniel2011}, 8) \cite{Philippov2013},
9) \cite{Mayer2016}, 10) \cite{Hog2000}, 11) \cite{Gonzales2000}, 12) \cite{Bakis2011}, 13) \cite{Martins2017}, 14) \cite{Pavlovski2005},
15) \cite{Garcia2014}, 16) \cite{Pavlovski2009A}, 17) \cite{Southworth2004}, 18) \cite{Lacy2004}, 19) \cite{Sestito2003}, 20) \cite{Villanova2009}}
\end{deluxetable*}
\begin{figure*}
\includegraphics[width=\textwidth,clip]{fig4.eps}
\caption{Histograms for main orbital parameters of 38 main sequence eclipsing binaries with B-type components.}
\label{hist}
\end{figure*}
\section{Age from accurate masses and radii: theoretical uncertainties}
As was demonstrated in the previous section the majority of binaries considered here have short orbital periods and low eccentricities.
This is very likely a manifestation of the tidal interaction between the components that leads to synchronisation and circularisation \citep[e.g.,][]{Siess2013}.
Thus in most cases the interaction can occur but without any signatures of mass transfer.
Therefore, in this paper we assume that, in the first approximation, each star evolves separately and we rely on single-star evolutionary code.
However, at some point the results should be confronted with advanced approach based on binary evolution models.
A grid of evolutionary models was computed using the Warsaw-New Jersey code \citep[e.g.,][]{Pamyatnykh1999}.
We adopted OPAL opacity tables \citep{Iglesias1996}, the initial hydrogen abundance by mass $X_0=0.70$
and the AGSS09 solar mixture of heavy elements.
The Warsaw-New Jersey code includes the effects of solid rotation, in particular the mean effects of centrifugal force,
assuming that global angular momentum is conserved during evolution.
Here, we neglected the effect of mass loss, because even for the hottest main sequence B stars, the mass-loss rate resulting
from line-driven winds is of the order of $10^{-9}~M_{\sun}/{\rm yr}$ \citep[e.g.,][]{Krticka2014}.
Thus, assuming the mean age of early B-type stars on the main sequence of the order of $10^7$\,yr one gets that a star loses
a mass of the order of hundredths of solar mass.
Overshooting from the convective core is included according to \cite{Dziembowski2008}.
This is a two-parameter prescription that allows for non-zero gradient of the hydrogen abundance inside
the partly mixed region above the convective core. The extent of overshooting is measured by $\alpha_{\rm ov}H_p$
where $H_p$ is the pressure height scale and $\alpha_{\rm ov}$ is a free, unknown parameter.
In the considered sample of stars the energy transport by convection in envelope can be neglected.
Also, as has been mentioned, in the considered range of masses ($\sim$3-15$M_\odot$), there is no need to include the effect of mass loss.
Thus, we get rid of the uncertainty associated with two additional free parameters.
The dependency between radius and mass at various ages is the classical diagram used for the age determination.
In Fig.\,2, we plotted the isochrones computed for the standard chemical composition from ZAMS to TAMS.
Only the mass range corresponding to the B-type stars was considered.
However, using the mass-radius (MR) diagram demands a number of interpolations to derive an accurate age.
The radius-age diagrams at the fixed values of masses are much simpler to use \citep[e.g.,][]{Higl2017}.
Any of these approaches contains uncertainties associated with the values of the parameters used for theoretical calculations.
In Fig.\,5, we show the value of radius, $R$, as a function of age, $\log (t/{\rm yr})$, for the four values
of masses, $M=3$, 5, 9 and 15\,$M_{\sun}$.
There are shown also the effects of various parameters in each panel, namely the effect of metallicity, $Z$, (the left$-$top panel),
the effect of the initial hydrogen abundance, $X_0$, (the right$-$top panel), the effect of core overshooting, $\alpha_{\rm ov}$, (the left$-$bottom panel)
and the effect of rotation, $V_{\rm rot}$, (the right$-$bottom panel).
The following values were adopted for these comparisons: $Z=0.014,~0.020$, $X_0=0.70,~0.75$, $V_{\rm rot}=0, ~0.4 V_{\rm rot}^{\rm crit}$,
where $V_{\rm rot}^{\rm crit}\approx\sqrt\frac{GM}{R}$ is the critical rotation, and $\alpha_{\rm ov}=0.0, ~0.2$.
The plots are from the age $\log (t/{\rm yr})=5.25$, which corresponds to $t=0.2$ Myr, because below this value the radius is approximately constant.
The upper age corresponds to an overall contraction phase.
Here, we present results obtained with the OPAL opacities, but as we have checked, the effect of the adopted opacity tables is negligible.
\begin{figure*}
\includegraphics[width=\textwidth]{fig5a.eps}
\includegraphics[width=\textwidth]{fig5b.eps}
\caption{The radius-age diagrams computed for the four values of masses. The top panels show the effect of metallicity
and initial hydrogen abundance, whereas the bottom panels - the effect of overshooting from convective core and rotation.
The tracks are depicted from $\log t =5.25$ ($t\approx 0.2$ Myr) up to the overall contraction phase.}
\label{fig5}
\end{figure*}
To give a quantitative estimate of theoretical uncertainties, we selected four models with masses 3, 5, 9 and 15 $M_\odot$ (see Table\,2)
and determined the range of age assuming the errors of 3\% both in mass and radius. Then, we investigated the effects of metallicity, $Z$, initial hydrogen abundance, $X_0$,
overshooting from the convective core, $\alpha_{\rm ov}$, and rotation, $V_{\rm rot}$.
We selected the evolutionary stages where differences in the age determination are significant, i.e., all models are roughly in the middle of their main sequence lifespan.
The reference models were computed with the following parameters: $Z=0.014$, $X_0=0.70$, $\alpha_{\rm ov}=0.0$ and $V_{\rm rot}=0$.
\begin{table*}
\centering
\caption{{\bf Age estimates with the errors} for models with masses and radii given in the first and second columns, respectively. In the third column, we give the reference age
determined at metallicity $Z=0.014$, initial hydrogen $X_0=0.70$, without overshooting from the convective core and zero-rotation rate.
The following columns contain the age estimated when one of these parameters has been changed.}
\begin{tabular}{c|c|cccccc}
\hline
$M$ [$M_{\odot}$] & $R$ [$R_{\odot}$] & & $\log t$ & $\log t_Z$ & $\log t_X$ & $\log t_{\rm ov}$ & $\log t_{\rm rot}$ \\
$\pm$ 3\%$M$ & $\pm$ 3\%$R$ & & std & $Z=0.020$ & $X_0=0.75$ & $\alpha_{\rm ov}=0.2$ & $V_{\rm rot}^0 = 0.4 V_{\rm crit}$\\
\hline
3 & 2.3 & & $8.01\pm0.15$ & $7.93\pm0.19$ & $8.13\pm0.14$ & $8.04\pm0.14$ & $7.37\pm0.46$ \\
5 & 3.1 & & $7.47\pm0.19$ & $7.38\pm0.24$ & $7.58\pm0.18$ & $7.50\pm0.19$ & $6.93\pm0.48$ \\
9 & 4.4 & & $6.97\pm0.12$ & $6.89\pm0.16$ & $7.07\pm0.12$ & $7.00\pm0.12$ & $6.65\pm0.24$ \\
15 & 6.7 & & $6.80\pm0.07$ & $6.76\pm0.07$ & $6.88\pm0.07$ & $6.82\pm0.07$ & $6.66\pm0.09$ \\
\hline
\end{tabular}
\end{table*}
In Table\,2, we give the range of age at various sets of parameters for these four selected models. As one can see, increasing metallicity ($Z=0.020$)
will result in a younger age. However, it should be noted that for models close to Terminal Age Main Sequence (TAMS) the effect is reverse, i.e.,
the higher metallicity gives an older age (see the top$-$left panel of Fig.\,5).
In turn, increasing the abundance of initial hydrogen ($X_0=0.75$) will make always the star older.
Including overshooting from the convective core ($\alpha_{\rm ov}=0.2$) works in the same direction as increasing hydrogen.
On the other hand, increasing the velocity of rotation causes that we will determine the younger age of the star.
The age estimates in Table\,2 can be treated as indicative errors expected from the adopted parameters in evolutionary computations.
At the values of parameters mentioned above and the 3\% errors in mass and radii,
the estimated uncertainties of the age are: $\Delta\log t_Z\approx 0.04-0.09$ for metallicity, $\Delta\log t_X\approx0.11$ for hydrogen abundance,
$\Delta\log t_{\rm ov}\approx 0.02-0.04$ for convective core overshooting and $\Delta\log t_{\rm rot}\approx 0.14-0.64$ for rotation.
However, one has to be aware that there are some correlations between these parameter.
These correlations can be easily discerned when fitting evolutionary tracks to the position of a star in the HR diagram.
Changing the bulk metallicity $Z$ mimics well the effect of changing a mass, in such a way that increasing the metallicity gives a larger mass
for a fixed position in the HR diagram.
In a similar direction works an increase of the hydrogen abundance $X$ but, in addition, the main sequence is expanded. These effect
will be illustrated in the next section where the results for individual stars will be presented.
The main sequence can be extended also by adding overshooting from the convective core or increasing rotation (cf. Fig\,1).
Higher overshooting shifts evolutionary tracks towards higher effective temperatures and luminosities, thus
a lower mass will be determined for a given position on the HR diagram.
On the contrary, the higher rotation shifts evolutionary tracks towards lower effective temperatures and luminosities.
In this respect, the increase in rotation works in a similar way as increasing the metallicity.
The change of $(X,~Z)$ is limited by the requirement of identical chemical composition for both components
as well as by observational determinations if they exist.
Moreover, the metallcity for most of the stars should not be much different than $Z=0.014$ as determined by \cite{Nieva2012} for galactic B-type stars.
The projected rotational velocities is usually derive
for each component of double-lined eclipsing binaries with a good accuracy (see Table\,1).
Overshooting is described by the free parameter which, of course, can be different for each component.
However, even with these constraints, it may happen that one effect can be compensated with combination of other effects.
\section{The results}
We determine the age assuming that both components are formed at the same time and have the same chemical
composition $(X, Z)$. We based our age determination on the radius-age diagrams plotted for the mass range derived for both components.
Such diagrams are very convenient for the age determination and often used, e.g., recently by \cite{Higl2017}.
The age was determined for each component separately and then the common age was adjusted.
In the next step, the positions of both stars on the HR diagram were confronted with the corresponding evolutionary tracks.
For the vast majority of selected binary stars, the range of the rotational velocity is well constrained by observations
and we use these values. The determination of metallicity [m/H] is available only for seven stars. Usually, this is the value of [Fe/H]
because iron is a good indicator of metallicity and iron lines are prominent and easy to measure.
These determinations are usually not very accurate; with large errors and no errors at all.
Therefore for each binary we searched the values of {\bf the metal abundance by mass} $Z$ in the range [0.005, 0.030],
which is enough wide for galactic B-type stars with no-peculiar chemical composition. The conversion from [m/H] to $Z$
is given by a simple formula, e.g., \cite{Salaris2005} (Section 8.1, page 239). See also \cite{Eker2018} where there are given
a table and figure allowing to convert easily between $Z$ and [m/H] or [Fe/H].
As overshooting from the convective core strongly affects the lifetime of a star, we also changed the value
of the overshooting parameter, $\alpha_{\rm ov}$. We considered the range $\alpha_{\rm ov}\in[0.0,~0.5]$
and $\alpha_{\rm ov}$ was adjusted for each value of the metallicity, separately for the component A and B.
We changed the parameters $(Z, \alpha_{\rm ov})$ until the values of mass, radius, effective temperature and luminosity were agreed
with the observational values. In a few cases, the solution has not been found.
The accuracy of the age determination depends also on the evolutionary stage of the stars. This is because the stellar radius increases slowly
at the beginning of main-sequence evolution, and increases much faster when a star is approaching the end of main sequence.
Therefore, despite of the same accuracy in mass and radius, various accuracy in the age estimation may be reached.
Below, we describe the results for some representative systems for which a consistent solution was obtained
as well as a few problematic cases. The results for all studied systems can be found in Appendix\,A, where
in Table\,3 we give the consistent values of the system age as well as the separate range for each component.
We provide the solution for $Z=0.014$ and $\alpha=0.0$ if it exists. If not, we give the solution for the values of $(Z,\alpha_{\rm ov})$
that are closest to $(Z,~\alpha_{\rm ov})=(0.014,~0.0)$.
Table \,4 contains the whole range of the parameters $Z$ and $\alpha_{\rm ov}$ for which it was possible to get a common age
from the radius-age dependence and the agreement on the HR diagram. The common age from the radius-age relation
and for a specific values of $Z$ and $\alpha_{\rm ov}$ for each component is {\bf listed} in Appendix\,B.
\subsection{Consistent solutions}
For the vast majority of 38 B-type main-sequence binaries it was possible to determine the common age of the two components
as well as to agree their location on the HR diagram. These are the following 33 systems:
AH Cep, V578 Mon, V453 Cyg, V380 Cyg, CW Cep, NY Cep, V346 Cen, DW Car, V379 Cep, QX Car, V399 Vul, V497 Cep,
V539 Ara, CV Vel, LT CMa, AG Per, U Oph, DI Her, EP Cru, V760 Sco, MU Cas, GG Lup, V615 Per, BD+03 3821, V1665 Aql,
$\zeta$ Phe, YY Sqr, V398 Lac, $\chi^2$ Hya, V906 Sco, $\eta$ Mus, V799 Cas, PV Cas.
In the case of V380 the consistent solution was possible only if the primary was in the post-main sequence phase after an overall contraction.
For some binaries we had to change the metallicity or add overshooting from the convective core
to agree the star's positions on the HR diagram. In some cases, both of these parameters had to be changed.
For example, for MU Cas the observed metallicity derived by \cite{Lacy2004} is $Z=0.025$.
With this value of $Z$ it was possible to determine the common age from the radius-age relation but it gave a disagreement in the position
of the stars on the HR diagram. It appeared that the solution with $Z=0.014$ is fully consistent.
The other cases which required changing metallicity were: V453 Cyg, DW Car, V379 Cep, V399 Vul, LT CMa, DI Her, V615 Per, BD+03\,3821,
V1665 Aql, $\zeta$ Phe, V398 Lac, $\chi^2$ Hya, V906 Sco, $\eta$ Mus and PV Cas.
Another example is the binary V615 Per. This is a member of the young open cluster h Persei (NGC 869) for which the estimated age from photometric observations
is $13 -14$ Myr \citep[e.g.,][]{Capilla2002, Currie2010}. However, this value was obtained assuming a solar metallicity.
For the metallicity higher than $Z=0.009$ and $\alpha_{\rm ov}=0.0$ it is possible to determine only the upper limit of the age
because the curves $R(\log t)$ do not intersect the line $R=R_{\rm min}$
On the other hand, the value of at least $Z=0.009$ was needed to agree the positions of both components on the HR diagram.
We got the age range $0.8 - 32.0$ My. In order to narrow down this range, it would be necessary to reduce the errors of radius determinations,
which amount to about 6\%.
The metallicity of $Z\approx 0.01$ for V615 Per was suggested by \cite{Southworth2004A} from reproducing masses and radii with the evolutionary computations. But the authors reduced the abundance of hydrogen to $X=0.63$ to keep the age of 13 Myr.
The subsolar metallicity was derived also from fitting disentangled spectra of V615 Per by \cite{Tamajo2011}.
Below we give details of the subsequent steps of the age determination for the three systems. We selected binaries with masses around
10\,$M_{\sun}$, 6\,$M_{\sun}$ and 3\,$M_{\sun}$.
\vspace{0.25cm}
{\bf V578 Monocerotis}
\vspace{0.15cm}
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig6.eps}
\caption{The top and middle panels show evolutionary changes of the radius for the minimum and maximum mass of the primary and secondary component
of V578 Mon, respectively. There is shown the effect of metallicity and the effect of core overshooting for the primary.
The bottom panel shows the HR diagram with the location of the two components.
}
\label{fig6}
\end{figure}
V578 Mon is one of the few double-lined eclipsing binaries with masses of both components above 10\,$M_\odot$
and with masses and radii determined with an accuracy below 3\%. Therefore, the object has been already studied by several groups.
The recent determination of basic stellar parameters was done by \cite{Garcia2014} and we used their values of masses and radii in this work, i.e.,
$M_A=14.54(8)~M_{\sun}$, $R_A=5.41(4)~R_{\sun}$ for the primary and $M_B=10.29(6)~M_{\sun}$, $R_B=4.29(5)~R_{\sun}$ for the secondary.
Comparing to the values of \cite{Torres2010}, the errors given by \cite{Garcia2014} are about 30\% lower in mass and 50\% lower in radius.
The star is a member of the OB cluster NGC\,2244 which drives the star-forming region called the Rosette nebula \citep[e.g.,][]{Roman-Zuniga2008}.
The metallicity of V578 Mon from disentangled component spectra is [m/H]=-0.30(13) \citep{Pavlovski2005} and for the whole cluster NGC\,2244
\cite{Tadross2003} estimated [Fe/H]$\approx$-0.46. The age of the binary derived from disentangled component spectra by \cite{Hensberge2000}
is 2.3(2) Myr ($\log (t/{\rm yr})=6.36(4)$.
The estimated age of NGC 2244 from 2MASS photometry by \cite{Bonatto2009} is in the range $1 - 6$ Myr ($\log (t/{\rm yr})=6.00 - 6.78$).
On the other hand, the determination of age of V578 Mon involving more advanced evolutionary modelling did not give consistent result.
\cite{Garcia2014} did not find a common age of both components and invoked a large overshooting from the convective core
for the primary to lower the discrepancy. Recently, \cite{Higl2017} obtained similar results for this binary.
They also showed that the assumption of solar metallicity, rather than [Fe/H]=-0.3 as determined from observations, reduces the age gap between components.
This age discrepancy for the V578 Mon system was obtained also by \cite{Schneider2014} in the framework of the BONNSAI project.
Contrary to the previous works, our models allowed to find the consistent age the V578 Mon system,
i.e., the common age from the radius-age relation and the agreement in the star's positions on the HR diagrams.
At the metallicity $Z=0.011$, which corresponds approximately to [m/H]=-0.2, and the overshooting of at least $\alpha_{\rm ov}=0.3$
for the more massive component, we determined the age of about 3.89 Myr ($\log (t/{\rm yr})=6.59$). At the metallicity $Z=0.014$
there is no need to include overshooting and the age of the system is 3.02(7) Myr ($\log (t/{\rm yr})=6.47-6.49$).
The consistent age exists for $Z\in[0.011,0.016]$ and various combinations of the overshooting parameters for both components
which are in the ranges: $\alpha_{ov}(A)\in[0.0,~0.5]$ and $\alpha_{ov}(B)\in[0.0,~0.2]$.
The method of determining the age is illustrated in Fig.\,6, where we depicted the evolution of the radius for the minimum and maximum mass allowed
by observations. The top and middle panels correspond to the primary and secondary component, respectively.
The horizontal grey belts mark the observed ranges of the radii.
As one can see both solutions, with $Z=0.011$ and $Z=0.014$, are consistent with the positions of the components on the HR diagrams shown
in the bottom panel of Fig.\,6. With dotted line we depicted also the isochrone $\log (t/{\rm yr})=6.48$ for $Z=0.014$.
The case of V578 Mon demonstrates very well the correlation between
the overshooting parameter and metallicity. Therefore in Table\,3, we give as an example both solutions, for $Z=0.011$ and $Z=0.014$.
The first solution is more consistent with the metallicity determined from spectroscopy by \cite{Pavlovski2005}. The larger parameter of overshooting
for the more massive primary could be also explained by the need of some additional mixing which results not necessarily from overshooting itself
but, for example, from diffusion or/and rotation.
The second solution, without overshooting, requires higher metallicity than the observed value, i.e., about $Z=0.014$,
which is, in turn, a typical value for galactic B-type stars \citep{Nieva2012}.
Both our estimates of age are more or less consistent with the previous determinations for the cluster NGC 2244.
\vspace{0.25cm}
{\bf CV Velorum}
\vspace{0.15cm}
The system of CV Vel consists of two B-type stars with almost the same masses determined with a very high accuracy: $M_A=6.086(44)~M_{\sun}$ and $M_B=5.982(35)~M_{\sun}$,
what gives the errors below 0.8\% in mass. The values of radii are determined with a similar accuracy: $R_A=4.089(36)~R_{\sun}$ and $R_B=3.950(36)~R_{\sun}$.
Several estimates of age can be found in the literature for this binary. The oldest determination by \cite{Clausen1977} gives about 30 Myr ($\log (t/{\rm yr})=7.48$).
From the membership of CV Vel in IC2391, \cite{Gimenez1996} estimated the age between 33 and 82 Myr ($\log (t/{\rm yr})=7.72\pm 0.20$), whereas,
based on evolutionary models, \cite{Yakut2007} derived 40 Myr ($\log (t/{\rm yr})=7.60$).
The most recent determination by \cite{Schneider2014} is about 35 Myr.
In the top and middle panels of Fig,\,7, we show the evolution of radius for the two components considering two values of metallicity, $Z=0.014$ and $0.010$.
The lower value of $Z$ was suggested by \cite{Yakut2007}. For both metallicities it was possible to determine the common age of the system from such diagrams, but, as one can see in the HR diagram (the bottom panel of Fig.\,7), only the higher metallicity solution is consistent with the observational error boxes.
We derived $t=31.6 - 33.9$ Myr ($\log (t/{\rm yr})=7.50 -7.53$) for $Z=0.014$. The isochrone $\log (t/{\rm yr})=7.515$ for $Z=0.014$ is marked in Fig.\,7.
The consistent solution was found for the metallicity $Z\in[0.011,~ 0.018]$ and the overshooting parameters $\alpha_{ov}\in[0.0,~0.2]$ for both components.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig7.eps}
\caption{The similar plots as in Fig.\,6 but for the CV Vel system.}
\label{fig7}
\end{figure}
CV Vel was also a subject of the BANANA project by \cite{Albrecht2014}. They found the misalignment of the rotation axis of the primary star
and the changes of the projected rotation velocity of both stars. The latter finding can be interpreted as precession of the rotation axes \citep{Albrecht2014}.
It is worth to mention also that \cite{Yakut2007} reported line profile variability which could be related with pulsations.
This is very plausible because both components are located in the instability strip of Slowly Pulsating B-type stars.
Detection of pulsational frequencies would allow for more in-depth studies of the system by mean of asteroseismic modelling.
\vspace{0.25cm}
{\bf V906 Scorpii}
\vspace{0.15cm}
V906 Sco is a triple system in the open cluster NGC 6475 (M7) \citep{Alencar1997}. The eclipsing binary is composed of late B-type stars with masses:
$M_A=3.378(71)~M_{\sun}$ and $M_B=3.253(69)~M_{\sun}$, and radii: $R_A=4.521(35)~R_{\sun}$ and $R_B=3.515(39)~R_{\sun}$. The third component has a similar spectral type (B9V)
and is on the wide orbit with the period of the order of hundred years \citep{Alencar1997}. Its contribution to the total light is about 5\%.
The metallicity of NGC\,6475 estimated by \cite{Sestito2003}, is [Fe/H] = $+0.14(6)$ whereas \cite{Villanova2009} derived
[Fe/H] = $+0.03(2)$ and suggested oversolar helium abundance $Y=0.33(2)$. The age of the cluster is about 220 Myr according to \cite{Meynet1993}
and 200$\pm$50 Myr according to \cite{Villanova2009}.
Fig.\,8 shows the run of $R(\log t)$ for the allowed mass range of the two components of the V906 Sco binary (the top and middle panels)
and their location on the HR diagram (the bottom panel). Following the estimates in the literature, we considered three compositions:
($X_0=0.70,~Z=0.014$), ($X_0=0.70,~Z=0.017$) and ($X_0=0.65,~Z=0.015$).
The primary component is quite evolved stars; it is already close to TAMS and overshooting of at least $\alpha_{\rm ov}=0.1$
is required for the star to be on main sequence for each values of ($X_0,~Z$). In Fig.\,8 we depicted the evolution of the radius for $\alpha_{\rm ov}=0.2$.
No overshooting is needed for the secondary star.
The common age from the radius-age diagrams
was determinable in each case and we obtained 208.5(1.5) Myr ($\log (t/{\rm yr})=8.316 -8.322$) for ($X_0=0.70,~Z=0.014$),
$\sim$219 Myr ($\log (t/{\rm yr})=8.340 - 8.341$) for ($X_0=0.70,~Z=0.017$)
and 164(1) Myr ($\log (t/{\rm yr})=8.213 - 8.217$) for ($X_0=0.65,~Z=0.015$).
Here, we got a younger age for lower metallicity because the primary star is close to TAMS.
The evolutionary tracks used to determine the age are depicted on the HR diagram in the bottom panel of Fig.\,8 together with error boxes for both components. The isochrone $\log (t/{\rm yr})=8.319$ for $X_0=0.70,~Z=0.014$ and $\alpha_{\rm ov}=0.2$ is depicted as the dotted line.
As one can see, only solutions with the initial hydrogen abundance $X_0=0.70$ are consistent and the corresponding evolutionary tracks
are within the error boxes of both stars.
Moreover, in these cases our estimates of the age for $\alpha_{\rm ov}(A)=0.2$ ($\sim209$ Myr for $Z=0.014$ and $\sim219$ for $Z=0.017$)
are in agreement with the age of NGC6475.
In turn, the solution with the hydrogen abundance of $X_0=0.65$, as suggested by \cite{Villanova2009}, gives inconsistent values
of luminosity and the system is too young.
The similar result was obtained by \cite{Higl2017} who derived the age of 236(13) Myr, which is slightly higher that our values.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig8.eps}
\caption{The similar plots as in Fig.\,6 but for the V906 Sco binary.}
\label{fig8}
\end{figure}
\subsection{Problematic cases}
There were a few systems for which it was impossible to get consistent solutions, i.e., to agree the age of both components from the radius-age relation
and their positions on the HR diagram. These are the binaries: HI Mon, V380 Cyg, V1331 Aql, V1388 Ori, IM Mon and V413 Ser.
We did not determine the age for HI Mon because of a disagreement on the HR diagram.
To determine the common age for both components from the radius-age diagram a large overshooting $\alpha_{\rm ov}>0.5$ for the primary was indispensable,
but then the primary was underluminous for its mass.
HI Mon is a massive binary ($M_A=14.2(3)~M_{\sun}$ and $M_B=12.2(2)~M_{\sun}$) with both components rotating
quite fast ($V_{\rm rot}\approx 150$ km/s). Thus, maybe other effects, like differential rotation or rotation-induced mixing
have to be taken into account.
For one system, V1331 Aql, we were unable to determine even a common age of both components from the radius-age relation.
There is also a strong disagreement in the position on the HR diagram for the secondary,
which is oversized and overluminous for its mass.
Changing any parameter in a reasonable range does not lead to a consistent solution.
As show by \cite{Lorenz2005}, the secondary of V1331 Aql reaches about 96\% of the Roche radius.
Thus, V1331 Aql is a detached configuration but the secondary is about to fill its Roche lobe.
It was not possible to get the consistent age for the binary V413 Ser because of a disagreement on the HR diagram
for all searched values of $Z$ and $\alpha_{\rm ov}$. V413 Ser is located in the Serpens star-forming region \citep{Chavarria1988}.
The Serpens dark cloud contains an extremely young star cluster and most probably both components of V413 Ser are pre-main sequence objects \citep{Cakirli2008}, and not the main-sequence ones as we assumed in our age determination.
Results for IM Mon, Ori V1388 and V380 Cyg will be presented below.
\vspace{0.25cm}
{\bf IM Monocerotis}
\vspace{0.15cm}
Till now, IM Mon was studied in details only in two papers, i.e., by \cite{Bakis2010} and \cite{Bakis2011}.
In \cite{Bakis2011}, the analysis of all available photometric and spectroscopic observations allowed to determine
the absolute values of masses: $M_A=5.35(24)~M_{\sun}$ and $M_B=3.32(16)~M_{\sun}$,
and radii: $R_A=3.15(4)~R_{\sun}$ and $R_B=2.36(3)~R_{\sun}$.
Moreover, it has been shown that IM Mon is a member of Ori OB1a association
and the age of the system is about 11.5 Myr. The age was estimated from a comparison of the location of stars
on the diagram $\log T_{\rm eff} - \log g$ with the isochrones of \cite{Girardi2000}.
A comparison of the observed high-resolution spectrum with Kurucz atmosphere models gave atmospheric parameters and
metallicity [Fe/H]$=+0.2$.
The age of IM Mon determined by \cite{Bakis2011} is in agreement with earlier determinations for Ori OB1a association
by \cite{Blaauw1991} and \cite{Brown1999}.
Using the masses and radii, we tried to re-determine the age of the system, as illustrated in Fig.\,9. We found that metallicity of at least $Z=0.025$
is needed to find a common age of the components which is in line with the values of [Fe/H] obtained by \cite{Bakis2011}.
Then, the age of the system is about 9.7(2.9) Myr, which is in agreement with the previous determinations.
However, when we put the stars on the HR diagram (the bottom panel of Fig.\,9),
it turned out that the secondary component is overluminous for its mass.
The evolutionary tracks for the metallicity typical for galactic B-type stars, $Z=0.014$, would be consistent, although marginal,
but in this case there is no common age for both components even if the large overshooting ($\alpha_{\rm ov}=0.5$) is assumed for the primary star.
One reason of that could be too hight effective temperature. The lower value of $T_{\rm eff}$ would move the star
to the right and down in the HR diagram.
The second explanation could be a more complicated configuration of this binary, i.e, maybe the system is not fully detached
or there is a third light.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig9.eps}
\caption{The similar plots as in Fig.\,6 but for components of IM Mon.}
\label{fig9}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig10.eps}
\caption{The similar plots as in Fig.\,6 but for V1388 Ori. }
\label{fig10}
\end{figure}
\vspace{0.25cm}
{\bf V1388 Orionis}
\vspace{0.15cm}
This binary is located close to Galactic open cluster NGC 2169 but because of significant age difference its cluster membership is unlikely.
The absolute values of masses and radii of the components of V1388 Ori were determined by \cite{Williams2009} and they are
$M_A=7.42(16)~M_{\sun}$, $M_B=5.16(6)~M_{\sun}$, and $R_A=5.60(8)~R_{\sun}$, $R_B=3.76(6)~R_{\sun}$.
From the position on the HR diagram, \cite{Williams2009} set
the age of the system at about 25 Myr, whereas the age of NGC 2169 is about $9\pm 2$ Myr as estimated by \cite{Jeffries2007}.
Fitting the spectral energy distribution to the photometric measurements in the passband $U BVJHK_s$ transformed to fluxes
allowed to estimate the distance $d=832\pm 89$ pc.
Up to now, no common age was determined for the components of the V1388 Ori system \citep[e.g.,][]{Schneider2014}.
We have attempted to agree the age of both components by changing the metallicity and the parameter of core overshooting.
Adopting the metallicity $Z=0.014$ did not lead to a consistent solution; always the primary was much younger than the secondary.
Even the high values of the overshooting parameter $\alpha_{\rm ov}>0.4$ for the primary did not shift the age to the enough higher values.
The corresponding curves $R(\log t)$ are plotted in the top and middle panels of Fig.\,10.
It was possible to make the primary older and the secondary younger by increasing the metallicity up to $Z=0.03$. This is because the primary
is close to TAMS and the effect of metallicity is reverse, i.e., the higher the metallicity the older the age of a star.
Comparing the positions of the V1388 Ori stars on the HR diagram with evolutionary tracks confirmed the previous results in the literature
that in the case of this binary both components are much overluminous for their masses \citep{Williams2009}. The discrepancy is stronger
for higher metallicity because for a given mass the evolutionary tracks are shifted to lower effective temperatures and lower luminosities.
\vspace{0.25cm}
{\bf V380 Cygni}
\vspace{0.15cm}
The binary V380\,Cyg consists of two early B-type stars with a significant mass difference ($M_B/M_A\approx0.6$).
There are two most recent determinations of masses and radii of these system components.
\cite{Pavlovski2009} derived the following values of masses: $M_A=13.13(24)~M_{\sun}$ , $M_B=7.78(10)~M_{\sun}$,
and radii: $R_A=16.22(26)~R_{\sun}$ and $R_B=4.060(84)~R_{\sun}$. Lower masses and radii were derived by \cite{Tkachenko2014}
who used high-precision photometry obtained by the Kepler space mission and high-resolution ground-based spectroscopy.
Below, we first provide details of the age determination adopting masses and radii of \cite{Pavlovski2009}.
Assuming the metallicity of $Z=0.014$ and no-overshooting from the convective core, the primary is beyond the main sequence,
as can be seen from the top and bottom panel of Fig.\,11.
Besides, the primary is much overluminous comparing with the corresponding evolutionary tracks.
To catch the massive component on the main sequence or just after it,
the overshooting from the convective core of at least $\alpha_{\rm ov}=0.5$ was indispensable. Such value was
suggested for the first time by \cite{Guinan2000}.
We tried to determined the age of V380 Cyg in the three phases of evolution: 1) main-sequence, 2) post-main sequence in an overall contraction,
and 3) post-main sequence after an overall contraction. In each case, it was possible to find the common age from the radius-age relation.
However, only in the third case (after the overall contraction) the agreement in the positions on the HR diagram has been reached.
In Table\,3, we provide this solution.
Adopting the masses and radii determined by \cite{Tkachenko2014}, it was not possible to derive the consistent age for any phase of evolution
because of a disagreement on the HR diagram for the primary.
Based on current observations, we cannot resolve what is the evolutionary stage of the V380 Cyg system and how our solution after the overall contraction is reliable.
Of course, observing a star in the main-sequence phase is much more likely due to time scales, but the probability of catching even a quite massive star in the post-main sequence
phase is not completely zero.
All scenarios demand a large overshooting parameter. As in the case of V578 Mon, the question is whether such effective overshooting
from the convective core has any physical origin. In particular: is the mixing at the core boundary really so large, or maybe a large overshooting parameter
compensates for the effects of other mixing processes?
More observations and analysis {\bf are} needed to answer the question about the evolutionary stage of the V380 Cyg primary.
Here, the studies of pulsations of the primary star would help but contrary to \cite{Tkachenko2014}, we did not find any oscillation frequencies in the Kepler light curve \citep{Miszuda2018}.
Similarly, computations with the more advanced evolutionary code MESA
did not resolve the evolutionary stage of the massive component of V380 Cyg \citep{Tkachenko2014, Miszuda2018}.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig11.eps}
\caption{The similar plots as in Fig.\,6 but for V380 Cyg.}
\label{fig11}
\end{figure}
\subsection{Discussion}
For most binaries, an equal age solution was found from the mass-radius-age diagram that is also compatible with the effective temperature
and luminosity of individual stellar components. In some case, the additional adjustment of metallicty or the overshooting parameter was necessary.
There are also the systems where an equal age solution is unsatisfactory because
it was not possible to agree positions of individual stellar components in the HR diagram.
For example, in the binary IM Mon, the secondary was overluminous and
in the binary V1388 Ori both components were overluminous. Another, completely different, problematic case was the V380 Cyg system
in which the evolutionary stage of the primary is uncertain. The primary star can be still on the main-sequence or has just entered a post-main sequence evolution
but the consistent age exists only if the primary is the post-main sequence star after the overall contraction.
Besides, as mentioned already, the accuracy of the derived age depends on the evolutionary stage and even if masses and radii are determined
with very small errors (below 3\%) the age cannot be determined accurately; an example is the binary DI Her.
The ages of the primary and secondary stars of binaries are plotted against each other in Fig.\,12.
As one can see, only V1331 Aql is an outlier taking into account the errors.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig12.eps}
\caption{The age of the primary and secondary component from the radius-age relation for all 38 binaries from our sample.}
\label{fig12}
\end{figure}
A comparison of the observed values of the effective temperatures, $T_{\rm eff}$ and luminosities, $\log L/L_\odot$, with those derived
from evolutionary computations ("evol") is presented in Fig.\,13. Only 33 binaries with the consistent age was included.
The left-hand panel correspond to the effective temperature, $T_{\rm eff}$, and the right-hand panel to the luminosities, $\log L/L_\odot$.
As one can see the agreement between the observed and evolutionary values is very good. This indicates that the predicted theoretical stellar tracks
and isochrones would also agree with the star's position on the HR diagram.
\begin{figure*}
\includegraphics[width=\columnwidth, height=7.5cm, clip]{fig13a.eps}
\includegraphics[width=\columnwidth, height=7.5cm, clip]{fig13b.eps}
\caption{A comparison of the values of the effective temperature (the left panel) and luminosity (the right panel)
determined from observations with the corresponding values derived from evolutionary models reproducing the consistent age
of the binary systems.}
\label{fig13}
\end{figure*}
Fig.\,14 shows a comparison of our age determination with the values gathered from the literature.
These values are also listed in the Appendix\,C in Table\,5 with the references to their sources.
There is one clear outlier, AG Per, which is generally accepted as a member of the Per OB2 association \citep{Blaauw1952}.
The age determined by \cite{Gimenez1994} is about 50(10) Myr which is much above our value, 15.6(2.5) Myr.
Our estimation is, however, compatible (within the error) with the age of Per OB2
which, according to \cite{Gimenez1994}, amounts to about 12.5(2.5) Myr.
Some disagreement is also for the four more stars: GG Lup, NY Cep, $\zeta$ Phe and PV Cas.
It can results from different approach to the age determination.
In these cases, the literature values are rather rough estimates, without errors given.
For other stars our values of the age are consistent with those found in the literature,
even if the latter are just rough estimates as in cases of V359 Ara, CV Vel, U Oph, V1665 Aql, YY Ser, $\eta$ Mus and V799 Cas.
\begin{figure}
\includegraphics[width=\columnwidth, clip]{fig14.eps}
\caption{Comparison of the age of B-type binaries determined in this paper with the literature values.}
\label{fig14}
\end{figure}
\section{Summary and future studies}
The goal of this paper was to determine the age for a homogeneous sample of stars. To this aim we chose main-sequence B-type stars in double-lined eclipsing binaries.
There are 38 suitable systems and for most of them the masses and radii are determined with the accuracy below 3\%.
These stars are relatively simple objects because of the lack of subsurface convection and no significant mass loss.
Thus, in evolutionary modelling we get rid of two additional free parameters describing these phenomena.
On the other hand, the B-type stars are very important for evolution of chemical elements and structure of galaxies.
Moreover, the properties of main-sequence objects are fundamental for further evolution and ultimate fate of stars.
In all cases but one it was possible to determine the common age for both components from the radius-age diagrams.
The exception is the system V1331 Aql in which the secondary is about to fill its Roche lobe.
There were also five systems, HI Mon, IM Mon, V1388 Ori, V380 Cyg and V413 Ser, for which the determination of the common age was possible but it was inconsistent with
the positions of the stars on the HR diagram.
In case of HI Mon a large overshooting $\alpha_{\rm ov}>0.5$ for the primary is necessary to get a common range of the age but then the primary is underluminous.
To agree the age of the components of the IM Mon system, it was necessary to adopt the higher metallicity, $Z=0.025$. However, with such high value of $Z$,
the secondary star of IM Mon is much overluminous.
In the case of V1388 Ori, establishing a common age required the large overshooting parameter $\alpha_{\rm ov}>0.5$ for the primary.
However, in the HR diagram both components are much overluminous.
More observations and analysis {\bf are} needed for these systems to reveal their actual state and formation history.
The binary V413 Ser is a completely different case, because most probably both components of this system are pre-main sequence stars
and here we did not considered this phase of evolution.
In the case of V380 Cyg, evolutionary computations with no-overshooting and metallicty $Z=0.014$, locate the primary beyond the main sequence
and no common age is determinable. If one wants to keep the primary star on MS or just after it, a large overshooting, $\alpha_{\rm ov}=0.5$,
has to be included.
We cannot prove if such large core overshooting has a physical meaning.
From asteroseismic studies usually lower values, $\alpha<0.4$, are derived \citep[e.g.,][]{Pamyatnykh2004, Aerts2003, JDD2017}.
The important results is that usually for more massive components which rotate fast, we had to increase overshooting
to adjust the common age, e.g., V578 Mon, HI Mon, V380 Cyg, V1388 Ori.
It is possible that this large value of the overshooting parameter compensates for other mixing effect \citep{Jermyn2018}.
In particular it can be a signature of the rotational-induced mixing which we did not take into account.
To distinguish between various mixing processes, we need more observables which can be directly compared with theoretical counterparts.
It seems that most suitable systems for such studies are binaries with pulsating components.
The combined binary and asteroseismic modelling has the greatest potential to provide the most stringent constraints
on various parameters of a model and theory. This type of studies is still waiting for its golden age.
\acknowledgments
The work was financially supported by the Polish NCN grants 2015/17/B/ST9/02082 and 2018/29/B/ST9/02803.
|
2,877,628,089,536 | arxiv | \section{Introduction}\setcounter{equation}{0}
The Skyrme model is a soliton model for nuclear physics \cite{Sk,book,BR}. The soliton solutions are called Skyrmions, and the conserved integer-valued topological charge of the Skyrmions is interpreted as the baryon number of nuclei. The Skyrme Lagrangian is
\begin{equation}
L= \int \left\{ -\frac{F^2_\pi}{16} \Tr(R_{\mu}R^{\mu}) + \frac{1}{32e^2} \Tr([R_{\mu},R_{\nu}][R^{\mu},R^{\nu}]) + \frac{m^2_\pi F^2_\pi}{8} \Tr(U-{\boldsymbol I}) \right\} \, d^3x \, ,
\label{eqn:SkyLag}
\end{equation}
\noindent where $U(x)$ is the $SU(2)$-valued scalar Skyrme field, $R_{\mu} = \partial _{\mu} U U^{\dagger}$ is the right current, and $m_\pi$ is the pion mass. The free parameters of the model ($F_\pi$, $e$, $m_\pi$) can be absorbed by using the Skyrme mass unit $\frac{F_\pi}{4e}$ and length unit $\frac{2}{eF_\pi}$. The scaled pion mass is $\frac{2m_\pi}{eF_\pi}$, and in this paper, a value of unity is used. The Skyrme field $U$ can be written in terms of the pion fields $\mbox{\boldmath $\pi$} = (\pi_1, \pi_2, \pi_3)$ and sigma field $\sigma$ as
\begin{equation}
U(x) = \sigma(x) \, {\boldsymbol I} + i \mbox{\boldmath $\pi$}(x) \cdot \mbox{\boldmath $\tau$} \,,
\end{equation}
where $\mbox{\boldmath $\tau$}$ are the Pauli matrices. The Skyrme field is subject to the constraint $UU^{\dagger} = (\sigma^2 + \mbox{\boldmath $\pi$} \cdot \mbox{\boldmath $\pi$}){\boldsymbol I} = {\boldsymbol I}$. Hence, $\sigma^2 + \mbox{\boldmath $\pi$} \cdot \mbox{\boldmath $\pi$} =1$ and $\sigma$ and $\mbox{\boldmath $\pi$}$ are not independent.
The baryon number, the topological charge of a Skyrmion, is the integral of the baryon density
\begin{equation}
{\cal B} = -\frac{1}{24\pi^2}\epsilon_{ijk}{\rm Tr}(R_iR_jR_k) \,.
\label{eq:bardens}
\end{equation}
The angular structure of Skyrmions is not known precisely except for the $B=1$ ``hedgehog''. For others with higher baryon number, solutions can be found numerically with the help of the rational map ansatz \cite{HMS}. An initial field configuration is constructed from a rational map, which encodes the angular structure, and a radial profile function, and it is then numerically relaxed to the stable Skyrmion. A geometrical construction of rational maps using a cubic grid was developed in \cite{FLM}, and a range of new Skyrmions was found. Most of the new solutions retain the symmetry of the rational maps, which are subgroups of the cubic symmetry group $O_h$.
The $T_d$- and $O_h$-symmetric Skyrmions are of particular interest to us, because of various theoretical studies of possibly tetrahedral and cubic nuclei \cite{DGSM,DCDDPOS}. The idea of tetrahedrally-symmetric nuclei is supported by some experimental results. The characteristic $3^-$ state, which is the first allowed rotational excitation of a tetrahedral rigid body with $0^+$ ground state, can be seen experimentally in the spectra of $^{16}$O and $^{40}$Ca \cite{TWC,CS}. The next excited state is a $4^+$ state.
The rotational spectra of Skyrmions can be calculated using semi-classical quantization \cite{ANW,BC}. The spectra of isospin excitations can also be calculated. A tetrahedrally-symmetric $B=16$ Skyrmion is known \cite{BMS} and several others have been found recently. The $3^-$ quantum state together with the second rotational state, $4^+$, which we will calculate using the Cartesian method in Section 4, can be used to identify potential $T_d$-symmetric nuclei from experiments. The ratio of the excitation energies between two rotational states with spins $l_2$ and $l_1$, respectively, is $\frac{l_2(l_2+1)}{l_1(l_1+1)}$. This equals $\frac{5}{3}\sim1.67$ for the $4^+$ and $3^-$ states. The existence of these two states as the lowest rotational excitations with a ratio of energies approximately $\frac{5}{3}$ suggests that a nucleus is tetrahedrally-symmetric. Such states can be seen clearly in $^{16}$O. The energy of the lowest $3^-$ and $4^+$ states of $^{16}$O are $6.130$ and $10.356$ MeV, respectively, with a ratio of $1.69$ \cite{TWC}. The procedure is reviewed and applied to the $B=20$ $T_d$-symmetric Skyrmion below. We show more generally that the quantum states of $T_d$-symmetric Skyrmions constructed from the cubic grid, which all have baryon number a multiple of 4, are classified into three classes.
In Section 2, we briefly review semi-classical Skyrmion quantization. In Section 3, we recall the cubic grid method, based on the Skyrme crystal structure, for
constructing rational maps. The $B=20$ $T_d$-symmetric Skyrmion and its rational map are described here. Section 4 discusses the semi-classical quantization of this Skyrmion. We introduce a method for writing down quantum states using Cartesian coordinates rather than Euler angles. This simplifies the analysis of states with $T_d$ and $O_h$ symmetry. Energy levels of the $B=20$ $T_d$-symmetric Skyrmions are calculated and presented. We also describe the use of the cubic grid to construct the parity operator pictorially. Sections 5 and 6 discuss the extension of the Cartesian method to a large class of Skyrmions with $T_d$ and $O_h$ symmetries, respectively. Concluding remarks are in Section 7, and our numerical methods are discussed in the Appendix.
\section{Semi-classical Skyrmion quantization}
In the semi-classical quantization method, a Skyrmion is treated as a rigid body free to rotate in ordinary space and isospace. The Skyrmion is restricted to have rotational and isorotational degrees of freedom only \cite{ANW,BC} and is parametrized as
\begin{equation}
U( \mathbf{x} , \mathbf{A}, \mathbf{B}) \equiv \mathbf{A} U_0(\mathbf{R}(\mathbf{B})\mathbf{x})\mathbf{A}^{\dagger} \, .
\label{eqn:U_rot}
\end{equation}
$U_0(\mathbf{x})$ is the prepared static Skyrmion solution in some convenient fixed orientation, $\mathbf{A}$ is the $SU(2)$ isospatial rotation, and $\mathbf{R(B)}$ is the spatial rotation represented by an $SU(2)$ matrix $\mathbf{B}$. Classically, $\mathbf{A}$ and $\mathbf{B}$ are time-dependent.
The semi-classical quantization of Skyrmions generalizes the quantization of a rigid body of normal matter. The classical rotational energy of an isotropic rigid body is
\begin{equation}
E = \frac{L^{2}}{2I} \, ,
\end{equation}
where $I$ is the moment of inertia of the body. To quantize the rigid body, we promote the squared angular momentum $L^2$ to a quantum operator $\hat{L}^2$. The eigenvalue of $\hat{L}^2$ is $L(L+1)$ in a state of angular momentum $L$. This formalism can be applied to Skyrmions, but the situation is more complicated. Skyrmions possess both spin and isospin, and the inertia tensors are not generally isotropic.
Substituting (\ref{eqn:U_rot}) into the Skyrme Lagrangian (\ref{eqn:SkyLag}), one can show after some rearrangements that the kinetic part of the Lagrangian is
\begin{equation}
H_{\textrm{kin}} = {\frac{1}{2}} a_{i} U_{ij} a_{j} - a_{i} W_{ij} b_{j} + {\frac{1}{2}} b_{i} V_{ij} b_{j} \, ,
\label{eqn:kin}
\end{equation}
\noindent where $a_j$ and $b_j$ are the angular velocities in isospace and ordinary space defined by
\begin{equation}
a_{j} = - i \Tr \left( \mbox{\boldmath $\tau$} _{j} A ^{\dagger} \dot{A}\right) \, , \quad b_{j} = i \Tr \left( \mbox{\boldmath $\tau$} _{j} \dot{B} B^{\dagger}\right) \, .
\end{equation}
\noindent $U_{ij}$, $W_{ij}$ and $V_{ij}$ are the moment of inertia tensors, which can be written in terms of the Skyrme field $U_0$, its right current $R_i$, and the further current $T_{i} = \frac{i}{2} [\mbox{\boldmath $\tau$} _i , U_0] U_0^{\dagger}$ as \cite{BC}
\begin{align}
U_{ij} &= - \int \, \Tr \left( T_i T_j + \frac{1}{4} [R_{k} , T_{i}][R_{k} , T_{j}] \right) \, d^{3}x \, , \label{eqn:U} \\
V_{ij} &= - \int \, \epsilon _{ilm} \epsilon _{jnp} x_{l} x_{n} \Tr \left( R_{m} R_{p} + \frac{1}{4} [R_{k} , R_{m}] [R_{k} , R_{p}] \right) \, d^{3}x \, , \label{eqn:V} \\
W_{ij} &= \int \, \epsilon _{jlm} x_{l} \Tr \left( T_{i} R_{m} + \frac{1}{4} [R_{k},T_{i}][R_{k} , R_{m}] \right) \, d^{3}x \, . \label{eqn:W}
\end{align}
\indent In order to calculate the nuclear spectra from the Skyrme model, one must first calculate numerically all the moments of inertia ($U_{ij}, W_{ij}, V_{ij}$) of a given Skyrmion. The details of the calculations can be found in the Appendix.
Skyrmions can be in any orientation with respect to the underlying coordinate system. One can introduce the body-fixed isospin ($\mathbf{K}$) and spin ($\mathbf{L}$), and the space-fixed isospin ($\mathbf{I}$) and spin ($\mathbf{J}$). The body-fixed isospin and spin are the conjugate momenta to the isospatial rotation $\mathbf{A}$ and the spatial rotation $\mathbf{B}$, derived from $H_{\textrm{kin}}$; the explicit relations in terms of the angular velocities $a_j$ and $b_j$ are
\begin{equation}
K_i = U_{ij} a_{j} - W_{ij} b_{j} \,, \quad L_{i} = - W^{T}_{ij} a_{j} + V_{ij} b_{j} \, .
\label{eqn:ang}
\end{equation}
The space-fixed isospin and spin can be obtained from the body-fixed isospin and spin by suitable rotations
\begin{equation}
I_{i} = - R_{ij} (\mathbf{A}) K_{j} \,, \quad J_{i} = -R_{ij} (\mathbf{B})^{T} L_{j} \, .
\end{equation}
\indent The four sets of quantized angular momentum operators mutually commute and satisfy the usual $SU(2)$ commutation relations
\begin{align}
[\hat{I}_{i},\hat{I}_{j}] &= i \epsilon _{ijk} \hat{I}_{k} \,, \quad \ \ [\hat{J}_{i},\hat{J}_{j}] = i \epsilon _{ijk} \hat{J}_{k} \, , \\
[\hat{K}_{i},\hat{K}_{j}] &= i \epsilon _{ijk} \hat{K}_{k} \,, \quad [\hat{L}_{i},\hat{L}_{j}] = i \epsilon _{ijk} \hat{L}_{k} \, .
\end{align}
The total angular momentum operators $\hat{I}^2$, $\hat{J}^2$, $\hat{K}^2$ and $\hat{L}^2$ are independent of orientations. As a result, the total isospin and total spin are equal in the body-fixed and space-fixed frames,
\begin{equation}
\hat{I}^2 = \hat{K}^2 \,, \quad \hat{J}^2 = \hat{L}^2 \, .
\end{equation}
\indent The quantized kinetic Hamiltonian of the Skyrmion can be written in the standard way in terms of the body-fixed angular momenta, $\hat{K}_i$ and $\hat{L}_i$. The Hamiltonian of a general Skyrmion with no symmetry is complicated, because of the cross term $W_{ij}$, which mixes isospin and spin. The expression is simple for symmetric Skyrmions. For example, the Hamiltonian of the $B=20$ $T_d$-symmetric Skyrmion is
\begin{equation}
\hat{H}=\frac{1}{2v}\hat{L}^2 + \frac{1}{2U_{11}}(\hat{K}^2-\hat{K}_3^2) +\frac{1}{2U_{33}}\hat{K}_3^2 \,.
\label{eqn:TdHam}
\end{equation}
\noindent Because of the $T_d$ symmetry, $V_{ij}=v \delta_{ij}$, $U_{11}=U_{22}$ and the isospin and spin contributions decouple because the cross term $W_{ij}$ vanishes. The symmetry also puts constraints on the spin and isospin quantum numbers. These are called Finkelstein-Rubinstein constraints \cite{FR}.
The wavefunction of the Skyrmion can be expressed as a tensor product of spin and isospin Wigner $D$-functions, $D^L_{J_3, L_3}(\phi, \theta, \psi) \otimes D^K_{I_3, K_3}(\alpha, \beta, \gamma)$ (or in quantum state notation $\left | L, L_3, J_3 \right \rangle \otimes \left | K, K_3, I_3 \right \rangle$), where $J_3$ and $I_3$ take all values in the standard range. The energy eigenvalues do not depend on the values of $J_3$ and $I_3$. For Skyrmions of even baryon number (which they all are in this paper), the spin and isospin are integer-valued. We can then simplify the quantum state by setting $J_3=I_3=0$. A $U(1)$ subgroup of the $SU(2)$ group is thereby quotiented out, and the manifold where the $D$-functions live is $S^3/S^1 \sim S^2$. The 2-sphere is parametrized by the remaining two angles, and the Wigner functions can be expressed in terms of the more familiar spherical harmonics. The product of Wigner functions $D^L_{0, L_3}(\phi,\theta,\psi) \otimes D^K_{0, K_3}(\alpha,\beta,\gamma)$ is proportional to $(-1)^{L_3}Y_{L, L_3}^*(\theta,\psi) \otimes (-1)^{K_3}Y_{K, K_3}^*(\beta,\gamma)$; the quantum state simplifies to $\left | L, L_3 \right \rangle \otimes \left | K, K_3 \right \rangle$.
The quantum state $\left | L, L_3 \right \rangle \otimes \left | K, K_3 \right \rangle$ is an eigenstate of the Hamiltonian operator (\ref{eqn:TdHam}). The energy of the state can be easily calculated, by replacing the operators $\hat{L}^2$, $\hat{K}^2$ and $\hat{K}_3$ with their respective eigenvalues $L(L+1)$, $K(K+1)$ and $K_3$. This energy is in Skyrme units; we can convert it back to physical units by a conversion factor. The rotational energy of the Skyrmion has the form $\frac{\hat{L}^2}{2v}$, where $\hat{L}^2$ is proportional to $\hbar^2$, but we have set $\hbar=1$. The moment of inertia $v$ has dimension $\text{[Mass][Length]}^2$. By inserting the Skyrme mass and length units, one Skyrme unit of moment of inertia is converted to $(\frac{F_\pi}{4e})(\frac{2}{eF_\pi})^2 = \frac{1}{e^3F_\pi}$ in physical units, which corresponds to an energy conversion factor $e^3F_\pi$ with a typical value of $O(10^3)$ MeV \cite{BMSW}. In this paper, a value $e^3F_\pi = 4 \times 10^3$ MeV is used.
\section{Rational maps from the cubic grid}\setcounter{equation}{0}
Finding rational maps with $T_d$ and $O_h$ symmetry becomes difficult as the degree of the maps becomes large. The cubic grid method for constructing them was first proposed in \cite{MSkySD} and developed in detail in \cite{FLM}. It relies on the observation that $\hat{\mbox{\boldmath $\pi$}}_3$ takes the values $\pm 1$ at the lattice points of the Skyrme crystal, corresponding to the zeros or poles of the rational map. A cubic chunk of the Skyrme crystal can be visually simplified to a cubic grid, and rational maps can be constructed by picking specific points from the grid. In general, large baryon number Skyrmions are found using a multi-layer rational map ansatz \cite{MP}. In the cubic grid method, the $n$-th layer rational map is constructed from a $2n \times 2n \times 2n$ grid. The first three layers of the grid allow a maximum number of 8, 56 and 152 points, respectively, and lead to rational maps of maximum degree 4, 28 and 76 if one restricts to zeros and poles of multiplicity one. The degree of the rational map is the baryon number of the layer. The example in Figure \ref{fig:cube_grid} shows the cubic grid method applied to the outer layer of the $B=32$ Skyrmion, which is the second layer of the cubic grid. The circles and squares indicate the 28 zeros and poles of the rational map, respectively. By labelling points with scaled Cartesian coordinates, ($x_1, \,x_2,\,x_3$), the rational map can be expressed in terms of the complex Riemann sphere coordinate, $\textrm{z}=\frac{x_1+ix_2}{r+x_3}$, with $r^2=x_1^2+x_2^2+x_3^2$. The locations of the zeros and poles can be found with the same formula.
\begin{figure}[ht]
\centering
\hspace{4cm}
\includegraphics[width=8.5cm]{Cube_4.pdf}
\caption{Second layer of the cubic grid}
\label{fig:cube_grid}
\end{figure}
With the aid of the cubic grid, the symmetry of the rational map can be easily visualized too. The $B=32$ Skyrmion possesses cubic symmetry $O_h$, which is the full symmetry group of the grid. The points on the grid can be separated into smaller subsets of points still preserving the $O_h$ symmetry (see Figure \ref{fig:grid}).
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{R_F.pdf}
\caption{Points on face interiors}
\label{fig:R_F}
\end{subfigure}
~ \,
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{R_E.pdf}
\caption{Points on edges}
\label{fig:R_E}
\end{subfigure}
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{R_v.pdf}
\caption{Points on vertices}
\label{fig:R_v}
\end{subfigure}
\caption{Subsets of the $4 \times 4 \times 4$ cubic grid}
\label{fig:grid}
\end{figure}
\noindent $O_h$-symmetric rational maps of various degrees can be constructed by taking combinations of the subsets. One can relax the symmetry requirement, and obtain many further rational maps.
The $B=20$ $T_d$-symmetric Skyrmion is found by using a double-layer rational map ansatz. The inner degree 4 rational map is constructed using all eight vertices of the first layer of the cubic grid. It is expressed in terms of the quartic Klein polynomials with zeros on alternating vertices, denoted by $p_+$ and $p_-$, respectively; these are
\begin{align}
p_+(\mathrm{z}) &= \left(\mathrm{z} + \frac{1-i}{\sqrt{3} + 1}\right) \left(\mathrm{z} - \frac{1-i}{\sqrt{3} + 1}\right) \left(\mathrm{z} + \frac{1+i}{\sqrt{3} - 1}\right) \left(\mathrm{z} - \frac{1+i}{\sqrt{3} - 1}\right) = \mathrm{z}^4 + 2 \sqrt{3}i \mathrm{z}^2 + 1 \,, \\
p_-(\mathrm{z}) &= \left(\mathrm{z} + \frac{1+i}{\sqrt{3} + 1}\right) \left(\mathrm{z} - \frac{1+i}{\sqrt{3} + 1}\right) \left(\mathrm{z} + \frac{1-i}{\sqrt{3} - 1}\right) \left(\mathrm{z} - \frac{1-i}{\sqrt{3} - 1}\right) = \mathrm{z}^4 - 2 \sqrt{3}i \mathrm{z}^2 + 1 \,.
\label{eqn:Klein}
\end{align}
The inner map is simply $\rfrac{p_+}{p_-}$. The outer degree 16 rational map is constructed by taking zeros on the faces, poles on the edges, and both zeros and poles on the vertices. The points for the outer rational map are shown in Figure \ref{fig:B_20}, and the rational map is given by \cite{FLM}
\begin{equation}
R_{20,T_d} = \left( \frac{1+c_2}{1+c_1} \right) \frac{p_+}{p_-}
\left(\frac{c_1 p_+^3 + p_-^3}{p_+^3 + c_2 p_-^3}\right) \,,
\label{eqn:B_20map}
\end{equation}
\noindent where $c_1=-2.873$ and $c_2=0.178$.
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{4x4x4_Grid_Td.pdf}
\caption{Outer rational map for the $B=20$ $T_d$-symmetric Skyrmion}
\label{fig:B_20}
\end{figure}
These rational maps give an initial ansatz. On relaxation we obtain the $B=20$ Skyrmion shown in Figure \ref{fig:B_20_Td} using the same colouring scheme as in \cite{FLM}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{B_20_T.jpg}
\caption{$B=20$ $T_d$-symmetric Skyrmion}
\label{fig:B_20_Td}
\end{figure}
\section{Quantization of $B=20$ $T_d$-symmetric Skyrmion}
In order to quantize the $B=20$ Skyrmion, we use the tensor product states $\left| L , L_3 \right \rangle \otimes \left| K, K_3 \right \rangle$ introduced in Section 2. The Finkelstein-Rubinstein (F-R) constraints are operator constraint equations which encode the symmetry and topology of the Skyrmion. For the $B=20$ $T_d$-symmetric Skyrmion, the F-R constraints are
\begin{equation}
e^{\frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, , \qquad \qquad e^{i \pi \hat{L}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, .
\label{eqn:FR}
\end{equation}
\noindent The operators here are two generators of the tetrahedral rotation group, $T$. The reflection elements in $T_d$ will play a role when we consider the parity of states. The second constraint is simple; the eigenvalue of $\hat{L}_3$ must be even. The first constraint is more complicated and harder to solve. Two different ways of solving these constraints will be discussed in the following subsections.
Besides the isospin and spin, quantum states are labelled by parity. To construct the parity operator of the Skyrmion, the explicit form of the rational map $R(z)$ can be used. The parity operation on any Skyrmion is defined as a combined inversion $I$ in the complex coordinate, $\mathrm{z}$, and the isospace complex coordinate, $R$, i.e. a combination of $\mathrm{z} \rightarrow - \, \rfrac{1}{\bar{\mathrm{z}}}$ and $R \rightarrow - \, \rfrac{1}{\overline{R}}$.
Applying the combined inversion \textit{I} to the $B=20$ $T_d$-symmetric outer rational map (\ref{eqn:B_20map}), one finds
\begin{equation}
R(\mathrm{z}) \xrightarrow{\quad \textit{I} \quad } -\frac{1}{\overline{R} (-\frac{1}{\bar{\mathrm{z}}})} = -\frac{1}{R(i\mathrm{z})} \, .
\end{equation}
\noindent This shows that the parity operation on this Skyrmion is equivalent to a $\frac{\pi}{2}$-rotation about the $x_3$-axis ($\mathrm{z} \rightarrow i\mathrm{z}$) together with a $\pi$-rotation about the isospace 2-axis ($R \rightarrow - \, \rfrac{1}{R}$). The effect of $I$ on the inner rational map is the same. The quantum parity operator is therefore
\begin{equation}
\hat{P} = e^{i \frac{\pi}{2} \hat{L}_3} e^{i \pi \hat{K}_2} \,,
\label{eqn:par_op}
\end{equation}
whose eigenvalue $\pm 1$ can be directly calculated for any state $\left| \Psi \right \rangle$. The parity operator can be understood more geometrically using the cubic grid, as will be shown in Subsection 4.4.
\subsection{Angular momentum basis method}
The allowed quantum states can be found by solving the F-R constraints, in an angular momentum basis. $L$ and $K$ are good quantum numbers, so we fix these. Dealing with the F-R constraints is a difficult task as it involves exponentiating matrices. With the help of \textit{Mathematica}, we found and solved the F-R constraints up to $L=4$, which requires exponentiating $9 \times 9$ matrices.
As an example, we apply this method to find the non-trivial quantum state with $L=3$ and $K=0$. For $K=0$, the F-R constraints simplify to
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, , \qquad \qquad e^{i \pi \hat{L}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, .
\label{eqn:B20_FRK0}
\end{equation}
\noindent We use the standard angular momentum basis, and write $\left | \Psi \right \rangle = \sum_{L_3=-L}^L C_{L_3} \left | L , L_3 \right \rangle$. The F-R constraints are
\begin{equation}
\begin{pmatrix}
-\frac{i}{8} & - \frac{i}{4} \sqrt{\frac{3}{2}} & -\frac{i}{8} \sqrt{15} & -\frac{i}{4}\sqrt{5} & -\frac{i}{8} \sqrt{15} & - \frac{i}{4}\sqrt{\frac{3}{2}} & -\frac{i}{8} \\
\frac{1}{4} \sqrt{\frac{3}{2}} & \frac{1}{2} & \frac{1}{4} \sqrt{\frac{5}{2}} & 0 & -\frac{1}{4} \sqrt{\frac{5}{2}} & -\frac{1}{2} & -\frac{1}{4} \sqrt{\frac{3}{2}} \\
\frac{i}{8}\sqrt{15} & \frac{i}{4} \sqrt{\frac{5}{2}} & - \frac{i}{8} & -\frac{i}{4}\sqrt{3} & - \frac{i}{8} & \frac{i}{4} \sqrt{\frac{5}{2}} & \frac{i}{8} \sqrt{15} \\
-\frac{1}{4} \sqrt{5} & 0 & \frac{1}{4}\sqrt{3} & 0 & -\frac{1}{4}\sqrt{3} & 0 & \frac{1}{4}\sqrt{5} \\
-\frac{i}{8}\sqrt{15} & \frac{i}{4} \sqrt{\frac{5}{2}} & \frac{i}{8} & -\frac{i}{4}\sqrt{3} & \frac{i}{8} & \frac{i}{4} \sqrt{\frac{5}{2}} & -\frac{i}{8}\sqrt{15} \\
\frac{1}{4} \sqrt{\frac{3}{2}} & -\frac{1}{2} & \frac{1}{4} \sqrt{\frac{5}{2}} & 0 & -\frac{1}{4} \sqrt{\frac{5}{2}} & \frac{1}{2} & -\frac{1}{4} \sqrt{\frac{3}{2}} \\
\frac{i}{8} & - \frac{i}{4}\sqrt{\frac{3}{2}} & \frac{i}{8} \sqrt{15} & -\frac{i}{4}\sqrt{5} & \frac{i}{8} \sqrt{15} & - \frac{i}{4}\sqrt{\frac{3}{2}} & \frac{i}{8}
\end{pmatrix}
\begin{pmatrix}[1.4]
C_{3} \\
C_{2} \\
C_{1} \\
C_{0} \\
C_{-1} \\
C_{-2} \\
C_{-3} \\
\end{pmatrix}
=
\begin{pmatrix}[1.4]
C_{3} \\
C_{2} \\
C_{1} \\
C_{0} \\
C_{-1} \\
C_{-2} \\
C_{-3} \\
\end{pmatrix} \, ,
\end{equation}
\noindent and
\begin{equation}
\begin{pmatrix}
-1 & 0 & 0 & 0 & 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 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 \\
\end{pmatrix}
\begin{pmatrix}
C_{3} \\
C_{2} \\
C_{1} \\
C_{0} \\
C_{-1} \\
C_{-2} \\
C_{-3}
\end{pmatrix}
=
\begin{pmatrix}
C_{3} \\
C_{2} \\
C_{1} \\
C_{0} \\
C_{-1} \\
C_{-2} \\
C_{-3}
\end{pmatrix} \, .
\end{equation}
\indent The second can be solved without explicit calculation; only the states with even $L_3$ are allowed. These are $\left | 3 , 2 \right \rangle$, $\left | 3 , 0 \right \rangle$ and $\left | 3 , {-2} \right \rangle$ only. The first equation implies that there is only one allowed combination, namely
\begin{equation}
\left| \Psi \right\rangle = \left( \left| 3 , 2 \right\rangle - \left| 3 , {-2} \right\rangle \right) \otimes \left| 0 , 0 \right\rangle \, ,
\end{equation}
\noindent where $\left | 0 , 0 \right \rangle$ is the isospin state. Applying the parity operator (\ref{eqn:par_op}), one finds that $\left| \Psi \right \rangle$ is a negative parity state, $3^-$.
\subsection{Cartesian method}
The angular momentum basis method works well up to $L=4$ but becomes computationally intractable for higher spin. To overcome this, an alternative method using the Cartesian representation of spherical harmonics is developed. This method uses Cartesian coordinates ($x$,$y$,$z$) in ordinary space and ($X,Y,Z$) in isospace to form $T$-invariant polynomials. The Cartesian coordinates are set up with respect to the body-frames and are related to the Euler angles ($\theta,\psi$) and ($\beta,\gamma$) by
\begin{align}
x&=\rho\sin\theta \cos \psi & &,& \quad y&=\rho\sin\theta \sin \psi & &,& \quad z&=\rho\cos \theta &\, &; \\
X&=\lambda\sin\beta \cos \gamma & &,& \quad Y&=\lambda\sin\beta \sin \gamma & &,& \quad Z&=\lambda\cos \beta &\, &,
\end{align}
where $\rho^2=x^2+y^2+z^2$ and $\lambda^2=X^2+Y^2+Z^2$.
The F-R constraints (\ref{eqn:FR}) can be understood geometrically using the Cartesian coordinates. The operator $e^{i\pi \hat{L}_3}$ of the second constraint rotates ordinary space by $\pi$ about the $z$-axis (the ($0,0,1$)-direction) and there is no rotation in isospace. The Cartesian coordinates transform as
\begin{align}
x &\rightarrow -x & &, & \ y &\rightarrow -y & &, & \ z &\rightarrow z &\, &; \\
X &\rightarrow X & &, & Y &\rightarrow Y & &, & \ Z &\rightarrow Z &\, &.
\end{align}
The first constraint mixes both ordinary space and isospace. The operator $e^{i\frac{2\pi}{3\sqrt{3}}(\hat{L}_1+\hat{L}_2+\hat{L}_3)}$ rotates ordinary space by $\frac{2 \pi}{3}$ about the $(1,1,1)$-direction, and the operator $e^{i\frac{2\pi}{3}\hat{K}_3}$ rotates isospace by $\frac{2 \pi}{3}$ about the $Z$-axis. The effect on isospace is more transparent when we change variables from $X$ and $Y$ to $X+iY$ and $X-iY$. The Cartesian coordinates transform as
\begin{align}
x &\rightarrow y & &, & \ y &\rightarrow z & &, & \ z &\rightarrow x &\, &; \label{eqn:spin_perm} \\
(X+iY) &\rightarrow \omega (X+iY) & &, & (X-iY) &\rightarrow \omega^2 (X-iY) & &, & \ Z &\rightarrow Z &\, &,
\end{align}
\noindent where $\omega = e^{i \frac{2 \pi}{3}}$.
The $3^-$ state with isospin zero has a simple Cartesian expression,
\begin{align}
\left| \Psi \right\rangle &= \left( \left| 3 , 2 \right\rangle - \left| 3 , {-2} \right\rangle \right) \otimes \left| 0 , 0 \right\rangle \, , \nonumber \\
&\propto (Y_{3,2}(\theta,\psi) - Y_{3,-2}(\theta,\psi))^* \otimes Y^*_{0,0}(\beta,\gamma) \, , \nonumber \\
&\propto \frac{1}{\rho^3} \left[ (x+iy)^2z - (x-iy)^2z \right] ^* \otimes 1 \, , \nonumber\\
&\propto \frac{1}{\rho^3} (x^2z+2ixyz- y^2z - x^2z+2ixyz + y^2z)^* \otimes 1 \, , \nonumber \\
&\propto \frac{1}{\rho^3} xyz \otimes 1 \, .
\end{align}
The state is proportional to a monomial of degree three, and is clearly invariant under (\ref{eqn:spin_perm}). $xyz$ is one of the $T$-invariant generating polynomials.
\subsubsection{Pure spin states}
We look here at the general F-R allowed states with isospin $K$ equal to zero. The $T$-invariant generating polynomials are listed in Table \ref{tab:spin}. These are further classified using $\hat{P}$, the parity operator (\ref{eqn:par_op}). The Cartesian coordinates transform under $\hat{P}$ as
\begin{align}
\hat{P}: \quad x &\rightarrow y & &,& \ y &\rightarrow {-x} & &,& \ z &\rightarrow z & \, &; \\
X &\rightarrow -X & &,& \ Y &\rightarrow Y & &,& \ Z &\rightarrow -Z & \, &.
\end{align}
The first polynomial $f_2$ is $\rho^2$ and has zero spin. Dividing by $\rho^2$ gives $1$. The polynomial $f_3$ is the $L=3$ spin state that we presented, and $f_4$ is cubically-symmetric. The last polynomial $f_6$ is different from the others because it is not invariant under any reflection of $T_d$; however, it satisfies the F-R constraints and must be included.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Degree & & $T$-invariant polynomials & Parity \\ \hline
$2$ & $f_2$ & $x^2 + y^2 + z^2$ & $+$ \\ \hline
$3$ & $f_3$ & $xyz$ & $-$ \\ \hline
$4$ & $f_4$ & $x^4 + y^4 + z^4$ & $+$ \\ \hline
$6$ & $f_6$ & $x^4(y^2-z^2) + y^4(z^2-x^2) + z^4(x^2-y^2)$ & $-$ \\
\hline
\end{tabular}
\end{center}
\caption{$T$-invariant polynomials of pure spin states}
\label{tab:spin}
\end{table}
To find a quantum state of spin $L$, a degree $L$ polynomial is constructed using the generating polynomials. As an example, an $L=4$ together with an $L=0$ state can be constructed using $f_2^2$ and $f_4$. The general degree 4 polynomial is
\begin{equation}
F = a f_2^2 + b f_4 \, .
\label{eqn:F4}
\end{equation}
With arbitrary constants $a$ and $b$, this is a state of mixed spin, but the ratio of the constants can be determined using
\begin{equation}
\hat{L}^2 \left | \Psi \right \rangle = L(L+1) \left | \Psi \right \rangle \, .
\label{eqn:L2}
\end{equation}
The operator $\hat{L}^2$ is
\begin{align}
\hat{L}^2=& \, \hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2 \nonumber \\
=& - [ (y^2+z^2) \partial^2_x + (x^2+z^2) \partial^2_y + (x^2 + y^2) \partial^2_z \nonumber \\
& \ \ - 2(xy \partial_x \partial_y + xz \partial_x \partial_z + yz \partial_y \partial_z) -2(x \partial_x + y \partial_y + z \partial_z)] \, . \label{eqn:L2op}
\end{align}
where $(\hat{L}_x,\hat{L}_y,\hat{L}_z)$ are expressed in the Cartesian coordinates, for example, $\hat{L}_z= -i( x\partial_y-y \partial_x)$. For the polynomial (\ref{eqn:F4}), one obtains
\begin{equation}
\hat{L}^2 F = 8 b x^4 - 24 b x^2 y^2 + 8 b y^4 - 24 b x^2 z^2 - 24 b y^2 z^2 + 8 b z^4 \,
\end{equation}
and
\begin{align}
L(L+1)F = L(L+1)[&( a + b )x^4 + (a + b) y^4 + (a + b) z^4 \nonumber \\
& + 2 a x^2 y^2 + 2 a x^2 z^2 + 2 a y^2 z^2] \, ,
\end{align}
and to satisfy (\ref{eqn:L2}), two independent linear equations need to be satisfied, namely
\begin{align}
8b &= L(L+1) (a+b) \, , \\
-24b &= L(L+1) (2a) \, \label{eqn:L4K0} .
\end{align}
The solutions are $\frac{b}{a}=-\frac{5}{3}$ for $L=4$ and $b = 0$ for $L=0$. The pure spin states are
\begin{equation}
F_{L=4,K=0}=3f_2^2 - 5f_4\, , \qquad F_{L=0,K=0}=f_2^2 \, .
\end{equation}
\indent An alternative but equivalent way of fixing the constants of our polynomial quantum states is to use the Laplace equation, $\nabla^2f(x,y,z)=0$. The condition (\ref{eqn:L2}) is satisfied automatically by a degree $L$ polynomial $f$, if it is a solution to the Laplace equation. The proof is as follows. Let us consider a degree $L$ polynomial $f(x,y,z)= \sum_{i,j,k}a_{ijk} x^iy^jz^k$, where $a_{ijk}$ are constants, $i+j+k=L$ and $i,j,k\geq0$. In terms of the spherical polar coordinates, $f(x,y,z)$ is $\rho^Lg(\theta,\psi)$, where $g(\theta,\psi)$ is some angular function. Substituting this expression into the Laplace equation, expressed in spherical polars, one obtains
\begin{equation}
\nabla^2f = \frac{1}{\rho^2}\left[\partial_\rho(\rho^2\partial_\rho f) - \hat{L}^2 f\right] = \ 0 \, .
\end{equation}
Because of the $\rho^L$ factor, $\partial_\rho(\rho^2\partial_\rho f) = L(L+1)f$, so $\nabla^2 f=0$ implies that
\begin{equation}
L(L+1)g - \hat{L}^2 g = 0 \, ,
\end{equation}
which means that $f$ is a state of spin $L$. The advantage of using the Laplace operator over (\ref{eqn:L2op}) is that it has a much simpler expression in terms of Cartesian coordinates,
\begin{equation}
\nabla^2f=(\partial_x^2 + \partial_y^2 + \partial_z^2) f(x,y,z) \,,
\end{equation}
and the calculation is easier.
\indent We can repeat the calculation for the $L=4$, $K=0$ state here using the Laplace operator. For $F=af_2^2+bf_4$,
\begin{align}
\nabla^2F&=\nabla^2\left[a(x^2+y^2+z^2)^2 + b(x^4+y^4+z^4) \right] \nonumber \\
&= (20a + 12b)(x^2 + y^2 + z^2) \nonumber \\
&= 0
\end{align}
if $\frac{b}{a}=-\frac{5}{3}$. The Laplace operator can only project out the quantum state with angular momentum equal to the degree of the polynomial; the other solution with $L=0$ cannot be obtained using this method. There is no $L=2$ state, because $f_2$, the only degree 2 polynomial, does not satisfy the Laplace equation.
States can be converted back to the angular momentum basis using the coefficients
\begin{equation}
C_{L_3}= \iint F_{L,K}(\theta , \psi) Y^*_{L, L_3}(\theta,\psi) \sin \theta \, d \theta d \psi \, .
\label{eqn:ang_conv}
\end{equation}
$F_{L,K}$ is expressed in spherical polar coordinates here and $Y_{L ,L_3}$ is the standard spherical harmonic. The spin 4 state $F=3f_2^2 - 5f_4$ in spherical polar coordinates is
\begin{equation}
F_{L=4,K=0}= 3 - 5 \Big( \sin^4 \theta \cos ^4 \psi + \sin^4 \theta \sin ^4 \psi + \cos ^4 \theta \Big) \, .
\end{equation}
Using equation (\ref{eqn:ang_conv}), the state can be expressed in the angular momentum basis as
\begin{align}
\left | \Psi \right \rangle &= \sum_{L_3=-L}^L C_{L_3} \left | L , L_3 \right \rangle \otimes \left| 0,0 \right \rangle \nonumber \\
&= \left(\left | 4 , -4 \right \rangle + \sqrt{\frac{14}{5}} \left | 4, 0 \right \rangle + \left | 4 , 4 \right \rangle \right) \otimes \left | 0, 0 \right \rangle \, ,
\end{align}
which involves an irrational coefficient. This is to be compared with the neat rational coefficients of the Cartesian form of the state.
\subsubsection{Pure isospin states}
We now move to the case of pure isospin with spin $L$ set to zero. The approach is very similar, and the invariant generating polynomials are presented in Table \ref{tab:isospin}.
\begin{table}[!h]
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Degree & & & Parity \\ \hline
$1$ & $g_1$ & $Z$ & $-$ \\ \hline
$2$ & $g_2$ & $X^2 + Y^2$ & $+$ \\ \hline
$3$ & $g_{3-}$ & $X(X^2-3Y^2)$ & $-$ \\ \hline
$3$ & $g_{3+}$ & $Y(3X^2-Y^2)$ & $+$ \\
\hline
\end{tabular}
\end{center}
\caption{Invariant generating polynomials of pure isospin states}
\label{tab:isospin}
\end{table}
As an example, we construct the $K=3$ states using degree 3 polynomials. For degree 3, four polynomials are allowed, ($g_1^3$, $g_2g_1$, $g_{3-}$ and $g_{3+}$), compared to only two, ($f_2^2,f_4$), for degree 4 in the case of pure spin. The calculation can be simplified with the help of parity. The only positive parity polynomial is $g_{3+}$, and it is the only possible positive parity $K=3$ state. The negative parity polynomials are $g_1^3$, $g_2g_1$ and $g_{3-}$. To find pure isospin states, the same procedure as in the case of pure spin is repeated: one formulates and solves simultaneous equations by comparing the polynomials term by term. The $K=3$ states are
\begin{align}
F^+_{0,3}&=g_{3+} \, , \\
F^-_{0,3}&=g_1^3-\frac{3}{2}g_2g_1 \, .
\end{align}
\subsubsection{Mixed spin and isospin}
The cases of pure spin and pure isospin have been covered. We wish to construct mixed quantum states too. Here we present all states up to spin 4 and isospin 2. $L_3$ must be even, as before. The F-R constraint
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \,
\label{eqn:FReqn_2}
\end{equation}
\noindent allows the spin and isospin operators each to have a complex eigenvalue of unit magnitude. The constraint equation can be satisfied as long as the phases cancel out. This allows two more generating polynomials for both spin and isospin. They are
\begin{equation}
\omega(x^2+\omega y^2 +\omega^2 z^2), \ \omega^2(x^2 + \omega^2 y^2 + \omega z^2), \ (X+iY) \text{ and } \ (X-iY) \, ,
\end{equation}
\noindent where $\omega = e^{i \frac{2\pi}{3}}$ as before. Notice that they are not eigenstates of the parity operator but transform into each other under parity (see Table \ref{tab:s_and_iso}).
\begin{table}[ht]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Degree & & & Under F-R (\ref{eqn:FReqn_2}) & Under parity \\ \hline
$2$ & $f_{\omega}$ & $\omega^2(x^2 + \omega^2 y^2 + \omega z^2)$ & $\omega \, f_{\omega}$ & $f_{\omega^2}$ \\ \hline
$2$ & $f_{\omega^2}$ & $\omega(x^2+\omega y^2 +\omega^2 z^2)$ & $\omega^2 f_{\omega^2}$ & $f_{\omega}$ \\ \hline
$1$ & $g_{\omega}$ & $(X+iY)$ & $\omega \, g_{\omega}$ & $-g_{\omega^2}$ \\ \hline
$1$ & $g_{\omega^2}$ & $(X-iY)$ & $\omega^2 g_{\omega^2}$ & $-g_{\omega}$ \\
\hline
\end{tabular}
\end{center}
\caption{Generating polynomials for mixed spin and isospin states}
\label{tab:s_and_iso}
\end{table}
We can now start constructing quantum states satisfying the F-R constraints with both non-zero spin and isospin; the allowed terms are tensor products of the spin and isospin generating polynomials where $\omega$ and $\omega^2$ factors cancel. $f_\omega$ and $f_{\omega^2}$ satisfy the Laplace equation, hence they are states of pure spin with $L=2$. As an example, the allowed terms for $L=2,K=1$ states are
\begin{equation}
f_{\omega} \otimes g_{\omega^2} \text{ and } f_{\omega^2} \otimes g_{\omega} \, .
\end{equation}
\noindent Note that the zero spin polynomial $f_2$ is not needed. We obtain the following two states with definite parities
\begin{align}
F^{+}_{L=2,K=1} &= f_{\omega} \otimes g_{\omega^2} - f_{\omega^2} \otimes g_{\omega} \, ; \\
F^{-}_{L=2,K=1} &= f_{\omega} \otimes g_{\omega^2} + f_{\omega^2} \otimes g_{\omega} \, .
\end{align}
The list of the allowed quantum states for the $B=20$ $T_d$-symmetric Skyrmion up to $L=4$, $K=2$ is presented in Table \ref{tab:quantum}. The states are arranged according to the spin and do not reflect the actual ordering of energy levels. In order to evaluate the energy of each state, the moments of inertia are required. The energy calculation and the list of energy levels are discussed in the following subsection.
\begin{table}[ht]
\begin{center}
\hspace*{-0.8cm}
\begin{tabular}{ | c | c | p{8.4cm} | p{4cm} | }
\hline
Spin, $L^{P}$ & Isospin, $K$ & $\left| \Psi \right \rangle$ in angular momentum basis & $\left| \Psi \right \rangle$ in Cartesian basis\\ \hline
$0^{+}$ & $0$ & $\left | 0, 0 \right \rangle \otimes \left | 0, 0 \right \rangle$ & $1$ \\ \hline
$3^{-}$ & $0$ & $\left | 3, \pm 2 \right \rangle_{-} \otimes \left | 0, 0 \right \rangle $ & $f_3$ \\ \hline
$4^{+}$ & $0$ & $\left( \left | 4 ,\pm 4 \right \rangle_{+} + \sqrt{\frac{14}{5}} \left | 4, 0 \right \rangle \right) \otimes \left | 0, 0 \right \rangle$ & $f_2^2-\frac{5}{3}f_4$ \\ \hline
$0^{-}$ & $1$ & $\left | 0, 0 \right \rangle \otimes \left | 1 , 0 \right \rangle $ & $g_1$ \\ \hline
$2^{+}$ & $1$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 1 , -1 \right \rangle - \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 1 , 1 \right \rangle$ & $f_{\omega}g_{\omega^2}-f_{\omega^2}g_{\omega}$ \\ \hline
$2^{-}$ & $1$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 1 , -1 \right \rangle + \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 1 , 1 \right \rangle$ & $f_{\omega}g_{\omega^2}+f_{\omega^2}g_{\omega}$ \\ \hline
$3^{+}$ & $1$ & $\left | 3 , \pm 2 \right \rangle_{-} \otimes \left | 1 , 0 \right \rangle $ & $f_{3}g_{1}$ \\ \hline
$4^{+}$ & $1$ & $\left( \left | 4 , \pm 4 \right \rangle_{+} + \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 1 , 1 \right \rangle \newline + \newline \left( \left | 4 , \pm 4 \right \rangle_{+} - \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 1 , -1 \right \rangle$ & $f_2(f_{\omega}g_{\omega^2} - f_{\omega^2}g_{\omega}) + \frac{7}{4} (f_{\omega}^2 g_{\omega} - f_{\omega^2}^2 g_{\omega^2})$ \\ \hline
$4^{-}$ & $1$ & $\left( \left | 4 , \pm 4 \right \rangle_{+} + \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 1 , 1 \right \rangle \newline - \newline \left( \left | 4 , \pm 4 \right \rangle_{+} - \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 1 , -1 \right \rangle$ & $f_2(f_{\omega}g_{\omega^2} + f_{\omega^2}g_{\omega}) - \frac{7}{4} (f_{\omega}^2 g_{\omega} + f_{\omega^2}^2 g_{\omega^2})$ \\ \hline
$4^{-}$ & $1$ & $\left( \left | 4 ,\pm 4 \right \rangle_{+} + \sqrt{\frac{14}{5}} \left | 4, 0 \right \rangle \right) \otimes \left | 1, 0 \right \rangle$ & $f_2^2g_1-\frac{5}{3}f_4 g_1$ \\ \hline
$0^{+}$ & $2$ & $\left | 0, 0 \right \rangle \otimes \left | 2 , 0 \right \rangle $ & $g_1^2 - \frac{1}{2}g_2$ \\ \hline
$2^{+}$ & $2$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , -1 \right \rangle + \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , 1 \right \rangle$ & $f_{\omega}g_{\omega^2}g_1 + f_{\omega^2}g_{\omega}g_1$ \\ \hline
$2^{+}$ & $2$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , 2 \right \rangle - \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , -2 \right \rangle$ & $f_{\omega}g_{\omega}^2 + f_{\omega^2}g_{\omega^2}^2$ \\ \hline
$2^{-}$ & $2$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , -1 \right \rangle - \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , 1 \right \rangle$ & $f_{\omega}g_{\omega^2}g_1 - f_{\omega^2}g_{\omega}g_1$ \\ \hline
$2^{-}$ & $2$ & $\left( \left | 2 , \pm 2 \right \rangle_{+} + \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , 2 \right \rangle + \newline \left( \left | 2 , \pm 2 \right \rangle_{+} - \sqrt{2}i \left | 2 , 0 \right \rangle \right) \otimes \left | 2 , -2 \right \rangle$ & $f_{\omega}g_{\omega}^2 - f_{\omega^2}g_{\omega^2}^2$ \\ \hline
$3^{-}$ & $2$ & $\left | 3 , \pm 2 \right \rangle_{-} \otimes \left | 2 , 0 \right \rangle $ & $f_{3}(g_{1}^2-\frac{1}{2}g_2)$ \\ \hline
$4^{+}$ & $2$ & $\left( \left | 4 ,\pm 4 \right \rangle_{+} + \sqrt{\frac{14}{5}} \left | 4, 0 \right \rangle \right) \otimes \left | 2, 0 \right \rangle$ & $(f_2^2-\frac{5}{3}f_4)(g_1^2-\frac{1}{2}g_2)$ \\ \hline
$4^{+}$ & $2$ & $\left( \left | 4 , \pm 4 \right \rangle_{+} + \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 2 , -2 \right \rangle \newline - \newline \left( \left | 4 , \pm 4 \right \rangle_{+} - \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 2 , 2 \right \rangle$ & $f_2(f_{\omega}g_{\omega}^2 + f_{\omega^2}g_{\omega^2}^2) - \frac{7}{4} (f_{\omega}^2 g_{\omega^2}^2 + f_{\omega^2}^2 g_{\omega}^2)$ \\
\hline
$4^{-}$ & $2$ & $\left( \left | 4 , \pm 4 \right \rangle_{+} + \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 2 , -2 \right \rangle \newline + \newline \left( \left | 4 , \pm 4 \right \rangle_{+} - \sqrt{\frac{12}{7}}i \left | 4 , \pm 2 \right \rangle_{+} - \frac{10}{7} \left | 4 , 0 \right \rangle \right) \otimes \left | 2 , 2 \right \rangle$ & $f_2(f_{\omega}g_{\omega}^2 - f_{\omega^2}g_{\omega^2}^2) + \frac{7}{4} (f_{\omega}^2 g_{\omega^2}^2 - f_{\omega^2}^2 g_{\omega}^2)$ \\
\hline
\end{tabular}
\end{center}
\caption{Allowed quantum states $\left| \Psi \right \rangle$ of the $B=20$ $T_d$-symmetric Skyrmion. The notation used is $\left | L , \pm L_3 \right \rangle_{\pm} = \left | L , L_3 \right \rangle \pm \left | L , -L_3 \right \rangle$.}
\label{tab:quantum}
\end{table}
\clearpage
\subsection{Energy calculation}
The energy levels of the $B=20$ $T_d$-symmetric Skyrmion can be computed using the moments of inertia given in Table \ref{tab:B_20} in the Appendix. Recall that the Hamiltonian of the Skyrmion is
\begin{equation}
\hat{H}=\frac{1}{2v}\hat{L}^2 + \frac{1}{2U_{11}}(\hat{K}^2-\hat{K}_3^2) +\frac{1}{2U_{33}}\hat{K}_3^2 \, .
\label{eqn:20Ham}
\end{equation}
From Table \ref{tab:B_20}, we extract the values $v=12854$, $U_{11}=759$ and $U_{33}=820$.
We illustrate the energy calculation using the $3^-$ state, $ \left| \Psi \right \rangle = \left| 3, \pm 2 \right \rangle _{-} \otimes \left| 0, 0 \right \rangle$. It is an eigenstate of the operators, $\hat{L}^2$, $\hat{L}_3^2$, $\hat{K}^2$ and $\hat{K}_3^2$, hence an eigenstate of the Hamiltonian. The energy $E$, in Skyrme units, can be calculated by replacing the operators with their eigenvalues,
\begin{align}
E_{3,0}&= \frac{1}{2v}L(L+1) + \frac{1}{2U_{11}}(K(K+1) - K_3^2) + \frac{1}{2U_{33}} K_3^2 \nonumber \\
&= \frac{3(3+1)}{2v} + 0 + 0 \nonumber \\
&= \frac{6}{v} \nonumber \\
&= 4.6677 \times 10^{-4} \,.
\end{align}
Using the conversion factor $e^3F_\pi=4000$ MeV, the state has an energy of $1.87$ MeV above the ground state. This is the correct order of magnitude for the excitation energy of a nucleus of this size.
A list of energy levels of the $B=20$ $T_d$-symmetric Skyrmion up to spin $4$ and isospin $2$ is presented in Table \ref{tab:energy}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Spin, $L^{P}$ & Isospin, $K$ & Energy, $E$ & $E$ in Skyrme units & $E$ in MeV \\ \hline
$0^{+}$ & $0$ & $0$ & $0$ & $0$ \\ \hline
$3^{-}$ & $0$ & $\frac{6}{v}$ & $4.668 \times 10^{-4}$ & $1.87$ \\ \hline
$4^{+}$ & $0$ & $\frac{10}{v}$ & $7.780 \times 10^{-4}$ & $3.11$ \\ \hline
$0^{-}$ & $1$ & $\frac{1}{U_{11}}$ & $13.184 \times 10^{-4}$ & $5.27$ \\ \hline
$2^{+}$ & $1$ & $\frac{3}{v}+\frac{1}{2U_{11}}+\frac{1}{2U_{33}}$ & $15.024 \times 10^{-4}$ & $6.01$ \\ \hline
$2^{-}$ & $1$ & $\frac{3}{v}+\frac{1}{2U_{11}}+\frac{1}{2U_{33}}$ & $15.024 \times 10^{-4}$ & $6.01$ \\ \hline
$3^{+}$ & $1$ & $\frac{6}{v}+\frac{1}{U_{11}}$ & $17.851 \times 10^{-4}$ & $7.14$ \\ \hline
$4^{+}$ & $1$ & $\frac{10}{v}+\frac{1}{2U_{11}}+\frac{1}{2U_{33}}$ & $20.470\times 10^{-4}$ & $8.19$ \\ \hline
$4^{-}$ & $1$ & $\frac{10}{v}+\frac{1}{2U_{11}}+\frac{1}{2U_{33}}$ & $20.470\times 10^{-4}$ & $8.19$ \\ \hline
$4^{-}$ & $1$ & $\frac{10}{v}+\frac{1}{U_{11}}$ & $20.963\times 10^{-4}$ & $8.39$ \\ \hline
$0^{+}$ & $2$ & $\frac{3}{U_{11}}$ & $39.551\times 10^{-4}$ & $15.82$ \\ \hline
$2^{+}$ & $2$ & $\frac{3}{v}+\frac{1}{U_{11}}+\frac{2}{U_{33}}$ & $39.910\times 10^{-4}$ & $15.96$ \\ \hline
$2^{-}$ & $2$ & $\frac{3}{v}+\frac{1}{U_{11}}+\frac{2}{U_{33}}$ & $39.910\times 10^{-4}$ & $15.96$ \\ \hline
$2^{+}$ & $2$ & $\frac{3}{v}+\frac{5}{2U_{11}}+\frac{1}{2U_{33}}$ & $41.391\times 10^{-4}$ & $16.56$ \\ \hline
$2^{-}$ & $2$ & $\frac{3}{v}+\frac{5}{2U_{11}}+\frac{1}{2U_{33}}$ & $41.391\times 10^{-4}$ & $16.56$ \\ \hline
$3^{-}$ & $2$ & $\frac{6}{v}+\frac{3}{U_{11}}$ & $44.219\times 10^{-4}$ & $17.69$ \\ \hline
$4^{+}$ & $2$ & $\frac{10}{v}+\frac{1}{U_{11}}+\frac{2}{U_{33}}$ & $45.356\times 10^{-4}$ & $18.14$ \\
\hline
$4^{-}$ & $2$ & $\frac{10}{v}+\frac{1}{U_{11}}+\frac{2}{U_{33}}$ & $45.356\times 10^{-4}$ & $18.14$ \\
\hline
$4^{+}$ & $2$ & $\frac{10}{v} + \frac{3}{U_{11}}$ & $47.331\times 10^{-4}$ & $18.93$ \\ \hline
\end{tabular}
\end{center}
\caption{Energy levels of $B=20$ $T_d$-symmetric Skyrmion}
\label{tab:energy}
\end{table}
\subsection{Construction of parity operator}
In this subsection, a pictorial method of constructing the parity operator $\hat{P}$ is presented. The effect of combined inversions in both ordinary space and isospace on a rational map,
\begin{equation}
R(\mathrm{z}) \rightarrow - \frac{1}{\overline{R}(-\frac{1}{\bar{\mathrm{z}}})} \, ,
\end{equation}
is first studied algebraically, using a simple example, then we demonstrate that it can be equivalently represented pictorially, using the cubic grid, and the corresponding parity operator can be read off from the grid without any algebraic calculation.
The simple example is a rational map constructed from two pairs of zero/pole points, ($a,- \, \rfrac{1}{\bar{a}}$) and ($\rfrac{1}{a},-\bar{a}$). The two pairs are related by a $\pi$-rotation about the $x_1$-axis ($z \rightarrow \rfrac{1}{z}$), and the points of each pair are antipodal. There is no tetrahedral symmetry here. The rational map is
\begin{equation}
R(\mathrm{z})=\frac{(\mathrm{z}-a)(\mathrm{z}-\frac{1}{a})}{(\mathrm{z}-(-\frac{1}{\bar{a}}))(\mathrm{z}-(-\bar{a}))} \, ,
\label{eqn:sim_rat}
\end{equation}
and is transformed under the combined inversion $I$ to
\begin{align}
R(\mathrm{z}) \xrightarrow{\quad \textit{\large{$I$}} \quad} -\frac{1}{\overline{R}(-\frac{1}{\bar{\mathrm{z}}})} &= -\frac{\overline{(-\frac{1}{\bar{\mathrm{z}}}-(-\frac{1}{\bar{a}}))(-\frac{1}{\bar{\mathrm{z}}}-(-\bar{a}))}}{\overline{(-\frac{1}{\bar{\mathrm{z}}}-a)(-\frac{1}{\bar{\mathrm{z}}}-\frac{1}{a})}} \nonumber \\
&= -\frac{(-a\bar{a})(\mathrm{z}-a)(\mathrm{z}-\frac{1}{a})}{(-a\bar{a})(\mathrm{z}-(-\frac{1}{\bar{a}}))(\mathrm{z}-(-\bar{a}))} \nonumber \\
&= -R(\mathrm{z}) \, .
\label{eqn:sim_alg}
\end{align}
$R(\mathrm{z})$ is almost invariant, but gains a minus sign, so $I$ is equivalent here to a $\pi$-rotation about the $Z$-axis. The parity operator is $\hat{P}=e^{i \pi \hat{K}_3}$ in this case.
\begin{figure}[ht]
\noindent\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{parity_1.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; \text{ \; \large{$I_{ \textrm{s} }$ } \;}} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{parity_2.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\text{ \large{$\sigma_{ \textrm{iso} }$} }} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{parity_3.pdf}
\end{minipage}
\caption{Effect on the grid under $I_{ \textrm{s} }$ followed by $\sigma_{ \textrm{iso} }$}
\label{fig:parity_example}
\end{figure}
To explain our pictorial method, the points above are, for illustrative purposes only, assumed to be on the edges of the grid. The effect of the combined inversion on the rational map (\ref{eqn:sim_rat}) is represented in Figure \ref{fig:parity_example}. A spatial inversion $I_{\textrm{s}}$ ($\mathrm{z} \rightarrow - \, \rfrac{1}{\bar{\mathrm{z}}}$) is applied first; it moves the points to the antipodal points and exchanges zeros and poles. Second, an isospatial inversion is applied; this is a combined reflection $\sigma_{\textrm{iso}}$ in the $X-Y$ plane with a $\pi$-rotation, $C_{2 \, \textrm{iso}}$, about the $Z$-axis. The reflection $\sigma_{\textrm{iso}}$ exchanges zeros and poles ($R \rightarrow \rfrac{1}{\overline{R}}$); the isorotation $C_{2 \, \textrm{iso}}$ multiplies the rational map by $-1$ ($R \rightarrow -R$), but it does not change the positions of zeros and poles; as a result, the operation $C_{2 \, \textrm{iso}}$ cannot be represented in the Figure.
From Figure \ref{fig:parity_example}, one sees that the rational map returns to itself exactly after the operations $I_{\textrm{s}}$ and $\sigma_{\textrm{iso}}$. Including the factor $-1$ from the isorotation $C_{2 \, \textrm{iso}}$, we arrive at the same expression as shown in the algebraic calculation (\ref{eqn:sim_alg}),
\begin{equation}
R(\mathrm{z}) \xrightarrow{\quad \text{\large{$I_{ \textrm{s}}$ , $\sigma_{\textrm{iso}}$ }} \quad} \frac{1}{\overline{R}(-\frac{1}{\bar{\mathrm{z}}})} = R(\mathrm{z}) \xrightarrow{\quad \text{\large{$C_{2 \, \textrm{iso}}$}} \quad} -R(\mathrm{z}) \, ,
\end{equation}
and the parity operator $\hat{P}=e^{i \pi \hat{K}_3}$ is recovered.
The algebraic calculation becomes messy for rational maps of high degree. The pictorial method is then particularly effective. We now apply this method to the rational map for the $B=20$ $T_d$-symmetric Skyrmion; the operations are shown in Figure \ref{fig:B20parity}.
\begin{figure}[ht]
\noindent\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{4x4x4_Grid_Td.pdf}
\label{fig:Td_grid_1}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; \text{ \large{$I_{\textrm{s}}$} } \;} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{4x4x4_Grid_Td_2.pdf}
\label{fig:Td_grid_2}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\text{ \large{$\sigma_{\textrm{iso}}$} }} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{4x4x4_Grid_Td_3.pdf}
\label{fig:Td_grid_3}
\end{minipage}
\caption{$B=20$ rational map transformed by $I_{\textrm{s}}$ followed by $\sigma_{\textrm{iso}}$}
\label{fig:B20parity}
\end{figure}
The rational map does not return to its original form after $I_{\textrm{s}}$ and $\sigma_{\textrm{iso}}$; the transformed map is related to the original map by a $\frac{\pi}{2}$-rotation about the $x_3$-axis ($\textrm{z} \rightarrow i\textrm{z}$) and a $\pi$-rotation about the $X$-axis ($R \rightarrow \rfrac{1}{R}$). We deduce that, including the isorotation $C_{2 \, \textrm{iso}}$, the inversions have the following algebraic effect
\begin{equation}
R(\textrm{z}) \xrightarrow{\quad \text{\large{$I_{ \textrm{s}}$ , $\sigma_{\textrm{iso}}$ }} \quad} \frac{1}{\overline{R}(-\frac{1}{\bar{\textrm{z}}})} = \frac{1}{R(i\textrm{z})} \xrightarrow{\quad \text{\large{$C_{2 \, \textrm{iso}}$}} \quad} -\frac{1}{R(i\textrm{z})} \, ,
\end{equation}
and the parity operator $\hat{P}=e^{i \frac{\pi}{2} \hat{L}_3}e^{i \pi \hat{K}_2}$ can be read off. ($C_{2 \, \textrm{iso}}$ has the effect of replacing the $\pi$-rotation about $X$-axis by a $\pi$-rotation about the $Y$-axis.)
\section{Further $T_d$-symmetric Skyrmions}
\subsection{Quantization, F-R constraints and parity operator}
Although the Cartesian method was used above specifically to study the $B=20$ $T_d$-symmetric Skyrmion, it can be generalized to other Skyrmions with the same symmetry. The first question of the generalization is whether there is a modification of the F-R constraints, as this would have a great impact on the allowed states. This can be resolved with some symmetry arguments together with the help of the cubic grid. The F-R constraints only involve the 12 even elements (rotations) of the $T_d$ point group ($g_{\text{even}} \in T \subset T_d$). A set of 12 points can be generated by acting with the even elements on a generic point of the cubic grid (see Figure \ref{fig:T_element}).
\begin{figure}[ht]
\centering
\noindent\begin{minipage}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_1.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; g_{ \text{even} } \;} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_mI.pdf}
\end{minipage}
\caption{Points generated under the action of even elements of $T_d$}
\label{fig:T_element}
\end{figure}
A rational map can be constructed by taking such a set of points as zeros and a similar set of 12 points as poles. This results in a degree 12 rational map, which gives a $B=12$ Skyrmion upon relaxation. Rational maps of higher degree can be constructed similarly, by using more such sets of 12 points. The degree of a rational map, constructed by this method, is a multiple of 12; the baryon number of the resulting Skyrmion has the general form $B=12n$, $n \in \mathbb{N}$, when $n$ sets of points are used for the zeros and $n$ sets for the poles. The points on the vertices of the grid are special, because only four points are generated by the action of the even elements, $g_{\text{even}}$. If there are zeros on four tetrahedrally-related vertices, there need to be poles on the other four. This modifies the baryon number to $B=12n + 4m$, when zeros and poles of multiplicity $m$ on the vertices are included. The general rational map of this geometrical type has the form
\begin{equation}
R_{T_d(m,n)}= \left(\frac{p_+}{p_-}\right)^m \left(\frac{\prod_{i=1}^{n} p_i}{\prod_{i=1}^{n} q_i}\right) \,.
\label{eqn:Tdgen_rat}
\end{equation}
Here, $p_+$ and $p_-$ are the Klein polynomials given in (\ref{eqn:Klein}); each $p_i$ and $q_i$ is a degree 12 polynomial constructed from a generic set of 12 $T$-related points. $p_i$ and $q_i$ are Klein polynomials having the following general forms \cite{FLM},
\begin{equation}
p_i = \frac{1}{1+a_i}(a_i p_+^3 + p_-^3) \, , \qquad q_i = \frac{1}{1+b_i} ( b_i p_+^3 + p_-^3 ) \,,
\end{equation}
where $a_i$ and $b_i$ are mutually distinct constants.
Under the $\frac{2\pi}{3}$-rotation associated to the operator $e^{i\frac{2\pi}{3\sqrt{3}}(\hat{L}_1+\hat{L}_2+\hat{L}_3)}$, the rational map (\ref{eqn:Tdgen_rat}) picks up a phase factor which depends on the value $m$, but not $n$,
\begin{equation}
R_{T_d(m,n)} \rightarrow e^{i\frac{2m\pi}{3}} R_{T_d(m,n)} \, .
\end{equation}
This implies that the corresponding F-R constraint is dictated by the multiplicity $m$ of the vertex points; it is
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2m\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = (-1)^{\mathcal{N}_1} \left | \Psi \right \rangle \,.
\label{eqn:FRmod}
\end{equation}
\noindent Note that $m$ appears in the operator on the left hand side. $\mathcal{N}_1$ can be calculated using the formula \cite{Krusch}
\begin{equation}
\mathcal{N} = \frac{B}{2 \pi} (B \alpha - \beta) \, , \label{eqn:FRsign}
\end{equation}
where $B=12n+4m$ is the baryon number, and $\alpha=\frac{2\pi}{3}$ and $\beta=\frac{2m\pi}{3}$ are the angles of rotation in space and isospace occurring in (\ref{eqn:FRmod}). So
\begin{align}
\mathcal{N}_1 &= \frac{(12n+4m)}{2\pi}\left((12n+4m)\frac{2 \pi}{3} - \frac{2 m \pi}{3}\right) \nonumber \\
&= \frac{1}{3}(12n+4m)(12n+4m-m) \nonumber \\
&= 4(3n+m)(4n+m) \, . \label{eqn:Tdsign}
\end{align}
$\mathcal{N}_1$ is always a multiple of $4$, hence $(-1)^{\mathcal{N}_1} = 1$.
The rational map (\ref{eqn:Tdgen_rat}) is invariant under the $\pi$-rotation associated to the operator $e^{i\pi\hat{L}_3}$,
\begin{equation}
R_{T_d(m,n)} \rightarrow R_{T_d(m,n)} \, ,
\end{equation}
hence the second F-R constraint is not modified and is independent of $m$ and $n$; it is
\begin{equation}
e^{i \pi \hat{L}_3} \left | \Psi \right \rangle = (-1)^{\mathcal{N}_2} \left | \Psi \right \rangle \, .
\label{eqn:TdFR2mod}
\end{equation}
As $\alpha=\pi$ and $\beta=0$, formula (\ref{eqn:FRsign}) gives
\begin{align}
\mathcal{N}_2 &= \frac{(12n+4m)}{2\pi}\left((12n+4m)\pi - 0 \right) \nonumber \\
&= (6n+2m)(12n+4m) \nonumber \\
&= 8(3n+m)^2 \, , \label{eqn:Tdsign_2}
\end{align}
which is again even, so $(-1)^{\mathcal{N}_2} = 1$. The conclusion is that all the $T_d$-symmetric rational maps constructed from the grid have positive F-R signs for both F-R constraints, but the first constraint equation depends on $m$.
\begin{figure}[ht]
\noindent\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_2.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; \text{ \large{$I_{\textrm{s}}$} } \;} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_Is.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\text{ \large{$\sigma_{\textrm{iso}}$} }} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_mI.pdf}
\end{minipage}
\caption{$T_d$-symmetric rational map under $I_{\textrm{s}}$ followed by $\sigma_{\textrm{iso}}$}
\label{fig:Tdinv}
\end{figure}
Using the cubic grid, one can also show that all $T_d$-symmetric rational maps have the same parity operator. Figure \ref{fig:Tdinv} shows the effect of applying $I_{\textrm{s}}$ followed by $\sigma_{\textrm{iso}}$ on a set of 12 generic points generated by the even elements of $T_d$. The effect is the same as for the $B=20$ $T_d$-symmetric Skyrmion; it is a $\frac{\pi}{2}$-rotation about the $x_3$-axis combined with a $\pi$-rotation about the $X$-axis, that is,
\begin{equation}
R(\textrm{z}) \xrightarrow{\quad \text{\large{$I_{ \textrm{s}}$ , $\sigma_{\textrm{iso}}$ }} \quad} \frac{1}{\overline{R}(-\frac{1}{\bar{\textrm{z}}})} = \frac{1}{R(i\textrm{z})} \xrightarrow{\quad \text{\large{$C_{2 \, \textrm{iso}}$}} \quad} -\frac{1}{R(i\textrm{z})} \,.
\end{equation}
The parity operator is therefore the same as before, $\hat{P}=e^{i \frac{\pi}{2} \hat{L}_3} e^{i \pi \hat{K}_2}$. The trichotomy of F-R constraints allows us to classify the quantum states of $T_d$-symmetric Skyrmions, constructed from the cubic grid, into three classes. The classification is discussed in the following subsection.
\subsection{Classification of $T_d$-symmetric quantum states}
The isospin operator in the F-R constraint (\ref{eqn:FRmod}) depends on the value $m$. The polynomials $g_\omega$ and $g_\omega^2$ pick up different phase factors under this operator (see Table \ref{tab:FRm}), and this leads to three classes of allowed states.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Polynomials & $m=0 \pmod 3$ & $m=1 \pmod 3$ & $m=2 \pmod 3$ \\ \hline
$g_\omega$ & $g_{\omega}$ & $\omega \, g_{\omega}$ & $\omega^2 \, g_{\omega}$ \\ \hline
$g_{\omega^2}$ & $g_{\omega^2}$ &$\omega^2 g_{\omega^2}$ & $\omega \, g_{\omega^2}$ \\
\hline
\end{tabular}
\end{center}
\caption{Transformation of $g_{\omega}$ and $g_{\omega^2}$ under the isospin operator in (\ref{eqn:FRmod})}
\label{tab:FRm}
\end{table}
\noindent For $m=0 \pmod 3$, the F-R constraints simplify to
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} \left | \Psi \right \rangle = \left | \Psi \right \rangle \,, \qquad e^{i \pi \hat{L}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \,.
\end{equation}
They are the same as (\ref{eqn:B20_FRK0}), which are the F-R constraints for the $B=20$ $T_d$-symmetric Skyrmion with $K=0$. The value of isospin is not constrained, and the allowed values of spin are those given in Table \ref{tab:quantum} with $K=0$, i.e. $L=0,3,4$. States of spin $L$ and isospin $K$ form an isospin ($2K+1$)-plet; the example of states with $L=3$ and $K=1$ is shown in Table \ref{tab:m0L3K1}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{ | c | c | p{5cm} | p{3cm} | }
\hline
Spin, $L^{P}$ & Isospin, $K$ & Angular momentum basis & Cartesian basis\\ \hline
$3^{+}$ & $1$ & $\left | 3 , \pm 2 \right \rangle_{-} \otimes \left | 1 , 0 \right \rangle $ & $f_{3}g_{1}$ \\ \hline
$3^{+}$ & $1$ & $\left | 3 , \pm 2 \right \rangle_{-} \otimes \left( \left | 1 , 1 \right \rangle + \left | 1 , -1 \right \rangle \right)$ & $f_{3}(g_{\omega}-g_{\omega^2})$ \\ \hline
$3^{-}$ & $1$ & $\left | 3 , \pm 2 \right \rangle_{-} \otimes \left(\left | 1 , 1 \right \rangle - \left | 1 , -1 \right \rangle \right)$ & $f_{3}(g_{\omega}+g_{\omega^2})$ \\
\hline
\end{tabular}
\end{center}
\caption{$L=3$ and $K=1$ triplet for $m=0 \pmod{3}$}
\label{tab:m0L3K1}
\end{table}
For $m=1 \pmod 3$, the F-R constraints are the same as those for the $B=20$ $T_d$-symmetric Skyrmion,
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \,, \qquad e^{i \pi \hat{L}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \,.
\end{equation}
The allowed states are identical to the states given in Table \ref{tab:quantum}.
For $m=2 \pmod 3$, the states in Table \ref{tab:quantum} that involve $g_\omega$ and $g_{\omega^2}$ no longer satisfy the F-R constraint (\ref{eqn:FRmod}). The roles of $g_\omega$ and $g_{\omega^2}$ are exchanged; for example, the $L=2$ and $K=1$ positive parity state $F^+_{2,1}=f_\omega g_{\omega^2} - f_{\omega^2} g_\omega$ for $m=1$ is replaced by $F^+_{2,1}=f_\omega g_{\omega} - f_{\omega^2} g_{\omega^2}$ for $m=2$. The list of states for $m=2$ can be obtained from Table \ref{tab:quantum} by exchanging $g_\omega$ by $g_{\omega^2}$.
\subsection{$B=56$ $T_d$-symmetric Skyrmion}
An application of the classification is to the quantum states of a $T_d$-symmetric Skyrmion with baryon number $B=56$. This Skyrmion is found by using a double-layer rational map ansatz. The first layer of the cubic grid is left out, and the inner, degree 28 rational map comes from the second layer of the cubic grid (see Figure \ref{fig:cube_grid}). The outer, degree 28 rational map is constructed from the third layer of the cubic grid; the points used in the construction are shown in Figure \ref{fig:cubic_grid_56}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{6x6x6_grid_Td.pdf}
\caption{Outer rational map for the $B=56$ $T_d$-symmetric Skyrmion}
\label{fig:cubic_grid_56}
\end{figure}
\noindent The outer rational map is
\begin{equation}
R_{56,T_d} = \left( \frac{1+d_2}{1+d_1} \right) \frac{p_+}{p_-}
\left(\frac{d_1 p_+^3 + p_-^3}{d_2 p_+^3 + p_-^3}\right) \left( \frac{d_3 p_+^3 + p_-^3}{p_+^3 + d_3 p_-^3} \right) \, ,
\end{equation}
\noindent where $d_1=2.715$, $d_2=-22.253$ and $d_3=-0.670$. For a more detailed discussion of the interpretation of such constants, see ref.\cite{FLM}. The $B=56$ Skyrmion obtained by relaxing these rational maps preserves the $T_d$ symmetry, and it is a higher baryon number analogue of the $B=20$ Skyrmion. The Skyrmion is shown in Figure \ref{fig:B_56}. It is worth noting that if one uses a triple-layer rational map ansatz, including the first layer of the cubic grid as the innermost, degree 4 rational map, one finds a $B=60$ $T_d$-symmetric Skyrmion that looks similar.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{B_56.jpeg}
\caption{$B=56$ Skyrmion with $T_d$ symmetry}
\label{fig:B_56}
\end{figure}
The primary colours, red, green and blue, are cyclically permuted when a $\frac{2\pi}{3}$-rotation is applied about a vertex, and the corresponding isorotation is a $\frac{2\pi}{3}$-rotation about the $Z$-axis. It implies that the $B=56$ Skyrmion has the same $T_d$ symmetry realization as the $B=20$ Skyrmion. They therefore have the same set of F-R constraints with $m=1$ and the same F-R signs. The $B=56$ Skyrmion has the same parity operator as the $B=20$ Skyrmion too, which can be shown using the pictorial method. They therefore have the same set of allowed spin/isospin states. The states only differ in their excitation energies. The moments of inertia of the $B=56$ Skyrmion are larger than those of the $B=20$ Skyrmion because of its size, and the excitation energy of each state is lower.
\section{$O_h$-symmetric Skyrmions}
The $T_d$ symmetry group, which we have been studying, is a normal subgroup of the cubic group $O_h$, and the $O_h$ group is the full symmetry group of the cubic grid. Thus, it is natural to extend our techniques to $O_h$. $O_h$ ($48$ elements) is twice as big as $T_d$ ($24$ elements). This imposes more constraints on the states, and fewer states are allowed. Using the $B=32$ $O_h$-symmetric Skyrmion as an example, the corresponding F-R constraints are
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, , \qquad
e^{i \frac{\pi}{2} \hat{L}_3} e^{i \pi \hat{K}_1} \left | \Psi \right \rangle = \left | \Psi \right \rangle \, .
\label{eqn:OhFR}
\end{equation}
24 points will be generated from a generic point under the action of the 24 even elements of $O_h$, hence one may think that the baryon number is of the form $B=24n$, $n \in \mathbb{N}$; however, this is incorrect because of the second F-R constraint. The operators of the constraint turn zeros into poles, and vice versa. The set of $24$ points consists of $12$ zeros and $12$ poles; therefore, the baryon number is of the form $B=12n$, when $n$ sets are included. Including the points on the vertices, we obtain the same formula as in the $T_d$ case, $B=12n+4m$, $n,m \in \mathbb{N}$. The only difference between the $T_d$ and $O_h$ symmetry is the extra constraint that relates the zeros and poles (see Figure \ref{fig:O_element}).
\begin{figure}[ht]
\centering
\noindent\begin{minipage}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_elements_1.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; g_{ \text{even} } \;} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{O_elements.pdf}
\end{minipage}
\caption{Points generated under the action of even elements of $O_h$}
\label{fig:O_element}
\end{figure}
To derive the generalized F-R constraints for $O_h$-symmetric Skyrmions, constructed from the cubic grid, we consider a general $O_h$-symmetric rational map given by
\begin{equation}
R_{O_h(m,n)}=\left( \frac{p_+}{p_-} \right)^m \prod_{i=1}^n \left(\frac{a_i p_+^3 + p_-^3}{p_+^3 + a_i p_-^3}\right) \,,
\label{eqn:Ohgen_rat}
\end{equation}
where $a_i$ is constant. The $T_d$-symmetric rational map (\ref{eqn:Tdgen_rat}) can be specialized to an $O_h$-symmetric rational map by imposing the extra conditions $b_i = \rfrac{1}{a_i}$,
\begin{align}
R_{T_d(m,n)}\Bigg|_{b_i=\frac{1}{a_i}}&= \left(\frac{p_+}{p_-}\right)^m \left(\frac{\prod_{i=1}^{n} p_i}{\prod_{i=1}^{n} q_i}\right)\Bigg|_{b_i=\frac{1}{a_i}} \nonumber \\
&=\left(\frac{p_+}{p_-}\right)^m \left(\frac{\prod_{i=1}^{n} \frac{1}{1+a_i}(a_i p_+^3 + p_-^3)}{\prod_{i=1}^{n} \frac{1}{1+b_i}(b_i p_+^3 + p_-^3)}\right)\Bigg|_{b_i=\frac{1}{a_i}} \nonumber \\
&= \left(\frac{p_+}{p_-}\right)^m \prod_{i=1}^{n} \left(\frac{a_i p_+^3 + p_-^3}{p_+^3 + a_i p_-^3} \right) \nonumber \\
&= R_{O_h(m,n)} \, .
\end{align}
\indent The $O_h$-symmetric rational map transforms under the rotation associated to the operator $e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)}$ as
\begin{equation}
R_{O_h(m,n)} \rightarrow e^{i\frac{2m\pi}{3}}R_{O_h(m,n)}
\end{equation}
and under the operator $e^{i\frac{\pi}{2}\hat{L}_3}$ as
\begin{equation}
R_{O_h(m,n)} \rightarrow \frac{1}{R_{O_h(m,n)}} \,.
\end{equation}
This leads to the following generalized $O_h$ F-R constraints,
\begin{equation}
e^{i \frac{2 \pi}{3 \sqrt{3}} (\hat{L}_1 + \hat{L}_2 + \hat{L}_3)} e^{i \frac{2m\pi}{3} \hat{K}_3} \left | \Psi \right \rangle = (-1)^{\mathcal{N}_1}\left | \Psi \right \rangle \, , \qquad
e^{i \frac{\pi}{2} \hat{L}_3} e^{i \pi \hat{K}_1} \left | \Psi \right \rangle = (-1)^{\mathcal{N}_2}\left | \Psi \right \rangle \, .
\label{eqn:OhFR}
\end{equation}
The first F-R constraint is the same as the $T_d$-symmetric case, while the second F-R constraint is a ``square root'' of the $T_d$ F-R constraint (\ref{eqn:TdFR2mod}).
$\mathcal{N}_1$ is identical to (\ref{eqn:Tdsign}) and is even; $\mathcal{N}_2$ can be calculated using (\ref{eqn:FRsign}) with $B=12n+4m$, $\alpha=\frac{\pi}{2}$ and $\beta=\pi$,
\begin{align}
\mathcal{N}_2 &= \frac{(12n+4m)}{2\pi}\left((12n+4m)\frac{\pi}{2} - \pi \right) \nonumber \\
&= (6n+2m)(6n+2m - 1) \nonumber \\
&= 2(3n+m)(6n+2m-1) \, ,
\label{eqn:Ohsign_2}
\end{align}
\noindent which is still even, hence $(-1)^{\mathcal{N}_2}=1$. The operators of the second F-R constraint transform the Cartesian coordinates as
\begin{align}
x &\rightarrow y & &, & y &\rightarrow -x & &, & \ z &\rightarrow z & \, &; \\
X &\rightarrow X & &, & Y &\rightarrow -Y & &, & \ Z &\rightarrow -Z & \, &.
\end{align}
The parity operator is different from the $T_d$ case and can be extracted with the pictorial method as shown in Figure \ref{fig:Oh_inv}.
\begin{figure}[ht]
\noindent\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{Oh_general_grid.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\; \text{ \large{$I_{\textrm{s}}$} } \;} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{Oh_general_grid_2.pdf}
\end{minipage}
\begin{minipage}{.1\textwidth}
\begin{equation}
\xrightarrow{\text{ \large{$\sigma_{\textrm{iso}}$} }} \nonumber
\end{equation}
\end{minipage}
\begin{minipage}{.25\textwidth}
\centering
\includegraphics[width=\textwidth]{Oh_general_grid.pdf}
\end{minipage}
\caption{$O_h$-symmetric rational map under $I_{\textrm{s}}$ followed by $\sigma_{\textrm{iso}}$}
\label{fig:Oh_inv}
\end{figure}
\noindent The rational map $R(\mathrm{z})$ returns to its original form after the transformation $I_{\textrm{s}}$ and $\sigma_{\textrm{iso}}$. Including the isorotation $C_{2 \, \textrm{iso}}$, one finds that $R(\mathrm{z})$ transforms to $-R(\mathrm{z})$. The parity operator is therefore $\hat{P}=e^{i \pi \hat{K}_3}$.
\begin{table}[ht]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Degree & & $T$-invariant polynomials & $1^{\text{st}}$ F-R & $2^{\text{nd}}$ F-R & Parity\\ \hline
$2$ & $f_2$ & $x^2 + y^2 + z^2$ & $f_2$ & $f_2$ & $+1$ \\ \hline
$3$ & $f_3$ & $xyz$ & $f_3$ & $-f_3$ & $+1$ \\ \hline
$4$ & $f_4$ & $x^4 + y^4 + z^4$ & $f_4$ & $f_4$ & $+1$ \\ \hline
$6$ & $f_6$ & $x^4(y^2-z^2) + y^4(z^2-x^2) + z^4(x^2-y^2)$ & $f_6$ & $-f_6$ & $+1$ \\ \hline
$1$ & $g_1$ & $Z$ & $g_1$ & $-g_1$ & $+1$ \\ \hline
$2$ & $g_2$ & $X^2 + Y^2$ & $g_2$ & $g_2$ & $+1$ \\ \hline
$3$ & $g_{3-}$ & $X(X^2-3Y^2)$ & $g_{3-}$ & $g_{3-}$ & $-1$ \\ \hline
$3$ & $g_{3+}$ & $Y(3X^2-Y^2)$ & $g_{3+}$ & $-g_{3+}$ & $-1$ \\ \hline
$2$ & $f_{\omega}$ & $\omega^2(x^2 + \omega^2 y^2 + \omega z^2)$ & $\omega \, f_{\omega}$ & $f_{\omega^2}$ & $+1$ \\ \hline
$2$ & $f_{\omega^2}$ & $\omega(x^2+\omega y^2 +\omega^2 z^2)$ & $\omega^2 f_{\omega^2}$ & $f_{\omega}$ & $+1$ \\ \hline
$1$ & $g_{\omega}$ & $(X+iY)$ & $\omega^m g_{\omega}$ & $g_{\omega^2}$ & $-1$ \\ \hline
$1$ & $g_{\omega^2}$ & $(X-iY)$ & $(\omega^2)^m g_{\omega^2}$ & $g_{\omega}$ & $-1$ \\
\hline
\end{tabular}
\end{center}
\caption{$T$-invariant polynomials under $O_h$ F-R operators}
\label{tab:cubic}
\end{table}
The $O_h$-symmetric states can be found by using the $T$-invariant polynomials. Table \ref{tab:cubic} shows the effect on the $T$-invariant polynomials of the operators of the $O_h$ F-R constraints (\ref{eqn:OhFR}). The polynomials $f_3$, $f_6$, $g_1$ and $g_{3+}$ are no longer invariant. They gain a minus sign under the operator occurring in the second constraint. We have to combine an even number of these polynomials to form $O_h$-invariants, for example, $f_3^2$ and $g_1g_{3+}$. In Table \ref{tab:quantum}, some of the $T_d$-symmetric states consist of an odd number of these polynomials; these are not allowed as states for an $O_h$-symmetric Skyrmion.
The allowed quantum states of $O_h$-symmetric Skyrmions belong to a subset of the $T_d$-symmetric states for each value of $m$. For example, the $3^-$ state is forbidden while the $4^+$ state is allowed for a quantized Skyrmion with $O_h$ symmetry.
\section{Conclusions}
In this paper, we systematically studied the semi-classical quantization of the families of $T_d$- and $O_h$-symmetric Skyrmions constructed using the cubic grid method. A Cartesian method of solving the Finkelstein-Rubinstein (F-R) constraints was developed and applied to $T_d$-symmetric Skyrmions and $O_h$-symmetric Skyrmions. The action of the $T_d$ and $O_h$ groups on the quantum states of Skyrmions is more transparent in the Cartesian coordinates, and the F-R constraints are easier to solve.
The $T_d$- and $O_h$-symmetric rational maps constructed from the cubic grid have similar structures, and they lead to Skyrmions with baryon number a multiple of four. The symmetries of the Skyrmions are realized in almost the same way. The F-R constraints of the Skyrmions differ in the contribution of the vertex points of the cubic grid to the rational maps, and this difference is characterized by an integer $m \pmod{3}$. We classified all quantum states of the Skyrmions into three classes depending on this integer $m$. All the $T_d$- and $O_h$-symmetric Skyrmions in each class have the same set of spin and isospin quantum states.
We applied this classification to the quantum states of the $B=20$ and newly found $B=56$ $T_d$-symmetric Skyrmions. The class which these Skyrmions belong to was identified easily by looking at the F-R constraints (they have $m=1$), and the quantum states for these Skyrmions are listed in Table 4. The energies of the states of the $B=20$ Skyrmion are given in Table 5, and for $B=56$, they would be similar.
The next step would be to relate the results presented here to other studies of possibly tetrahedrally- and cubically-symmetric nuclei \cite{DGSM,DCDDPOS}; another possible step is to extend the classification method to other normal subgroups of $O_h$, for example $D_{2h}$, which is the symmetry group of another $B=20$ Skyrmion.
\clearpage
|
2,877,628,089,537 | arxiv | \section{Introduction} \label{intro}
The broad and blueshifted \ion{C}{4}$\lambda\lambda 1548,1551$ lines
observed in
broad absorption-line quasars (BALQs) reveal the unambiguous signature
of outflows. Therefore, BALQs may be important sources of quasar
feedback in galaxy evolution. In this context, an important parameter
is the ratio of the kinetic luminosity ($L_\mathrm{KE}$) to bolometric
luminosity ($L_\mathrm{Bol}$), because it has been shown that if
$L_\mathrm{KE}/L_\mathrm{Bol}$ exceeds 0.5--5\%
\citep{dimatteo05,he10} then sufficient energy is available to
regulate star formation and therefore produce the distribution of
galaxies that we see today.
Determining $L_\mathrm{KE}/L_\mathrm{Bol}$ in the general population
of BALQs is challenging. The measurement of $L_\mathrm{KE}$ requires
constraint of key physical parameters of the outflowing gas including
the column density $N_H$, the velocity, and the location of the
outflow, as well as an estimate of $\Omega$, the fraction of the full $4\pi$
sterradians that is covered by the outflow. The velocity can be
estimated from the blueshift of the absorption lines, and $\Omega$ is
estimated from the incidence of BAL quasars in the population. The
column density and location require measurement of the
photoionization properties of the outflowing gas. These properties
can be inferred directly from measurements of the optical depths of
absorption lines in the spectrum, but only when the lines are
relatively narrow and line blending is not severe. This approach has
been used on some tens of spectra \citep[e.g.,][and references
therein]{arav20,miller20}.
However, this type of analysis cannot be done on spectra in which
line blending is significant, i.e., most of the $\sim$30,000 BAL quasar
spectra present in the SDSS archive \citep[DR14Q;][]{paris18}.
It is easy to understand why we cannot generalize results from 10s of
spectra with narrow absorption lines to the quasar
population in general. The kinetic luminosity depends
principally on the outflow velocity, since that factor enters the
equation for $L_\mathrm{KE}$ to the cubed power. Moreover,
observations show that BAL velocities have a tremendous range, from
nearly zero velocity or inflow in a few cases to more than 30,000$\rm
\, km\, s^{-1}$. At high velocities, blending is usually significant.
Samples of BAL quasars repeatedly show a relationship
between the BAL outflow velocity and the luminosity of the quasar
\citep[e.g.,][]{laor02, ganguly07}, but the dependence is complicated,
with the relationship showing an upper limit envelope rather than a
one-to-one dependence. So, we can expect that a powerful quasar
located at $z=2$--3 that may have a bolometric luminosity exceeding
$10^{48}\rm \, erg s^{-1}$ will have much different outflow properties
and $L_\mathrm{KE}/L_\mathrm{Bol}$ than a Seyfert luminosity object at
$z=0.5$--1.5.
It is also very plausible that $L_\mathrm{KE}/L_\mathrm{Bol}$
depends on the Eddington ratio $L_\mathrm{Bol}/L_\mathrm{Edd}$
of the quasar. The terminal velocity of an accelerated outflow
depends on the magnitude of the acceleration but also on the
deceleration due to gravity provided by the supermassive black hole.
Also, we expect the geometry and physical conditions of the central
engine to change with this parameter \citep[e.g.,][]{gp19}. Finally,
we infer profound dependence of the broad emission lines on
$L_\mathrm{Bol}/L_\mathrm{Edd}$ and ask why the absorption lines
should not also depend on this factor.
The bottom line is that
it is probably not reasonable to expect that a single
value of $L_\mathrm{KE}/L_\mathrm{Bol}$ should be applicable for all UV
outflows, but rather that this parameter should depend on (at least)
the luminosity (equivalently the black hole mass) and the Eddington
ratio.
A first step in making progress on the relationships between
fundamental quasar parameters and outflow properties would be to
analyze broad absorption-lines from many more objects representing the
full range of the BAL phenomena. This step requires a method to
handle line blending.
Here, we use the term ``line
blending'' to mean that the velocity width of the broad absorption-lines is sufficiently large that different absorption lines blend
together regardless of the spectral resolution.
That problem is now approachable using the
novel spectral synthesis code {\it SimBAL}.
The {\it SimBAL} methodology is described in \citet{leighly18}. {\it SimBAL} uses a forward modeling method.
It creates synthetic spectra parameterized by photoionization conditions of the outflowing gas, and then compares them with the observed spectrum using a Markov Chain Monte Carlo
method \citep[{\tt emcee};][]{emcee}. Additional examples of the use
of {\it SimBAL} can be found in \citet{leighly19} and \citet{choi20}.
The {\it SimBAL} analysis of the objects described in this paper is
found in \citet{choi22}.
The second thing we can do is to try to understand the physics of BAL
outflows. If that could be done, then we could predict the outflow
properties in a given quasar, given its (say) luminosity and Eddington
ratio.
Despite fifty years of study, it is still not known how
these outflows are accelerated, what confines the BAL ``clouds'', or
what the origin of the absorbing gas is. In short, the same questions
that were posed in the '80s and '90s are still unanswered and are
still relevant today.
Some of these questions can be addressed by studying the rest-frame
optical emission lines in BAL quasars. First, the $H\beta$$\lambda 4863$\AA\/ emission
line region yields an estimate of the black hole mass. Black hole
mass estimates are also available using the \ion{Mg}{2}$\lambda\lambda
2796,2804$
\citep[e.g.,][]{bahk19} and \ion{C}{4} \citep[e.g.,][]{coatman17}, but
those lines may be significantly absorbed in BAL quasars, a fact that
adds significant uncertainty.
In contrast, it is rare to observe absorption in Balmer lines and even when they are present in the spectra, the Balmer emission lines (e.g., H$\beta$) generally are not significantly absorbed and can be easily studied\citep[e.g.,][]{schulze18}.
The H$\beta$ region also includes
[\ion{O}{3}]$\lambda 4960, 5008$ and many thousands of lines in the
\ion{Fe}{2} pseudocontinuum emission, and together with H$\beta$,
these parameters are thought to reflect the physical conditions of the
central engine through a pattern of behavior known as Eigenvector 1
\citep{bg92} that is widely considered to be a probe of the
Eddington ratio.
Eigenvector 1 is repeatedly found in principal
components analysis (PCA) of optical rest-frame quasar spectra
\citep[e.g.,][]{grupe04,ludwig09,wolf20}.
Thus, we might expect that, just as we observed the
BAL outflow velocity to depend on the quasar luminosity, we should
observed BAL properties to depend on the Eigenvector 1 properties.
BAL quasars are divided into three types based on the absorption lines
present in the spectrum.
\ion{C}{4} is the most commonly observed line, observed in 10-26\% of optically
selected quasars \citep{tolea02, hf03, reichard03, trump06, knigge08, gibson09}.
Objects that only have this line
plus other high-ionization lines such as \ion{N}{5}$\lambda\lambda
1239,1242$ and \ion{Si}{4}$\lambda\lambda 1394, 1403$
are called high-ionization broad absorption-line quasars (HiBALQs). About 1.3\% of quasars have broad
\ion{Mg}{2} absorption \citep{trump06}; these are called
low-ionization broad absorption-line quasars (LoBALQs). About 0.3\% of
quasars also have absorption from \ion{Fe}{2}, and these are called
iron low-ionization broad absorption-line quasars
\citep[FeLoBALQs,][]{trump06}.
These objects are observed much less frequently than the other two
categories, but to some extent, their rarity can be attributed to the
difficulties in detection these objects, since they may lack strong
emission lines due to absorption. Furthermore, their spectral energy
distributions (SEDs) show the reddest colors among BAL quasars which
suggests that they may represent a short-lived stage in quasar
evolution where the quasar expels its cocoon of gas and dust
(“blowout” phase; e.g., Sanders et al. 1988), transitioning from the
ultraluminous infrared galaxy (ULIRG) phase \citep[e.g.,][]{urrutia09,
glikman17, glikman18}. Finally, the physical conditions of the
outflow gas can be particularly well constrained using the thousands
of \ion{Fe}{2} absorption lines.
This paper is the third in a sequence of four papers reporting the
results of a comprehensive analysis of a sample of FeLoBAL
quasars. Paper I \citep{choi22} describes the sample of 50
low redshift ($0.66 \leq z \leq 1.63$) FeLoBALQs and the {\it
SimBAL} spectral synthesis
analysis of their absorption lines. That work represents an increase
by a factor of five of FeLoBALQs with detailed photoionization
analysis. We measured the velocity and velocity width, the ionization
parameter, density, column density and covering fraction directly
from the spectra. We extracted the location of the outflow, as well
as the mass outflow rate and energetics of the outflow. We found that
FeLoBAL outflows cover a large range of outflow locations in a
quasar, spanning $\log R $ between 0 and 4.4 [pc]. While many of the
troughs were well described by a single outflow component
characterized by a single ionization parameter and density, about 20\%
of the objects showed evidence for multiple troughs, where the higher
velocity components generally had higher ionization parameters. Among
these objects, several special classes of BAL outflows were found.
Overlapping trough objects \citep[e.g.,][]{hall02} show broad
absorption troughs that blanket the near-UV shortward of \ion{Mg}{2}.
All of these objects were found to have compact outflows with $\log R
< 1$ [pc]. Among these objects, we discovered a new type of FeLoBALQ.
Dubbed ``loitering'' outflow FeLoBALQ, these objects generally have
narrow lines and low velocities, and are also compact with $\log
R < 1$ [pc]. They are distinguished from the other overlapping trough
objects not only by their velocity width but also their tendency to
have higher ionization parameter and higher density gas
\citep[Fig.\ 6][]{choi22}, which leads
to opacity from many high excitation states. The outflowing gas in
about half of these objects occults only the continuum emission
region, but not the broad emission lines. The remaining objects were
located farther from the quasar and generally did not have extreme
properties. We also found that there was a significant
correlation between the color of the quasar UV--NIR SED and the
outflow strength where the quasars with redder SEDs have more
energetic outflows.
Finally, we discussed how the potential
acceleration mechanisms and the origins of the FeLoBAL winds may
differ for outflows at different locations in the quasars.
Thirty of the fifty FeLoBALQs analyzed in \citet{choi22} have
sufficiently low redshift that the H$\beta$ emission-line region is
present at the red end of the SDSS spectra. Paper II \citet{leighly22}
describes the rest-frame optical emission-line analysis
of this subsample, along with a 132-object
comparison sample of non-BAL quasars. The principal result of that
paper is that the FeLoBALQs are divided into two classes based on
their emission-line properties, and their emission line properties are
distributed differently than those of the comparison sample.
Specifically, FeLoBALQs were characterized by either weak
\ion{Fe}{2} relative to H$\beta$ and strong [\ion{O}{3}], or strong
\ion{Fe}{2} relative to H$\beta$ and weak [\ion{O}{3}], and
intermediate values were avoided.
Further analysis revealed that the emission line properties reflect the
accretion rate relative to the Eddington limit. Therefore,
FeLoBAL quasars at low redshift are characterized by either high
Eddington ratios (typically greater than 1), a result that agrees with previous analysis of BAL
quasars \citep{yw03, boroson02, runnoe13}, or low Eddington ratios (typically less than 0.1),
a new result, but are uncommon at the intermediate Eddington ratios that
are the most prevalent in the comparison sample. The fact that their
emission-line properties are different than unabsorbed quasars shows
that among low-redshift and low-luminosity objects, FeLoBAL and
non-BAL quasars do not have the same parent sample. This result is
different than has been recently reported for high-ionization broad
absorption line (HiBAL) quasars which have higher luminosities
\citep{rankin20}.
This paper combines the {\it SimBAL} results from \citet{choi22} and
emission-line analysis results of the 30-object subsample from
\citet{leighly22} to search for relationships between the
properties of the outflowing gas and the properties of the central
engine. \citet{choi22} demonstrated that the BAL outflow
velocity is related to the bolometric luminosity in the 50-object
sample, as has been found previously
\citep[e.g.,][]{laor02,ganguly07,fiore17}. However, {\it SimBAL}
delivers
quite a bit more information characterizing the outflow than typical
BAL analyses, including the parameters describing the ionization
state, density, column density, and covering fraction, but also the
location of the outflow. We therefore take the first steps in
tackling the question posed above and examine how the location,
geometry, confinement and other properties of the outflowing gas
depend on the global quasar properties such as the luminosity and
Eddington ratio.
The final paper in the series, Paper IV (Leighly et al.\ in prep.),
includes an analysis of the the broad-band optical/IR properties and
discusses the potential implications for quasar evolution scenarios.
This paper is organized as follows. In \S\ref{data} we briefly
describe the data extracted from \citet{choi22} and \citet{leighly22}.
We principally focused on the 30-object $z<1$ subsample
for which the H$\beta$ region is available and the analysis described
in \citet{leighly22}, although we also explored the volume
filling factor in the full sample. In \S\ref{dist_comp}, we used the
$E1$ parameter defined in \citet{leighly22} to divide the
FeLoBAL quasars into high and low accretion rate objects and then
compared the {\it SimBAL} properties of the two classes. We also
correlated the {\it SimBAL} parameters with one another, and with the
optical emission-line and global parameters. Finally, we investigated
the relationship between the properties of the observed [\ion{O}{3}]
emission lines and the [\ion{O}{3}] emission predicted to originate in
the BAL gas. \S\ref{discussion} presents a summary of the results.
We used cosmological parameters $\Omega_\Lambda$=0.7,
$\Omega_M=0.3$, and $H_0=70\rm\, km\, s^{-1}\, Mpc^{-1}$, unless
otherwise specified.
\section{Data} \label{data}
The data used in this paper are described in detail in Papers I
\citep{choi22} and
II \citep{leighly22}l. The {\it SimBAL} model-fitting results
were drawn from Paper I, and the optical emission-line modeling and
global properties were taken from Paper II. Those parameters are
described briefly in the next sections.
\subsection{{\it SimBAL} Parameters} \label{simbal}
The {\it SimBAL} model fits for the sample are given in
\citet{choi22}, and the details can be found in that paper
(Tables 2 and 3). We extracted the following parameters from those
results: the ionization parameter $\log U$, the gas density $\log n \, \rm
[cm^{-2}]$, the broad absorption line velocity $V_\mathrm{off}\rm \,
(km\, s^{-1})$, the broad absorption line velocity width
$V_{width}\rm \, (km\, s^{-1})$\footnote{The representative offset
velocity is the median value of the MCMC chain after weighting by
the true opacity. The true opacity is distinguished from
the apparent opacity in that it takes into account partial
covering of continuum and emission-line emitting region. The
apparent opacity is extracted directly from the spectrum.; see \citet{choi22} \S~4.1.3 for the
definition. The minimum and maximum velocities and the velocity
width were estimated from the
90\% transmittance level from the model \ion{Mg}{2}$\lambda 2796$
velocity profile.}, the total column density integrated over
the BAL feature $\log N_{H}\, \rm [cm^{-2}]$,
the covering fraction parameter $\log a$, and the radius of the
outflow $\log R$ [pc].
Several additional and derived parameters were also produced that are
not reported in tables in \citet{choi22}. These include
the largest and smallest outflow velocities $V_{max}$, $V_{min}$,
force multiplier (the ratio of the total opacity to the electron
scattering opacity; FM), the thickness of the outflow $\Delta R$ [pc],
the filling factor $\log \Delta R/R$, the net outflow rate $\log
\dot{M}\rm \, [M_{\odot}\, yr^{-1}]$ (per component), the net
kinetic luminosity $\log L_\mathrm{KE}\, \rm [erg\, s^{-1}]$, and the
ratio of the net kinetic luminosity to the bolometric luminosity. The
thickness of the outflow $\Delta R$ is the ratio
of the total column density and the density. Making the simple
assumption that the azimuthal size of the outflowing gas is comparable
to the thickness of the gas (i.e., the gas is distributed into
individual clouds and the clouds are approximately spherical), and
using the radius of the accretion
disk at 2800\AA\/ described in \citet{leighly22}, we
computed the log of the number of clouds required to cover the
continuum emission region (see \S\ref{covfrac} for more details).
Finally, we used the parameters from the {\it SimBAL} best fitting
solution to compute the luminosity of the predicted [\ion{O}{3}]
emission line assuming an emission-line global covering fraction of $0.1$
(discussed in \S~\ref{oiiiemission}). We also computed a parameter
called the covering fraction correction, described in that section,
which parameterizes the comparison of the predicted [\ion{O}{3}]
luminosity from the wind with the observed [\ion{O}{3}] luminosity.
\subsection{Optical and Global Parameters} \label{opt_glob}
The following optical emission-line parameters were taken from Table 1
of \citet{leighly22}. We used the H$\beta$ FWHM and
equivalent width to parameterize the H$\beta$ line. \citet{leighly22}
also defined a
parameter called the H$\beta$ deviation that measures the
systematically broader H$\beta$ FWHM observed among the FeLoBAL
quasars compared with unabsorbed quasars. \ion{Fe}{2} was
parameterized using R$_\mathrm{FeII}$ defined as the ratio of the
\ion{Fe}{2} equivalent width in the range 4434--4684 \AA\/ to the broad
H$\beta$ equivalent width \citep[e.g.,][]{shen_ho_14}. The
[\ion{O}{3}] emission line was parameterized using the equivalent
width and luminosity, along with profile parameters $v_{50}$ and
$w_{80}$ defined according to the prescription of
\citet{zg14}. Briefly, from the normalized cumulative function of the
broad [\ion{O}{3}] model profile, the velocities at 0.1, 0.5, and 0.9
were identified. The velocity at 0.5 is assigned to $v_{50}$, and
$w_{80}$ is the difference between the velocities at 0.1 and 0.9.
\citet{leighly22} also fit the FeLoBAL objects and the unabsorbed
objects with eigenvectors created from the continuum-subtracted
spectra of the unabsorbed objects \citep[\S2.3,][]{leighly22}.
The fit coefficients for the first
four eigenvectors (SPCA1--4) serve as a parameterization of the
spectra. The first eigenvector displays the relationship between
the strength of \ion{Fe}{2} and [\ion{O}{3}] that is commonly found
\citep[e.g.,][]{grupe04, ludwig09, wolf20}, and none of the other
ones display any particular anomalies. We estimated the
bolometric luminosity using the rest-frame
flux density at 3 microns and a bolometric correction of 8.59
\citep{gallagher07}.
BAL quasars tend to be reddened
\citep[e.g.,][]{krawczyk15}, and evidence for reddening is present
in this sample \citep[$0\lesssim E(\bv)\lesssim0.5$, Figure~20 in][]{choi22,leighly_prep_paper4}.
Therefore, we used the 3-micron luminosity
density as representative, rather than the luminosity in the
optical or UV.
The black hole mass and Eddington ratio were
computed using standard methods and as described in \citet[][\S
2.1]{leighly22}, and the calculation of the location of the
2800\AA\/ emission from the accretion disk follows the method used in
\citet{leighly19}.
Finally, we continue to use the $E1$ parameter, which was defined in
\S 3.1 of \citet{leighly22}. This parameter is a function
of the measured values of R$_\mathrm{FeII}$ and the [\ion{O}{3}]
equivalent width.
As described in \S 3.1 of \citet{leighly22},
we normalized and scaled R$_\mathrm{FeII}$ and the
[\ion{O}{3}] equivalent width of the 132-object comparison sample,
and then derived the bisector line.
We performed a coordinate rotation so that $E1$ is a parameter
that lies along the bisector. Our $E1$ parameter is therefore
related to the \citet{bg92} Eigenvector 1. \citet{bg92} performed a
principal components analysis of emission line properties in the
vicinity of H$\beta$ and found that the variance is dominated by an
anticorrelation between R$_\mathrm{FeII}$ and the
[\ion{O}{3}] equivalent width. Moreover, Eigenvector 1 has been
shown to dominate the variance in quasar emission-line properties
\citep[e.g.,][]{francis92,bg92,brotherton94,corbin96,wills99,sulentic00,
grupe04,yip04,wang06,ludwig09,shen16}.
$E1$ is very strongly correlated with SPCA1
($p=6.5\times 10^{-15}$ and $p=10^{-50}$ for the FeLoBALQs and
unabsorbed comparison sample that was analyzed in parallel,
respectively). It is also correlated
with the Eddington ratio ($p=1.2\times 10^{-5}$ and $p=9.3\times
10^{-7}$ for the FeLoBALQs and unabsorbed comparison sample,
respectively).
Here, $p$ is a measure of the statistical
significance of the correlation. More specifically, it is a
probability that the observed correlation could have been produced
from draws from two uncorrelated samples \citep[e.g.,][]{bevington}.
A low (negative) value of the $E1$ parameter corresponds to a low
accretion rate, while a high (positive) value of the $E1$ parameter
corresponds to a high accretion rate. The 90\% range of $\log
L_\mathrm{bol}/L_\mathrm{Edd}$ among the sample is -1.4 to 0.84
\citep[Fig.\ 9][]{leighly22}. As discussed in \S 3.3 of
\citet{leighly22}, the apparent bimodal
distribution of FeLoBALQs in the $E1$ shows that there are two
types of FeLoBALQ: objects with $E1<0$ are characterized by low
Eddington ratio, and objects with $E1>0$ are characterized by high
Eddington ratio. We divided the FeLoBALQs into two groups based on
this parameter, where the dividing line $E1=0$ corresponds to $\log
L_\mathrm{Bol}/L_\mathrm{Edd}=-0.5$, i.e., an Eddington ratio of about
0.3. Throughout this paper, we use a consistent coloring scheme to
denote $E1$: red (blue) corresponds to $E1<0$ and low Eddington ratio
($E1>0$ and high Eddington ratio).
\section{Distributions and Correlations Between Optical and BAL Outflow Parameters}\label{dist_comp}
\subsection{Distributions}\label{distributions}
\citet{leighly22} presented comparisons of the distributions
of the emission-line and derived properties of the FeLoBAL quasars
with those of the comparison sample of unabsorbed quasars using
cumulative distribution plots. We applied the two-sample
Kolmogorov-Smirnov (KS) test, and the two-sample Anderson-Darling (AD)
test.
The KS test reliably tests the difference
between two distributions when the difference is large at the median
values, while the AD test is more reliable if the differences lie
toward the maximum or minimum values (i.e., the median can be
the same, and the distributions different at larger and smaller
values)\footnote{E.g.,
https://asaip.psu.edu/articles/beware-the-kolmogorov-smirnov-test/}.
We also compared the samples divided by the sign of the $E1$
parameter. Note that the values of the $E1$ parameter and plots of the
spectral model fits are given in \citet{leighly22}. In this paper,
we present the comparisons of the {\it SimBAL} parameters for the
FeLoBAL quasars segregated by $E1$ parameter (Table~\ref{tab_distributions}).
\startlongtable
\begin{deluxetable}{lCC}
\tablecaption{{\it SimBAL} $E1<0$ versus $E1>0$ Parameter Comparisons\label{tab_distributions}}
\tablehead{
\colhead{Parameter Name} & \colhead{KS\tablenotemark{a}} &
\colhead{AD\tablenotemark{b}} \\
& Statistic / Probability & Statistic / Probability \\
}
\startdata
$\log U$ & 0.21 / 0.28 & 0.64 / 0.18 \\
$\log n$ [cm$^{-3}$] & 0.24 / 0.57 & -0.07 / >0.25 \\
V$_\mathrm{offset}$ (km s$^{-1}$) & 0.50 / 0.016 & 4.4 / 5.9\times10^{-3} \\
V$_\mathrm{max}$ (km s$^{-1}$) & 0.50 / 0.014 & 4.2 / 7.0\times 10^{-3} \\
V$_\mathrm{min}$ (km s$^{-1}$) & 0.55 / 4.7\times
10^{-3} & 5.0 / 3.6\times 10^{-3} \\
V$_\mathrm{width}$ (km s$^{-1}$) & 0.30 / 0.33 & 0.67 /
0.17 \\
$\log a$ & 0.36 / 0.14 & 1.1 / 0.11 \\
Net $\log$ N$_\mathrm{H}$ [cm$^{-2}$] & 0.33 / 0.21 &
0.094 / > 0.25 \\
$\log$ Force Multiplier & 0.35 / 0.17 & 0.32 / >0.25 \\
$\log$ R [pc] & 0.26 / 0.45 & -0.10 / >0.25 \\
$\Delta$ R [pc] & 0.46 / 0.032 & 3.4 / 0.013 \\
$\log$ Volume Filling Factor & 0.51 / 9.8\times 10^{-3} &
5.7 / 2.0\times 10^{-3} \\
$\log$ Number of Clouds & 0.46 / 0.032 & 4.7 / 4.6\times 10^{-3} \\
Net \.M (M$_\odot$ yr$^{-1}$) & 0.31 / 0.28 &
0.084 / > 0.25 \\
L$_\mathrm{KE}$ [erg s$^{-1}$] & 0.42 / 0.10 & 1.4 / 0.09 \\
L$_\mathrm{KE}$/L$_\mathrm{Bol}$ & 0.30 / 0.41 & 0.37 / 0.24 \\
Predicted [OIII] Luminosity & 0.26 / 0.49 & 0.56 / 0.19 \\
Covering Fraction Correction & 0.58 / 2.1\times 10^{-3} & 9.3 / < 0.001 \\
\enddata
\tablenotetext{a}{The Kolmogorov-Smirnov Two-sample test. Each entry
has two numbers: the first is the value of the statistic, and the
second is the probability $p$ that the two samples arise from the same
parent sample. Bold type indicates entries that yield $p<0.05$.}
\tablenotetext{b}{The Anderson-Darling Two-sample test. Each entry
has two numbers: the first is the value of the statistic, and the
second is the probability $p$ that the two samples arise from the same
parent sample. Note that the implementation
used does not compute a probability larger than 0.25 or smaller than
0.001. Bold type indicates entries that yield $p<0.05$.}
\end{deluxetable}
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=5.5truein]{simbal_distributions_new_4_new.pdf}
\caption{The cumulative distributions of seventeen of the {\it SimBAL}
parameters segregated by their $E1$ parameter value. Distributions
that are significantly different ($p<0.05$;
Table~\ref{tab_distributions}) are shown in dark red and dark blue,
while distributions that are not significantly different are
shown in a pale hue of the same color. All of the
outflow velocity parameters ($V_\mathrm{off}$, $V_\mathrm{max}$,
$V_\mathrm{min}$) are shown in the top-right panel. Of the {\it
SimBAL}-related parameters, those three, the thickness of the
absorber, the log filling factor, the number of clouds covering
the continuum emission-line region, and the covering fraction
correction (discussed in \S~\ref{oiiiemission})
show statistically different distributions between the $E1>0$ and
$E1<0$ subsets. \label{simbal_dist}}
\end{center}
\end{figure*}
The results for the {\it SimBAL} parameters are shown in
Fig.~\ref{simbal_dist}, and the statistics are given in
Table~\ref{tab_distributions}. The
parameters that exhibit statistically significant differences between
the $E1>0$ and $E1<0$ groups are all three velocity outflow
parameters ($V_\mathrm{off}$, $V_\mathrm{max}$, and
$V_\mathrm{min}$), the thickness of the outflow $\Delta R$, the log
filling factor, the log of the number of clouds covering the continuum
emitting source, and the covering fraction correction factor. The
$E1>0$ objects show systematically larger opacity-weighted outflow
velocities than the $E1<0$ objects, with median values $-1520\rm \,
km\, s^{-1}$ and $-500\rm \, km\, s^{-1}$, respectively. For the
$E1<0$ objects, the outflows are thicker, have larger filling
fractions, and fewer\footnote{A median number of
clouds less than zero might be interpreted as an approximately
continuous outflow. Alternatively, a single cloud, if in the line
of sight, would completely cover the continuum source.} are
required cover the continuum source (median
$\log \Delta R= -1.8$ [pc], $\log$ volume filling fraction $=-3.8$, and
$\log$ number of clouds $=-0.54$). The
$E1>0$ objects have
thinner outflows, smaller volume filling fractions, and more clouds
are necessary to cover the continuum emission ($\Delta R= -2.4$ [pc],
$\log$ volume filling fraction $=-5.6$, and $\log$ number of
clouds $=0.34$). These properties are discussed further in
\S~\ref{covfrac}. The covering fraction correction factor is lower
for the $E1<0$ than for the $E1>0$ objects (median values of 0.6 and
1.5, respectively); this parameter was defined in \S\ref{simbal} and
is discussed in detail in \S~\ref{oiiiemission}.
The ionization parameter, $\log a$, and the force multiplier show
suggestions of differences. Specifically, while the highest
ionization parameter represented by $E1>0$ objects is $\log U=-0.5$,
25\% of $E1<0$ objects have ionization parameters larger than that
value (Fig.~\ref{simbal_dist}). Likewise, lower values of force
multiplier are dominated by $E1<0$ objects (Fig.~\ref{simbal_dist}). In
addition, the kinetic luminosity $L_\mathrm{KE}$ and the ratio of the
kinetic to bolometric luminosity are consistently lower for the $E1<0$
objects (Fig.~\ref{simbal_dist}).
\subsection{Correlations}\label{correlations}
In \citet{leighly22}, we examined the relationships among
the optical and derived parameters such as the bolometric luminosity. In this
paper, we compare those parameters with the {\it SimBAL} parameters,
both for the full sample, and for the sample segregated by $E1$
parameter. We used the Spearman-rank correlation for our comparisons.
We first correlated the {\it SimBAL} parameters with one another. A
correlation analysis of the {\it SimBAL} results for the full
50-object sample is given \S~6.1 of \citet{choi22}. Here
we considered only the low-redshift subsample. The results are shown
in Fig.~\ref{correlation_simbal}. The plots represent the log of
the $p$ value for the correlation, where the sign of the value gives the
sense of the correlation. That is, a large negative value implies a
highly significant anticorrelation.
Parameter uncertainties were propagated through the correlations
using a Monte Carlo scheme. We made 10,000 normally distributed draws
of each parameter, where the distribution was stretched to the size of
the error bar. Asymmetrical errors were accounted for by using a
split-normal distribution (i.e., stretching the positive draws
according to the positive error, and the negative draws according to
the negative error). We chose $p<0.05$ as our threshold for
significance. In most cases, taking the errors into account did not
dramatically change the significance of a correlation, if present.
There are several extremely strong correlations among the {\it
SimBAL} parameters (Fig.~\ref{correlation_simbal}) that were also
observed for the full sample \citet{choi22}. Since we
discuss only FeLoBALQs in this paper, most gas columns extend beyond
the hydrogen ionization front\footnote{The hydrogen ionization front is
the location in a slab of photoionized gas where the
hydrogen-ionizing photon flux is exhausted. In the context of
\ion{H}{2} regions, it is the location of the Str\"omgren sphere.} in
order to include sufficient Fe$^+$ ions to create an observable
absorption line.
The thickness of the \ion{H}{2} region increases with ionization
parameter, a fact that explains the strong correlation between the
column density and the ionization parameter. The net mass outflow
rate is a function of the velocity, explaining the strong correlation
between those two parameters. The force multiplier depends inversely
on the ionization parameter.
We also correlated the {\it SimBAL} parameters for the $E1$-divided
samples (Fig.~\ref{correlation_simbal}). As noted above, $E1$ is
related to the \citet{bg92} Eigenvector 1, which dominates the
variance in quasar properties. The motivation for looking for
correlations among the $E1$-divided samples is that by removing that
dominant dependence, more subtle parameter
dependencies may be revealed.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=3.00truein]{correlation_simbal_jan9_new.pdf}
\caption{The results of the Spearman rank correlation analysis for the
eighteen {\it SimBAL} parameters. The stars show the results for a
Monte Carlo
scheme to estimate the effects of the errors; see \citet{leighly22}
for details. The top plot shows the results for the
whole sample, while the middle and bottom plots show the results
divided by $E1$ parameter. Of particular interest is the weak
but significant anticorrelation between the velocity parameters and
location of the outflow for $E1<0$ and the correlation for the same
parameters for $E1>0$. \label{correlation_simbal}}
\end{center}
\end{figure*}
Finally, Fig.~\ref{corr_optical_simbal} shows the correlations between
the seventeen optical and global parameters and the eighteen {\it
SimBAL} parameters. A special technique was used to handle
these data because there are thirty objects and 36 outflow
components. Five of the objects showed multiple outflow
components\footnote{SDSS~J025858.17$-$002827.0,
SDSS~J103903.03$+$395445.8, SDSS~J104459.60$+$365605.1,
SDSS~J112526.12+002901.3, and SDSS~J144800.15+404311.7.
SDSS~J144800.15+404311.7 has three outflow components; the other
objects have two.}; \citet[][\S6.2,]{choi22} discussed the
need for multiple outflow components in these objects.
We assumed that physically
one of the components is more representative than the other one
or two. For example, [\ion{O}{3}] may be produced by one
component in the outflow but not another. Likewise, the
outflow location may be correlated with the outflow velocity for
components that share some fundamental property (i.e., perhaps they
are the main outflow in the system), but other, subsidiary outflows
do not obey this trend. Therefore, we
computed correlations among all combinations and present the
statistically most significant one.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=2.75in]{optical_simbal_correlation_plots_new_jan9.pdf}
\caption{The results of the Spearman rank correlation analysis for the
17 optical emission line parameters and global properties and the 18
{\it SimBAL} absorption-line and derived properties. The
symbols have the same meaning as in
Fig.~\ref{correlation_simbal}. The top plot shows the results for
30-object low-redshift sample, while the middle
and bottom plots show the results for the $E1<0$ and $E1>0$
subsamples respectively. Among the optical parameters, $E1$ and
related parameters $R_\mathrm{FeII}$, [\ion{O}{3}] equivalent
width, SPCA1, $E1$, $L_\mathrm{Bol}$ and
$L_\mathrm{Bol}/L_\mathrm{Edd}$, are most strongly
correlated with the {\it SimBAL} parameters. Among the {\it
SimBAL} parameters, the offset velocities, and parameters
associated with the volume filling fraction are the most strongly
correlated with the optical parameters. \label{corr_optical_simbal}}
\end{center}
\end{figure*}
\section{Analysis}\label{analysis}
\subsection{The Location of the Outflow}\label{logr}
An interesting set of patterns observed among the {\it SimBAL}
parameters is shown in Fig.~\ref{logr_voffset}. We plot the velocity
of the outflow as a function of the log of the radius divided by the
dust sublimation radius \citep[$R_d=0.16 L_{45}^{1/2}\rm \,
pc$,][]{en16}; note that $R_\mathrm{d}$ is close to 1 parsec in
these samples. It is immediately apparent that among the compact
outflows $\log R/R_d < \sim 2$, the $E1>0$ objects have
systematically larger velocity outflows than the $E1<0$ objects.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.5truein]{apo_proposal_plot_new_2_separated.pdf}
\caption{The location of the outflow determined using
{\it SimBAL} modeling \citep{choi22} and normalized by the dust
sublimation radius
as a function of the outflow velocity. Negative velocities denote
outflows. The circles and squares show
the column-density weighted velocity, while the left and right
triangles show the maximum and minimum speeds, respectively. There
is an anticorrelation (correlation) between $\log R/R_\mathrm{d}$
and the outflow velocity for the $E1 <0$ ($E1>0$) FeLoBALQs.
These different behaviors imply a difference in the formation
and acceleration of outflows for the two classes of FeLoBAL
quasars. \label{logr_voffset}}
\end{center}
\end{figure*}
While there is no correlation between the location of the outflow
$\log R/R_\mathrm{d}$ and the outflow velocity for the sample as a
whole, we found a tentative or marginal correlation for the $E1>0$
subsample and an anticorrelation for the $E1<0$ subsample (p-values are reported later in the text).
$E1<0$ objects with outflows close to the central engine generally have no
net outflow velocity, while those located far from the central engine
show modest outflows. In contrast, some of the largest velocities in
the sample are found in $E1>0$ object outflows located close to the
central engine, while at larger distances, the velocities tend to
be lower.
The correlation between the outflow velocity and $\log R/R_\mathrm{d}$
observed among the $E1>0$ objects might
be expected from a high accretion rate quasar. The terminal velocity
for a radiatively driven outflow is larger at smaller radii because of
the higher available photon momentum, which
means that larger velocity at small radii would be needed for a
sustained (i.e., not failed) wind.
\citet{choi22} identified a new class of FeLoBAL quasars
called ``loitering'' outflows as objects that have $\log R < 1$ [pc] and
velocity offset of the excited state \ion{Fe}{2}$\lambda 2757$
$v_\mathrm{off,FeII excited} < 2000\rm \, km\, s^{-1}$
\citep[Fig.\ 18][]{choi22}. They
found that the loitering outflow objects had distinct photoionization
properties too: they have larger ionization parameters
and densities compared with the full sample.
\citet{leighly22} found that in the low-redshift subsample,
almost all loitering outflow objects had $E1<0$ and were
therefore categorized as low accretion rate objects.
We next try to understand the origin of the low velocities among
the loitering outflows. The high ionization parameter
\citep[$-2 < \log U < 0.5$, Fig.\ 6][]{choi22} and accompanying
large column density
\citep[$\log N_H > 21.5$ (cm$^{-2}$), Fig.\ 7][]{choi22}
yield a low force multiplier; basically, the slab has very large
column of gas where the illuminated face and a significant
fraction of the total column density is too ionized to contribute
much to resonance scattering \citep[e.g.,][]{arav94b}. The
$E1<0$ objects are characterized by a low Eddington ratio
($\log L_\mathrm{Bol}/L_\mathrm{Edd} < -0.5$), which means that the
radiative flux is small relative to the gravitational binding of the
black hole and therefore it is less able to accelerate the outflow
gas. Thus, the combination of the large outflow column and low
radiative flux compared with gravity may explain the low velocities
\citep[$V_\mathrm{off} < -2000\rm \, km\, s^{-1}$,
Fig.\ 18][]{choi22}.
However, not all $E1<0$ objects have loitering outflows; at larger
radii, near $\log R/R_\mathrm{d}=3$, the $E1<0$ objects merge with the
$E1>0$ objects, and outflows of both $E1$ groups have velocities near
$-1000\rm \, km\, s^{-1}$. The combination of near-zero velocity
for small $\log R/R_\mathrm{d} < 2$ and outflows for larger
$\log R/R_\mathrm{d} > 2$ results in the anticorrelation
between the outflow velocity and $\log R/R_\mathrm{d}$ among the
$E1<0$ objects shown in Fig.~\ref{logr_voffset}.
As seen in Fig.~\ref{simbal_dist}, it is clear that the $E1>0$
objects have systematically larger velocity outflows than the $E1<0$
objects; the probability that they are drawn from the same population
is less than 1.6\% (Table~\ref{tab_distributions}). However,
correlations between the outflow velocity and the location of the
outflow are barely or arguably not significant, and might be construed
to fall in the realm of p-hacking. For example, if we examine $\log
R$ as a function of V$_{off}$, then the $E1<0$ correlation is significant
($p=0.025$) but the $E1>0$ is not ($p=0.076$), while for $\log
R/R_d$, the $E1<0$ correlation is not significant ($p=0.052$) while
$E1>0$ is significant ($p=0.050$). On the other hand, these
correlations are principally driven by the differences in the velocity
distributions at $\log R < 2$ [pc], which are very clear from
Fig.~\ref{logr_voffset}. Regardless, these differences in the
behavior in outflow velocity among the $E1<0$ and $E1>0$ objects may
point to a difference in formation and acceleration of outflows among
those two classes of objects. In particular, if the $E1$ parameter is
considered to be a proxy for $L_\mathrm{Bol}/L_\mathrm{Edd}$, and
these differing behaviors may point to a difference in formation and
acceleration of outflows as a function of that parameter.
The relationship between potential acceleration mechanisms and the
origin and location of FeLoBAL winds is discussed extensively in
\citet{choi22} \S7.2.
Using the black hole masses estimated in \citet{leighly22}
we computed the Keplerian velocities at the location of the outflow.
We then computed the ratio of the Keplerian
velocity to the outflow velocity $V_\mathrm{off}$, as well as the
ratio to $V_\mathrm{min}$ and $V_\mathrm{max}$, and plot the results
in Fig.~\ref{vkep_rat}. This plot shows
several interesting features. Large values of
$V_\mathrm{Kep}/\lvert V_\mathrm{off} \rvert$ indicate absorption features
that have line-of-sight velocities
much lower than the local Keplerian velocity (e.g., SDSS~J1125+0029,
SDSS~J1321+5617). A number of $E1<0$ objects fall into this
category. Because the continuum emission region is so small
compared with the outflow location (\S\ref{volumn_filling}), this
phenomenon could be achieved if the outflow velocity vector lies
strictly along the line of sight to the nucleus, and the local
Keplerian motion is principally tangential.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.5truein]{vkep_ratio.pdf}
\caption{The ratio of the Keplerian velocity at the location of the
outflow to the absolute value of the outflow velocities. The stars
denote $V_\mathrm{off}$. The triangles show $V_\mathrm{min}$
and $V_\mathrm{max}$, and can be offset from the stars if the
trough includes both inflow and outflow. Very large values of this
ratio show absorption features that have velocities much lower
than the Keplerian velocity. \label{vkep_rat}}
\end{center}
\end{figure*}
On the other hand, very low values of
$V_\mathrm{Kep}/\lvert V_\mathrm{off} \rvert$ indicate objects with
outflow velocities very much larger than the local Keplerian velocity
(e.g., SDSS~J1039+3954, SDSS~J1044+3656). Small values
suggest that the outflow is kinematically decoupled from the
gravitational potential of the black hole. It is possible that these
small ratios signal a distinct acceleration mechanism for their
outflows, such as the ``cloud-crushing'' mechanism suggested by
\citet{fg12}, rather than radiative line driving or dust acceleration,
where the magnitude of those mechanisms scales with the location of the
outflow.
Finally, a number of the outflows show
$V_\mathrm{Kep}/\lvert V_\mathrm{off} \rvert$ commensurate with unity.
Most of the $E1>0$ objects fall into this category. Of particular
interest is SDSS~J1448+4043. \citet{choi22} found that the
{\it SimBAL} solution required three separate outflow components in
this object (\S~6.2 of that paper). The lowest velocity component
could be seen to be kinematically distinct, since it is narrow and
shows a prominent ground-state \ion{Mg}{1}$\lambda 2853$ line. The
other two features are kinematically blended in this
overlapping-trough object and were inferred to be distinct based on
their photoionization properties \citep[Fig.\ 11][]{choi22}. They
found that the outflows lie at dramatically different distances ($\log
R=0.84$, 2.05, and $>3.1$
[pc]), yet their outflow velocities are commensurate with Keplerian
velocity at those locations. These outflows could be
considered kinematically coupled in some way to the gravitational
potential of the black hole.
\subsection{Bolometric Luminosity}\label{lbol}
The plot illustrating the results of the correlation analysis
(Fig.~\ref{corr_optical_simbal}) reveals the most
frequently observed relationship among BAL quasar outflows:
the correlation of the luminosity and Eddington ratio with the outflow
speed ($p=5\times 10^{-4}$ and $p=1.2\times 10^{-3}$, respectively).
The relationship between the BAL outflow velocity and
Eddington ratio is shown in Fig.~\ref{lum_vel}. These relationships
have been previously reported for HiBAL quasars \citep{laor02,
ganguly07,gibson09}, and are generally found among objects with
outflows \citep[e.g.,][]{fiore17}. \citet{ganguly07} noted that the
terminal velocity of an outflow should scale with Eddington ratio as
$v_\mathrm{terminal} \propto (L_\mathrm{bol}/L_\mathrm{Edd})^{1/2}$
\citep{hamann98a, misawa07}; this dependence arises from the solution
of the conservation of momentum equation \citep[e.g.,][]{leighly09}
assuming that the outflow is accelerated by
radiation. Thus the outflow velocity is predicted to be
correlated with the luminosity relative to the Eddington value.
However, a linear relationship between these two quantities is not what
is observed; rather, there is usually an upper-limit envelope of
velocity as a function of luminosity or
$L_\mathrm{bol}/L_\mathrm{Edd}$. That is, at
any luminosity, there is a range of outflow velocities up to some
upper limit value, and those upper limit values are correlated with
luminosity. For example, see Fig.\ 6 in \citet{laor02} and Figs.\ 6
and 7 in \citet{ganguly07}. This upper-limit relationship is seen in
our data too (Fig.~\ref{lum_vel}, top).
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.5truein]{simbal_lum_vel_all_final_revised_new.pdf}
\caption{The BAL outflow velocity as a function of Eddington ratio for
the sample of low-redshift FeLoBAL quasars. {\it Top:} The sample
shows the $V\propto L_\mathrm{bol}/L_\mathrm{Edd}$ upper-limit
envelope behavior commonly found in BALQs \citep{laor02,
ganguly07}; see text for details. The scatter below the
envelope can be characterized by the parameter $V_\mathrm{ledd=1}$, the velocity
any object would have if $L_\mathrm{bol}/L_\mathrm{Edd}=1$ (see
text for details). {\it Bottom Left:} An anticorrelation
between the force multiplier and $V_\mathrm{ledd=1}$
indicates that the scatter in the $E1<0$ objects may be caused by
the scatter in $FM$. {\it Bottom Right:} A correlation of $R/R_S$ with $V_\mathrm{ledd=1}$
suggests that the scatter in the $E1>0$ objects may be caused by
the scatter in $R/R_S$. \label{lum_vel}}
\end{center}
\end{figure*}
Theoretically, the speed of a radiatively driven outflow should also
depend on two factors: 1.) the launch radius (because the flux
density of the radiation drops as $R^{-2}$) and 2.) the force
multiplier (because the ability of an outflow to use the photon
momentum depends on which ions are available that can scatter the
photons). Using the
momentum conservation equation it can be shown that $$v \sim FM^{1/2}
(L/L_\mathrm{Edd})^{1/2} (R_\mathrm{launch}/R_\mathrm{S})^{-1/2},$$
where $R_\mathrm{S}$ is the Schwarzschild radius. The launch radius is
difficult to determine, but we can estimate it by assuming that the
measured velocity is some representative fraction of the terminal
velocity. For a $\beta$ velocity law\footnote{A $\beta$ velocity
law is expressed as $v(r)=v_\infty (1-R_\mathrm{Launch}/r)^{\beta}$
where $v_\infty$ is the terminal velocity and $R_\mathrm{Launch}$
is the launch radius. It is often used for modeling winds from
hot stars.}, the launch radius is a
constant fraction of the radius at which the outflow is observed, i.e.,
$R_\mathrm{launch}/R_\mathrm{S}\approx R/R_\mathrm{S}$. While this
approximation would be inadequate to solve an equation of motion for a
single object \citep[e.g.,][]{choi20}, it may be sufficient to
compare a sample of objects. The force multiplier, defined to be the
ratio of the total opacity to the electron scattering opacity, is an
output of {\it Cloudy}; it includes both line and continuum opacity.
Armed with these relationships and approximation, we can investigate
whether the scatter in outflow velocity for a given
$L_\mathrm{bol}/L_\mathrm{Edd}$ could be a consequence of intrinsic
scatter, whether there are trends in either the launch radius or the
force multiplier, or whether both cases may apply. To put it another
way, if the velocity upper-limit envelope describes the behavior of an
optimal outflow, perhaps the objects below that optimal level may be
deficient in either their force multiplier or their launch radius.
We note that there is another factor that could be important: the
angle of the line of sight to the velocity vector. We do not discuss
that factor here, which means that there should be additional
intrinsic scatter associated with that parameter. In addition, dust
scattering may be important in accelerating quasar outflows
\citep[e.g.,][]{thompson15, ishibashi17}, and our use of the force
multiplier means that we are only considered continuum and resonance
scattering for the acceleration mechanism. Additional discussion
of the potential role of dust scattering in acceleration in FeLoBAL
quasar outflows can be found in \citet{choi22} \S7.2.
We investigated these questions by first defining a parameter $V_\mathrm{ledd=1}$
that describes how far below the optimal outflow velocity an object
lies on the Eddington ratio versus velocity plot shown in
Fig.~\ref{lum_vel}. That is, we solved $V_\mathrm{off}=V_\mathrm{ledd=1}
(L_\mathrm{bol}/L_\mathrm{Edd})^{1/2}$ for $V_\mathrm{ledd=1}$, where
$V_\mathrm{off}$ is plotted as the $y$ axis in the top panel of
Fig.~\ref{lum_vel}. Then, for any
particular object, $V_\mathrm{ledd=1}$ is the velocity that it would have if
$L_\mathrm{bol}/L_\mathrm{Edd}=1$. In essence, we have derived a
parameter that can be used to compare objects as though they have the
same $L_\mathrm{bol}/L_\mathrm{Edd}$ (similar to the concept of absolute
magnitude). Traces for representative values
of $V_\mathrm{ledd=1}$ are shown in Fig.~\ref{lum_vel}. We then
plotted this
parameter against the force multiplier and $R/R_S$ (Fig.~\ref{lum_vel}).
We first considered the force multiplier. We found that the force
multiplier is not correlated with $V_\mathrm{ledd=1}$ for the whole sample, but it
is marginally anticorrelated for the $E1<0$ objects ($r_s=-0.46$, $p=0.047$), and
correlated for the $E1>0$ objects ($r_s=0.62$, $p=6.7\times
10^{-3}$). Keeping in mind that $V_\mathrm{ledd=1}$ is
negative for outflows, this means that $E1<0$ objects with larger $FM$
have higher outflow velocities. We interpret this behavior to mean
that unfavorable $FM$ (too small) is responsible for non-optimal (i.e.,
below the envelope) outflows in the $E1<0$ objects. In contrast, the
correlation for the $E1>0$ objects means that the $E1>0$ objects with
low $FM$ have higher outflow velocities, opposite of the physical
expectation. We interpret this to mean that $E1>0$ objects produce
their outflows {\it despite} non-optimal $FM$ values, and therefore
another factor is causing the below-envelope scatter in the $E1>0$
objects.
Turning to $R/R_S$, we found that, while there is no correlation between
this parameter and $V_\mathrm{ledd=1}$ for the sample as a whole, there is an
anticorrelation for the $E1<0$ objects ($r_s=-0.52$, $p=0.023$), and a
correlation for the $E1>0$ objects ($r_s=0.51$,
$p=0.035$). Again keeping in mind that $V_\mathrm{ledd=1}$ is negative for
outflows, the correlation between $R/R_S$ and $V_\mathrm{ledd=1}$ means that objects
with smaller $R/R_S$ have larger outflow speeds, as predicted by
the solution to the momentum conservation equation above. This
correlation suggests that the below-envelope scatter among the $E1>0$
objects is caused by unfavorable (too large) $R/R_S$
(Fig.~\ref{lum_vel}, bottom panel). The anticorrelation seen among
the $E1<0$ objects can be interpreted as implying that $R/R_S$ is less
important in determining their outflow velocity.
Finally, we can compare the $V_\mathrm{ledd=1}$ values for the $E1<0$
and the $E1>0$ objects. It turns out that the $V_\mathrm{ledd=1}$
distributions for the two classes are statistically indistinguishable.
That is, both types of objects lie along the same set of
$V_\mathrm{ledd=1}$ traces in Fig.~\ref{lum_vel}. The $E1<0$ objects
have both systematically lower outflow velocities
(Fig.~\ref{simbal_dist}) and systematically lower
$L_\mathrm{bol}/L_\mathrm{Edd}$ \citep[Fig.~7,][]{leighly22}.
This
result implies that $E1>0$ objects reach a larger velocity because of
their larger $L_\mathrm{bol}/L_\mathrm{Edd}$, i.e.,
$L_\mathrm{bol}/L_\mathrm{Edd}$ is primary for both classes of
objects.
We have shown that in the low-redshift sub-sample of FeLoBAL quasars
the outflow velocity depends on the $L_\mathrm{bol}/L_\mathrm{Edd}$.
This is not a new result; it has been seen before in other samples
\citep[e.g.,][]{ganguly07}. Such a result is relatively simple to
extract from any set of BAL quasars, depending as it does only on
estimation of the outflow velocity (for example, from the \ion{C}{4}
trough), and an estimate of the Eddington luminosity. That requires
an estimate of the black hole mass, which is arguably most reliably
extracted from H$\beta$ but can also be estimated using \ion{Mg}{2}
and \ion{C}{4}, in principal, although difficult in FeLoBAL quasars
due to the heavy absorption. The difference in our analysis is that
because {\it SimBAL} yields the physical conditions of the outflow
(including $\log U$, $\log n$, $\log N_{H}$) we can investigate this
relationship in more detail and parse the dependence on the subsidiary
parameters: force multiplier and estimated launch radius. We
discovered a difference in velocity dependence on these two parameters
among the two accretion classes. So while
$L_\mathrm{bol}/L_\mathrm{Edd}$ is principally responsible for
determining the outflow velocity, the force multiplier (launch radius)
is responsible for producing the scatter in the velocity at a
particular value of $L_\mathrm{bol}/L_\mathrm{Edd}$ in the $E1 <0$
($E1>0$) objects. This result provides additional evidence for
differences in acceleration mechanism.
\subsection{Geometry Properties of the Outflow}\label{volumn_filling}
\subsubsection{The Full Sample}\label{full}
The volume filling factor ($\Delta R/R,\ \Delta R=N_H/n_H$) gives us
information about the physical size scales of the outflowing gas. It
is most directly interpreted as the fractional volume of space
occupied by the outflow. Typically, using the values for column
density, density, and radius derived using excited state absorption
lines, small log volume filling factors, mostly ranging
between $-6$ to $-4$ are found
\citep[e.g.,][]{korista08,moe09,dunn10}. The volume occupied by the
absorbing clouds ranges from 0.01\% to 1\%.
The volume filling
factor tells us how thin or extended in the radial
direction the BAL cloud structure is and provides us with
information about the BAL physical conditions. A small volume
filling factor ($\log \Delta R/R\sim -5$) for BALs may imply a
pancake- or shell-like geometry that is very thin in the radial
direction \citep[e.g.,][]{gabel06,hamann11,hamann13}. These BAL
absorbers with $\log \Delta R/R\lesssim-3$ may be composed of smaller
gas clouds \citep[e.g.,][]{waters19} that are potentially supported by
magnetic confinement in order to avoid dissipation
\citep[e.g.,][]{dekool95}. In contrast, \citet{murray97} proposed
that a continuous flow from the accretion disk is the origin of broad
emission lines and BAL
features; such a flow would have a volume filling factor of
1. We emphasize that our results are not consistent with
a direct observation of a disk wind because the size scales that we
measure are too large. The minimum distance of the outflow from the
central engine found in our sample is $R\sim 1$ pc in SDSS~J1125$+$0029,
whereas reasonable size scales for disk wind outflows should be
comparable to the size of the accretion disk ($R\ll 0.01$ pc).
That does not imply that disk winds do not exist but rather that we do
not find them to have rest-UV BAL outflow signatures.
This result is consistent with the literature; among the FeLoBAL quasars
previously subjected to detailed analysis, typical outflow distances
lie between 0.4 and 700 parsecs \citep[e.g.,][]{dekool01, dekool02a,
dekool02b, moe09, dunn10, aoki11, lucy14, shi16, hamann19b,
choi20}, i.e., no closer than the broad line region.
\begin{figure*}[t]
\includegraphics[width=.48\linewidth]{a_logdRR_markers_4_1.pdf}
\includegraphics[width=.51\linewidth]{logR_logdRR_markers_4_1.pdf}
\caption{The volume filling factor ($\log \Delta R/R$) as a function
of the ionization parameter ($\log U$) and the location of the
outflow ($\log R$). We found a wide range of $\log \Delta R/R$,
with the overlapping trough and loitering BALs having higher values
of $\log \Delta R/R$. {\it Left panel}: $\log \Delta R/R$ increases
with $\log U$ following the slope of $\sim1.5$. This tight
correlation is expected given the relationship between $\log U$ and
$\log N_H$ for FeLoBALs \citep[\S~5.3,][]{choi22}. One of the
main sources of the scatter along the $\log \Delta R/R$ can be
ascribed to the range of $\log n$ observed in the sample. {\it
Right panel}: We found a wider range of $\log \Delta R/R$
distribution for the FeLoBALs that are located close to the central
black holes. The green (dashed) and blue (dotted-dashed) diagonal
lines represent the locations of the constant physical thickness of
the BAL winds at $\log \Delta R=-5$ and 0 [pc], respectively. The
error bars show 2$\sigma$ uncertainties and the grey shaded bars
represent the range of the values among the tophat model bins for
each BAL.
\label{fig:logdRR}}
\end{figure*}
We calculated the volume filling factors for the full sample and
examined the dependence on BAL properties (Fig.~\ref{fig:logdRR}). The
strong correlation seen between $\log \Delta R/R$ and $\log U$ can be
explained by the mathematical relationship between the parameters as
follows. First, the BAL physical thickness ($\Delta R$) is
proportional to the hydrogen column density ($N_H$) which is also
proportional to the ionization parameter $U$ since $\log N_H-\log U$
is nearly constant in the FeLoBALQ sample, and the distance of the
outflow from the central SMBH ($R$) is inversely proportional to
$U^{1/2}$, both for a fixed density. Dividing the BAL thickness by
its distance from the center, we obtain the volume filling factor
$\Delta R/R\propto U^{1.5}$, and we find a slope of $\sim1.5$ in the
left panel of Figure~\ref{fig:logdRR}. We observe a scatter around
that line because of the range of $\log N_H-\log U$ and $\log n$ for
the FeLoBALs in our sample \citep[Fig.~4,][]{choi22}. There
is also a range in photoionizing flux $Q$ which we assume to be
proportional to $L_\mathrm{Bol}$. This parameter enters through
$U=Q/4\pi R^2 nc$. Therefore, larger values of $\log N_H-\log U$
(thicker outflows), smaller density, or smaller $\log L_{bol}$
correspond to a larger value of $\log \Delta R/R$.
The distribution of $\log \Delta R/R$ is not uniform
across $\log R$. At large radii, corresponding to $\log U\lesssim-1$,
the volume filling factors mostly range between $-6$ to $-4$. These
values are similar to those reported in the literature for samples of
high-ionization BAL quasars
\citep[e.g.,][]{gabel06,hamann11,hamann13}. In contrast, the outflows
that are found at $\log R\lesssim1$ have a very wide range of $\log
\Delta R/R$, ranging from $-6$ to nearly almost zero. These are
mostly the special types of BALs that were identified in
\citet{choi22}, including the overlapping trough and loitering BALs.
The analysis for the full sample shows significant differences in
$\log \Delta R/R$ as a function of radius. This result suggests that
BAL winds may favor different models at different radii
\citep[][Leighly et al.\ in prep.]{choi22}. The compact
winds at $\log R\lesssim1$ [pc] showed a wide range of $\log \Delta
R/R$ that agrees with the predictions of the various BAL physical
models that explain either thin shell-like outflows (small volume
filling factor) or stream-like outflows (large volume filling factor).
On the other hand, the properties of distant BAL winds only favor the
physical model with thin pancake-like BAL geometry. Specifically,
\citet{fg12} proposed that FeLoBALs with large $\log R\gtrsim3$ [pc]
and small $\log \Delta R/R\sim -5$ are formed by ``cloud crushing''
where the ambient ISM is shocked by the supersonic energy-conserving
quasar outflow and the FeLoBALs are formed in-situ at kiloparsec
scales rather than formed near the accretion disk. In addition, in
order for distant BALs to have large filling factors, the BAL clouds
would need to have large physical radial thicknesses proportional to
their distances from the central engine ($\Delta R\gtrsim10$
pc). Maintaining such large structure is physically challenging due to
cloud destructive processes \citep[e.g.,][]{proga15}.
\subsubsection{The $z<1$ FeLoBALQs and the {\it E1} Dependence}\label{covfrac}
In this section, we discuss the relationship between the BAL
outflow parameters involving the geometry of the outflow and the
optical-band emission line properties. We have emission-line
properties only for the 30-object $z<1$ subsample, so this
discussion only involves that subsample. In particular, we
investigated how the parameters that describe the geometrical
properties of the outflow dependon the $E1$ parameter, and by
extention, the accretion rate.
We first investigated the physical thickness of the gas $\Delta R$.
We showed in \S~\ref{distributions} that the $\Delta R$ parameter is
significantly different for the $E1<0$ and $E1>0$ subsamples
(Fig.~\ref{simbal_dist} and Table~\ref{tab_distributions}); the median
$\log$ thickness is about 1 dex larger for the $E1<0$ objects. This
result arises because although there is no statistical difference in
$\log U$ between the $E1<0$ and $E1>0$, there is a tendency for $E1<0$
objects to have larger $\log U$ and therefore thicker outflows. The
thickness is also anticorrelated with both $R_\mathrm{FeII}$ and the
E1 parameter (Fig.~\ref{corr_optical_simbal}) for the same reason.
We next considered the volume filling factor for the low-redshift
subsample. We found that $E1<0$ objects have significantly
larger volume filling factors than $E1>0$ objects because
of the significant difference in thickness $\log \Delta R$ but also in
the tendency for $E1<0$ objects to have larger $\log U$ and therefore
smaller $\log R$, i.e., to be located closer to the central engine.
Because we have black hole mass and accretion rate estimates
\citep{leighly22}, we can estimate the number of spherical clouds
required to completely cover the continuum emission region.
This
parameter is useful to visualize the BAL absorption region in
the quasar. The first ingredient in this computation is the size of
the emission region $R_{2800}$. We calculated the size of the
2800\AA\/ continuum emission region using the procedure described in
\S~6.1 of \citet{leighly19}. To summarize, we used a simple
sum-of-blackbodies accretion disk model \citep{fkr02} and assigned the
2800\AA\/ radius to be the location where the radially-weighted
brightness dropped by a factor of $e$ from the peak value. Among the
objects in this sample, this parameter spans a rather small
range of values with 90 percent of the objects having $-2.9 < \log
R_\mathrm{2800} < -2.3$ [pc] \citep[Fig.~8, ][]{leighly22}.
This is likely a consequence of the $T^4$ dependence of the accretion
disk.
The second consideration is the wide range of angular diameters that
the continuum emission region will subtend at the location of the
outflows. For example, the continuum emission region will subtend an
angular diameter that is 1000 times larger to a wind located at one
parsec than to one located at 1000 parsecs. Folding in the small
difference in size of the continuum emission region we found that the
largest angular diameter is presented to the higher velocity
component of SDSS~J1125+0029 at 18 arc minutes\footnote{For reference,
the angular diameter of the full moon is 31 arc minutes.}, and the
smallest is the higher velocity component of SDSS~J1044+3656 at
$4.5\times 10^{-4}$ arc minutes.
The final ingredient is the transverse size of the absorber. This
parameter cannot be measured directly from these data since absorption
is a line-of-sight measurement. Instead, we make the simplifying
order-of-magnitude assumption that the absorbing gas occurs in clouds
and the clouds are approximately spherical. Making this assumption
yields a transverse size that is equal to $\Delta R$, the thickness of
the absorbing gas.
Employing these three ingredients (the size of the continuum emission
region $R_\mathrm{2800}$, the angular diameter that continuum emission
region will subtend at the location of the outflow, and the transverse
size of the absorbing cloud), we can determine the number of clouds
required to cover the continuum emission region. For example, if the
angular size of the cloud from the perspective of an observer located
at the continuum emission region is the same as the angular size of
the continuum emission region at the location of the cloud, then only
one cloud is required to cover the continuum emission
region.\footnote{We note in passing that this is the same concept that
is applied to the measurement of the size of the quasar continuum
emission region using gravitational microlensing. The gravitational
lens caustics of a single star are very small, but if the continuum
emission region has a commensurate angular size (because the star is
located in the quasar host galaxy), its light can be
differentially magnified.} Conversely, if the angular size of the
cloud from the perspective of an observer located at the continuum
emission region is much smaller than the angular size of the continuum
emission region from the perspective of a viewer located at the
outflow, then many clouds are required. The number of clouds is the
ratio of the area of the continuum emission region and the transverse
area of the cloud. For this order-of-magnitude computation, we assume
that the continuum emission region is viewed face on.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=4.5truein]{numclouds_logr_new.pdf}
\caption{Relationship between the number of clouds required to occult
the 2800\AA\/ continuum emission region and the location of the
outflow normalized by the dust sublimation radius. The size of the
points is scaled with the angle subtended by the continuum emission
region as viewed from the location of the outflow. Compact outflows
(those with $\log R/R_\mathrm{D} < 2$ are
segregated by the $E1$ parameter, with $E1<0$ objects having
macroscopic clouds and $E1>0$ objects requiring 100s--1000s of
cloudlets to cover the source. \label{numclouds_logr}}
\end{center}
\end{figure*}
The resulting number of clouds spans a huge range for the sample.
Ninety percent of the values fall between 0.016 (which means that the
inferred size of the cloud is about 60 times larger than the emission
region) to $\sim 1000$, a factor of more than 60,000. Moreover, there
is a significant difference between the $E1<0$ and $E1>0$ objects
(Fig.~\ref{simbal_dist}, Table~\ref{tab_distributions}); the median
values of the log of the number of clouds are 0.3 and 2.2,
respectively.
The $E1>0$ outflows located close to the central
engine (lower right in Fig.~\ref{numclouds_logr}) require a large
number of clouds (100s to 1000s) to cover the continuum
emission region. Their outflows are characterized by a lower
ionization parameter (typically less than $-1.5$) compared with the
$E1<0$ objects ($\log U\sim 0$), and therefore physical thickness of
the outflow is smaller in these objects. Physically, this scenario
suggests that the outflow is a fine mist of cloudlets.
Objects in which the number of clouds is less than 1 are split
between those at large distances from the central engine ($\log
R/R_\mathrm{D} > 2$) and those at small distances ($\log
R/R_\mathrm{D} < 2$). In these objects, the angular size of the BAL
cloud structure is comparable or larger than the projected angular
size of the continuum emission region. One possibility is that our
simplifying estimate that the clouds are approximately spherical is
wrong. For example, the number of clouds required to cover the
continuum emission region could be much larger if each cloud were long
and needle-like, with the long axis pointed toward the central engine.
This scenario might be somewhat reasonable physically if the clouds
are confined magnetically along field lines that are bent radially by
radiation pressure \citep[e.g.,][]{dekool95} or sheared by radiation
pressure.
Another possibility is that the nature of the partial covering is
different in some of the objects. The {\it SimBAL} model includes a
power-law partial covering parameter $\log a$ that may parameterize a
mist of clouds uniformly covering the continuum emission region
\citep[see ][and references therein for discussion and
visualization]{leighly19}. The presence of power-law partial
covering does not preclude the presence of step-function partial
covering as well. The step-function partial covering can be
understood as a partial occultation of the continuum emission region.
There is some evidence for the presence of step-function partial
covering in objects lying in the lower left corner of
Fig.~\ref{numclouds_logr}. In several of these objects, the {\it
SimBAL} models required that a portion of the continuum and/or
emission lines be unabsorbed by the outflow \citep[\S6.4,
Fig.\ 17][]{choi22}. Physically, this result might be expected when
the continuum emission region has a large angular size from the
perspective of the absorber, i.e., an absorber with small $\log
R/R_\mathrm{d}$. It may mean that the outflow is not a mist of clouds
uniformly covering the source, but a distribution of nearly continuous
gas.
\subsection{[\ion{O}{3}] Emission from the BAL Gas}\label{oiiiemission}
Outflows in quasars are also seen in ionized emission lines. For
example, kiloparsec-scale outflows observed in emission lines such as
[\ion{O}{3}] are known to be common among luminous AGN
\citep[e.g.,][]{harrison14,bischetti17,vayner21b}. More compact
ionized emission-line outflows have been resolved in nearby objects;
for example, ionized gas outflows have been found 0.1--3~kpc from the
nucleus \citep{revalski21}. It is possible that ionized emission-line
outflows and BAL outflows are related. The outflowing broad
absorption line gas is photoionized, and therefore it must produce
line emission. In particular, [\ion{O}{3}]$\lambda 5007$ line
emission is an important coolant in photoionized gas
\citep[e.g.,][]{of06}. It is possible that in some objects the same
gas produces absorption lines along the line of sight as well as
emission lines from all lines of sight.
There are several fundamental problems that make finding a connection
difficult. While absorption is a line-of-sight effect, an emission
line is an aggregate of many lines of sight, so that emission from gas
not associated with the BAL would be included in any observed line
emission. In other words, the covering fraction of the emitting gas
needs not be the same as that of the absorbing gas. [\ion{O}{3}]
emission is observed to have a very large range of equivalent widths
\citep[6--84
\AA\/,][]{shen11} potentially originating in a range of gas covering
fractions \citep{bl05b, ludwig09,stern12}. Moreover, line emissivity
depends on density squared below the critical density, i.e.,
$n_{cr}=6.8\times 10^5\rm \, cm^{-2}$ for [\ion{O}{3}]
$\lambda 5007$], \citep{of06}, and on the density above the critical
density. Thus, the line emission might not be seen against the
continuum if the density is too low. Finally, if the absorption lines are
broad, then the line emission may be distributed over a large range of
velocities, and the line may be too broad to be seen against the
continuum.
We do not have information about the extent of the line
emission for our objects. However, it is interesting to see if there
is a correspondence or relationship between the predicted [\ion{O}{3}]
emission from the BAL gas and the observed [\ion{O}{3}] emission.
\citet{xu20} tackled this problem using a sample of seven $z\sim 2$
quasars.
Those objects were chosen to have $r$-band magnitude
$\lesssim 18.8$ and deep \ion{Si}{4}$\lambda\lambda 1393.76, 1402.77$
troughs. They found evidence for a link between the [\ion{O}{3}]
emission and the BAL absorption. However, they
presented a conceptual error: they suggested that [\ion{O}{3}] emission is
suppressed at high densities. In fact, [\ion{O}{3}] emission always
increases with density. Rather than a decrease or suppression of
[\ion{O}{3}] at high densities, the property that decreases is the
ratio of the [\ion{O}{3}] emission with respect to lines with higher
critical densities. This physics is the basis of the
\ion{Si}{3}]$\lambda 1893$/\ion{C}{3}]$\lambda 1909$ density
diagnostic used in the near UV \citep[e.g.,][]{leighly04}.
We used {\it Cloudy} to predict the [\ion{O}{3}] line emission from the
outflowing gas in each of the 36 BAL components of the thirty $z<1$
objects. Each component is characterized by a single ionization
parameter and density, but is generally split into multiple bins with
different column densities \citep[][]{leighly18,choi22}. The
[\ion{O}{3}] emission was computed for
each bin and then summed. The local covering
fraction $\log a$ for the BAL outflows was not taken into account in
the computation of the [\ion{O}{3}] flux, since it is not clear how
it would manifest in the observations of line
emission. The luminosity of the [\ion{O}{3}] emission was then
computed for each component assuming a global covering
fraction of 10\% for the line-emitting gas.
We expect the luminosity of the predicted [\ion{O}{3}] could depend on
several parameters.
The [\ion{O}{3}] flux density should be larger for higher
densities. It should be larger for higher ionization parameters as a
consequence of the larger column density required to include the
hydrogen ionization front in FeLoBAL outflows. Both of these
conditions are met in outflows closer to the central engine. But the
volume included increases as $R^2$ for the absorbing material for
a fixed emission-line-region covering fraction.
Since $R^2\propto \frac{1}{nU}$, the effects cancel out, leaving the
{\it Cloudy} L$_\mathrm{[OIII]}$ uncorrelated with $n$, $U$, or $R$
(Fig.~\ref{correlation_simbal}).
The left panel of Fig.~\ref{lum_vs_lum} shows the {\it
Cloudy} predicted [\ion{O}{3}] luminosity as a function of the
observed [\ion{O}{3}] luminosity. All of the predicted line emission
from objects with multiple BAL outflow components are plotted with
respect to the single observed [\ion{O}{3}]
luminosity. There is clearly no relationship between these two
luminosities. Moreover, while both the observed and predicted
[\ion{O}{3}] luminosities each span about 1.7 dex, the ratio of the
two spans almost 2.7 dex. In other words, the difference between the
observed and predicted [\ion{O}{3}] emission is not subtle.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.5truein]{oiii_lum_vs_lum_both.pdf}
\caption{The inferred [\ion{O}{3}] luminosity from the {\it SimBAL}
models as a function of the
observed [\ion{O}{3}] luminosity. The several objects with more
than one outflow component are plotted more than once. The marker
color denotes the corresponding E1 parameter and the size of the
marker corresponds to the location $\log R$ parameter. {\it
Left:} The predicted values are generally larger than the
observed values, potentially implying a smaller emission-line gas
covering fraction than the assumed 10\%. {\it Right:} The same as
the left plot, with covering fraction correction determined by
the regression analysis (see text). The data were plotted using
regression results for each of the 48 combinations. \label{lum_vs_lum}}
\end{center}
\end{figure*}
It is possible that the observed and predicted [\ion{O}{3}]
luminosities could be reconciled if the global covering fraction of
outflowing line-emitting gas were not constant for all outflows or for
all emission-line regions. For example, the location of the outflow in
this sample spans three orders of magnitude, and it is conceivable
that the emission-line covering fraction is different in the vicinity
of the torus ($R\sim 1$ pc) compared with on galaxy scales ($\log R
\sim 3 $ [pc]).
We parameterized the profound difference between the observed and
predicted [\ion{O}{3}] luminosities by defining the covering fraction
correction factor. The covering fraction correction factor is the
difference between the log of the {\it Cloudy} predicted [\ion{O}{3}]
luminosity and the log of the observed [\ion{O}{3}] luminosity. The
covering fraction correction factor measures how much larger
or smaller than the assumed value of 0.1 that the covering fraction
needs to be to reconcile the observed and predicted [\ion{O}{3}]
luminosities. In \S~\ref{distributions} we showed that the covering
fraction correction factor is significantly different for $E1<0$
versus $E1>0$ objects. The median values were 0.6 (1.5) for the $E1<0$
($E1>0$) objects respectively, implying that, on average, the
emission-line region covering fraction needs to be 4 (32) times
smaller than 0.1.
We explored the
possibility that the covering fraction depends on other parameters
using a multiple regression analysis. We considered seven independent
variables, and our reasons for choosing these parameters follows. We
included $\log R$ for the reasons outlined
above. We also considered Seyfert type using the E1 parameter, and
the log of the estimated bolometric luminosity because there is
evidence for a reduction in [\ion{O}{3}] emission at larger
luminosities \citep{bl05b}. Objects with a high Eddington ratio might
produce more powerful winds and thereby evacuate a larger fraction of
their reservoir of gas, so we also consider $\log
L_\mathrm{Bol}/L_\mathrm{Edd}$.
There is evidence that the BAL partial covering fraction $\log a$ is
local \citep{leighly19}, but it is possible there are global trends as
well. In any particular outflow component, $\log a$ is a function of
velocity. For this experiment, we chose the
representative $\log a$ to be the one in the bin with the deepest
ground-state \ion{Fe}{2} absorption. If the velocity of the outflow
is very large, then the resulting line may be very broad and blend
with the continuum, so we also considered the opacity-weighted
$v_{off}$. Finally, the observed [\ion{O}{3}] velocity offset might
reveal a connection between the outflow and the observed emission.
The result forms a multiple regression problem with the seven
independent variables listed above. As in \S~\ref{correlations}, we
accounted for the
multiple components in five of the objects by running the regression
analysis for all 48 combinations. We used {\tt
mlinmix\_err} \citep{kelly07} which accounts for measurement errors
in both the dependent variable (the covering fraction) and the
independent variables (the design matrix) using the Pearson
correlation coefficient. To determine which
independent variables can best reproduce the variance in the covering
fraction, the multiple regression procedure was iterated, each time
removing the independent variable with the largest $p$ value until all
of the remaining variables showed a $p$ value no larger than the cutoff
which was chosen to be 0.025.
A statistically significant correlation was found between the E1
parameter and the covering fraction correction factor for all 48
combinations. The next most significant parameter was $\log R$ (39),
followed by the velocity offset of [\ion{O}{3}] (23). Since more than
half of the combinations found significant regression with the E1
parameter and $\log R$, we proceeded to extract the regression
parameters for these two parameters and all combinations.
The right panel of Fig.~\ref{lum_vs_lum} shows the results of the
regression analysis between the covering fraction correction factor
and the independent parameters E1 and $\log R$. The {\it Cloudy}
covering fraction was adjusted using the best-fitting regression
parameters. Points are shown for each of the
48 combinations. The relationship between the observed and {\it
SimBAL}-predicted [\ion{O}{3}] emission is now linear, although
considerable scatter remains.
We next examine how large the log covering fraction correction factors
are, and how they depend on the regression variables. The results
from the regression are seen in Fig.~\ref{cov_frac_corr}. The points
are plotted for all 48 combinations, and the grey shaded region shows
the 90\% confidence regions from the {\tt mlinmix\_err} procedure.
The left panel shows the results in three dimensions, while the
right two panels show the two dimensional projections for each of
the regression variables.
The assumed emission-line covering fraction was 0.1, so, for example,
a log covering fraction correction value of 1 would imply that the
covering fraction of 0.01 is needed to reconcile the observed and
predicted [\ion{O}{3}] values. Many of the $E1>0$ objects
have large covering fraction correction factors (1.5--2), which would
seem to imply that the emission-line gas covering fraction needs to be be very small
(0.001--0.003) in order to reconcile the observed and predicted
[\ion{O}{3}] emission.
In contrast, several of the loitering outflow
objects \citep[$E1<0$ with a low-velocity and compact
outflow][\S6.5, Fig.\ 18]{choi22} show very low covering fraction
correction factors near zero, which means that the
observed [\ion{O}{3}] emission is consistent with being produced in
the outflow. Fig.~\ref{oiii_tau_vel} compares the [\ion{O}{3}]
emission profile and the absorption opacity profiles \citep{choi22}
for the six objects with $\log R< 1$ and $\log$
covering fraction correction less than 0.5. In these objects, the
data are roughly consistent with the line emission and absorption
being produced in the same gas. The profiles are not identical, but there
are consistent trends: objects with broader [\ion{O}{3}] emission lines
show broader absorption profiles.
For example, in SDSS~J1128+0113, the [\ion{O}{3}] line has a velocity width of
$w_{80}=830\rm \, km\, s^{-1}$ \citep{leighly22} and the \ion{Mg}{2}
absorption line has a velocity width of $3000 \rm \, km\, s^{-1}$
\citep{choi22}. In contrast, in SDSS~J0916$+$4534, the [\ion{O}{3}]
line has a velocity width of
$w_{80}=460\rm \, km\, s^{-1}$ \citep{leighly22} and the \ion{Mg}{2}
absorption line has a velocity width of $500 \rm \, km\, s^{-1}$
\citep{choi22}. SDSS~J1321$+$5617 is particularly
interesting: both the [\ion{O}{3}] emission-line and the
absorption-line optical-depth profiles are narrow with a blue wing.
However, there is a flaw in this analysis. We showed in \S~\ref{logr}
that the BAL outflow velocity in some objects was much smaller than
the local Keplerian velocity (Fig.~\ref{vkep_rat}). The
[\ion{O}{3}] emission-line width in the same objects is also much less
than the Keplerian velocity, by factors of 5 to 24. While we could
explain the small outflow velocity if the orientation is directly
along the line of sight, the same argument does not work for the
emission lines, since they are composed of emission from all lines of
sight.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.5truein]{oiii_covering_fraction_corrections.pdf}
\caption{The covering fraction corrections inferred from the
regression analysis necessary to reconcile the observed [\ion{O}{3}]
emission with that predicted to be emitted by the absorption line gas.
The assumed emission-line covering fraction was 0.1, so a log covering
fraction correction value of 1 would imply that the covering
fraction of 0.01 would be needed to reconcile the values. The point
color represents $E1$ and the point size represents $\log R$. {\it
Left:} The multiparameter regression is a function of the E1
parameter and $\log R$ and is therefore most accurately represented
in 3D. The bowtie
surface shows the inferred errors from the {\tt mlinmix\_err}
procedure; the results from all 48 combinations have been
plotted. {\it Middle:} The results projected onto the $E1$ parameter
plane. {\it Right:} The results projected onto the $\log R$ plane.
\label{cov_frac_corr}}
\end{center}
\end{figure*}
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=4.5truein]{plot_oiii_vel_3.pdf}
\caption{Comparison of the [\ion{O}{3}]$\lambda 5008$ emission profile
with the absorption profile for objects with compact outflows
($\log R < 1$ [pc]) and low log covering fraction correction
factor ($<0.5$). {\it Top panel in each frame:} The
[\ion{O}{3}]$\lambda 5008$ profile from the normalized and
continuum-subtraction spectrum. {\it Bottom
panel in each frame:} The {\it SimBAL} derived opacity profiles
for ground-state \ion{Mg}{2}, ground-state \ion{Fe}{2}$\lambda
2883$, and excited state \ion{Fe}{2}$\lambda 2757$ taken from
\citet[Fig.\ 13][]{choi22}.
\label{oiii_tau_vel}}
\end{center}
\end{figure*}
The correlation analysis discussed in \S~\ref{correlations}
indicates significant correlations between the BAL outflow velocity
offset and the [\ion{O}{3}] velocity offset and width
(Fig.~\ref{corr_optical_simbal}). Fig.~\ref{bal_vel_vs} (left and
middle) explores these relationships. In $E1>0$
objects, the [\ion{O}{3}] emission line is sometimes very small and
difficult to discern amid the sometimes strong and broad \ion{Fe}{2}
emission; therefore, points with the smallest symbols are less robustly
measured. The correlation between the BAL outflow velocity and
[\ion{O}{3}] velocity offset ($p=1.9\times 10^{-4}$) and the
anticorrelation between BAL outflow velocity and [\ion{O}{3}] velocity
width ($p=2.7\times 10^{-4}$) are apparent.
\begin{figure*}[!t]
\epsscale{1.0}
\begin{center}
\includegraphics[width=6.9truein]{bal_velocity_offset_vs_new.pdf}
\caption{Comparison of the BAL velocity offset with the [\ion{O}{3}]
properties. Note that negative BAL velocities denote outflows. In
the left and middle panels, the squares show
the column-density weighted velocity, while the left and right
triangles show the maximum and minimum speeds, respectively. The
size of the square scales with the log of the [\ion{O}{3}]
equivalent width. {\it
Left:} There is a correlation between the [\ion{O}{3}] velocity
offset and the BAL velocity offset ($p=1.9\times 10^{-4}$).
{\it Middle:} There is an
anticorrelation between the [\ion{O}{3}] velocity
width and the BAL velocity offset ($p=2.7\times 10^{-4}$). {\it
Right:} The covering fraction correction factor is
anti-correlated with the BAL velocity offset for our sample
($p=7.8\times 10^{-3}$), but possibly also for the \citet{xu20}
sample. Larger velocity offsets may produce line emission that is
distributed over a range of wavelength where it may be blended
with the continuum or hidden under strong \ion{Fe}{2} emission.
\label{bal_vel_vs}}
\end{center}
\end{figure*}
Our regression analysis above explored the relationships between the
observed and predicted [\ion{O}{3}] luminosities for seven selected
measurements. We also looked for relationships with other measured
parameters via correlation analysis between the covering fraction
correction factor and the {\it SimBAL} parameters
(Fig.~\ref{correlation_simbal}) and the optical parameters
(Fig.~\ref{corr_optical_simbal}). Interestingly, although the
predicted [\ion{O}{3}] emission must increase with density, we found
no correlation between the covering fraction correction factor and the
BAL $\log n$. This is explained by the fact that lower density BAL
gas is found at larger radii, where, for a fixed global covering
fraction of the line-emitting gas, the volume of emitting gas is
larger.
One correlation stands out: the outflow velocity is anticorrelated
with the covering fraction correction factor. This anticorrelation is
significant ($p<0.025$) in 27 of 48 cases using Spearman Rank; it did not
appear to be significant in the multiple regression analysis, as {\tt
mlinmix\_err} uses Pearson's R. This dependence offers a way to
reconcile the large covering fraction correction factors required by
the $E1>0$ objects: the [\ion{O}{3}] emitted by the BAL could be so
broadened that it blends with the continuum, especially in low
signal-to-noise spectra or amidst strong \ion{Fe}{2} emission.
\section{Summary and Future Work}\label{discussion}
\subsection{Summary}\label{summary}
This is the third in a sequence of four papers that discuss the
properties of low-redshift FeLoBAL quasars. Taken together, they
build a picture of the properties and physical conditions of the BAL
gas, and explore links between these properties to the accretion and
emission-line properties of the quasars.
This paper combines the {\it SimBAL} \citep{choi22} and
optical emission-line analysis \citep{leighly22}. The most
significant result is the discovery that the $E1$ parameter division
discovered by \citet{leighly22}
carries over to the outflow properties (\S~\ref{logr}). Among the
$E1>0$ high accretion rate objects, the outflow velocity {\it
decreases} with distance from the central engine
(Fig.~\ref{logr_voffset}). This is consistent with the expectation of
radiative line driving acceleration. Among the $E1<0$ low accretion
rate objects, the outflow velocity {\it increases} with the distance
from the central engine.
We confirmed the relationship between the outflow velocity and
Eddington ratio previously reported by \citet{ganguly07}, i.e., at a
particular Eddington ratio, a range of velocities are observed up to a
maximum velocity which is itself a function of the Eddington ratio
(\S\ref{lbol}). Using the physical properties of the outflows obtained
from the {\it SimBAL} analysis, we investigated whether the scatter
in the velocity at a particular Eddington ratio could be a consequence
of a scatter in the force multiplier or the launch radius. We found that
among $E1>0$ objects, the scatter in the velocity could plausibly be
attributed to a scatter in the launch radius, while among $E1<0$
objects, the scatter in the velocity could be attributed to scatter in
the force multiplier.
We investigated the volume filling factor of the outflows, both for
the full sample (\S~\ref{full}) and for the $z<1$ subsample
(\S~\ref{covfrac}). The full sample reveals a large range of $\log$
volume filling factors, from $-6$ to $-1$. At large distances from
the central engine, the $\log$ volume filling factors were less than
$-3$, similar to those inferred in HiBAL quasars
\citep[e.g.,][]{gabel06,hamann11,hamann13}. Closer to the black hole,
for $\log R \lesssim 1$, the full range of volume filling factors
was found. We also found that the special BAL classes identified by
\citet{choi22} (the loitering outflows and the other overlapping
trough objects) are also divided by their $E1$ parameter, and by
extension, their Eddington ratio. The loitering outflow objects have
$E1<0$ and low Eddington ratios, while the other overlapping trough
objects have $E1>0$ and high Eddington ratios. Moreover, although
both special types of
FeLoBAL quasar outflows are compact, with typical location $\sim 10$
parsecs from the central engine, a dramatically different number of
assumed spherical clouds would be required to occult the continuum
emission region (Fig.~\ref{numclouds_logr}). For the loitering
outflows, a single cloud (or continuous outflow) would be sufficient.
For the other overlapping trough absorbers, 100s to 1000s of clouds
would be required. Of course, the outflow may not have the structure
of discrete spherical clouds; the point is that the differing physical
conditions of these two categories of outflows tell us that the
structure of the outflows is dramatically different.
We also investigated the relationship between the observed
[\ion{O}{3}] emission line and the [\ion{O}{3}] emission predicted to
be produced by the BAL outflow gas (\S\ref{oiiiemission}). This
analysis first underlines the very large range of equivalent widths
observed in this sample. The observed [\ion{O}{3}] luminosity and the
estimated bolometric
luminosity both span about 1.6 dex, but the ratio of the two
luminosities spans 2.1 dex. We found that in order to
reconcile the observed and predicted $\log$ [\ion{O}{3}] luminosities,
the
emission-line gas global covering fraction may depend on the $E1$
parameter and the location of the outflow $\log R$ (Fig.~\ref{lum_vs_lum}).
At the same time, the covering fraction correction factor (defined as
the difference between the observed and predicted [\ion{O}{3}]
luminosity) was observed to be correlated with the outflow velocity,
which may imply that the predicted strong [\ion{O}{3}] emission in
$E1>0$ objects is broadened and hidden under their typically strong
\ion{Fe}{2} emission (Fig.~\ref{bal_vel_vs}). Most intriguing were
the six $E1<0$ objects with the lowest covering fraction correction
factors, whose [\ion{O}{3}] profiles resembled the BAL absorption
profiles (Fig.~\ref{oiii_tau_vel}).
The final paper in this series of four papers (Leighly et al.\ in
prep.) includes an analysis of the the broad-band optical/IR
properties and discusses the potential implications for quasar
evolution scenarios.
\subsection{Future Work}\label{future}
This sample was limited to objects with redshifts less than 1.63 for
the full 50-object sample from \citet{choi22} and
less than 1.0 for the optical emission-line analysis from
\citet{leighly22}. FeLoBAL quasars can be observed up to $z\sim 3$
in ground-based optical-band spectra. Because the SDSS is a
flux-limited survey, we generally
expect higher-redshift objects to be more luminous; a sample currently
being analyzed has bolometric luminosities about one order of magnitude
larger than the $z<1$ sample. Assuming a similar distribution of
Eddington ratios, the higher-redshift objects will have larger black
hole masses, and therefore a softer (more UV-dominant) SED. The
softer SED could influence the properties of the BAL outflows in two
ways: 1.\ the ions present in the outflowing gas would be different,
e.g., tend toward lower ionization species \citep[e.g.,][]{leighly07},
and 2.\ the velocities might be larger, since the SED will produce a
relatively larger number of UV photons that can transfer momentum by
resonance scattering in the outflowing gas. Preliminary analysis of a
higher-redshift sample that is being observed in the near infrared
shows evidence of higher outflow velocities \citep{voelker21}.
Another interesting feature of the high-redshift
sample is that the larger redshifts provide access to the
high-ionization lines (e.g., \ion{C}{4}, \ion{Si}{4}) that these
objects share with the much more common HiBAL quasars. Preliminary
analysis shows that HiBAL quasars seem to be much different than the
FeLoBAL quasars \citep{hazlett19, leighly_aas19}; in particular, the
high ionization lines sometimes have complicated velocity structure
and certainly extend to higher velocities. The
potential link between the low-ionization line and high-ionization
line properties may also prove to be very illuminating.
Our investigation of whether there could be a relationship between the
ionized outflows manifest in [\ion{O}{3}] emission and the BAL
outflows showed that in most objects such a relationship would be
possible only if the emission-line covering fraction is extremely low,
or if the [\ion{O}{3}] is broad and blended with the \ion{Fe}{2}
emission. An exception was several of the loitering outflow objects,
where the emission line and absorption line profiles appeared to
resemble one another. All of these objects had compact outflows, so
the possibility that there is a direct relationship might be able to be
tested with spatially-resolved observations of the [\ion{O}{3}]
emission; unlike many quasars
\citep[e.g.,][]{harrison14,bischetti17,vayner21b}, the [\ion{O}{3}]
emission in these objects should be unresolved if it does indeed
originate in the BAL outflow. Proving a relationship would be very
difficult in general. However, we might be able to falsify one. At
a redshift of 0.9, the angular scale is about 7.9 kpc per arc second,
so it would be possible to determine whether the [\ion{O}{3}] emission
was extended or compact on the scale of a kilo-parsec. Many of the
{\it SimBAL} solutions indicate BAL outflows with size scales much
less than one kiloparsec. The [\ion{O}{3}] emission should be
unresolved in such objects.
\vskip 1pc
Regardless of the details, the analysis presented here has shown that
again, key physical properties of the outflows differ as a
function of the location in the quasar, and as a function of accretion
rate as probed by the $E1$ parameter. These patterns and differences
have broad potential implications. We may finally be able to
understand the acceleration mechanisms that operate in quasars. In
addition, we may be able to use the accretion properties measured from
the emission lines to statistically infer the outflow properties in
objects that do not show outflows along the line of sight. These
ambitious goals will require much more work and analysis of many more
objects, but at least we have identified a promising path forward.
\acknowledgements
We thank the current and past {\it SimBAL} group members and the anonymous referee for useful discussions and comments on the manuscript.
Support for {\it SimBAL} development and analysis is provided by NSF
Astronomy and Astrophysics Grants No.\ 1518382 and 2006771. This work
was performed in part at Aspen Center for Physics, which is supported
by National Science Foundation grant PHY-1607611. SCG thanks the
Natural Science and Engineering Research Council of Canada.
Long before the University of Oklahoma was established, the land on
which the University now resides was the traditional home of the
“Hasinais” Caddo Nation and “Kirikiris” Wichita \& Affiliated
Tribes. This land was also once part of the Muscogee Creek and
Seminole nations.
We acknowledge this territory once also served as a hunting ground,
trade exchange point, and migration route for the Apache, Comanche,
Kiowa and Osage nations. Today, 39 federally-recognized Tribal nations
dwell in what is now the State of Oklahoma as a result of settler
colonial policies designed to assimilate Indigenous peoples.
The University of Oklahoma recognizes the historical connection our
university has with its Indigenous community. We acknowledge, honor
and respect the diverse Indigenous peoples connected to this land. We
fully recognize, support and advocate for the sovereign rights of all
of Oklahoma’s 39 tribal nations.
This acknowledgement is aligned with our university’s core value of
creating a diverse and inclusive community. It is our institutional
responsibility to recognize and acknowledge the people, culture and
history that make up our entire OU Community.
Funding for the SDSS and SDSS-II has been provided by the Alfred
P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the
Max Planck Society, and the Higher Education Funding Council for
England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the
American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve
University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Joint Institute for Nuclear Astrophysics, the
Kavli Institute for Particle Astrophysics and Cosmology, the Korean
Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington.
Funding for SDSS-III has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science
Foundation, and the U.S. Department of Energy Office of Science. The
SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven
National Laboratory, Carnegie Mellon University, University of
Florida, the French Participation Group, the German Participation
Group, Harvard University, the Instituto de Astrofisica de Canarias,
the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins
University, Lawrence Berkeley National Laboratory, Max Planck
Institute for Astrophysics, Max Planck Institute for Extraterrestrial
Physics, New Mexico State University, New York University, Ohio State
University, Pennsylvania State University, University of Portsmouth,
Princeton University, the Spanish Participation Group, University of
Tokyo, University of Utah, Vanderbilt University, University of
Virginia, University of Washington, and Yale University.
\software{Cloudy \citep{ferland13}, mlinmix\_err
\citep{kelly07}, SimBAL \citep{leighly18}}
|
2,877,628,089,538 | arxiv | \section{Introduction}
\label{sec:intro}
Atomic nuclei are known to exhibit a wide range of collective excitations~\cite{Boh75}.
Amongst them, giant resonances (GRs) are of particular interest in our understanding of collective motion in nuclei.
Macroscopically, they are associated to high energy modes of small amplitude vibration of the entire nucleus.
Microscopically, they can be described, in a first approximation, by coherent superpositions of one-particle
one-hole (1p1h) excitations~\cite{Har01}.
Giant resonances lie usually at energies above the light particle emission threshold. Different kinds of decay are
commonly distinguished. For instance, direct decay occurs when one particle is emitted leaving the daughter nucleus
in a single hole state. However, the decay is sequential when two-particle two-hole (2p2h) configurations are populated
by the residual interaction. The direct decay by nucleon emission is of particular interest as the emitted nucleon can
give a glimpse on the microscopic structure of the GR. As such, it is an experimental tool of choice to study GRs~(see,
e.g., Refs.~\cite{Str00,Har01,Zeg03,Hun07}).
Collective motion in nuclei can be investigated with microscopic theories based on time-dependent Energy-Density-Functional
(TDEDF) approaches. In particular, the time-dependent Hartree-Fock(-Bogoliubov) equations in their linearized version
(Q)RPA~\cite{Rin80,Bla86}, have been widely used to study various GR strength distributions (see~\cite{Kha02,Eba10} and
references therein). When a proper treatment of the continuum is accounted for, those methods are also capable of describing
direct neutron and proton decay.
The goal of the present work is to investigate the link between the 1p1h structure of giant resonances and their direct-decay
pattern. The scalar-isoscalar Giant Monopole Resonance (GMR) is studied in several closed-shell nuclei. The response to a
scalar monopole excitation in the small amplitude regime is computed using the time-dependent Hartree-Fock (TDHF) formalism
\cite{Dir30}, based on a modern Skyrme functional~\cite{Sky56}. To study GMR direct-decay, the calculations are performed in
a large spherical grid, allowing for a spatial separation of the nucleus from the emitted nucleons to the continuum. Similar
TDHF calculations have been performed in the past with schematic energy density functionals to study GMR widths and lifetimes
from direct decay~\cite{Cho87,Pac88}. These quantities were extracted from the time evolution of the number of emitted nucleons.
In the present paper, new informations are obtained by computing energy spectra of the nucleons which are emitted in the
direct-decay of the GMR. The structure of these spectra is then analyzed in terms of single-hole configurations of the
$A-1$ daughter nucleus.
Theoretical framework and numerical details are presented in section~\ref{sec:framework}.
Results obtained on several closed-shell nuclei are discussed in section~\ref{sec:results}
before concluding in section~\ref{sec:conclusion}.
\section{Theoretical framework and numerical details}
\label{sec:framework}
\subsection{Time-dependent Hartree-Fock approach for collective vibrations}
\label{subsec:tdhf}
The TDHF equation reads~\cite{Rin80}
\begin{equation}
i \hbar \frac{d}{dt} \rho = \left[h\left[\rho\right],\rho\right],
\label{eq:tdhf}
\end{equation}
where $\rho$ is the one-body density matrix and $h\left[\rho\right]$ is the single-particle hamiltonian derived from the energy
density functional $E[\rho]$ using
\begin{equation}
h_{ij}\left[\rho\right]=\frac{\delta E\left[\rho\right]}{\delta \rho_{ji}}.
\label{eq:HFhamiltonian}
\end{equation}
In the TDHF theory, the many-body state is constrained to be an independent particle state (e.g., a Slater determinant of the
occupied single-particle states) which implies $\rho^2=\rho$ at all time.
The one-body density matrix is used to compute expectation value of one-body operators for which TDHF is optimized~\cite{bal81}.
Information on vibrational modes such as energy can then be obtained through the evolution of their corresponding multipole moment.
For instance, the excitation of a monopole mode $\ket{\nu}$ on top of the ground state induces an oscillation of the expectation value
of the monopole moment~\cite{Rin80,Bla86}
\begin{equation}
\hat{F}=\sum_{\sigma,q}\int r^2 \hat{a}^\dagger_{\mathbf{r}\sigma{q}} \hat{a}_{\mathbf{r}\sigma{q}} d\mathbf{r}
\label{eq:operatorF}
\end{equation}
where $\hat{a}^\dagger_{\mathbf{r}\sigma{q}}$ ($\hat{a}_{\mathbf{r}\sigma{q}}$) creates (annihilates) a nucleon in $\mathbf{r}$ with spin
$\sigma$ and isospin $q$. In this mode, $\langle{\hat{F}}\rangle(t)$ oscillates at a frequency
$\omega_\nu=(E_\nu-E_0)/\hbar$, where $E_0$ is the ground state energy and $E_\nu$ the energy of the excited state.
The linearization of the TDHF equation leads to the RPA~\cite{Rin80}.
As a consequence, TDHF calculations in the small amplitude limit contain all the RPA residual interaction which is responsible for the collectivity of giant resonances.
Using TDHF codes ensures that the same EDF is used to determine both the HF ground-state and the dynamics thanks to the structure of the TDHF equation.
It is then equivalent to fully self-consistent RPA codes.
In addition to energies of GRs, TDHF has been used to study their strength distribution~\cite{Blo79,Uma05,Mar05,Nak05,Alm05,Nak07,Ste07,Ste10},
their life-time and escape width~\cite{Cho87,Pac88}, and their anharmonicities~\cite{Sim03,Rei07,Sim09}. Escape width and Landau
damping are accounted for in TDHF, however, two-body correlations responsible for the spreading width are neglected. Inclusion of
pairing correlations is possible in the TDHF-Bogoliubov theory~\cite{Rin80,Bla86}, however its numerical applications~\cite{Ave08,Eba10}
are time-consuming and prevent the use of large spatial grid as in the present study.
\subsection{Linear response theory}
The nucleus is assumed to be in its ground state at initial time. An external field $\hat{V}_{{ext}}=\epsilon\eta(t)\hat{F}$
is applied, where $\epsilon$ is the intensity of the perturbation and $\eta(t)$ is its time profile. We choose a delta-function
for the latter: $\eta(t)=\delta(t)$. The evolution of an observable $\delta Q(t)=\expec{\hat{Q}}(t)-\expec{\hat{Q}}_0$ in the linear
regime, where $\expec{\hat{Q}}_0$ is the expectation value on the ground state $|0\rangle$, can be written
\begin{equation}
\delta Q(t) = \frac{\epsilon \Theta(t)}{i \hbar} \sum_{\nu} \bra{0}\hat{Q}\ket{\nu}\bra{\nu}\hat{F}\ket{0} e^{-(\lambda_\nu+i \omega_\nu) t} +c.c.,
\label{eq:Oop_evol}
\end{equation}
where $\Theta(t)=0$ if $t<0$ and 1 if $t\ge0$, $\{|\nu\rangle\}$ are the eigenmodes, and $c.c.$ stands for {\it complex conjugated}.
For bound states, $\lambda_\nu=0$, while unbound states decay with a lifetime $1/\lambda_\nu$.
To analyze the spectral response of $\delta{Q}$, we introduce the quantity
\begin{equation}
R_{Q}(\omega) = \frac{-\hbar}{\pi\epsilon}\int_{0}^{\infty}\delta Q(t)\sin({\omega t})dt. \label{eq:FT_respfunc}
\end{equation}
which, in the particular case where $\hat{Q}=\hat{F}$, reduces to the strength function. For bound states,
the latter reads
\begin{eqnarray}
R_{F}(\omega) &=& \sum_{\nu}\left|\bra{0}\hat{F}\ket{\nu}\right|^2 \delta\left(\omega-\omega_\nu\right), \label{eq:Strength}
\end{eqnarray}
while a Lorentzian width appears for finite values of $\lambda_\nu$ in the continuum.
\subsection{Monopole vibrations in spherical symmetry}
The monopole moment of Eq.~(\ref{eq:operatorF}) is used in the external field. Assuming spherical symmetry of the ground state,
only monopole modes are excited. It is then convenient to decompose the response of the nucleus in terms of the usual quantum
numbers $l$ and $j$ using the operators
\begin{equation}
\hat{F}_{lj}=\sum_{n,n',m_j}\bra{nljm_j}\hat{F}\ket{n'ljm_j}\hat{a}^\dagger_{nljm_j} \hat{a}_{n'ljm_j},
\label{eq:operatorFqlj}
\end{equation}
where the isospin is omitted to simplify the notation. The spectral response, $R_{F_{lj}}(\omega)$, defined in Eq.~(\ref{eq:FT_respfunc}),
can then be used to analyze microscopically the strength distribution of the GMR.
\subsection{Spectra of emitted nucleons}
\label{subsec:env}
The energy spectra of emitted (free) nucleons can be computed by evaluating overlaps between the single-particle wave-functions
$\varphi_{nlj{m_j}}(\mathbf{r}\sigma)$ (with degeneracy $2j+1$ in spherical symmetry) and spherical plane waves $\phi_{klj{m_j}}(\mathbf{r}\sigma)$ of energy
$E=\frac{\hbar^2 k^2}{2m}$, where $m$ is the nucleon mass.
The latter read
\begin{equation}
\phi_{klj{m_j}}\left(\mathbf{r}\sigma\right)=\frac{1}{\mathcal{N}} j_l(kr)\Omega^\sigma_{lj{m_j}}(\theta,\phi),
\label{eq:spherical}
\end{equation}
where $j_l(kr)$ is a spherical Bessel function for angular momentum $l$ and momentum $k$, $\Omega^\sigma_{lj{m_j}}(\theta,\phi)$ is a
spherical spinor~\cite{Var88}, and $\mathcal{N}$ is a normalization constant. The overlaps between a single-particle wave-function
and a spherical plane wave with the same $l,j$ and ${m_j}$ thus reads
\begin{equation}
c^{n}_{lj{m_j}}(k)=\sum_{\sigma}\int \phi^*_{klj{m_j}}(\mathbf{r}\sigma)\varphi_{nlj{m_j}}\left(\mathbf{r}\sigma\right)d\mathbf{r}.
\label{eq:ck}
\end{equation}
The probability to find a nucleon with energy $E$ and quantum numbers $l$ and $j$ is given by
\begin{equation}
\rho^{\uparrow}_{lj}\left(E\right)=\sum_{n}\sum_{{m_j}=-j}^j\left|c^{n}_{lj{m_j}}(k)\right|^2.
\label{eq:proba_nucl}
\end{equation}
\subsection{Numerical details}
\label{subsec:numerics}
The evolution of the occupied single-particle wave-function is determined from the TDHF equation in a spherical mesh of radius
$R_{box}=600$~fm, with a radial spacing $\delta r=0.3$~fm and hard boundary conditions. The SLy4 parametrization~\cite{Cha98} of the Skyrme
EDF~\cite{Sky56} is used. In a first step, the HF initial condition is computed in a $R_0=30$~fm box with the same radial
spacing. At initial time, all single-particle wave-functions vanish for $r>R_0$.
In principle, the nucleon spectrum should be determined at large distances from the nucleus to avoid any long-range interaction
with the latter. In practice, the integral in Eq.~(\ref{eq:ck}) is performed from $R_0$ to $R_{box}$.
Note that the upper limit of the integral in the definition of the monopole moment in Eq.~(\ref{eq:operatorF})
is chosen to be $R_0$ to avoid a divergence of $\delta{F}$ due to emitted nucleons.
To account for the finite size of the box, only spherical plane waves vanishing at $R_{box}$ are considered in
Eq.~(\ref{eq:spherical}). (Their normalization constant $\mathcal{N}$ is determined assuming that they vanish
for $r\ge R_{box}$). As a consequence, the momentum of each partial wave with quantum number $l$ can take only
discrete values.
To avoid spurious peaks from this discretization, $\rho^{\uparrow}_{lj}\left(E\right)$ is convoluted
with a normalized Gaussian distribution in energy with a standard deviation of $150$~keV.
\section{Results}
\label{sec:results}
\subsection{Detailed analysis of the GMR in $\Ox{16}$}
To illustrate the method, let us start with a microscopic analysis of the GMR direct decay in $\Ox{16}$. The
evolution of the expectation value $\delta{F}(t)=\langle{F}\rangle(t)-\langle{F}\rangle_0$ of the observable
$\hat{F}$ given in Eq.~(\ref{eq:operatorF}), where the integral is performed up to $R_0$, after a monopole
excitation in linear regime is shown in Fig.~\ref{fig:O16_evol}.
\begin{figure}[!h]
\begin{center}
\includegraphics{r2evolve_sktdhfb_Oxygen16.eps}
\end{center}
\caption{Evolution of the monopole moment in $\Ox{16}$ after a monopole boost in linear regime.}
\label{fig:O16_evol}
\end{figure}
A damped oscillation associated to the GMR is observed, with an average period $T_{GMR}\simeq57.9$~fm/$c$
corresponding to an energy $E_{GMR}=2\pi\hbar/T_{GMR}\simeq21.4$~MeV.
In a coherent state picture (see, e.g., \cite{sim01}), the number of excited phonons is proportional
to the square of the oscillation amplitude. The damping of the oscillation is then a signature for GMR decay.
Here, the decay occurs by nucleon emission as $E_{GMR}$ is greater than the proton and neutron separation
thresholds in $^{16}$O. The description of nucleon emission is possible if the continuum is properly
accounted for. This is ensured, here, by the large size of the spherical mesh to prevent spurious effects
coming from reflected flux on the edge of the box~\cite{Rei06}.
The strength function associated to $\delta{F}(t)$ and obtained from Eq.~(\ref{eq:FT_respfunc}) with $Q=F$ is plotted
in Fig.~\ref{fig:S0_O16} (black solid line). The main peak at $\sim{E}_{GMR}$ is surrounded by two shoulders
at $\sim17$ and $\sim31$~MeV. In order to get a deeper insight into the microscopic origin of these structures, the
spectral responses of the $\delta{F}_{lj}(t)$ [see Eq.~(\ref{eq:operatorFqlj})] following the same monopole boost are
computed from Eq.~(\ref{eq:FT_respfunc}) and plotted with colored lines in Fig.~\ref{fig:S0_O16}. We have checked
numerically that the sum of these spectral functions is equal to the total strength function $R_F$, as expected. This
decomposition shows that the low-energy shoulder and the main peak are mainly due to $\ljp{1}$ and $\ljp{3}$ orbitals,
respectively.
The structure of the high-energy shoulder is more complicated. It involves a constructive contribution of the $\ljs$
orbitals while the $\ljp{3}$ ones reduce the strength in this region. The dominant role of the $\ljs$ neutron and
proton contributions in the high-energy shoulder has already been noted using similar techniques but other EDF
parametrizations~\cite{Pac88,Ste10}.
\begin{figure}[!h]
\begin{center}
\includegraphics{gle_r2strength_qlj_Oxygen16.eps}
\end{center}
\caption{\emph{(Color online)} Strength function of the GMR in $\Ox{16}$ (black solid line) computed from the time evolution
of the monopole moment $\delta{F}(t)$. The spectral responses $R_{F_{lj}}$ associated to the single-particle quantum
numbers $l$ and $j$ and labelled by their spectroscopic notation are plotted in colored solid (dashed) lines for neutrons
(protons).}
\label{fig:S0_O16}
\end{figure}
The next step of our analysis is to study the decay properties of the GMR and to investigate their relationship with
the previous microscopic decomposition. The spectra of emitted nucleons are computed 2250~fm/$c$ after the boost, e.g.,
when the monopole oscillation is fully damped (see Fig.~\ref{fig:O16_evol}). We use the method described in
Sec.~\ref{subsec:env}. The proton and neutron spectra for each set of $l$ and $j$ are plotted in
Fig.~\ref{fig:spec_decay_O16}. We have checked numerically that, in the linear regime, these spectra are
quadratic in the intensity of the boost~$\epsilon$~\cite{Cho87}.
First we note that proton and neutron spectra have similar global features, as expected for light $N=Z$ nuclei,
although the proton spectra are shifted by $\sim3$~MeV due to the Coulomb repulsion.
In addition, the Coulomb barrier, which is $B_c\simeq1.7$~MeV in $^{16}$O, reduces strongly the emission of
protons at energy $E<B_c$ due to the exponentially decreasing tunneling probability. We also note that there
is no relevant emission of $\ljs$ orbitals. In fact, the single-particle energy of the $1s_{1/2}$ orbitals,
which are the only occupied $\ljs$ single-particle states in the HF ground-sate of $^{16}$O, are $-32.4$~MeV
for protons and $-36.2$ for neutrons (see table~\ref{tab:16Oesp}). As a result, the 1p1h states with a hole in
a $1s_{1/2}$ orbital, and contributing to the GMR microscopic structure, are bound for neutrons and below $B_c$ for protons. Indeed,
the different structures in the GMR spectrum (see Fig.~\ref{fig:S0_O16}) are located at energies below $\sim33$~MeV,
which is not sufficient to emit a $1s_{1/2}$ single-particle.
\begin{table}
\caption{\label{tab:16Oesp} Energies (in MeV) of the occupied single-particle states in the HF ground state of $^{16}$O.}
\begin{ruledtabular}
\begin{tabular}{ccc}
s.p. state & proton & neutron\\
\hline
$1s_{1/2}$ &-32.4& -36.2\\
$1p_{3/2}$ &-17.1& -20.6\\
$1p_{1/2}$ &-11.2& -14.5\\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[!h]
\begin{center}
\includegraphics{gle_decayspec_qlj_Oxygen16.eps}
\end{center}
\caption{\emph{(Color online)} Neutron (solid lines) and proton (dashed lines) direct-decay spectra of the GMR in $\Ox{16}$ for
each set of single-particle quantum numbers $l$ and $j$, given in spectroscopic notation.
\label{fig:spec_decay_O16}
\end{figure}
Let us now attempt to reconstruct the GMR strength function from the spectra of emitted nucleons shown in
Fig.~\ref{fig:spec_decay_O16}. In a pure harmonic picture, the GMR is a coherent sum of 1p1h states with a
difference $\sim{E}_{GMR}$ between the particle and the hole energies. However, the nucleon emitted by direct decay
have only the energy of the particle-state, which is obviously smaller than $E_{GMR}$ because the energy of
the hole-state is negative. To reconstruct the GMR strength function, it is then necessary to shift the energy
of the emitted nucleons by the energy of their associated hole-state, that is, the energy of the initially
occupied single-particle state. The latter are given in table~\ref{tab:16Oesp} for $^{16}$O at the HF level.
The ''shifted'' spectra of emitted nucleons for each set of $l$ and $j$, noted $\rho^{\uparrow(s)}_{lj}$, obtained from this procedure, are
plotted in Fig.~\ref{fig:S0_decay_O16}-b, together with their sum.
It is striking to see that the shape of the latter is very similar to the strength function shown in
Fig.~\ref{fig:S0_O16} and recalled in Fig.~\ref{fig:S0_decay_O16}-a, although the two spectra have
been obtained from totally different quantities, i.e., the spectra of emitted nucleons and the time evolution
of the monopole moment.
Similarly to the microscopic decomposition of the strength function obtained from the time evolution of the
monopole moment (see Fig.~\ref{fig:S0_O16}), we see in Fig.~\ref{fig:S0_decay_O16}-b that the low energy
shoulder and the main peak are associated to $\ljp{1}$ and $\ljp{3}$ orbitals, respectively. Again, the
situation for the high energy shoulder is more complicated. Although it is mainly due to $\ljs$ orbitals
in the monopole response (see Fig.~\ref{fig:S0_O16}), it is in fact associated to the emission of $p$-orbitals
in Fig.~\ref{fig:S0_decay_O16}-b. As discussed before, this is because the energy of the GMR is not sufficient
to promote a $1s_{1/2}$ single-particle to the continuum. However, the RPA residual interaction couples the
responses associated to the different single-particle quantum numbers to produce a collective monopole vibration.
As a result, the vibration of $s$-orbitals can be coupled to $p$-unbound-states contributing to the direct decay.
Such coupling might be the reason for the complex competition between constructive and destructive contributions
in the high-energy shoulder observed in Fig.~\ref{fig:S0_O16}.
\begin{figure}[!h]
\begin{center}
\includegraphics{gle_decay_qlj_Oxygen16_big.eps}
\end{center}
\caption{\emph{(Color online)} (a) Strength function of the GMR in $^{16}$O obtained from the time evolution of the monopole moment.
(b) Spectra of emitted neutrons (colored solid lines) and protons (colored dashed lines) ''shifted'' by the energy of the initially
occupied single-particle state (see text). Their sum is shown in black solid line.
\label{fig:S0_decay_O16}
\end{figure}
\subsection{Tin isotopic chain}
Similar analyses have been performed for the GMR in $^{100,132}$Sn nuclei. The strength functions obtained from the time
evolution of the monopole moment are plotted with black solid lines in the upper panels of Fig.~\ref{fig:Tin100}
and \ref{fig:Tin132}. The reconstructed GMR strength, obtained from the emission spectra are plotted with black solid lines
in the lower panels. As in the $^{16}$O case, the agreement between the two methods is excellent.
The gross feature of the strength is similar for the two isotopes, with a single peak centered at $E_{GMR}=17.2$~MeV for
$\Sn{100}$ and $15.8$~MeV for $\Sn{132}$ (see table~\ref{tab:Sn}), although their widths vary more importantly. It is
striking, however, that the decompositions in terms of single-particle quantum numbers exhibit some important differences
between the two techniques. Although both proton and neutron single-particle orbitals participate to the vibration (see
upper panels in Fig.~\ref{fig:Tin100} and \ref{fig:Tin132}), the decay only occurs by neutron emission for $\Sn{132}$,
whereas $\Sn{100}$ decays through proton emission only. This can be understood by the fact that the proton (neutron) separation
energy increases (decreases) with the number of neutrons. These quantities are given in table~\ref{tab:Sn}, together with the
Coulomb barrier for protons and the position of the peak energy associated to the GMR energy. Indeed, the binding energy of
neutrons in $^{100}$Sn is of the order of $E_{GMR}$ and proton emission is favored, while the combined effects of an increasing $S_p$
and the Coulomb barrier $B_c$ prevent proton emission in $\Sn{132}$.
\begin{table}
\caption{\label{tab:Sn} Neutron and proton separation energies and Coulomb barriers in Sn isotopes.
The GMR energy corresponds to the position of the peak in the strength function. All energies are in MeV.}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
& $S_p$ & $B_{c}$ & $S_p+B_{c}$ & $S_n$ & $E_{GMR}$\\
\hline
$^{100}$Sn & 3.1 & 8.1 & 11.4 & 16.9 & 17.2 \\
$^{132}$Sn & 15.6 & 7.4 & 23.0 & 7.7 & 15.8 \\
\end{tabular}
\end{ruledtabular}
\end{table}
As in the case of the high-energy shoulder in the $^{16}$O spectra of figures~\ref{fig:S0_O16} and~\ref{fig:S0_decay_O16},
the difference in the microscopic decomposition of the GMR strength with the two techniques is a signature of the RPA residual
interaction which couples the bound particle-hole states to the unbound ones, allowing their decay. In particular, the vibration
of the protons (neutrons) in $^{132}$Sn ($^{100}$Sn) decays via neutron (proton) emission thanks to the collective vibration
of both proton and neutron mean-fields.
\begin{figure}[!h]
\includegraphics{gle_decay_qlj_Tin100_big.eps}
\caption{\emph{(Color online)} (a) Strength function $R_F$ (black solid lines) of $\Sn{100}$. Its single-particle
decomposition, multiplied by a factor of three ($R_{F_{lj}}\times 3$), is represented with colored solid (dashed)
lines for neutrons (protons). (b) Spectra of emitted particles ''shifted'' from their single-particle energies
are plotted in colored solid (dashed) lines for neutrons (protons). Their sum appears in black.
\label{fig:Tin100}
\end{figure}
\begin{figure}[!h]
\includegraphics{gle_decay_qlj_Tin132_big.eps}
\caption{\emph{(Color online)} Same as Fig.~\ref{fig:Tin100} for $\Sn{132}$.
\label{fig:Tin132}
\end{figure}
\section{Conclusions}
\label{sec:conclusion}
Using a simple single-particle decomposition, the microscopic structure and (direct) decay of the GMR in some spherical nuclei
has been studied. The GMR strength was obtained by two techniques: $(i)$ from the real-time small amplitude monopole response
and $(ii)$ from the spectra of emitted nucleons. In $(ii)$, each microscopic contribution has to be shifted by the single-particle
energy of the hole state.
Although both techniques lead to almost identical monopole strengths, their microscopic structure (i.e., their decomposition onto the single-particle quantum numbers $l$ and $j$) may be very different. This is understood by the fact that, although GMR energies are usually above the nucleon emission thresholds, only part of the particle-hole states lie in the continuum. Those which are bound are coupled to unbound states via the RPA residual interaction which is responsible for the collectivity of the vibration. This is illustrated, e.g., in monopole spectra of neutron-rich tin isotopes: although both protons and neutrons contribute to the monopole vibration, only neutrons can be emitted because protons are more bound and have to overcome the Coulomb barrier.
A major improvement of the method would be to include two-body correlations, like pairing interaction and collision terms.
The latter are known to be responsible for the additional spreading width of giant resonances as well as a fragmentation of their strength function~\cite{Lac04}.
They are also expected to modify the structure of the spectra of emitted nucleons by coupling 1p1h states to 2p2h.
A first step would be to use the Extended-TDHF \cite{Won78,Ayi80,Lac99} or the time-dependent Density-Matrix~\cite{Wan85,Toy95,Ass09} formalisms. Because of computational limitations, spherical symmetry (and thus, only monopole vibrations) might be first considered.
\begin{acknowledgments}
We thank T. Lesinski for providing the routine for computation of Bessel functions.
We are also gratefull for comments and discussions with Ph. Chomaz and D. Lacroix at the
early stage of this work. Partial support from ARC Discovery grants DP06644077 and
DP110102858 is acknowledged.
\end{acknowledgments}
|
2,877,628,089,539 | arxiv | \section{The model}
We consider haploid populations with non-overlapping generations. In the absence of epistasis between alleles at a single locus, this analysis could easily be extended to consider diploid populations. We describe here the additive infinite population variants of the model (other variants are described in (SI) \S\ref{variants}). A slightly unusual feature of the model is that we do not assume alleles come from a pre-existent pool, but consider a (form of random walk mutation) model in which alleles are created by mutation as time passes, possibly without any bound on attainable fitness.
We shall assume that genes fitnesses take integer values, but one could also consider real valued fitnesses without substantial changes in behaviour. Most other features of the model, which we now describe in more detail, are essentially standard in the literature.
\
Each instance of the model is determined by three principal parameters: $\ell$, $D$ and $\mu$.
{First}, $\ell\in {\mathbb N}\ (>1)$ specifies the number of loci.
With each individual specified by $\ell$ genes, in the absence of epistasis we need only be concerned with the fitness values corresponding to those genes, and so each individual can be identified with a tuple $\boldsymbol{x}=(x_1,...,x_\ell)\in {\mathbb Z}^\ell$.
The {\em fitness} of $\boldsymbol{x}$ is $F(\boldsymbol{x})=\sum_{i=1}^\ell x_i$. (For the multiplicative model, one would define $F(\boldsymbol{x})=\prod_{i=1}^\ell x_i$ instead.)
{Second}, the {\em domain} $D\subset {\mathbb Z}^\ell$ determines which individuals are allowed to exist.
We will use three types of domains in this paper: The {\em ${\mathbb N}$-model} uses as domain $D={\mathbb N}^\ell$, where ${\mathbb N}=\{1,2,3,....\}$; the {\em ${\mathbb Z}$-model} uses $D=\{\boldsymbol{x}\in {\mathbb Z}^\ell: F(\boldsymbol{x})>0\}$; and the {\em bounded-model} uses $D=\{1,...,N\}^\ell$ for some upper bound $N\in{\mathbb N}$ on gene fitnesses.
In practice there is almost no difference between the ${\mathbb N}$- and ${\mathbb Z}$-models, but there are situations when it is simpler to consider one or the other.
{Third}, $\mu\colon{\mathbb Z}\to {\mathbb R}^{\geq 0}$, the {\em mutation probability function}, determines how mutation affects the fitness of genes: $\mu(k)$ is the probability that the fitness of a gene will increase by $k$.
For the sake of simplicity we assume this distribution to be identical for all loci.
While there is no clear canonical choice for $\mu$, the behaviour of the model is robust to changes in this parameter so long as negative mutations are more likely than positive ones, both being possible.
This is because any such choice of $\mu$ will approximate a Gaussian distribution over multiple generations.
The simplest mutation distributions one may consider are those taking non-zero values only on $\{-1,0,1\}$.
Unless stated otherwise, it should be assumed that from now on mutations are of this form and that $\mu(0)>\mu(-1)>\mu(1)$ (giving a form of stepwise-mutation model \cite{OK}).
By a {\em population} we mean a probability distribution ${\boldsymbol{\phi}}\colon {\mathbb Z}^\ell\to {\mathbb R}^{\geq 0}$, where ${\boldsymbol{\phi}}(\boldsymbol{x})$ is the proportion of individuals that have `genotype' $\boldsymbol{x}\in {\mathbb Z}^\ell$. For a population ${\boldsymbol{\phi}}$, we shall also use $X=(X_1,...,X_{\ell})$, where the $X_i$'s take values in ${\mathbb Z}$, to denote a random variable that picks an individual with gene fitness values $X_1$,...,$X_\ell$ according to the distribution given by ${\boldsymbol{\phi}}$. We let $M({\boldsymbol{\phi}})$ denote the {\em mean fitness} of the population ${\boldsymbol{\phi}}$, namely $E(F(X))$. It should be assumed throughout that all populations considered have finite means, variances, and that all cumulants are finite (as is the case, for example, for distributions ${\boldsymbol{\phi}}$ with finite support, i.e. with finitely many $\boldsymbol{x}\in {\mathbb Z}^\ell$ such that ${\boldsymbol{\phi}}(\boldsymbol{x})\neq 0$).
For a sexual population, the next generation is obtained by application of three operations: selection, mutation and recombination.
We refer to the consecutive application of these operations over multiple generations as the {\em sex process}.
For the {\em asex process}, the operations applied are selection and mutation, and the recombination phase is omitted.
With a much less significant effect, at the end of each generation we will also apply a truncation operation that erases individuals falling outside the domain.
\noindent \textbf{Selection}.
The probability of survival for an individual is proportional to its fitness value.
If ${\boldsymbol{\phi}}$ is the population prior to selection then the resulting population, $\mathtt{Sel}({\boldsymbol{\phi}})$, is given by:
\[
\mathtt{Sel}(\phi)(\boldsymbol{x})=\frac{F(\boldsymbol{x})}{M({\boldsymbol{\phi}})}{\boldsymbol{\phi}}(\boldsymbol{x}),
\quad \mbox{ for } \boldsymbol{x}\in \mathbb{{\mathbb Z}}^\ell.
\]
The factor $1/M({\boldsymbol{\phi}})$ normalises the probability distribution.
\noindent \textbf{Mutation}.
Let $C_i$ be i.i.d.\ random variables taking values in ${\mathbb Z}$ with distribution $\mu$.
If we apply mutation to a random variable $X=(X_1,...,X_\ell)$ we get $(X_1+C_1,...,X_\ell+C_\ell)$.
Equivalently, if ${\boldsymbol{\phi}}$ is the population prior to mutation then, for $ \boldsymbol{x}\in \mathbb{{\mathbb Z}}^\ell$:
\[
\mathtt{Mut}({\boldsymbol{\phi}})(\boldsymbol{x})=\sum_{\boldsymbol{y}\in D} {\boldsymbol{\phi}}(\boldsymbol{y})\cdot \boldsymbol{\mu}(\boldsymbol{y}-\boldsymbol{x}),
\]
where $\boldsymbol{\mu}$ is the extension of $\mu$ to a function on $\mathbb{Z}^\ell$ according to the assumption that mutations act independently on distinct loci (i.e., $\boldsymbol{\mu}(a_1,...,a_\ell)=\prod_{i=1}^\ell\mu(a_i)$).
\noindent \textbf{Recombination}. For the sake of simplicity we assume that the $\ell$ loci are unlinked, so that they either correspond to loci on distinct chromosomes (one may consider that we are choosing a `representative' from each chromosome), or else lie at sufficient distances when they share a chromosome. In general the effect of recombination is to leave the distributions at individual loci unchanged, while bringing the population towards linkage equilibrium. We make the simplifying assumption (for the infinite models) that the effect of a single application of recombination is to bring the population immediately to linkage equilibrium. (A population is at {\em linkage equilibrium} if the random variables $X_i$ are independent.)
If $\phi_i(x)\colon {\mathbb Z}\to {\mathbb R}^{\geq 0}$ is the distribution at locus $i$, (i.e.\ $\phi_i(x)=\sum_{\boldsymbol{y}\in D, y_i=x}{\boldsymbol{\phi}}(\boldsymbol{y})$) then the resulting population is given by:
\[
\mathtt{Rec}({\boldsymbol{\phi}})(\boldsymbol{x})=\prod_{i=1}^\ell \phi_i(x_i),
\quad \mbox{ for }\boldsymbol{x}=(x_1,...,x_\ell)\in{\mathbb Z}^\ell.
\]
Recombination as we consider it here is thus equivalent to multiple applications of recombination in its standard form.
\def\mathtt{Tru}{\mathtt{Tru}}
Mutation and recombination may create individuals that fall outside the domain $D$. At the end of each generation, we therefore perform {\em truncation} to remove those outlying individuals. $\mathtt{Tru}({\boldsymbol{\phi}})(\boldsymbol{x})$ is defined to be ${\boldsymbol{\phi}}(\boldsymbol{x})/s$ if $\boldsymbol{x}\in D$, and $0$ otherwise, where $s$ is the normalising factor $s=\sum_{\boldsymbol{x}\in D}{\boldsymbol{\phi}}(\boldsymbol{x})$.
We will see (Tables 1-3, \S 4 Extended Data) that the proportion of the population moving outside the bounds of $D$ in each generation is negligible, and that truncation along the lower bounds will have an insignificant effect on the whole process.
\section{Analysing the model}\label{se: Basic Analysis}
The objective now is to show that mean fitness increases more rapidly for sexual populations (reduction to selection at the gene level can then be achieved in a standard fashion, by consideration of the effect of selection on genes which code for sexual rather than asexual reproduction). Proofs of all claims in this section appear in (SI).
Each generation sees two forces acting on the mean fitness $M=M({\boldsymbol{\phi}})$.
On the one hand, mutation causes a fixed decrease in $M$ by an amount that depends only on $\mu$. (Recall that deleterious mutations are more likely than beneficial ones.) Selection, on the other hand, can be shown to increase mean fitness by $V/M$ (a form of Fisher's `fundamental theorem' \cite{Fish}), where $V=V({\boldsymbol{\phi}})=\mathop{\mathrm{Var}}(F(X))$ is the {\em variance of the fitness} of ${\boldsymbol{\phi}}$. Recombination does not affect $M$ directly. Thus, for fixed $\mu$, the increase in mean fitness at each generation is determined by the variance.
The difference between the sex and asex processes will be seen to stem from the effect of recombination on variance, which then results in an increase to the change in mean fitness for the sex process during the selection phase.
The effect of mutation on the variance is a fixed increase at each generation (again entirely determined by $\mu$).
The effect of selection on variance is given by:
\[
V(\mathtt{Sel}({\boldsymbol{\phi}}))-V({\boldsymbol{\phi}}) = \frac{\kappa_3}{M}- \left(\frac{V}{M}\right)^2,
\]
where $\kappa_3$ is the third cumulant of $F(X)$.
Our first theorem shows that for the sex process, the effect of recombination on variance is positive, giving an advantage of sex over asex.
\begin{theorem}\label{thm: effect rec}
If ${\boldsymbol{\phi}}^{\ast}=\mathtt{Sel}({\boldsymbol{\phi}})$ was obtained by an application of selection to a population ${\boldsymbol{\phi}}$ at linkage equilibrium, then
the effect of recombination on fitness variance is given by:
\[
V(\mathtt{Rec}({\boldsymbol{\phi}}^{\ast}))-V({\boldsymbol{\phi}}^{\ast}) = \frac{\sum_{i\neq j} V_iV_j}{M^2},
\]
where $V_i=\mathop{\mathrm{Var}}(\phi_i)$ and $M=M({\boldsymbol{\phi}})$. This effect is therefore non-negative.
\end{theorem}
This theorem applies to the sex process because a previous application of recombination would bring the population ${\boldsymbol{\phi}}$ to linkage equilibrium.
Linkage equilibrium is then preserved by mutation.
Our second theorem shows that recombination has a positive effect on variance in a much more general situation, as for instance, during an asex process where we suddenly apply recombination.
It establishes that for a population initially at linkage equilibrium, \emph{any} subsequent applications of recombination during later generations always give an increase in variance and so a corresponding increase in the rate of change of mean fitness.
\begin{theorem}\label{thm: rec positive}
For the ${\mathbb Z}$-model, starting with a population at linkage equilibrium, suppose we iterate the operations of mutation, selection and recombination in any order (possibly applying only mutation and selection over multiple generations, and of course applying truncations when relevant).
Then any non-trivial application of recombination has a positive effect on variance.
\end{theorem}
\noindent By a {\em trivial} application of recombination we mean one acting on a population which is already at linkage equilibrium, and so which has no effect at all. This is the case, for instance, if one applies recombination twice in a row: the second application is trivial.
The theorem is stated only for the ${\mathbb Z}$-model because truncation creates technical difficulties when producing a proof for the other models. With the effect of truncation being so small, however, the claim of the theorem is, in fact, verified in all simulations we have run for any of the models.
To explain what is behind Theorem \ref{thm: rec positive}, we need to introduce two new key terms: the linkage disequilibrium term $LD_2$ and the {\em flat variance}.
We define $LD_2({\boldsymbol{\phi}})$ to be the decrease in variance produced by recombination:
\[
LD_2({\boldsymbol{\phi}}) = V({\boldsymbol{\phi}}) - V(\mathtt{Rec}({\boldsymbol{\phi}})).
\]
$LD_2$ can be shown to be equal to the covariance term $\sum_{i\neq j} E(X_i X_j)-E(X_i)E(X_j)$.
Theorem \ref{thm: rec positive} states that $LD_2({\boldsymbol{\phi}})$ is negative at all stages of the process, unless the population is at linkage equilibrium, in which case $LD_2({\boldsymbol{\phi}})=0$.
A more geometric way of understanding $LD_2$ is through the notion of flat variance.
Let $\boldsymbol{M}=(E(X_1),E(X_2),...,E(X_\ell))\in {\mathbb R}^\ell$; this vector represents the average individual in the population.
The {\em global variance} of a population is defined as $GV({\boldsymbol{\phi}})=E(\|X-\boldsymbol{M}\|^2)$.
Recombination does not affect the global variance, $GV({\boldsymbol{\phi}})$, at all.
However, it changes the shape of the population by increasing the variance in the direction that is useful for selection, namely the fitness variance.
Consider the diagonal line $d=\{(x_1,...,x_\ell)\in{\mathbb R}^\ell:x_1=x_2=\cdots=x_\ell\}$ and its $(\ell-1)$-dimensional orthogonal complement $P=\{(x_1,...,x_\ell)\in{\mathbb R}^\ell:x_1+x_2+\cdots+x_\ell=0\}$, and let $\pi_d$ and $\pi_P$ be the projection functions onto $d$ and $P$ respectively.
Using that $F(X)$ is the inner product of $X$ and $(1,1,...,1)$, one can show that:
\[
V({\boldsymbol{\phi}})=\ell\cdot \mathop{\mathrm{Var}}(\|\pi_d(X)\|).
\]
We define the {\em flat variance} of a population to be the variance of its projection onto $P$ multiplied by a correcting factor:
\[
FV({\boldsymbol{\phi}})=\frac{\ell}{\ell-1}\cdot E(\|\pi_P(X-\boldsymbol{M})\|^2).
\]
Informally, $V({\boldsymbol{\phi}})$ measures how \emph{tall} a population is along the vector $(1,1,...1)$, while $FV({\boldsymbol{\phi}})$ measures how \emph{fat} it is.
The effect of recombination on variance and flat variance satisfies a simple formula:
\[
V(\mathtt{Rec}({\boldsymbol{\phi}})) = FV(\mathtt{Rec}({\boldsymbol{\phi}})) = \frac{V({\boldsymbol{\phi}})+(\ell-1)FV({\boldsymbol{\phi}})}{\ell},
\]
and hence
\[
LD_2 = \frac{\ell-1}{\ell}\Big(V({\boldsymbol{\phi}})-FV({\boldsymbol{\phi}}) \Big).
\]
Thus, $LD_2$ being negative is equivalent to $FV$ being greater than $V$, or, more informally, the population being fatter than it is tall along $d$.
The dynamics of this interaction are explained in Figure 2, and the effects for unbounded and bounded domains are illustrated in Figures 3 and 4 respectively.
\begin{figure}[!ht] \label{process}
\begin{adjustbox}{addcode={\begin{minipage}{\width}}
{\caption{This illustration shows the level curves for 2-locus sex (red) and asex (blue) populations which begin with the same Gaussian type distribution (loci distributions are i.i.d.\ with $\kappa_3=0$). The $x$ and $y$ axes indicate fitnesses at the first and second loci respectively, fitness increasing along the up-right diagonal.
The reason the last application of selection gives a greater increase in mean fitness for sex is that, {\em as opposed to asex, the sex process capitalises on the increase in flat variance due to mutation}.
The first phase is selection which decreases the fitness variance (given $\kappa_3=0$), and does not interact with flat variance since fitness (measured along the $\diagup$-diagonal) is the only factor influencing the ability of an individual $\boldsymbol{x}$ to survive selection, while flat variance is measured along the planes where fitness is constant (the $\diagdown$-diagonal).
Selection thus causes the flat variance to be greater than the variance, or equivalently, causes negative $LD_2$, as we show in Theorem \ref{thm: rec positive}.
Mutation then increases both flat variance and variance by the same amount.
Recombination, only occurring in the sex process, averages out the fitness variance and flat variance, increasing the variance and decreasing the flat variance, as seen by the rounded form of the level curves at that phase in the figure.
Notice that recombination does not increase global variance; it just transforms the flat variance, which is useless for selection, into fitness variance.
Finally, selection now has an increased effect on the mean fitness of the sexual population due to the larger fitness variance produced by recombination.
For the asex population, mutation will keep on increasing the flat variance, but, in the absence of recombination, selection will not capitalise on this growth.}
\end{minipage}},rotate=90,center}
\includegraphics[scale=.215]{Fig2_lowq.png}
\end{adjustbox}
\end{figure}
\begin{figure}[!ht]
\begin{adjustbox}{addcode={\begin{minipage}{\width}}
{\caption{The benefit of recombination. This figure shows the progress of two 2-locus populations (for the ${\mathbb N}$ model), one of which is sexual and the other of which is \emph{initially} asexual. The $x$ and $y$ axes correspond to fitnesses at the first and second loci respectively, the $z$ axis corresponding to probability density. Both populations begin with initial allele fitnesses of 5, and progress with mutation rate $10^{-4}$, the probability any given mutation is beneficial being $0.1$. During the first 2000 generations the sexual population quickly achieves greater mean fitness, and one can clearly see the increased flat variance of the asexual population. At stage 2000, a single application of recombination is made to the previously asexual population, converting that flat variance into variance and greatly increasing the rate of increase in mean fitness. Without any subsequent applications of recombination, however, the asexual population will eventually have smaller variance than the sexual one and will once again fall behind in mean fitness.
}\end{minipage}},rotate=90,center}
\includegraphics[scale=.215]{Fig3_lowq.png}
\end{adjustbox}
\end{figure}
\begin{figure}
\includegraphics[scale=.2]{Fig4_lowq.png}
\caption{This figure shows the level curves for 2-locus populations proceeding according to the bounded model, with maximum allele fitness 400, mutation rate 0.2, and with the probability any given mutation is beneficial being $10^{-4}$. All alleles initially have fitness 50. The probability density level curves are depicted at stages 500, 1500, 2500, 3500, 4500 and 5500.
We can again observe the increase in flat variance and decrease in variance for the asexual population, and also that the sexual population does not necessarily have a higher global variance.}
\end{figure}
\
Theorems \ref{thm: effect rec} and \ref{thm: rec positive} show an important advantage that sex has over asex.
In comparing sex and asex populations evolving independently, however, these theorems do not suffice to entirely specify how the variances of the two populations differ at any given generation. To make this comparison we would need to understand the evolution of the third cumulant, which behaves differently in each process.
The evolution of the third cumulant depends on the fourth, which depends on the fifth, and so on.
Rather than analysing further the evolution of populations over time, we now study what happens to the sexual and asexual populations in the long term.
We prove that, for the bounded model, whatever the initial populations are, sex outperforms asex in the long run.
We state the following theorem in terms of a mixed population containing both sexual and asexual individuals competing for resources.
Thus the population distribution ${\boldsymbol{\phi}}$ now has domain $D\times \{\mathtt{s,a}\}$, the second coordinate indicating whether the individual is sexual or asexual.
Mutation acts exactly as before among each type of individual.
Selection is also the same, now using $M({\boldsymbol{\phi}})=\sum_{\boldsymbol{x}\in D\times \{\mathtt{s,a}\}}F(\boldsymbol{x}) {\boldsymbol{\phi}}(\boldsymbol{x})$ to normalise.
Recombination acts only among the sexual individuals.
\begin{theorem}\label{main bounded alleles}
Given $\mu$ and $\ell$, for all sufficiently large bounds $N$ and for any initial population in which the proportion of sexual individuals is non-zero, the proportion of sexual individuals converges to 1 and the proportion of asexual ones converges to 0.
\end{theorem}
This is the longest and most complicated proof of the paper: the proof appears in (SI \S\ref{ss: theorem 3}).
Applying the Perron-Frobenius Theorem suffices to prove that the asex distribution converges to a limit, and using other techniques we are also able to find a good approximation for the mean fitness at that limit.
We do not prove that the sex process converges to a limit, but still get a good estimate of the geometrical average of the mean fitness over generations.
Such ideas would not work for the ${\mathbb N}$- and ${\mathbb Z}$-models as in those cases the mean fitness diverges to infinity in both the sex and the asex processes. Figure 5 shows the manner in which sexual and asexual populations converge to their respective fixed points over time (while we do not prove that convergence to a fixed point always occurs for sexual populations, such convergence was observed in all simulations).
\begin{figure}
\includegraphics[scale=.27]{Fig5_lowq.png}
\caption{Each of the six plots shows the trajectory of the centre of mass for various sexual and asexual 2-locus populations over multiple generations, for a number of different initial populations and for the bounded model. Each point represents the centre of mass of a population at a single generation, and the populations were then allowed to evolve for sufficiently many generations that an equilibrium point was reached.
The bottom-left plot shows intermediate steps in the evolution towards the middle plot in the bottom row.
For that plot, we have 40 different initial populations, half sexual (red), half asexual (blue). The bound, $N$, on gene fitness is 50 for all plots except for the top-centre and bottom-right, where $N=301$.
The probability of mutation is $0.5$ except for the top-left plot, where the probability of mutation is $0.9$.
The probability that a mutation is beneficial is $0.001$ in all cases.
Starting from the top-left and moving clockwise, the original populations are Gaussian distributions with standard deviations $5$, $25$, $6$, $8$, $6$ and $6$ respectively.
}
\end{figure}
\section{Discussion}
In nature one must surely expect a variety of mechanisms to be of significance in determining the most efficient methods of reproduction. As well as those factors mentioned in the introduction, sex may provide advantages for species not subject to random mating by strengthening selection \cite{GY}, for example, or may provide a straightforward advantage in providing two parents to care for young offspring \cite{DL}. Such arguments, however, do not suffice to explain the prevalence of sex in species for which random mating is a good approximation or without parental care. Our aim here has been to rigorously establish a fundamental and underlying mechanism conferring strong advantages to sex.
We have seen that independence between loci allows for more rapid growth in mean fitness.
In the absence of such independence, \emph{the selection of fitter alleles at a particular locus will be stronger when other genes have lower fitness values}. A simple analogy may be given in terms of the comparative value of improvements to sensory abilities: If an organism has little sight, a small improvement in hearing may be more important than it is for an organism with excellent vision.
Thus, in the asex process, the result is that individuals which have high fitness on a gene, tend to have low fitness on another -- this is essentially what negative $LD_2$ means, and what is behind the proof of Theorem \ref{thm: rec positive}. The effect of the sex process is to break down these negative associations, but not to increase or decrease the \emph{global} variance of a population. The key role of recombination is to transform the variance produced by negative associations -- the flat variance -- into the form of variance which can then be acted upon by selection -- the fitness variance.
\
Of course a natural question, having considered the infinite populations case, is the extent to which this analysis carries over to the finite populations model. The principal difference in moving to finite populations is that the process is no longer deterministic. The equations governing the change in mean fitness and variance due to selection and mutation for the infinite population model would now perfectly describe the \emph{expected} effect of mutation and selection for finite populations, and the finite populations model could be seen simply as a stochastic approximation to the infinite case, were it not for the loss in variance and higher cumulants due to sampling
(since picking $n$ individuals from a distribution with variance $v$ produces a population with expected variance $v(n-1)/n$).
For large populations this effect will be very small on a stage by stage basis, and so our analysis for infinite populations can be seen as a good approximation over a number of generations which is not too large. Ultimately, however, sampling will have the effect that mean fitness for the population no longer increases without limit: once variance is sufficiently large the expected loss in variance due to sampling balances the increase that one would see for an infinite population with the same cumulants. Larger populations are thus able to sustain much higher mean fitnesses than small ones.
\
While sexual reproduction has been seen here to confer strong advantages in the absence of epistasis, i.e.\ in the setting of simplistic and entirely modular fitness landscapes, we have said nothing about how this picture changes in the presence of epistasis. Assuredly, the task of efficiently navigating fitness landscapes (i.e.\ optimisation) is one that, beyond its relevance here, is of fundamental significance across large areas of applied mathematics and computer science (hence the recent interest of computer scientists in the benefits of sexual reproduction\cite{LP}). However large the role of epistasis in the biological context, it is certainly true that in most of these applications epistasis (in one guise or another) plays a crucial role, and so the interesting question becomes that as to whether sexual reproduction continues to offer these substantial benefits in the face of more complex fitness landscapes. It may be the case that as well as capitalising more efficiently on existing modularity, sex plays a fundamental role in \emph{finding} modularity\cite{LP2}. One would expect a proper analysis to require classification of fitness landscapes in terms of their amenability to different forms of population based search (see, for example, the work of Prugel-Bennet\cite{PB}).
|
2,877,628,089,540 | arxiv | \section{Introduction}
Recent advances in natural language generation (NLG) have greatly improved the diversity, control, and quality of machine-generated text. However, this increased ability to quickly and efficiently create unique, manipulable, human-like text also presents new challenges for detecting the abuse of NLG models in phishing attacks.
Machine-generated texts can pose various risks depending on the context and how they are used. For example, in the case of NLG models, the ability to generate legitimate texts atht looks like emails can lead to attacks like phishing, where the attacker tricks the victim into disclosing sensitive information by impersonating someone else.
Another effect of machine generated text is mass disinformation campaigns. With the ability to generate large amounts of text automatically and quickly, it is possible for malicious actors to create fake news, hoaxes, and other forms of false or misleading information that can harm individuals, organizations, and even entire societies.
Moreover, machine-generated texts can also raise ethical concerns, such as the impact on employment and the potential for bias and discrimination. For example, the use of NLG models to automate certain writing tasks may lead to job losses for human writers, and the algorithms used in NLG may reflect and amplify the biases and stereotypes present in the data they are trained on.
Abuses of NLG models, such as phishing \cite{baki2017scaling,giaretta2018community},disinformation\cite{shu2020mining,stiff2022detecting,zellers2019defending} has been on the rise.
Email is a common method used by phishers to deliver malicious links and attachments to victims. Anti-Phishing Working Group found over 121860 phishing email incidents in march 2017 and in 2016, the APWG received more than 1313771 unique phishing reports. In the first quarter of 2017, around 870 organizations were targeted by W2-based phishing scams, a significant increase from the 100 organizations in 2016. These attacks are becoming more sophisticated and difficult to detect.
Phishers often use techniques such as bulk mailing, spamming, and including action words and links in phishing emails to increase their chances of success. However, these techniques can be easily detected by improved statistical detection models. Another popular method is email masquerading, where the attacker gains access to the victim's email inbox or outbox and studies the content and nature of the emails to create a synthetic malicious email that resembles a benign one. This reduces the chances of detection by automated classifiers and increases the likelihood of a successful attack.
Modern large language models have enabled users to generate text based on context. These models can be trained to generate text using predefined grammars, such as the Dada Engine\cite{baki2017scaling}, or by leveraging deep learning neural networks, such as recurrent neural networks (RNNs)\cite{yao2017automated}, to learn and emulate the input to the system.
NLG systems that use advanced deep learning neural networks (DNNs) can be used by phishers to generate coherent and convincing sequences of text. These systems have been shown to be effective for generating text in various genres, from tweets\cite{sidhaye2015indicative} to poetry\cite{ghazvininejad2016generating}. It is likely that phishers and spammers will soon start using email datasets, both legitimate and malicious, in conjunction with DNNs to create deceptive malicious emails that mimic the properties of legitimate emails. This makes it harder for pre-trained email detectors to identify and block these attacks.
In this report, we try to show a class of attacks where existing large-scale language models have been trained on both legitimate and malicious (phishing and spam) email data. We also aim to show how the generated emails can bypass existing production-level email protection mechanisms and propose a future work to detect such attacks.
\section{Related Work}
Phishing detection is a well-studied area in cybersecurity, but many victims still fall for these attacks. In their work, Drake et al \cite{drake2004anatomy} provide a detailed analysis of the structure and tactics used in phishing emails. In this section, we review previous research on natural language generation, deep learning, and their applications in generating and detecting phishing attacks.
Natural language generation techniques have been widely used to synthesize unique pieces of text. Previous work by Reiter and Dale et al \cite{covington2001building} relied on pre-constructed templates for specific purposes, while the fake email generation system in Baki et al\cite{baki2017scaling} used manually constructed rules to define the structure of fake emails. Recent advances in deep learning have enabled the generation of creative and equitable text with enough training data. RNN(Recurrent Neural Networks) language models are used to generate a range of genres, including poetry by Ghazvininejad et al \cite{ghazvininejad2016generating}, fake reviews by Yao et al \cite{yao2017automated}, tweets \cite{sidhaye2015indicative}, and geographical information by Turner et al \cite{turner2009generating}, among others.
\section{Experimental Methodology}
The section is divided into four subsections. The first subsection (Section 3.1) describes the nature and source of the training and evaluation data. The second subsection (Section 3.2) discusses the pre-processing steps applied to the data. The third subsection (Section 3.3) presents the system setup and experimental settings used in the study.
\subsection{Data Description}
To create a legitimate looking phishing email we first need to start from actually benign and legitimate emails. The text generation algorithms must be trained in legitimate emails. Hence it was imperative to have valid benign emails in the dataset used for training. However, since the goal here is to create emails that even though can serve as a phishing email, should still look like legitimate emails, a mix of legitimate and bad emails was used as a dataset for training and augmenting the models.
For legitimate datasets, instead of using one dataset on our own, we use pre-trained models from Meta and Google to create benign emails. The pre-trained models utilized are Roberta, The Pile, and PushShift.io Reddit. Since training these large language models is almost impossible in normal infrastructure, we utilize \cite{zhang2022opt} to generate the texts. This has been augmented with \cite{zhang2019email} to have email generation capabilities. Python clean text \cite{cleantextPyPI-2022-02-02} has been used to remove email, and phone numbers from the dataset.
For malicious datasets, we primarily use two datasets to augment the benign email data. Notably, the Phishing emails from Jose Nazario’s Phishing corpus \cite{gonzalez2011phishing} and \cite{monkeyorg404-2000-07-24} along with the Enron email dataset \cite{shetty2004enron}.
\subsection{Data Processing}
Most of the pre-processing was done by trying to remove personal information using Python clean text \cite{cleantextPyPI-2022-02-02}. As well as Removal of special characters like \@, \#, \$, \% as well
as common punctuations from the email body.
However, as we have realized later generating emails was not perfect.
\subsection{Experimental Setup}
The experimental setup has been designed with certain different methods in mind. We primarily focused on
\begin{itemize}
\item Using GPT-2 to generate emails. Augmented with email dataset \cite{emailblogKaggle-2022-12-08}
\item GPT-3 to generate emails without any training
\item Contextual support for GPT-3 with da-vinci-beta which has been trained in email by openai
\item The DADA engine \cite{baki2017scaling}
\item Word based RNN's proposed by Xie et al \cite{xie2017neural}, Das et al \cite{das2019automated}
\item Augmenting Open Pre-trained Transformer Language Models\cite{zhang2022opt} on \cite{zhang2019email}
\end{itemize}
\begin{tcolorbox}
While using the general large language models were interesting in trying to produce emails.
The spam and phishing email datasets used for training the models to produce malicious looking email produced better results. The Jose Nazario dataset has 32,000 spams and 415 phishing email. These are all in Unix mbox formatted dataset which were cleaned using clean-text.
The Enron corpus was email dataset from Enron Corporation. It has been used in email research for quite some time and was made public during their legal investigation. This however gives us a rare glimpse in the working of legitimate email data of a big corporation. It consists of 619446 emails from 158 users. It has folders like "discussion threads, "notes inbox" and more in its outlook inboxes. These were created by default for most of the users but were nota ctively used for tagging. The "All Documents" folder present also had large number of duplicates present from inbox. To further clean our dataset, these all were removed.
The cleaned corpus has 200399 emails from 158 users.
We have tried to validate our ways with existing findings available here https://github.com/egmp777/basic \_data\_cleaning\_enron\_case\_study. But our purpose was different than exploratory analysis.
\end{tcolorbox}
\subsection{Experiment}
A PoC was done to evaluate if we can auto-detect any emails from the generated text. GPT3 API, GPT2, OPT, Chatbot and our trained models were all used to generate the texts. Some samples have been provided below. In our test we had generated 100 samples using each. However some of these gave us inconsistent result for a same prompt. We also had sample fo real phishing email (only one for that topic) to compare it with.
A variety fo prompts were used to see what kind of emails the AI's are more comfortable generating.
We applied the GPT-3 davinci model with default sampling parameters (temperature=1, top\_p=1, presence\_penalty=0, frequency\_penalty=0) to predict the log probabilities for the first 100 tokens in each sample using all the preceding tokens as the prompt. We then calculated the actual token's probability, absolute rank, contextual entropy, and whether it matched the predicted token by the GPT-3 API. To account for the maximum of 100 log probabilities returned by the API, we masked out any outliers with an absolute rank greater than 100. We also normalized the results if there were less than 100 tokens in a sample.
\section{Analysis and Results}
\subsubsection{Example Results generated}
\textbf{ChatGPT}:
The following was generated with the prompt \textit{Can you generate an email that will be from The President of a University asking its students to fill up course completion survey form}
\begin{tcolorbox}
Dear Students,
As the academic year comes to a close, we would like to gather your feedback on your experiences and courses at the University. To help us improve our programs and support your learning, we encourage you to take a few minutes to complete the course completion survey form.
Your honest and thoughtful responses will be greatly appreciated and will help us better understand your needs and expectations. The survey form is available online and can be accessed through your student portal.
We thank you in advance for your participation and look forward to hearing from you.
Best regards,
[Name], President
[University]
\end{tcolorbox}
\textbf{GPT3 da-vinci}:
The following was generated with the prompt \textit{Can you generate an email that will be from The President of a University asking its students to fill up course completion survey form} \\
\begin{tcolorbox}
Hello,
As the President of the University, I am writing to request your help in completing a course completion survey form. This form will help us to understand how well our students are doing in their courses and how we can improve our offerings.
I would greatly appreciate it if you could take a few minutes to fill out the form. You can access the form at the following link:
[insert form URL here]
Thank you for your time and assistance.
Sincerely,
[Your Name]
\end{tcolorbox}
\textbf{GPT2}:
The following was generated with the prompt of \textit{Hello, As the President of the University, I am writing to request your help in completing a course completion survey form.} \\
\begin{tcolorbox}
Hello,
As the President of the University, I am writing to request your help in completing a course completion survey form.
As requested, we would like your answer by May NUMBER, NUMBER.
The survey should only take a few minutes of your time (it takes only about NUMBER minutes to complete the survey). All survey responses are anonymous and confidential, and will not be shared outside of our University community. If you have
\end{tcolorbox}
\textbf{OPT}:
The following was generated with the prompt of \textit{Hello, As the President of the University, I am writing to request your help in completing a course completion survey form.} \\
\begin{tcolorbox}
(<PROMPT>
Hello,
As the President of the University,
I am writing to request your help in completing a course completion survey
form.<END-OF-PROMPT>
The purpose of this survey is to gather feedback from <NUMBER> current and
former students on their experiences at UofH.
Please complete the survey no later than Friday, October 00th by sending it
back to me via fax no.
<PHONE>.
Thank you very much for your time.
Best Regards,
UofH Office of
Admissions <EMAIL> (See attached file: uofh0000\_sample.pdf))
\end{tcolorbox}
The most interesting part of this prompt was the ending where it signed off as UofH Office of admissions and a sample pdf. Whereas our prompt never indicates those details\cite{rabimbaemailgennlg-2022-12-08}.
Further testing with the GPT2 dataset produced the following output from the same prompt \cite{rabimbaemailgennlg-2022-12-08} \\
\begin{tcolorbox}
Hello,
As the President of the University,
I am writing to request your help in completing a course completion survey form.
By completing this form, you are welcome to access the COMPANY Club website and ask additional questions about the UCLA Club and our events.
To access the website you may click the link at the top of this page.If you prefer to not complete this form at this time, please let me know and I will
\end{tcolorbox}
Notably, UCLA was not present in the prompt. This shows us that with enough clever prompt discovery it is probably possible to extract meaningful information from the trained dataset even with safeguards in place.
\subsubsection{Training Parameters}
The training parameters used for the HF opt model was
\begin{itemize}
\item learning\_rate: \(6e^-5\)
\item train\_batch\_size: 8
\item eval\_batch\_size: 8
\item seed: 42
\item distributed\_type: GPU
\item gradient\_accumulation\_steps: 16
\item total\_train\_batch\_size: 128
\item optimizer: Adam with betas = (0.9, 0.999) and epsilon = \(1e^-8\)
\item lr\_scheduler\_type: cosine
\item lr\_scheduler\_warmup\_ratio: 0.03
\item num\_epochs: 8
\end{itemize}
And the training parameters used for HF postbot GPT2
\begin{itemize}
\item learning\_rate: 0.001
\item train\_batch\_size: 16
\item eval\_batch\_size: 16
\item seed: 42
\item distributed\_type: multi-GPU
\item gradient\_accumulation\_steps: 8
\item total\_train\_batch\_size: 128
\item optimizer: Adam with betas = (0.9, 0.999) and epsilon = \(1e^-8\)
\item lr\_scheduler\_type: cosine
\item lr\_scheduler\_warmup\_ratio: 0.02
\item num\_epochs: 3
\end{itemize}
\section{Future Work}
Research on the risks of using natural language generation (NLG) models suggests that being able to detect machine-generated text is useful for reducing the harm caused by abuse of these models. When we want to detect machine-generated text, it can be treated as a binary classification problem. We train a classifier to differentiate between machine-generated and human-generated text \cite{crothers2022adversarial}.
We can use generative models without fine-tuning to detect their own outputs or the outputs of other similar models. Autoregressive generative models like GPT-2, GPT-3 are unidirectional, where each token has an embedding that depends on the embeddings of the tokens that come before it. This shows us that an embedding can be created if we add a token at the end of an input sequence, thus creating a sequence of tokens. This now can be used as a new feature vector. Now once we have these newly created features, they can be utilized along with human data to train a layer of neurons for classification.
Research on how to detect machine-generated text has looked at the problem of detecting text when a different dataset was used to train RoBERTa than GPT-2. But here, it was observed that just tuning the detection model with couple of hundred different attack samples provided by domain esperts had a significant effect on the detector's performance on different domains\cite{rodriguez2022cross}.
One another possibility is when an attacker decides to generate the attack from an existing hand-written content. Much like how we have started in this email generation problem. Using human like sample but tweaking the generating parameters to closely meet his goals. Analysis showed that making these targeted changes to texts reduces the effectiveness of GPT-2 or RoBerta-based detectors \cite{bhat2020effectively}.
A generalized solution to this is trying to differentiate between human and machine generated text. Giant Language Model Test Room is a software developed to improve the detection of machine-generated text by adding human review in the pipeline. The tool helps humans classify text by highlighting texts based on how likely of them being chosen by the Transformer model. However, this tool was designed to target GPT-2, which was found to be easier for untrained human evaluators to detect. In addition, GLTR uses "top-k" sampling to determine the likelihood of a word being selected, but this method has been largely replaced by nucleus sampling, which is used in GPT-3 and other works that build on the GPT-2 architecture. While highlighting words based on sampling likelihood may improve human classification ability, it is clear that it still will pose a problem when they have to detect the more advanced models and sampling methods of today.
In long term, we want to propose a framework that can differentiate NLG-generated emails from human-generated emails. Prior work has already been done trying to determine machine-generated text, however specifically for email and malicious emails, there are distinct characteristics we have observed that can be exploited to augment prior works to be more effective. Few of these are homogeneous to what we have seen in language models \cite{DBLP:journals/corr/abs-1906-04043}, but some are significantly distinct and should be explored more.
\section{Conclusion}
The more we experimented with large language models and prior works by Das et al \cite{das2019automated}, Baki et al \cite{baki2017scaling} it became clear that prior RNN-based models and DIDA engines, even though show some malicious intent in their generation, don't actually pose threat to be understood as real malicious email. All of them went past Gmail and outlook when sent from a legitimate email id. The emails generated by GPT3 and OPT significantly pose a larger threat to be believed as real emails when generated in mass using tools and bulk emailed with targeted intent. Especially with targeted email dataset training and keywords in prompts, the models generated very convincing-looking emails. Even with safeguards in place for GPT3, we were able to generate these emails and chatGPT was a very interesting contender in the tests. Even though chatgpt didn't let us generate the email directly in one go, we were able to find creative ways by 'conversing' with it and giving it a plausible context to overcome its barriers. Here we identify how these new language models can be weaponized to be used as phishing and scamming tools which gets past our present email systems like Gmail and Outlook. However, that's hardly surprising considering they look legitimate. We want to further this work by integrating it with tools like PhEmail\cite{dionachPhEmailPhEmailisapythonopensourcephishingemailtoolthatautomatestheprocessofsendingphishingemailsaspartofasocialengineeringtest-2022-12-08} which makes sending NLG generated emails to targeted bulk userbase a keypress away.
\bibliographystyle{IEEEtran}
|
2,877,628,089,541 | arxiv | \section{Introduction}
The second planet, Venus, is similar to our own planet in terms of mass and size.
In contrast, however, Venus has a massive atmosphere composed mostly of CO$_2$,
and is shrouded by a thick layer of clouds,
which blow around the planet with a period of about four days.
This cloudy atmosphere causes a runaway greenhouse effect,
leading to surface temperatures of $\sim$740 K (Hunten, 1999).
The planet is quite different from ours in another way:
it is a slow, retrograde rotator, with obliquity of $\sim177.3^\circ$
and an inertial rotation rate of one revolution every 243.0 Earth days (see Table 1 for Venusian parameters).
Adding the latter to its orbital mean motion of one revolution every 224.7 Earth days
yields the length of Venus' solar ``day'': one diurnal cycle every 116.8 Earth days.
[INSERT TABLE 1]
\subsection{Satellites of Venus}
Another difference from the Earth is that Venus has no known moons.
A recent search using the Baade-Magellan 6.5 meter telescope at Las Campanas observatory
likely rules out moons larger than a few hundred meters (Sheppard and Trujillo, 2009).
Many authors have noted that strong gravitational perturbations from the Sun
place severe limits on the long-term stability of Venus orbiters. For example,
Buchar (1969) found that prograde moons of Venus had a limiting semi-major axis of 581,000 km,
but that retrograde satellites had a larger limiting semi-major axis of 655,000 km. In contrast,
Rawal (1986) found a prograde limit of 698,000 km, and a retrograde limit of 1,452,000 km.
For comparison, Venus’ Hill radius (sometimes called its sphere of influence) is $\sim$1,011,000 km.
Inside the above limits, numerous papers (McCord, 1966, 1968; Counselman, 1973; Burns, 1973; Ward \& Reid, 1973;
Kumar, 1977; Donnison, 1978) all showed that within the age of the Solar system, tidal friction inside Venus
would have forced large moons either to escape into heliocentric orbit, or to crash into Venus itself.
Likewise, several papers (McCord, 1968; Singer, 1970; Malcuit \& Winters, 1995)
have shown that the tidal crash of a retrograde moon could have caused Venus' slow retrograde spin.
In an interesting variation, Alemi and Stevenson (2006) suggest that a prograde moon of Venus
may have escaped, but subsequently impacted Venus, reversing its rotation.
However, satellites are not the only possible companions for a planet; it also may have ``co-orbital'' companions.
There are now four recognized types of co-orbital companions: the best known are Trojan companions,
which librate about a planet's equilateral Lagrange points L4 and L5 in tadpole-shaped orbits.
Currently thousands of asteroids are known as Trojan companions
of Neptune, Uranus, Mars, Earth, and Venus, as well as Jupiter.
``Horse-shoe'' companions librate in wide arcs about the L3 equilibrium point,
enclosing both L4 and L5 as well. Several horse-shoe companions of Earth are known,
and Saturn's small moons Janus and Epimetheus currently share a horse-shoe orbit.
``Quasi-satellites'' appear to be orbiting a planet outside of its Hill radius
in the retrograde direction (contrary to the the planet's orbital motion);
however, they are really not energetically bound to the planet, but orbiting the Sun in the prograde direction.
Such orbits also are called Distant Retrograde Orbits (DRO);
St. Cyr {\it et al.} (2000), Stramacchia {\it et al.} (2016), and Perozzi {\it et al.} (2017)
provide good descriptions of DRO-type missions associated with Earth.
Currently several quasi-satellites of Earth, Venus, Neptune, and Pluto are known.
Furthermore, many asteroids are in hybrid co-orbital resonances,
which alternate among tadpole, horse-shoe, and quasi-satellite configurations.
Finally, ``counter-orbital'' resonances are also possible,
when two objects orbit with the same period, but in nearly opposite directions
(Dobrovolskis, 2012; Morais \& Namouni, 2013, 2016).
To date, only one counter-orbital companion of Jupiter has been found (Wiegert {\it et al.}, 2017):
asteroid 2015 BZ 509, also known as (514107) Ka'epaoka'awela.
While Venus has no known moons, it does have natural co-orbital companions in high-eccentricity orbits,
namely the quasi-satellite 2002 VE68 (Mikkola {\it et al.}, 2004),
the horseshoe/quasi-satellite hybrid 2001 CK32 (Brasser {\it et al.}, 2004),
the Trojan/horseshoe hybrid 2012 XE133 (de la Fuente Marcos \& de la Fuente Marcos, 2013),
the Earth-crossing Trojan 2013 ND15 (de la Fuente Marcos \& de la Fuente Marcos, 2014),
and the possible transient Trojan 2015 WZ12 (de la Fuente Marcos \& de la Fuente Marcos, 2017).
The next section describes the Circular Restricted 3-Body Problem (CR3BP),
and analyzes its usefulness as a model for a satellite of Venus affected by the Sun's gravity;
while Section 3 applies Poincar\'e's surface of section technique to understand global satellite dynamics.
Then Section 4 describes periodic orbits around Venus in general,
while Section 5 describes a unique Venus-synchronous orbit.
Finally, Section 6 summarizes our findings, and discusses possible applications.
\section{The Circular Restricted 3-Body Problem}
Perturbed two-body dynamics is adequate to describe the behavior of co-orbitals,
but for a fuller understanding we must turn to the CR3BP,
where the behavior of a body of infinitesimal mass is affected by the gravitational force
of two massive bodies orbiting each other. For our purposes, the two massive bodies are the Sun and Venus,
and the test particle can represent a co-orbital, a quasi-satellite, or even a Venus orbiter/satellite.
\subsection{Dynamical Description}
The CR3BP is well described in most Celestial Mechanics/Astrodynamics textbooks
({\it e.g.}, Moulton, 1914; Blanco and McCuskey, 1961; Kaplan, 1976; Danby, 1992; Murray and Dermott, 1999).
However, in this work we will follow Szebeheley (1967), which the interested reader may consult for details.
Consider two bodies of masses $m_1$ and $m_2$ revolving about their common center of mass
in circular orbits due to their mutual attraction, and define the dimensionless mass ratio
\begin{equation}
\mu \equiv \frac{m_2}{m_1+m_2} .
\end{equation}
We choose the unit of mass such that $m_1 + m_2 = 1$;
then $m_1 = 1-\mu$ and $m_2 = \mu$. The distance between $m_1$ and $m_2$ is the unit of length.
Finally the unit of time is such that the gravitational constant $G$ is unity as well; then by Kepler's third law,
the mutual orbital period $P$ of $m_1$ and $m_2$ becomes $2\pi$. Now consider a third body of infinitesimal mass,
which does not influence the motion of the two massive bodies, but is influenced by them;
the study of the CR3BP is really the study of the motion of this particle.
It is convenient to use a rotating Cartesian coordinate system wherein the coordinates
of $m_1$ and $m_2$ remain fixed at ($\mu$, 0, 0) and ($\mu-1$, 0, 0), respectively.
Then this synodic frame originates at the center of mass of the system,
such that the (rotating) $x$-axis connects the two massive bodies,
and the $z$ axis lies along the direction of the angular velocity vector of their mutual orbit,
while the (rotating) y-axis completes a right-handed triad $(x,y,z)$.
The conversion between this synodic $(x,y,z)$ frame and a non-rotating (sidereal) coordinate system $(x',y',z')$
with the same origin is then
\begin{equation}
x'= x \cos(t) - y \sin(t) ,
\end{equation}
\begin{equation}
y'= x \sin(t) + y \cos(t) ,
\end{equation}
\begin{equation}
{\rm and} \; \; z' = z ,
\end{equation}
where $t$ is the time. Note that the orbital angular velocity of the two massive bodies is unity
because of our choice of the unit of time.
In the synodic (rotating) frame, the equations of motion for the test particle are
\begin{equation}
\ddot{x} - 2 \dot{y} = \partial U/\partial x ,
\end{equation}
\begin{equation}
\ddot{y} + 2 \dot{x} = \partial U/\partial y ,
\end{equation}
\begin{equation}
{\rm and} \; \; \ddot{z} = \partial U/\partial z,
\end{equation}
where
\begin{equation}
U = x^2/2 + y^2/2 + (1-\mu)/r_1 +\mu/r_2 .
\end{equation}
Here $r_1$ and $r_2$ are the distances of the test particle from $m_1$ and $m_2$ respectively, and are given by
\begin{equation}
r_1 = \sqrt{ (x-\mu)^2 +y^2 +z^2},
\end{equation}
\begin{equation}
r_2 = \sqrt{ (x-\mu +1)^2 +y^2 +z^2} .
\end{equation}
To within an arbitrary constant, the CR3BP has only one integral of motion, namely the Jacobi constant
\begin{equation}
C \equiv 2U - \dot{x}^2 -\dot{y}^2 -\dot{z}^2 +\mu[1-\mu].
\end{equation}
Note that by these conventions, $U$ is minus the usual potential energy (centrifugal plus gravitational),
while $C$ is --2 times the usual total energy (potential plus kinetic), plus a constant term $\mu[1-\mu]$.
This constant is chosen so that $C$ = 3 when $r_1 = r_2$ = 1 and $\dot{x} = \dot{y} = \dot{z}$ = 0.
\subsection{The Lagrange Points}
The three-dimensional CR3BP has three degrees of freedom, and since only one constant of motion exists,
there is no general solution in closed form to the equations of motion (5) through (7).
However, specific solutions do exist.
The equations of motion contain five equilibria, named the {\it Lagrange points}, and labeled L1, L2, L3, L4, and L5.
The first three are called the collinear Lagrange points because they all lie on the $x$ axis;
in our coordinates, L2 lies between the primary mass $m_1$ and the secondary mass $m_2$,
but L1 lies on the side of $m_2$ opposite $m_1$, while L3 lies on the side of $m_1$ opposite $m_2$.
The two remaining points L4 and L5 are called the triangular Lagrange points,
because they form equilateral triangles with $m_1$ and $m_2$ in the plane $z = 0$;
in our coordinates, L5 lies $60^\circ$ ahead of $m_2$ in its orbit about $m_1$,
while L4 lies $60^\circ$ behind $m_2$. It can be shown that the collinear points are always unstable,
but the triangular points are stable provided that $\mu < (1-\sqrt{23/27})/2 \approx$ 0.0385208965 (Szebehely, 1967).
\subsection{The Zero Velocity Curves}
It can be seen from Definition (11) of the Jacobi constant $C$ that there are constraints on the motion
of the test particle; clearly motion is only possible when $0 < \dot{x}^2 +\dot{y}^2 +\dot{z}^2 = 2U -C +\mu[1-\mu]$.
Then for a given value of $C$,
setting $C < 2U +\mu[1-\mu]$ gives Zero Velocity Curves (ZVCs) which define regions where motion is allowed.
There are good descriptions and sketches of these curves in the aforementioned references, especially Szebehely (1967).
Very briefly, at high enough values of $C$ the ZVCs form one closed oval around $m_1$,
and another around $m_2$, as well as a larger outer oval encompassing both masses;
motion of the test particle is only possible inside the inner smaller ovals or outside the larger one
(in this situation either $r_1$ or $r_2$ is very small, or $r_0 \equiv \sqrt{x^2+y^2+z^2}$ is fairly large).
As the value of $C$ is lowered to a critical value $C_2$, the two inner ovals meet at the L2 point;
and as $C$ is lowered even further, the L2 point provides an avenue through which the particle
can move between the two inner ovals. As we lower $C$ even more to another critical value $C_1$,
the oval surrounding $m_2$ ``touches'' the L1 point; and as $C$ is lowered even further,
this point provides an opening through which the particle can escape to infinity.
Continuing with this process, we reach another critical value $C = C_3$ where the ZVCs meet at the L3 point.
As we lower $C$ even further towards $C_4 = C_5$ = 3, the ZVCs shrink to smaller and smaller ovals
surrounding the triangular Lagrange points L4 and L5, and vanish completely for $C < 3$.
Note that the ZVCs are symmetric about the (rotating) $x$ axis and that,
the lower the Jacobi constant, the more space is available for the test particle to explore.
\subsection{Applicability to the Sun-Venus-satellite case}
The CR3BP assumes that the two massive bodies orbit each other in perfectly circular orbits.
Venus orbits the Sun in a slightly elliptical orbit; however, Venus' orbital eccentricity
$e_V \approx 0.000677$ is the smallest of any planet in the Solar System (Murray and Dermott, 1999),
so to first order in $e_V$ we are justified in neglecting its eccentricity.
A second effect we ignore is that Venus is not perfectly spherical.
Its dynamical oblateness dominates solar perturbations for orbits inside
about 2.6 Venus radii (Capderou, 2005).
Another perturbation we neglect is the gravity of the other planets.
We find that the perturbation of the Earth on satellites of Venus is roughly four orders of ten smaller
than the solar perturbation. The other planets have even smaller influences on Venus satellites.
A massive satellite also would be subject to tidal perturbations.
We estimate that even a close moon of Venus would have to have a mass on the order
of $3 \times 10^{17}$ kg (about 30 times the mass of Mars' larger moon Phobos)
for tides to be as important as solar perturbations;
for more remote satellites, this mass increases as the eighth power of the distance.
Therefore we ignore tidal perturbations as well.
[INSERT TABLE 2]
\section{Surfaces of Section}
The solution to the CR3BP may be viewed as a flow in a six-dimensional phase space.
However, the system may be confined to only four dimensions by setting $z$ and $\dot{z}$ to zero
in Eqs. (5) through (7); thus motion takes place in the $(x,y)$ plane.
Then this becomes the so-called Plane Circular Restricted 3-Body Problem. Furthermore,
in the Plane CR3BP the existence of the Jacobi constant constrains the phase space motion to take place
on a three-dimensional manifold. We now consider the intersection of a two-dimensional plane with this manifold.
Following Smith and Szebehely (1992) and Alvarellos (1996), we choose $y=0$, {\it i.e.} the $(x,\dot{x})$ plane;
furthermore we investigate only the positive crossings ($\dot{y} > 0$).
This is the concept of the Poincar\'e surface of section (Lichtenberg and Lieberman, 1992).
We used many different initial conditions to construct our surfaces of section,
and wrote a program to propagate each initial condition forward in time
with a fifth-order Runge-Kutta (Cash-Karp) variable step-size integrator (Press {\it et al.}, 1992).
Figure 1 shows a surface of section for $C$ = 3.0010,
a relatively high value of the Jacobi constant compared to $C_1$ and $C_2$ (see Table 2),
so that the Zero Velocity Curves around Venus are closed.
In this plot the location of Venus is approximately (--1.0, 0.0);
on either side of the vertical line at $x \approx -1$,
we can see sets of concentric curves known as ``invariant curves'' (Lichtenberg and Lieberman, 1992).
Each of these curves represents a quasi-periodic orbit,
while the point at the center of each family of concentric curves
represents a simply periodic orbit, a topic which we will revisit later.
[INSERT FIGURE 1: C = 3.0010]
In general, those concentric curves on the right-hand side of Fig. 1 ($x > -1$) represent prograde orbits;
these objects orbit Venus in the same direction as Venus orbits the Sun. Those on the left ($x < -1$)
represent retrograde orbits, which orbit Venus in the opposite direction as Venus orbits the Sun.
The dash-dot curves enveloping all orbits represent the projection of the ZVC
on the $(x,\dot{x})$ plane (H\'enon, 1970).
For this relatively high value of the Jacobi constant, objects revolve about Venus in small orbits, close to the planet.
Note that we see a somewhat random scattering of points near the central line $x = -1$;
these points represent chaotic orbits, so here we see the onset of chaos.
For comparison, Fig. 2 plots a surface of section for the critical value $C_1$ = 3.000776... (Table 2).
For this value, the ZVCs around Venus are just barely closed at the L1 point, located at (--1.00937..., 0);
but the ZVCs are open at the L2 point, at (--0.99068..., 0), so that satellites can escape to inner heliocentric orbits.
Indeed this is an effective mechanism for Venus to lose prograde satellites,
since chaotic orbits tend to ``explore'' all of the phase space available to them (Lichtenberg and Lieberman, 1992).
[INSERT FIGURE 2: C = C1 = 3.000776...]
Finally, Fig. 3 plots a surface of section for $C$ = 3.0006,
a relatively low value of the Jacobi constant relative to $C_1$ and $C_2$,
so that the ZVC around Venus are now wide open at both the L1 and L2 points.
Note how empty the right side of the plot is,
as most prograde orbits have turned chaotic and escaped to heliocentric space;
but most retrograde orbits still show signs of stability, although we see some hints of developing chaos.
These surfaces of section demonstrate the well-known fact that retrograde orbits are more stable than prograde ones.
[INSERT FIGURE 3: C= 3.0006]
\section{Periodic Orbits}
As mentioned in the previous section, as the invariant curves in the surface of section plots shrink down to a point,
that point itself represents a (simply) periodic orbit. A periodic orbit has the property that
$x$, $y$, $z$, $\dot{x}$, $\dot{y}$, and $\dot{z}$ at time $t$ are all the same as at time $t +T$,
where $T$ is the period of that orbit.
Note that $T$ represents the period only in a synodic frame of reference rotating with period $P$; in a non-rotating
(sidereal) frame, a prograde (or ``direct'') orbit completes one revolution during a time interval $T_D$, where
\begin{equation}
1/T_D = 1/T +1/P .
\end{equation}
In contrast, a retrograde orbit of period $T$ in a synodic frame completes
one revolution in a non-rotating frame during a time interval $|T_R|$, where
\begin{equation}
1/T_R = 1/T -1/P .
\end{equation}
In what follows we search for periodic orbits about the secondary mass, namely Venus.
We have obtained our periodic orbit solutions via a manual/iterative surface-of-section method.
A more rigorous approach would be to use continuation methods, such as those described by Davis (1962).
\subsection{Prograde Periodic Orbits (g and g' families)}
Inspection of Figure 1 reveals the concentric invariant curves on both sides, and from the right-hand side
we can obtain approximate initial conditions for a simply-periodic prograde orbit: $x(0) \approx$ --0.997 and $\dot{x}(0)$ = 0 (this
is where the curves shrink down to a point). From the definition of the surface of section, $y(0)$
vanishes, while the value of $\dot{y}(0)$ can be obtained from Equation 11. Using this information
we manually iterate until we find the periodic orbit (i.e., the trajectory exactly repeats).
The ``measured'' period $T$ of this orbit is about 0.318; recall that the orbital period of Venus around
the Sun is 224.7 days, equivalent to $2\pi$, from which we get $T$ = 11.4 days.
This prograde, periodic orbit around the secondary body (Venus in our case)
belongs to the g-family of orbits, which originate as close prograde satellite orbits (H\'enon 1969).
The same procedure allows us to obtain initial conditions for other prograde periodic orbits
corresponding to the aforementioned surface of section plots.
As we move from high to low values of $C$ and reach the critical value $C_2$,
an interesting thing occurs: the prograde periodic orbits ``split''.
In other words, for a given value of $C < C_2$ we find \textit{two} simply-periodic orbits rather than just one.
We see hints of this in the two islands floating in the chaotic region along the x-axis seen on the right-hand side of Fig. 2.
This bifurcation phenomenon was already noted by H\'enon (1969), and he assigned these prograde orbits to the g'-family.
There is a unique correspondence between a given value of $C$ and a unique g-family prograde periodic orbit,
while this is not true for the g'-family (H\'enon, 1969).
Detailed initial conditions for these orbits are shown in Table 3.
[INSERT TABLE 3]
Figure 4 plots the g and g' families of orbits around Venus from Table 3, with orbital periods in the range 11.4 $\le T \le$ 60.7 days.
For comparison, the dotted circle represents Venus' Hill sphere, with radius
\begin{equation}
R_H = (\mu/3)^{1/3} \approx 0.00934 ;
\end{equation}
while each of the Lagrange points L1 and L2 is marked with an X.
Note that the distance from Venus to either L1 or L2 is approximately equal to its Hill radius.
[INSERT FIGURE 4]
The innermost g orbit in Fig. 4 corresponds to a Jacobi constant $C$ = 3.0015. To obtain smaller, tighter orbits we would
need to go to even higher values of $C$. Such orbits would resemble Keplerian orbits as $C$ increases still further.
At the other end of the spectrum, we can obtain larger periodic g' orbits at lower Jacobi constant values;
these become even more distorted until we reach a collision orbit.
Thereafter more prograde periodic orbits are found until again
we reach a second collision orbit, and so on (Szebehely 1967; H\'enon 1969).
\subsection{Retrograde Periodic Orbits (f family)}
By applying the same approach from the previous section to the retrograde orbits,
we obtain the initial conditions shown in Table 4.
Figures 5 through 7 display the orbits thus obtained;
these represent members of the f family of periodic orbits,
which originate as close retrograde satellite orbits (H\'enon, 1969).
[INSERT TABLE 4]
Figure 5 shows the inner periodic f orbits; note that in contrast to the prograde case,
there is no splitting, and the orbits shown here are very close to circular.
Again, the orbits become small and nearly Keplerian for high values of $C$.
Surfaces of section are useful to obtain initial conditions,
but they are not strictly necessary: by carefully extrapolating from existing periodic orbits we can obtain further
periodic orbits without them.
[INSERT FIG. 5]
Figure 6 shows larger periodic f orbits; as in Fig. 4, the dotted circle represents Venus' Hill sphere,
for comparison, while again each of the Lagrange points L1 and L2 is marked with an X.
Note that the orbit labeled 8 is comparable in size to Venus' Hill sphere.
At some point between the orbits labeled 7 and 8, trajectories become noticeably non-circular,
with their $y$ diameter growing larger than their $x$ diameter.
[INSERT FIG. 6]
Finally, Fig. 7 shows the evolution of even larger retrograde periodic f orbits.
As we lower the value of $C$, trajectories become even more distorted and kidney-shaped.
Note how orbit 13 crosses the dashed circle of radius 1.3825 representing the Earth's orbit around the Sun,
as well as the dotted circle of radius 0.5352 representing Mercury's orbital semi-major axis;
such an orbit cannot be expected to be stable in the real Solar system.
The same applies to orbit 14; eventually we reach orbit 15, which collides with the Sun.
This only represents the end of the first phase of retrograde periodic orbits;
beyond the collision orbit we can find further periodic orbits that loop around the Sun
and return to ``orbit'' the whole system (Szebehely, 1967).
[INSERT FIG. 7]
\section{Synchronous Satellites}
Now reconsider the otherwise undistinguished retrograde f orbit number 9, just outside Venus' Hill sphere in Fig. 6.
Table 4 shows that its period $T$ in a synodic frame is 116.8 days,
while its period $T_R$ in a non-rotating (sidereal) frame is 243 days.
These are just the same as Venus' rotation period in those frames;
thus a particle in this orbit would appear to hover almost stationary over Venus' surface,
like a synchronous satellite.
\subsection{Dynamical Considerations}
Figure 8 plots both the orbit of Venus and the synchronous orbit in a sidereal frame with the Sun at the origin.
To produce Fig. 8, we compute the particle's position in the sidereal frame via Eqs. (2) and (3) (we confine ourselves
to the $xy$ plane), while the Sun's sidereal position as a function of time is given by ($\mu \cos t$, $\mu \sin t$);
then we determine the relative position of the particle with respect to the Sun, obtaining
\begin{equation}
x'_{\rm rel} = (x-\mu) \cos(t) - y \sin(t)
\end{equation}
and
\begin{equation}
y'_{\rm rel} = y \cos(t) + (x-\mu) \sin(t) .
\end{equation}
[INSERT FIGURE 8]
In Fig. 8a we see the two orbits are very close at this scale. Starting at the leftmost position at time $t$ = 0,
the particle is farther away from the Sun than Venus (Fig. 8b); but moving counter-clockwise, at $t = P/4$ (56 days)
the particle is now closer to the Sun (Fig. 8c). The pattern then repeats, since at $P/2$ (112 days)
the particle is again farther away from the Sun; while at $3P/4$ (169 days), the particle is closer again.
The particle's distance $r_2$ from the center of Venus
ranges from $\sim1.2 \times 10^6$ km to $\sim1.6 \times 10^6$ km ($\sim200$ to $\sim260$ Venus radii),
with an oscillation period of only $T/2 \approx$ 58.3 day. In contrast,
the particle's distance $r_1$ from the Sun varies between 107.0 $\times 10^6$ and 109.4 $\times 10^6$ km,
but with an average value approximately equal to Venus' semi-major axis (108.2 $\times 10^6$ km),
and an oscillation period $T$ of $\sim$116.8 days. For comparison,
Venus' Hill radius $R_H \approx$ 1.011 $\times 10^6$ km in physical units.
The particle's sidereal speed $V$ is
\begin{equation}
V^2 = ( dx'_{\rm rel}/dt )^2 + ( dy'_{\rm rel}/dt )^2
\end{equation}
\[
= (x^2+y^2) + (\dot{x}^2+\dot{y}^2) + 2(x\dot{y}-y\dot{x}) - 2\mu(x+\dot{y}) + \mu^2 .
\]
Converting from CR3BP units to physical units, we find that $V$ varies between 34.4 and 35.6 km/s
for a synchronous satellite of Venus, with an average value approximately equal to Venus' orbital speed (35.0 km/s);
the period of these oscillations is again $T \approx 116.8$ days.
The particle's heliocentric semi-major axis $a$ can be obtained from the vis-viva equation
\begin{equation}
\frac{(1-\mu)}{2a} = \frac{1-\mu}{r_1} -V^2/2
\end{equation}
(Danby, 1992). For a synchronous satellite of Venus, $a$ varies between 106.6 and 109.9 $\times 10^6$ km;
its average value is approximately equal to Venus' semi-major axis, while its period of oscillations is again $T$.
To obtain the particle's heliocentric eccentricity $e$,
we also need the orbit's specific angular momentum $h$ with respect to the Sun.
Given the heliocentric position (Eqs. 15 and 16) and velocity, after some algebra we obtain
\begin{equation}
h = r^2_1 + (x-\mu)\dot{y} - y \dot{x} .
\end{equation}
Then the eccentricity $e$ can be obtained from
\begin{equation}
h^2 = a(1-e^2)
\end{equation}
(Danby, 1992). For a synchronous satellite of Venus, $e$ varies between 0.0204 and 0.0267,
with an average of $\sim$0.0237 and an oscillation period of just $T/2$.
Table 5 summarizes the characteristics of this synchronous orbit.
[INSERT TABLE 5]
Figure 9 shows orbit number 9 again, but this time in a non-rotating (sidereal) frame centered on Venus.
In this frame, note how the quasi-satellite circulates slowly clockwise about Venus, in a nearly square trajectory.
For comparison, Fig. 10 shows orbit number 9 once more, but this time in a rotating frame centered on Venus,
and fixed in its body. In this frame,
the particle circulates counter-clockwise twice during each $T$ = 116.8-day period around an apparent oval path
of longitudinal (east-west) radius $\sim2.5 \times 10^5$ km, and a radius of $\sim1.6 \times 10^5$ km
in the radial direction, centered at a point fixed in this frame $\sim1.4 \times 10^6$ km from the center of Venus.
[INSERT FIG. 9 - SYNCHROSAT IN NON-ROTATING FRAME]
[INSERT FIG. 10 - SYNCHROSAT IN BODY FRAME]
Thus a spacecraft in this orbit would appear to hover indefinitely above a nearly fixed point on Venus' equator,
like a synchronous satellite, while librating in longitude by $\sim \pm11^\circ$, with a period of 58.4 days.
Its libration in latitude due to Venus' obliquity is only $\sim \pm3^\circ$, with a period of 243 days.
Synchronous satellites have long been considered impossible for both Mercury and Venus,
because they were thought to be unstable ({\it e.g.}, Capderou, p. 467),
unlike synchronous satellites of the Earth or Mars.
A straightforward application of Kepler's Third Law to synchronous orbits around Venus
gives a semi-major axis of 1,530,500 km (Wertz, 2011).
This lies outside the usual stability limit of $\sim R_H/2$,
corresponding to an orbital period $p$ of only $\sim$0.35 times
the planet's heliocentric orbital period $P$ (Hamilton \& Burns, 1991).
However, the above limit applies only to {\it prograde} satellites.
We have seen from the surface of section plots that retrograde orbits are more stable than prograde ones.
Alvarellos (1996) found the same stability limit as Hamilton and Burns (1991) did for prograde satellites,
while he found a limit of $\sim R_H$ for retrograde satellites {\it in circular orbits}.
When the assumption of circular orbits is relaxed, however
{\it there is no formal outer limit to retrograde satellites}
({\it e.g.}, Jackson, 1913; H\'enon, 1969, 1970; Benest, 1971, 1979).
Specifically, the f family of retrograde, periodic orbits are very stable;
indeed, Figs. 5 through 7 indicate very large, retrograde trajectories.
Henon and Guyot (1970) state that if $0 \leq \mu \leq$ 0.0477
(a condition certainly met by the Sun-Venus system),
then ``... essentially all retrograde periodic orbits around the lighter $m_2$ body are stable''.
Thus Mercury, with a prograde obliquity of only $0.033^\circ$, an orbital period $P$ of 87.97 days, and a long
rotation period of 58.65 days = $2P/3$, cannot retain synchronous satellites against the tidal pull of the Sun.
In contrast, because Venus is a slow {\it retrograde} rotator, it can retain synchronous quasi-satellites indefinitely.
Of course, a spacecraft in a synchronous orbit is not really affected by Venus' rotation;
its synodic period just happens to match the planet's diurnal period.
The principal effect of Venus' gravity is just to keep the object's epicenter
(the center of its epicycle in the synodic frame) librating about Venus' location.
\subsection{Stability Considerations}
We have tested the stability of this venusynchronous orbit with the freely-available NASA software
called the General Mission Analysis Tool, or GMAT (version R2018a)\footnote{https://sourceforge.net/projects/gmat/}.
Advantages of using GMAT as a propagator are that it is very easy
(a) to switch central bodies; (b) to turn forces on or off; and (c) to find the spacecraft groundtrack.
To begin, we included only the point-mass gravitational forces of the Sun and Venus.
We transformed the initial conditions of orbit 9 from Table 4 to a heliocentric,
Cartesian state vector in the J2000 frame at its Epoch (1-Jan-2000, 12:00:00.0 UTC),
and we used the Runge-Kutta 89 integrator to propagate them forward in time for 10 Earth years.
The resulting orbit is co-planar with Venus' orbit, and the putative spacecraft
always remains just outside Venus' Hill radius for the duration of the integration.
Over this time period, the sub-satellite latitude stays within $\sim2.7^\circ$ of Venus' equator,
while its longitude spans less than $60^\circ$. Over a full rotation of Venus (243 days),
the sub-satellite longitude spans only $\sim26^\circ$.
Next we took the GMAT-baseline case and turned on the different individual forces one by one,
to investigate their effects on the groundtrack.
We found that solar radiation pressure has negligible effects on a ten-year time-scale,
using values typical for a geosynchronous telecommunications satellite:
a Solar Radiation Pressure (SRP) coefficient of 1.8,
along with an area-to-mass ratio value of 0.04 m$^2$/kg.
The gravitational effects of Mercury and Mars were negligible as well; however,
we found that the gravitational effect of the Earth is to drift the groundtrack noticeably eastward,
while the gravitational effect of Jupiter is to drift the groundtrack in the opposite direction.
Figure 11 plots this groundtrack, taking into account all realistic forces (point gravitational effects
of the Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune, as well as SRP);
their overall effect is to drift the pattern westward while the sub-satellite latitude always remains near the equator.
The groundtrack drifts westward, regardless of the starting Epoch;
if it is deemed necessary to keep the groundtrack constrained to a certain small region,
the spacecraft may be equipped with a propulsion system to counteract the main perturbative effects
({\it i.e.}, Earth and Jupiter). Finally, Fig. 12 plots the evolution of the heliocentric orbital elements
with all relevant forces previously mentioned turned on; the orbit is highly stable.
[INSERT FIGURE 11]
[INSERT FIGURE 12]
\section{Discussion/Conclusions}
We investigated co-orbital companions of the planet Venus using the planar CR3BP as a model,
with the Sun and Venus as the massive bodies. We used surfaces of section
to find an interesting orbit whose sidereal period matches Venus' rotational period of 243 days;
an object in this orbit would in principle remain nearly stationary above a point on the equator of the planet,
akin to a stationary orbiter. In practice the groundtrack is not strictly stationary,
but oscillates about a fixed longitude on the equator of Venus (it also oscillates slightly in latitude).
This orbit lies beyond Venus` Hill radius, and a spacecraft located there is more properly said to orbit the Sun,
and not Venus itself.
Since this particular distant retrograde orbit (DRO) is stable in the CR3BP,
we also investigated its stability in a more realistic ephemeris model with all additional relevant forces included.
We have also found the orbit to be stable in the more realistic model.
Note that a simple application of Kepler's Third Law to compute the distance to a stationary satellite of Venus
also gives a (different) value beyond the Hill radius,
but placing a spacecraft in a circular orbit at that distance is not viable
because the satellite would be quickly lost to heliocentric space.
The only way to have a spacecraft in a stable venusynchronous orbit
is to place it in the specific orbit which we have found in this paper;
it is a heliocentric orbit located in Venus' orbital plane, perturbed by the gravity of Venus itself.
Venusynchronous satellites would be rather far away from the planet;
from such a vantage point, Venus' angular diameter ranges from $0.4^\circ$ to $0.6^\circ$,
about as big as our Moon looks from Earth ($0.5^\circ$).
In contrast, the Earth spans a diameter of $17.4^\circ$ from a geostationary satellite (Soop, 1994).
Despite this distance, one prime advantage of venusynchronous satellites
is that of continuous monitoring of the full planetary disk at a nearly fixed longitude;
there are several other advantages as well (Montabone {\it et al.}, 2020).
Venusynchronous satellites could be equipped with appropriate imaging equipment,
since Venus' surface is invisible to telescopic eyes and its cloudy atmosphere
is transparent in only a few spectral windows in the near-infrared (Hunten, 1999).
They could also be used to maintain constant contact for long intervals with long-lived landers,
such as those NASA is trying to develop (Kremic, Hunter, \& Tolbert, 2019);
or continously to monitor specific surface features, such as Idunn Mons,
a young volcano located at $46^\circ$ south latitude in the Imdr Regio volcanic province (D'Incecco {\it et al.}, 2017),
especially given recent reports of possible present-day volcanism on Venus
(Filiberto {\it et al.}, 2020; G\"{u}lcher {\it et al.}, 2020; see also Esposito, 1984 and Marcq {\it et al.}, 2013).
One advantage of venusynchronous satellites over geostationary ones is that of power:
in the vicinity of the Earth, $\sim$1.4 kW/m$^2$ of solar flux are available for spacecraft operations (Soop, 1994);
but for Venus spacecraft, the equivalent value is $\sim$2.6 kW/m$^2$, almost twice as much.
For the latter, and assuming that the synchrosat bus is similar to geostationary ones,
the solar array drive assemblies (SADAs) need to complete a full rotation in 116.8 days to track the Sun.
However, these Venus synchrosats would experience more challenging thermal issues than the typical Earth satellite.
Geosynchronous satellites require batteries to store energy for periodic intervals in Earth's shadow;
they undergo two eclipse seasons centered around each equinox, with a maximum eclipse duration of about 72 minutes.
In addition, operators need to account for less-frequent but occasionally deep lunar eclipses.
In contrast, Venus synchrosats would never be in total eclipse, because Venus' umbra
(total shadow, of length = $a_V R_V/(R_{Sun} -R_V) \approx$ 949000 km) lies inside its Hill sphere.
Because of perturbations, geostationary spacecraft typically need to perform periodic maneuvers
to maintain their orbital slots, {\it i.e.}, ``stationkeeping'' (Soop, 1994).
An uncontrolled geosynchronous satellite will oscillate back and forth in longitude
around the nearest of the two stable equilibrium points with a period of more than two years.
The main perturbations affecting mission operations are luni-solar gravities (affecting mainly the spacecraft's
inclination); the asymmetric geopotential, the main terms being $J_2$ (oblateness, affecting inclination)
and $J_{22}$ (equatorial ellipticity, affecting the longitude); and SRP (affecting the eccentricity).
The dynamical environment for a putative venusynchronous satellite is quite different, though.
Since such a spacecraft is far away from Venus, the $J_2$ and $J_{22}$ perturbations would be minimal.
Regarding luni-solar perturbations, of course Venus has no moon, and the Sun-Venus arrangement
is in fact what keeps the spacecraft in its synchronous orbit; hence there is no need to correct for these effects.
The only significant perturbation a spacecraft in this orbit may need to correct for are the effects of SRP
on the eccentricity. Note that SRP is strongly dependent on the spacecraft's area-to-mass ratio;
therefore the smaller this value, the less propellant will be needed.
Orbit insertion requirements for a venusynchronous satellite
also should be much less demanding than those for a close satellite.
A small Venus synchrosat might be dropped off by a vehicle on its way to Venus,
such as NASA's VERITAS and DAVINCI+ missions, or ESA's EnVision mission,
or by one flying past Venus {\it en route} to another destination.
For comparison, a retrograde satellite of Venus with a period of only 4 days
(synchronous with the ``super-rotation'' of Venus' cloud layer) could be useful
for long-term monitoring of cloud features, trace gases, or a long-lived balloon.
A simple calculation indicates that such a satellite would have an orbital
semi-major axis of $\sim100,000$ km $\approx$ 16.5 Venus radii,
much more comparable to synchronous satellites of Earth or Mars.
\section{Acknowledgments}
We would like to thank L. Dones for some references, as well the GMAT, Tex Maker, and GNU Octave teams.
Thanks also to L. Plice, N. Alem and E. Tapio. JLA thanks his family for their patience
(Alejandra; Jose Jr; Isabella and Daniel; my `pichus'). Dedicated to the memories of Y. Geremia and N.E.P.
JLA also would like to thank Ravi Mathur of Emergent Space for previewing the manuscsript;
we also thank Giovanni Valsecchi and an anonymous referee for reviewing it.
\section{References}
\setlength{\parindent}{-0.2in}
Alemi, A., and D. Stevenson (2006). Why Venus has no moon (abstract). {\it B.A.A.S.} {\bf 38}, 491.
Alvarellos, J. L. (1996). Orbital Stability of Distant Satellites of Jovian Planets.
M.Sc. thesis, San Jose State University.
Benest, D. (1971). Elliptic restricted problem for Sun-Jupiter:
Existence of stable retrograde satellites at large distance.
{\it Astronomy \& Astrophysics} {\bf 13}, 157--160.
Benest, D. (1979). Large satellite orbits of Jupiter
and Earth in the perturbed restricted three-body problem.
Pp. 225--230 in {\it Natural and Artificial Satellite Motion},
ed. P. Nacozy and S. Ferraz-Mello. U. of Texas Press, Austin.
Blanco V. M. and S. W. McCuskey (1961). Basic Physics of the Solar System.
Addison-Wesley Publishing Co., Massachussetts.
Brasser, K., K. A. Innanen, M. Connors, C. Veillet, P. Weigert, S. Mikkola, and P. W. Chodas (2004).
Transient co-orbital asteroids. {\it Icarus} {\bf 171}, 102-109.
Buchar, E. (1969). A short study on the motion of artificial satellites
of Mercury and Venus (abstract). P. 41 in COSPAR-IAU-IAG/IUGG-IUTAM
Symposium {\it Dynamics of Satellites}, Prague; ed. B. Morando.
Burns, J. A. (1973). Where are the satellites of the inner planets ?
{\it Nature Physical Science} {\bf 242}, 23--25.
Capderou, M. (2005). Satellites: Orbits and Missions. Springer-Verlag, France.
Counselman, C. C. III (1973). Outcomes of tidal evolution. {\it Astrophysical J.} {\bf 180}, 307--314.
Danby, J. M. A. (1992). Fundamentals of Celestial Mechanics. Willman-Bell, Inc. Richmond.
Davis, H. T. (1962). Introduction to Nonlinear Differential and Integral Equations. Dover, New York.
de la Fuente Marcos, C., and R. de la Fuente Marcos (2013).
Asteroid 2012 VE133: A transient companion to Venus. {\it M.N.R.A.S.} {\bf 432}, 886--893.
de la Fuente Marcos, C., and R. de la Fuente Marcos (2014).
Asteroid 2013 ND15: Trojan companion to Venus, PHA to the Earth.
{\it M.N.R.A.S.} {\bf 439}, 2970--2977.
de la Fuente Marcos, C., and R. de la Fuente Marcos (2017).
Transient co-orbitals of Venus: An update.
{\it Research Notes of the AAS} {\bf 1:1}.
D'Incecco, P., N. M\"uller, J. Helbert, M. D'Amore (2017).
Idunn Mons on Venus: Location and extent of recently active lava flows.
{\it Planet. Space Sci.} {\bf 136}, 25–-33.
Dobrovolskis, A. R. (2012). Counter-orbitals: another class of co-orbitals.
{\it AAS Division for Planetary Sciences Meeting} {\bf 44}, abstract 112.22 .
Donnison, J. R. (1978). The escape of natural satellites from Mercury and Venus.
{\it Astrophysics and Space Science} {\bf 59}, 499--501.
Esposito, L. W. (1984). Sulfur Dioxide: Episodic injection shows evidence for active Venus volcanism.
{\it Science} {\bf 223}, 1072--1074.
Filiberto, J., D. Trang, A. H. Treiman, and M. Gilmore (2020). Present-day volcanism on Venus
as evidenced from weathering rates of olivine. {\it Science Advances} {\bf 6}, 1--4.
G\"{u}lcher, A. J. P., T. V. Gerya, L. G. J. Montési and J. Munch (2020).
Corona structures driven by plume-lithosphere interactions and evidence
for ongoing plume activity on Venus. {\it Nature Geoscience} {\bf 13}, 547--554.
Hamilton, D. P., and J. A. Burns (1991). Orbital stability zones about asteroids. {\it Icarus} {\bf 92}, 118--131.
H\'enon, M. (1969). Numerical exploration of the restricted problem, V.
Hill's case: periodic orbits and their stability. {\it Astronomy and Astrophysics} {\bf 1}, 223--238.
H\'enon, M. (1970). Numerical exploration of the restricted problem, VI.
Hill's case: non-periodic orbits. {\it Astronomy and Astrophysics} {\bf 9}, 24--36.
H\'enon, M., Guyot, M. (1970). Stability of periodic orbits in the restricted problem.
In Periodic Orbits, Stability and Resonances, ed. G. E. O. Giacaglia.
Dodrecht-Holland: D. Reidel Publishing Company.
Hunten, D. (1999). Venus: Atmosphere. Pp. 147--159, in Encyclopedia of the Solar System,
ed. P. R. Weissman, L. McFadden, and T. V. Johnson. Academic Press.
Jackson, J. (1913). Retrograde satellite orbits. {\it M.N.R.A.S.} {\bf 74}, 62--82.
Kaplan, M. H. (1976). Modern Spacecraft Dynamics and Control. Wiley and Sons, New York.
Kremic, T., G. Hunter, and C. Tolbert (2019). Status of developments
toward long-lived Venus landers. {\it EPSC Abstracts} {\bf 13}, 2068.
Kumar, S. S. (1977). The escape of natural satellites from Mercury and Venus.
{\it Astrophysics and Space Science} {\bf 51}, 235--238.
Lichtenberg, A. J., and M. A. Lieberman (1992). Regular and Chaotic Dynamics (2d Ed). Springer-Verlag, New York.
Malcuit, R. J., and R. R. Winters (1995). Numerical simulation of retrograde gravitational capture
of a satellite by Venus: Implications for the thermal history of the planet (abstract).
{\it L.P.S.C.} {\bf 26}, 885--886.
Marcq, E., J.-L. Bertaux, F. Montmessin, and D. Belyaev (2013).
Variations of sulphur dioxide at the cloud top of Venus's dynamic atmosphere.
{\it Nature Geoscience} {\bf 6}, 25--28.
McCord, T. B. (1966). Dynamical evolution of the Neptunian system. {\it Astronomical J.} {\bf 71}, 585--590.
McCord, T. B. (1968). The loss of retrograde satellites in the solar system.
{\it J. Geophysical Research} {\bf 73}, 1497--1500.
Mikkola, S., R. Brasser, P. Wiegert, and K. Innanen (2004).
Asteroid 2002 VE 68, a quasi-satellite of Venus. {\it M.N.R.A.S.} {\bf 351}, L63--L65.
Montabone, L., N. Heavens, and 38 co-authors (2020). Observing Mars from areostationary orbit:
benefits and applications. White paper submitted to the Planetary Science and Astrobiology Decadal Survey 2023--2032.
Morais, M. H. N., and F. Namouni (2013).
Retrograde resonance in the planar three-body problem. {\it Cel. Mech. Dyn. Astron.} 125, 91–106.
Morais, M. H. N., and F. Namouni (2016).
A numerical investigation of coorbital stability and libration in three dimensions.
{\it Cel. Mech. Dyn. Astron.} {\bf 125}, 91–106.
Moulton, F. R. (1914). An Introduction to Celestial Mechanics (2d Ed).
MacMillan. Reprinted in 1970 by Dover, New York.
Murray, C. D., and S. F. Dermott (1999). Solar System Dynamics. Cambridge U. Press, Cambridge.
Perozzi, E., M. Ceccaroni, G. B. Valsecchi, and A. Rossi (2017).
Distant retrograde orbits and the asteroid hazard. {\it The European Physical Journal Plus} {\bf 132}, 1--9.
Press, W., {\it et al.} (1992). Numerical Recipes in FORTRAN. Cambridge U. Press, Cambridge.
Rawal, J. J. (1986). Possible satellites of Mercury and Venus.
{\it Earth, Moon, and Planets} {\bf 36}, 135--138.
Sheppard, S. S., and C. A. Trujillo (2009). A survey for satellites of Venus. {\it Icarus} {\bf 202}, 12--16.
Singer, S. F. (1970). How did Venus lose its angular momentum? {\it Science} {\bf 170}, 1196--1198.
Sjogren, W. L., W. B. Banerdt, P. W. Chodas, A. S. Knopliv, G. Balmino, J. P. Barriot, J. Arkani-Hamed,
T. R. Colvin, and M. E. Davis (1997). The Venus gravity field and other geodetic parameters.
Pp. 1125--1161 in Venus II, ed. S. W. Bougher, D. M. Hunten, and R. J. Phillips. U. of Arizona Press.
Smith, R. H., and V. Szebehely (1992).
The onset of chaotic motion in the restricted problem of three bodies.
{\it Cel. Mech. Dyn. Astron.} {\bf 56}, 409--425.
Soop, E. M. (1994). Handbook of Geostationary Orbits. Springer/Kluwer.
St. Cyr, O. C., M. A. Mesarch, H. M. Maldonado, D. C. Folta, A. D. Harper, J. M. Davila, and R. R. Fisher (2000).
Space Weather Diamond: a four spacecraft monitoring system.
{\it J. of Atmospheric and Solar-Terrestrial Physics} {\bf 62}, 1251--1255.
Standish, E. M. (1998). JPL Planetary and Lunar Ephemerides, DE405/LE405, JPL IOM 312.F-98-048.
Stramacchia, M., C. Colombo, and F. Bernelli-Zazzera (2016).
Distant retrograde orbits for space-based Near Earth Objects detection.
{\it Advances in Space Research} {\bf 58}, 967--988.
Szebehely, V. G. (1967). Theory of Orbits. Academic Press, New York.
Ward, W. R., and M. J. Reid (1973). Solar tidal friction and satellite loss. {\it M.N.R.A.S.} {\bf 164}, 21--32.
Wertz, J. R. (2011). Orbits and Astrodynamics.
Pp. 197--234 in Space Mission Engineering: The New SMAD,
eds. J. R. Wertz, D. F. Everett, and J. J. Puschell. Microcosm Press.
Wiegert, P., M. Connors, and C. Veillet (2017).
A retrograde co-orbital asteroid of Jupiter.
{\it Nature} {\bf 543}, 687–689.
\vspace*{0.2in}
\begin{quotation}
\noindent
Table 1. Properties of Venus.
\end{quotation}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Parameter & Symbol & Value & Source \\
\hline
& & 1.082089 $\times 10^8$ km & \\
Semi-major axis & $a_V$ & (0.72333199 AU) & Murray \& Dermott (1999) \\
Eccentricity & $e_V$ & 0.00677323 & " \\
Obliquity & & $177.3^\circ$ & " \\
Inclination to Ecliptic & & $3\dotdeg39$ & \\
Gravitational Parameter & $Gm_p$ & 324858.601 & \\
& & $\pm$ 0.014 km$^3$/s$^2$ & Sjogren {\it et al.} (1997) \\
Equatorial Radius & $R_p$ & 6051.0 km & " \\
Hill Radius & $R_H$ & 1011000 km & \\
Orbital Period & $P$ & 224.7 days & \\
Rotation Period & & 243.0 days & \\
\hline
\end{tabular}
\end{center}
\newpage
\noindent
Table 2. Sun-Venus CR3BP Parameters. The value of $\mu$ is computed using the $GM$ values for Venus (see Table 1)
and the Sun (1.3271244002 $\times 10^{11}$ km$^3$/s$^2$ Standish, 1998).
Locations of the Lagrange points are given in the synodic (rotating) frame.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Parameter & Value & Page ref. in Szebehely (1967) \\
\hline
Mass Parameter $\mu$ & 2.44783236410728 $\times 10^{-6}$ & p. 217 \\
\hline
Location of the L1 point & (-1.0093710166, 0) & " \\
Location of the L2 point & (-0.9906822994, 0) & p. 221 \\
Location of the L3 point & (+1.0000010199, 0) & p. 225 \\
Location of the L4 point$^{\dagger}$ & (-0.4999975522, +0.8660254038) & \\
Location of the L5 point$^{\dagger}$ & (-0.4999975522, -0.8660254038) & \\
\hline
$C_1$ & 3.0007768995 & p. 217 \\
$C_2$ & 3.0007801633 & p. 221 \\
$C_3$ & 3.0000048957 & p. 225 \\
$C_4$ & 3 & \\
$C_5$ & 3 & \\
\hline
\end{tabular}
\end{center}
{$\dagger$ The locations of the L4 and L5 points are ($\mu -0.5$, $\pm\sqrt{3}/2$). }
\vspace*{0.2in}
\noindent
Table 3. Prograde periodic orbits around Venus: Initial conditions in CR3BP units
(in addition note that $y(0) = z(0) = \dot{x}(0) =\dot{z}(0) = 0$).
$T$ and $T_D$ are the periods in the synodic and sidereal frames, respectively.
Note that orbits 1-3 have unique Jacobi constants (g-family),
but orbits 4 and 5 have the same value of the Jacobi constant (g'-family); see Fig. 4.
\begin{center}
\begin{tabular}{|c|c||c|c||c|c|c||c|}
\hline
& & & & $T$ & $T$ & $T_D$ & \\
Orbit & $C$ & $x(0)$ & $\dot{y}(0)$ & (CR3BP) & (days) & (days) & Notes \\
\hline
1 & 3.0015 & -0.998229599... & 0.035653318... & 0.318 & 11.4 & 10.8 & \\
2 & 3.0010 & -0.997092625... & 0.026520158... & 0.718 & 25.7 & 23.0 & Fig. 1$^{\dagger}$ \\
3 & 3.0009 & -0.996670046... & 0.024438066... & 0.926 & 33.1 & 28.9 & \\
\hline
4 & $C_2$ & -0.998216072... & 0.044385974... & 1.636 & 58.5 & 46.4 & \\
5 & $C_2$ & -0.993100000... & 0.008102525... & 1.698 & 60.7 & 47.8 & \\
\hline
\end{tabular}
\end{center}
{$\dagger$~Right side of the surface of section.}\\
\newpage
\noindent
Table 4. Retrograde periodic orbits around Venus: Initial Conditions in CR3BP units
(in addition note that $y(0) = z(0) = \dot{x}(0) =\dot{z}(0) = 0$).
These represent the f-family of periodic orbits.
$T$ and $T_R$ are the periods in the synodic and sidereal frames, respectively.
Orbits 1--6 are shown in Fig. 5, Orbits 6--10 in Fig. 6, and Orbits 7--15 in Fig. 7.
\begin{center}
\begin{tabular}{|c|c||c|c||c|c|c||c|}
\hline
& & & & $T$ & $T$ & $T_R$ & \\
Orbit & $C$ & $x(0)$ & $\dot{y}(0)$ & (CR3BP) & (days) & (days) & Notes \\
\hline
1 & 3.0015 & -1.001497449... & 0.041992835... & 0.226 & 8.08 & 8.38 & \\
2 & 3.0010 & -1.002120439... & 0.036225577... & 0.374 & 13.37 & 14.22 & Fig. 1$^{\dagger}$ \\
3 & 3.0009 & -1.002312066... & 0.034984347... & 0.422 & 15.09 & 16.18 & \\
4 & $C_2$ & -1.002580804... & 0.033580263... & 0.494 & 17.67 & 19.17 & \\
5 & $C_1$ & -1.002580048... & 0.033636896... & 0.496 & 17.74 & 19.26 & Fig. 2$^{\dagger}$ \\
6 & 3.0006 & -1.003114808... & 0.031500033... & 0.646 & 23.10 & 25.75 & Fig. 3$^{\dagger}$ \\
\hline
7 & 3.0002677... & -1.0050000 & 0.0297900 & 1.212 & 43.3 & 53.7 & \\
8 & 2.9999879... & -1.0090000 & 0.0281100 & 2.554 & 91.3 & 153.9 & \\
9 & 2.9999046... & -1.0111475 & 0.0299500 & 3.266 & 116.8 & 243. & See Text \\
10 & 2.9997765... & -1.0150000 & 0.0348000 & 4.328 & 154.8 & 498. & \\
\hline
11 & 2.9974926... & -1.0500000 & 0.0993000 & 6.190 & 221.37 & 14922 & \\
12 & 2.9596138... & -1.2000000 & 0.3835000 & 6.274 & 224.37 & 153142 & \\
13 & 2.7323333... & -1.5000000 & 0.9225000 & 6.276 & 224.44 & 195710 & \\
14 & 2.3284617... & -1.7500000 & 1.3700000 & 6.282 & 224.66 & 1170804 & \\
15 & 1.0597786... & -2.0000000 & 1.9850000 & $2\pi$ & 224.70 & $\infty$& Collision \\
16 & -1.6662078... & -2.2500000 & 2.7600000 & 12.560 & 449.17 & -449.63 & Circum- \\
& & & & & & & binary \\
\hline
\end{tabular}
\end{center}
{$\dagger$~Left side of the surface of section.} \\
\noindent
Table 5. Venusynchronous orbit characteristics.
The heliocentric orbital elements are specified in a J2000 Ecliptic frame for 01-Jan-2000, 12:00:00 UTC (J2000 Epoch).
Specific initial conditions in CR3BP units and other characteristics are shown in Table 4, orbit 9.
\begin{center}
\begin{tabular}{|l|c|c|c}
\hline
Parameter & Value & Notes \\
\hline
Semi-major axis (km) & 106590220.95 & \\
Eccentricity & 0.022717 & \\
Inclination (deg) & 3.39471 & Same as Venus \\
Longitude of Asc. Node (deg) & 76.68069 & " \\
Argument of Perihelion (deg) & 298.94917 & \\
True Anomaly (deg) & 166.95154 & \\
\hline
Average heliocentric semi-major axis, $a$ & 1.0823 $\times 10^8$ km (0.7235 AU) & See Fig. 12a \\
Average heliocentric eccentricity, $e$ & 0.0237 & See Fig. 12b \\
\hline
Average distance from Venus & 1.414 $\times 10^6$ km & See Fig. 10 \\
\hline
\end{tabular}
\end{center}
\clearpage
\section{Figure Captions}
\begin{description}
\item[Figure 1] Surface of section for the Jacobi constant value $C$ = 3.0010;
the dashed curves represent the zero velocity curves (ZVCs).
\item[Figure 2] Surface of section for the Jacobi constant value $C = C_1$ (3.000776...).
The L1 point is located at $x \approx$ --1.0094, $dx/dt$ = 0, where the ZVCs meet.
The Sun (not shown) is located to the right at $(\mu, 0)$.
Note the widespread evidence of chaos on the right-side of the plot (prograde orbits).
A particle in a prograde orbit is free to escape the vicinity of Venus towards the Sun through the L2 point.
\item[Figure 3] Surface of section for the Jacobi constant value $C$ = 3.0006 . Most prograde orbits
have turned chaotic and escaped the vicinity of Venus to escape either to the Sun through the L2 point,
or to infinity through the L1 point. In contrast, retrograde orbits stay in the vicinity of Venus.
\item[Figure 4] Prograde periodic orbits around Venus (shown as the black dot; not to scale) in the synodic frame.
The direction of motion is counterclockwise. The numbers correspond to the initial conditions shown in Table 3.
Orbits 1--3 belong to the g-family, while orbits 4 and 5 belong to the g'-family (left and right branches, respectively).
The dotted circle represents Venus' Hill sphere, while the Lagrange points L1 and L2 are marked with crosses.
The Sun (not shown) is located to the right at $(\mu, 0)$.
\item[Figure 5] Inner retrograde periodic orbits around Venus (shown as the black dot; not to scale)
in the synodic frame. The direction of motion is clockwise. The numbers correspond to the initial conditions
shown in Table 4. Orbits 4 and 5 are indistinguishable at this scale.
The Sun (not shown) is located to the right at $(\mu, 0)$.
\item[Figure 6] Middle retrograde periodic orbits around Venus (shown as the black dot; not to scale) in the synodic
frame. The direction of motion is clockwise. The numbers correspond to the initial conditions shown in Table 4.
The dotted circle represents Venus' Hill sphere, while the Lagrange points L1 and L2 are marked with crosses. The Sun
(not shown) is located to the right at $(\mu, 0)$. For scale, orbit 6 is shown in both this and the previous figure.
\item[Figure 7] Outer retrograde periodic orbits around Venus (shown as the black dot; not to scale)
in the synodic frame. The direction of motion is clockwise.
The numbers correspond to the initial conditions shown in Table 4. The outermost orbit shown is a collision orbit.
The locations of the Lagrange points L3, L4 and L5 are marked with crosses. The Sun is located at $(\mu, 0)$ (shown as
the yellow dot; not to scale). For comparison, the dashed circle represents the orbit of the Earth around the Sun,
while the dotted circle represents the orbital semi-major axis of Mercury.
\item[Figure 8] (a) Orbits of Venus (solid) and its synchronous satellite (dotted; orbit 9 in Figure 6/Table 4)
in a non-rotating (sidereal) frame centered at the Sun. The length scale is in km.
On this scale, the orbits of Venus and its synchrosat are almost indistinguishable;
for context, the ellipse represents the orbit of Mercury.
In this frame, all three bodies move counter-clockwise;
locations of Venus and its synchrosat are shown at four different times, in days.
(b) Close-up at $t$ = 0; the synchrosat is farther from the Sun than Venus.
(c) Close-up at $t$ = 56.2 days; the synchrosat is closer to the the Sun than Venus.
\item[Figure 9] Orbit 9 viewed in a non-rotating (sidereal) frame centered on Venus, over an interval of 715 days.
In this frame, the Sun completes a revolution ``around'' Venus in 224.7 days.
The dotted circle represents Venus' Hill sphere.
\item[Figure 10] Orbit 9 viewed in a reference frame centered on Venus, and fixed in its body.
The polar angle represents longitude as measured on the surface of Venus.
The dotted circle represents Venus' Hill sphere.
\item[Figure 11] Orbit 9 groundtrack on a Venus cloud background: 10-year simulation performed with GMAT
(SRP taken into account, as well as gravitational forces of the Sun and all the planets).
The groundtrack shifts westwards with time. The inset shows the groundtrack in detail.
\item[Figure 12] Evolution of the heliocentric orbital elements of a putative venusynchrosat over 10 years
(GMAT simulations with all relevant forces included). The starting time corresponds to 1-Jan-2000, 12:00:00 UTC,
and the reference frame is J2000 Ecliptic (see Table 5). The orbit is stable.
\end{description}
\clearpage
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{Ceq3pt0010}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{CeqC1}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{Ceq3pt0006}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{gfamily_orbits}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{ffamily_orbits_inner}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{ffamily_orbits_middle}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{ffamily_orbits_outer}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{heliocentric_orbits}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{square_orbit.jpg}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{synchronous_BodyFrame.jpg}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{groundtrack_fullmodel.jpg}
\caption{}
\end{center}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=160mm]{heliocentric_elements_10yrs_allForces.jpg}
\caption{}
\end{center}
\end{figure}
\end{document}
UNWANTED STUFF CAN BE STORED DOWN HERE
|
2,877,628,089,542 | arxiv | \section{Introduction} \label{sec:intro}
The X-ray nova GS 2023+338 was discovered on 1989 May 22 by the Ginga X-ray satellite ~\citep{1}.
Soon after the object was identified with the variable star V404 Cyg, which had been seen to brighten by seven magnitudes
in the optical in 1938 ~\citep{2,3}. It is now well established ~\citep{4} that V404 Cyg is a binary system consisting of a
giant secondary star, older and less massive than the Sun
and a primary star which is an accreting black hole candidate \citep{BestBH,casares}. The secondary (optical companion) is believed to undergo episodic
enhanced mass loss every few decades, losing part of its mass to the black hole. This matter emits
X-ray radiation while heating up to several hundred million degrees in the accretion disk. The detection of
nonthermal radio and soft X-ray radiation ~\citep{6}, typically found in quasars, that are millions of times more
massive and with powerful black holes residing in nuclei of active galaxies (i.e. AGN), led researchers to call this black hole a
microquasar, ejecting relativistic particles in the form of jets. In this paper we report the detection, for the first
time, of variable optical polarization arising in the relativistic jet generated by the black hole.
On 2015 June 15 the Swift space observatory found that one of the most securely identified Galactic stellar mass black hole candidates, the binary system V404 Cyg, began undergoing another active X-ray phase after 25 years of inactivity ~\citep{7}.
The twin optical telescope, MASTER-Tunka of the MASTER Global Robotic Net ~\citep{8}, located near Lake Baikal (Lomonosov Moscow State University and Irkutsk State University Tunka astrophysical center), was the first ground-based observatory to point to V404 Cyg after the Swift alert. This was 22 sec after the notice time \citep{9} of the burst alert, via space communication system (i.e. socket messages \textbf{GCN: The Gamma-ray Coordinates Network alert \citep{Barthelmy94,Barthelmy95}})
and after 150 seconds we detected a bright optical flare ~\citep{9}, reaching magnitude 14, with the star brightening by a factor of 2.5 in a mere half-hour. In the next few weeks six MASTER twin robotic telescopes performed
about 20 pointings. Dozens of telescopes worldwide subsequently observed the object at various wavelengths
(~\citet{4,10,11,12,13,14,15,16}), but MASTER is the first to report on the discovery of variable optical linear polarization arising in the relativistic jet generated by the black hole.
\section{MASTER Global Robotic Network observation}
The MASTER robotic telescopes\footnote{\url{http://observ.pereplet.ru}} are a network of identical instruments deployed over several continents
and equipped with identical CCD cameras capable of performing B,V,R$_J$,I (Johnson/Bessell) photometric and linear polarimetric observations ~\citep{8,25,17}. Each MASTER-II observatory has twin 0.40-m f/2.5 wide field reflectors (Hamilton design; see ~\cite{25}) and a prime focus 4k x 4k CCD camera, which provide a total 8 square-degree field and reaches to a white-light magnitude of 20-21 (in 180s exposure). In addition, the mounts for these two telescopes also have two very wide-field (VWF) optical cameras, with 800 square degrees field of view, reaching to magnitude of 15 \citep{8,25}. The observatories are located as follows: operating in Russia, from east to west, the MASTER-Amur, MASTER-Tunka, MASTER-Ural, and MASTER-Kislovodsk; the MASTER-SAAO in South Africa; the MASTER-IAC in the Canary Islands and MASTER-OAFA in Argentina.
The observations with MASTER-Net can be performed in three different modes: alert, survey and inspection. Alert mode is aimed at automated observations of rapid events, like Gamma Ray Bursts (GRBs) from SWIFT, Fermi, IPN, MAXI, INTEGRAL and gravitational wave or neutrino events. The primary goals of the MASTER Global Robotic Network are the rapid response to alerts (first, GRB alerts). The alert mode is triggered if a transient position has good accuracy (when the error-box size is less than 2$^\circ\times$2$^\circ$ MASTER FOV) and is typically used to observe GRBs upon receiving notices from the Gamma-ray Coordinates Network\footnote{\url{http://gcn.gsfc.nasa.gov/}} . In this alert mode MASTER observes with parallel 0.4-m telescopes, each covering the same field, and with polarizers \citep{Lipunov2016a}, pointing just after the notice time. Exposure times follow the relation $t_{exp} = (T_{start}-T_{0})/5$, where $T_{0}$ is the trigger time (UT), $T_{start}$ is the time of the beginning of exposure (UT). The exposure time is rounded to an integer with a step of 10 s and does not exceed 3 min. All images are reduced automatically in real-time mode via our own software developed by the MASTER team over the past ten years. If we observe in survey or inspection mode, we usually observe in the white (unfiltered) light to increase the limiting magnitude. The corresponding internal photometric magnitudes can be described fairly well by the equation $W=0.8*R2+0.2*B2$, where $R2$ and $B2$ are the second epoch DSS red and blue magnitudes, respectively, adopted from USNO-B1.0 catalog ~\citep{monet,25,gor2012}.
The technique of polarimetric measurements used on MASTER network telescopes is considered in detail in \citep{17}. It allows detecting the \textit{linear} polarization at 1-2\% level for objects brighter then 14mag. We stress that MASTER-II consists of twin wide field telescopes each with a 2$^\circ\times$2$^\circ$ field. It gives us the possibility to determine the polarization for thousands stars from 12 to 16 magnitudes simultaneously. To estimate an error of the object's polarimetric measurement we select a few tens of field stars with similar brigthness in the same frame and calculate the standard deviations of their polarizations, assuming that the stars have the same polarization produced by the ISM.
Therefore, we calculate the error of polarization measurements as a chance fluctuations of the background stars,
assuming that they have no variable polarization.
\section {Polarimetry technique}
\label{section:obs}
The MASTER Net was designed with the objective to deliver polarization information as early as possible after GRB triggers. More than 100 observations of GRBs were made by the MASTER global robotic net. Optical emission was detected for $\sim$ 20 GRBs \citep{lipunov2007,gor2012,17}. The GRB\,100906A, GRB\,110422A, and GRB\,121011A events deserved attention because their optical observations were carried out during the prompt gamma-ray emission.
The technique of polarimetric measurements used on MASTER network telescopes is considered in detail in \citep{17} and although primarily designed for GRB followup, the same techniques can be used for polarimetry of other objects.
We only utilize linear polarizers for MASTER, so cannot determine the circular polarization of objects.
If we let $I_1$ and $I_2$ be the signals on the CCD detector with orthogonal polarizing filters (for example oriented with their axes at $45^o$ and $135^o$), then a lower limit of the linear polarization is obtained from $P_{low}=(I_1-I_2)/(I_1+I_2)$.
To derive the degree of linear polarization, rather than a limit, one would need to perform observations with polarizers positioned at three angles, for example, at $0^o$, $45^o$, and $90^o$ with respect to the reference direction. Since we
have two fixed orthogonal filters at each of the MASTER-II telescopes, each telescope alone observes only a lower limit on polarization level --- $P_{low}$.
If the value of $P_{low}$ is less than measurement error $\sigma_\mathrm{p}$ , then formally we
have zero as a minimum estimate for the degree of linear polarization. Even with a 100-percent linearly polarized source, it is still possible, if the polarizers were so positioned that linear polarization position was inclined exactly by $45^o$ to
both of them, to register zero polarization for the object.
The two synchronous frames (taken with different cameras) used to measure $P_{low}$ are mutually calibrated so that the average $P_{low}$ for comparison stars would be $ Avg (P_{low}) = 0$. This is achieved at the stage of the photometric calibration. We use the same reference stars for frames in both polarizations from the USNO B-1 catalog (with no polarization measuremtns). This implies, for unpolarized sources, that $ Avg (I_{45} - I_{135}) = 0 $ and consequently $ Avg (P_{low})=0 $. If the reference stars are polarized (e.g. due to interstellar polarization, then they will show non-zero $P_{low}$. For the purpose of this paper, the exact measurement of the linear polarization value is not the aim, but rather the purpose is to establish the polarization variability of V404Cyg. Since the Galactic polarization is constant, it does not affect to the determination of polarization variability.
The MASTER polarization band is determined by the response curves of the CCD camera and a transmission curve of polarizing filter, which have been reviewed here \citep{17}.
For the determination of errors, we analyzed the distribution of $P_{low}$ of the field reference stars, depending on the magnitude. Then we selected only the stars with magnitudes in the interval $m_{V404} \pm 0.5 $, where $m_{V404}$ is a V404 magnitude on a given frame. The dispersion of comparison stars $P_{low}$ in this interval are used to define the $1 - \sigma $ error of object $P_{low}$. We use the reference stars in a radius of $0.5^o$ for V404 Cyg. All errors that are reported in this paper are $1 \sigma$.
\section{MASTER V404 Cyg results}
In the direction of V404 Cyg, strong interstellar polarization is observed due to scattering by
Galactic dust, which is aligned with the Galactic magnetic field. Naturally this polarization is essentially constant, as
noted in the very first reported observations of the new outburst of the microquasar
\citep{18,19}. The magnitude and position angle measured for V404 Cyg in ~\citep{19} was found to be consistent with the interstellar linear polarization, at a level of about 8$\%$ and at a position angle of $\sim15^\circ$, respectively.
The MASTER-Net observations
were performed throughout 2015 June at the MASTER-Tunka and MASTER-Kislovodsk nodes. They reveal not only the constant interstellar polarization component during the earliest epochs of optical observations, but also polarization variations at certain epochs in a number of cases. The full description of the data reduction process, polarization standards, photometry and polarization measurements are decribed in \citep{17}.
We observed in two orthogonally related polarizers we detect the value $Y=(I_{1}-I_{2})/(I_{1}+I_{2})$. It's evident that Y is the low limit of linear polarization of the object. But the Y change (time evolution of Y) authentically proves the existence of an additional variable linear polarization, that is not connected with the interstellar dust.
\textbf{As the Y value is measured in 2 polarizers with their fast axes orthogonal to each other, its variability is a result of the change of the E-vector, i.e. the degree of linear polarization and its position angle. The variability of flux in different polarizers can be explained by the variation of the contribution of polarized flux to the total flux. The degree of polarization increases when the unpolarized component of the optical flux of the system decreases relative to the contribution of the polarized flux. Therefore we expect the polarization to increase when the total optical brightness of system falls, say due to less X-ray irradiation of the secondary star. The polarized jet component then begins to dominate over the unpolarized reprocessed component, increasing the level of polarization.}
We found at least two cases of bona fide linear polarization variability. The first event lasted a
little more than an hour and was detected during alert observations on 2015 June 18. These observations were
triggered by an INTEGRAL Alert (GCN, trigger Number 7029) following the detection of an X-ray burst. The MASTER-Tunka
telescope automatically started observing the microquasar with the two parallel aligned telescope tubes, with mutually perpendicular polaroids,
positioned along the North-South and East-West directions, 42 seconds after the trigger. At 15:36:00 UT MASTER-Tunka
recorded the onset of an abrupt factor of six decrease of optical flux (Fig.1), followed by its recovery at an
even higher level than the initial level. At the same time the polarization increased by $\sim4\%$, reaching
$\sim 12\%$, while it was of about $\sim 8\%$ before and after this event (one of the axes of our polarizer is aligned practically parallel to the position angle of the interstellar polarization \citep{20}.
\begin{figure}
\figurenum{1}
\plotone{figure.eps}
\caption{
The variation of the $Y=(I_{1}-I_{2})/(I_{1}+I_{2})$ value of V404 Cyg microquasar based on MASTER-Tunka and MASTER-Kislovodsk observations. The data from MASTER-Tunka are up to 19:00 U.T, and from MASTER-Kislovodsk, after 19:00 U.T. There polarization of many field stars are also included. The behaviour of V404 light curve at the time of polarization variability is demonstrated by the magenta curve.
}
\end{figure}
For the polarization measurements, we used the field stars with corresponding similar stellar magnitudes as the source, as follows. The error of the measurement is the standard deviation for the several measurements of the background stars. We use MASTER's wide fields (4 square degrees) to have measurements for thousands of field stars, many with magnitudes similar to V404 Cyg.
We observed the Y value
to increase substantially, up to 12\%, during V404 Cyg's fading.
The error was estimated from reference stars where we found the chi-square statistical significance of this event to be
99.99$\%$.
The second polarization variation event seen was recorded after receiving INTEGRAL alert N7035, on 2015 June. Twenty-seven seconds later, at 17:29:05UT, MASTER-Kislovodsk started photometric observations of V404 Cyg
in two mutually perpendicular polarizations parallel and perpendicular to the meridian (i.e. $0^o$ and $90^o$.
After two hours MASTER recorded strong and fast flux increases, accompanied by a decrease of polarization from 14\% down to 8\%. This result
qualitatively corroborates the behavior found earlier in our MASTER-Tunka observations. The Tables of MASTER Y value and photometry data for V404 are available at http://master.sai.msu.ru/static/V404tables.pdf
\section{Interpretation and discussion} \label{sec:floats}
The main source of the optical emission in V404 Cyg is the
$0.7^{+0.3}_{-0.2}M_\bigodot$ K3 III-type subgiant companion ~\citep{21,22,23}, including X-ray
radiation reprocessed to optical on the illuminated side, facing the black hole. The intrinsic emission of the secondary (low-mass) star is
practically undetectable during the time of the outburst, in which there is a factor of $\sim 10^8$, increase of the microquasar
luminosity. Thus the rise of the optical flux during this outburst is primarily due
to the heating of the low-mass star and, possibly, of the outer layers of the accretion disk. The accretion disk
forms as a result of the mass loss from the giant star, through the L1 point of its Roche lobe, into the gravitational potential well
of the black
hole. While approaching the black hole the matter captured by it heats up to a temperature of several million degrees
and emits the observed X-ray radiation.
However, there is yet another possible source of radiation in addition to the two mentioned above, namely a
relativistic jet generated by the black hole and only observed until now in non-thermal radio and hard X-ray emission. We
believe that this jet (or, more precisely, two symmetric bipolar jets) are responsible for the polarized optical emission.
Here is our proposed scenario for the observed 2015 June 18 polarization variation in the microquasar:
\textbf{As the Y value is measured in two mutually perpendicular polarizers, its variability can be connected with variability of size of linear polarization or with turn of a vector of the polarization plane.
The first explanation seems more preferably as it is difficult to imagine an essential variability of jet orientation on the times about tens of minutes.
The variability of flux in different polarizers can be most possibly explained by the variability of the contribution of strongly polarized light in the general flux of an optical emission. Not incidentally, the polarization increases when the general optical flux of the system falls and the contribution polarization component grows. Anti-correlation between the value of polarization and brightness of an optical flux is apparently connected with the increase of jet contribution during the decreasing of the effect of warming up of an optical companion by x-ray emission arising near the black hole.}
At 15:36:00 (UT) the accreting
mass inflow began to decrease rapidly near the black hole, the region of maximum energy release. The X-ray flux
also decreased and, as a result, so did the intensity of the heating of the secondary (giant) star. At the same time the active
emission of the jet continued, because it is not directly associated with the instantaneous accretion rate but
rather with the acceleration of relativistic particles near the black hole. In other words, the jet became visible
because its flux contribution increased with respect to the decrease of the luminosity of the irradiated secondary star. The jet emission is due to
non-thermal synchrotron radiation and is therefore polarized, and hence its increased contribution resulted in the
increase of the polarization of the total emission.
Strong linear polarization (up to $\sim 30\%$) is observed in the jets of BL Lac type objects, or blazars, namely quasars with
relativistic jets pointing in the direction of the Earth. This is due to the synchrotron mechanism of their
emission. Thus the discovery of non-thermal polarized optical component provides the direct evidence that the
nature of one of the most secure black hole candidates, namely V404 Cyg, is similar to that of the millions of times more massive
quasars (blazars). If we assume that the total jet polarization is equal to $\simeq$30 $\%$, as observed in blazar type objects, then
the jets in V404 Cyg contribute about K = $L_{j} /L_{o}$ $ \sim 15\%$ to the total luminosity, implying a jet
optical flux of about $2\cdot10^{-11}$ ${\rm erg} {\rm s}^{-1} {\rm cm}^{-2}$. For the adopted distance of 2.39 $\pm$ 0.14 kpc ~\citep{24} we
obtain a total jet luminosity of $\sim$ 4 $\cdot 10^{34}$ erg/s. The detection of unprecedentedly powerful
non-thermal radio emission amounting to 6 Jy during outbursts ~\citep{14} provides independent evidence confirming
the likely existence of a non-thermal jet during this last outburst. This fact agrees quite well with jet flux estimates inferred in
terms of the synchrotron emission mechanism.
During this paper's preparation, information about independent polarimetry observations was published \citep{20}, where the authors wrote that they didn't see polarization variability. We stress that we have a more extensive data set and also don't see polarization variability, except unequivocally at two diferent times, during rapid and deep optical flux decreases, which is the subject of the paper. The optical flux decrease was over a short timescale (compared to the scale of the several dozens days in total of the MASTER monitoring observations) and has the character of a rapid decrease of the optical flux of the system. The relative rarity of such events explains why were no other polarization variations have been seen in V404 Cyg (e.g. by \cite{20}).
\section*{Acknowledgments}
MASTER Global Robotic Net is supported in part by the Development Programm of Lomonosov Moscow State University.
This work was also supported in part by RFBR 15-02-07875 grant, Russian Science Foundation 16-12-00085 and National Research Foundation of South Africa.
We are grateful to Professor Tanaka for the information in his astro-ph paper, prior to publication. We are gratefull to referee for the number of remarks and suggestions that have improved the paper.
|
2,877,628,089,543 | arxiv | \section{\label{sec:intro}Introduction}
Quantum algorithms are designed to outperform the best classical ones \cite{NC10}. Many nondeterministic NP-hard problems still have only the exhaustive search way to solve them \cite{BBBV96}. The one-way function (oracle) $f(x)$ ($f:\{0,1\}^n\rightarrow \{0,1\}$) can identify the solution state: if $t$ is the solution (target state), then $f(t)=1$; otherwise the one-way function output is zero. The classical way to execute the exhaustive search is by querying each state in the database (of $N$ items) by the one-way function. In the worst case, the total number of queries to the oracle is $N-1$. The principle of quantum superposition provides a superior way to perform the exhaustive search. Suppose that $N=2^n$, where $n$ is the number of qubits to represent the database. Grover's algorithm can find one target state with oracle complexity $\mathcal O(\sqrt N)$, which quadratically outperforms the classical algorithm \cite{Grover97,GK17}. The oracle in Grover's algorithm is $U_f$: $U_f|x\rangle|y\rangle=|x\rangle|f(x)\oplus y\rangle$ with $x\in\{0,1\}^n$ and $y\in\{0,1\}$.
Quantum computers have been vastly developed over the last ten years \cite{Barends14,BHLSL16,FMLLDM17,Arute19}. Still shallow-depth algorithms can be realized on real quantum computers (for the noisy intermediate-scale quantum (NISQ) era, see Ref. \cite{Preskill18}). The width (the number of physical qubits) represents the size of quantum computers. The algorithm's depth (the number of consecutive gate operations) represents the physical implementation time for the algorithm. Multiplying the width and depth we get the quantum volume, which gives a metric for NISQ computers \cite{CBSNG19}. Coherence time is limited in NISQ computers. A set of gates which can approximate any unitary operation is called the universal quantum gate set (Solovay–Kitaev theorem) \cite{NC10}. We assume that the quantum computer is equipped with a universal quantum gate set. So, the depth is counted by universal quantum gate operations.
The quantum oracle $U_f$ is realized by quantum gates from the universal quantum gate set. We assume that the depth of the quantum oracle scales polynomially with $n$ \cite{FMLLDM17}. The oracle complexity would be equivalent to the depth complexity if the quantum oracle would be the only operation realized in Grover's algorithm. However, it is not true. Another unitary operation (diffusion operator) is required for Grover's algorithm \cite{Grover97,GK17}. How to choose the diffusion operator is related to the initial state preparation \cite{Grover98,BHMT00}. The unstructured population space $\{0,1\}^n$ (database) can be prepared in an equal superposition state on a quantum computer polynomial efficiently:
\begin{equation}
\label{def s n}
|s_n\rangle=H^{\otimes n}|0\rangle^{\otimes n}
\end{equation}
with single-qubit Hadamard gate $H$ \cite{NC10}. Note that the initial state $|s_n\rangle$ can be efficiently prepared with a depth of one circuit. The diffusion operator has the constraint that the state $|s_n\rangle$ is the eigenvector of the diffusion operator with eigenvalue 1 \cite{Tulsi12,Tulsi15}.
Grover's algorithm is the only threat to postquantum cryptography. The postquantum cryptography standardization proposed by NIST in 2016 introduced the depth bound. Recently, more studies focused on the resource estimation, such as width and depth, for Grover's algorithm instead of the traditional oracle complexity \cite{KHJ18,JNRV19}. Grover's algorithm is optimal in oracle complexity \cite{BBHT98,Zalka99}. However, no research addressed the depth of the quantum search algorithm. Surprisingly, the depth of the diffusion operator can be reduced to one \cite{Kato05,JRW17}. However, these algorithms have 1/2 maximal successful probability, and the expected depth is not as efficient as the original Grover's algorithm. Inspired by the quantum partial search algorithm (QPSA) \cite{GR05,KG06,Korepin05,KL06}, we introduce a new depth optimization for the quantum search algorithm. Our algorithm can have lower depth than Grover's algorithm. To further lower the depth, we can apply a divide-and-conquer strategy (combined with depth optimization). The divide-and-conquer strategy means that the search algorithm is realized by several stages. Each stage can find a partial address of the target state. The next-stage initial state is the rescaled version of the last-stage initial state. The divide-and-conquer strategy naturally allows the parallel running of the quantum search algorithm.
If the oracle takes much more depths than diffusion operator depth, then the oracle complexity will be approximately equivalent to the depth complexity. We can define the ratio between oracle depth and diffusion operator depth. Above a critical ratio, Grover's algorithm is optimal in depth. Based on the depth optimization method proposed in this paper, we show that the critical ratio is $\mathcal O(n^{-1}2^{n/2})$. If we divide the algorithm into two stages, the critical ratio is a constant.
The paper is organized as follow. In Sec. \ref{sec:Grover}, we briefly review quantum search algorithms. The first one is Grover's original algorithm and the other is QPSA. We also set up notations. In Sec. \ref{sec:dep_opt}, we introduce the depth optimization method for the quantum search algorithm. We also show how to combine the divide-and-conquer strategy with depth optimization. In Sec. \ref{sec:alpha}, we talk about the critical ratios. Below the critical ratio, we can have a search algorithm which has lower depth compared to Grover's algorithm. Parallel running of the quantum search algorithm is briefly discussed in Sec. \ref{sec:parallel}. Section \ref{sec:conclusion} gives conclusions and outlook. We wrote three Appendixes. Appendix \ref{Appendix examples 6} provides detailed examples of the $n=6$ search algorithm with depth optimizations; Appendix \ref{Appendix opt examples} lists the numerical details provided in the main text; Appendix \ref{Appendix alpha} shows the numerical values of critical ratios.
\section{\label{sec:Grover}Review of Quantum Search Algorithms}
\subsection{\label{subsec:Grover}Grover's Algorithm}
The quantum oracle $U_f$ flips the ancillary qubit, if the target state $|t\rangle$ is fed in. The ancillary qubit can be prepared in the superposition state $H|1\rangle = (|0\rangle-|1\rangle)/\sqrt 2$. Then the oracle gives a sign flip acting on the target state:
\begin{equation}
U_f(1\!\!1_{2^n}\otimes H)|x\rangle\otimes|1\rangle = (-1)^{f(x)}(1\!\!1_{2^n}\otimes H)|x\rangle\otimes|1\rangle
\end{equation}
Here $1\!\!1_{2^n}$ is the identity operator on the $2^n$ dimensional Hilbert space. For convenience, we denote the oracle $U_f$ as
\begin{equation}
\label{def U t}
U_t=1\!\!1_{2^n}-2|t\rangle\langle t|
\end{equation}
if the ancillary qubit $H|1\rangle$ is prepared. The general phase flip can be constructed as follows: $U_{t,\phi}=1\!\!1_{2^n}-(1-e^{-i\phi})|t\rangle\langle t|$ with complex unit $i=\sqrt{-1}$. The generalized oracle $U_{t,\phi}$ has applications in the sure success search algorithm \cite{BHMT00,MTB18} and the fixed point search algorithm (for an unknown number of target states) \cite{YLC14}. Note that the operator $U_{t,\phi}$ ($\phi\neq \pi$) can be realized by two quantum oracles $U_f$ \cite{YLC14}. In this paper, we do \textit{not} consider the generalized oracle $U_{t,\phi}$ (low depth consideration). We concentrate on the one-target-state case. The depth optimization method in Sec. \ref{sec:dep_opt} can be easily generalized to multitarget cases.
The oracle $U_t$ reflects the state over the plane perpendicular to the target state. The most efficient diffusion operator (unstructured database search) is
\begin{equation}
\label{def I n}
D_n = 2|s_n\rangle\langle s_n|-1\!\!1_{2^n}
\end{equation}
Note that $|s_n\rangle$ defined in Eq. (\ref{def s n}) is the equal superposition of all items in the database. The operator $D_n$ can be viewed as a reflection of the amplitude in the average. The diffusion operator $D_n$ does not query the oracle. Therefore, the oracle complexity does not include the resource cost by $D_n$. The diffusion operator $D_n$ is single-qubit-gate-equivalent to the generalized $n$-qubit Toffoli gate $\Lambda_{n-1}(X)$ \cite{NC10}. Here $X$ is the NOT gate (Pauli-X gate). The notation $\Lambda_{n-1}(X)$ implies the $n-1$ control qubits NOT gate. When $n=3$, $\Lambda_{2}(X)$ is the Toffoli gate. When $n=2$, $\Lambda_{1}(X)$ is the controlled-NOT (CNOT) gate. How to realize the $\Lambda_{n-1}(X)$ gate on a real quantum computer is highly nontrivial. It is well known that an $n$-qubit $\Lambda_{n-1}(X)$ gate can be constructed with linear $n$ depth or quadratic $n^2$ depth from the universal gate set (CNOT gate plus single-qubit gates) \cite{BBCDMSSSW95}. Recent works also show that the $n$-qubit $\Lambda_{n-1}(X)$ gate can be realized in $\log n$ depth if $n$-qubit ancillary qubits are provided \cite{HLZWW17} or qutrit states are applied \cite{GBDBRC19}.
One query to oracle $U_t$ defined in Eq. (\ref{def U t}) combined with the diffusion operator $D_n$ defined in Eq. (\ref{def I n}) is called the Grover iteration or Grover operator:
\begin{equation}
\label{def G n}
G_n = D_n U_t
\end{equation}
See Fig. \ref{fig G n} for the quantum circuit diagram of $G_n$. The diffusion operator $D_n$ reflects the average of the whole database. The operator $G_n$ is also called the {\it global Grover iteration} ({\it global Grover operator}). One Grover operator $G_n$ uses one query to oracle $U_f$. Applying $G_n$ iteratively on the initial state $|s_n\rangle$, the amplitude of the target state will be amplified. After $j$ Grover iterations, the success probability $P_n(j)$ is
\begin{equation}
\label{def P n}
P_n(j) = |\langle t|G_n^j|s_n\rangle|^2 = \sin^2((2j+1)\theta)
\end{equation}
with $\sin\theta=1/\sqrt N$. When $j$ reaches $j_\text{max}=\lfloor \pi\sqrt N/4\rfloor$, the probability of finding the target state approaches 1. The maximal iteration number $j_\text{max}$ is the square root of $N$. Clearly, Grover's algorithm provides a quadratic speedup compared with the classical algorithm (in oracle complexity). The idea behind Grover's algorithm can be generalized into the amplitude amplification algorithm \cite{BHMT00}.
The success probability (finding the target state) does not scale linearly with the number of iterations. It suggests that Grover's algorithm becomes less efficient when $j$ approaches $j_\text{max}$. Previous works argued that the expected number of iterations $j/P_n(j)$ has the minimum at $j_\text{exp}=\lfloor 0.583\sqrt N \rfloor$, which is smaller than $j_\text{max}$ \cite{BBHT98,GWC00}. When $j$ is $j_\text{exp}$, the success probability is around 0.845. In practice, the iteration number $j_\text{exp}$ has a high probability to find the target state. The measurement result can be verified in classical ways. If the result fails, one has to run the algorithm again. The expected number of oracles is minimized at $j_\text{exp}$.
\begin{figure}
\subfloat[\label{fig G n} $G_n = D_nU_t$.]{%
\includegraphics[width=0.48\columnwidth]{G_n.pdf}%
}\hfill
\subfloat[\label{fig G m} $G_m = D_mU_t$.]{%
\includegraphics[width=0.5\columnwidth]{G_m.pdf}%
}
\caption{Quantum circuits of global Grover operator $G_n$ defined in Eq. (\ref{def G n}) and local Grover operator defined in Eq. (\ref{def G m}). The diffusion operator $D_n$ ($D_m$) is single-qubit-gate equivalent to the $n$-qubit Toffoli gate $\Lambda_{n-1}(X)$ ($m$-qubit Toffoli gate $\Lambda_{m-1}(X)$) \cite{NC10}. Here $X$ and $Z$ are Pauli gates, and $H$ is the Hadamard gate. The subspace where $D_m$ acts can be chosen arbitrarily.}
\label{fig G n m}
\end{figure}
\subsection{\label{subsec:QPSA}Quantum Partial Search Algorithm}
The QPSA was introduced by Grover and Radhakrishnan \cite{GR05}. Since Grover's algorithm is optimal (in oracle complexity), the QPSA trades accuracy for speed. A database of $N$ items is divided into $K$ blocks: $N=bK$. Here $b$ is the number of items in each block. We can assume that the number $b$ is also a power of 2: $b=2^m$. And the number of blocks is $K=2^{n-m}$. The QPSA can find the block which has the target state. In other words, the QPSA finds the partial $(n-m)$-bit of the target state (which is $n$ bits long). The optimized QPSA can win over Grover's algorithm a number scaling as $\sqrt b$ \cite{GR05,Korepin05,KG06}. A larger block size (less accuracy) gives a faster algorithm.
Suppose that the address of the target state $|t\rangle$ is divided into $|t\rangle=|t_1\rangle\otimes|t_2\rangle$. Here $t_1$ is $(n-m)$ bits long and $t_2$ is $m$ bits long. The task is to find $t_1$ instead of the whole $t$. Besides the diffusion operator $D_n$ in Eq. (\ref{def I n}), the QPSA introduces a new diffusion operator $D_{n,m}$:
\begin{equation}
\label{def I n m}
D_{n,m} = 1\!\!1_{2^{n-m}}\otimes (2|s_m\rangle\langle s_m|-1\!\!1_{2^m})
\end{equation}
The diffusion operator $D_{n,m}$ reflects around the average in a block (simultaneously in each block). The diffusion operator $D_{n,m}$ can be viewed as the rescaled version of $D_n$ in Eq. (\ref{def I n}): the database with size $2^n$ is rescaled into size $2^{m}$. We can define a new Grover operator as
\begin{equation}
\label{def G m}
G_{n,m}=D_{n,m}U_t
\end{equation}
See Fig. \ref{fig G m} for the quantum circuit diagram of $G_{n,m}$. The diffusion operator $D_{n,m}$ reflects the average of block items. The operator $G_{n,m}$ is also called the {\it local Grover iteration} ({\it local Grover operator}). For simplicity, we shorten the notations to $D_{m}\equiv D_{n,m}$ and $G_{m}\equiv G_{n,m}$ in the rest of paper.
The QPSA is realized by applying operators $G_{m}$ and $G_{n}$ on the initial state $|s_n\rangle$. Then partial bits $t_1$ can be found with high probability (computational basis measurement on the final state). In the QPSA, the amplitudes of all nontarget items in the target block are the same, and the amplitudes of all items in the nontarget blocks are the same. Therefore, we can follow only three amplitudes. Let us introduce a basis:
\begin{subequations}
\begin{align}
\label{basis t} &|t\rangle = |t_1\rangle\otimes |t_2\rangle, \\
\label{basis ntt} &|ntt\rangle = \frac 1 {\sqrt{b-1}}\sum_{j\neq t_2}|t_1\rangle\otimes |j\rangle, \\
\label{basis u} &|u\rangle = \frac{1}{\sqrt{N-b}}\left(\sqrt N|s_n\rangle-|t\rangle-\sqrt{b-1}|ntt\rangle\right)
\end{align}
\end{subequations}
The state $|ntt\rangle$ is the normalized sum of all nontarget states in the target block. The state $|u\rangle$ is the normalized sum of all items in the nontarget blocks. At the new basis, the initial state $|s_n\rangle$ in Eq. (\ref{def s n}) can be rewritten as
\begin{equation}
\label{def s n rewrite}
|s_n\rangle = \sin \gamma \sin\theta_2|t\rangle + \sin \gamma \cos\theta_2|ntt\rangle+\cos\gamma |u\rangle
\end{equation}
The angle $\theta_2$ is defined as $\sin\theta_2 = 1/\sqrt b$. The angle $\gamma$ is defined as $\sin\gamma = 1/\sqrt K$. The global Grover operator $G_n$ defined in Eq. (\ref{def G n}) and the local Grover operator $G_m$ defined in Eq. (\ref{def G m}) can be reformulated as elements in the $O(3)$ group \cite{KV06}. Operators $G_{m}$ and $G_{n}$ have highly nontrivial commutation relations \cite{KV06}. The order of application of these operators is the key in the QPSA. Extensive studies have suggested that the optimal sequence (in oracle complexity) is $G_nG_m^{j_2}G_n^{j_1}$ \cite{KL06,KV06}. One can minimize the number of queries to the oracle (minimize $j_1+j_2+1$) given by a threshold success probability. The QPSA requires less number of oracles (the saved oracle number scales as $\sqrt b$) than Grover's algorithm. The QPSA can also be generalized into multitarget cases \cite{CK07,ZK18}. Interestingly, the QPSA can be performed in a hierarchical way: each time the QPSA finds several bits of the target bits $t$ \cite{KX07}.
\section{\label{sec:dep_opt}Depth Optimization}
\subsection{\label{subsec:MED}Minimal Expected Depth}
Depth is defined as the number of consecutive parallel gate operations. For example, the initial state $|s_n\rangle$ can be prepared with one depth circuit, see (\ref{def s n}). Suppose that the diffusion operator $D_n$ in Eq. (\ref{def I n}) has depth $\text{d}(D_n)$, which is the same as the depth of the $n$-qubit generalized Toffoli gate $\Lambda_{n-1}(X)$ \cite{NC10}. Different search tasks have different oracle realizations. We denote the ratio of oracle depth $U_t$ and diffusion operator depth $D_n$ as $\alpha$:
\begin{equation}
\label{def alpha}
\alpha = \frac{\text{d}(U_t)}{\text{d}(D_n)}
\end{equation}
It is an important parameter for depth optimization. For the one-item search algorithm, the practical minimal value for $\alpha$ is 1: $\alpha\geq1$ \cite{FMLLDM17}. The ratio $\alpha$ maybe different for the same problem with a different database size. We fix $n$; then the ratio $\alpha$ is a constant for one problem. The design for a low-depth generalized Toffoli gate can also be a benefit for oracle depth \cite{GBDBRC19}.
Given by $\text{d}(D_n)$ and $\alpha$, Grover's algorithm can be mapped to depth complexity directly. We define the \textit{minimal expected depth} (MED) of Grover's algorithm as:
\begin{equation}
\label{def d G}
\text d_\text{G}(\alpha) = \min_j \frac{\text d(G_n^j)}{P_n(j)}
\end{equation}
Here $P_n(j)$ defined in Eq. (\ref{def P n}) is the success probability of finding the target state (with $j$ Grover iterations). The numerator denotes the depth $\text d(G_n^j)=(\alpha+1)j\text d(D_n)$. The above optimization is the same as the expected iteration number optimization $j/P_n(j)$ \cite{BBHT98,GWC00}, up to a constant factor. Therefore, we can use $j_\text{exp}=\lfloor 0.583\sqrt N \rfloor$ in the MED. Note that we have $P_n(j_\text{exp})\approx 0.845$. Then we have
\begin{equation}
\label{eqa d G}
\text d_\text{G}(\alpha) \approx 0.69\times 2^{n/2}(\alpha+1)\text d(D_n)
\end{equation}
If the oracle can be constructed in polynomial depth $\text{d}(U_t)=\mathcal O(n^k)$, then the MED of Grover's algorithm scales as $\mathcal O(n^{k}2^{n/2})$ (assume that $k>1$). Grover's algorithm is optimal in oracle complexity \cite{BBHT98,Zalka99}. The minimal expected iteration number $j_\text{exp}$ is optimal. The scale $\mathcal O(n^{k}2^{n/2})$ is also optimal for depth complexity. However, we show that the number $\text d_\text{G}(\alpha)$ in Eq. (\ref{eqa d G}) is {\it not} optimal (if $\alpha$ in Eq. (\ref{def alpha}) is finite).
\subsection{\label{subsec: opt method} Optimization Method}
The local diffusion operator $D_m$ defined in Eq. (\ref{def I n m}) has lower depth than the global diffusion operator $D_n$ in Eq. (\ref{def I n}). The optimization idea is \textit{to replace the global diffusion operator by the local diffusion operator}. The global Grover operator $G_n$ defined in Eq. (\ref{def G n}) does not commute with the local Grover operator $G_m$ in Eq. (\ref{def G m}) \cite{KV06}. The order of $G_n$ and $G_m$ is important. Suppose that we have the sequence
\begin{equation}
\label{def S n m}
S_{n,m}(j_1,j_2,\ldots,j_q) = G_n^{j_1}G_m^{j_{2}}\cdots G_n^{j_{q-1}}G_m^{j_q}
\end{equation}
Here $\{j_1,j_2,\ldots,j_q\}$ are some non-negative integers. We have
\begin{equation}
j_\text{tot} = \sum_{p=1}^q j_{p}
\end{equation}
total number of queries to the oracle. To remove the ambiguity in the notation $S_{n,m}(j_1,j_2,\ldots,j_q)$, we require that \textit{the last number $j_q$ is always the number of local Grover operators}. For example, $S_{6,4}(1,2) = G_6G^2_4$ and $S_{6,4}(1,1,0) = G_4G_6$. Note that $S_{n,m}(j,0)=G^j_n$ is the original Grover algorithm. Since the sequence $S_{n,m}(j,0)=G^j_n$ does not have any local Grover operators, the number $m$ is irrelevant. As convention, we choose the notation $S_{n}(j,0)=S_{n,m}(j,0)$. The sequence $S_{n,m}(j_1,j_2,\ldots,j_q)$ can find the target state with probability:
\begin{equation}
P_{n,m}(j_1,j_2,\ldots,j_q) = |\langle t|S_{n,m}(j_1,j_2,\ldots,j_q)|s_n\rangle|^2
\end{equation}
Then we can define the expected depth of the $S_{n,m}(j_1,j_2,\ldots,j_q)$ algorithm. We want to minimize the expected depth, like for Grover's algorithm (\ref{def d G}). Define a new MED:
\begin{equation}
\label{d 1 G}
{\rm d}_1(\alpha) = \min_{m,j_1,j_2,\ldots,j_q} \frac{{\rm d}(S_{n,m}(j_1,j_2,\ldots,j_q))}{P_{n,m}(j_1,j_2,\ldots,j_q)}
\end{equation}
The minimization goes through non-negative integers $\{j_1,j_2,\ldots,j_q\}$. We also optimize the number $m$ (positive integer), which is $m<n$. The minimal value for $m$ is 2. The subscript 1 defined in $\text d_1(\alpha)$ suggests that we find the target state in one stage, i.e., no measurement within the algorithm until the end. In the quantum circuit model, a one-stage algorithm means only three steps: initialization, unitary operations and measurements. We can define multistage algorithms, which have several rounds of initializations, unitary operations, and measurements. Later we define the MED of multistage search algorithms.
Let us see one example. For $n=6$, Grover's algorithm has the MED when $j=4$:
\begin{equation}
P_6(4) = |\langle t|G_6^4|s_6\rangle|^2\approx 0.816
\end{equation}
Consider a new sequence:
\begin{equation}
S_{6,4}(1,1,2) = G_4G_6G_4^2
\end{equation}
and $S_{6,4}(1,1,2)$ gives the success probability
\begin{equation}
P_{6,4}(1,1,2) = |\langle t|S_{6,4}(1,1,2)|s_6\rangle|^2\approx 0.755
\end{equation}
Note that both sequences $G_6^4$ and $G_4G_6G_4^2$ have four oracles. According to \cite{BBCDMSSSW95}, six-qubit and four-qubit Toffoli gates can be decomposed into 64 and 16 depth circuits (with single- and two-qubit gates). We suppose that $\text d(D_6)=64$ and $\text d(D_4)=16$. One can find that if the ratio $\alpha$ in Eq. (\ref{def alpha}) is $\alpha<2.029$, then the new sequence $G_4G_6G_4^2$ has a lower expected depth. More examples (about the $n=6$ search algorithm) with quantum circuit diagrams can be found in Appendix \ref{Appendix examples 6}.
We can go back to Grover's algorithm if the number of $G_m$ is zero. We always have
\begin{equation}
\text d_1(\alpha)\leq\text d_\text{G}(\alpha)
\end{equation}
The choice of subspace (acted upon by local diffusion operators $D_m$ defined in (\ref{def I n m})) can be arbitrary, such as qubits with high connectivity in real quantum computers. But all local diffusion operators $D_m$ should act on the same qubits. For example, the sequence $S_{6,4}(1,1,2)$ has three local Grover operators. The three local diffusion operators are acting on the same four qubits. Making the wrong choice of the subspace can dramatically increase the number of invariant amplitude subspaces. Such a strategy may have some advantages in search algorithms, but it is beyond the scope of this paper.
The minimization results will depend on: the size of the database (the number $n$), the ratio between oracle depth $\text d(U_t)$ and diffusion operator depth $\text d(D_n)$ (the value of $\alpha$ defined in (\ref{def alpha})); how $\text d(D_n)$ scales with $n$ (logarithmic, linear, or quadratic with $n$). In numerical optimizations, we can set some constraints which rule out the possibility $\text d_1(\alpha)<\text d_\text{G}(\alpha)$. For example, we can set the total number of $G_n$ to less than $\lfloor0.69\sqrt N\rfloor$; if the number of $G_n$ is $j$, then the number of $G_m$ should be less than $\lfloor(0.69\sqrt N-j)(\alpha+1)/\alpha\rfloor$. As examples, we find the optimal sequence for $n=4,5,\ldots,10$ with $\alpha=1$ (assuming $\mathcal O(n)$ depth of the $\Lambda_{n-1}(X)$ gate \cite{BBCDMSSSW95}). The estimated depths are plotted in Fig. \ref{fig 2}. Details of the corresponding optimal sequences and success probabilities can be found in Appendix \ref{Appendix opt examples}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{depth.pdf}
\caption{(a) Estimated $\text d_\text{G}(\alpha)$ (MED of Grover's algorithm is defined in Eq. (\ref{def d G})), $\text d_1(\alpha)$ defined in Eq. (\ref{d 1 G}) and $\text d_2(\alpha)$ defined in Eq. (\ref{d 2 G}) with $\alpha=1$. Depth $\text d(D_n)$ is counted using the optimal results in Ref. \cite{BBCDMSSSW95}. The corresponding optimal sequences and success probabilities are listed in Appendix \ref{Appendix opt examples}. (b) Depth of the optimal sequence. The left (red) bar is Grover's algorithm. The right (green) bar is the optimal sequence from $\text d_1(1)$ defined in Eq. (\ref{d 1 G}). $\text d_2(1)$ has two stages: the bottom of the middle bar is the depth of the first stage circuit and the top of the middle bar is the depth of the second stage circuit. }\label{fig 2}
\end{center}
\end{figure}
\subsection{\label{subsec:divded and conquer} Depth Optimizations for Multistage Quantum Search Algorithms}
In the NISQ era, errors can be suppressed if a long algorithm is divided into shorter pieces (by new initializations and measurements). Inspired by the hierarchy QPSA \cite{KX07}, we propose depth optimizations for the multistage quantum search algorithm. For simplicity, we consider the two-stage quantum search algorithm firstly.
Suppose that the target state is divided into two-parts:
\begin{equation}
\label{eqa t t1}
|t\rangle=|t_1\rangle\otimes|t_2\rangle
\end{equation}
Suppose that the bit length of $t_1$ is $m_1$ and the bit length of $t_2$ is $m_2$. Note that we have $m_1+m_2=n$. After first stage, the search algorithm can find $|t_1\rangle$ with high probability. Based on the result on the first stage, we can rescale the database. After the second stage, the algorithm can find $|t_2\rangle$ with high probability (if $|t_1\rangle$ is found in the first stage). The algorithm has the following steps:
\begin{enumerate}[Step 1:]
\item Initialize the state to $|s_n\rangle$ defined in Eq. (\ref{def s n}).
\item Perform the sequence
\begin{equation}
S^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q) = G_n^{j_1}G_{m_2}^{j_{2}}\cdots G_n^{j_{q-1}}G_{m_2}^{j_q}
\end{equation}
on the initial state $|s_n\rangle$. The local diffusion operator $D_{m_2}$ (defined in $G_{m_2}$) is acting on $m_2$ qubits.
\item Measure the qubits (computational basis measurements) which do \textit{not} have the local diffusion operator $D_{m_2}$ acting on them. Suppose that we get the classical results: $t'_1\in\{0,1\}^{m_1}$. The probability that $t'_1=t_1$ is denoted as $P^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q)$.
\item Initialize the state to
$$ |t_1'\rangle\otimes|s_{m_2}\rangle
$$
Here $|s_{m_2}\rangle$ is the rescaled initial state:
\begin{equation}
|s_{m_2}\rangle = H^{\otimes m_2}|0\rangle^{\otimes m_2}
\end{equation}
\item Perform the sequence
\begin{equation}
S^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q) = G_{m_2}^{j'_1}G_{m'}^{j'_2}\cdots G_{m_2}^{j'_{q-1}}G_{m'}^{j'_q}
\end{equation}
on the new initial state. We have $m'<m_2$. The diffusion operator $D_{m_2}$ (defined in $G_{m_2}$) is acting on $|s_{m_2}\rangle$. And the diffusion operator $D_{m'}$ is acting on the subspace of $|s_{m_2}\rangle$.
\item Measure the qubits (computational basis measurements) which have the initial state $|s_{m_2}\rangle$. Suppose that we get the classical results: $t'_2\in\{0,1\}^{m_2}$. The probability that $t'_2=t_2$ is denoted as $P^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q)$.
\item Verify the solution $|t'\rangle = |t_1'\rangle\otimes|t_2'\rangle$ by classical oracle. If the solution is the target item, then stop; if not, back to step 1.
\end{enumerate}
Steps 1-3 are the first stage: we find $t_1$ with high probability. Steps 4-6 are the second stage: we find the remaining bits of the target state. Step 7 is used to verify. Different sequences $S^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q)$ and $S^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q)$ give different success probabilities $P^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q)$ and $P^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q)$. We want to find the MED. The MED of the two-stages search algorithm is
\begin{widetext}
\begin{equation}
\label{d 2 G}
{\rm d}_2(\alpha) = \min_{m_2,m',j_1,\ldots,j_q,j'_1,\ldots,j'_q} \frac{{\rm d}(S^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q))+{\rm d}(S^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q))}{P^{(1)}_{n,m_2}(j_1,j_2,\ldots,j_q)P^{(2)}_{m_2,m'}(j'_1,j'_2,\ldots,j'_q)}
\end{equation}
\end{widetext}
We optimize the total expected depth. We do not optimize the expected stage depth, because we cannot verify the partial bit by neither classical nor quantum oracle. Note that $m_2$ is the bit length of $t_2$. We can either fix $m_2$ or optimize different choices of $m_2$. In the definition of ${\rm d}_2(\alpha)$, we optimize the choices of $m_2$. The second-stage algorithm is a rescaled version of the full search algorithm. Such a two-stage quantum search algorithm (with depth optimization) can be easily generalized to the multi-stage quantum search algorithm.
As an example, let us consider the $n=4$ two-stage search algorithm. Grover's algorithm (one-stage search algorithm) has the success probability
\begin{equation}
P_4(3)=|\langle t|G_4^3|s_4\rangle|^2\approx 0.961
\end{equation}
In a two-stage search algorithm, we divide the target state into two parts: $|t\rangle=|t_1\rangle|t_2\rangle$. We choose the first-stage sequence as $S^{(1)}_{4,2}(1,1)=G_4 G_2$. Then we measure the two qubits which do \textit{not} have $D_2$ (defined in $G_2$) acting on them. The probability that the measurement results reveal $|t_1\rangle$ is
\begin{equation}
P^{(1)}_{4,2}(1,1)\approx 0.953
\end{equation}
Suppose that the measurement results are $|t'_1\rangle$ after the first stage. Then we rescale the initial state as $|t'_1\rangle\otimes|s_2\rangle$. We choose the second stage sequence as $S^{(2)}_{2}(1,0)=G_2$. Recall that the two-qubit Grover's algorithm can find the target state with 100\% probability with one Grover operator. Therefore, the second-stage success probability is
\begin{equation}
P^{(2)}_{2}(1,0)=1
\end{equation}
Then the total success probability is
\begin{equation}
P^{(1)}_{4,2}(1,1)P^{(2)}_{2}(1,0)\approx 0.953
\end{equation}
The result is quite close to Grover's algorithm with the same number of oracles, but the depth in each stage is less than in Grover's algorithm.
Another interesting example (two-stage $n=4$ search algorithm) is that the sequence $S^{(1)}_{4,2}(1,2)$ gives probability $1$ for finding $t_1$. Combined with the second-stage sequence $S^{(2)}_{2}(1,0)$, we find a new approach for the $n=4$ exact search algorithm \cite{Diao10}. We estimate $\text d_2(\alpha)$ with $\alpha=1$ for the $n=4,5\ldots,10$ search algorithms, see Fig. \ref{fig 2}. The corresponding optimal sequences are listed in Appendix \ref{Appendix opt examples}. See Appendix \ref{Appendix examples 6} for more examples (with quantum circuit diagrams) on two-stage quantum search algorithms.
\section{\label{sec:alpha}Critical Ratios}
\subsection{\label{sec: one stage alpha}The Critical Ratio for the One-stage Algorithm}
Grover's algorithm is optimal in the number of queries to the oracle \cite{BBHT98,Zalka99}. Grover's algorithm is a one-stage search algorithm: no measurement occurs within the algorithm until the end. When $\alpha\rightarrow\infty$, we expect ${\rm d}_1(\alpha)=\text d_\text{G}(\alpha)$ (no local diffusion operators). Here ${\rm d}_1(\alpha)$ is defined in Eq. (\ref{d 1 G}). And $\text d_G(\alpha)$ defined in Eq. (\ref{def d G}) is the MED of Grover's algorithm. We define the critical alpha $\alpha_{c,1}$ for the one-stage search algorithm:
\begin{equation}
\label{def alpha c}
\alpha_{c,1}=\max\{\alpha|{\text d}_1(\alpha) <{\text d}_{\text G}(\alpha)\}
\end{equation}
The subscript 1 in $\alpha_{c,1}$ denotes the one-stage search algorithm. Below $\alpha_{c,1}$, the depth of Grover's algorithm is {\it not} optimal. Based on the depth optimization method proposed in Sec. \ref{subsec: opt method}, we can give an estimation of $\alpha_{c,1}$:
\begin{theo}
\label{theorem 1} $\alpha_{c,1}=\mathcal O(n^{-1}2^{n/2})$.
\end{theo}
\begin{proof}
The MED $\text d_1(\alpha)$ defined in Eq. (\ref{d 1 G}) is a search algorithm with two different diffusion operators. One is the local diffusion operator $D_m$, see (\ref{def I n m}). The other is the global diffusion operator $D_n$, see (\ref{def I n}). The local diffusion operator $D_m$ is only acting on the subspace of the database. We can follow a three-dimensional subspace: the target state $|t\rangle$ defined in Eq. (\ref{basis t}); the normalized sum of nontarget states in the target block $|ntt\rangle$ defined in Eq. (\ref{basis ntt}); the normalized sum of rest states in the database $|u\rangle$ defined in Eq. (\ref{basis u}). The notations are taken from the QPSA, see Sec. \ref{subsec:QPSA} and \cite{Korepin05,KG06}.
Operators $G_n$ and $G_m$ only change the relative amplitudes of states $|t\rangle$, $|ntt\rangle$, and $|u\rangle$. Therefore, operators $G_n$ and $G_m$ are elements of the $O(3)$ group \cite{KV06}. It is interesting to see that operator $G_m$ can be viewed as a rescaled version of $G_n$. In the new basis $\{|t\rangle,|ntt\rangle,|u\rangle\}$, the sequence $S_{n,m}(j)=G^j_m$ (which only has local Grover operators $G_m$) has the representation
\begin{equation}
S_{n,m}(j)=G^j_m=\left(\begin{array}{ccc}
\cos(2j\theta_2) & \sin(2j\theta_2) & 0\\
-\sin(2j\theta_2) & \cos(2j\theta_2) & 0\\
0 & 0 & 1
\end{array}
\right)
\end{equation}
For example, the matrix element $\sin(2j\theta_2)$ is obtained from
\begin{equation}
\sin(2j\theta_2) = \langle t|S_{n,m}(j)|ntt\rangle
\end{equation}
The angle is defined as
\begin{equation}
\sin\theta_2 = 1/\sqrt b,\quad b=2^m
\end{equation}
We want to estimate the critical ratio $\alpha_{c,1}$. We consider the sequence:
\begin{equation}
S_{n,n-1}(1,1,1) = G_{n-1}G_{n}G_{n-1}
\end{equation}
Here we choose $m=n-1$. It means that the database is divided into two blocks. At the basis $\{|t\rangle,|ntt\rangle,|u\rangle\}$ defined in Eqs. (\ref{basis t})-(\ref{basis u}), the sequence $S_{n,n-1}(1,1,1)$ has the matrix representation
\begin{widetext}
\begin{equation}
\label{def S 111}
S_{n,n-1}(1,1,1)=\left(
\begin{array}{ccc}
c^2(c^2-3s^2) & cs(3c^2-s^2)(c^2-3s^2) & s(3c^2-s^2) \\
-cs(3c^2-s^2)(c^2-3s^2) & s^2(s^2-3c^2) & c(c^2-3s^2) \\
-s(3c^2-s^2) & c(c^2-3s^2) & 0
\end{array}
\right)
\end{equation}
\end{widetext}
with short notations $c=\cos\theta_2$ and $s=\sin\theta_2$. Note that $\sin\theta_2=\sqrt{2/N}$ since we choose $m=n-1$. The matrix $S_{n,n-1}(1,1,1)$ has the eigenvalues:
\begin{equation}
\lambda_0 = -1,\quad\quad \lambda_\pm = e^{\pm i\gamma}
\end{equation}
with
\begin{equation}
\tan \gamma = \frac{\Delta}{1+\cos\theta_2},\quad
\Delta = \sqrt {3-2\cos(6\theta_2)-\cos^2(6\theta_2)}
\end{equation}
The corresponding normalized eigenvectors are denoted as $|v_0\rangle$ (with eigenvalue $\lambda_0$) and $|v_\pm\rangle$ (with eigenvalue $\lambda_\pm$). States $|v_0\rangle$ and $|v_\pm\rangle$ have the form:
\begin{subequations}
\begin{align}
\label{eigen lambda 0} |v_0\rangle =& \frac 1 {\mathcal N_0}\left(0,1,\cos\theta_2(1-4\cos^2\theta_2)\right)^T,\\
\label{eigen lambda +-} |v_\pm\rangle =& \frac 1 {\mathcal N_\pm}\left(\mp i \sqrt{\frac{3+\cos 6\theta_2}{2}},\cos 3\theta_2,1\right)^T
\end{align}
\end{subequations}
The notation $T$ means transpose. $\mathcal N_0$ and $\mathcal N_\pm$ are normalizations. Note that the eigenvector $|v_0\rangle$ (with eigenvalue $-1$) is orthogonal to the target state, i.e., $\langle t|v_0\rangle=0$. We can view the operator $S_{n,n-1}(1,1,1)$ as rotation combined with reflection. Rotation is around an axis perpendicular to $|t\rangle$. The rotation angle is $\gamma$. Reflection is around a plane perpendicular to $|t\rangle$. Iteration $S_{n,n-1}(1,1,1)$ on the initial state gives
\begin{multline}
\label{tilde G s n}
\langle t|S^{\tilde j}_{n,n-1}(1,1,1)|s_n\rangle= \\ \lambda_+^{\tilde j}\langle t|v_+\rangle\langle v_+|s_n\rangle+\lambda_-^{\tilde j}\langle t|v_-\rangle\langle v_-|s_n\rangle
\end{multline}
We have $\langle t|v_\pm\rangle = \mp i /\sqrt 2$. Because $N=2^n$ is a large number, the angle $\theta_2$ is a small number. We can expand:
\begin{subequations}
\begin{align}
&\gamma= 3\sqrt 2 \theta_2+\mathcal O\left(\theta^2_2\right),\\
&\langle v_\pm|s_n\rangle = \frac 1 {\sqrt 2} +\mathcal O\left(\theta_2\right)
\end{align}
\end{subequations}
We substitute the above relations into Eq. (\ref{tilde G s n}). After some algebra, we can get the success probability of finding the target state:
\begin{equation}
|\langle t|S^{\tilde j}_{n,n-1}(1,1,1)|s_n\rangle|^2 = \sin^2\left(3\sqrt 2\tilde j\theta_2\right)+\mathcal O(\theta_2)
\end{equation}
Because the sandwich sequence $S_{n,n-1}(1,1,1)$ has three oracles, we set $\tilde j=3j$. Then the probability difference between $S^{\tilde j}_{n,n-1}(1,1,1)$ and Grover's algorithm (with the same number of oracles) is
\begin{equation}
|\langle t|G^j_n|s_n\rangle|^2 -|\langle t|S^{\tilde j}_{n,n-1}(1,1,1)|s_n\rangle|^2=\delta>0
\end{equation}
Here $\delta$ is a small number:
\begin{equation}
\delta = \mathcal O(2^{-n/2})
\end{equation}
Grover's algorithm (with $j$ Grover iterations) has success probability $P_n(j)$, see Eq. (\ref{def P n}).
Then the success probability for the $S^{\tilde j}_{n,n-1}(1,1,1)$ sequence (with $\tilde j=j/3$ iterations) is $P_n(j)-\delta$. If we want the new sequence $S^{\tilde j}_{n,n-1}(1,1,1)$ to have lower expected depth than Grover's algorithm, we can set
\begin{equation}
\frac{3(\alpha+1)\text d(D_n)}{P_n(j)}>\frac{(3\alpha+1)\text d(D_n)+2\text d(D_{n-1})}{P_n(j)-\delta}
\end{equation}
The left-hand side (times $j/3$) is the expected depth of Grover's algorithm. The right-hand side (times $\tilde j=j/3$) is the expect depth of the $S^{\tilde j}_{n,n-1}(1,1,1)$ algorithm. The above inequality gives
\begin{equation}
\alpha < \frac{2(\text d(D_n)-\text d(D_{n-1}))P_n(j)}{3\text d(D_n)\delta}
\end{equation}
The diffusion operator $D_n$ has the depth $\text d(D_n) = \mathcal O(n)$ or $\text d(D_n) = \mathcal O(n^2)$ \cite{BBCDMSSSW95}. Then we have
\begin{equation}
\alpha_c = \mathcal O(n^{-1}2^{n/2})
\end{equation}
This is the end of the proof.
\end{proof}
As examples, we numerically estimate $\alpha_{c,1}$ defined in Eq. (\ref{def alpha c}) for $n=4,5,\ldots,10$ based on the linear depth of $D_n$, see Appendix \ref{Appendix alpha} and Table \ref{Table 4}. Below the critical ratio $\alpha_{c,1}$, at least two-third of the global diffusion operators $D_n$ can be replaced by $D_{n-1}$ (to have lower expected depth). The saved depth scales as $\mathcal O(2^{n/2})$.
\subsection{\label{sec: two stage alpha}The Critical Ratio for the Two-stage Algorithm}
Similar to the one-stage search algorithm, we can define the critical ratio for the two-stage algorithm:
\begin{equation}
\label{def alpha c 2}
\alpha_{c,2}=\max\{\alpha|{\text d}_2(\alpha) <{\text d}_{\text G}(\alpha)\}
\end{equation}
Here $\text d_2(\alpha)$ is the MED of the two-stage search algorithm, defined in Eq. (\ref{d 2 G}). The two-stage search algorithm has two measurements. After the first measurement, we reinitialize the state in the rescaled database. The amplified amplitude of the target state $|t\rangle$ is lost in the new initialization. One can argue that
\begin{equation}
{\rm d}_2(\alpha)>{\rm d}_{1}(\alpha),
\end{equation}
and it implies that $\alpha_{c,2}<\alpha_{c,1}$. Analytically, we can prove the following theorem.
\begin{theo} \label{theorem 2}
$\lim_{N\rightarrow\infty}\alpha_{c,2} = 1+\sqrt 3 \approx 2.732$.
\end{theo}
\begin{proof}
Similar to the proof of Theorem \ref{theorem 1}, we construct a special sequence. Then we compare the expected depth of such a sequence with the expected depth of Grover's algorithm. Since we consider the two-stage search algorithm, we need two sequences for two stages. First, we assume that the target state $|t\rangle$ has two parts $|t\rangle=|t_1\rangle\otimes|t_2\rangle$, the same as in Eq. (\ref{eqa t t1}). And the bit length of $t_2$ is 2. For the first stage, we consider the sequence:
\begin{equation}
S^{\tilde j}_{n,2}(1,1) = \left(G_nG_2\right)^{\tilde j}
\end{equation}
In the first stage (by the sequence $S^{\tilde j}_{n,2}(1,1)$), we find $t_1$ with high probability. The probability is denoted as $P^{(1)}_{n,2}$. In the second stage, we have a rescaled two-qubit search algorithm. One Grover operator $G_2$ can find the target state with $100\%$ probability. Therefore, the second stage has the sequence:
\begin{equation}
S_{2}(1,0) = G_2
\end{equation}
The probability of finding $t_2$ is $P^{(2)}_{2}=1$.
In the basis $\{|t\rangle,|ntt\rangle,|u\rangle\}$ defined in Eqs. (\ref{basis t})-(\ref{basis u}), the sequence $S_{n,2}(1,1)$ has the matrix representation
\begin{equation}
S_{n,2}(1,1) = \frac 1 2\left(
\begin{array}{ccc}
\cos 2\gamma & \sqrt 3 & \sin 2\gamma \\
\sqrt 3\cos 2\gamma & -1 & \sqrt 3 \sin 2\gamma \\
-2\sin 2\gamma & 0 & 2\cos 2\gamma
\end{array}
\right)
\end{equation}
with $\sin \gamma = 2/\sqrt N$. We can easily find eigenvalues and eigenvectors of $S_{n,2}(1,1)$. Then we can have a matrix expression for $S^{\tilde j}_{n,2}(1,1)$. Applying $S^{\tilde j}_{n,2}(1,1)$ on the initial state $|s_n\rangle$ (Eq. \ref{def s n rewrite}), \begin{equation}
|\langle u|S^{\tilde j}_{n,2}(1,1)|s_n\rangle|^2 = \cos^2(\sqrt 3 \tilde j\gamma)+\mathcal O(\gamma)
\end{equation}
Note that $|\langle u|S^{\tilde j}_{n,2}(1,1)|s_n\rangle|^2$ is the probability of finding the state in the nontarget block. In other words, we have
\begin{equation}
P^{(1)}_{n,2} = 1-|\langle u|S^{\tilde j}_{n,2}(1,1)|s_n\rangle|^2
\end{equation}
The second stage has probability 1 (the two-qubit Grover's algorithm with one Grover operator has probability 1). Then $P^{(1)}_{n,2}$ is also the probability of finding the target state.
The two stages designed above have a total of $2\tilde j+1$ queries to the oracle. In order to compare with Grover's algorithm, we set $j=\sqrt 3 \tilde j$ (where $j$ is the number of queries to the oracle in Grover's algorithm). Grover's algorithm with $j$ iterations has a success probability $P_n(j)$ of finding the target state, see Eq. (\ref{def P n}). Then the two-stage search algorithm (with sequences $S^{\tilde j}_{n,2}(1,1)$ and $S_{2}(1,0)$) can find the target state with probability $P_n(j)+\delta$. Here $\delta$ is a small number in order $\delta = \mathcal O(2^{-n/2})$. If we want the two-stage search algorithm to have lower expected depth than Grover's algorithm, we need
\begin{equation}
\frac{(\alpha+1)\text d(D_n)}{P_n(j)}>\frac{(2\alpha+1)\text d(D_n)+3}{\sqrt 3(P_n(j)+\delta)}
\end{equation}
The left-hand side (times $j$) is the expected depth of Grover's algorithm (with $j$ iterations). The right-hand side (times $j$) gives the expected depth of the designed two-stage search algorithm. Note that the second-stage circuit only contributes order $\mathcal O(2^{-n/2})$ to the critical value $\alpha_{c,2}$; therefore, we can neglect it here. Then we can solve the inequality
\begin{equation}
\alpha > 1+\sqrt 3 -\frac 3 {\text d(D_n)}+\mathcal O\left(2^{-n/2}\right)
\end{equation}
For large $N$, we have the critical ratio
\begin{equation}
\lim_{N\rightarrow\infty}\alpha_{c,2} = 1+\sqrt 3 \approx 2.732
\end{equation}
This ends of the proof.
\end{proof}
Theorem $\ref{theorem 2}$ suggests that the two-stage search algorithm can have lower expected depth than Grover's algorithm, only when the oracle can be realized as efficiently as the global diffusion operator. The real advantage of the two-stage algorithm is to mitigate the error accumulations for long circuits. For examples, see Fig. \ref{fig 2} and Appendixes \ref{Appendix examples 6} and \ref{Appendix opt examples}. We numerically estimate the value $\alpha_{c,2}$ ($n=4,5,\ldots,10$) based on a linear scale depth of $\text d(D_n)$, see Appendix \ref{Appendix alpha} and Table \ref{Table 4}.
\section{\label{sec:parallel}Parallel Running of Quantum Search Algorithm}
Now we discuss how to run the quantum search algorithm on several quantum computers in parallel. The simplest idea is running a low-success-probability search algorithm on different quantum computers. We verify the result by classical oracle and continue the algorithm until one of the quantum computers finds the target state \cite{GWC00}. First we can set a threshold success probability. Then we find the optimal sequence which gives the MED (the success probability is lower than the threshold success probability). We can run such a sequence on several quantum computers.
Another parallel running method is to combine the random guess with search algorithm, as mentioned in Ref. \cite{Korepin05} for the QPSA. For example, the target state is divided into two parts: $|t\rangle=|t_1\rangle\otimes|t_2\rangle$, the same as in Eq. (\ref{eqa t t1}). One can randomly guess the bits $t_1$. Then one performs the search algorithm on bits $t_2$. Each quantum computer can pick up one guess. However, if more than half of the bits are chosen randomly, the quadratic speedup is lost. Such a strategy is more efficient if some of the bits have higher probability (prior information about the target state).
If we want near-deterministic (the fail probability is $\mathcal O(2^{-n/2})$) parallel running of the search algorithm, then we can apply the multistage search algorithm on different quantum computers. Suppose the target state has length $n$. The target state is divided into $p$ parts, and each part has equal $n/p$ length. Then we can assign the search algorithm on $p$ quantum computers. Each quantum computer finds one part of the target state. Combining all the results from each quantum computers, we can piece together the whole solution $t$ at one time. The sequence running on each quantum computer can be found by maximizing the number of local Grover operators $G_m$ defined in Eq. (\ref{def G m}), based on some threshold success probability ($\mathcal O(1-2^{-n/2})$). It requires at most $n$ quantum computers. Each quantum computer finds one bit of the target state. However, the most efficient way to find one bit of the target state is by running the random-guess one-bit search algorithm \cite{Korepin05}.
\section{\label{sec:conclusion}Conclusion and Outlook}
In this paper, we propose a new way to optimize the depth of quantum search algorithms. The quantum search algorithm can be realized by global and local diffusion operators. The ratio of the depth of the oracle and global diffusion operator is important. The ratio is denoted by $\alpha$, and defined in Eq. (\ref{def alpha}). The minimal practical value for $\alpha$ is 1 (in one target search algorithm). When $\alpha$ is below a threshold, we can design a new algorithm (new sequence) which has a lower expected depth than Grover's algorithm. We gave examples for $\alpha=1$. In examples, our algorithm has around $20\%$ lower depth than Grover's algorithm. We also study the depth optimization in the multi-stage quantum search algorithm. In each stage, the circuit has lower depth than in Grover's algorithm. The multistage quantum search algorithm gives a natural way for parallel running of the quantum search algorithm.
Ideas in this work can be easily generalized to the multitarget solution search \cite{BBHT98}. However, the exact number of target states is required in order to find the optimal sequence. In this paper, we only consider two kinds of diffusion operators (at each stage). Further improvement is possible if more diffusion operators are working together. It will be interesting to optimize the depth of the amplitude amplification algorithm \cite{Grover98,BHMT00}. Grover's algorithm is only optimal in the oracle measure. Our search algorithm has lower depth than Grover's algorithm.
\begin{acknowledgments}
The authors are grateful to Professor Jin Wang and Yulun Wang. V.K. is supported by SUNY Center for Quantum Information Science at Long Island Project No. CSP181035.
\end{acknowledgments}
|
2,877,628,089,544 | arxiv | \section{Introduction}\label{sec:introduction}
Despite the media hype, %
collaborative security approaches are seldom implemented as they
raise several important challenges.
Security data, such as firewall logs or attack intelligence, might expose confidential and/or sensitive information, challenge
corporations' competitiveness, or even reveal negligence.
As a consequence, previous work proposed to sanitize data prior to %
sharing~\cite{lincoln2004privacy,porras2006large,xu2002prefix}.
However, this makes data less useful~\cite{lakkaraju2008evaluating} and still prone to de-anonymization~\cite{coull2007playing}.
One alternative is to let entities contribute encrypted data to a semi-trusted central repository that obliviously aggregates contributions~\cite{pets10}, or use distributed data aggregation protocols based on secure multi-party computation~\cite{burkhart2010sepia}.
While aggregation can help compute traffic statistics, it only identifies most prolific attack sources and yields global models. As shown in~\cite{soldo,zhang2008highly}, however, generic attack models miss a significant number of attacks, especially when attack sources choose targets strategically and focus on a few known vulnerable networks.
In theory, Fully Homomorphic Encryption (FHE)~\cite{gentry2009fully} could be used
to compute personalized recommendations, however, FHE is still far from being practical and
it remains unclear whether complex machine learning algorithms needed for the prediction could effectively run over encrypted data.
\descr{Intuition.}
This paper explores a novel approach to collaborative threat mitigation where organizations find suitable collaboration partners in a distributed and privacy-preserving way, and organize into coalitions prior to sharing. This way, sharing takes place within groups of related victims, i.e., that share relevant sources of information.
In our model, parties first identify a set of {\em potential} partners from a larger pool of organizations, e.g., corporations in the same sector, and then select the {\em best} partners. In practice, this can be repeated over time to ensure relevant and near real-time protection.
We introduce the {\em Sharing is Caring (SIC)} framework, which supports two types of algorithms: one for estimating the benefits of sharing in a privacy-preserving way (i.e., without disclosing plaintext data), and the other for sharing agreed-upon datasets with selected partners, e.g., only common attacks.
We focus on data sharing for predictive blacklisting, namely forecasting attack sources based on logs generated by different organizations' firewalls and/or intrusion detection systems. %
As shown in previous work~\cite{katti2005collaborating,soldo,zhang2008highly}, collaboration improves
defense accuracy, as attackers tend to target victims in similar ways.
\descr{Experiments.}
One of our main goals is to investigate which collaboration strategies work best, in terms of the resulting improvement in prediction accuracy. %
To this end, we conduct several experiments on a real-world dataset of 2 billion suspicious IP addresses collected by DShield.org~\cite{dshield} over 2 months. %
This dataset contains a large variety of contributors, as confirmed by our analysis, which allows us to test the effectiveness of data sharing among diverse groups of victims.
\descr{Main Results.}
Our analysis yields several key findings, as we observe that: (1) the more information is available about attackers, the better the prediction, as intuitively expected; (2) different collaboration strategies yield a large spectrum of performances, in fact, with some strategies, sharing does not actually help much; (3) sharing information only about common attackers is almost as useful as sharing everything. This highlights both the importance of selecting the right partners and the usefulness of controlled data sharing.
\descr{Summary of Contributions.}
Our work is the first to provide a privacy-enhanced solution for collaborative predictive blacklisting. %
We demonstrate that data sharing does not have to be an ``all-or-nothing'' process: by relying on efficient cryptographic protocols for privacy-preserving information sharing, it is possible to share relevant data, and only when beneficial.
Compared to prior work, our approach has several advantages:
(1) it helps privately identify entities with good partnership potential,
(2) it minimizes information disclosure, and
(3) it increases speed of malicious activity detection, leading to near real-time mitigation.
Our work could also be applied to other security-related applications that benefit from data sharing, such as spam filtering~\cite{damiani2004p2p}, malware detection~\cite{hailpern2001collaborative}, or DDoS mitigation~\cite{oikonomou2006framework}.
\descr{Paper Organization.}
The rest of the paper is organized as follows.
Next section presents some preliminary notions. Section~\ref{sec:framework} introduces the Sharing is Caring (SIC) framework, while Section~\ref{sec:dshield} presents a measurement-based analysis of a real-world dataset of security logs. Section~\ref{sec:experiments} covers an experimental evaluation of proposed techniques, followed by related work, in Section~\ref{sec:related}. The paper concludes with Section~\ref{sec:conclusion}.
\section{Preliminaries}\label{sec:preliminaries}
This section presents our system assumptions and some relevant background information.
\subsection{System Model}\label{sec:system}
We assume a network of entities $\mathcal{V} = \{V_i\}_{i=1}^n$.
Each $V_i$ maintaining a dataset $S_i$ of suspicious events, such as suspicious IP addresses observed by a firewall
$\langle$IP, time, port$\rangle$. We denote this list of events as $L_i$ (for each entity $V_i$). Hence, $S_i=\{L_i\}$.
Each entity $V_i$ aims to predict and block (i.e., blacklist) future attacks.
\descr{Existing Approaches.} Thus far, two main approaches have been used for predictive blacklisting: (1) no collaboration, i.e., each entity $V_i$ independently performs the prediction based only its own dataset $S_i$, or (2) community-based, i.e., each entity $V_i\in\mathcal{V}$ submits its dataset $S_i$ to a central repository, which returns a customized blacklist for $V_i$, also based on all entities' datasets.
The latter provides increased accuracy~\cite{zhang2008highly,soldo}
but requires entities to reveal their datasets to a central repository.
\descr{Our Novel Model.}
We introduce a privacy-friendly collaborative model for predictive blacklisting, whereby entities identify good collaboration partners via pairwise secure computations (without the need for a trusted third-party), and then share data.
This way,
data sharing takes place in groups of related victims. Each entity performs predictions based not only on its own dataset but also on an augmented dataset that comprises information possibly shared by the counterpart, aiming to %
improve prediction and, at the same time, avoiding the wholesale disclosure of datasets.
\descr{Threat Model.} We denote with $\mathcal{A} \in \mathcal{V}$ an adversary attempting to learn information about other entities' datasets.
(External adversaries are not considered, since their actions can be mitigated via standard network security techniques.)
In the worst case, $\mathcal{A}$ may try to collaborate with all other entities and collect available information after each data sharing attempt. $\mathcal{A}$ obtains network traces that allow inference of strategic information. Hence, we aim to protect data confidentiality for each $V_i \in \mathcal{V}$.
We assume adversary $\mathcal{A}$ to be semi-honest (or honest-but-curious): $\mathcal{A}$ follows protocols' specifications and does not misrepresent any of its inputs, but, during or after protocol execution, it might attempt to infer additional information about other parties' inputs.
\subsection{Privacy-preserving Information Sharing}
We now review some cryptographic primitives used through the rest of the paper.
\descr{Secure Two-Party Computation (2PC)} allows two parties, on input $x$ and
$y$, respectively, to privately compute the output of any public function $f$ over
$(x,y)$. Both parties learn nothing beyond what can be inferred from the output of the
function. For more details on 2PC refer to~\cite{Yao,HEKM11}.
\descr{Private Set Intersection (PSI)} allows two parties, a server, on input a set $S$, %
and a client, on input a set $C$, %
to interact in such a way that the latter only learns $S\cap C$, and the former
learns nothing beyond the size of $C$.
State-of-the-art instantiations, include both garbled-circuit based techniques~\cite{HEK12,dong2013private,PSZ14} and specialized protocols~\cite{FNP,KS05,DT10,JL10,DT12}.
In our experiments, we use the PSI construction presented in~\cite{DT10},
secure under the One-More-RSA assumption~\cite{onemore} in the Random Oracle Model (ROM), with computational and communication complexities linear in set sizes. Note, however, that one can select any PSI construction, without affecting our design.
\descr{Private Set Intersection Cardinality (PSI-CA)}
allows two parties, a server, on input a set $S$,
and a client, on input a set $C$,
to interact in such a way that the latter only learns $|S\cap C|$, while
the former learns nothing beyond $|C|$.
PSI-CA is a more ``stringent'' variant than PSI, as it only reveals the magnitude of the intersection,
but not the actual contents.
There are several instantiations of PSI-CA~\cite{FNP,sigmod03,hw2006,DGT12},
and, in our experiments, we use the construction presented in \cite{DGT12}, which has linear complexities, with security under the One-More-DH assumption~\cite{onemore} in the Random Oracle Model (ROM). Again, note that any PSI-CA construction can be employed.
\descr{Private Jaccard Similarity (PJS)} allows two parties, a server, on input a set $S$, and a client, on input a set $C$, to interact in such a way that
the client only learns $J(\ensuremath{S},\ensuremath{C})=(|S\cap C|)/(|S\cup C|)$,
where $J(C,S)$ denotes the Jaccard Similarity index~\cite{jaccard} between sets $S$ and $C$.
Blundo et al.~\cite{espresso} slightly relax the above definition and shows how to privately compute the Jaccard Similarity index using only PSI-CA. Since
$J(\ensuremath{S},\ensuremath{C})=(|S\cap C|)/(|S| + |C| - |S\cap C|)$,
parties can obtain $J(S,C)$ without disclosing the actual sets' content. %
\subsection{Predictive Blacklisting}\label{subsec:predictive}
As mentioned earlier, we focus on predictive blacklisting, i.e., forecasting future malicious sources based on past attacks.
\descr{Algorithm.} Let $t$ denote the day an attack was reported and $T$ the current time, so $t = 1, 2, ..., T$. We partition $T$ into two windows of consecutive days: a training window, $T_\text{\em train}$ and a testing window, $T_\text{\em test}$. Prediction algorithms rely on information in the training data, $t \in T_\text{\em train}$, to tune their model and validate the predictions for the testing data, $t \in T_\text{\em test}$.
The Global Worst Offender List (GWOL) is a basic prediction algorithm that selects top attack sources from $T_\text{\em train}$, i.e., highest number of globally reported attacks~\cite{zhang2008highly}.
Local Worst Offender List (LWOL) is the local version of GWOL and operates on a local network based entirely on its own history~\cite{zhang2008highly}. %
LWOL fails to predict on attackers not previously seen, while GWOL tends to be irrelevant to small victims.
Thus, machine learning algorithms were suggested to improve GWOL and LWOL~\cite{soldo,zhang2008highly}.
We use the {\em Exponentially Weighted Moving Average} (EWMA) algorithm, as proposed by Soldo et al.~\cite{soldo}, to perform blacklisting prediction. EWMA uses time series aggregation: it consists in aggregating attack events from $T_\text{\em train}$ to predict future attacks. Other features one could consider include the historical malicious activity of an IP address, the clustering of IP addresses with similar malicious behavior, and the network centrality of a target. It is out of the scope of this paper to improve on existing prediction algorithms -- rather, we focus on how to help organizations identify useful partners in a privacy-preserving way, and how different collaboration strategies perform in comparison to each other.
\descr{Accuracy Metrics.} As commonly done with prediction algorithms, we measure accuracy with {\em True Positives} (TP), which is the number of predictions that correctly match future events. In the blacklisting scenario, TP correspond to the number of attacks in the blacklist that are correctly predicted.
In practice, sources might not be blacklisted at once and blacklisting algorithms might rely on several observations over time before blacklisting a source, such as the rate at which the source is attacking, the payload of suspicious packets, etc. %
It is important to distinguish between the \emph{prediction algorithm}, which identifies potential malicious sources and/or creates a watch-list from the \emph{blacklisting algorithm}, which actually blocks sources. Blacklisting algorithms are site-specific and need to optimize, among others, false negative and false positive ratios. The prediction algorithm enables the identification of suspicious IP addresses that deserve further scrutiny and improve the effectiveness of blacklisting algorithms.
Therefore, just like prior work~\cite{soldo,zhang2008highly}, we focus on measuring the TP of the prediction algorithm, i.e., the ability to identify potential sources of attacks, and do not consider false positives as it is out of the scope of our work.
\descr{Upper Bounds.} A future attack can be predicted if it already appeared in the logs of some victims. Traditional upper-bounds on collaboration algorithms capture this and we use them to evaluate the performance of our collaboration algorithms. The Global Upper Bound $GUB(V_i)$ measures, for every target $V_i$, the number of attackers that are both in the training window \emph{of any victim} and in $V_i$'s testing window. For every $V_i$, we define the Local Upper Bound $LUB(V_i)$, as the number of attackers that are both in $V_i$'s training and testing windows.
\section{The SIC Framework}\label{sec:framework}
\begin{figure}[t]
\centering
\includegraphics[trim=0 0 0 0, width=0.9\linewidth] {graphs/diagram-4.pdf}
\vspace{-0.2cm}
\caption{Illustration of two entities operating in the SIC framework. (1) Entity $V_i$ starts interacting with entity $V_j$ and they jointly and privately estimate the benefits of collaboration; (2) Entities decide whether or not to partner; (3) Partners decide how to merge their datasets.}
\label{fig:diagram}
\end{figure}
\subsection{Overview}
We now describe, in details, the logics behind our intuition for privacy-enhanced collaborative predictive blacklisting by introducing the Sharing Is Caring (SIC) framework. It involves two types of algorithms: one supporting the secure {\em selection} of collaboration partners, and another for the privacy-preserving {\em merging} of (i.e., sharing) datasets among partners.
As discussed in Section~\ref{sec:preliminaries}, we assume a network of
$\{V_i\}_{i=1}^n$ entities and define $S_i$ to be the set of unique IP addresses held by $V_i$: $S_i=\{\mbox{IP} \in L_i\}$.%
A high-level sketch of the SIC framework is presented in Fig.~\ref{fig:diagram}.
In (1), potential partner entities $V_i$ and $V_j$ estimate the benefits they would receive from sharing their security data with each other. They could do so by securely computing one or multiple metrics.
such as size of intersection, Jaccard similarity, Pearson correlation, or cosine similarity between their datasets.
In (2),
Then, based on the estimated benefits, entities decide whether to partner or not. For instance, $V_i$ and $V_j$ become partners if the expected benefit is above a certain threshold; alternatively, each entity might partner with $k$ other entities that yield the maximum benefits.
Finally, in (3),
partners merge their datasets, e.g., by only sharing common attacks.
\begin{table}[t]
\centering
\small
\begin{tabular}{ | c | c | c | }
\hline
{\bf Benefit Esti-} & {\bf Operation} & {\bf Private} \\
{\bf mation Metric} & & {\bf Protocol}\\
\hline
\emph{Intersection-} & \multirow{2}[2]{*}{$|S_i\cap S_j|$} & \multirow{2}[2]{*}{PSI-CA~\cite{DGT12}}\rule{0pt}{2ex}\\
\emph{Size} & &\\[0.5ex]
\hline
\emph{Jaccard} & $\dfrac{|S_i \cap S_j|}{|S_i \cup S_j|}$ & PJS~\cite{espresso}
\rule{0pt}{4ex}\rule[-0.9ex]{0pt}{0pt}\\[2ex]
\hline
\multirow{2}[2]{*}{\emph{Pearson}} & \multirow{2}[2]{*}{$\sum_{l=1}^{N}\dfrac{(s_{i_l} - \mu_i)(s_{j_l}-\mu_j)}{N \sigma_i \sigma_j}$} & Garbled\rule{0pt}{2.5ex}\\
& & Circuits~\cite{HEKM11}
\rule[-0.9ex]{0pt}{0pt}\\[1ex]
\hline
\multirow{2}[2]{*}{\emph{Cosine}} & \multirow{2}[2]{*}{$\dfrac{\vec{S_i}\vec{S_j}}{\|\vec{S_i}\| \|\vec{S_j}\|}$} & Garbled\rule{0pt}{2.5ex}\\
& & Circuits~\cite{HEKM11}
\rule[-0.9ex]{0pt}{0pt}\\[1ex]
\hline
\end{tabular}
\vspace{-0.15cm}
\caption{Metrics for estimating potential benefits of data sharing between $V_i$ and $V_j$, along with corresponding protocols for their secure computation. $\mu_i,\mu_j$ and $\sigma_i,\sigma_j$ denote, resp., mean and standard deviation of $\vec{S_i}$ and $\vec{S_j}$.}
\label{tab:metric-1}
\vspace{-0.15cm}
\end{table}
\subsection{Select}\label{subsec:select}
Entities select partners by privately evaluating, {\em in pairwise interactions}, the benefits of sharing their data with each other.
\descr{Supported Metrics.} We consider several similarity metrics for partner selection. Metrics are reported in Table~\ref{tab:metric-1}, along with the corresponding protocols for their privacy-preserving computation.
We consider similarity metrics since previous work~\cite{katti2005collaborating, zhang2008highly} showed that collaborating with correlated victims works well. Victims are correlated if they are targeted by correlated attacks, i.e., attacks mounted by the same source IP against different networks around the same time. Intuitively, correlation arises from attack trends; in particular, correlated victim sites may be on a single hit list or might be natural targets of a particular exploit (e.g., PHP vulnerability). Then, collaboration helps re-enforce knowledge about an on-going attack and/or learn about an attack before it hits.
\descr{Set-based and Correlation-based Similarity.}
We consider two set-based metrics: \emph{Intersection-Size} and \emph{Jaccard}, which measure set similarity and operate on unordered sets. We also consider \emph{Pearson} and \emph{Cosine}, which provide a more refined measure of similarity than set-based metrics, as they also capture statistical relationships.
The last two metrics operate on data structures representing attack events, such as a binary vector, e.g., $\vec{S_i}=[s_{i_1}~s_{i_2}\cdots s_{i_N}]$, of all possible IP
addresses with 1-s if an IP attacked at least once and 0-s otherwise. This can make
it difficult to compute correlation in practice, as both parties need to agree on the
range of IP addresses under consideration to construct vector $\vec{S_i}$.
Considering the entire range of IP addresses is not reasonable (i.e., this would
require a vector of size 3.7 billion, one entry for each routable IP address).
Instead, parties could either agree on a range via 2PC or fetch predefined ranges from a public repository.
In practice, entities could decide to compute any combination of metrics. %
Note that the list in Table~\ref{tab:metric-1} is non-exhaustive and other metrics could be considered,
as long as it is possible to efficiently support their privacy-preserving computation.
\descr{Establishing Partnerships.} After assessing the potential benefits of data sharing, entities make an informed decision as to whether or not to collaborate, based, e.g., on:\vspace{-0.08cm}
\begin{enumerate}
\item {\em Threshold:} $V_i$ and $V_j$ partner up if the estimated benefit of sharing is above a certain threshold; \vspace{-0.08cm}
\item {\em Maximization:} $V_i$ and $V_j$ independently enlist $k$ potential partners to maximize their overall benefits (i.e., $k$ entities with maximum expected benefits); \vspace{-0.08cm}
\item {\em Hybrid:} $V_i$ and $V_j$ enlist $k$ potential partners to maximize their overall benefits, but also partner with entities for which estimated benefits are above a certain threshold.\vspace{-0.08cm}
\end{enumerate}
While in practice entities could refuse to collaborate with other entities, one could rely on well-known collaboration algorithms that offer stability (e.g., Stable Marriage/Roommate Matching~\cite{gusfield1989stable}). Without loss of generality, we leave this for future work and assume cooperative parties, i.e., entities systematically accept collaboration requests.
\descr{Symmetry of Benefits.}
Some of the protocols used for secure computation of benefits, such as PSI-CA~\cite{DGT12} and PJS~\cite{espresso}, reveal the output of the protocol to only one party. Without loss of generality, we assume that this party always reports the output to its counterpart. We operate in the semi-honest model, thus parties are assumed not to prematurely abort protocols. %
Metrics discussed above are \emph{symmetric}, i.e., both parties obtain the same value, and facilitate partner selection as both parties have incentive to select each other. %
\subsection{Merge}\label{subsec:merge}
After the Select stage, entities are organized into coalitions, i.e., groups of victims that agreed to share data with each other.
Entities can now merge their datasets with selected partners.
\descr{Strategies.} Partners could share their datasets in several ways: e.g., they can disclose their whole data or only share which IP addresses they have in common, or transfer all attack events associated to common addresses and/or a selection thereof.
\descr{Privacy-preserving Merging.}
Our goal is to ensure that nothing about datasets is disclosed to partners beyond what is agreed. For instance, if partners agree to only share information about attackers they have in common, they should not learn any other information.
Possible merging strategies, along with the corresponding privacy-preserving protocols, are reported in Table~\ref{tab:metric-3}. Again, we assume that the output of the merging protocol is revealed to both parties.
Strategies denoted as \emph{Intersection/Union with Associated Data} mean that
parties not only compute and share the intersection (resp., union), but also all events related to items in the resulting set.
Obviously, Union with Associated Data does not yield any privacy, as all events are mutually shared.
Organizations could also limit the information sharing in time, e.g., by only disclosing data older than a month or of the last week, and previously proposed sanitization techniques~\cite{adar2007user,lincoln2004privacy,slagell2005sharing} could be used on top of SIC's merging strategies.
\begin{table}[ttt]
\centering
\small
\begin{tabular}{ | c | c | c | }
\hline
{\bf Sharing Strategy} & {\bf Operation} & {\bf Private Protocol} \\
\hline
\emph{Intersection} & $S_i\cap S_j$ & ~PSI~\cite{DT10}
\rule{0pt}{2.5ex}\rule[-0.9ex]{0pt}{0pt}\\[1ex]
\hline
\emph{Intersection with} & $\{\langle\mbox{IP,time,port}\rangle|$ & PSI with~\rule{0pt}{3ex}\\
\emph{Associated Data} & $\mbox{IP}\in S_i\cap S_j\} $ & Data Transfer~\cite{DT10} \\[1ex]
\hline
\emph{Union with} & $\{\langle\mbox{IP,time,port}\rangle|$ & \multirow{2}[2]{*}{-- No Privacy --}\rule{0pt}{3ex}\\
\emph{Associated Data} & $\mbox{IP}\in S_i\cup S_j\} $ & \\[1ex]
\hline
\end{tabular}
\caption{Strategies for merging datasets among partners $V_i$ and $V_j$, along with corresponding protocols for their secure computation.}
\label{tab:metric-3}
\end{table}
\subsection{Properties}
\descr{Privacy.}
Our approach guarantees privacy through limited information sharing. Only data explicitly authorized by parties %
is actually shared.
Data sharing occurs by means of secure two-party computation techniques, thus, security follows, provably, from that of underlying cryptographic primitives.
\descr{Authenticity.} %
Recall that we assume semi-honest adversaries, i.e., entities do not alter their input datasets.
If one relaxes this assumption, then it would become possible for
a malicious entity to inject fake inputs or manipulate datasets to violate counterpart's privacy.
Nonetheless, we argue that assuming honest-but-curious entities is realistic in our model. First, organizations can establish long-lasting relations and reduce the risk of malicious inputs as misbehaving entities will eventually get caught. Also, one could also leverage peer-to-peer techniques to detect malicious behavior~\cite{raya2008data}.
\descr{Incentives and Competitiveness.} Since data exchanges are bi-directional, each party directly benefits from participation and can quantify the contribution of its partners. If collaboration metrics do not indicate high potential, each entity can deny collaboration. That is, the incentive to participate is immediate as benefits can be quantified before establishing partnerships.
\descr{Trust.}
SIC relies on data to establish trust automatically. %
If multiple entities report similar data, then it is likely correct and contributors can be considered as trustworthy. SIC enables entities to estimate each others' datasets and potential collaboration value. This increases awareness of the contribution value and enables automation of trust establishment.
\descr{Speed.} Due to the lack of a central authority and vetting processes, data sharing in SIC is instantaneous, thus, entities can interact as often and as fast as they like.
\begin{figure*}[ht]
\centering
\hspace{-0.25cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.245] {figures/total.attacks.per.target.hist.rel.pdf}
\label{fig:average:subfig1}
}
\hspace{-0.25cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.245] {figures/total.attacks.per.source.hist.rel.pdf}
\label{fig:average:subfig2}
}
\hspace{-0.25cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.245] {figures/sources.per.target.cdf.pdf}
\label{fig:average:subfig3}
}
\hspace{-0.25cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.245] {figures/targets.per.source.cdf.pdf}
\label{fig:average:subfig4}
\hspace{-0.3cm}
}
\label{fig:average}
\vspace{-0.2cm}
\caption{Number of attacks per day: (a) on all targets, and (b) by all sources. CDF of the daily number of common and unique: (c) sources per active victims, and (d) victims per active sources. Active refers to the fact that we ignore victims/sources that do not contribute attacks on that specific day to avoid strong bias towards $0$. }
\end{figure*}
\section{The DShield Dataset}\label{sec:dshield}
In order to assess the effectiveness of our approach, we should ideally obtain
security data from real-world organizations. Such datasets are hard to come by
because of their sensitivity. Therefore, we turn to DShield.org~\cite{dshield} and obtain
a dataset of firewall and IDS logs mostly contributed by individuals and small organizations.
DShield contains data contributors are willing to report, however, as in previous work~\cite{soldo,zhang2008highly}, we can assume strong correlation between the amount of reporting and the amount of attacks.
In this section, we show that DShield dataset contains data from a large variety of contributors (in terms of the amount of contributions) and provides a reasonable alternative to experiment with our privacy-enhanced collaborative approach.
\subsection{The Dataset}
We obtained two months' worth of logs from the DShield repository. Each entry in the logs includes a Contributor ID, a source IP address, a target port number, and a timestamp -- see Table~\ref{tab:illustrationDShield}.
\begin{table}[hhh]
\resizebox{0.99\columnwidth}{!}{
\centering
\begin{tabular}{ | c | c | c | c | c | }
\hline
{\bf Contributor ID} & {\bf Source IP} & {\bf Target port} & {\bf Timestamp} \\
\hline
44cc551a & 211.144.119.042 & 1433 & 2013-01-01 11:48:36 \\
\hline
\end{tabular}
}
\vspace{-0.1cm}
\caption{Example of an entry in the DShield dataset.}
\label{tab:illustrationDShield}
\vspace{-0.15cm}
\end{table}
%
The \emph{source} of an attack refers to the attacker and \emph{target} (or contributor) refers to a victim ($V_i$).
Note that DShield anonymized the ``Contributor ID'' field by replacing it with
a random yet {\em unique} string that maps to a single victim.
Data obtained from DShield consists of about 2 billion entries, from $800$K unique contributors, including more than $16$M malicious IP sources, for a total of $170$GB. %
We pre-processed the dataset in order to reduce noise and erroneous entries, following the same methodology adopted by previous work on DShield data~\cite{soldo,zhang2008highly}. We removed approximately 1\% of of all entries, which belonged to invalid, non-routable, or unassigned IP addresses, or referred to non-existent port numbers. %
\subsection{Measurements \& Observations}
We now present a measurement analysis of the DShield dataset, aiming to better understand characteristics of attackers and victims.
Overall, our observations are in line with prior work~\cite{dacier2009wombat,pouget2005vh,soldo} and highlight how attackers tend to hit victims in a coordinated fashion, thus confirming the potential of collaboration.
\descr{General Statistics.}
We observe that $75\%$ of targets contribute less than $10\%$ of the time, while $6\%$ of targets ($50,000$ targets) contribute daily. We describe, at the end of this section how we filter out targets that seldom contribute. For more details and statistics, we refer to the Appendix.
\descr{Victims' Profile.}
Fig.~\ref{fig:average:subfig1} shows the number of attacks per day on targets, with mean number of daily attacks on targets of $58.46$ and median of $1$. We observe three distinct victims' profiles: (1) rarely attacked victims: $87\%$ of targets get less than $10$ attacks day, indicating many victims seldom attacked; (2) lightly attacked victims: $11\%$ of victims get $10$ to $100$ attacks a day; (3) heavily attacked victims: only $2\%$ of targets are under high attack (peaking at $11$M a day). In other words, most attacks target few victims.
\descr{Attackers' Profile.} Fig.~\ref{fig:average:subfig2} shows the number of victims attacked by each source per day, with mean number of daily attacks of $45.85$ and median of $2$. We observe that $80\%$ of sources initiate less than $10$ attacks a day, i.e., most sources appear stealthy. A small number of sources generates most attacks (up to $10$M daily). This indicates two main categories of attackers: stealth and heavy hitters. In our data set, we observe that several of top heavy attackers (more than $20$M attacks) come from IP addresses owned by ISPs in the UK.
\descr{Attacks' Characteristics.} Fig.~\ref{fig:average:subfig3} shows the CDF of the number of unique sources seen by each active target a day. We focus on active victims: victims that did report an event on that particular day because, as previously discussed, many victims report attacks rarely thus creating a strong bias towards $0$ otherwise. The figure contains attackers shared with other targets (common attackers) and attackers unique to a specific victim. 90\% of victims are attacked by at most $40$ unique sources and $60$ shared sources. This shows that, from the victim's perspective, targets observe more shared sources than unique ones. Compared to previous work~\cite{soldo, katti2005collaborating}, this reinforces the past trend of targets having many common attackers. %
Fig.~\ref{fig:average:subfig4} shows that $90\%$ of sources attack $30$ common victims and $60$ unique victims. Although attackers share a large number of common victims, they also uniquely attack specific victims. Note that in Fig.~\ref{fig:average:subfig3} and Fig.~\ref{fig:average:subfig4}, we observe again three types of victims and two types of attackers.
\descr{Observations.} A significant proportion of victims ($\sim$70\%) contributes a single event overall. After thorough investigation, we find that these \emph{one-time contributors} can be grouped into clusters all reporting the same IP address within close time intervals (often within one second). Many contributors share only one attack event, at the same time, about the same potentially malicious IP address. %
Similarly, many contributors only contribute one day out of the two months. %
These contributors correlate with the aforementioned one-time contributors.
We remove victims that do not share much, specifically, we remove victims that (1) share one event overall, and (2) contribute only one day and less than 20 events over the two month (i.e., 10\% of mean total contributions per victim $2$,$263$). This data processing maintains properties identified in this section and reduces the number of considered victims from $800$,$000$ to $188$,$522$, corresponding to the removal of about 2 million attacks. This filtering maintains a high diversity of contributors, and seeks to model real-world scenarios (as opposed to focusing on large contributors).
\section{Experimental Evaluation}\label{sec:experiments}
We now present an experimental evaluation of the SIC framework
focused on (1) investigating which {\em select metrics} work best to estimate the benefits of sharing (measured as the resulting improvement in prediction accuracy),
and (2) measuring what {\em merging strategies} (i.e., what data to share) provide the best privacy/accuracy trade-off. To do so, we use the DShield dataset built in Section~\ref{sec:dshield}.
Experiments involve 188,522 contributing entities, each reporting an average of 2,000 attacks,
for a total of 2 billion attacks.
\subsection{Experimental Setup}
Experiments are implemented in R. %
Source code is available upon request.
\descr{General Parameters.}
For the prediction algorithm, we use a one-week window for training ($T_\text{\em train}=7$) and aim to predict attacks for the next day ($T_\text{\em test}=1$). %
As previously discussed, organizations do not run SIC with all possible other organizations, but focus on a few potential partners. To model this, we take a {\em sampling} approach: For each iteration, we select $100$ victims at random from the set of all $188$,$522$ possible victims and run our select/merge algorithms. We average our results over $100$ iterations.
\descr{Select Algorithms.} %
We %
analyze how well different collaboration metrics (i.e., select strategies) perform in comparison to each other, where performance is measured in terms of resulting improvement in prediction accuracy.
SIC supports both set-based ({\em Intersection-Size} and {\em Jaccard}) and correlation-based ({\em Pearson} and {\em Cosine}) metrics.
With the former, the input of each entity $V_i$ is a set of unique attacking IP addresses $S_i$. {\em Intersection-Size} returns the number of IP addresses attacking both parties, while {\em Jaccard} is the ratio between the size of set intersection and the size of the union.
By contrast, for correlation to work between two entities $V_i$ and $V_j$, they need to agree on the range of IPs captured in $\vec{S_i}$ and $\vec{S_j}$.
We assume that both parties know the global list of suspicious IP addresses. In practice, parties can agree on the range via secure computation or fetch known malicious IP address lists from repositories such as DShield.
Metrics are computed pairwise, thus, we obtain a matrix estimating data sharing benefits among all possible pairs. %
We assume that parties select partners by maximizing their potential benefits in the collaboration matrix. Typically, each party picks the list of partners with the largest potential benefits. W.l.o.g. we consider that the $50$ largest collaboration pairs are selected (i.e., only 1\% of $100*99/2 = 4~950$ possible pairs as we consider 100 victims). Such a small number provides a high degree of privacy and takes a conservative stance by limiting the possible improvement in the prediction accuracy. Recall that the goal of our experimental evaluation is to understand {\em which} metrics work {\em better}, not to establish the optimal size of collaboration pools.
\begin{figure*}[t!]
\centering
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale=0.35]{figures/prediction.alpha.pdf}
\label{fig:prediction:alpha}
}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.35] {figures/prediction.visual.pdf}
\label{fig:prediction:visual}
}
\vspace{-0.3cm}
\label{fig:alpha}
\caption{Evaluation of baseline prediction (with no collaboration). (a) Number of true positives for different values of prediction algorithm parameter $\alpha$. (b) Visualization of a victim's predictions over time for a series of attackers with $\alpha=0.9$ on y-axis.
}
\end{figure*}
\descr{Merge Algorithms.} We consider two types of algorithms, \emph{Union with Associated Data} and \emph{Intersection with Associated Data} (see Section~\ref{subsec:merge}). With the former, partners share all data known by each party prior to current time $t$ and share it with each selected partner. It is a generous strategy that enriches others' datasets rapidly. With the latter, partners only share events from those IP addresses that belong to the intersection (i.e., that attacked both partners) and thus is a more conservative option. This approach can help reinforce knowledge about given adversaries, and thus help better predict attacks.
\descr{Accuracy.}
As discussed in Section~\ref{subsec:predictive}, we measure the prediction success by computing the number of True Positives (TP), as in prior work~\cite{soldo,zhang2008highly}, i.e., successfully predicted attacks. Specifically, we measure improvement as $I = (\mbox{TP}_{c}-\mbox{TP})/\mbox{TP}$,
where $\mbox{TP}$ is the number of True Positives before collaboration and $\mbox{TP}_c$ is the number of True Positives after collaboration.
We note that improvement can be measured over all entities, or for specific entities. In the following, we give both improvement measures.
\subsection{Results}
\begin{figure*}[t!]
\centering
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.33] {figures/correlation.tp.known.pdf}
\label{fig:collaboration:subfig1}
}
\hspace{-0.3cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.33] {figures/prediction.TP.pdf}
\label{fig:collaboration:subfig2}
}
\hspace{-0.3cm}
\subfigure[]{
\includegraphics[trim=0 0 0 0, scale =0.33] {figures/prediction.TP.colla.pdf}
\label{fig:collaboration:subfig3}
}
\label{fig:collaboration}
\vspace{-0.3cm}
\caption{Prediction Analysis. (a) Correlation between number of events known by targets, and their ability to predict attacks. The blue curve shows the linear regression (note the log-log scale). (b) Average sum of True Positives over time for different select methods. (c) Average sum of True Positives over time among collaborators selected by \emph{Intersection-Size} including upper bounds (LUB and GUB).
}
\end{figure*}
\smallskip\noindent {\bf Determining the Value of $\alpha$.}
Before testing the performance of select/merge algorithms, we need to identify appropriate $\alpha$ values for the EWMA prediction algorithm by evaluating the performance of the prediction.
For small values of $\alpha$, the prediction algorithm aggregates past information uniformly across the training window to craft predictions. In other words, events in the far past have a similar weight to events in the short past and the algorithm has a long memory. On the contrary, with a large $\alpha$, the prediction algorithm focuses on the most recent past events; it has short memory.
Fig.~\ref{fig:prediction:alpha} shows the evolution of the baseline prediction for different values of $\alpha$, plotting the True Positives (TP) sum of all $100$ victims averaged over $100$ iterations. Values between $\alpha=0.5$ and $\alpha=0.9$ perform best. This can be explained by remembering the ``bursty nature'' of web attacks, as discussed in Section~\ref{sec:dshield}. Prediction algorithms that react fast to the apparition of new attackers perform better.
We set $\alpha = 0.9$.
\descr{Visualizing Predictions.} Fig.~\ref{fig:prediction:visual} shows a visualization of the prediction. When an attack occurs (blue square), the algorithm systematically predicts an attack (red cross) in the next time slot. Because $\alpha=0.9$, the last attack event has a larger weight.
\descr{Baseline Prediction.}
We verify the effectiveness of the prediction algorithm by correlating the information known prior to collaboration with the ability to predict attacks. We obtain that, as expected, targets that know more about past attacks (large $S_i$), successfully predict more future attacks. We measure correlation $R>0.9$ on average, which indicates strong correlation. This, once again, suggests that collaboration increases prediction success. We visualize this correlation for a specific simulation in Fig.~\ref{fig:collaboration:subfig1}.
\descr{Select Strategies.}
Fig.~\ref{fig:collaboration:subfig2} shows the accuracy of predictions for different select methods over the course of one week, fixing the merge algorithm to \emph{Intersection with Associated Data}, as it provides the strongest privacy protection. We sum the total number of TP for ``collaborators'' (i.e., entities that do share data) and ``non-collaborators'' (entities that do not share data, thus performing as in the baseline). We observe that \emph{Intersection-Size} performs best, followed by \emph{Jaccard}, %
and \emph{Cosine/Pearson}. %
The overall decrease in sum of True Positives after day $10$ is due to the decrease of attacks
on those days
as discussed in the Appendix (see Fig.~\ref{fig:general:subfig1}).
\descr{Improvement Over Baseline.} In Fig.~\ref{fig:collaboration:subfig3}, we compare the prediction accuracy of upper-bounds, baseline, and collaboration using \emph{Intersection-Size} as the select metric and merging data using \emph{Intersection with Associated Data}.
We sum the total number of TP for collaborators selected by the \emph{Intersection-Size} metric.
Remind that with the Global Upper Bound (GUB), every victim shares with every other victim and predicts perfectly. With the Local Upper Bound (LUB), organizations do not share anything but still predict perfectly.
The accuracy of \emph{Intersection-Size} predictions tends to match LUB, showing that collaboration helps perform as well as a local perfect predictor.
Note that prediction performance can be significantly improved (thus, reducing the ``gap'' with GUB) by enabling more collaboration pairs than the conservative $50$ (1\% of all pairs) considered in our experiments.
\begin{figure}[t]
\vspace{-0.1cm}
\centering
\includegraphics[trim=0 0 0 0, scale =0.4] {figures/collaboration.setsize.boxplot.pdf}
\vspace{-0.1cm}
\caption{Boxplot of number of events known by collaborators given different select metrics. The bottom and top of the box correspond to first and third quartiles. The band inside the box is the second quartile (the median). Outliers are shows with small circles.
}
\label{fig:boxplot}
\vspace{-0.3cm}
\end{figure}
\descr{Effects of Selective Collaboration.}
Table~\ref{tab:improvements} summarizes prediction improvements for collaborators given different select metrics, reporting the mean, max, and min improvement, as well as number of collaborators.
Correlation-based metrics provide a less significant prediction improvement than set-based metrics. Mean $I$ for \emph{Pearson} and \emph{Cosine} is about $40\%$. Notably, the \emph{Intersection-Size} has a $105\%$ mean improvement. Also, mean $I$ for \emph{Jaccard} is about $60\%$. Naturally, the improvement can also be measured for each entity: $I$ for \emph{Intersection-Size} is up to $700\%$.
Differences between select metrics are due to several reasons. First, metrics that use a normalization factor (i.e., all but \emph{Intersection-Size}) tend to create partnerships of small collaborators. By contrast, \emph{Intersection-Size} leads to better performance because it promotes collaboration with larger victims. To confirm this hypothesis, we measure the set size of collaborators according to different metrics (Fig.~\ref{fig:boxplot}) and confirm that metrics with a normalization factor tend to pick collaborators that know less.
Second, correlation-based metrics tend to select partners that are \emph{too} similar: maximum correlation values are close to $1$, whereas maximum \emph{Jaccard} values get to $0.5$ only. Although this implies that targets learn to better defend against specific adversaries, it also leads to little acquired knowledge.
Third, depending on the select metric, at each time step, victims may partner with previous collaborators, or with new ones. We find that \emph{Intersection-Size}, \emph{Pearson}, and \emph{Cosine} lead to stable groups of collaborators with about 90\% reuse over time, whereas \emph{Jaccard} %
has larger diversity of collaborators over time. This is because about 20\% of victims have high \emph{Jaccard} similarity versus only 4\% for correlation-based metrics providing a larger pool of potential collaborators. Hence, if \emph{Intersection-Size} helps a few learn a lot, \emph{Jaccard} helps many victims over time.
\descr{Statistical Analysis.} A t-test analysis shows that the mean of the number of events known by collaborators differs significantly ($p < 0.0005$) across all pairs of select metrics but \emph{Cosine} and \emph{Pearson}.
If one categorizes collaborators as ``large'' if they know more than $500$ events, and ``small'' otherwise, and consider \emph{Cosine} and \emph{Pearson} as one (given the t-test result), we obtain a $3$X$2$ table of select metrics and size categories. A $\chi^2$-test shows that categorization differences are statistically significant: \emph{Intersection-Size} tends to select larger collaborators, but also more collaborators than \emph{Pearson/Cosine} (see Table~\ref{tab:improvements}). Other metrics tend to select small collaborators. We obtain $\chi^2(2, N=448)=191.99, p < 0.0005$, where $2$ is the degrees of freedom of the $\chi^2$ estimate, and $N$ is the total number of observations.
\descr{Coalitions.} Recall that, at each time step, entities can decide to partner with a number of other entities. Table~\ref{tab:improvements} shows the mean, standard deviation, and median number of collaborators per party for different collaboration metrics. We observe that with \emph{Jaccard}, entities tend to select less collaborators. Other metrics tend to have similar behavior and have entities to collaborate with about $5$ other entities out of $100$. This is in line with previous work~\cite{katti2005collaborating}, which showed the existence of small groups of correlated entities.
We also observe that, after a few days (usually $2$), \emph{Intersection-Size}, \emph{Pearson}, and \emph{Cosine} converge to a relatively stable group of collaborators. From one time-step to another, parties continue to collaborate with about 90\% of entities they previously collaborated. In other words, coalitions are relatively stable over time. Comparatively, \emph{Jaccard} has a larger diversity of collaborators over time.
\descr{Merge Algorithms.}
The next step is to compare the average prediction improvement $I$ for different merge algorithms. As showed in Fig.~\ref{fig:union}, \emph{Intersection with Associated Data} performs almost as good as {\em Union with Associated Data} with all select strategies. Actually, it performs better with {\em Jaccard}.
Merging using the union entails sharing more information, thus, one would expect it to always perform better.
However, using \emph{Union with Associated Data}, organizations quickly converge to a stable set of collaborators, and obtain a potentially lower diversity of insights over time. With most metrics, the set of collaborators is stable over time anyways, and so union does perform better than intersection. But, as previously discussed, \emph{Jaccard} tends to yield a larger diversity of collaborators over time and thus benefits more from \emph{Intersection with Associated Data} as it re-enforces such diversity of insights.
\begin{table}[t]
\centering
\scriptsize
\begin{tabular}{ | c | c | c | c | c | c | c | c | c |}
\hline
{\bf Select} & \multicolumn{3}{c |}{\textbf{Improvement}} & \multicolumn{2}{c |}{\textbf{Collaborators}} & \multicolumn{3}{c |}{\textbf{Coalitions}} \\
\cline{2-4} \cline{5-6} \cline{7-9}
{\bf Metric} & Mean & Max & Min & Mean & SD & Mean & SD & Med \\
\hline
\hline
\emph{Int-Size} & 1.05 & 7 & 0 & 19.47 & 2.24 & 5.09 & 4.09 & 4 \\
\hline
\emph{Jaccard} & 0.58 & 8 & 0 & 30.17 & 4.44 & 3.16 & 2.74 & 2 \\
\hline
\emph{Pearson} & 0.37 & 8 & 0 & 18.08 & 1.40 & 5.20 & 3.15 & 5 \\
\hline
\emph{Cosine} & 0.39 & 8 & 0 & 17.98 & 1.29 & 5.26 & 3.14 & 5 \\
%
\hline
\end{tabular}
\vspace{-0.15cm}
\caption{Fraction of Prediction Improvements $I$ for Collaborators, number of collaborators, and size of coalitions.}
\label{tab:improvements}
\end{table}
\subsection{Performance}\label{sec:performance}
We now estimate the operational cost of our techniques and show that it is appreciably low.
Specifically, we evaluate the overhead introduced by the privacy protection layer.
Excluding correlation-based metrics (due to lower accuracy improvement), the protocols for privately selecting partners ({\em Intersection-Size} and {\em Jaccard}) can be realized based on Private Set Intersection Cardinality (PSI-CA), and we choose the instantiation proposed in \cite{DGT12}, which incur computation and communication overhead linear in sets size. Privacy-preserving merging relies on the Private Set Intersection (PSI) with Data Transfer protocol from~\cite{DT10} in order to realize {\em Intersection with Associated Data}.
We implemented protocols from~\cite{DGT12} and~\cite{DT10} in C, %
and conducted experiments on Intel Xeon desktops with 3.10GHz CPU, connected by a 100Mbps Ethernet link. Fig.~\ref{fig:average:subfig3} shows that 98\% of targets are attacked by about $200$ sources. Using sets of size $200$, it takes approximately $400ms$ to execute PSI from~\cite{DT10} and $550ms$ for PSI-CA from~\cite{DGT12}. Assuming that $n$ organizations contribute to our framework, we have a total of $n-1$ interactions per entity, and a total of $(n-1)n/2$ pairwise executions.
\begin{figure}[t]
\centering
\fbox{\includegraphics[scale=0.35]{figures/union.intersection.pdf}}
\caption{Number of True Positives (TP) for two different merge algorithms: \emph{Union/Intersection with Associated Data}.}
\label{fig:union}
\end{figure}
Naturally, it is not reasonable to consider all possible partnerships in a large pool of organizations.
Parties first identify a set of {\em potential} partners, such as organizations within an industry, and then select the {\em best} partners within.
Realistically, we can thus assume $n=100$. We obtain that the running time amounts to $54s$ for one entity to estimate benefits, using PSI-CA, with all other ($99$) parties. Following a conservative stance, i.e., assuming that entities select and share with all possible $99$ partners, privacy-preserving merging via PSI-DT takes $40s$ (in the worst case scenario).
Pairwise executions can obviously be performed in parallel, at least, between different pairs. Even if we assumed a worst-case scenario, where data sharing occur in a sequential manner across all organizations, the total computation overhead (again, assuming $n=100$ partners and merging with all partners) would amount to $45$ minutes for benefit estimation and $33$ minutes for dataset merging, which is still reasonable for computations that are performed, e.g., once a day. %
Thus, we conclude that overhead introduced by the privacy protection layer is appreciably low and does not impede the deployment and the adoption of our techniques.
\subsection{Summary of Results}\label{sec:discussion}
\descr{Knowing More Means Predicting More.}
Our experiments show that targets that know more tend to successfully predict more attacks. This confirms our hypothesis about the opportunity to collaborate with targets exposed to numerous attacks. %
However, the simple ``more-data-the-better'' approach conflicts with privacy, thus, the challenge consists in identifying partners that help most.
Choosing partners based on higher values of \emph{Intersection-Size} works best and provides convenient privacy properties since it only discloses information about attackers entities already know of. %
\descr{Sometimes Sharing Does Not Help Much.}
In some cases, data sharing does not yield significant improvements:
we show that differences in similarity definition may lead to significant variations in accuracy. %
When considering correlation-based similarity between victims' profiles, small contributors are paired together, leading to small overall improvements. By contrast, set-based metrics favor larger contributors and thus yield larger overall improvement.
\descr{Sharing Only Common Attacks Is Almost As Useful As Sharing Everything.}
When merging datasets, organizations sharing only information about common attacks (i.e., using {\em Intersection with Associated Data}) achieve a good trade-off between privacy and utility
as the improvement is almost as good as when sharing everything.
%
Intuitively, merging using intersection helps because it reinforces knowledge of a particular attacker, while using union helps victims targeted by varying group of attackers. Thus, victims benefit as much from improving their knowledge of current attackers, as learning about sources that attack them next. In other words, learning information about attackers targeting a victim in the past is similar to learning about attackers that might target a victim in the future. %
\subsection{Limitations}
We acknowledge that the DShield dataset used in our experiments might be biased toward small organizations voluntarily reporting data,
thus it might not be directly evident how to generalize our results.
However, our findings show strong statistical evidence that collaboration metrics affect data sharing performance in interesting ways. Our proposed algorithms and methodology can serve as the basis for further experiments that explore the concept of privacy-enhanced sharing of security-relevant data.
Also, as in previous work~\cite{soldo,zhang2008highly}, we do not consider false positives but focus on measuring algorithm's TP rate. Nonetheless, as discussed in Section~\ref{subsec:predictive}, this is reasonable as the {\em prediction} algorithm identifies suspicious IP addresses that deserve further scrutiny and that are subsequently processed by {\em blacklisting} algorithms, which actually block sources, even though false positives might increase the computational load and complexity of the blacklisting algorithm by providing larger inputs.
Finally, note that this paper does not aim to present a finished product, but to demonstrate the viability and effectiveness of privacy-enhancing technologies on collaborative threat mitigation. While the overhead introduced by our peer-to-peer approach is still non-negligible, it is significantly lower than existing alternatives such as FHE. Also, a few improvements could be explored in future work to improve performance, including parallelization, centralization, and/or sampling.
\section{Related Work}\label{sec:related}
\subsection{Collaborative Security Initiatives}\label{sec:comm-gov}
\descr{Public Sector.} In 1998, U.S. President Clinton initiated a national program on Critical Infrastructure Protection~\cite{clinton1998presidential}, which promoted collaboration between government and private sector, and created the Financial Sector Information Sharing and Analysis Center (FS-ISAC). In 2003, this was extended to virtual systems and IT infrastructures with the Homeland Security Presidential Directive 7 (HSPD-7), and recently reinforced~\cite{obama-state}. In 2013, the US House of Representatives passed the Cyber Intelligence Sharing and Protection Act (CISPA), which met huge opposition as it granted broad immunity to data sharing entities, and took generous views on what data could be shared and with whom. The bill was not voted on by the Senate and the debate is still ongoing with similar proposals~\cite{uscongress}.
Standardization bodies also push collaborative frameworks and established appropriate data formats (IDMEF, IODEF RFC 5070~\cite{IODEF}), collaboration protocols (the Real-time Inter-network Defense RFC 6545~\cite{RID}), and guidelines (ISO 27010, ITU-T SG17).
\descr{Private Sector.} %
The RedSky Alliance~\cite{redsky} helps security professionals share intelligence after a vetting process for trust establishment. Another example is TF-CSIRT (Task Force of Computer Security Incident Response Teams)~\cite{terena}, which improves coordination among European Community Emergency Response Teams (CERTs).
Besides DShield~\cite{dshield}, other community-based initiatives focus on sharing and correlating security data. %
DOMINO (Distributed Overlay for Monitoring InterNet Outbreaks)~\cite{DOMINO} provides distributed intrusion detection promoting collaboration among nodes. In Europe, the Worldwide Observatory of Malicious Behaviors and Attack Threats (WOMBAT) gathers security related data in real-time. Symantec also introduced a data sharing platform, WINE. Finally, the MITRE Corporation~\cite{stix} developed file formats (STIX), collaboration protocols (TAXII), and repository formats (CAPEC, MAEC) for structure threat information exchange.
\descr{Barriers to Adoption.} These initiatives have had little success, as pointed out by the Federal Communications Commission's Working Group on Communications Security, Reliability and Interoperability Council's (CSRIC)~\cite{CSRIC}.
Existing solutions rely on manual out-of-band channels to establish \emph{trust}. For instance, the RedSky alliance relies on a long and costly vetting process that requires manual labor to verify the trustworthiness of potential partners. %
Furthermore, organizations need to reveal their datasets to a centralized third-party and rely on it to for security. Thus, they have limited control over how their data is shared with other participants. %
These solutions have a turnover of a few days for RedSky alliance, to a few weeks for ISACs. Feedback is significantly slower than the spread of malware. It is difficult for companies to quantify how much others are contributing, and the lack of transparency discourages contributions.
\subsection{Collaborative Threat Mitigation}\label{sec:academic}
Most of previous works for collaborative predictive blacklisting~\cite{katti2005collaborating,soldo,pouget2005vh,zhang2008highly} rely on central repositories and provide no privacy protection.
Katti et al.~\cite{katti2005collaborating} show that correlated attacks, i.e., mounted by same sources against different victims, are prevalent on the Internet.
They cluster victims that share common attacks and find that: (1) correlations among victims are persistent over time, and (2) collaboration among victims from correlated attacks improves malicious IP detection time.
Pouget at al.~\cite{pouget2005vh} also obtain similar results using distributed honeypots for observation of malicious online activities.
Then, Zhang et al.~\cite{zhang2008highly} experiment with predictive blacklisting, suggesting that victims can predict future attackers, with significantly improved accuracy, based on their logs and those of other similar victims.
Soldo et al.~\cite{soldo} also aim to forecast attack sources based on shared attack logs, using an implicit recommendation system %
and improve on prediction accuracy as well as robustness against poisoning attacks.
\subsection{Privacy-Preserving Data Sharing}
As data sharing raises important confidentiality and privacy concerns,
the security community has suggested, in slightly different contexts, to
use anonymization or cryptographic techniques to protect privacy.
Lincoln et al.~\cite{lincoln2004privacy} suggest sharing sanitized security data for collaborative analysis of security threats. Specifically, they remove, prior to sharing, sensitive data such as IP addresses. Other mechanisms include prefix-preserving anonymization of IP addresses~\cite{slagell2005sharing,xu2002prefix} and statistical obfuscation~\cite{adar2007user}.
However, inference attacks can de-anonymize network traces~\cite{coull2007playing},
and it is quite difficult to maintain data utility~\cite{kenneally2010dialing,lakkaraju2008evaluating,nikita,titan}.
Applebaum et al.~\cite{pets10} introduce privacy-preserving data aggregation protocols geared for anomaly detection. Their approach requires a semi-trusted proxy aggregator and only provides participants with aggregated counts of common data points across multiple entities.
Burkhart et al.~\cite{burkhart2010sepia} explore a distributed solution, based on secure multi-party computation and secret sharing, that supports aggregation of security alerts and traffic measurements among peers, e.g., to estimate global traffic volume.
These protocols are secure as long as the majority of peers do not collude, %
assume a reliable infrastructure to distribute key shares, and incur a large number of rounds and high communication overhead.
While aggregation can help compute traffic statistics, it mainly identifies most prolific attack sources and yields global models. However, as shown in~\cite{soldo,zhang2008highly}, generic attack models miss a significant number of attacks, especially when attack sources choose targets strategically and focus on a few known vulnerable networks.
In theory, Fully Homomorphic Encryption (FHE)~\cite{gentry2009fully} could be used
to compute personalized recommendations, but FHE is still far from being practical and
it is unclear whether complex prediction algorithms could effectively be run over encrypted data.
\section{Conclusion}\label{sec:conclusion}
This paper presented a novel privacy-friendly approach to collaborative threat mitigation.
We showed how
organizations can quantify expected benefits in a privacy-preserving way (i.e., without disclosing their datasets)
before deciding whether or not to collaborate. Based on these benefits, they can then organize into coalitions and decide what/how much to share.
We focused on collaborative predictive blacklisting, evaluated our techniques on a real-world dataset, and observed a significant improvement in prediction accuracy (up to 105\%, even when only 1\% of all possible partners collaborate).
Our analysis showed that some collaboration strategies work better than others. The number of common attacks provides a good estimation of the benefits of sharing, as it drives entities to partner with more knowledgable collaborators. Interestingly enough, only sharing information about common attacks proves to be almost as useful as sharing everything.
This suggests that victims benefit as much from improving their knowledge about entities that currently attack them, as from learning about entities that do not attack them now, but might in the future.
We demonstrated the benefits of privacy-preserving information sharing on collaborative threat mitigation %
and established that data sharing does not have to be an ``all-or-nothing'' process: by relying on efficient secure computation, %
it is possible to only share relevant data, and only when beneficial. Privately assessing whether or not, and how, entities should partner up prompts interesting challenges, which our work is really the first to tackle.
As part of future work, we intend to study other metrics for partner selection (e.g., dissimilarity) and experiment with other prediction algorithms and incentive mechanisms. %
We will also explore how to adapt our approach to other collaborative security problems, e.g., spam filtering~\cite{damiani2004p2p}, virus detection~\cite{hailpern2001collaborative}, or DDoS mitigation~\cite{oikonomou2006framework}. %
\input{arxiv.bbl}
|
2,877,628,089,545 | arxiv | \section{Introduction} We first recall that for $c>0$, the classical prolate spheroidal wave functions (PSWFs) were first discovered and studied by D. Slepian and his co-authors, see \cite{Slepian1, Slepian2}. These PSWFs are defined as the solutions of the following energy maximization problem :
\begin{equation*}
\mbox{ Find } g=\arg\max_{f\in B_c}\frac{\int_{-1}^1|f(t)|^2 dt}{\int_{\mathbb{R}}|f(t)|^2 dt}
\end{equation*}
Where $B_c$ is the classical Paley-Winer space, defined by
\begin{equation}
\label{Bc}
B_c=\{ f\in L^2(\mathbb R),\,\, \mbox{Support } \widehat f\subseteq [-c,c]\}.
\end{equation}
Here, $\widehat f$ is the Fourier transform of $f\in L^2(\mathbb R),$ defined by ${\displaystyle \widehat f(\xi)=\lim_{A\rightarrow +\infty}\int_{[-A,A]} e^{-i x\xi} f(x)\, dx.}$
Moreover, it has been shown in \cite{Slepian1}, that the PSWFs are also the eigenfunctions of the integral operator ${\displaystyle \mathcal Q_c}$ defined on $L^2(-1,1)$ by $${\displaystyle \mathcal Q_c f(x)=\int_{-1}^1 \frac{\sin(c(x-y))}{\pi(x-y)} f(y)\, dy} $$ as well as the eigenfunctions of a commuting differential operator $ \mathcal{L}_c$ with $ \mathcal{Q}_c$ and given by
$$ \mathcal L_c f(x)= (1-x^2)f''(x)-2x f'(x) - c^2 x^2 f(x) $$
In this paper we are interested in a weighted family of PSWFs called the generalized prolate spheroidal wave functions (GPSWFs), recently given in \cite{Wang2} and \cite{Karoui-Souabni1}. They are defined as the eigenfunctions of the Gegenbauer perturbed differential operator,
\begin{equation*}
\mathcal L^{(\alpha)}_c \varphi(x)= (1-x^2)\varphi''(x)-2\left(\alpha+1\right)x\varphi'(x) - c^2 x^2 \varphi(x)=
\mathcal L_0 \varphi(x)-c^2 x^2 \varphi(x), \mbox{ where } \alpha>-1; c>0,
\end{equation*}
as well as the eigenfunctions of the following commuting weighted finite Fourier transform integral operator,
\begin{equation} \label{integraloperator}
{\displaystyle \mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha}\, dy,\, \alpha > -1.}
\end{equation}
The importance of PSWFs is due to their wide range of applications in different scientific research area such as signal processing, physics and applied mathematics. The question of quality of approximation by PSWFs has attracted some interests. In fact the authors in \cite{chen} have given an estimate of the decay of the PSWFs expansion of $f \in H^s(I)$. Here $ H^s(I)$ is the Sobolev space over $I=[-1,1]$ and of exponent $s>0$. Later in \cite{Wang2} authors studied the convergence of the expansion of functions $f \in H^s(I)$ in a basis of PSWFs. We should mention that the problem of the best choice of the value of the bandwidth $c>0$ arises here. Numerical answers were given in \cite{Wang2}. Recently in \cite{Bonami-Karoui4}, the authors have given a precise answer to this question.
Note that the convergence rate of the projection ${\displaystyle S_N \cdot f = \displaystyle \sum_{k=0}^N <f,\ps> \ps }$ to f where $ f \in B_c^{(\alpha)}$ or $ f \in H^s_{\alpha,per}(I),$ a periodic Sobolev space. This is done by using an estimate for the decay of the eigenvalues $(\lambda^{(\alpha)}_n(c)$ of the self-adjoint operator ${\displaystyle \mathcal Q_c^{(\alpha)} =\frac{c}{2\pi} \mathcal F_c^{(\alpha)^*} \mathcal F_c^{(\alpha)},}$ that is
$ \lambda_n^{\alpha}(c) = \frac{c}{2\pi}{|\mu_n^{\alpha}(c)|^2}$, where $\mu_n^{\alpha}$ is the n-th eigenvalues of the integral operator \eqref{integraloperator}. That's why the first part of this work will be devoted to the study of the spectral decay of the eigenvalues of the integral operator $(\lambda_n^{\alpha}(c)$. \\
On the other hand, in \cite{Landau}, the authors have shown that the PSWFs are well adapted for the approximation of functions that are band-limited and and also almost time-limited. That is for an $\varepsilon >0,$ this space is defined by
$$E_c(\varepsilon)=\{ f \in B_c;\,\ \|f\|_{L^2(\mathbb R)} =1,\,\,\, \|f\|^2_{L^2(I)}\geq 1-\varepsilon\}.$$
Here $I=[-1,1]$ and $B_c$ is the Paley-Wiener space given by \eqref{Bc}. In particular, the approximate dimension of the previous space of functions
has been given in \cite{Landau}. Moreover, in \cite{Moumni}, the author has extended this result in the context of the circular prolate spheroidal wave functions (CPSWFs). These laters are defined as the different eigenfunctions of the finite Hankel transform operator.
In this paper we give the extension of the approximate dimension of the space of band-limited and almost time-limited function, in the context of the GPSWFs. \\
This work is organized as follows. In section 2, we give some mathematical preliminaries concerning the GPSWFs. Note that the extra weight function $ \omega_{\alpha}(y)=(1-y^2)^{\alpha} $ appearing in the weighted finite Fourier transform operator, generates new difficulties concerning the orthogonality of GPSWFs over $ \mathbb{R}.$ Unlike the classical PSWFs, the GPSWFs are orthogonal over $[-1,1],$ but their analytic extension to the real line are not orthogonal with respect to the Lebesgue measure. Hence, based on the properties of the GPSWFs, we give a procedure for recovering the orthogonality over $\mathbb R,$ of these laters. In section 3, we give some further estimates of the GPSWFs and some estimates of the eigenvalues of both integral and differential operators that define GPSWFs which will be useful later on in the study of the quality of approximations by GPSWFs. In section 4, we first extend to the case of the GPSWFs, the result given in \cite{Bonami-Karoui4}, concerning the quality of approximation by the PSWFs
of functions from the periodic Sobolev space $H^s_{per}(I).$ Then, we show that the GPSWFs best approximate almost time-limited functions and belonging
to the restricted Paley-Wiener space $ B_c^{(\alpha)},$ given by
$$B_c^{(\alpha)}=\{ f\in L^2(\mathbb R),\,\, \mbox{Support } \widehat f\subseteq [-c,c],\, \, \widehat f\in
L^2\big((-c,c), \omega_{- \alpha}(\frac{\cdot}{c})\big)\}.$$ Finally, in section 5 we give
some numerical examples that illustrate the different results of this work.
\section{Mathematical Preliminaries}
In this section, we give some mathematical preliminaries concerning some properties as well as the computation of the GPSWFs. These mathematical preliminaries are used to describe the different results of this work.
We first mention that some content of this paragraph has been borrowed from \cite{Karoui-Souabni1}. Since, the GPSWFs are defined as the eigenfunctions of
the weighted finite Fourier transform operator $\mathcal F_c^{(\alpha)},$ defined by
\begin{equation}\label{Eq1.1}
\mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,\omega_{\alpha}(y)\, dy,\quad \omega_{\alpha}(y)=(1-y^2)^{\alpha},\quad \alpha >0,
\end{equation}
then, they are also the eigenfunctions of the
operator ${\displaystyle \mathcal Q_c^{(\alpha)}=\frac{c}{2\pi}\mathcal F_c^{({\alpha})^*} \circ \mathcal F_c^{(\alpha)}},$ defined on $L^2{(I, \omega_{\alpha})}$ by
\begin{equation}\label{EEq0}
\mathcal Q_c^{(\alpha)} g (x) = \int_{-1}^1 \frac{c}{2 \pi}\mathcal K_{\alpha}(c(x-y)) g(y) \omega_{\alpha}(y) \, dy,
\quad \mathcal K_{\alpha}(x)=\sqrt{\pi} 2^{\alpha+1/2}\Gamma(\alpha+1) \frac{J_{\alpha+1/2}(x)}{x^{\alpha+1/2}}.
\end{equation}
Here $ J_{\alpha}(\cdot)$ is the Bessel function of the first kind and order $ \alpha>-1.$
Moreover, the eigenvalues $\mu_n^{\alpha}(c)$ and $\lambda_n^{\alpha}(c)$ of $\mathcal F_c^{\alpha}$ and
$\mathcal Q_c^{\alpha}$ are related to each others by the identity
${\displaystyle \lambda_n^{\alpha}(c)= \frac{c}{2\pi } |\mu_n^{\alpha}(c)|^2}.$
Note that the previous two integral operators commute with the following Gegenbauer-type Sturm-Liouville operator $\mathcal L_c^{(\alpha)},$ defined by
$$
\mathcal L_c^{(\alpha)} (f)(x)= -\frac{1}{\omega_{\alpha}(x)} \frac{d}{dx}\left[ \omega_{\alpha}(x) (1-x^2) f'(x)\right] +c^2 x^2 f(x),\quad \omega_{\alpha}(x)= (1-x^2)^{\alpha}.$$
Also, note that the $(n+1)-$th eigenvalue $\chi_n^{\alpha}(c)$ of $\mathcal L_c^{(\alpha)}$ satisfies the following classical inequalities,
\begin{equation}
\label{boundschi}
n (n+2\alpha+1) \leq \chi_n^{\alpha}(c) \leq n (n+2\alpha+1) +c^2,\quad \forall n\geq 0.
\end{equation}
For more details, see \cite{Karoui-Souabni1}. We will denote by$(\ps)_{n\geq 0},$ the set of the eigenfunctions of $\mathcal F_c^{(\alpha)}, \mathcal Q_c^{(\alpha)}$ and $\mathcal L_c^{(\alpha)}.$ They are called generalized prolate spheroidal wave functions (GPSWFs).
It has been shown that $ \{ \ps , n\geq 0 \} $ is an orthogonal basis of $ L^2(I,\omega_{\alpha}), I=[-1,1].$
We recall that the restricted Paley-Wiener space of weighted $c-$band-limited functions has been defined in \cite{Karoui-Souabni1} by
\begin{equation}\label{GBc}
B_c^{(\alpha)}=\{ f\in L^2(\mathbb R),\,\, \mbox{Support } \widehat f\subseteq [-c,c],\, \, \widehat f\in
L^2\big((-c,c), \omega_{- \alpha}(\frac{\cdot}{c})\big)\}.
\end{equation}
Here, $L^2\big((-c,c), \omega_{- \alpha}(\frac{\cdot}{c})\big)$ is the weighted $L^2(-c,c)-$space with norm given by
$$\| f\|^2_{L^2\big((-c,c), \omega_{- \alpha}(\frac{\cdot}{c})\big)}= \int_{-c}^c |f(t)|^2 \omega_{-\alpha}\left(\frac{t}{c}\right)\, dt.$$
Note that when $\alpha=0,$ the restricted Paley-Wiener space $B_c^{(0)}$ is reduced to the usual space $B_c.$ Also,
it has been shown that
\begin{equation}
\label{inclusionPaley}
B_c^{(\alpha)} \subseteq B_c^{(\alpha')},\qquad \forall\, \alpha \geq \alpha' \geq 0.
\end{equation}
As a consequence, it has been shown in \cite{Karoui-Souabni1} that the eigenvalues
$\lambda_n^{\alpha}(c)$ decay with respect to the parameter $\alpha,$ that is for $c>0,$ and any $n\in \mathbb N,$
\begin{equation}\label{decayalpha}
0 < \lambda_n^{\alpha}(c) \leq \lambda_n^{\alpha'}(c) < 1,\quad \forall \, \alpha \geq \alpha' \geq 0.
\end{equation}
Hence, by using the precise behaviour as well as the sharp decay rate of the $\lambda_n^{0}(c),$ given in
\cite{Bonami-Karoui3}, one gets a first result concerning the decay rate of the $(\lambda_n^{\alpha}(c))_n,$ for any $\alpha \geq 0.$ Moreover, since in
\cite{Bonami-Karoui3}, the authors have shown that for any $c\geq 1$ and any $ 0<b<4/e$, there exists $N_b \in \mathbb{N}$ such that :
\begin{equation} \label{decaylambda0}
\lambda_n^{0}(c) < e^{-2n \log(\frac{bn}{c})} \qquad \forall n \geq N_b,
\end{equation}
then by combining (\ref{decayalpha}) and (\ref{decaylambda0}), one gets
\begin{equation} \label{decaylambda1}
0 < \lambda_n^{\alpha}(c) \leq \lambda_n^{0}(c) < e^{-2n \log(\frac{bn}{c})} \qquad \forall n \geq N_b,\quad \alpha \geq 0.
\end{equation}
Note that a second decay rate of the $ \lambda_n^{\alpha}(c)$ and valid for $0<\alpha <\frac{3}{2},$ has been recently given in \cite{Karoui-Souabni2}. More precisely, it has been shown in \cite{Karoui-Souabni2}, that if $c>0$ and $0<\alpha <\frac{3}{2}$, then there exist
$N_{\alpha}(c)\in \mathbb N$ and a constant $C_{\alpha}>0$ such that
\begin{equation}
\label{decay_lambda2}
\lambda_n^{\alpha}(c) \leq C_{\alpha} \exp\left(-(2n+1)\left[ \log\left(\frac{4n+4\alpha+2}{e c}\right)+ C_{\alpha} \frac{c^2}{2n+1}\right]\right),\quad
\forall\, n\geq N_{\alpha}(c).
\end{equation}
Also, by using the fact that
\begin{equation*}
\psi_{n,c}^{(\alpha)}(x)= \frac{1}{\mu_n^{\alpha}}\int_{-1}^1\psi_{n,c}^{(\alpha)}(t)(1-t^2)^{\alpha}e^{itcx}dt=\frac{1}{c\mu_n^{\alpha}}\int_{\mathbb{R}}\psi_{n,c}^{(\alpha)}(t/c)(1-(t/c)^2)^{\alpha}e^{itx}dt,
\end{equation*}
then, we have
\begin{equation}\label{fourier}
\widehat{\psi_{n,c}^{(\alpha)}}(x)= \frac{\sqrt{2\pi}}{ \sqrt{ c \lambda_{n}}} \psi_{n,c}^{(\alpha)}\Big(\frac{x}{c}\Big) \Bigg( 1-\Big(\frac{x}{c}\Big)^2 \Bigg)^{\alpha} \mathbf 1_{[-c,c]}(x).
\end{equation}
Note that the GPSWFs are normalized by the following rule,
\begin{equation}\label{normalisation1}
\| \ps \|^2_{L^2(I, \omega_{\alpha})}=\int_{-1}^1 \left(\psi_{n,c}^{(\alpha)}(t)\right)^2 \, \omega_{\alpha}(t)\, dt = \lambda^{(\alpha)}_n(c).
\end{equation}
The extra weight function generates new complications concerning the orthogonality of $ \ps$ over $ B_c^{(\alpha)}$. To solve this problem, we propose the following procedure. We define on $B_c^{(\alpha)},$ the following inner product
\begin{equation}\label{innerproduct}
<f,g>_{\alpha}=\frac{1}{2\pi}\int_{\mathbb{R}} \widehat{f}(x)\widehat{g}(x)\omega_{-\alpha}\left(\frac{x}{c}\right)\, dx
\end{equation}
By using (\ref{fourier}) and (\ref{normalisation1}), we have :
\begin{equation}\label{orthogonalite}
<\ps,\psi_{m,c}^{(\alpha)}>_{\alpha}= \frac{1}{c \lambda_n^{\alpha}(c)}\int_{-c}^c \ps\left(\frac{x}{c}\right) \psi_{m,c}^{(\alpha)}\left(\frac{x}{c}\right)\omega_{\alpha}\left(\frac{x}{c}\right)dx
= \delta_{n m}
\end{equation}
Here $\delta_{n m}$ is the Kronecker's symbol. For more details, see \cite{Karoui-Souabni1}.
Now for $ f \in B_c^{(\alpha)}$, we have $\widehat{f}(x)=g(x)\omega_{\alpha}(x/c)$ for some $g\in L^2((-c,c),\omega_{\alpha}(\frac{.}{c}))$. By using the Fourier inversion formula and the fact that $\{\ps,n\in \mathbb{N} \} $ is an orthogonal basis of $ L^2(I,\omega_{\alpha}),$ one gets
\begin{eqnarray}\label{gen}
f(x)&=& \frac{1}{2\pi} \int_{-c}^c e^{ixy} \widehat{f}(y)dy = \frac{1}{2\pi} \int_{-c}^c e^{ixy} g(y) \omega_{\alpha}(\frac{y}{c}) dy \nonumber \\
&=& \frac{c}{2\pi} \int_{-1}^1 e^{icxy} g(cy) \omega_{\alpha}(y) dy = \frac{c}{2\pi} \int_{-1}^1 e^{icxy} \Big( \sum_{n=0}^{\infty}\alpha_n \ps(y) \Big) \omega_{\alpha}(y) dy \nonumber \\ &=& \frac{c}{2\pi} \sum_{n=0}^{\infty} \alpha_n \int_{-1}^1 \ps(y)e^{icxy} \omega_{\alpha}(y)dy = \frac{c}{2\pi} \sum_{n=0}^{\infty} \alpha_n \mu_n^{\alpha}(c) \ps(x)
\end{eqnarray}
where $ \alpha_n = \int_{-1}^1 g(cy) \ps(y) \omega_{\alpha}(y)\, dy.$
Consequently, the set $ \{ \ps , n\in \mathbb{N} \} $ is an orthogonal basis of $L^2(I,\omega_{\alpha})$ and an orthonormal basis of $ B_c^{(\alpha)},$ when this later is equipped with the inner product $<\cdot, \cdot>_{\alpha}.$
Also, in \cite{Karoui-Souabni1}, the authors have proposed the following scheme for the computation of the GPSWFs. In fact, since $\psi^{(\alpha)}_{n,c}\in L^2(I, \omega_{\alpha}),$ then its series expansion
with respect to the Jacobi polynomials basis is given by
\begin{equation}\label{expansion1}
\psi^{(\alpha)}_{n,c}(x) =\sum_{k\geq 0} \beta_k^n \wJ_k (x),\quad x\in [-1,1].
\end{equation}
Here, $\wJ_k$ denotes the normalized Jacobi polynomial of degree $k,$ given by
\begin{equation}\label{JacobiP}
\wJ_{k}(x)= \frac{1}{\sqrt{h_k}}\J_k(x),\quad h_k=\frac{2^{2\alpha+1}\Gamma^2(k+\alpha+1)}{k!(2k+2\alpha+1)\Gamma(k+2\alpha+1)}.
\end{equation}
The expansion coefficients $\beta_k^n$ as well as the eigenvalues $\chi_n^\alpha(c)$ this system is given by
\begin{eqnarray}\label{eigensystem}
\lefteqn{\frac{\sqrt{(k+1)(k+2)(k+2\alpha+1)(k+2\alpha+2)}}{(2k+2\alpha+3)\sqrt{(2k+2\alpha+5)(2k+2\alpha +1)}} c^2 \beta_{k+2}^n
+ \big( k(k+2\alpha+1)+c^2 \frac{2k(k+2\alpha+1)+2\alpha-1}{(2k+2\alpha+3)(2k+2\alpha-1)} \big)
\beta_k^n}\nonumber \\
&&\hspace*{2cm} + \frac{\sqrt{k(k-1)(k+2\alpha)(k+2\alpha-1)}}{(2k+2\alpha-1)\sqrt{(2k+2\alpha+1)(2k+2\alpha-3)}} c^2
\beta_{k-2}^n= \chi_n^{\alpha}(c) \beta_k^n, \quad k\geq 0.
\end{eqnarray}
\section{Further Estimates of the GPSWFs and their associated eigenvalues.}
In this section, we first give some new estimates of the eigenvalues of the eigenvalues $\mu_n^\alpha(c)$ associated with the
GPSWFs. Then, we study a local estimate of these laters.
\subsection{Estimates of the eigenvalues.}
We first prove in a fairly simple way, the super-exponential decay rate of the eigenvalues of the weighted Fourier transform operator.
We should the techniques used in this proof is inspired from the recent paper \cite{Bonami-jamming-Karoui},
where a similar result has been given in the special case $ \alpha = 0$.
\begin{proposition}
For given real numbers $c > 0$,$ \;\; \;\alpha>-1$ and for any integer $n > \frac{ec+1}{2},$ we have
\begin{equation}\label{Eq3.1}
| \mu_n^{\alpha}(c)| \leq \frac{k_{\alpha}}{c^{\frac{\alpha+3}{2}}\log\left(\frac{2n-1}{ec}\right)}
\left(\frac{ec}{2n-1}\right)^{n+\frac{\alpha+1}{2}},\;\;\; k_{\alpha}=\left(\frac{2}{e}\right)^{\frac{3+\alpha}{2}}\pi^{5/4}(\Gamma(\alpha+1))^{1/2}.
\end{equation}
\end{proposition}
\noindent
{\bf Proof:} We first recall the Courant-Fischer-Weyl Min-Max variational principle concerning the eigenvalues
of a compact self-adjoint operator $T$ on a Hilbert space $\mathcal G,$ with positive eigenvalues arranged in the decreasing order $\lambda_0\geq \lambda_1\geq \cdots\geq \lambda_n\geq \cdots ,$ then we have
$$\lambda_n = \min_{f\in S_n}\,\,\, \max_{f\in S_n^{\perp},\|f\|_{\mathcal G}=1} < Tf, f>_{\mathcal G},$$
where $S_n$ is a subspace of $\mathcal G$ of dimension $n.$ In our case, we have $T=\mathcal F_c^{\alpha^*}\mathcal F_c^{\alpha},$ $\mathcal G= L^2(I,\omega_{\alpha}).$ We consider the special case of $$S_n=\mbox{Span}\left\{\J_0, \J_1,\ldots,\J_{n-1}\right\}$$ and
$$f=\ds\sum_{k\geq n} a_k \J_k \in S_n^{\perp},\qquad
\parallel f\parallel_{L^2_{(I,\omega_\alpha)}}=\ds\sum_{k\geq n}|a_k|^2=1.$$
From [\cite{NIST} page 456], we have
\begin{equation}\label{Eq3.2}
\| \mathcal F_c^{\alpha} \J_k\|_{L^2(I,\omega_{\alpha})} = \parallel \sqrt{\pi} \left(\frac{2}{cx}\right) ^{\alpha+1/2} \frac{\Gamma(k+\alpha+1)}{\Gamma(k+1)}J_{k+\alpha+1/2}(cx)
\parallel_{L^2(I,\omega_{\alpha})}.
\end{equation}
and
\begin{equation}\label{5}
\parallel x^k
\parallel_{L^2(I,\omega_{\alpha})}^2=\beta(k+1/2,\alpha+1)\leq \sqrt{\frac{\pi}{e}}\frac{\Gamma(\alpha+1)}{k^{\alpha+1}}.
\end{equation}
By using the well-known bound of the Bessel function given by
\begin{equation}\label{1}
|J_{\alpha}(x)|\leq \frac{|x|^{\alpha}}{2^{\alpha}\Gamma(\alpha+1)}\;\;\;\;\;\;\;\;\forall\alpha>-1/2, \;\; \forall x \in \mathbb{R},
\end{equation}
one gets,
\begin{eqnarray}
\| \mathcal F_c^{\alpha} \wJ_k\|_{L^2(I,\omega_{\alpha})}
&\leq & \frac{\sqrt{\pi}\Gamma(k+\alpha+1)}{\Gamma(k+1)\Gamma(k+\alpha+3/2)}\left(\frac{c}{2}\right) ^{k} \parallel x^k
\parallel_{L^2(I,\omega_{\alpha})}.\\
&\leq & \frac{\sqrt{\pi}\Gamma(k+\alpha+1)}{\Gamma(k+1)\Gamma(k+\alpha+3/2)}\left(\frac{c}{2}\right) ^{k}\left(\frac{\pi}{e}\right)^{1/4}\frac{\sqrt{\Gamma(\alpha+1)}}{k^{\frac{\alpha+1}{2}}}
\end{eqnarray}
Next, we use the following useful inequalities for the Gamma function, see \cite{Batir}
\begin{equation}\label{3}
\sqrt{2e}\left(\frac{x+1/2}{e}\right)^{x+1/2}\leq\Gamma(x+1)\leq \sqrt{2\pi}\left(\frac{x+1/2}{e}\right)^{x+1/2}\;,\;\;\; x>0.
\end{equation}
Then, we have
\begin{eqnarray}
\| \mathcal F_c^{\alpha} \wJ_k\|_{L^2(I,\omega_{\alpha})} &\leq &\frac{\pi ^{5/4}\sqrt{\Gamma(\alpha+1)}}{\sqrt{c}e^{3/4}}\frac{1}{k^{\frac{\alpha+2}{2}}}\left(\frac{ce}{2k+1}\right) ^{k+1/2}.
\end{eqnarray}
Hence, for the previous $f\in S_n^{\perp},$ and by using H\"older's inequality, combined with the Minkowski's inequality for an infinite sums, and taking into account that $\|f\|_{L^2(I,\omega_{\alpha})}= 1,$ so that $|a_k|\leq 1,$ for $k\geq n,$ one gets:
\begin{eqnarray}
|< \mathcal F_c^{\alpha^*}\mathcal F_c^{\alpha} f,f>_{L^2(I,\omega_{\alpha})}|=|< \mathcal F_c^{\alpha} f,\mathcal F_c^{\alpha}f>_{L^2(I,\omega_{\alpha})}| &\leq &\sum_{k\geq n} |a_k|^2 \| \mathcal F_c^{\alpha} \wJ_k\|^2_{L^2(I,\omega_{\alpha})}\\
\label{Eq3.3}
&\leq&\left(\frac{\pi ^{5/4}\sqrt{\Gamma(\alpha+1)}}{\sqrt{c}e^{3/4}} \sum_{k\geq n} \left(\frac{ec}{2k+1}\right)^{k+1/2}\frac{1}{k^{\frac{\alpha+2}{2}}}\right)^2
\end{eqnarray}
The decay of the sequence appearing in the previous sum, allows us to compare this later with its integral counterpart, that is
\begin{equation}\label{Eq3.4}
\sum_{k\geq n} \left(\frac{ec}{2k+1}\right)^{k+1/2}\frac{1}{k^{\frac{\alpha+2}{2}}}
\leq \int_{n-1}^{\infty}\frac{ e^{-(x+1/2) \log(\frac{2x+1}{ec})}}{x^{\frac{\alpha+2}{2}}}\, dx\leq
\int_{n-1}^{\infty}\frac{ e^{-(x+1/2)\log(\frac{2n-1}{ec})}}{(n-1)^{\frac{\alpha+2}{2}}}\, dx.
\end{equation}
Hence, by using (\ref{Eq3.3}) and (\ref{Eq3.4}), one concludes that
\begin{eqnarray}
\max_{f\in S_n^{\perp},\, \|f\|_{L^2(I,\omega_{\alpha})=1}} < \mathcal F_c^{\alpha} f, \mathcal F_c^{\alpha} f>^{1/2}_{L^2(I,\omega_{\alpha})}&\leq &\frac{\pi ^{5/4}\sqrt{\Gamma(\alpha+1)}}{\sqrt{c}e^{\frac{3}{4}}} \frac{1}{\log(\frac{2n-1}{ec})} \left(\frac{ec}{2n-1}\right)^{n-1/2}\frac{1}{(n-1)^{\frac{\alpha+2}{2}}}\\
&\leq&2^{\frac{\alpha+2}{2}} \frac{\pi ^{5/4}\sqrt{\Gamma(\alpha+1)}}{(ec)^{\frac{\alpha+3}{2}} } \frac{1}{\log(\frac{2n-1}{ec})} \left(\frac{ec}{2n-1}\right)^{n+\frac{\alpha+1}{2}}\label{Eq3.5}
\end{eqnarray}
To conclude the proof of the theorem, it suffices to use the Courant-Fischer-Weyl Min-Max variational principle.$\qquad \Box $
\begin{remark} \label{newdecay}
By the fact that $ \lambda_n^{\alpha}(c) = \frac{c}{2\pi} |\mu_n^{\alpha}(c)|^2 $ and straightforward computations one gets for $c,\alpha>0$ and for all $ n > \frac{ec+1}{2}$ :
$$ \lambda_n^{\alpha}(c) \leq \frac{K_\alpha}{c^{\alpha+2}\log^2(\frac{2n-1}{ec})}\Big(\frac{ec}{2n-1}\Big)^{2n+\alpha+1} \quad \mbox{with} \quad K_{\alpha} = \frac{\pi^{3/2}}{2}\big(2/e \big)^{\alpha+3} \Gamma(\alpha+1) $$
\end{remark}
\begin{remark}
The decay rate given by the last proposition improve the two results mentioned above \eqref{decaylambda1} and \eqref{decay_lambda2}. Indeed, we have a more precise result furthermore the proof given in \cite{Karoui-Souabni2} contains two serious drawbacks. It is only valid in the particular case $0 < \alpha < 3/2 $ and clearly more sophisticated than the previous one.
\end{remark}
\subsection{Local estimates of the GPSWFs}
In this paragraph, we give a precise local estimate of the GPSWFs, which is valid for $0\leq \alpha \leq 1/4.$ Then, this local estimate will be used to provide us with a new lower bound for the eigenvalues $\chi_{n}^\alpha(c)$ of the differential operator $\mathcal L_c^\alpha,$ for $0\leq \alpha \leq 1/4. $ For this purpose, we first recall that $\ps $ are the bounded solutions of the following ODE :
\begin{equation}\label{eq2_diff}
\omega_{\alpha}(x)\mathcal L_c^{(\alpha)} \psi(x)+w_{\alpha}(x)\chi_{n,\alpha}\psi(x)= ( w_{\alpha}(x)\psi'(x)(1-x^2))'+ w_{\alpha}(x)(\chi_{n}^{\alpha}-c^2 x^2)\psi(x)=0,\quad x\in [-1,1].
\end{equation}
As it is done in \cite{Bonami-Karoui2}, we consider the incomplete elliptic integral
\begin{equation}\label{Liouvile_transform1}
S(x)=\int_x^1 \sqrt{\frac{1-qt^2}{1-t^2}}\, dt,\qquad q=\frac{c^2}{\chi_n^\alpha(c)}<1.
\end{equation}
Then, we write $\psi_{n,c}^{(\alpha)}$ into the form
\begin{equation}\label{Liouvile_transform2}
\psi(x)=\phi_{\alpha}(x)V(S(x)),\qquad \phi_{\alpha}(x)=(1-x^2)^{(-1-2\alpha)/4} (1-qx^2)^{-1/4}.
\end{equation}
By combining (\ref{eq2_diff}), (\ref{Liouvile_transform2}) and using straightforward computations,
it can be easily checked that $V(\cdot)$ satisfies the following second order differential equation
\begin{equation}
V''(s)+\left(\chi_{n,\alpha}+\theta_{\alpha}(s)\right)V(s)=0,\quad s\in [0, S(0)]
\end{equation}
with $$ \theta_{\alpha}(S(x))=(w_{\alpha}(x)(1-x^2)\phi_{\alpha}'(x))'\frac{1}{\phi_{\alpha}(x)w_{\alpha}(x)(1-qx^2)}.$$
Define $ Q_{\alpha}(x)=w_{\alpha}(x)^2(1-x^2)(1-qx^2),$ then we have
${\displaystyle \frac{\phi_{\alpha}'(x)}{\phi_{\alpha}(x)}=-1/4 \frac{Q_{\alpha}'(x)}{Q_{\alpha}(x)}. }$
It follows that $\theta_{\alpha}(s)$ can be written as
\begin{equation}
\theta_{\alpha}(S(x))=\frac{1}{16(1-qx^2)}\Big[\Big(\frac{Q_{\alpha}'(x)}{Q_{\alpha}(x)}\Big)^2(1-x^2)-4\frac{d}{dx}\Big((1-x^2)\frac{Q_{\alpha}'(x)}{Q_{\alpha}(x)}\Big)-4(1-x^2)\frac{Q_{\alpha}'(x)}{Q_{\alpha}(x)}
\frac{w_{\alpha}'(x)}{w_{\alpha}(x)}\Big].
\end{equation}
Since $ Q_{\alpha}(x)=w_{\alpha}^2(x)Q_0(x),$ then we have
${\displaystyle \frac{Q_{\alpha}'(x)}{Q_{\alpha}(x)}=2\frac{w_{\alpha}'(x)}{w_{\alpha}(x)}+\frac{Q_0'(x)}{Q_0(x)}} $ and ${\displaystyle \frac{w_{\alpha}'(x)}{w_{\alpha}(x)}=-\frac{2\alpha x}{1-x^2}.} $
Hence, we have
\begin{eqnarray}
\theta_{\alpha}(S(x)) &=&\theta_0(S(x))+\frac{-1}{4(1-qx^2)}\Big[ \big(\frac{w_{\alpha}'(x)}{w_{\alpha}(x)}\big)^2 (1-x^2)+2\frac{d}{dx}[(1-x^2)\frac{w_{\alpha}'(x)}{w_{\alpha}(x)}] \Big] \nonumber \\
&=&\theta_0(S(x))+ \frac{1}{(1-x^2)(1-qx^2)}\Big( -\alpha^2 x^2 + \alpha (1-x^2) \Big) \nonumber \\
&=&\theta_0(S(x))+ \frac{\alpha(1+\alpha)}{(1-qx^2)}-\frac{\alpha^2}{(1-q x^2)(1-x^2)}.
\end{eqnarray}
The previous equality allows us to prove the following lemma.
\begin{lemma}
For any $0\leq \alpha\leq \frac{1}{4}$ and $0<q=c^2/\chi_n^{\alpha}(c) <\frac{3}{17}$, we have $\theta_{\alpha}(S(x))$ is increasing on $ [0.1]$.\\
\end{lemma}
\noindent
{\bf Proof:}
We use the notation $u=1-x^2$ and with Straightforward computations, we have
\begin{eqnarray*}
(\theta_{\alpha}(S(x)))'&=&(\theta_{0}(S(x)))'+\frac{2x}{(1-qx^2)^2(1-x^2)^2}\left[q\alpha(\alpha+1)(1-x^2)^2
-q\alpha^2(1-x^2)-\alpha^2(1-qx^2)\right] \\
&=& \frac{2xH(u)}{4u^2(1-q+qu)^4}
\end{eqnarray*}
where
\begin{eqnarray*}
H(u) &=& G(u)+4(1-q+qu^2)^2[q(\alpha^2+\alpha)u^2-\alpha^2(1-q+2qu)]\\
&\geq&(1-q+qu^2)^2[G(u)+4[q(\alpha^2+\alpha)u^2-\alpha^2(1-q+2qu)]]
\end{eqnarray*}
with $G(u)\geq\frac{(1-q)^2}{4}(4-4q+u(17q-3))\geq0,\;\;\forall q>0,\;\;\alpha>0$ given by [\ref{e}]
then
\begin{eqnarray*}
H(u) &\geq&(1-q+qu^2)^2\left[ \frac{(1-q)^2}{4}(4-4q+u(17q-3))+4[-q\frac{\alpha^3}{\alpha+1}-(1-q)\alpha^2]\right] \\
&\geq& (1-q+qu^2)^2\left[(1-q)^3-4\alpha^2(1-q\frac{1}{\alpha+1})+\frac{(1-q)^2}{4}(17q-3)u\right].
\end{eqnarray*}
Then if $0\leq q\leq\frac{3}{17}$ and $\alpha\leq \frac{1}{4}$ we have
\begin{eqnarray*}
H(u) &\geq& (1-q+qu^2)^2 \left[(1-q)^3-4\alpha^2(1-q\frac{1}{\alpha+1})+\frac{(1-q)^2}{4}(17q-3)\right]\nonumber\\
&\geq &(1-q+qu^2)^2\left[\frac{(1-q)^2}{4}(1+13q)-4\alpha^2\right]\geq0
\end{eqnarray*}
$\qquad \Box $
As a consequence of the previous lemma, one gets.
\begin{lemma}
For any $0\leq \alpha\leq \frac{1}{4}$ and for any integer $n\in \mathbb N,$ with $0<q=\frac{c^2}{\chi_n^\alpha(c)} <\frac{3}{17}$, we have
\begin{equation}\label{estimation}
\ds{\sup_{x\in[0,1]}}\sqrt{(1-x^2)(1-qx^2)}\omega_{\alpha}(x)|\psi_{n,c}^{(\alpha)}(x)|^2\leq|\psi_{n,c}^{(\alpha)}(0)|^2+
\frac{|{\psi_{n,c}^{(\alpha)}}'(0)|^2}{\chi_n^{\alpha}(c)}= A^2 \leq 2 \alpha + 1 \;
\end{equation}
\end{lemma}
\begin{proof}
For $0\leq \alpha\leq \frac{1}{4}$ and $0<q<\frac{3}{17}$ ,we have $ \theta_{\alpha} \circ S $ is increasing. Since $S$ is decreasing then $ \theta $ is also decreasing. We define $K(s)= |V(s)|^2 + \frac{1}{\chi_n^{\alpha}(c)+\theta(s)}|V'(s)|^2 $
By the fact that $K(s)$ and $ \frac{1}{\chi_n^{\alpha}(c)+\theta(s)} $ has the same monotonicity and $\theta$ is decreasing , we conclude that $ K $ is increasing.
Then $$ |V(S(x))|^2 \leq |K(S(x)) | \leq |K(S(0)) | $$
We remark easily that $ V(S(0)) = \psi_{n,c}^{(\alpha)}(0) $ and $ V'(S(0)) ={ \psi_{n,c}^{(\alpha)}}'(0) $ then we conclude for the first inequality of \eqref{estimation}. For the second inequality see remark \ref{rq}
\end{proof}
Next, we give some lower and upper bounds for the eigenvalues $\chi_n^\alpha(c),$ the $n+1$th eigenvalues of the differential operator. This is given by the following proposition.
\begin{proposition}
For c and n such that $0\leq \alpha\leq \frac{1}{4}$ and $0<q<\frac{3}{17}$,
we have :
$$ n(n+2\alpha+1)+C_\alpha c^2 \leq \chi_n^{\alpha}(c) \leq n(n+2\alpha+1)+c^2, $$
where $ C_{\alpha} = 2(2\alpha+1)^2+1 - 2(2\alpha+1)\sqrt{1+(2\alpha+1)^2}.$
\end{proposition}
\begin{proof}
By differentiating with respect to $c$ the differential equation satisfied by GPSWFs , one gets
$$ (1-x^2) \partial_c {{\psi_{n,c}^{(\alpha)}}'}'(x) - 2(\alpha+1)x \partial_c {\psi_{n,c}^{(\alpha)}}'(x)
+ (\chi_n^{\alpha}(c)-c^2x^2) \partial_c \psi_{n,c}^{(\alpha)}(x) + \Big( \partial_c \chi_n^{\alpha}(c)-2cx^2 \Big) \psi_{n,c}^{(\alpha)}(x) = 0 $$
Hence, we have
$$ \Big(\mathcal{L}^{(\alpha)}_c + \chi_n^{\alpha}(c)Id\Big).\partial_c \psi_{n,c}^{(\alpha)} + \Big( \partial_c \chi_n^{\alpha}(c)-2cx^2 \Big) \psi_{n,c}^{(\alpha)}(x) = 0 $$
It's well known that $\mathcal{L}^{(\alpha)}_c $ is a self-adjoint operator and have $ \psi_{n,c}^{(\alpha)}$ as eigenfunctions . Hence, we have
$$ < \Big(\mathcal{L}^{(\alpha)}_c + \chi_n^{\alpha}(c)Id\Big).\partial_c \psi_{n,c}^{(\alpha)}; \psi_{n,c}^{(\alpha)}>_{\omega_{\alpha}} = 0 $$
It follows that
$$ \int_{-1}^{1} \Big( \partial_c \chi_n^{\alpha}(c)-2cx^2 \Big) {\psi_{n,c}^{(\alpha)}} ^{2}(x) dx = 0. $$
From the fact that $ \norm{\psi_{n,c}^{(\alpha)}}_{L^2(I,\omega_\alpha)} = 1 $, one gets
\begin{equation}\label{derivee}
\partial_c \chi_n^{\alpha}(c) = 2c \int_{-1}^{1} x^2 {\psi_{n,c}^{(\alpha)}}^2(x) \omega_{\alpha}(x) dx
\end{equation}
As it is done in \cite{Bonami-Karoui1}, we denote by $ A= \Bigg[\psi_{n,c}^{(\alpha)}(0) + \frac{{\psi_{n,c}^{(\alpha)}}'(0)}{\chi_n^{\alpha}(c)} \Bigg]^{1/2} $ and consider the auxiliary function $$ K_n(t) = - (1-t^2)^{2\alpha+1} {\psi_{n,c}^{(\alpha)}}^2 - \frac{(1-t^2)^{2\alpha+2}}{\chi_n^{\alpha}(c)(1-qt^2)}({\psi_{n,c}^{(\alpha)}}')^2 $$
Straight forward computations give us $$ K'_n(t) = 2(2\alpha+1)t(1-t^2)^{2\alpha} {\psi_{n,c}^{(\alpha)}}^2(t) - {H(t) {\psi_{n,c}^{(\alpha)}}'(t)}^2 $$ with $ H(t) \geq 0 $ for $t\in[0,1]$. Hence
$$ K'_n(t) \leq 2(2\alpha+1)t\omega_{\alpha}^2(t) {\psi_{n,c}^{(\alpha)}}^2(t) \leq 2(2\alpha+1)t\omega_{\alpha}(t) {\psi_{n,c}^{(\alpha)}}^2(t). $$
So one has the inequality
\begin{equation}
\label{Estimate0}
K_n(1) - K_n(0) = A^2 \leq 2(2\alpha+1) \int_{0}^{1} t{|\psi_{n,c}^{(\alpha)}(t) |}^2 \omega_{\alpha}(t) dt \leq (2\alpha+1) \Bigg[\int_{-1}^{1}t^2 |\psi_{n,c}^{(\alpha)}(t) |^2 \omega_{\alpha}(t) dt \Bigg] ^{1/2}.
\end{equation}
That is
\begin{equation}\label{11}
A^2 \leq (2\alpha+1) B^{1/2}.
\end{equation}
Remark that \eqref{estimation} implies that
\begin{equation} \label{12}
1-B \leq 2 A^2
\end{equation}
By combining \eqref{11} and \eqref{12} we conclude that $B^{1/2} $ is bounded below by the largest solution of the equation
$ X^2 + 2(2\alpha+1) X -1 = 0 $
\end{proof}
\begin{remark} \label{rq}
By using \eqref{11} and since $B\leq1$, one concludes that the constant $A,$ given in \eqref{estimation} satisfies $A^2\leq 2\alpha+1.$
\end{remark}
\section{Qualities of approximation by GPSWFs}
\subsection{Approximation by the GPSWFs in Weighted Sobolev spaces}
In this section, we study the issue of the quality of spectral approximation of a function $f \in H^s_{\alpha}([-1,1])$ by its truncated GPSWFs series expansion. \\
Note that a different spectral approximation result by GPSWFs has already given in \cite{Wang2}. It is important to mention here that this approximations are given in a different approach. More precisely, by considering the weighted Sobolev space associated with the differential operator defined by
$$ \widetilde{H}^{r}_{\omega_{\alpha}}(I) = \{ f\in L^2(I,\omega_{\alpha}): \|f\|_{\widetilde{H}^{r}_{\omega_{\alpha}}(I)} = \|(\mathcal{L}^{(\alpha)}_c)^{r/2}.f\|^2 = \sum_{k=0}^{\infty}(\chi_n^{\alpha})^r |f_k^{\alpha}|^2 <\infty \} $$
where $f_k$ are expansions coefficients of $f$ in the GPSWFs's basis. Then it has been shown that: \\
For any $f \in \widetilde{H}^{r}_{\omega_{\alpha}}(I) $ with $ r\geq 0 $
$$ \| S_{N,c}.f-f \|_{L^2(I,\omega_{\alpha})} \leq \Big( \chi^{(\alpha)}_{N+1} \Big)^{-r/2} \|f\|_{\widetilde{H}^{r}_{\omega_{\alpha}}(I)} $$
For more details, we refer the reader to \cite{Wang2}. \\
In the following, we need a theorem of [\cite{Karoui-Souabni1}]
Let $c > 0$, be a fixed positive real number. Then, for all positive integers
$n,\; k$ such that $q <1$ and $ k(k+2\alpha+1)+C'_{\alpha}c^2\leq \chi_n^{\alpha}(c)$ , we have
\begin{eqnarray}
|\beta_k^n|&\leq& C_{\alpha}\left(\frac{2\sqrt{\chi_{n}^{\alpha}(c)}}{c} \right)^k |\mu_{n}^{\alpha}(c)|.\label{9}
\end{eqnarray}
With $C'_{\alpha}$ a constant depends only on $\alpha$, and $ {\displaystyle C_{\alpha}=\frac{2^{\alpha}(3/2)^{3/4}(3/2+2\alpha)^{3/4+\alpha}}{e^{2\alpha+3/2}}}$
\begin{remark}
The condition $ k(k+2\alpha+1)+C_{\alpha}c^2\leq \chi_n^{\alpha}(c)$ of previous theorem can be replaced with the following more explicit condition. $n\geq cA$ and $k\leq n/B$, for any real $A, B$ such that\\
If $0\leq\alpha \leq 1/4$, just take $\;\;\;\;A^2=B^2=2.18$\\
If $-1<\alpha<0$ or $1/4\leq\alpha $, just take $\;\;\;\;A^2=B^2=2.8$
\end{remark}
\begin{lemma}
Let $c>0$ and $\alpha >-1$, then for all positive integers $n,\; k$ such that, $k\leq n/1.7$ and
$n\geq (75+46\alpha)^{0.7}c$
we have
\begin{eqnarray}
|\beta_k^n| &\leq & k_{\alpha,c}e^{-an}\label{10}
\end{eqnarray}
with $k_{\alpha,c}$ and $a$ positive constants .
\end{lemma}
\begin{proof}
By $(\ref{9})$ ,$(\ref{Eq3.1})$ , the inequality $\chi_{n}^{\alpha}(c)\leq n(n+2\alpha+1)+c^2$ and the previous remark.
One concludes that for $c>0$ and $\alpha >-1$, and for all positive integers $n,\; k$ such that, $k\leq n/A$ and
$n\geq Ac$ with $A\geq 1.7$, we have,
\begin{eqnarray}
|\beta_k^n| &\leq &C_{\alpha}\left(\frac{2\sqrt{\chi_{n}^{\alpha}(c)}}{c} \right)^k |\mu_{n}^{\alpha}(c)|\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}\left(\frac{ec}{2n-1}\right)^{n+\frac{\alpha+1}{2}}
\left(\frac{2\sqrt{n(n+2\alpha+1)+c^2}}{c} \right)^{n/A+1} \nonumber\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\left(\frac{ec}{2n-1}\right)^{2A}\right)^{\frac{\alpha+1}{4A}-\frac{1}{2}}
\left(\frac{4 e^{2A}c^{2(A-1)}{(n(n+2\alpha+1)+c^2)}}{(2n-1)^{2A}} \right)^{\frac{n+A}{2A}}\nonumber\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\frac{4 e^{2A}c^{2(A-1)}{(n(n+2\alpha+1)+c^2)}}{(2n-1)^{2A}} \right)^{\frac{2n+\alpha+1}{4A}}\nonumber\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\frac{e^{\frac{A}{A-1}}c{\left(1+\frac{2\alpha+1}{n} +\frac{c^2}{n^2} \right)^{\frac{1}{2(A-1)}}} } {2n\left(1-\frac{1}{2n}\right)^{\frac{A}{A-1}}} \right)^{\frac{(2n+\alpha+1)(A-1)}{2A}}
\end{eqnarray}
For the appropriate value of value of $A=1.7$ , we have
\begin{eqnarray}
|\beta_k^n| &\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\frac{c{\left(1.35+\frac{2\alpha+1}{1.7c} \right)^{0.7}} } {2n\left(0.36-\frac{0.1}{c}\right)^{2.4}} \right)^{0.4n+0.2(\alpha+1)}\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\frac{c(1.94+1.18\alpha )^{0.7}}{0.078n} \right)^{0.4n+0.2(\alpha+1)}\\
&\leq &C_{\alpha}\frac{1}{c^{\frac{({\alpha+3})}{2}}\log(\frac{2n-1}{ec})}
\left(\frac{c}{n}(74.6+45.4\alpha)^{0.7} \right)^{0.4n+0.2(\alpha+1)}.
\end{eqnarray}
Then for all $n\geq (75+46\alpha)^{0.7}c$. We have \eqref{10}
\end{proof}
\begin{lemma}
for $c\geq0$, $\alpha>-1$, for all positif real $n,k$ such that $k\leq 0.14n$ and $n\geq (75+46\alpha)^{0.7}c$.
There exist $C_{\alpha,c},\delta>0$ such that
\begin{eqnarray}
|\langle e^{ik\pi x}, \psi_{n,c}^{\alpha}\rangle _{L^2(I,\omega_{\alpha})}| &\leq& C_{\alpha,c} e^{-\delta n}
\end{eqnarray}
\end{lemma}
\begin{proof}
We have
\begin{eqnarray}
|\langle e^{ik\pi x}, \psi_{n,c}^{\alpha}\rangle _{L^2(I,\omega_{\alpha\alpha})}| &\leq& \displaystyle{ \sum_{m\geq 0} } |\beta_m^n||\langle e^{ik\pi x}, \tilde{P}_{m}^{(\alpha,\alpha)}\rangle _{L^2(I,\omega_{\alpha\alpha})}|\nonumber\\
&\leq& \displaystyle {\sum_{m= 0} ^{[n/A]}}|\beta_m^n||\langle e^{ik\pi x}, \tilde{P}_{m}^{(\alpha,\alpha)}\rangle _{L^2(I,\omega_{\alpha\alpha})}| \nonumber \\
&+ & \displaystyle{ \sum_{m\geq {[n/A]}+1}} |\beta_m^n||\langle e^{ik\pi x}, \tilde{P}_{m}^{(\alpha,\alpha)}\rangle _{L^2(I,\omega_{\alpha\alpha})}| =I_1 +I_2
\end{eqnarray}
For $I_2$ since,$|\beta_m^n|\leq1$ and by using $(\ref{Eq3.2}), \; (\ref{1})$ , one gets
\begin{eqnarray}
I_2&\leq& \ds{ \sum_{m\geq {[n/A]}+1}} |\langle e^{ik\pi x}, \tilde{P}_{m}^{(\alpha,\alpha)}\rangle _{L^2(I,\omega_{\alpha\alpha})}|\nonumber\\
&\leq& \ds{ \sum_{m\geq {[n/A]}+1}}\sqrt{\pi}(\frac{2}{k\pi})^{\alpha+1/2}\frac{\Gamma(m+\alpha+1)}
{\Gamma(m+1)}|J_{m+\alpha+1/2}(k\pi)|\nonumber\\
&\leq& \ds{ \sum_{m\geq {[n/A]}+1}}\sqrt{\pi}\frac{\Gamma(m+\alpha+1)}
{\Gamma(m+1)}(\frac{({k\pi})^m}{2^m \Gamma(m+\alpha+3/2)})\nonumber\\
&\leq& \ds{ \sum_{m\geq {[n/A]}+1}}\sqrt{\frac{\pi}{2m+1 }}\left(\frac{{k\pi e}}{2m +1} \right)^m
\leq \ds{ \sum_{m\geq {[n/A]}+1}}\sqrt{\frac{\pi}{2[\frac{n}{A}]+3 }}\left(\frac{{k\pi e}}{2[\frac{n}{A}]+3} \right)^m \nonumber\\
&\leq &K\sqrt{\frac{\pi}{2c+1 }}\left(\frac{{k\pi e}}{2\frac{n}{A}+1} \right)^{\frac{n}{A}}
\end{eqnarray}
Where $K$ is a positive constant. \\
It is clear that the appropriate value of $A$ is $1.7$ then
for $k\leq \frac{1.2n}{e\pi}\simeq 0.14n$ there is $b>0 $ such as
\begin{eqnarray}
I_2 &\leq &K\sqrt{\frac{\pi}{2c+1 }}\left(\frac{{k\pi e}}{1.2n+1} \right)^{\frac{n}{1.7}}
\leq K e^{-bn}
\end{eqnarray}
For $I_1 $, we have $|\langle e^{ik\pi x}, \tilde{P}_{m}^{(\alpha,\alpha)}\rangle _{L^2(I,\omega_{\alpha\alpha})}|\leq 1$, so we use (\ref{10}). For $n\geq (75+46\alpha)^{0.7}c$. we have
\begin{eqnarray}
I_1 &\leq & \ds {\sum_{m= 0} ^{[n/1.7]}}|\beta_m^n|\leq K_{\alpha} e^{-an}
\end{eqnarray}
\end{proof}
We first recall that if $f\in H^s_{\alpha,per} $ then $f(x)=\ds{\sum_{k\geq0}}<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}\widetilde{P}_{n}^{(\alpha,\alpha)}$ and \\ $\|f\|^2_{H^s_{\alpha,per}}=\ds{\sum_{k\geq0}}|<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}|^2(1+k^2)^s$.\\
\begin{proposition}
Let $c>0$ and $\alpha >-1$, then there exist constants $K>0$ and $a>0$ such that, when
$N > (75+46\alpha)^{0.7}c$ and $f \in H^s_{\alpha,per}$; $s > 0$, we have the inequality
\begin{eqnarray}
\|f-S_N(f)\|_{L^2_{\alpha}(I)} &\leq& \left(1+ \left(\frac{N}{2}\right)^2\right)^{\frac{-s}{2}} \|f\|_{H^s_{\alpha,per}} +K.e^{-a N}\|f\|_{L^2_{\alpha}(I)}\label{8}
\end{eqnarray}
Where $S_N(f)(t)=\ds{\sum_{n<N}}<f,\psi_{n,c}^{\alpha}>_{L^2_{\alpha}(I)}\psi_{n,c}^{\alpha}$
\end{proposition}
\begin{proof}
Assume that $f\in H^s_{\alpha,per} $ such that $\|f\|_{L^2_{\alpha}(I)}=1$
and
\begin{eqnarray}
f(x)&=&g(x)+h(x)=
\ds{\sum_{k\geq\left[\frac{N}{2}\right]}}<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}\widetilde{P}_{n}^{(\alpha,\alpha)}
+\ds{\sum_{k<\left[\frac{N}{2}\right]}}<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}
\widetilde{P}_{n}^{(\alpha,\alpha)} \nonumber
\end{eqnarray}
then
\begin{eqnarray}
\|g\|^2_{L^2_{\alpha}(I)}&=& \ds{\sum_{k\geq\left[\frac{N}{2}\right]}}|<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}|^2
\frac{(1+k^2)^s}{(1+k^2)^s}\leq\frac{1}{(1+\left(\frac{N}{2}\right)^2)^s}\|f\|^2_{H^s_{\alpha,per}}\label{6}
\end{eqnarray}
On the other hand by (\ref{10}) there exist $a>0$ and $K>0$ such that,\\
for all $N\geq (74.6+45.4\alpha)^{0.7}c$
\begin{eqnarray}
\|h-S_N(h)\|^2_{L^2_{\alpha}(I)} &=& \ds{\sum_{n\geq N}}|<f,\psi_{n,c}^{\alpha}>_{L^2_{\alpha}(I)}|^2 =\ds{\sum_{n\geq N}}\ds{\sum_{k<\left(\frac{N}{2}\right)}}|\beta_k^n|^2\;|<f,\widetilde{P}_{n}^{(\alpha,\alpha)}>_{L^2_{\alpha}(I)}|^2
\nonumber \\
&\leq & Ke^{-aN} \|f\|^2_{L^2_{\alpha}(I)} \label{7}
\end{eqnarray}
finally by combining $(\ref{6})\;and (\ref{7})$ on gets $(\ref{8})$
\end{proof}
\begin{remark}
The previous proposition is considered as the generalization of a similar result given in \cite{Bonami-Karoui4}, in the special case $ \alpha = 0 $
\end{remark}
\subsection{Approximate dimension of the space of almost time-limited functions from the restricted Paley-Wiener space.}
The aim of this paragraph is to show that the Landau-Pollak approximate dimension of the space of band-limited and almost time-limited functions \cite{Landau} initially proved in the context of the classical PSWFs, can be generalized in the context of the GPSWFs. That is for functions from $B_c^{(\alpha)},$ that are almost time-limited.
In this case, we have some additional difficulties, compared to the classical case. In fact and unlike the PSWFs, the GPSWFs do not have the the double orthogonality property over $[-1,1]$ and over $\mathbb R.$
To overcome this problem, we will use the new inner product $<\cdot,\cdot>_{\alpha},$ given by
(\ref{innerproduct}).
In the sequel, for $ \varepsilon>0, $ we let $ \EE $ denote the space of almost time-limited functions from the restricted Paley-Wiener space $ B_c^{(\alpha)},$ given by
$$ \EE = \{ f \in B_c^{(\alpha)};\,\, \| f \|_{\alpha}=1,\,\,\, \| f \|^2_{L^2(I, \omega_{\alpha})} =1-\varepsilon^2 \}.$$
\begin{definition}
Let $ \phi_{0}$ ,$ \phi_{1}$,...,$ \phi_{N-1} \in B_c^{(\alpha)}$ and let us denote by $ S_{\phi}^{N}$ the subspace spanned by the $ \phi_i$ , $0\leq i \leq N-1 $.
The deflection $ \delta_N (\EE,S_{\phi}^{N}) $ of $\EE$ from $ S_{\phi}^{N}$ , is defined as follows:
\begin{equation}
\label{deflection}
\delta_N (\EE,S_{\phi}^{N}) = \sup_{f\in \EE} \Big\| f-P_N(\phi)\cdot f \Big\|_{\alpha}.
\end{equation}
Here $P_N(\phi)$ is the projection operator over $S_{\phi}^{N}.$
\end{definition}
We first recall the following result from \cite{Karoui-Souabni1} that gives us a first result of approximation of band-limited functions by GPSWFs over $[-1,1].$
\begin{proposition}\label{band-limited} Let $c>0,\,\, \alpha \geq 0$ be two real numbers and let $f\in B_c^{(\alpha)}.$
For any positive integer $N> \frac{2c}{\pi},$ let
$$S_N(f)(x) = \sum_{k=0}^{N} <f,\Psi_{k,c}^{(\alpha)}>_{L^2(I,\omega_{\alpha})} \Psi_{k,c}^{(\alpha)}(x).$$
Here, ${\displaystyle \Psi_{k,c}^{(\alpha)}(x)=\frac{1}{\sqrt{\lambda_k^{(\alpha)}(c)}}\psi_{k,c}^{(\alpha)}(x).}$
Then, we have
\begin{equation}\label{approx1}
\left(\int_{-1}^{1}|f(t)- S_N f(t)|^2 \omega_{\alpha}(t) dt\right)^{1/2}\leq C_1 \sqrt{\lambda_N^{(\alpha)}(c)}\, (\chi_N(c))^{(1+\alpha)/2}\, \| f\|_{L^2(\mathbb R)},
\end{equation}
and
\begin{equation}\label{approx2}
\sup_{x\in [-1,1]}|f(x)- S_N f(x)|\leq C_1 \sqrt{\lambda_N^{(\alpha)}(c)}\, (\chi_N(c))^{1+\alpha/2}\, \| f\|_{L^2(\mathbb R)}.
\end{equation}
for some uniform constant $C_1$ depending only on $\alpha.$
\end{proposition}
Recall that in the previous proposition $\chi_n^{\alpha}(c)$ is the $n-$th eigenvalue of the differential operator $\mathcal L_c$ and satisfying the bounds given by (\ref{boundschi}).
The following proposition improves the result of the previous one, in the case where the function
$f$ belongs to $B_c^{(\alpha)}.$
\begin{proposition}
Let $c>0,\,\, \alpha \geq 0$ be two real numbers and let $f\in B_c^{(\alpha)}.$
For any positive integer $N\geq \frac{2c}{\pi},$ let
$$S_N(f)(x) = \sum_{k=0}^{N} <f,\Psi_{k,c}^{(\alpha)}>_{L^2(I,\omega_{\alpha})} \Psi_{k,c}^{(\alpha)}(x).$$
Then, we have
\begin{equation}\label{Approxx1}
\left(\int_{-1}^{1}|f(t)- S_N f(t)|^2 \omega_{\alpha}(t) dt\right)^{1/2}\leq C_1 \sqrt{\lambda_N^{(\alpha)}(c)}\, \| f\|_{\alpha}.
\end{equation}
Moreover, for any $N\geq \frac{2c}{\pi},$ we have
\begin{equation}\label{Approxx2}
\sup_{x\in [-1,1]}|f(x)- S_N f(x)|\leq C_1 \sqrt{\lambda_N^{(\alpha)}(c)}\, (\chi_N(c))^{1/2+\alpha/2}\, \| f\|_{\alpha}.
\end{equation}
for some uniform constant $C_1$ depending only on $\alpha.$
\end{proposition}
\noindent
{\bf Proof:} We first note that since the $\psi_{k,c}^{(\alpha)}$ form an orthogonal basis of
$L^2(I,\omega_{\alpha})$ with ${\displaystyle \|\psi_{k,c}^{(\alpha)}\|_{L^2(I,\omega_{\alpha})}=\sqrt{\lambda_k^{(\alpha)}(c)},}$ then the
${\Psi_{k,c}^{(\alpha)}}$ form an orthonormal basis of $L^2(I,\omega_{\alpha}).$ Hence, for any
$f\in B_c^{(\alpha)},$ we have
\begin{equation}\label{EEq1}
f(x)= \sum_{n=0}^{\infty} \beta_n \Psi_{n,c}^{(\alpha)}(x)= \sum_{n=0}^{\infty} \beta_n
\frac{\psi_{n,c}^{(\alpha)}(x)}{\sqrt{\lambda_n^{\alpha}(c)}}.
\end{equation}
Moreover, from Parseval's equality, one gets
\begin{equation}\label{approxx1}
\int_{-1}^{1}|f(t)- S_N f(t)|^2 \omega_{\alpha}(t) dt= \sum_{n=N+1}^{\infty} |\beta_n|^2.
\end{equation}
On the other hand, we have already shown that the $\psi_{n,c}^{(\alpha)}(x)$ form an orthonormal basis of
$B_c^{(\alpha)},$ hence, we have
\begin{equation}\label{EEq2}
f(x)= \sum_{n=0}^{\infty} \gamma_n \psi_{n,c}^{(\alpha)}(x),\quad \forall\, x\in \mathbb R.
\end{equation}
Since both equalities (\ref{EEq1}) and (\ref{EEq2}) agree on $[-1,1],$ then one concludes that
$$ \beta_n= \sqrt{\lambda_n^{\alpha}(c)} \gamma_n,\quad \forall\, n\geq 0.$$
Also, Parseval's equality applied to (\ref{EEq2}) gives us
$$ \sum_{n=0}^{\infty} |\gamma_n|^2 = \|f\|_{\alpha}^2.$$
By combining the previous two equalities, one concludes that
$$ \sum_{n=N+1}^{\infty} |\beta_n|^2 \leq {\lambda_N^{(\alpha)}(c)} \| f\|_{\alpha}^2.$$
Finally, by using the fact that the $\lambda_n^{\alpha}(c)$ have a fast decay starting from $n= 2c/\pi,$ one gets
the error bound (\ref{Approxx1}). Finally, to get the uniform error given by (\ref{Approxx2}), it suffices to combine the previous analysis together with the following bound the $\Psi_{n,c}^{(\alpha)},$ given in \cite{Karoui-Souabni1}
$$\sup_{x\in [-1,1]} |\Psi_{n,c}^{(\alpha)}(x)| \leq C_{\alpha} (\chi_n^{\alpha}(c))^{1/2+\alpha/2},\quad n\geq 2c/\pi.\qquad\qquad \Box$$
Next, we define the projection operator over $B_c^{(\alpha)},$ denoted by $\Pi^{(\alpha)}_c$ and given by
\begin{equation}
\label{eq3}
\Pi^{(\alpha)}_c\cdot f(x)=\int_{-c}^{c}\widehat{f}(t)e^{itx}\omega_{\alpha} \Big(\frac{t}{c} \Big) dt.
\end{equation}
It is clear that $ \Pi^{(\alpha)}_c \cdot f \in B_c^{(\alpha)}. $
By using the finite Fourier transform of the weight function
\begin{equation}
\label{fourier_weight}
\int_{-1}^1 e^{ixy} \omega_{\alpha}(y)\, dy = \sqrt{\pi} 2^{\alpha+1/2}\Gamma(\alpha+1) \frac{J_{\alpha+1/2}(x)}{x^{\alpha+1/2}}=
\mathcal K_{\alpha}(x),\quad x\in \mathbb R,
\end{equation}
one gets the expression of $\Pi^{(\alpha)}_c\cdot f $ in terms of $f,$ that is
\begin{equation}
\Pi^{(\alpha)}_c\cdot f(x)=c\int_{\mathbb{R}}\mathcal{K}_{\alpha}(c(x-u))f(u)du.
\end{equation}
Note that
\begin{equation}
\Pi^{(\alpha)}_c\, \mathbf 1_{[-1,1]}=\frac{1}{2\pi} Q_c^{(\alpha)},
\end{equation}
where $Q_c^{(\alpha)}$ is as given by (\ref{EEq0}).
The following theorem tells us that the $ \ps $ are actually the best basis for the approximation of almost-time limited
functions from the space $B_c^{(\alpha)}.$
\begin{remark}
By using \ref{newdecay} and straightforward computations one gets
\begin{equation}
\left(\int_{-1}^{1}|f(t)- S_N f(t)|^2 \omega_{\alpha}(t) dt\right)^{1/2}\leq C_{\alpha} \frac{1}{\log(\frac{2N-1}{ec})}\Big(\frac{ec}{2N-1}\Big)^{N+\frac{\alpha+1}{2}} \|f\|_{\alpha}
\end{equation}
\end{remark}
\begin{theorem}
For every positive integer N, $S_{\psi}^N $ best approximate $ \EE$ in the sense that the deflexion of $ \EE$ for $\alpha \geq 0 $ from that subspace is smaller than from any other subspace of dimension N. Moreover, for any
$N \geq [\frac{2c}{\pi}]+1,$ we have
\begin{equation}
\delta_N (\EE,S_{\psi}^{N}) \leq \frac{\varepsilon^2}{1-\lambda_N^{(\alpha)}} .
\end{equation}
\end{theorem}
\begin{proof}
We first compute the deflection of $ \EE$ from $ S_{\psi}^N $. As it is done in \cite{Landau}, we have to consider two cases : \\
\noindent
{\bf Case 1:} If $ 1-\varepsilon^2 \leq \lambda_N^{(\alpha)}(c).$
Since $\| \ps \|^2_{\alpha} =1 $ and $ \| \ps \|^2_{L^2(I,\omega_{\alpha})} =\lambda_n^{\alpha}(c),$ then $ \delta_N (\EE,S_{\psi}^{N})=1.$ To prove this it suffices to consider
the function $$f=a_N \psi_{N,c}^{(\alpha)} + a_{N+1}\psi_{N+1,c}^{(\alpha)},$$ where
$$a_N^2+a_{N+1}^2=1\mbox{ and }\lambda_n^{\alpha} a_N^2 + \lambda^{(\alpha)}_{N+1} a_{N+1}^2=1-\varepsilon^2.$$
\noindent
{\bf Case 2:} If $ 1-\varepsilon^2 \geq \lambda_N^{(\alpha)}(c),$ then $ \delta_N (\EE,S_{\psi}^{N})= \frac{ \lambda_0^{(\alpha)}(c)-(1-\varepsilon^2)}{\lambda_0^{(\alpha)}(c)-\lambda_N^{(\alpha)}}.$ In fact, let $ f \in \EE,$ then by the fact that $ \{ \ps ,n\in \mathbb{N} \} $ is an orthonormal basis of $B_c^{(\alpha)},$ one gets
$$ f=\sum_{n=0}^{\infty} a_n \ps \qquad a_n = <f,\ps >_{\alpha}$$
with
\begin{equation}\label{aa}
\sum_{n=0}^{\infty} a_n^2=1 \qquad and \qquad \sum_{n=0}^{\infty} \lambda_n^{\alpha} a_n^2 = {1-\varepsilon^2}.
\end{equation}
Note that $$ \Big\| f -\sum_{n=0}^{N-1} a_n \ps \Big\|^2_{\alpha} = \sum_{k=N}^{\infty} |a_k|^2 $$
From (\ref{aa}), one can easily see that
$$ \lambda_0^{(\alpha)}(c) - (1-\varepsilon^2) = \sum_{k=0}^{N-1} (\lambda_0^{(\alpha)}(c)-\lambda_k^{(\alpha)}(c))a_k^2 + \sum_{k=N}^{\infty} (\lambda_0^{(\alpha)}(c)-\lambda_k^{(\alpha)}(c))a_k^2.$$
Hence, by using the monotonicity of the $ \lambda_n^{\alpha}, $ one gets
\begin{equation}
\Big\| f -\sum_{n=0}^{N-1} a_n \ps \Big\|^2_{\alpha} \leq \frac{ \lambda_0^{(\alpha)}(c)-(1-\varepsilon^2)}{\lambda_0^{(\alpha)}(c)-\lambda_N^{\alpha}(c)}.
\end{equation}
To conclude for the proof, it suffices to consider $ f = a_0 \psi_{0,c}^{(\alpha)}+a_N \psi_{N,c}^{(\alpha)} $ , where
$$ a_0^2+a_N^2=1 \quad \mbox{and} \quad a_0^2 \lambda_0^{(\alpha)}(c) + a_N^2 \lambda_n^{\alpha}(c) = 1-\varepsilon^2. $$
Consequently, we have
\[\delta_N (\EE,S_{\psi}^{N}) = \begin{cases}
1 & \mbox{ if $ 1-\varepsilon^2 \leq \lambda_N^{(\alpha)}(c) $ }
\\
\frac{\lambda_0^{(\alpha)}(c)-(1-\varepsilon^2)}{\lambda_0^{(\alpha)}(c)-\lambda_N^{(\alpha)}(c)} & \mbox{ if $1-\varepsilon^2 \geq \lambda_N^{(\alpha)}(c)$}
\end{cases}.\]
Now, we consider the following map :
$$
\begin{array}{ccccc}
\Phi & : & B_c^{(\alpha)} & \longrightarrow & \mathbb{R}^2 \\
& & f & \longmapsto & \Big( x=\frac{\|\mathbf 1_{[-1,1]} f\|^2}{\|f\|_{\alpha}^2},y=\frac{\|f\|_{\alpha}^2-\|P_N f\|_{\alpha}^2}{\|f\|_{\alpha}^2} \Big) \\
\end{array}
$$
Note that the $x-$coordinate of points in $R(\alpha)$ satisfy $ 0\leq x \leq \lambda_0^{(\alpha)}(c),$ while the $y-$coordinates satisfy $ 0\leq y \leq 1.$ Also, $x= \lambda_0^{(\alpha)}(c) $ is achieved by $ \psi_{0,c}^{(\alpha)}$ and $y=1$ is achieved by functions orthogonal to $ S_{\psi}^N.$
Therefore by Weyl-Courant lemma, we have $$ \sup_{y=1}x \geq \lambda_N^{(\alpha)}(c).$$
To prove that $ R(\alpha) = \Phi(B_c^{\alpha})$ is convex, we use the same techniques developed in \cite{Landau}. We should only mention that for $a,b\in\mathbb{R},$ the operator
$ a\, P^{(\alpha)}_c \mathbf 1_{[-1,1]} P^{(\alpha)}_c+b P_N$ is completely continuous. Here, $P_N $ is the orthogonal projection over $S_{\psi}^N.$ In fact, $P^{(\alpha)}_c\cdot
\bf 1_{[-1,1]}$ is a Hilbert-Schmidt operator and $P^{(\alpha)}_c$ is a bounded operator. On the other hand, $P_N$ is a projection over a subspace of finite dimension so it is completely continuous.
The rest of the proof in \cite{Landau} is independent of the operators $P^{(\alpha)}_c$ and $\mathbf 1_{[-1,1]}.$
By combining the convexity of $R(\alpha)$ with the fact that $(\lambda_N,1)$,$(0,\lambda_0) \in R(\alpha)$ one gets
\[\delta_N (\EE,S_{\varphi}^{N})=\sup_{x=1-\varepsilon_T^2} y \begin{cases}
=1 & \mbox{ if $ 1-\varepsilon^2 \leq \lambda_N^{(\alpha)}(c) $ }
\\ \geq
\frac{\lambda_0^{(\alpha)}(c)-(1-\varepsilon^2)}{\lambda_0^{(\alpha)}(c)-\lambda_N^{(\alpha)}(c)} & \mbox{ if $1-\varepsilon^2 \geq \lambda_N^{(\alpha)}(c)$}
\end{cases}.\]
Hence,
for $ \lambda_N^{(\alpha)}(c) < 1-\varepsilon^2$ we have $ \frac{\lambda^{(\alpha)}_0(c)-(1-\varepsilon^2)}{\lambda^{(\alpha)}_0(c)-\lambda^{(\alpha)}_N(c)}<\frac{\varepsilon^2}{1-\lambda^{(\alpha)}_N(c)}$ and for $ \lambda^{(\alpha)}_N(c) \geq 1-\varepsilon^2$, that is $1 \leq \frac{\varepsilon^2}{1-\lambda^{(\alpha)}_N}$, we have $ \delta_N \leq \frac{\varepsilon}{\sqrt{1-\lambda^{(\alpha)}_N(c)}}.$
\end{proof}
\section{Numerical results}
In this section, we give
some numerical examples that illustrate the different results of this work.\\
\noindent
{\bf Example 1: } In this second example, we illustrate the quality of approximation of a band-limited functions by the GPSWFs. To do this, we have considered $ \alpha=1, c=50$ and the band-limited function
${\displaystyle f(x)=\frac{\sin( 40 x)}{40 x}.}$
Then, we have computed
the projection $ P_N f$ given by :
$$S_N(f)(x) = \sum_{k=0}^{N} <f,\psi_{k,c}^{(\alpha)}>_{L^2(I,\omega_{\alpha})} \psi_{k,c}^{(\alpha)}(x),\quad x\in I=[-1,1].$$ In Figure 2, we have plotted the graphs of $f,$ together with the approximation errors $E_N= f-S_Nf,$ with $N=20, 30.$ Note that as predicted by Proposition 1, Figure 2 indicate that our GPSWFs based approximation method provides us with high accurate approximation of the band-limited functions.
\begin{figure}![h]
\centering
{\includegraphics[width=18.05cm,height=6.5cm]{Graph_example2}}
\caption{(a) Graph of $f(x)=\sin(40 x)/(40 x)$, (b) graph of the error $E_N= f(x)-P_N f(x),$ with $N=20.$
(c) same as (b) with $N=30.$}
\end{figure}
\noindent
{\bf Example 2: } In this example, we illustrate the quality of approximation of a band-limited functions by GPSWFs, given by Proposition 2. For this purpose, we have considered $ \alpha=1, c=50$ and the function
$ g(x)=c \sqrt{\pi} \Big( \frac{2}{cx} \Big)^{\alpha+1/2} \Gamma(\alpha+1)J_{\alpha+1/2}(cx).$
Using (\ref{fourier_weight}), we have $ \widehat{g}(x)=\Big(1-(\frac{x}{c}) ^2\Big)^{\alpha} \chi_{[-c,c]}(x),$
so that $g\in B_c^{(\alpha)}.$ Then, we have computed
the projection $ P_N g$ with $N=32, 40.$ In Figure 3, we have plotted the graphs of $g,$ together with the approximation errors $E_N(x)= f(x)-S_Nf(x),$ with $N=30, 40$ and for $x\in [0,1].$ By symmetry, the graphs of $g$ and the associated approximation errors inside $[-1,0]$ are similar to those obtained in $[0,1].$ It is interesting to note the high precision with which the GPSWFs approximate functions from the space $B_c^{(\alpha)}.$
\begin{figure}[!h]
\centering
{\includegraphics[width=18.05cm,height=6.5cm]{Graph_example3}}
\caption{(a) Graph of $g(x)= c \sqrt{\pi} \Big( \frac{2}{cx} \Big)^{\alpha+1/2} \Gamma(\alpha+1)J_{\alpha+1/2}(cx),\,\, c=50,\, \alpha=1, $ (b) graph of the error $E_N= g(x)-P_N g(x),$ with $N=32.$
(c) same as (b) with $N=40.$}
\end{figure}
\noindent
{\bf Example 3:} In this last example, for any positive real number $s$, we consider the random function
$$ B_s(x) = \sum_{k=1}^{\infty} \frac{X_k}{k^s} \cos(k \pi x) \quad -1 \leq s \leq 1 $$
Here $X_k$ is a sequence of independent standard Gaussian random variable.\\
We compute for $ c= 5\pi $, the truncated series expansion of $B_s$ in the GPSWFs basis to the order $N$.
The graph of $B_{s}$ and the graph of its approximation $B_{s}$ are given by the figure \ref{brownian}
\begin{figure}[h]\label{brownian}
\centering
{\includegraphics[width=18.05cm,height=6.5cm]{brownian_function}}
\caption{(a) Graph of $B_s(x) $ (b) graph of $B^{(N)}_s(x) $ (c) graph of the error $E_N= B_s(x)-B_s^{(N)},$ with $N=90.$}
\end{figure}
|
2,877,628,089,546 | arxiv | \section{Introduction}
\label{sec:introduction}
\begin{figure*}
\centering
\begin{tabular}{c}
\includegraphics*[scale=0.14]{plot/1998dk.pdf}
\includegraphics*[scale=0.14]{plot/2003cq.pdf}
\includegraphics*[scale=0.14]{plot/2006ef.pdf}
\includegraphics*[scale=0.14]{plot/2007gi.pdf}
\includegraphics*[scale=0.14]{plot/2008cl.pdf}
\includegraphics*[scale=0.14]{plot/ptf09djc.pdf}\\
\includegraphics*[scale=0.14]{plot/ptf09dnp.pdf}
\includegraphics*[scale=0.14]{plot/2010ii.pdf}
\includegraphics*[scale=0.14]{plot/ptf10lot.pdf}
\includegraphics*[scale=0.14]{plot/ptf10tce.pdf}
\includegraphics*[scale=0.14]{plot/ptf10ygu.pdf}
\includegraphics*[scale=0.14]{plot/ptf12giy.pdf}
\end{tabular}
\caption{Some examples of high \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ velocity SN~Ia
(high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SN~Ia; defined as $\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\geq12000\,\mathrm{km\,s^{-1}}$)
host galaxies in this work. The position of the SN is indicated by the purple triangle.
Images are all generated from SDSS with a size of $80\arcsec\times80\arcsec$.
North is up and east is left.
}
\label{host-cutout}
\end{figure*}
Type Ia supernovae (SNe Ia) are believed to be the result of the thermonuclear explosion of an accreting carbon-oxygen white dwarf (WD) star in a close binary system \citep{2011Natur.480..344N,2012ApJ...744L..17B}. However, the nature of the companion star that donates material is not yet clear. The various possibilities include the single degenerate \citep{1973ApJ...186.1007W} and double degenerate \citep{1984ApJS...54..335I,1984ApJ...277..355W} scenarios, as well as more variations on these themes \citep[for recent reviews, see][]{2000ARA&A..38..191H, 2013FrPhy...8..116H, 2014ARA&A..52..107M}.
The host galaxies of SNe Ia has long been a profitable route to probe the SN Ia population, with the observed properties of SNe Ia known to correlate with the physical parameters of their host galaxies. Previous studies have found significant correlations between SN Ia light curve parameters and luminosities, and the properties of their host galaxies \citep[e.g.,][]{2010ApJ...715..743K,2010ApJ...722..566L,2010MNRAS.406..782S,2011ApJ...743..172D,2013MNRAS.435.1680J,2013ApJ...770..108C,2014MNRAS.438.1391P}. SNe Ia with faster light curves are preferentially resided in massive and metal-rich galaxies than those in lower-mass and metal-poor systems. Galaxies with stronger star-formation and younger populations tend to host slower and brighter SNe Ia than passive and older galaxies.
Previous studies also showed some evidence that the SN~Ia spectral features correlate with the host properties. By dividing SNe~Ia into two sub-groups according to their photospheric velocities, \citet{2013Sci...340..170W} found SNe Ia with high \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ velocities \citep[high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia, defined as $\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}} \ga 12,000$\,km\,s$^{-1}$;][]{2009ApJ...699L.139W} tend to be more concentrated in the inner regions of their host galaxies, whereas the normal-velocity events (normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia; defined as $\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}} < 12,000$\,km\,s$^{-1}$) span a wider range of radial distance. Given the metallicity gradients observed in both the Milky Way and many external galaxies \citep[e.g.,][]{1999PASP..111..919H}, they suggested the metallicity could be important in driving this relation. \citet[][hereafter P15]{2015MNRAS.446..354P} further supported this idea by finding some evidence that high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to reside in more massive galaxies than the normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ counterparts, although their results are not statistically significant due to the small sample size. These host studies together suggested there could be at least two distinct populations of SNe~Ia in terms of their ejecta velocities.
High-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SN~Ia has long been suspected to have a different origin from normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SN~Ia. This was firstly proposed by \citet{2009ApJ...699L.139W}, where they found high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to be redder and prefer a lower extinction ratio ($R_{V}$) than normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. \citet{2011ApJ...729...55F} argued the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia actually do not have a different reddening law but are intrinsically redder than normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. Recently, \citet{2019ApJ...882..120W} found some evidence that these high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to show blue excess in their late-time light-curves and variable Na\,\textsc{i} absorption lines in their spectra. These observations were attributed as the circumstellar dust surrounding the SNe. They claimed the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are likely associated with the single degenerate systems.
Theoretical study also suggested the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia may originate from unique explosions. Using 1D WD explosion models, \citet{2019ApJ...873...84P} showed the sub-Chandrasekhar explosions could produce SNe~Ia with a wide range of \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}. Their results further indicated the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia could be primarily produced by sub-Chandrasekhar type of explosions. The red color found for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia can also be explained by the line blanketing effect due the ashes of helium shell in their models.
In this paper, we revisit the relation between SN~Ia \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and host-galaxy properties with a parent sample of \about280 SNe~Ia ($\gtrsim$2 times larger than that in P15), with the purpose to differentiate the progenitor properties and explosion mechanisms between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. A plan of the paper follows. In Section~\ref{sec:data} we introduce our SN Ia spectral sample and the determination of host parameters. We show the results in Section~\ref{sec:analysis}. The discussion and conclusions are presented in Section~\ref{sec:discussion} and Section~\ref{sec:conclusions}, respectively. Throughout this paper, we assume $\mathrm{H_0}=70$\,km\,s$^{-1}$\,Mpc$^{-1}$ and a flat universe with $\ensuremath{\Omega_{\mathrm{M}}}=0.3$.
\section{Data and method}
\label{sec:data}
\begin{figure*}
\centering
\begin{tabular}{c}
\includegraphics*[scale=0.62]{plot/m-vsi6150-020.pdf}
\hspace{0.25cm}
\includegraphics*[scale=0.62]{plot/metal-vsi6150-020.pdf}
\end{tabular}
\caption{{\it Left}: The \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ velocities (\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}) as a function
of host-galaxy stellar mass (\ensuremath{M_{\mathrm{stellar}}}). The high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe Ia
are shown as red triangles, and the normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe Ia are
shown as blue circles. The vertical and horizontal dashed
lines represent the criterion used to split the
sample in velocity and \ensuremath{M_{\mathrm{stellar}}}\ space,
respectively. The bottom histograms show
the cumulative fractions of \ensuremath{M_{\mathrm{stellar}}}\ for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and
normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe Ia. {\it Right}: The same as left panel,
but with host-galaxy gas-phase metallicity instead.}
\label{mass-vsi6150}
\end{figure*}
\subsection{SN data}
\label{sec:sn-data}
In this work, we extend our analysis with the full spectroscopic sample studied in \citet[][hereafter M14]{2014MNRAS.444.3258M}. This is the parent sample of what was studied by P15, containing 264 spectroscopically normal SNe~Ia with \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ measurements near peak (e.g., within 5 days from the peak luminosity). They were all discovered by the Palomar Transient Factory (PTF). In addition to the PTF sample, we further include the SNe studied in \citet[][hereafter S15]{2015MNRAS.451.1973S} to increase our sample size. This added another \about150 SNe~Ia with near-peak \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ measurements from the Berkeley SN~Ia Program (BSNIP) after removing the duplicate objects from our PTF sample. This gives a sample of \about400 SNe~Ia at $z<0.2$. The description of the spectroscopic observation and data reduction can be found in detail in M14 and S15.
Since M14 and S15 used very similar techniques in measuring the spectral features, we do not perform new measurements but simply adopting their results in our analysis. A complete description of the line measurement can be found in M14. Briefly speaking, the SN spectrum is firstly corrected into the rest frame, define (by eye inspection) continuum regions on either side of the feature, and fit a straight line pseudo-continuum across the absorption feature. The feature is then normalised by dividing it by the pseudo-continuum. A Gaussian fit is performed to the normalised \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ line in velocity space. The resulting fit then gives the velocity and pseudo-equivalent widths (pEW) of the feature.
\subsection{Host-galaxy properties}
\label{sec:host}
The main purpose of this work is to investigate the relation between \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ of SNe~Ia and their host properties. The host stellar mass (\ensuremath{M_{\mathrm{stellar}}}) is derived by fitting the photometry of the host galaxy with the photometric redshift code \textsc{z-peg} \citep{2002A&A...386..446L}. The host photometry is provided from SDSS \protect\hbox{$ugriz$} catalog \citep{2018ApJS..235...42A}. The SDSS model magnitudes are used here. \textsc{z-peg} fits the observed galaxy colours with galaxy spectral energy distribution (SED) templates corresponding to 9 spectral types (SB, Im, Sd, Sc, Sbc, Sb, Sa, S0, and E). Here we assume a \citet{1955ApJ...121..161S} initial-mass function (IMF). The photometry is corrected for foreground Milky Way reddening with $R_{V} = 3.1$ and a \citet*[][CCM]{1989ApJ...345..245C} reddening law. Fig.~\ref{host-cutout} shows some SDSS color images of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SN~Ia host galaxies studied in this work.
We also measure the host gas-phase metallicity and star-formation rate (SFR). This is done by obtaining the optical spectra of the host galaxies, primarily with the SDSS spectrograph on the Sloan Foundation 2.5-m telescope and Gemini Multi-Object Spectrographs (GMOS) on the Gemini Observatory. We fit the emission lines and stellar continuum of the host spectrum using the Interactive Data Language (\textsc{IDL}) codes \textsc{ppxf} \citep{2004PASP..116..138C} and \textsc{gandalf} \citep{2006MNRAS.366.1151S}. A complete description of this process can be found in \citet{2014MNRAS.438.1391P}. Briefly, \textsc{ppxf} fits the line-of-sight velocity distribution (LOSVD) of the stars in the galaxy in pixel space using a series of stellar templates. Before fitting the stellar continuum, the wavelengths of potential emission lines are masked to remove any possible contamination. The stellar templates are based on the \textsc{miles} empirical stellar library \citep{2006MNRAS.371..703S, 2010MNRAS.404.1639V}. A total of 288 templates is selected with $[M/H]=-1.71$ to $+0.22$ in 6 bins and ages ranging from $0.063$ to $14.12$\,Gyr in 48 bins.
We correct all spectra for foreground Galactic reddening using the calibrations of \citet{2011ApJ...737..103S}. The host-galaxy extinction is corrected with the two-component reddening model in \textsc{gandalf}. The first component assumes a diffusive dust throughout the whole galaxy that affects the entire spectrum. It is determined by comparing the observed spectra to the un-reddened spectral templates. The second component measures the local dust around the nebular regions and affects only the emission lines. It is constrained only if the Balmer decrement (the H$\alpha$ to H$\beta$ line ratio) can be measured.
After the emission-line measurements from \textsc{ppxf} and \textsc{gandalf}, we determine the SFR by adopting the conversion of \citet{1998ARA&A..36..189K}, which used evolutionary synthesis models to relate the luminosity of the H$\alpha$ line to the SFR. We calculate the host gas-phase metallicity based on the diagnostics from \citet{2016Ap&SS.361...61D}. \citet{2016Ap&SS.361...61D} used the ratios of [\ensuremath{\mathrm{N}\,\textsc{ii}}]\,$\lambda6584$ to [\ensuremath{\mathrm{S}\,\textsc{ii}}]\,$\lambda\lambda6717,6731$ and [\ensuremath{\mathrm{N}\,\textsc{ii}}]\,$\lambda6584$ to H$\alpha$ to calibrate the gas-phase metallicity. This has the advantage of requiring a narrow wavelength range and therefore is less affected by the reddening correction. We further use BPT diagrams \citep*{1981PASP...93....5B} to check for potential contamination from active galactic nuclei (AGNs) in our host galaxies. The criteria proposed by \citet{2001ApJ...556..121K} are adopted to distinguish between normal and AGN host galaxies. The potential AGNs are excluded from our emission line analyses. A summary of our measurements can be found in Table~\ref{sample1}.
\section{Analysis}
\label{sec:analysis}
We firstly investigate the relation between \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and host \ensuremath{M_{\mathrm{stellar}}}. Ideally, one should distinguish between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia with the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ measured exactly at peak luminosity. In practice, previous studies generally used the spectra within a few days from the peak given that the spectral evolution is relatively mild at those epochs. M14 measured \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ using all the spectra observed within 5 days from the peak luminosity and did not apply phase corrections to their line measurements. Given a larger sample size, we adopt a more stringent phase criterion in this work; we use all the spectra observed within only 3 days from the peak luminosity. That way the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ of different SNe are compared at closer phases and reduces the uncertainties of phase evolution. This gives a final parent sample of 281 SNe (41 of them are high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia), which is still more than 2 times larger than that studied in P15. Releasing this criterion does not change our conclusion but only making our results less significant.
The result is shown in the left panel of Fig.~\ref{mass-vsi6150}. We find clear evidence that most of the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia reside in massive galaxies ($\log(\ensuremath{M_{\mathrm{stellar}}})>10\,M_{\odot}$), whereas the normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia can be found in both lower-mass and massive galaxies. This is consistent with the finding in P15. Here we confirm this trend with a much larger sample and our results are statistically more significant. Both Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) tests give a $p$-value of $\lesssim$0.02 that the \ensuremath{M_{\mathrm{stellar}}}\ distributions of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are drawn from the same underlying population. This value is \about10 times smaller than that studied in P15. It is also evident that this relation is neither linear nor monotonic. In fact, it implies the existence of multiple populations of SNe~Ia (see Section~\ref{sec:population} for a discussion).
We also note the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ of \about12\,000\,\,km\,s$^{-1}$\ is a fairly good criterion to distinguish between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. For SNe with \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ $>12\,000$\,\,km\,s$^{-1}$, \about84 percent of their host galaxies have $\log(\ensuremath{M_{\mathrm{stellar}}})>10\,M_{\odot}$. The ratio goes up to only \about90 percent if the criterion is raised to 13\,000\,\,km\,s$^{-1}$, but drops significantly to \about68 percent if the criterion is lowered to 11\,000\,\,km\,s$^{-1}$. All of the SNe with \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ $>12\,000$\,\,km\,s$^{-1}$\ have $\log(\ensuremath{M_{\mathrm{stellar}}})>9.6\,M_{\odot}$.
Next we investigate the relation with host gas-phase metallicity. The result is shown in the right panel of Fig.~\ref{mass-vsi6150}. Given the tight relation between \ensuremath{M_{\mathrm{stellar}}}\ and metallicity \citep[e.g.,][]{2004ApJ...613..898T}, it is reasonable to suspect that the metallicity is the underlying source to drive the relation we see with \ensuremath{M_{\mathrm{stellar}}}. In general, we find the relation between \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and host metallicity is consistent with what we have found for host \ensuremath{M_{\mathrm{stellar}}}. The host galaxies of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to be metal-rich, having metallicities mostly above solar value \citep[8.69;][]{2001ApJ...556L..63A}. The K-S and A-D tests give a $p$-value of 0.05 and 0.04, respectively, that the metallicity distributions of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are drawn from the same underlying population. This is less significant than the trend with host \ensuremath{M_{\mathrm{stellar}}}. However, our sample with host metallicity measurements is only one third the size of that with host \ensuremath{M_{\mathrm{stellar}}}. The progenitor metallicity is also expected to differ from nuclear metallicity measurements performed in this work. A larger sample with direct metallicity measurements near the SN location is critical to constrain the metallicity effect in the future.
\section{Discussion}
\label{sec:discussion}
\subsection{Silicon velocity and metallicity}
\label{sec:population}
\citet{2000ApJ...530..966L} showed that the observed \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ could vary with the C+O layer metallicity in SN~Ia. The blue-shifted velocity of the \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ feature increase with C+O layer metallicity due to the increasing opacity in the C+O layer moving the features blueward and causing larger line velocities. P15 determined a linear relation between the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and C+O metallicity of SN progenitor using the models of \citet{2000ApJ...530..966L} and showed the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ increase with metallicities with a slope of 435\,\,km\,s$^{-1}$\,$\rm dex^{-1}$. They claimed the observed relation between host \ensuremath{M_{\mathrm{stellar}}}\ and \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ is in qualitative agreement with that of \citet{2000ApJ...530..966L} models. However, fitting a linear relation between the observed \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and host gas-phase metallicity with our sample (i.e., right panel of Fig.~\ref{mass-vsi6150}) gives a slope of $87\pm427$\,\,km\,s$^{-1}$\,$\rm dex^{-1}$, which is consistent with no trend.
In fact, it is now evident that both high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia can be found in metal-rich host environments. If the high photospheric velocity is mainly caused by the increasing opacity (due to higher progenitor metallicity) in SN, we would expect a monotonic relation between \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and host metallicity, instead of a L-shaped distribution shown in both panels of Fig.~\ref{mass-vsi6150}. Thus, it is not precise to say that the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to reside in {\it more} metal-rich environments than that of normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. While the opacity could still have some effect here, we argue that there are likely at least two populations of SNe~Ia responsible for the observed trend. The high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia could be part of a unique population which is sensitive to the progenitor metallicity and can only be formed in metal-rich environments (see Section~\ref{sec:explosion} for a discussion).
\subsection{Implications on progenitor systems and explosion mechanisms}
\label{sec:explosion}
\begin{figure}
\centering
\begin{tabular}{c}
\includegraphics*[scale=0.5]{plot/ssfr_ha-cdf.pdf}
\end{tabular}
\caption{The cumulative fractions of specific star-formation rate (sSFR)
for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ (red) and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ (blue) SNe~Ia.}
\label{ssfr-cdf}
\end{figure}
\citet{2013Sci...340..170W} found high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are more concentrated in the inner and brighter regions of their host galaxies. They suggested the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia likely originate from younger and more metal-rich progenitors than those of normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. Recently, \citet{2019ApJ...882..120W} showed some evidence that these high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to present blue excess in their late-time light-curves and variable Na\,\textsc{i} absorption lines in the spectra. They attributed these observations to the circumstellar dust surrounding the SNe and concluded that the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are likely from the single degenerate and double degenerate systems, respectively.
Our results support that the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SN~Ia has a strong preference to occur in metal-rich environment. However, we argue that they may not come from particularly young populations. Fig.~\ref{ssfr-cdf} shows the cumulative fractions of specific star-formation rate (sSFR) for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe Ia. Here the sSFR is defined as the SFR per unit \ensuremath{M_{\mathrm{stellar}}}. Theoretically, the sSFR is a more appropriate indicator to measure the relative star-formation activity of a galaxy as it measures the star-formation relative to the underlying galaxy stellar mass \citep{1997ApJ...489..559G}. There is also a strong correlation between sSFR and age of the galaxies, in a sense that higher-sSFR galaxies tend to have younger stellar populations than lower-sSFR galaxies \citep[e.g.,][]{2004MNRAS.351.1151B}.
We find the difference between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe Ia is not statistically significant in terms of their host sSFR. The K-S test gives a $p$-value of 0.93 that the sSFR distributions of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are drawn from the same underlying population. We determine a mean $\rm\log(sSFR)$ of $-10.93\pm1.11$\,yr$^{-1}$ and $-11.15\pm1.16$\,yr$^{-1}$ for normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia, respectively. Thus, the host galaxies of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia do not tend to be younger than their normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ counterparts. This is consistent with the results in P15, where they showed the youngest populations are likely related to those SNe~Ia with dispatched high-velocity features (HVFs), not those with high photospheric velocities. They also found there is a significant number (more than 30\,percent) of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia in early-type galaxies. Our results imply the metallicity is probably the only important (or dominant) factor in forming high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia.
Theoretical studies also suggested the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia may have unique explosion mechanisms. Using 1D WD explosion models, \citet{2019ApJ...873...84P} showed the sub-Chandrasekhar class of explosions can produce SNe~Ia of a wide range of luminosities and photospheric velocities. In particular, their results indicated the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia could be primarily produced by sub-Chandrasekhar explosions, whereas normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia can be produced by both sub-Chandrasekhar and near-Chandrasekhar explosions. The significant line blanketing due to the ashes of helium shell in their models also explained the intrinsically red color of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia \citep[e.g.,][]{2011ApJ...729...55F}. However, it is not yet clear the contribution of progenitor metallicity on such models.
Progenitor metallicity is believed to have significant impact on explosions of SN~Ia. For example, it may affect the accretion onto the WD by changing the opacity in the wind for some single degenerate scenarios \citep[e.g.,][]{1998ApJ...503L.155K}. The mass of the WD is also expected to vary with metallicity. At a given mass, stars of higher metallicity generally produce less massive WDs \citep[e.g.,][]{1999ApJ...513..861U}. This implies they may be more difficult to reach the Chandrasekhar limit for explosions. Under the circumstances, the sub-Chandrasekhar class of explosions are probably more efficient and may account for some SNe~Ia having higher progenitor metallicities. The preference of metal-rich environments can be used as a strong constraint to discriminate between models for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia in the future.
\subsection{Implications on cosmology}
\label{sec:cosmology}
\begin{figure}
\centering
\begin{tabular}{c}
\includegraphics*[scale=0.5]{plot/z-vsi6150.pdf}
\end{tabular}
\caption{The \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ velocities (\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}) as a function of redshift. The dashed line represents the
the criterion used to split the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia
(i.e., $\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}=12000\,\mathrm{km\,s^{-1}}$).}
\label{z-vsi}
\end{figure}
Our results also have significant implications on cosmology. \citet*{2011ApJ...729...55F} found high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are intrinsically redder (in terms of $B-V$ at maximum light) than normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. By accounting for this color difference, they reduced the scatter in Hubble residuals (HRs). Also recently, \citet{2020MNRAS.tmp..548S} determined a \about3-$\sigma$ HR step between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia, with high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia having more negative HRs than the normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. These results are indicative that the ejecta velocity can be used to improve SN~Ia distances.
Moreover, if the fraction of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ to normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia changes with redshift, it could introduce significant bias on our cosmological analysis when assuming a single intrinsic color for SN~Ia \citep{2011ApJ...729...55F}. Given the strong preference of metal-rich environments for high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia, we would expect an evolution on the number of discovered high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia with redshift, with a decreased rate towards higher redshifts. Fig.~\ref{z-vsi} shows the \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ as the function of redshift with our sample. We find some evidence that the ratio of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ to normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to be higher at lower redshifts. At $z<0.1$, \about16 percent (13 percent if using untargeted PTF sample only) of SNe in our sample are high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe, while only \about11 percent of the SNe are high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe at $z>0.1$. It is also obvious that the extremely high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia (e.g., $\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\gtrsim13\,000$\,\,km\,s$^{-1}$) are hardly found at $z>0.1$.
However, the selection effect might play a role for SNe discovered at $z>0.1$ in our sample \citep[for a discussion, see][]{2014MNRAS.438.1391P}. For example, we would expect a bias if the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are significantly brighter or fainter than normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. Using $\ensuremath{\Delta m_{15}}$($B$) as the proxy of SN~Ia brightness, we determine a mean $\ensuremath{\Delta m_{15}}$($B$) of $1.12\pm0.33$\,mag and $1.12\pm0.22$\,mag for normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia, respectively. This indicates (on average) they are not different in brightness. Thus, there is unlikely a Malmquist bias (at least) on the fraction of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ to normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia at higher redshifts. We also note that the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to show less dispersion in $\ensuremath{\Delta m_{15}}$($B$) than that of normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. It is unclear if this trend is intrinsic to the explosion or simply due to the smaller sample size of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia.
Another caveat is the difficulty in finding SNe on very bright galaxy backgrounds, where the contrast of the SN over the host galaxy is low. Since high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to be found in massive galaxies and are more concentrated in the inner regions of their hosts, they may be more difficult to be fond at higher redshifts. However, this is only an issue with modern image subtraction techniques when the SN brightness drops to $<10\%$ of that of the host background \citep[e.g.,][]{2010AJ....140..518P}. The future analysis with data taken from higher-redshift surveys (e.g., Pan-STARRS1; Pan et al., in preparation) will be necessary to constrain the potential evolution of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia.
\section{Conclusions}
\label{sec:conclusions}
In this work, we investigate the relation between photospheric \ensuremath{\mathrm{Si}\,\textsc{ii}\,\lambda6355}\ velocities (\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}) and host-galaxy properties of SN~Ia. A more stringent criterion on the phase of SN spectra is adopted to distinguish between normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. We find the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia are likely formed from a distinct population which favors massive host environments. This is further supported by the direct measurements on host gas-phase metallicities. Although opacity may have some effect (due to progenitor metallicity), we argue the difference in photospheric velocities between high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia is mainly caused by different explosion mechanisms.
Theoretical studies suggested the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia may originate from sub-Chandrasekhar explosions. This is consistent with our results. At a given mass, stars of higher metallicities generally produce less massive WDs. This may increase the chance for them to explode under sub-Chandrasekhar mass. Nevertheless, the detailed investigation is still needed to evaluate the effect of progenitor metallicity on such models.
Previous studies also suggested the high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia could be formed via single degenerate and double degenerate scenarios, respectively. However, we find high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia do not tend to originate from younger populations than that of normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. We argue the metallicity is the only important factor in forming high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia.
Our results also imply potential evolution of high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia. We would expect less high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia to be discovered at higher redshifts when the Universe is more metal-poor than present. This evolution could introduce a bias on our cosmological analysis given that high-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ and normal-\ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ SNe~Ia tend to have intrinsically different colors. Future spectroscopic studies using higher-redshift dataset will be critical to measure this evolution effect.
\begin{table*}
\centering
\caption{Summary of our sample in this work.}
\begin{tabular}{lcccc}
\hline\hline
SN Name & Redshift & \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ & log\,$M_{\rm stellar} $ & $\rm12+log\,(O/H)$\\
& & (\,km\,s$^{-1}$) & (M$_{\odot}$) & \\
\hline
PTF09bai & 0.180 & $11385\pm 18$ & $10.518^{+0.104}_{-0.279}$ & \nodata \\
PTF09bj & 0.144 & $ 9777\pm 50$ & $10.495^{+0.324}_{-0.123}$ & \nodata \\
PTF09djc & 0.034 & $13013\pm 16$ & $10.771^{+0.008}_{-0.176}$ & \nodata \\
PTF09dlc & 0.067 & $10615\pm 9$ & $ 8.923^{+0.138}_{-0.433}$ & $7.874\pm0.023$ \\
PTF09dnl & 0.024 & $10955\pm 4$ & $ 8.056^{+0.062}_{-0.053}$ & \nodata \\
PTF09dnp & 0.037 & $12189\pm 7$ & $10.586^{+0.143}_{-0.033}$ & $9.121\pm0.013$ \\
PTF09dqt & 0.113 & $11146\pm 22$ & $ 9.357^{+0.055}_{-0.011}$ & \nodata \\
PTF09dxo & 0.052 & $10924\pm 10$ & $10.364^{+0.449}_{-0.157}$ & $8.865\pm0.014$ \\
PTF09e & 0.149 & $10349\pm 38$ & $ 9.312^{+0.154}_{-0.939}$ & \nodata \\
PTF09fox & 0.072 & $11560\pm 11$ & $10.309^{+0.033}_{-0.034}$ & $8.706\pm0.005$ \\
PTF09foz & 0.054 & $10190\pm 8$ & $10.540^{+0.039}_{-0.147}$ & \nodata \\
PTF09gn & 0.139 & $10880\pm 72$ & $ 9.558^{+0.124}_{-0.215}$ & \nodata \\
PTF09gul & 0.072 & $11352\pm 33$ & $10.988^{+0.145}_{-0.011}$ \nodata \\
PTF09h & 0.121 & $10979\pm 31$ & $10.859^{+0.074}_{-0.140}$ & \nodata \\
PTF09ib & 0.122 & $11216\pm 15$ & $ 9.854^{+0.177}_{-0.303}$ & \nodata \\
PTF09isn & 0.101 & $11463\pm 37$ & $ 9.315^{+0.128}_{-0.015}$ & \nodata \\
PTF09s & 0.046 & $ 8926\pm 14$ & $ 9.308^{+0.104}_{-0.355}$ & \nodata \\
PTF09v & 0.119 & $11386\pm 37$ & $ 8.526^{+0.154}_{-0.173}$ & \nodata \\
PTF10aaea & 0.160 & $ 9775\pm147$ & $10.923^{+0.232}_{-0.125}$ & \nodata \\
PTF10abjv & 0.076 & $10705\pm 27$ & $10.024^{+0.039}_{-0.079}$ & \nodata \\
PTF10acqp & 0.170 & $12347\pm132$ & $10.980^{+0.167}_{-0.085}$ & \nodata \\
PTF10bhw & 0.110 & $11233\pm 25$ & $11.423^{+0.049}_{-0.239}$ & \nodata \\
PTF10cmj & 0.112 & $10490\pm 30$ & $11.117^{+0.064}_{-0.057}$ & \nodata \\
PTF10cwm & 0.079 & $10525\pm 18$ & $10.575^{+0.072}_{-0.194}$ & $8.693\pm0.141$ \\
PTF10cxk & 0.018 & $10938\pm 13$ & $ 9.147^{+0.090}_{-0.083}$ & \nodata \\
PTF10duy & 0.079 & $ 9602\pm 15$ & $ 9.242^{+0.074}_{-0.047}$ & \nodata \\
PTF10duz & 0.064 & $11502\pm 33$ & $10.206^{+0.111}_{-0.044}$ & $8.459\pm0.010$ \\
PTF10fej & 0.110 & $12703\pm 82$ & $10.886^{+0.270}_{-0.096}$ & $8.897\pm0.054$ \\
PTF10fj & 0.050 & $10757\pm 19$ & $10.906^{+0.137}_{-0.009}$ & \nodata \\
PTF10fxe & 0.099 & $11152\pm155$ & $ 9.391^{+0.206}_{-0.149}$ & \nodata \\
PTF10fxl & 0.030 & $11227\pm 12$ & $10.740^{+0.013}_{-0.165}$ & $8.785\pm0.003$ \\
PTF10fxp & 0.104 & $11490\pm 17$ & $11.668^{+0.039}_{-0.117}$ & \nodata \\
PTF10fxq & 0.107 & $10914\pm 34$ & $10.971^{+0.079}_{-0.249}$ & $8.923\pm0.123$ \\
PTF10fyl & 0.055 & $10405\pm 9$ & $10.312^{+0.004}_{-0.180}$ & \nodata \\
PTF10glo & 0.075 & $11151\pm 31$ & $ 9.226^{+0.083}_{-0.029}$ & \nodata \\
PTF10gnj & 0.078 & $10889\pm 19$ & $10.273^{+0.098}_{-0.120}$ & $8.586\pm0.107$ \\
PTF10goo & 0.087 & $10651\pm 31$ & $10.763^{+0.106}_{-0.210}$ & $8.690\pm0.114$ \\
PTF10gop & 0.097 & $10080\pm 33$ & $ 8.307^{+0.753}_{-1.440}$ & \nodata \\
PTF10goq & 0.088 & $10910\pm 59$ & $10.557^{+0.086}_{-0.215}$ & \nodata \\
PTF10hdm & 0.165 & $ 9839\pm230$ & $ 9.743^{+0.106}_{-0.030}$ & \nodata \\
PTF10hld & 0.038 & $13477\pm 24$ & $ 9.964^{+0.037}_{-0.090}$ & \nodata \\
PTF10jab & 0.187 & $11126\pm 16$ & $11.346^{+0.132}_{-0.014}$ & \nodata \\
PTF10lot & 0.022 & $12802\pm 16$ & $ 9.902^{+0.079}_{-0.061}$ & \nodata \\
PTF10mtd & 0.079 & $11248\pm 15$ & $ 9.875^{+0.101}_{-0.065}$ & \nodata \\
PTF10mwb & 0.031 & $ 9909\pm 4$ & $ 9.276^{+0.208}_{-0.286}$ & $7.954\pm0.041$ \\
PTF10ncy & 0.130 & $ 9697\pm 45$ & $10.003^{+0.226}_{-0.031}$ & \nodata \\
PTF10ncz & 0.170 & $10429\pm 95$ & $ 9.364^{+0.266}_{-0.025}$ & \nodata \\
PTF10nda & 0.101 & $10985\pm 26$ & $11.374^{+0.049}_{-0.216}$ & \nodata \\
PTF10nhu & 0.153 & $ 9163\pm132$ & $10.212^{+0.252}_{-0.131}$ & \nodata \\
PTF10nnh & 0.150 & $10670\pm106$ & $10.856^{+0.261}_{-0.065}$ & \nodata \\
PTF10nvh & 0.068 & $10843\pm 8$ & $ 9.950^{+0.071}_{-0.117}$ & \nodata \\
PTF10oth & 0.145 & $11289\pm 35$ & $11.237^{+0.174}_{-0.103}$ & \nodata \\
PTF10pvh & 0.105 & $10504\pm 39$ & $ 9.833^{+0.179}_{-0.247}$ & \nodata \\
PTF10pvi & 0.080 & $11286\pm 46$ & $10.326^{+0.079}_{-0.178}$ & $8.668\pm0.066$ \\
PTF10qhp & 0.032 & $11488\pm 25$ & $10.889^{+0.025}_{-0.165}$ & $8.448\pm0.080$ \\
PTF10qjl & 0.058 & $11109\pm 5$ & $ 8.671^{+0.020}_{-0.299}$ & $7.926\pm0.052$ \\
PTF10qjq & 0.028 & $10813\pm 12$ & $10.059^{+0.087}_{-0.044}$ & $8.711\pm0.003$ \\
\hline
\label{sample1}
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{Summary of our sample in this work (continued).}
\begin{tabular}{lcccc}
\hline\hline
SN Name & Redshift & \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ & log\,$M_{\rm stellar} $ & $\rm12+log\,(O/H)$\\
& & (\,km\,s$^{-1}$) & (M$_{\odot}$) & \\
\hline
PTF10qkf & 0.080 & $10156\pm 55$ & $10.445^{+0.115}_{-0.161}$ & $8.428\pm0.026$ \\
PTF10qky & 0.074 & $10448\pm 13$ & $10.559^{+0.094}_{-0.116}$ & $8.663\pm0.005$ \\
PTF10qny & 0.033 & $11664\pm 12$ & $10.340^{+0.484}_{-0.160}$ & $8.547\pm0.014$ \\
PTF10qsc & 0.088 & $10441\pm 31$ & $ 9.762^{+0.137}_{-0.378}$ & $7.854\pm0.082$ \\
PTF10qwg & 0.068 & $13006\pm 16$ & $10.089^{+0.151}_{-0.157}$ & $8.749\pm0.006$ \\
PTF10qyq & 0.160 & $10592\pm 99$ & $ 9.445^{+0.212}_{-0.050}$ & \nodata \\
PTF10rab & 0.085 & $11296\pm 70$ & $ 7.920^{+1.043}_{-0.114}$ & $7.811\pm0.005$ \\
PTF10ran & 0.160 & $11127\pm 57$ & $11.185^{+0.088}_{-0.147}$ & \nodata \\
PTF10rhi & 0.085 & $11257\pm 35$ & $10.014^{+0.077}_{-0.364}$ & \nodata \\
PTF10tce & 0.041 & $12362\pm 12$ & $10.542^{+0.161}_{-0.155}$ & $8.745\pm0.053$ \\
PTF10tqy & 0.045 & $11568\pm 11$ & $10.101^{+0.162}_{-0.005}$ & \nodata \\
PTF10trs & 0.073 & $11248\pm 2$ & $10.824^{+0.062}_{-0.070}$ & \nodata \\
PTF10trw & 0.170 & $10355\pm 40$ & $ 9.387^{+0.259}_{-0.024}$ & \nodata \\
PTF10twd & 0.073 & $11646\pm 23$ & $ 9.884^{+0.089}_{-0.055}$ & $8.365\pm0.006$ \\
PTF10ucl & 0.080 & $10541\pm 15$ & $10.683^{+0.561}_{-0.020}$ & \nodata \\
PTF10ufj & 0.073 & $10134\pm 10$ & $10.789^{+0.428}_{-0.051}$ & \nodata \\
PTF10urn & 0.110 & $10972\pm 76$ & $ 9.543^{+0.262}_{-0.487}$ & \nodata \\
PTF10vfo & 0.088 & $12488\pm 63$ & $10.751^{+0.011}_{-0.022}$ & \nodata \\
PTF10viq & 0.034 & $12480\pm 10$ & $10.792^{+0.201}_{-0.417}$ & $8.900\pm0.013$ \\
PTF10wnm & 0.066 & $10910\pm 11$ & $10.548^{+0.045}_{-0.116}$ & $8.749\pm0.044$ \\
PTF10wnq & 0.069 & $11681\pm 56$ & $11.341^{+0.107}_{-0.543}$ & \nodata \\
PTF10wov & 0.096 & $10368\pm 27$ & $10.165^{+0.276}_{-0.076}$ & \nodata \\
PTF10wri & 0.120 & $11595\pm 20$ & $11.777^{+0.101}_{-0.204}$ & \nodata \\
PTF10xeb & 0.122 & $ 9132\pm 62$ & $ 7.316^{+0.665}_{-0.091}$ & \nodata \\
PTF10xir & 0.052 & $10692\pm 19$ & $11.025^{+0.428}_{-0.178}$ & \nodata \\
PTF10xtp & 0.102 & $10391\pm103$ & $10.419^{+0.099}_{-0.133}$ & \nodata \\
PTF10ygr & 0.115 & $10177\pm114$ & $ 9.698^{+0.200}_{-0.022}$ & \nodata \\
PTF10ygu & 0.026 & $15428\pm 11$ & $10.911^{+0.491}_{-0.161}$ & $8.672\pm0.003$ \\
PTF10yux & 0.058 & $11777\pm 19$ & $10.764^{+0.006}_{-0.066}$ & \nodata \\
PTF10zai & 0.036 & $ 8715\pm 59$ & $10.822^{+0.563}_{-0.067}$ & $9.005\pm0.203$ \\
PTF10zak & 0.040 & $11915\pm 10$ & $10.757^{+0.400}_{-0.157}$ & $8.997\pm0.034$ \\
PTF10zbn & 0.114 & $ 9957\pm249$ & $11.189^{+0.094}_{-0.160}$ & \nodata \\
PTF10zgy & 0.044 & $11056\pm 40$ & $11.238^{+0.000}_{-0.000}$ & \nodata \\
PTF11apk & 0.041 & $10310\pm 10$ & $10.914^{+0.007}_{-0.164}$ & \nodata \\
PTF11bjk & 0.140 & $10613\pm 61$ & $10.954^{+0.064}_{-0.203}$ & \nodata \\
PTF11blu & 0.068 & $10495\pm 35$ & $ 9.963^{+0.016}_{-0.081}$ & $8.605\pm0.165$ \\
PTF11byi & 0.039 & $ 8625\pm 25$ & $10.527^{+0.077}_{-0.581}$ & $8.808\pm0.094$ \\
PTF11ctn & 0.079 & $10156\pm 11$ & $ 9.312^{+0.313}_{-0.100}$ & \nodata \\
PTF11cyv & 0.115 & $10398\pm 46$ & $ 9.851^{+0.125}_{-0.095}$ & \nodata \\
PTF11deg & 0.063 & $11724\pm 31$ & $ 9.452^{+0.097}_{-0.074}$ & \nodata \\
PTF11dws & 0.150 & $12637\pm 45$ & $11.174^{+0.095}_{-0.308}$ & \nodata \\
PTF11dzm & 0.041 & $11232\pm 59$ & $10.090^{+0.007}_{-0.163}$ & \nodata \\
PTF11eot & 0.090 & $11421\pm 50$ & $ 8.600^{+0.105}_{-0.135}$ & \nodata \\
PTF11fjw & 0.193 & $10684\pm 66$ & $10.892^{+0.216}_{-0.136}$ & \nodata \\
PTF11for & 0.128 & $12106\pm 83$ & $ 9.638^{+0.033}_{-0.083}$ & \nodata \\
PTF11gdh & 0.026 & $ 9838\pm 13$ & $10.390^{+0.190}_{-0.099}$ & $8.743\pm0.025$ \\
PTF11gin & 0.163 & $11347\pm 42$ & $ 9.626^{+0.255}_{-0.546}$ & \nodata \\
PTF11gjb & 0.125 & $10601\pm 42$ & $10.926^{+0.052}_{-0.131}$ & \nodata \\
PTF11glq & 0.156 & $ 9916\pm166$ & $11.360^{+0.088}_{-0.019}$ & \nodata \\
PTF11gnj & 0.131 & $10341\pm 61$ & $10.387^{+0.114}_{-0.032}$ & \nodata \\
PTF11khk & 0.031 & $ 9224\pm 11$ & $10.704^{+0.058}_{-0.120}$ & $8.926\pm0.034$ \\
PTF11kod & 0.188 & $10519\pm297$ & $ 8.741^{+0.912}_{-0.220}$ & \nodata \\
PTF11kpb & 0.143 & $10339\pm 19$ & $11.675^{+0.091}_{-0.014}$ & \nodata \\
PTF11kqm & 0.126 & $10878\pm 21$ & $10.428^{+0.241}_{-0.030}$ & \nodata \\
PTF11kx & 0.047 & $11027\pm 40$ & $10.315^{+0.169}_{-0.152}$ & $8.505\pm0.007$ \\
PTF11lbc & 0.150 & $11260\pm 44$ & $ 7.967^{+1.320}_{-0.848}$ & \nodata \\
PTF11lmz & 0.061 & $13674\pm 21$ & $11.206^{+0.074}_{-0.203}$ & \nodata \\
\hline
\label{sample2}
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{Summary of our sample in this work (continued).}
\begin{tabular}{lcccc}
\hline\hline
SN Name & Redshift & \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ & log\,$M_{\rm stellar} $ & $\rm12+log\,(O/H)$\\
& & (\,km\,s$^{-1}$) & (M$_{\odot}$) & \\
\hline
PTF11opu & 0.065 & $ 9791\pm 47$ & $ 9.797^{+0.127}_{-0.094}$ & $8.076\pm0.050$ \\
PTF11qpc & 0.089 & $10632\pm 23$ & $10.312^{+0.217}_{-0.239}$ & $8.643\pm0.123$ \\
PTF11qvh & 0.133 & $11042\pm 12$ & $ 9.800^{+0.171}_{-0.013}$ & \nodata \\
PTF11rke & 0.094 & $12348\pm 35$ & $ 9.968^{+0.066}_{-0.136}$ & \nodata \\
PTF11rpc & 0.143 & $11907\pm 49$ & $11.315^{+0.225}_{-0.183}$ & \nodata \\
PTF11yp & 0.121 & $ 8812\pm 41$ & $11.447^{+0.068}_{-0.313}$ & \nodata \\
PTF12awi & 0.045 & $ 9727\pm 48$ & $10.503^{+0.133}_{-0.108}$ & $8.764\pm0.052$ \\
PTF12cdb & 0.120 & $12068\pm 20$ & $10.877^{+0.212}_{-0.056}$ & \nodata \\
PTF12cjg & 0.067 & $11080\pm 17$ & $ 9.916^{+0.105}_{-0.131}$ & $8.615\pm0.142$ \\
PTF12cyd & 0.170 & $11668\pm 12$ & $ 9.118^{+0.446}_{-0.073}$ & \nodata \\
PTF12czu & 0.145 & $10709\pm151$ & $ 9.005^{+0.021}_{-0.138}$ & \nodata \\
PTF12dgy & 0.180 & $ 9911\pm 23$ & $10.919^{+0.367}_{-0.175}$ & \nodata \\
PTF12dhb & 0.057 & $10701\pm 12$ & $10.373^{+0.030}_{-0.072}$ & $8.618\pm0.032$ \\
PTF12dhl & 0.057 & $10441\pm 38$ & $11.001^{+0.111}_{-0.012}$ & \nodata \\
PTF12dhv & 0.140 & $12260\pm 80$ & $10.442^{+0.144}_{-0.160}$ & \nodata \\
PTF12dst & 0.192 & $ 9385\pm 24$ & $ 9.939^{+0.079}_{-0.019}$ & \nodata \\
PTF12dwm & 0.053 & $ 9675\pm 10$ & $10.538^{+0.159}_{-0.200}$ & $8.765\pm0.074$ \\
PTF12dxm & 0.063 & $10505\pm 9$ & $11.033^{+0.005}_{-0.005}$ & \nodata \\
PTF12egr & 0.132 & $10490\pm 46$ & $11.520^{+0.126}_{-0.013}$ & \nodata \\
PTF12fhn & 0.125 & $10801\pm 76$ & $11.281^{+0.142}_{-0.011}$ & \nodata \\
PTF12fsd & 0.069 & $10862\pm 25$ & $10.107^{+0.098}_{-0.117}$ & $8.398\pm0.063$ \\
PTF12giy & 0.029 & $12093\pm177$ & $10.835^{+0.177}_{-0.008}$ & $8.826\pm0.096$ \\
PTF12gnw & 0.100 & $10077\pm 14$ & $ 8.816^{+0.187}_{-0.234}$ & \nodata \\
SN~1989M & 0.005 & $12330\pm 50$ & $10.539^{+0.627}_{-0.144}$ & \nodata \\
SN~1994S & 0.015 & $10400\pm 50$ & $11.113^{+0.036}_{-0.520}$ & $8.943\pm0.013$ \\
SN~1997Y & 0.016 & $10510\pm 50$ & $11.018^{+0.067}_{-0.045}$ & $9.023\pm0.012$ \\
SN~1998dk & 0.013 & $12380\pm 50$ & $ 9.982^{+0.033}_{-0.186}$ & \nodata \\
SN~1998es & 0.011 & $10240\pm 50$ & $10.299^{+0.052}_{-0.010}$ & \nodata \\
SN~1999aa & 0.014 & $10350\pm 50$ & $10.533^{+0.132}_{-0.013}$ & $9.035\pm0.027$ \\
SN~1999ac & 0.009 & $10400\pm 50$ & $10.254^{+0.504}_{-0.128}$ & \nodata \\
SN~1999da & 0.013 & $10660\pm 50$ & $10.886^{+0.039}_{-0.120}$ & \nodata \\
SN~1999dq & 0.014 & $10860\pm 50$ & $10.941^{+0.422}_{-0.118}$ & \nodata \\
SN~1999gd & 0.018 & $10420\pm 50$ & $10.361^{+0.168}_{-0.044}$ & \nodata \\
SN~2000cp & 0.034 & $11000\pm 50$ & $ 9.976^{+0.459}_{-0.071}$ & $8.841\pm0.026$ \\
SN~2000dn & 0.032 & $10190\pm 50$ & $11.081^{+0.010}_{-0.177}$ & \nodata \\
SN~2001bp & 0.095 & $10860\pm 50$ & $10.602^{+0.065}_{-0.165}$ & $8.795\pm0.032$ \\
SN~2001da & 0.017 & $11350\pm 50$ & $10.633^{+0.440}_{-0.185}$ & \nodata \\
SN~2001ep & 0.013 & $10160\pm 50$ & $10.215^{+0.023}_{-0.117}$ & \nodata \\
SN~2001fe & 0.014 & $11070\pm 50$ & $10.341^{+0.109}_{-0.077}$ & \nodata \\
SN~2002aw & 0.026 & $10210\pm 50$ & $10.931^{+0.051}_{-0.076}$ & $8.976\pm0.029$ \\
SN~2002bf & 0.024 & $13680\pm 50$ & $10.672^{+0.004}_{-0.004}$ & \nodata \\
SN~2002bo & 0.004 & $13070\pm 50$ & $10.779^{+0.054}_{-0.449}$ & \nodata \\
SN~2002cd & 0.010 & $15230\pm 50$ & $10.971^{+0.124}_{-0.010}$ & \nodata \\
SN~2002dk & 0.019 & $10430\pm 50$ & $11.656^{+0.048}_{-0.151}$ & \nodata \\
SN~2002eb & 0.028 & $10230\pm 50$ & $10.976^{+0.164}_{-0.423}$ & \nodata \\
SN~2002eu & 0.038 & $11020\pm 50$ & $10.648^{+0.493}_{-0.070}$ & \nodata \\
SN~2002ha & 0.014 & $10930\pm 50$ & $11.075^{+0.481}_{-0.198}$ & \nodata \\
SN~2003U & 0.026 & $11300\pm 50$ & $ 9.670^{+0.614}_{-0.377}$ & $8.828\pm0.021$ \\
SN~2003Y & 0.017 & $ 9860\pm 50$ & $10.728^{+0.153}_{-0.037}$ & \nodata \\
SN~2003cq & 0.033 & $12080\pm 50$ & $11.884^{+0.064}_{-0.111}$ & \nodata \\
SN~2003he & 0.025 & $11310\pm 50$ & $10.335^{+0.545}_{-0.069}$ & \nodata \\
SN~2004dt & 0.020 & $13540\pm 50$ & $10.875^{+0.020}_{-0.144}$ & \nodata \\
SN~2004gs & 0.027 & $10430\pm 50$ & $10.740^{+0.137}_{-0.032}$ & \nodata \\
SN~2005M & 0.022 & $10670\pm 60$ & $ 9.898^{+0.081}_{-0.030}$ & $8.504\pm0.006$ \\
SN~2005W & 0.009 & $10600\pm 50$ & $10.473^{+0.007}_{-0.154}$ & \nodata \\
SN~2005ao & 0.038 & $11460\pm 50$ & $10.915^{+0.161}_{-0.033}$ & \nodata \\
SN~2005ag & 0.079 & $11370\pm 50$ & \nodata & $9.250\pm0.021$ \\
SN~2005bc & 0.012 & $10700\pm 50$ & $10.383^{+0.018}_{-0.120}$ & $9.054\pm0.007$ \\
\hline
\label{sample3}
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{Summary of our sample in this work (continued).}
\begin{tabular}{lcccc}
\hline\hline
SN Name & Redshift & \ensuremath{v_{\mathrm{Si}\,\textsc{ii}}}\ & log\,$M_{\rm stellar} $ & $\rm12+log\,(O/H)$\\
& & (\,km\,s$^{-1}$) & (M$_{\odot}$) & \\
\hline
SN~2005cg & 0.032 & $11560\pm 50$ & $ 8.574^{+0.057}_{-0.025}$ & $7.947\pm0.039$ \\
SN~2005er & 0.026 & $ 8740\pm 50$ & $11.478^{+0.492}_{-0.114}$ & \nodata \\
SN~2005eq & 0.029 & $10090\pm 50$ & $11.424^{+0.029}_{-0.452}$ & \nodata \\
SN~2005hj & 0.058 & $10550\pm 50$ & $10.046^{+0.078}_{-0.038}$ & $8.519\pm0.038$ \\
SN~2005ki & 0.020 & $11030\pm 50$ & $11.146^{+0.035}_{-0.120}$ & \nodata \\
SN~2005ms & 0.025 & $11840\pm 50$ & $10.684^{+0.210}_{-0.054}$ & \nodata \\
SN~2006N & 0.014 & $11300\pm 50$ & $10.545^{+0.494}_{-0.143}$ & \nodata \\
SN~2006S & 0.032 & $10710\pm 50$ & $10.628^{+0.006}_{-0.184}$ & \nodata \\
SN~2006bt & 0.032 & $10510\pm 50$ & $11.124^{+0.027}_{-0.155}$ & \nodata \\
SN~2006bz & 0.028 & $10850\pm 50$ & $10.471^{+0.006}_{-0.020}$ & \nodata \\
SN~2006cm & 0.016 & $11150\pm 50$ & $10.913^{+0.069}_{-0.047}$ & \nodata \\
SN~2006cq & 0.048 & $10160\pm 50$ & $11.212^{+0.011}_{-0.141}$ & \nodata \\
SN~2006cs & 0.024 & $10730\pm 50$ & $10.913^{+0.490}_{-0.024}$ & \nodata \\
SN~2006or & 0.021 & $11340\pm 50$ & $11.114^{+0.034}_{-0.117}$ & \nodata \\
SN~2006sr & 0.024 & $12470\pm 50$ & $10.778^{+0.325}_{-0.163}$ & \nodata \\
SN~2007A & 0.018 & $10600\pm 50$ & $10.775^{+0.252}_{-0.241}$ & \nodata \\
SN~2007N & 0.013 & $10330\pm 50$ & $10.376^{+0.017}_{-0.141}$ & \nodata \\
SN~2007O & 0.036 & $10000\pm 50$ & $11.033^{+0.007}_{-0.117}$ & $9.009\pm0.011$ \\
SN~2007af & 0.005 & $10560\pm 50$ & $ 9.562^{+0.181}_{-0.096}$ & \nodata \\
SN~2007ba & 0.038 & $ 9630\pm 50$ & $11.118^{+0.105}_{-0.068}$ & \nodata \\
SN~2007bc & 0.021 & $ 9850\pm 50$ & $10.922^{+0.020}_{-0.000}$ & \nodata \\
SN~2007bz & 0.022 & $11700\pm 50$ & $ 9.504^{+0.085}_{-0.100}$ & $8.329\pm0.012$ \\
SN~2007ci & 0.018 & $11830\pm 50$ & $10.969^{+0.007}_{-0.150}$ & \nodata \\
SN~2007fr & 0.051 & $10740\pm 50$ & $11.770^{+0.032}_{-0.102}$ & \nodata \\
SN~2007gi & 0.005 & $14870\pm 50$ & $10.643^{+0.479}_{-0.151}$ & \nodata \\
SN~2007hj & 0.014 & $11710\pm 50$ & $10.580^{+0.484}_{-0.143}$ & \nodata \\
SN~2007s1 & 0.027 & $11370\pm 50$ & $ 9.998^{+0.143}_{-0.030}$ & \nodata \\
SN~2008Z & 0.021 & $11460\pm 50$ & $ 9.488^{+0.203}_{-0.064}$ & $8.359\pm0.060$ \\
SN~2008ar & 0.026 & $10340\pm 50$ & $10.880^{+0.071}_{-0.512}$ & $8.972\pm0.016$ \\
SN~2008s1 & 0.022 & $10560\pm 50$ & $10.559^{+0.129}_{-0.040}$ & \nodata \\
SN~2008dx & 0.023 & $ 8850\pm 50$ & $10.678^{+0.027}_{-0.136}$ & \nodata \\
SN~2008ec & 0.016 & $10750\pm 50$ & $10.930^{+0.124}_{-0.075}$ & \nodata \\
SN~2009an & 0.009 & $12680\pm 50$ & $10.981^{+0.084}_{-0.000}$ & $9.086\pm0.014$ \\
SN~2009fx & 0.048 & $10110\pm 50$ & $10.000^{+0.020}_{-0.170}$ & \nodata \\
SN~2009ig & 0.009 & $13660\pm 50$ & $10.454^{+0.000}_{-0.175}$ & \nodata \\
SN~2009no & 0.046 & $10030\pm 50$ & $10.319^{+0.041}_{-0.187}$ & $8.776\pm0.028$ \\
SN~2010ex & 0.023 & $10890\pm 50$ & $10.089^{+0.142}_{-0.030}$ & \nodata \\
SN~2010ii & 0.027 & $12240\pm 50$ & $11.142^{+0.013}_{-0.144}$ & \nodata \\
SN~2010iw & 0.022 & $10360\pm 50$ & $ 9.731^{+0.066}_{-0.092}$ & \nodata \\
SN~2011ao & 0.011 & $10340\pm 50$ & $ 9.877^{+0.141}_{-0.032}$ & $8.481\pm0.010$ \\
SN~2011by & 0.003 & $10270\pm 50$ & $ 9.354^{+0.014}_{-0.191}$ & $8.624\pm0.022$ \\
SN~2011hb & 0.029 & $11800\pm 50$ & $11.247^{+0.004}_{-0.051}$ & \nodata \\
SN~2011ia & 0.017 & $10680\pm 50$ & \nodata & $8.865\pm0.010$ \\
SN~2012bh & 0.025 & $10360\pm 50$ & $10.538^{+0.406}_{-0.187}$ & \nodata \\
SN~2012cg & 0.001 & $10580\pm 50$ & $ 8.467^{+0.035}_{-0.160}$ & $8.729\pm0.007$ \\
SN~2012da & 0.018 & $11110\pm 50$ & $ 9.580^{+0.025}_{-0.061}$ & \nodata \\
SN~2013di & 0.024 & $11460\pm 50$ & $11.008^{+0.530}_{-0.060}$ & \nodata \\
\hline
\label{sample4}
\end{tabular}
\end{table*}
\section*{acknowledgments}
Y.-C.P. is supported by the East Asian Core Observatories Association (EACOA) Fellowship.
\bibliographystyle{apj}
|
2,877,628,089,547 | arxiv | \section{Introduction}
V2051 Ophiuchi was discovered by Sanduleak (1972). It is a dwarf nova,
a compact binary in which mass is fed to a white dwarf (the primary
star) by a Roche lobe filling late-type star (the secondary) via an
accretion disk. V2051 Oph shows recurrent but sparse outbursts in
which the disk expands and brightens by 2-3 magnitudes during 1-3 days
($B\simeq 13 \;mag$ at maximum, Bateson 1980; Warner \& Cropper 1983;
Warner \& O'Donoghue 1987) as a consequence of a sudden increase in
mass accretion through the disk. In the quiescent, low-mass accretion
state, the white dwarf and the bright spot (formed by the impact of
the infalling gas stream with the outer edge of the disk) dominate the
light from the binary at optical and ultraviolet wavelengths, and the
optical spectrum shows double-peaked H\,I and He\,II emission lines
which exhibit the classical rotational disturbance effect during
eclipse (Cook \& Brunt 1983; Watts et al. 1986).
The binary is seen at a high inclination angle ($i=83\degr$), which
leads to deep eclipses ($\Delta B \simeq$ 2.5 mag) in the light curve
every 1.5 hours when the white dwarf, accretion disk and bright spot
are occulted by the mass-donor star. This allows the emission from the
different light sources to be distinguished and spatially resolved
studies to be performed, making V2051 Oph an excellent laboratory for
the study of accretion physics. In particular, it is well suited for
the application of indirect imaging techniques to resolve the disk
emission both in position (eclipse mapping, Horne 1985) and velocity
(Doppler tomography, Marsh \& Horne 1988).
V2051 Oph displays large amplitude flickering activity (up to 30 per
cent of the total light in the optical), which is responsible for a
variety of eclipse morphologies (e.g., Warner \& Cropper 1983) and
usually hampers the measurement of white dwarf and bright spot eclipse
phases. Baptista et al. (1998) caught the star in an unusual faint
brightness state ($B\simeq 16.2 \;mag$) in which the mass accretion
rate and flickering activity were significantly reduced, allowing a
clean view of the white dwarf at disk center. They derived a mass
ratio of $q=0.19 \pm 0.03$ and an inclination of $i=83.\degr3 \pm
1.\degr4$.
Vrielmann et al. (2002) applied eclipse mapping techniques to
multicolor data of V2051 Oph to find that the disk is brighter in the
front side (the hemisphere nearer to the secondary) and interpreted
this behavior as caused by enhanced emission from the bright
spot. They also estimated a distance of $d=146 \pm 20\,pc$ for the
system.
Kiyota \& Kato (1998) observed V2051 Oph in superoutburst and detected
superhumps in the light curve of the object, confirming its
classification as an SU UMa star -- a sub-class of dwarf novae with
occasional superoutbursts $\sim 0.7$ mag brighter and $\sim 5$ times
longer than an ordinary outburst. However, due to the complex behavior
observed in V2051 Oph, Warner (1996) suggested a possible alternative
classification of it as a polaroid, an intermediate polar with a
synchronized primary.
\begin{figure*}
\includegraphics[bb=3cm 1cm 20cm 16cm,angle=-90,scale=0.65]{f1.eps}
\caption{Average out-of-eclipse UV (G160L; top ) and optical (G400H;
bottom) HST spectra of V2051~Oph. The phase ranges are +0.10 to +0.20
and +0.075 to +0.080 cycles, respectively, for the UV and optical
spectra. The observations correspond to the eclipse cycles 109,988
(optical) and 109,989 (UV) according to the linear ephemeris of
Baptista et al. (2003). Emission (vertical dotted lines) and
absorption (horizontal lines) features are labeled. Horizontal tick
marks indicate the passbands used to extract light curves (34 in the
UV and 68 in the optical).}
\label{spec_med}
\end{figure*}
Here we report the results of an optical and UV spectral mapping
experiment of V2051 Oph in its faint brightness state of 1996
(e.g. Baptista et al. 1998). The details of the data analysis are
given in Section 2. In Section 3 we investigate the disk spatial
structure in the lines and continuum, and we present spatially
resolved spectra of the disk, gas stream and uneclipsed light. The
results are discussed in Section 4 and summarized in Section 5.
\section{Data analysis}
\subsection{Observations}
The Faint Object Spectrograph (FOS) onboard the Hubble Space Telescope
(HST) was used to secure high-speed spectroscopy covering two
consecutive eclipses of V2051 Oph on 1996 January 29 while the star
was in an unusual low brightness state ($B\simeq 16.2$ mag). The
observations are summarized in Table 1. The first run covers a short
phase range around the egress of an eclipse in the optical (G400H
grating), and the second run covers the next eclipse in the UV (G160L
grating). The reader is referred to Baptista et al. (1998) for a
detailed description of the observations and data reduction
procedures.
\begin{table*}
\begin{center}
\caption{Journal of the Observations.\label{tbl-2}}
\vspace{12pt}
\begin{tabular}{c c c c c c}
\tableline\tableline
Run & Start & Eclipse & Number of & Spectral Range & Phase Range \\
& (UT) & Cycle & Exposures & \AA & (cycles) \\
\tableline
H1 & 19:03 & 109,988 & 113 & $3226-4781$ & $+0.01, +0.08$ \\
H2 & 20:24 & 109,989 & 693 & $1150-2507$ & $-0.09, +0.34$ \\
\tableline
\end{tabular}
\end{center}
\end{table*}
The average out-of-eclipse UV spectrum is shown in
Fig.\,\ref{spec_med} (top). It displays emission lines of Ly$\alpha$
$\lambda 1216$ (mostly geo coronal), N\,V $\lambda 1240,1243$, Si\,II
$\lambda 1300$, C\,II $\lambda 1336$, Si\,IV $\lambda 1394,1403$,
C\,IV $\lambda 1549,1551$, Si\,III $\lambda 1892$, C\,III $\lambda
2297$, as well as broad absorption bands reminiscent of those seen in
OY Car (Horne et al. 1994), which were interpreted as a blend of
Fe\,II lines. There is no evidence of He\,II $\lambda 1640$
emission. The position of the second-order $\rm Ly\alpha$ emission is
illustrated in Figure \ref{spec_med} (top). The bottom panel of Figure
\ref{spec_med} shows the average out-of-eclipse optical spectrum. It
displays the Balmer continuum in emission, as well as He\,I, He\,II
and strong Balmer emission lines. The optical emission lines show a
clear double-peaked profile. The blue peak of the lines is stronger
than the red peak because part of the disk side contributing
redshifted line emission is still occulted by the secondary star at
the phase range used to compute the average spectrum.
\subsection{Light-curve construction}
The UV spectra were divided into a set of 34 narrow passbands, with 22
continuum passbands 19-60\,\AA\ wide and 12 passbands for 10 lines
(Fig.\,\ref{spec_med}). The blue end of the spectra (shortward of the
$Ly\alpha$ line) was not included in this analysis because of the very
low count rate [and, consequently, low signal-to-noise ratio (S/N)] of
the data in this spectral region. The emission lines were sampled in a
single velocity bin 5000 or 6000$\,km\,s^{-1}$ wide (centered at the
rest wavelength, $v=0\,km\,s^{-1}$), except for the C\,IV line, which
was separated in a bin centered at the rest wavelength and
velocity-resolved bins for the blue and red wings of the line (all
bins with $\Delta v= 5000\,km\,s^{-1}$)\footnote{The choice of the
number (or width) of the passbands was constrained by the
signal-to-noise (S/N) ratio of the resulting light curves. Narrower
passbands lead to light curves too noisy to produce reliable eclipse
maps.}. The optical spectra were divided into a set of 68 passbands,
with 32 continuum passbands 15-42\,\AA\ wide and 36 passbands for 11
lines. The emission lines were resolved in velocity bins
$1000\,km\,s^{-1}$ wide, with one bin centered at the rest
wavelength. The systemic velocity of V2051~Oph ($\gamma\simeq
40\,km\,s^{-1}$; Watts et al. 1986) is both rather uncertain and much
smaller than the width of the passbands and was neglected. For those
passbands including emission lines the light curve comprises the total
flux at the corresponding bin, with no subtraction of a possible
continuum contribution.
Light curves were extracted for each passband by computing the average
flux on the corresponding wavelength range and phase-folding the
results according to the linear ephemeris of Baptista et al. (2003),
\begin{equation}
\rm T_{mid}(BJDD)=2\,443\,245.97752+0.062\,427\,8634~E,
\end{equation}
where $\rm T_{mid}$ gives the inferior conjunction of the white dwarf.
Light curves at two selected continuum passbands and for the C\,IV and
H$\gamma$ lines can be seen in Figs.\,\ref{mapas_uv} and
\ref{mapas_op}, respectively. The incomplete eclipse phase coverage of
the H1 run is clear (Fig.\,\ref{mapas_op}).
\subsection{Eclipse mapping}
Maximum-entropy eclipse mapping techniques (Horne 1985, Baptista \&
Steiner 1993) were used to solve for a map of the disk brightness
distribution and for the flux of an additional uneclipsed component in
each band. The reader is referred to Baptista (2001) for a recent
review on the eclipse mapping method.
As our eclipse map we adopted a flat grid of $51\times51$ pixels
centered on the primary star with side $2\,R_{L1}$, where $R_{L1}$ is
the distance from the disk center to the inner Lagrangian point. The
eclipse geometry is defined by the inclination $i$ and the mass ratio
$q$. The mass ratio $q$ defines the shape and the relative size of
the Roche lobes. The inclination $i$ determines the shape and
extension of the shadow of the secondary star as projected onto the
orbital plane. In this paper we adopted the values derived by Baptista
et al. (1998) , $\rm q=0.19$ and $\rm i=83\degr$, which correspond to
an eclipse phase width of $\Delta\phi= 0.0662$\,cycle. This
combination of parameters ensures that the white dwarf is at the
center of the map.
For the reconstructions we adopted the default of limited azimuthal
smearing of Rutten et al. (1992), which is better suited for
recovering asymmetric structures than the original default of full
azimuthal smearing (e.g. Baptista et al. 1996). We used a radial blur
width $\Delta r = 0.0157 R_{L1}$ and an azimuthal blur width $\Delta
\theta = 30 \degr$.
The statistical uncertainties in the eclipse maps were estimated with
a Monte Carlo procedure (e.g., Rutten et al. 1992). For a given
narrow band light curve, a set of 20 artificial light curves was
generated in which the data points were independently and randomly
varied according to a Gaussian distribution with standard deviation
equal to the uncertainty at that point. The light curves were fitted
with the eclipse-mapping code to produce a set of randomized eclipse
maps. These were combined to produce an average map and a map of the
residuals with respect to the average, which yields the statistical
uncertainty at each pixel. The uncertainties obtained with this
procedure were used to estimate the errors in the derived radial
brightness distributions, as well as in the spatially resolved
spectra.
The Appendix presents the results of eclipse mapping simulations
addressing the reliability of disk surface brightness reconstructions
from light curves of incomplete phase coverage, such as those of run
H1.
Light curves and respective eclipse maps at selected passbands for the
continuum and for the strongest emission lines are shown in
Figs.\,\ref{mapas_uv} and \ref{mapas_op}, respectively. Contour curves
in these Figures enclose the regions of the eclipse maps at the
3$\,\sigma$ and 5$\,\sigma$ levels of statistical significance.
\section{Results}
\subsection{Disk structure}
Maps of the disk surface brightness distributions, calculated by the
maximum entropy eclipse mapping method, allow us to obtain spatially
resolved information about the disk emission. In this section we
discuss the structures in the eclipse maps of the strongest lines in
the spectrum, as well as in selected continuum passbands.
\begin{figure*}
\centering
\includegraphics[bb=5cm 2cm 20cm 14cm,angle=-90,scale=.65]{f2.eps}
\caption{Data and model light curves and eclipse maps for the
C\,IV line center ($\Delta{v}=5000\,km\,s^{-1}$) and for two selected
continuum passbands in the UV. Left: Data (dots with error bars) and
model (solid lines) light curves. Dotted lines mark the contact phases
of the white dwarf eclipse. Right: Corresponding eclipse maps in a
logarithmic gray scale. Brighter regions are shown in black, fainter
regions in white. Contour curves enclose the eclipse map regions at
the 3$\,\sigma$ and 5$\,\sigma$ levels of statistical significance. A
cross marks the center of the disk; dotted lines show the Roche lobe,
the gas stream trajectory, and a disk of radius $0.56\;R_{L1}$; the
secondary is to the right of each map and the stars rotate
counter-clockwise. The vertical bars indicate the logarithmic
intensity level of the gray scale in each map.}
\label{mapas_uv}
\end{figure*}
Fig.\,\ref{mapas_uv} shows the light curves and respective eclipse
maps for the C\,IV line center and for two selected UV continuum
passbands. The S/N ratio of the light curves decreases towards the
blue end of the spectrum. Accordingly, the statistical significance of
the eclipse maps is lower in this spectral region. Thus, the brightest
parts of the $1444$\,\AA~continuum map are only significant at the
$3\,\sigma$ level, while the eclipse maps in the red side of the
spectrum are typically significant at the $5\,\sigma$ confidence
level.
The C\,IV line center light curve shows an inverse ``Gaussian''-shaped
eclipse with a minor asymmetry at egress and no sharp breaks in
slope. Accordingly, the eclipse map shows a broad, smooth, and fairly
symmetrical brightness distribution centered on the white dwarf, with
no clear sign of the bright spot.
In contrast to what is observed in the C\,IV line center, the
continuum light curves show sharp breaks in slope at the ingress and
egress phases of the white dwarf plus a conspicuous asymmetric egress
shoulder, indicating that there is significant additional emission
from the disk side containing the bright spot. The corresponding
eclipse maps display pronounced emission from the compact white dwarf
at disk center plus an asymmetric structure in the disk side that is
moving away from the secondary star (the upper hemisphere of the
eclipse maps in Fig.\,\ref{mapas_uv}) that can be associated with the
gas stream emission.
The presence of bright compact structures in the continuum maps and
their absence in the C\,IV line map, in addition to the observed
spectrum of the uneclipsed component (see Section 4.4), are
indications of the vertical extension and large optical depth of the
gas from which the C\,IV line originates. Our interpretation is that
the C\,IV line emission arises in a large gas region surrounding the
disk. This region needs to be vertically extended and optically thick
at this wavelength in order to veil the strong emission from the
underlying white dwarf and bright spot. This is in line with
expectations for an emission line produced by resonant scattering in a
disk chromosphere. A similar result was found for the nova-like
variable UX UMa (Baptista et al. 1995).
\begin{figure*}
\centering
\includegraphics[bb=5cm 9.5cm 20cm 14cm,angle=-90,scale=.65]{f3.eps}
\caption{Left: Data (dots with error bars) and corresponding
model (solid lines) light curves for the $H\gamma$ line center
passband ($v=0\,km\,s^{-1}, \Delta{v}=1000\,km\,s^{-1}$) and for its
blue and red line wings ($\Delta{v}=1000\,km\,s^{-1}$). Second from
left: Corresponding eclipse maps (total emission). Third from left:
Eclipse maps of the net line emission. Right: Eclipse maps of the
net line absorption in each case. The gray scale is the same in each
column. The notation is similar to that of Fig.\,\ref{mapas_uv}.}
\label{mapas_op}
\end{figure*}
The eclipse map at $2251$\,\AA~is representative of most of the
continuum maps. Aside from the asymmetry caused by the bright spot and
gas stream emission, it shows that another brightness asymmetry,
namely, the disk hemisphere closest to the secondary star (the
``front'' disk side) is systematically brighter than the disk side
farther away from the secondary star (hereafter called the ``back''
disk side). A similar effect was observed by Vrielmann et al. (2002),
who interpreted it as a consequence of azimuthally-smeared bright spot
emission. Our results will lead to a different interpretation (see
Section 4.1).
Fig.\,\ref{mapas_op} shows light curves and eclipse maps for the
$H\gamma$ line center ($v=0\,km\,s^{-1}$) and the blue
($v=-1000\,km\,s^{-1}$) and red ($v=+1000\,km\,s^{-1}$) line wing
passbands. The velocity-resolved $H\gamma$ line light curves show the
expected behavior for the eclipse of gas rotating in the prograde
sense (rotational disturbance), with the blue wing of the line
reappearing from eclipse earlier than the red wing. The phase of
maximum derivative in the three light curves coincides with the white
dwarf egress phase, indicating that the maxima of the distributions
are at the disk center. The light curves show deep eclipses with
smooth and long lasting egresses. As a consequence, the resulting
maps show fairly broad brightness distributions centered at the white
dwarf position with an asymmetric bright source in the quadrant
containing the bright spot. Simulations show that, in spite of the
incomplete phase coverage of the optical light curves, it is possible
to derive a fairly good reconstruction of the position and intensity
of the white dwarf and bright spot in the corresponding eclipse maps
(see the Appendix).
Fig.\,\ref{mapas_op} (third and fourth columns) shows the net line
emission and the net line absorption in each case. Net line
emission/absorption maps are obtained by combining continuum eclipse
maps on the short- and long-wavelength sides of the target emission
line and by subtracting the derived average continuum map from each of
the velocity-resolved line maps. Positive intensities in the resulting
maps (third column, black and gray) signal the regions where the line
appears in emission, while negative intensities (fourth column, black
and gray) trace the regions where the line is in absorption. The net
emission/absorption maps reveal that the Balmer lines are in emission
over most of the accretion disk, but appear in absorption at the disk
center and also in the region of the bright spot at the disk rim.
\subsection{Spatially resolved spectra}
\begin{figure}
\centering
\includegraphics[bb=1cm 1cm 20cm 21cm,angle=-90,scale=.4]{f4.eps}
\caption{Schematic diagram showing the regions defined as
``front'', ``back'' and ``gas stream''. Dashed lines mark the annular
regions of width $0.1\,R_{L1}$ used to extract spatially resolved
spectra. Dotted lines show the projection of the primary Roche lobe
onto the orbital plane and the gas stream trajectory. Azimuths are
measured with respect to the line joining both stars and increase
counter-clockwise. Four reference azimuths are labeled.}
\label{mapa_esq}
\end{figure}
\begin{figure*}
\centering
\includegraphics[bb=6cm 12cm 20cm 14cm,angle=-90,scale=.74]{f5.eps}
\caption{UV (left) and optical (right) spatially resolved disk
spectra for the azimuthal sections defined as front $(270^{o}-90^{o})$
(gray) and back $(90^{o}-270^{o};$ black lines) for three selected
annular regions (labeled in units of $R_{L1}$). Vertical dotted lines
mark the major transitions of the spectra.}
\label{spec_disc}
\end{figure*}
Our set of monochromatic eclipse maps allows us to separate the
spectra emitted by different parts of the accretion disk. Motivated by
the distinct emission observed in the gas stream region and by the
systematic difference in the emission from the opposite disk
hemispheres, we sliced the disk into three distinct regions:
``front'', ``back'' and ``gas stream''. Azimuths are measured from the
line joining both stars and increase counter-clockwise. We define
``front'' as the disk section between $270^{o}$ and $90^{o}$, and
``back'' as the region between azimuths $90^{o}$ and $270^{o}$. The
region defined as ``gas stream'' is depicted in Fig.\,\ref{mapa_esq}
(gray). In order to separate the disk spectra at different distances
from the disk center, we further divided each of the disk regions in
concentric annular sections of width $\Delta{R}=0.1R_{L1}$ .
In order to minimize the possible contributions from the bright spot
and gas stream to the disk ``front'' and ``back'' spectra we
calculated the symmetric disk-emission component in each
hemisphere. The symmetric component is obtained by slicing the disk
into a set of radial bins and by fitting a smooth spline function to
the resulting set of medians of the lower quartile of the intensities
in each bin. The spline fitted intensity in each annular section is
taken as the symmetric component. This procedure essentially preserves
the baseline of the radial profile, removing all azimuthal
structure. The statistical uncertainties affecting the fitted
intensities are estimated with the Monte Carlo procedure described in
Section 2.2.
The spatially resolved disk (front and back) spectra are shown in
Fig.\,\ref{spec_disc}. The inner disk shows a flat continuum which
becomes progressively fainter and redder with increasing radius,
indicating the existence of a radial temperature gradient. The lines
and the Balmer jump are in emission at all disk radii. The Balmer
decrement becomes flatter and the lines are more prominent and
narrower with increasing radii. The UV spectrum of the ``front'' side
is perceptibly different from that of the ``back'' side. The
comparison of the UV ``front'' and ``back'' spectra shows that the
latter contains broad absorption bands, possibly due to Fe\,II, that
become more conspicuous in the outer parts of the disk -- suggesting
that it arises from absorption along the line of sight by a vertically
extended region (e.g., Horne et al. 1994).
\begin{figure*}
\centering
\includegraphics[bb=4cm 12cm 19cm 14cm,angle=-90,scale=.74]{f6.eps}
\caption{Ratio of the gas stream to the front disk spectra (see
Fig.\,\ref{mapa_esq}) as a function of radius. The notation is similar
to that of Fig.\,\ref{spec_disc}.}
\label{spec_spot}
\end{figure*}
In order to investigate the emission along the gas stream trajectory,
we calculate the ratio of the spectrum of the gas stream to the front
disk symmetric component at the same radius
(Fig.\,\ref{spec_spot}). The result reveals that the spectrum of the
gas stream is typically 1 order of magnitude brighter than the disk
spectrum in the intermediate and mainly in the outer disk regions
($0.3 \lesssim R/R_{L1} \lesssim 0.7$) and this difference is
significant at the $3\,\sigma$ level. This result suggests the
occurrence of gas stream overflow. This is in agreement with the
results of Baptista \& Bortoletto (2004), who found clear evidence of
gas stream overflow in B band eclipse maps of V2051 Oph in quiescence.
In contrast to what is observed in the disk spectra, in which the
lines appear in emission at all radii, in the ratio of the gas stream
to the disk spectrum the C\,IV, Si\,III $\lambda 1892$ and Balmer
lines appear in absorption. This suggests the presence of matter
outside of the orbital plane (e.g., chromosphere + wind) or that the
disk has a non-negligible opening angle (flared disk). Horne et
al. (1994) observed strong C\,IV emission out of the orbital plane in
the dwarf nova OY Car in quiescence and suggested that this emission
may arise from magnetic activity on the surface of the quiescent disk
(Horne \& Saar 1991) or from resonant scattering in a
vertically-extended region well above the disk.
The slope of the continuum, the strength of the Balmer jump and the
line intensities give a wealth of information about the physical
conditions in the emitting gas. It is clear that simple blackbody and
even stellar atmosphere models are not adequate to describe the strong
line emission spectra of the V2051 Oph accretion disk. A more
quantitative analysis of the spatially-resolved spectra demands
optically thin disk models. This is beyond the scope of the present
work and will be the subject of a forthcoming paper (A. Zabot et
al. 2006, in preparation).
\subsection{The emission lines}
In this section we analyze the radial behavior of selected emission
lines. Fig.\,\ref{ew_rad} shows the radial intensity distributions for
the lines and the adjacent continuum (top), the radial distributions
of the net line emission (second from top), the radial run of the
equivalent width (EW; third from top), and the radial run of the full
width at half-maximum (FWHM; bottom) for the H$\delta$, H$\gamma$,
He\,I 4471 and He\,II 4686 lines. The diagrams were computed for the
disk ``back'' region to avoid contamination by gas stream emission
and, as in Section 3.2, we used the symmetric disk-emission component
in this analysis. We remind the reader that our line eclipse maps
comprise the line emission plus any subjacent continuum
contribution. The line distributions were obtained from the average of
all eclipse maps along the line, while the continuum distributions
were obtained from the average of the nearest continuum maps on both
sides of each line. The net line emission distributions were computed
by the subtraction of the adjacent continuum from the corresponding
line distributions.
\begin{figure*}
\centering
\includegraphics[bb=6cm 10cm 19cm 14cm,angle=-90,scale=.7]{f7.eps}
\caption{Top: Radial intensity distributions of selected optical
lines (filled symbols) and of the corresponding adjacent continuum
(open symbols). Dashed lines indicate the 1$\,\sigma$ limit in each
case. Dotted lines mark the outer disk radius of $R=0.56R_{L1}$
(Baptista et al. 1998). Second from the top: Net line emission radial
distributions. Dotted lines show the radial dependency for a slope $I
\propto R^{-1.78}$. Third from top: EW as a function of
radius. Bottom: FWHM of the lines as a function of radius. Dashed
lines show the law $v \propto R^{-1/2}$ expected for a gas rotating in
Keplerian orbits.}
\label{ew_rad}
\end{figure*}
As expected, the intensity in the line maps is stronger than in the
adjacent continuum (indicating that the lines appear in emission),
except in the innermost disk regions, where the intensity of the
continuum reaches that of the lines. For H$\delta$, He\,I 4471 and
He\,II 4686, the intensity in the line is equal or smaller than that
of the adjacent continuum in the innermost annulus (indicating that
the lines are in absorption in the white dwarf at the disk center) and
were not plotted in the corresponding net line emission panels. The
slope of the continuum distributions is steeper than that of the
lines. The net line distributions decrease in strength with increasing
disk radius. The lines are in emission at all disk radii with a radial
dependency $I \propto R^{-1.78\pm 0.06}$. This is steeper than the
empirical $I \propto R^{-1.5}$ law derived by Marsh et al. (1990)
assuming a Keplerian distribution of velocities for the line emitting
gas in the dwarf nova U Geminorum.
As a consequence of the radial behavior of the line and continuum
intensity distributions, the lines are relatively weaker in the inner
disk regions and their EW increases with increasing disk
radius. H$\delta$ and H$\gamma$ are the dominant lines with an EW
$\simeq 300$\,\AA\ at the disk edge ($\simeq 0.5\,R_{L1}$), while
He\,I 4471 and He\,II 4686 have EW $\simeq 100$\,\AA\ at the same
radius. In the innermost disk region ($R <0.1\,R_{L1}$) the EW becomes
negligible (or negative) because the continuum intensity reaches (or
exceeds) the line intensities. Because of this, the FWHM in this
region is also negligible and is not plotted in the corresponding FWHM
panels.
H$\delta$ and He\,II 4686 show FWHM values comparable to those
expected for a gas rotating in Keplerian orbits around an M$_{1}=0.78$
M$_{\sun}$ white dwarf, but the slope of the radial distribution is
flatter than the $v \propto R^{-1/2}$ law. Because of the large
uncertainties in the FWHM values, the distributions are still
consistent with the Keplerian expectation. On the other hand, the
H$\gamma$ and He\,I 4471 distributions are clearly different from the
Keplerian expectation. Both lines show sub-Keplerian velocities (at
the $2-3\,\sigma$ confidence level). While the slope of the H$\gamma$
distribution is consistent with the $v \propto R^{-1/2}$ law, He\,I
4471 shows a flat distribution, with velocities of 1/2 FWHM $\simeq
600-700\,km\,s^{-1}$ at all radii.
\subsection{The uneclipsed component}
The uneclipsed component was introduced in the eclipse mapping method
to account for the fraction of the total light that is not coming from
the accretion disk plane (e.g. light from the secondary star or from a
vertically-extended disk wind; Rutten et al. 1992).
Fig.\,\ref{neclip} shows the spectrum of the uneclipsed component, as
well as its fractional contribution, as a function of wavelength. We
estimated the fractional contribution of the uneclipsed component to
the total flux by computing the ratio of the uneclipsed light to the
average out of eclipse level at the corresponding passband.
\begin{figure*}
\centering
\includegraphics[bb=3cm 12cm 18cm 14cm,angle=-90,scale=.6]{f8.eps}
\caption{Spectrum of the uneclipsed component. Top: total
contribution. Bottom: fractional contribution. The notation is
similar to that of Fig.\,\ref{spec_disc}.}
\label{neclip}
\end{figure*}
The uneclipsed component is dominated by the Balmer continuum
emission, strong Balmer and He emission lines in the optical, and
resonant emission lines in the UV, suggesting that this light arises
in a hot, vertically extended, optically thin gas region. $Ly\alpha$
is the strongest line in the uneclipsed spectrum ($\simeq 30$ per cent
of the total light at that wavelength). Nevertheless, most of this
contribution is of geocoronal origin and not intrinsic to V2051
Oph. The remaining strong lines in the optical and UV spectra, as well
as the Balmer continuum, account for $\simeq 10$ per cent of the total
light at the respective wavelength. The other continuum regions in the
uneclipsed spectrum yield negligible fractional contributions to the
total light. This indicates that the continuum emission comes mostly
from the orbital plane while part ($\simeq 10$ per cent) of the line
emission arises in a vertically extended region above the disk.
\section{Discussion}
\subsection{Evidences for chromospheric emission}
Our continuum eclipse maps, particularly in the UV
(Fig.\,\ref{mapas_uv}), show a front-back brightness asymmetry. This
becomes clear when we compare the spectrum of the front and back disk
sides (Fig.\,\ref{spec_disc}): the front side of the disk is brighter
than the back side. A similar result was found by Vrielmann et
al. (2002). They discussed a few possibilities (non-negligible disk
opening angle, warped disk, and bulge caused by impact of gas stream)
but found no compelling explanation for the effect.
Our front and back disk spectra were computed from a symmetric disk
component (see Section 3.2), which traces the baseline of the radial
intensity distribution and removes the contribution of asymmetric
brightness sources. Furthermore, the spectrum of the gas stream region
is markedly different from the disk spectrum at intermediate and large
radii ($R \gtrsim 0.3\,R_{L1}$). We are, therefore, confident that the
observed front-back asymmetry is not an artifact caused by azimuthal
smearing of bright spot emission, but it is a real effect.
Vrielmann et al. (2002) discussed the possibility that the observed
asymmetry could be caused by a non-negligible opening angle ($\alpha$)
in the accretion disk, but they discarded this hypothesis because it
would lead to an enhancement of the emission from the back side (seen
at the lower effective inclination $i_{eff}=i - \alpha$) in comparison
to the front disk side (seen at a higher effective inclination
$i_{eff}=i + \alpha$). This argument is only correct if the disk
atmosphere shows a stellar-type temperature decreasing with vertical
height. If the disk has a chromosphere, in which the temperature
actually increases with vertical height, the effect will be the
opposite. Because the front disk side is seen at higher angles, the
emerging photons will sample the uppermost (and hotter) chromospheric
layers and the resulting spectrum will be brighter. The back disk side
is seen through a lower inclination, allowing the emerging photons to
come from deeper (and cooler) layers and leading to fainter
intensities. Preliminary fits of the spatially resolved spectrum with
LTE atmosphere models lead to effective temperatures for the front
disk spectrum that are systematically higher than the temperatures of
the back disk spectrum at the same radius (A. Zabot et al. 2006, in
preparation), confirming the above scenario.
Therefore, the front-back asymmetry observed in the accretion disk of
V2051 Oph can be interpreted as evidence of chromospheric emission
from a disk with a non-negligible (but probably small) opening
angle. This chromosphere is the site of the emission lines and is
probably also responsible for the veiling of the continuum emission
from the compact underlying white dwarf and bright spot. It may also
account for the Fe\,II absorption features that become more pronounced
with increasing radius in the disk back side, because the photons
arising from these regions travel an increasingly larger path across
the vertically-extended disk chromosphere before leaving the binary
and reaching the observer.
\subsection{The white dwarf spectrum and the distance}
Here we fit white dwarf atmosphere models to the white dwarf spectrum
in order to estimate the temperature of the primary star and the
distance to the object. We adopted a white dwarf radius of
$R_{WD}=0.0103\,R_{\odot}$ and a Roche lobe size of
$R_{L1}=0.422\,R_{\odot}$ (Baptista et al. 1998). Therefore, our
eclipse maps (see Section 2.3) have a scale factor of
$0.0392\,R_{L1}\,pixel^{-1}$. Since the primary comprises a diameter
of $0.0488\,R_{L1}$, the central pixel of the eclipse map is fully
contained in the white dwarf surface. We thus obtain a good estimate
of the white dwarf spectrum by extracting the flux of the central
pixel of the eclipse maps at each wavelength. These fluxes are
multiplied by a factor
\begin{equation}
\rm \frac{A_{WD}}{A_{pix}}=\left(
\frac{0.0244R_{L1}}{0.0392R_{L1}}\right)^{2} \frac {\pi}{cos \theta}
\hspace{2pt},
\end{equation}
to scale the spectrum to the effective area of the white dwarf. The
resulting spectrum shows a continuum filled with broad and shallow
absorption lines plus a Balmer jump in absorption, resembling that of
a DA white dwarf (Fig.\,\ref{primary}).
We employed a grid of DA white dwarf spectra with $6500\,K < T_{WD} <
20000\,K$ and $log~ g=8$ (D. Koester 2000, private communication) in
order to fit the V2051 Oph white dwarf spectrum. Since the DA white
dwarf models only account for the continuum level and the H\,I lines,
we masked the other line regions for the fitting procedure. This
includes removing the spectral region above 2300\,\AA~in the UV
spectrum and below 3400\,\AA~in the optical spectrum, to avoid
contamination by absorption bands due to Fe\,II. We considered two
possibilities: (i) the inner disk is opaque and the visible part of
the white dwarf surface is the projected area of the upper half
hemisphere above the opaque disk, (ii) the white dwarf surface is
fully visible through an optically thin disk.
Our best fit leads to a temperature of $T_{WD} =
9500\,^{+2900}_{-1900}\,K$ for the white dwarf with a distance of $d =
67\,^{+22}_{-25}\,pc$ if the inner disk region is opaque and $d =
92\,^{+30}_{-35}\,pc$ if the inner disk is optically thin. Both the
fitted temperature and the resulting distance are significantly
different from previously reported values.
Catal\'an et al. (1998) made a preliminary fit to the G160L extracted
white dwarf spectrum (assuming a white dwarf atmosphere model plus an
intervening cool gas layer to account for the Fe\,II absorption bands)
to find a white dwarf temperature of $15000\,K$. Their solution
provides a reasonable fit in the UV but underestimates the
white dwarf flux in the optical by a factor of 5. The white dwarf must
be cooler (and, therefore, the distance must be smaller) to match the
slope of the combined UV-optical spectrum. On the other hand,
Vrielmann et al. (2002) fitted white dwarf atmosphere models to UBVRI
photometric measurements to find $T_{WD} = 19600\,K$ (if only the
upper half of the white dwarf is visible) or $T_{WD} = 15000\,K$ (if
the white dwarf is fully visible) and $d = 146\,pc$. However, their
result largely overestimates the white dwarf contribution in the
UV. For comparison, Fig.\,\ref{primary} shows DA white dwarf
models for the temperatures and respective distances obtained by the
works mentioned above.
\begin{figure*}
\centering
\includegraphics[bb=3cm 3cm 19cm 23cm,,angle=-90,scale=.68]{f9.eps}
\caption{Spectrum of the V2051 Oph white dwarf (histogram) and DA
white dwarf model fits. The dot-dashed line corresponds to a
model with $T=19600 K, d=146\,pc$ (Vrielmann et al. 2002), and
the dashed line correspond to a model with $T=15000 K, d=184\,pc$
(Catal\'an et al. 1998). The best fit model is displayed as a
solid line. The bottom panel shows the S/N of the the white dwarf
spectrum.}
\label{primary}
\end{figure*}
As a first step in trying to understand the difference, we investigate
the possibility that some systematic error in the eclipse-mapping
analysis (e.g. wavelength dependent blurring of bright central
sources) is affecting the slope of the extracted white dwarf
spectrum. To test this hypothesis, we extract the white dwarf spectrum
by measuring the height of the jump in flux at white dwarf egress in
the set of narrow band UV and optical light curves used for eclipse
mapping. This provides an estimate of the white dwarf flux as seen
during egress and is quite similar to the procedure used by Catal\'an
et al. (1998) to extract the UV white dwarf spectrum. The resulting
spectrum is consistent with that of Fig.\,\ref{primary} under the
uncertainties. We are therefore confident that the red slope of the
white dwarf spectrum of V2051 Oph at the time of the HST observations
is real. Thus, the main point is that, it is not possible to model the
observed white dwarf spectrum with a hot ($T_{WD} \gtrsim 14000\,K$)
DA atmosphere model.
The origin of the discrepancy between our result and the previous
analyses may be in the restricted spectral coverage of the data used
in those works. Catal\'an et al. (1998) used only the UV part of the
spectrum ($1150 - 2500\,\AA$) to fit the white dwarf temperature. It
is possible to match the observed UV intensities with a DA spectrum of
$15000\,K$ for a distance of $d = 184\,pc$ (Vrielmann et al. 2002) at
the expense of a large mismatch in the optical.
The slope of a DA white dwarf spectrum is very similar in the optical
($3500 - 5000\,\AA$) for a large range of temperatures. Therefore, the
optical spectrum alone does not provide a strong constraint to the
white dwarf temperature. We tested this by fitting white dwarf models
only to the optical part of the spectrum. This exercise shows that it
is possible to fit temperatures in the range $6500 - 16500\,K$ with a
$\chi ^{2}$ only $20\%$ larger than the $\chi ^{2}$ of the best-fit
model in this case, $T_{WD} = 11500\,K$. This lack of sensitivity
affects the white dwarf temperature determination of Vrielmann et
al. (2002), since they used only optical photometric data in their
analysis. Moreover, they mistakenly took the value
$R_{WD}=0.0244\,R_{\odot}$ as the white dwarf radius, while the value
quoted by Baptista et al. (1998) is $R_{WD}=0.0103\,R_{\odot}$. To
compensate for the larger radius assumed for the white dwarf, their
fit had to increase the distance by a similar amount (a factor of
$\sim 2.3$).
Steeghs et al. (2001) used optical spectroscopy $(4000 - 6800\,\AA)$
to find that a $(15000\,K)$ blackbody was a good match to the slope of
their extracted white dwarf spectrum. Their result faces the same
problem discussed above; the slope of the optical continuum is very
similar for a large range of temperatures.
It is important to notice that at the time of our observations V2051
Oph was in an unusual low brightness state, in which mass accretion
may have been considerably reduced (Baptista et al. 1998). With
reduced (or absent) accretional heating, it is possible for the white
dwarf to cool down. We therefore expect that our fitted white dwarf
temperature is lower than that during a normal, quiescent state.
Our larger spectral coverage, combining optical and UV data, allows a
better determination of the slope of the white dwarf spectrum, and,
therefore, its temperature and distance. Our results show that the
distance to V2051 Oph is smaller than previously found.
\section{Conclusions}
The main results of our spectroscopic study of V2051 Oph in a faint
brightness state during 1996 can be summarized as follows:
\begin{enumerate}
\item The presence of white dwarf and bright-spot strong emission in
the continuum maps and their absence in the line maps, coupled with
the significant extra absorption in the spectra of the back disk side,
are indications of the vertical extension and large optical depth of
the gas from which the lines originate.
\item Distinct emission along the stream trajectory suggests the
occurrence of gas stream overflow.
\item Spatially resolved spectra show that the lines are in emission
at all disk radii. The Balmer decrement becomes shallower with
increasing radius.
\item The FWHM of the emission lines differs from that expected for a
gas in Keplerian rotation and the line intensities drop with a radial
dependency of $I \propto R^{-1.78\pm 0.06}$.
\item The spectrum of the uneclipsed light is dominated by strong
emission lines and a Balmer jump in emission, indicating origin in a
hot, vertically extended, optically thin gas region above the
disk. The strongest uneclipsed lines contribute $\simeq 10$ per cent
of the total flux.
\item Strong absorption bands, possibly due to Fe II, are seen in the
spectra of the back disk side, suggesting it arises from absorption
by an extended gas region above the disk.
\item The front disk spectrum is systematically brighter than the back
disk spectrum at the same radius. This can be explained in terms of
chromospheric emission (higher temperatures at the uppermost
atmospheric layers) from an accretion disk with a non-negligible
opening angle (limb brightening effect).
\item We fit stellar atmosphere models to the extracted white dwarf
spectrum to find a temperature $T_{WD} =
9500\,^{+2900}_{-1900}\,K$ and a distance of $d =
67\,^{+22}_{-25}\,pc$ (if the inner disk is opaque) or $d =
92\,^{+30}_{-35}\,pc$ (if the inner disk is optically thin).
\end{enumerate}
\acknowledgments
{\it Acknowledgments.} We thank the anonymous referee for useful
comments and suggestions. The white dwarf atmosphere models used in
this work were kindly provided by Detlev Koester. This work was
partially supported by CNPq/Brazil through the research grant
62.0053/01-1 -- PADCT III/Milenio. RB acknowledges financial support
from CNPq/Brazil through grants n. 300.354/96-7 and
301.442/2004-5. RKS acknowledges financial support from CAPES/Brazil
and CNPq/Brazil.
|
2,877,628,089,548 | arxiv | \section{Introduction}
Core-collapse supernovae (SNe) are luminous explosions that mark the end of a massive star's life.
These SNe are differentiated from their thermonuclear counterparts (Type Ia SNe) and
are grouped into classes via spectral and photometric analyses \citep[e.g.,][]{1997ARA&A..35..309F}. The major
core-collapse subclasses include SNe~IIP and IIL, events which show strong hydrogen throughout their evolution
and respectively do or do not exhibit a hydrogen-recombination plateau in their light curves
\citep[though the distinction between SNe IIP and IIL may be less clear than previously thought; e.g.,][]{2012ApJ...756L..30A,2014ApJ...786...67A,2015ApJ...799..208S}.
SNe Ib and Ic (often called stripped-envelope SNe) are core-collapse events that show no hydrogen in their spectra and
(for SNe~Ic and broad-lined SNe~Ic) no helium; they (along with the intermediate SNe~IIb, which exhibit very little hydrogen) are commonly
understood to arise from progenitor stars that have lost all or most of their hydrogen (and perhaps helium) envelopes prior to
core collapse, though the detailed connections between progenitors and SN observables remain somewhat uncertain
\citep[e.g.,][]{2001AJ....121.1648M,2011ApJ...741...97D,2014ApJS..213...19B,2014AJ....147...99M,2015arXiv151008049L,2015MNRAS.453.2189D}.
A subset of SNe reveal signatures of a dense shroud of circumstellar material (CSM) surrounding their progenitors at the time of explosion.
For hydrogen-rich events, relatively narrow lines (full width at half-maximum intensity [FWHM] $\lesssim 1000$\,km\,s$^{-1}$)
from this interaction with the CSM are often detected; these objects have been dubbed SNe~IIn
\citep[e.g.,][]{1990MNRAS.244..269S,1991ESOC...37..343F}.
This interaction can provide a significant luminosity boost \citep[as has been observed for superluminous SNe~IIn; e.g.,][]{2012Sci...337..927G}.
CSM interaction occurs in some hydrogen-poor SNe as well.
SNe~Ia-CSM are thermonuclear events that exhibit relatively narrow features (usually H$\alpha$ emission)
caused by interaction with hydrogen-rich CSM \citep[though the underlying SN ejecta are hydrogen-poor; e.g.,][]{2013ApJ...772..125S,2013ApJS..207....3S}.
A small number of stripped-envelope SNe have been found to exhibit the spectral signatures of CSM interaction. SNe~Ibn
\citep[e.g.,][]{2007ApJ...657L.105F,2008MNRAS.389..113P} show relatively narrow helium lines in their spectra
but no hydrogen, and they can be quite heterogeneous in their photometric evolution \citep[e.g.,][]{2016MNRAS.456..853P},
while the remarkable SN~2014C appeared to be a normal SN~Ib at peak brightness but then began interacting with hydrogen-rich CSM only a few
months later \citep{2015ApJ...815..120M}.
In addition to the examples of interaction with dense CSM described above, indications of very short-lived interaction with
CSM have been discovered through ``flash spectroscopy'' of very young SNe of various types
\citep[e.g.,][]{2014Natur.509..471G,2015ApJ...806..213S,2016ApJ...818....3K}.
These examples exist on a continuum of CSM densities with the strongly interacting events described above; they require much less CSM
and their observables at peak brightness generally align with those of ``normal'' events.
Here we present the results of our observational campaign to study
SN~2015U, a remarkable and very well-monitored SN~Ibn.
It was discovered in NGC~2388 by the Lick Observatory Supernova Search (LOSS)
with the 0.76\,m Katzman Automatic Imaging Telescope \citep[KAIT;][]{2001ASPC..246..121F}
on 2015 Feb.\ 11 (all dates and times reported herein are UTC).
Note that, because the official International Astronomical Union name was not assigned until
November 2015 \citep{2015CBET.4164....1},
this event has also been discussed in the literature under the name PSN~J07285387+3349106.
\citet{2015ATel.7105....1O} classified it as a young SN~Ibn based upon
spectra obtained on Feb.\ 18, which showed a blue continuum and relatively narrow \ion{He}{1} emission
features \citep[see also][]{2015CBET.4164....1}.
\citet{2015IBVS.6140....1T} present {\it BVRI} photometry of SN~2015U\ starting Feb.\ 17,
showing that it has one of the fastest decline rates known
(similar to those of SNe~2002bj, 2005ek, and 2010X) and is remarkably luminous (though
SN~2015U\ is significantly obscured by an uncertain amount of dust in the host galaxy).
\citet{2015MNRAS.454.4293P} present additional photometry and low-resolution spectra of this event
and describe SN~2015U\ within the context of SNe~Ibn
\citep[e.g.,][]{2007ApJ...657L.105F,2008MNRAS.389..113P,2016MNRAS.456..853P}.
In this paper we present photometric, spectroscopic, and spectropolarimetric
observations of SN~2015U, including one epoch of relatively high-resolution Keck DEIMOS
spectroscopy ($R = \lambda/\delta \lambda \approx 5000$), enabling us to study the narrow-line features in detail.
We show that SN~2015U\ is similar to several other SNe from the heterogeneous SN~Ibn class,
and that it shares many features with the rapid and luminous transients discovered in the Pan-STARRS1 (PS1) archives \citep{2014ApJ...794...23D},
those found in the SuperNova Legacy Survey (SNLS) archives \citep{2016ApJ...819...35A}, and
a few other rapidly fading SNe from the literature. SN~2015U\ offers valuable insights into the physics of the poorly observed class of rapidly fading SNe.
\section{Observations}
\label{sec:obs}
\subsection{Photometry}
\label{sec:phot}
\begin{figure*}
\includegraphics[width=.48\textwidth]{finder_pre-hst.pdf}
\includegraphics[width=.48\textwidth]{finder_nickel.pdf} \\
\includegraphics[width=.48\textwidth]{finder_hst.pdf}
\includegraphics[width=.48\textwidth]{finder_hst_zoom.pdf}
\caption{Top left: a pre-detection {\it HST} NICMOS image of NGC~2388 through the {\it F110W} filter.
Top right: a detection of SN~2015U\ by the Lick Nickel 1\,m telescope through the $V$-band filter.
Bottom left: an {\it HST} WFC3 image of SN~2015U's location through the {\it F555W} filter at $\sim1$\,yr post-explosion.
Bottom right: a zoom-in of SN~2015U's location in the {\it HST} {\it F555W} filter.
All images show the location of the SN with orange arrows, and the size of our 3$\sigma$ position error is marked with a circle. \label{fig:finder}}
\end{figure*}
SN~2015U\ was first detected by KAIT on Feb.\ 11.24 at $18.06 \pm 0.15$\,mag in an unfiltered
image.\footnote{The KAIT {\it clear} passband is similar to the $R$ band but broader. For more details and transformations to standard passbands, see \citet{1999AJ....118.2675R} and \citet{2003PASP..115..844L}.}
An unfiltered image taken the night before (Feb.\ 10.30) shows no source to a limit
of $\sim18.4$\,mag.
We began acquiring multiband photometry ({\it BVRI} and {\it clear}) starting
Feb.\ 14 with KAIT and the 1\,m Nickel telescope at Lick Observatory.
We used a set of 50 Nickel and KAIT images with strong detections of SN~2015U\ and with
good astrometric solutions for the field to calculate an updated position:
$\alpha = 07^h28^m53.90^s$, $\delta = +33^{\degree}49'10.56''$ (J2000), offset from the centre
of the galaxy by $\sim6''$. We believe this position to
be accurate within $0.15''$ or better --- the positions we measured for the SN exhibit
a scatter of $0.09''$ in both right ascension and declination across 50 images.
\citet{2010ApJS..190..418G} describe our photometric observing program at Lick in detail,
along with our KAIT and Nickel image-reduction pipeline. Point-spread-function (PSF) photometry
was performed using DAOPHOT \citep{1987PASP...99..191S} from the IDL Astronomy
User's Library.\footnote{\url{http://idlastro.gsfc.nasa.gov/}}
Instrumental magnitudes were calibrated to several nearby
stars from the Sloan Digital Sky Survey, transformed into the
Landolt system using the empirical prescription
presented by Robert Lupton.\footnote{\url{http://www.sdss.org/dr7/algorithms/sdssUBVRITransform.html}}
We measure SN~2015U's date of peak of brightness in each passband of our Nickel+KAIT photometry by fitting
a low-order polynomial to the light curve within the first week (first 10 days for the $I$ band)
and use Monte Carlo Markov Chain (MCMC) methods to estimate our uncertainties. In Modified Julian Day (MJD), we
find $t^{\rm max}_{V} = 57071.1 \pm 0.2$, $t^{\rm max}_{R} = 57071.6 \pm 0.5$, and
$t^{\rm max}_{I} = 57072.4 \pm 0.2$ (our data do not constrain the $B$-band peak well).
Throughout our analysis we present phases relative to the $V$-band peak.
NGC~2388 is at a redshift of $z_{\rm host} = 0.013790 \pm 0.000017$ \citep[NED;][]{1991rc3..book.....D},
which (assuming cosmological parameters H$_0 = 71$\,km\,s$^{-1}$\,Mpc$^{-1}$, $\Omega_m = 0.27$,
and $\Omega_{\Lambda} = 0.73$) translates into a luminosity distance of 58.9\,Mpc \citep{2006PASP..118.1711W}
and a distance modulus of $33.85$\,mag that we adopt for all absolute-magnitude corrections.
NGC~2388 and the SN site (pre-explosion) were imaged on 2004 Sep. 10 with
the {\it Hubble Space Telescope (HST)} and the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS)
with the NIC2 aperture (scale $0{\farcs}076$ pixel$^{-1}$) in bands {\it F110W}, {\it F160W}, {\it F187N}, and {\it F190N}.
We determined the SN position in each band's mosaic data products based on the absolute SN position and the world coordinate system of the image ---
no stellar object was detected at this position in any of the mosaics. We quantified these nondetections in the {\it F110W} and {\it F160W} mosaics
using DAOPHOT by assuming a PSF
constructed from the brightest isolated star in the mosaics and inserting an artificial star at the SN position.
The artificial star was measured with ALLSTAR within DAOPHOT using photometric calibrations established from the
online cookbook\footnote{\url{www.stsci.edu/hst/nicmos/performance/photometry/cookbook.html}; corrections to infinite aperture and zeropoints at zero magnitude were obtained from \url{www.stsci.edu/hst/nicmos/performance/photometry/postncs_keywords.html}.} and with parameters appropriate for NIC2,
and then reduced in luminosity until it was detected at a signal-to-noise ratio (S/N) of $\sim 3$.
The corresponding upper limits are $> 26.0$ and $> 25.2$\,mag in {\it F110W} and {\it F160W}, respectively.
Under our Cycle 23 Snapshot program with {\it HST}'s Wide Field Camera 3 (WFC3; GO-14149, PI Filippenko),
we obtained images of the SN location on 2016 Feb.\,14 (1\,yr post explosion)
through the {\it F555W} (710\,s) and {\it F814W} (780\,s) filters (scale $0{\farcs}04$ pixel$^{-1}$).
Figure~\ref{fig:finder} shows a Nickel detection of SN~2015U\ alongside the pre-explosion NICMOS {\it F110W} image
and the 1\,yr WFC3 {\it F555W} image of the SN location.
We find that SN~2015U\ exploded on the trailing edge of NGC~2388's spiral arm, in a region with several
probable dust lanes and a generally clumpy appearance. One of those clumps falls within the
3$\sigma$ position error circle in both the {\it F555W} and {\it F814W} images and is not detected
in the pre-explosion NICMOS images. However, this clump appears to be
extended and smoothly connected with other bright regions, not point-like, and we therefore
attribute it to the host galaxy and not to SN~2015U.
Table~\ref{tab:phot} presents our photometry of SN~2015U\ before applying any dust reddening corrections,
and Figure~\ref{fig:phot} shows the light curves after correcting for Milky Way (MW) dust absorption.
The data are also available for download from
the Berkeley SuperNova DataBase \citep[SNDB;][]{2012MNRAS.425.1789S}.\footnote{\url{http://heracles.astro.berkeley.edu/sndb/}}
\begin{figure}
\includegraphics[width=\columnwidth]{phot.pdf}
\caption{The {\it BVRI} and {\it clear} passband light curves of SN~2015U\
after correcting for MW dust absorption but not for the
host absorption.
KAIT data are indicated by circles and Nickel data by diamonds.
The dates of our spectral observations are marked by dotted vertical lines.
The phase relative to the $V$-band peak
is shown along the top while the absolute magnitude is shown on
the right. Because these absolute magnitudes have not been corrected for
significant but uncertain dust absorption arising within the host galaxy, they can be taken as
lower limits on the luminosity of SN~2015U.
\label{fig:phot}}
\end{figure}
\input{phot_table}
\subsection{Spectroscopy}
\label{sec:spec_data}
We began a spectral monitoring campaign of SN~2015U\ on 2015 Feb. 15, obtaining six spectra with the
Kast double spectrograph \citep{kast} on the Shane 3\,m telescope at Lick Observatory
and one spectrum with the DEIMOS spectrograph \citep{2003SPIE.4841.1657F} on the Keck-II 10\,m telescope.
Details of our spectral observations are listed in Table~\ref{tab:spec},
and all reductions and calibrations were performed with standard tools and methods, including IRAF routines and custom Python and IDL
codes\footnote{\url{https://github.com/ishivvers/TheKastShiv}} \citep[e.g.,][]{2000AJ....120.1487M,PER-GRA:2007,2012MNRAS.425.1789S}.
All spectra were taken at or near the parallactic angle \citep{1982PASP...94..715F},
and the amount of galaxy flux falling into the slit varied as a function of seeing and the
slit orientation on the sky.
For the spectra which included a large amount of host-galaxy flux (Feb.\ 18, 21, and 24),
we extract a spectrum from a region of the galaxy that does not include any SN~2015U\ flux,
perform a median-filter smoothing to obtain the galaxy continuum, and then subtract it from our spectra,
after determining a best-fit galaxy scaling coefficient by comparing synthetic photometry to the
multiband photometry observed the same night. We do not attempt to remove the narrow (unresolved) galaxy emission features,
so our spectra show varying degrees of narrow-line contamination from the host (including H$\alpha$, H$\beta$, and [\ion{S}{2}]).
\begin{figure*}
\includegraphics[width=\textwidth]{spec.pdf}
\caption{The spectral sequence of SN~2015U.
This figure shows the spectral evolution after dereddening to
correct for MW dust absorption but not for the host-galaxy absorption.
On the bottom, we show an extraction of NGC~2388's nucleus, taken from a
Kast spectrum obtained on Feb.\ 15.
Contamination from the host galaxy's strong emission lines is apparent in some spectra;
we indicate H$\alpha$, H$\beta$, and [\ion{S}{2}]\,$\lambda\lambda6716$, 6731, as well as
the host's \ion{Na}{1}\,D absorption.
\label{fig:spec}}
\end{figure*}
We renormalise each spectrum to match our template-subtracted $V$ or $R$-band photometry
(depending on the spectral wavelength coverage) after linearly interpolating the photometry to the time of
spectral observation. Figure~\ref{fig:spec} shows our total-flux spectra of SN~2015U.
All spectra are available to download from the Berkeley SNDB
and the Weizmann Interactive Supernova Data REPository
\citep[WiseREP;][]{2012PASP..124..668Y}.\footnote{\url{http://wiserep.weizmann.ac.il/}}
Using narrow H$\alpha$ host-galaxy emission in our DEIMOS spectrum, we measure a line-of-sight redshift
at the SN's location: $z_{\rm SN} = 0.013161 \pm 0.000005$, a difference of $\sim200$\,km\,s$^{-1}$\ from
the published redshift of the host galaxy --- consistent
with the SN's position on the approaching arm of this spiral galaxy.
\subsection{Spectropolarimetry}
\label{sec:specpol}
Three epochs of spectropolarimetry were obtained of SN~2015U\ near peak brightness utilising
the dual-beam polarimetry mode of the Lick 3\,m Kast spectrograph. The orientation of the slit on the sky was always set to a position angle of
$180^{\circ}$ (i.e., aligned north-south), and exposures of 900\,s were obtained at each of four waveplate positions
($0\fdg0$, $45\fdg0$, $22\fdg5$, and $67\fdg5$). On each night, several waveplate sequences were performed and coadded.
Flatfield and arc-lamp spectra were obtained immediately after each sequence without moving the telescope.
For polarimetric calibrations, the low-polarization standard stars BD+32$^{\circ}$3739 and BD+05$^{\circ}$2618 were observed
to verify the low instrumental polarization of the Kast spectrograph. We constrained the average fractional Stokes $Q$ and $U$
values to $<0.1$\%. By observing the above unpolarized standard stars through a 100\% polarizing filter we determined that
the polarimetric response is so close to 100\% that no correction was necessary. Finally, we obtained the instrumental polarization
position-angle curve and used it to correct the data.
We observed the high-polarization stars HD\,19820, BD\,+59$^{\circ}$389,
and V1Cyg12 to obtain the zeropoint of the polarization position angle on the sky ($\theta$) and to determine the accuracy of
polarimetric measurements, which were generally consistent with previously published values within $\Delta P<0.05$\% and
$\Delta \theta<1^{\circ}$. All spectropolarimetric reductions and calculations were performed using the methods described
by \citet[and references therein]{2015MNRAS.453.4467M}, and we define the polarimetric parameters in the same manner as those authors
(Stokes parameters $q$ and $u$, debiased polarization $P$, and sky position angle $\theta$).
To probe the Galactic component of interstellar polarization (ISP) along the line of sight toward SN~2015U, we used the
method of spectroscopic parallax to select three stars within $1^\circ$ of separation from the source and measured
their integrated $V$-band polarization ($P$) and position angle ($\theta$). For
HD\,58221 we obtained $P = 0.28\%\pm$0.01\%, $\theta = 13.0^{\circ}\pm0.6^{\circ}$; for
HD\,58726, $P = 0.50\%\pm$0.01\%, $\theta = 24.8^{\circ}\pm0.3^{\circ}$;
and for HD\,59291, $P = 0.24\%\pm$0.01\%, $\theta = 16.3^{\circ}\pm0.7^{\circ}$ (uncertainties are statistical).
We interpret the significantly higher value of $P$ for HD\,58726 (A0~V spectral type)
as being caused by some intrinsic polarization for that star, so we used the average values of the other two stars (0.25\%, 14.3$^{\circ}$) to
calculate the associated Serkowski-Whittet form \citep{1975ApJ...196..261S,1992ApJ...386..562W}, assuming a total-to-selective extinction ratio of
$R_V=3.1$ and that the polarization peaks at 5500\,{\AA}, values appropriate for the MW. The resulting curve was subtracted
from the SN~2015U\ data to remove the small but nonnegligible Galactic component of ISP.
The above observations (along with calibration observations of the low-polarization standard star HD\,14069 and the high-polarization
standards HD\,245310 and HD\,25443) were taken with the Kast spectrograph on 2015 Sep.\ 21.
\begin{table*}
\caption{Journal of Spectroscopic Observations}
\label{tab:spec}
\begin{minipage}{0.9\linewidth}
\centering
\begin{tabular}{ l | c c c c c c c }
\hline
UT Date & Type\footnote{F: total flux; S: spectropolarimetry.} & Tel./Instr. & Wavelength & Resolution &
Exp. & Observer\footnote{\label{obsnote}Observers and data reducers are indicated with their initials. IS: Isaac Shivvers; WZ: WeiKang Zheng; JM: Jon Mauerhan; MG: Melissa Graham; PK: Patrick Kelly; IK: Io Kleiser; JS: Jeffrey Silveman.} &
Reducer\footref{obsnote} \\
& & & (\AA) & (\AA) & (s) & & \\
\hline
2015-02-15.256 & F & Shane/Kast & 3440--10,870 & 2 & 1200 & IS & IS \\
2015-02-18.211 & F & Shane/Kast & 4400--9880 & 2 & 2400 & WZ & IS \\
2015-02-21.207 & F & Shane/Kast & 3500--10,500 & 2 & 1200 & WZ & IS \\
2015-02-21.313 & S & Shane/Kast & 4500--10,000 & 16 & $4 \times 4 \times 900$\footnote{All spectropolarimetry was obtained by rotating multiple times through four waveplate positions; see \S\ref{sec:specpol}.} & WZ & JM \\
2015-02-24.224 & F & Shane/Kast & 3500--10,500 & 2 & 1200 & WZ & IS \\
2015-02-24.322 & S & Shane/Kast & 4500--10,000 & 16 & $3 \times 4 \times 900^c$ & WZ & JM \\
2015-02-24.414 & F & Shane/Kast & 3500--7875 & 1 & 1800 & WZ & IS \\
2015-02-25.171 & F & Shane/Kast & 3500--10,500 & 2 & 1200 & MG & MG \\
2015-02-25.301 & S & Shane/Kast & 4500--10,000 & 16 & $5 \times 4 \times 900^c$ & MG & JM \\
2015-02-27.34 & F & Keck/DEIMOS & 4850--7510 & 0.3 & 1200 & PK & IS \\
2010-03-022.17\footnote{A previously unpublished observation of SN~2010al; see Figure~\ref{fig:spec_comparisons}.} & F & Shane/Kast & 3450--10,790 & 3 & 2100 & IK & JS \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Analysis}
\label{sec:analysis}
\subsection{Dust Corrections}
\label{sec:dust}
All observations of SN~2015U\ are heavily affected by an obscuring screen of dust
present in the host galaxy, NGC~2388. Because the treatment of
dust corrections affects much of the following discussion, we start
by describing our efforts to understand, characterise, and correct for the effects
of the host-galaxy dust. First, however, we correct for MW dust along the line of sight using the dust maps of
\citet[$E(B-V) = 0.0498$\,mag;][]{2011ApJ...737..103S} and assuming $R_V=3.1$ \citep{1989ApJ...345..245C}.
\citet{2015ATel.7105....1O} note that a significant amount of host-galaxy reddening
is apparent in their classification spectrum, with an estimated $E(B-V) = 1.0$\,mag
based upon the \ion{Na}{1}\,D feature.
Our higher-resolution DEIMOS spectrum from Feb.\ 27 reveals complex
\ion{Na}{1}\,D absorption near the host redshift (we examine the structure of this feature in \S\ref{sec:csm}).
We measure the equivalent width (EW) of the entire absorption complex to be
$9.1 \pm 0.15$\,\AA\ --- well outside the empirical relations of \citet{2012MNRAS.426.1465P} and similar
previous efforts, which show that the \ion{Na}{1}\,D lines saturate and lose their predictive power above a total EW
of $\sim2$\,\AA\ and $E(B-V) \approx 1.0$\,mag. This indicates that the reddening toward SN~2015U\
likely exhibits $E(B-V) > 1.0$\,mag, if the dust in NGC~2388 is similar to that in our MW.
\citet{2015MNRAS.454.4293P} measure an \ion{Na}{1}\,D EW of $6.5 \pm 0.5$\,\AA\ from their spectra; we tentatively suggest
that the discrepancy between their value and ours is caused by different choices made when defining the local continuum.
Figure~\ref{fig:HeI} shows that the \ion{Na}{1}\,D components fall on the red wing of the \ion{He}{1} emission line, even at early
times. However, if one instead defines the local continuum based upon the overall flux-density level (a reasonable choice if the emission-line wings
were not detected), one obtains \ion{Na}{1}\,D EW values similar to those of \citet{2015MNRAS.454.4293P}.
We next examined the 5780.5\,\AA\ diffuse interstellar band (DIB). This feature is one of the strongest
DIBs; it has long been known to correlate with extinction in the MW when the \ion{Na}{1}\,D
feature is saturated
\citep[e.g.,][]{1995ARA&A..33...19H,2011ApJ...727...33F,2015MNRAS.452.3629L,2015MNRAS.447..545B}, and \citet{2013ApJ...779...38P}
show a clear correlation between this feature and the reddening toward
SNe~Ia produced by host-galaxy dust. We do not detect this feature, and our spectra have insufficient
S/N to place strong constraints on the dust using our nondetection.
We also examined modeled estimates for the amount of dust obscuring the bulk of stars in the host
galaxy NGC~2388 \citep{2015A&A...577A..78P}, and we determined the same value at the location of the SN
(within $\sim1''$ on the sky) by measuring the Balmer ratio implied by the H$\alpha$
and H$\beta$ galaxy lines present in our DEIMOS spectrum \citep[e.g.,][]{1971MNRAS.153..471B}.
We estimate $E(B-V) \approx 0.56$ and $1.1$\,mag, respectively, toward the average star in NGC~2388's core and
the average star near the explosion site of SN~2015U.
Although this does not provide a measure of the line-of-sight reddening toward SN~2015U\ specifically,
it does provide valuable context about the dust content of NGC~2388.
Additional information is available to us through the spectropolarimetry: as we describe in \S\ref{sec:specpolAnalysis},
the peak of the polarization spectrum ($\lambda_{\rm max}$) of NGC~2388's ISM is blueward of 4600\,\AA. The shapes of the polarization law and the
dust-extinction law are related within the MW \citep[e.g.,][]{1975ApJ...196..261S,1988ApJ...327..911C}.
Using $R_V = (-0.29 \pm 0.74) + (6.67 \pm 1.17)\,\lambda_{\rm max}$ from \citet[with $\lambda_{\rm max}$ in $\rm{\mu}$m]{1988ApJ...327..911C},
our measurement of $\lambda_{\rm max} < 0.46\,\rm{\mu}$m implies $R_V < 2.8 \pm 0.9$.
However, note that \citet{2015A&A...577A..53P} show the polarization properties of the host-galaxy dust
obscuring several well-studied and reddened SNe~Ia to be remarkably different from those of the dust in our MW.
\begin{figure}
\includegraphics[width=\columnwidth]{rv_vs_ebv.pdf}
\caption{In the top panel, we fit reddened blackbodies to our photometry of SN~2015U\ at three epochs
(blue, Feb.\ 14.35; green, Feb.\ 18.21; red, Feb.\ 27.21) after correcting for MW absorption
and assuming various values for $R_V$ and $E(B-V)$ within NGC~2388.
For many data points, the measurement errors are smaller than the plotted symbol.
The weighted sum of squared differences ($\Sigma\sigma^2$) between observed photometric points and synthetic photometry of a blackbody
is shown in the upper right of each plot; lower values indicate a better agreement across all epochs.
In the bottom panel, we show $\Sigma\sigma^2$ as a function of $E(B-V)$ for $R_V = 2.1$ in black as well as $\Sigma\sigma^2$ broken down
by individual epochs, coloured as above.
$R_V = 2.1$ and $E(B-V) = 0.94^{+0.1}_{-0.4}$\,mag (marked with a dashed line in the bottom panel) are the preferred values.
In many panels a blue feature is introduced into the
reddened blackbody spectrum by the wavelength dependency of the MW's reddening law, but our spectral data provide no clear evidence for or against
such a feature in the spectra of SN~2015U.
\label{fig:bbody_phot}}
\end{figure}
Finally, we model SN~2015U\ as a blackbody emitter obscured behind a simple screen of dust in the host galaxy and fit for the parameters of that dust
via comparisons with our multiband photometry at three epochs (pre-maximum, maximum, and post-maximum brightness).
Our spectra show that the emission from SN~2015U\ is roughly that of a blackbody obscured by dust, at least at optical wavelengths
(unlike most SNe, which display spectra consisting of prominent and overlapping features with sometimes strong line blanketing),
and in \S\ref{sec:temp} we discuss why one would expect roughly blackbody emission from SN~2015U.
We use PySynphot\footnote{\url{http://ssb.stsci.edu/pysynphot/docs/}} and
assume the dust is described by the empirical extinction law of \citet{1989ApJ...345..245C} --- by varying the parameters
$R_V$ and $E(B-V)$ we are able to explore
a wide variety of dust populations and search for the parameters that produce the best blackbody fits to our photometric data.
Figure~\ref{fig:bbody_phot} shows a comparison between synthetic photometry from PySynphot and the photometry
assuming four different values for $E(B-V)$ ranging from 0.0\,mag to 1.0\,mag and three different reddening laws parameterised
by $R_V$ from 2.1 to 4.0 (note that PySynphot only includes a few built-in options for $R_V$; we explore the most relevant of them here).
For every combination of $R_V$ and $E(B-V)$, we perform a weighted least-squares fit to determine the best-fit temperature and luminosity of the
source blackbody and we list the weighted sum of squared differences ($\Sigma\sigma^2$) for each.
Figure~\ref{fig:bbody_phot} shows that a nonzero amount of host-galaxy reddening correction makes the fits worse for $R_V = 3.1$ and 4.0 ---
in those cases the photometric data prefer a blackbody with no dust.
However, the spectra clearly show a nonthermal rolloff at blue wavelengths (see Figure~\ref{fig:spec}), and so
there must be a significant amount of dust absorption within NGC~2388.
If we instead adopt $R_V = 2.1$, a grid search over $E(B-V)$ values indicates a best fit at $E(B-V) \approx 0.94$\,mag.
As Figure~\ref{fig:spec_correct} shows, dereddening the spectra for this reddening law appears to correct the blue rolloff.
Figure~\ref{fig:bbody_phot} shows that $\Sigma\sigma^2$ increases above $E(B-V) = 1.0$ and below $E(B-V) = 0.6$\,mag, but
between those values $\Sigma\sigma^2$ is only weakly dependent upon $E(B-V)$.
We therefore use Figure~\ref{fig:bbody_phot} to estimate our error bars: $E(B-V) = 0.94^{+0.1}_{-0.4}$\,mag.
A similar approach was taken by \citet{2015MNRAS.454.4293P}, who compared the colour curves of SN~2015U\
to the intrinsic colour curves of other SNe~Ibn to obtain $E(B-V)_{\rm tot} = 0.81 \pm 0.21$\,mag (assuming
$R_V = 3.1$). However, as they note, the SN~Ibn subclass is remarkably heterogeneous, and it is not clear
whether the physics governing those colour curves is the same for all members.
Given the above discussion, we adopt $R_V = 2.1$ and $E(B-V) = 0.94^{+0.1}_{-0.4}$\,mag.
The large EWs measured from the \ion{Na}{1}\,D and the odd absorption-feature complex indicate that a simple analysis
of those features cannot be trusted. Studies of the integrated galaxy flux and the Balmer
decrement measured from our spectrum show that the bulk of NGC~2388's stellar mass is strongly obscured,
and our spectropolarimetric data indicate that SN~2015U\ itself exploded behind a significant dust screen.
The low $\lambda_{\rm max}$ value in the polarization spectrum and our blackbody
fits to the light curves together prefer a relatively wavelength-dependent dust extinction law for NGC~2388 (i.e.,\ $R_V < 3.1$) and
$E(B-V) = 0.94^{+0.1}_{-0.4}$\,mag --- we find this line of reasoning most convincing.
The above result is consistent with the $E(B-V) = 0.99 \pm 0.48$\,mag adopted by \citet{2015MNRAS.454.4293P}, though they assume $R_V = 3.1$.
Our determination of the dust properties toward SN~2015U\ remains uncertain, so we consider a range of plausible reddening corrections
throughout the rest of this paper. We find that the extinction toward \S\ref{sec:csm} does not appear to change over the course of our observations,
and we discuss and reject the possibility that a significant fraction of the extinction toward SN~2015U\
arises from the SN's CSM rather than the host galaxy's ISM.
\subsection{Spectra}
\label{sec:spectra_analysis}
\begin{figure*}
\includegraphics[width=\textwidth]{spec_host_corrected.pdf}
\caption{The spectral sequence of SN~2015U\ after dereddening to
correct for MW dust absorption and the (uncertain) host absorption. Phases are
listed relative to the $V$-band peak brightness.
We have masked regions of our low-resolution spectra where the host's narrow
emission lines dominate, but we do not mask the \ion{Na}{1}\,D absorption features from the MW
and the host.
\label{fig:spec_correct}}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.45\textwidth]{He_I_NaD_evolution.pdf}
\includegraphics[width=0.45\textwidth]{He_I_evolution.pdf}
\caption{ The evolution of the \ion{He}{1}\,$\lambda$5876 (left) and $\lambda$7065 (right) lines. The spectra increase in time downward on the plot
as in Figure~\ref{fig:spec}. The rest wavelength of each line is shown with a dotted line and the absorption component is marked
at a blueshift of $v = 789$\,km\,s$^{-1}$\ (the mean blueshift of \ion{He}{1} lines measured from our DEIMOS spectrum).
In the left panel we also mark the host and MW \ion{Na}{1}\,D absorption features
(see \S\ref{sec:csm} for a discussion of the host galaxy's remarkable \ion{Na}{1}\,D absorption complex).
\label{fig:HeI} }
\end{figure*}
The optical spectra of SN~2015U\ show a strong continuum overlain with narrow and intermediate-width emission features.
Though our data cover only 12 days of SN~2015U's evolution, they range from a pre-maximum spectrum to
one taken after the SN had faded from peak by $\sim1$\,mag (see Figure~\ref{fig:phot}).
No broad ejecta features became apparent in our spectra throughout that timespan
--- markedly different behaviour from that of the well-observed, rapidly fading
SNe 2002bj, 2005ek, and 2010X \citep{2010Sci...327...58P,2013ApJ...774...58D,2010ApJ...723L..98K}
and similar to that of the interacting SN~2010jl \citep[e.g.,][]{2014ApJ...797..118F}.
Though broad features do not emerge, the continuum temperature shows significant evolution, the
narrow emission and absorption features evolve, and the relative ionisation states of the detected elements change.
While there are no strong H lines,
multiple ionisation levels of helium, nitrogen, iron, and perhaps oxygen and carbon are detected, some in both emission and absorption and some in absorption only.
\citet{2015MNRAS.454.4293P} also present a series of spectral observations of SN~2015U. Their data are in good
agreement with ours and extend to later epochs, while our (higher-resolution) spectra illustrate new details not apparent previously.
\subsubsection{Evolving Line Profiles}
\label{sec:lineprofiles}
The most obvious features in our spectra are emission lines of \ion{He}{1}. These lines
are centred at the rest wavelength, exhibit a relatively broad base
with a full width near zero intensity (FWZI) of $\sim9000$--10,000\,km\,s$^{-1}$, and are overlain with P-Cygni
absorption features blueshifted by $\sim745$\,km\,s$^{-1}$~and with widths of 400--500\,km\,s$^{-1}$.
The wavelengths and widths of these features do not significantly change during the $\sim12$ days covered by our spectra.
We find that the \ion{He}{1} line profiles can be parameterised by a ``modified Lorentzian'' emission component (where the exponent is allowed to vary)
with a suppressed blue wing and an overlain Gaussian absorption component, indicative of recombination lines broadened
via electron scattering within the CSM. SN~2015U's line profiles are very similar to those observed in many SNe~IIn but with a faster inferred CSM velocity
\citep[see, e.g., the spectra of SN~1998S;][]{2000ApJ...536..239L,2001A&A...367..506A,2001MNRAS.325..907F,2001MNRAS.326.1448C,2015ApJ...806..213S}.
Figure~\ref{fig:HeI} shows our best fit to the \ion{He}{1}\,$\lambda$5876 line using this parameterisation.
We also show the unsuppressed Lorentzian feature to emphasise the asymmetry, which arises from the effects of Compton
scattering within a wind-like CSM \citep[e.g.,][]{1972ApJ...178..175A,1991A&A...247..455H,2001MNRAS.326.1448C}.
The magnitude of the effect scales with the expansion velocity, explaining the strongly asymmetric emission lines of SN~2015U\ compared to
those observed in most SNe~IIn.
After \ion{He}{1} (and host-galaxy H$\alpha$ and [\ion{N}{2}]), the most clearly detected features in our spectra are the \ion{N}{2} lines.
As \citet{2015MNRAS.454.4293P} note,
SN~2015U\ appears to be the first SN~Ibn to show these features, though they are commonly
found in the spectra of hot stars. In our DEIMOS spectrum we are able to identify nine \ion{N}{2} absorption lines
(see \S\ref{sec:hires}), and for the strongest of these we track their
evolution throughout our spectral series. In our $-2.9$ and +0.1\,d spectra,
the 5667\,\AA\ and 5676\,\AA\ features are apparent in emission
around $v \approx 0$\,km\,s$^{-1}$, but they transition into absorption at $v \approx 680$\,km\,s$^{-1}$\ after maximum light --- see Figure~\ref{fig:NaII}.
The same trend is apparent in the \ion{N}{2} features at 6482\,\AA\ and 7215\,\AA\ (Figure~\ref{fig:spec}).
\begin{figure}
\includegraphics[width=\columnwidth]{NII_evolution.pdf}
\caption{The evolution of the \ion{N}{2} lines. The spectra increase in time downward
as in Figure~\ref{fig:spec}. Identified \ion{N}{2} lines are marked at rest using dotted lines and with
a blueshift of $v = 680$\,km\,s$^{-1}$\ using dashed lines (the mean blueshift of \ion{N}{2} lines measured from our DEIMOS spectrum).
The emission features marginally detected at $v = 0$\,km\,s$^{-1}$~in our earliest spectra evolve into blueshifted
absorption lines post-maximum brightness.
\label{fig:NaII} }
\end{figure}
We find a strong emission feature centred around 4650\,\AA\ in
our earliest spectrum. In Figure~\ref{fig:spec} we label it as a blend of \ion{N}{3}, \ion{C}{3}, and \ion{He}{2}, a blend that
has been observed and modeled in the spectra of a few other young, interacting SNe \citep[e.g.,][]{2014Natur.509..471G,2015ApJ...806..213S,2015MNRAS.449.1921P}.
The 4650\,\AA\ emission feature fades rapidly. Given the clear detection of \ion{N}{2} at all epochs we believe
the identification of \ion{N}{3} to be very robust, but whether the blue wing of that emission feature is caused by \ion{C}{3} or \ion{He}{2}
is more difficult to determine. The \ion{He}{2}\,$\lambda$5411 line is extremely weak in SN~2015U, which argues that
\ion{C}{3} must be contributing to the 4650\,\AA\ feature, but discerning between these ions likely requires detailed modeling and
is beyond the scope of this paper.
\subsubsection{A Trace of Hydrogen?}
\label{sec:hydrogen?}
Inspection of our two-dimensional (2D) spectral images reveals host-galaxy emission features at H$\alpha$, H$\beta$, and H$\gamma$,
and there is no indication of either absorption or emission features intrinsic to SN~2015U~for any of those lines.
However, our spectra at $-2.1$\,d and +3.1\,d do show an absorption feature near the expected wavelength of H$\delta$,
decelerating over those 5 days from $-1060 \pm 75$\,km\,s$^{-1}$~(blueshifted) to $160 \pm 65$\,km\,s$^{-1}$~(redshifted relative to the rest frame of SN~2015U,
consistent with the rest frame of NGC~2388) and then disappearing by +6.1\,d.
Though this feature is difficult to disentangle from the significant noise at these wavelengths,
it appears to show weak P-Cygni emission in our first observation.
NGC~2388 shows some H$\delta$ absorption, and the feature from +3.1\,d is probably associated with the galaxy (this spectrum
has undergone host-galaxy subtraction as described in \S\ref{sec:spec_data}). However, in our $-2.1$\,d spectrum SN~2015U~was
well-separated from the host and we performed no galaxy subtraction.
Along with the $\sim$1000\,km\,s$^{-1}$\ Doppler shift and faint P-Cygni feature, this indicates that the weak H$\delta$
feature in the earliest spectrum of SN~2015U~arose within the same CSM as the \ion{He}{1} and other features described above.
Several SNe have been found to exist between the IIn and Ibn subclasses, exhibiting weak hydrogen features
\citep[e.g., SNe 2005la, 2010al, and 2011hw;][]{2008MNRAS.389..131P,2015MNRAS.449.1921P}.
Remarkably, SN~2010al had an H$\delta$ feature of comparable strength to its H$\alpha$ emission while showing only faint traces
of H$\beta$ and H$\gamma$ emission --- perhaps similar physics governs the hydrogen emission lines in SN~2015U.
Regardless of this putative H$\delta$ feature, the amount of hydrogen in the CSM surrounding
SN~2015U~must be vanishingly small.
\subsubsection{Higher Resolution}
\label{sec:hires}
\begin{figure*}
\includegraphics[width=\textwidth]{deimos_lineIDs.pdf}
\caption{ The DEIMOS spectrum from Feb.\ 27 in detail. A simple polynomial continuum fit has been removed.
The full spectrum is shown in grey in the background while the black line displays the spectrum smoothed with a 10\,\AA\ Gaussian kernel.
Identified emission lines are shown with grey dashed lines and text; absorption lines are in black.
All absorptions are indicated with a 745\,km\,s$^{-1}$\ blueshift but emission lines are marked at rest velocity.
\label{fig:lineIDs}}
\end{figure*}
Our DEIMOS spectrum from Feb. 27 ($R \approx 5000$) reveals
fully resolved absorption and emission lines from a host of ions in the spectra of SN~2015U; see Figure~\ref{fig:lineIDs}
and Table~\ref{tab:lines}. We include measured positions and widths for several lines for which we have no good identification;
it is especially difficult to differentiate between \ion{Fe}{3} ($E_i = 30.651$\,eV) and \ion{Fe}{2} ($E_i = 16.20$\,eV).\footnote{All
ionisation energies and line wavelengths were found through the NIST Atomic Spectra Database (ASD): \url{http://www.nist.gov/pml/data/asd.cfm}.}
In our spectrum we find at least four Fe absorption lines between 4900 and 5200\,\AA\ (in the SN rest frame).
The strongest two were identified as \ion{Fe}{3} by \citet{2015MNRAS.454.4293P}, but several known \ion{Fe}{2} lines occur at very similar wavelengths,
and we also identify weak features at 4912.5 and 5180.3\,\AA\ which are most plausibly interpreted as \ion{Fe}{2}\,$\lambda$4924 and \ion{Fe}{2}\,$\lambda$5195.
However, these identifications are tentative at best, given the nondetection of other expected \ion{Fe}{2} lines and the odd relative strengths of these
features, and so we caution the reader against overinterpreting the Doppler velocities presented in Table~\ref{tab:lines}.
\begin{table}
\caption{Absorption Lines in the DEIMOS Spectrum}
\begin{minipage}{\columnwidth}
\begin{center}
\label{tab:lines}
\begin{tabular}{ l | c c c c }
\hline
Ion & Rest $\lambda$ & Observed $\lambda$ & Offset & FWHM \\
& (\AA) & (\AA) & (km\,s$^{-1}$) & (km\,s$^{-1}$) \\
\hline
\ion{He}{1} & 5015.7 & 5002.6 & 778.8 & 586.4 \\
\ion{He}{1} & 5047.7 & 5034.6 & 783.2 & 267.8 \\
\ion{He}{1} & 5875.6 & 5860.2 & 789.2 & 488.7 \\
\ion{He}{1} & 6678.2 & 6661.7 & 737.2 & 470.1 \\
\ion{He}{1} & 7065.2 & 7047.9 & 733.7 & 432.9 \\
\ion{He}{1} & 7281.3 & 7265.4 & 655.9 & 404.0 \\
\ion{N}{2} & 5666.6 & 5653.0 & 721.4 & 232.5 \\
\ion{N}{2} & 5676.0 & 5665.4 & 559.2 & 433.1 \\
\ion{N}{2} & 5710.8 & 5699.9 & 570.4 & 407.2 \\
\ion{N}{2} & 5747.3 & 5734.8 & 651.6 & 367.5 \\
\ion{N}{2} & 6482.1 & 6468.1 & 643.4 & 428.1 \\
\ion{N}{2} & 5941.6 & 5927.5 & 711.5 & 600.7 \\
\ion{N}{2} & 6796.6 & 6781.8 & 654.2 & 337.9 \\
\ion{N}{2} & 7214.7 & 7195.9 & 783.5 & 173.7 \\
\ion{N}{2} & 6384.3 & 6366.7 & 829.6 & 253.7 \\
\ion{Fe}{2}\footnote{\label{modref}It is difficult to determine exactly which \ion{Fe}{2} or
\ion{Fe}{3} transition is responsible for these lines, and so the listed Doppler offsets are only tentative.} & 4923.9 & 4912.5 & 698.0 & 677.2 \\
\ion{Fe}{2}\footref{modref} & 5126.2 & 5114.9 & 658.1 & 514.5 \\
\ion{Fe}{2}\footref{modref} & 5157.3 & 5143.2 & 819.4 & 584.9 \\
\ion{Fe}{2}\footref{modref} & 5195.5 & 5180.3 & 873.2 & 545.7 \\
? & --- & 6197.2 & --- & 166.9 \\
? & --- & 6207.3 & --- & 152.7 \\
? & --- & 6285.9 & --- & 153.8 \\
? & --- & 7206.5 & --- & 190.5 \\
\hline
\end{tabular}
\end{center}
\par
Identifications and properties of the absorption lines identified
in our DEIMOS spectrum (after correcting for $z = 0.013161$). All lines were fit by a Gaussian with
a linear approximation to the nearby continuum, and
rest wavelengths were downloaded from the NIST ASD. Characteristic error bars are $\sim0.6$\,\AA\
(30\,km\,s$^{-1}$) for the observed wavelengths and $\sim1.6$\,\AA\ for the Gaussian FWHM (85\,km\,s$^{-1}$), as estimated
by calculating the 95\% confidence intervals for the \ion{N}{2}\,$\lambda$5711 line using MCMC techniques.
\end{minipage}
\end{table}
We also find three unidentified emission features in our DEIMOS spectrum that are not from host-galaxy contamination,
falling near 5290, 6165, and 6340\,\AA\ and exhibiting widths similar to those of the \ion{He}{1} lines.
None of these is clearly detected in our other spectra or the spectra published
by \citet[though generally they exhibit a S/N too low to rule them out]{2015MNRAS.454.4293P}.
The feature at 6340\,\AA\ is suggestive of the [\ion{O}{1}]\,$\lambda\lambda$6300, 6364 doublet that regularly arises in
nebular spectra of stripped-envelope SNe \citep[Types Ib/c; e.g.,][]{1997ARA&A..35..309F,2001PASP..113.1155M},
but this identification is only tentative.
\subsubsection{The Na\,D Lines}
\label{sec:csm}
\begin{figure}
\includegraphics[width=\columnwidth]{NaD.pdf}
\caption{The Na\,D features in our DEIMOS spectrum of SN~2015U\ and NGC~2388. The top panel plots the observed spectrum of SN~2015U\
with our best-fit absorption profiles while the middle panel shows the result after introducing
the additional constraint that D$_2$/D$_1$ $\geq 1.0$ (see \S\ref{sec:csm}).
The shorter-wavelength doublet is shown in blue and the longer-wavelength one in red, with the central wavelengths indicated by
dashed and dotted vertical lines for D$_2$ and D$_1$, respectively. Their sum is given in grey.
The bottom panel shows the 2D frame of our DEIMOS observation in this region, including both the light from the host and the SN,
as well as a cutout of the H$\alpha$ emission line (plotted with the same Doppler $\delta v$ per pixel) to indicate the shape
of the galaxy's rotation curve.
\label{fig:nad} }
\end{figure}
Our DEIMOS spectrum of SN~2015U\ reveals a remarkable set of Na\,D absorption lines --- see Figure~\ref{fig:nad}.
In addition to the
MW doublet we find several overlapping lines from NGC~2388 along our sight line, though as we show
below, there is no evidence that any of these lines originate within the CSM immediately surrounding SN~2015U.
The astronomical Na\,D doublet has been studied in great detail (it was one of the original Fraunhofer lines)
and the relative strength of the doublet has long been used to measure the interstellar abundance
of neutral sodium: the Na\,D$_2$ line is generally observed to be stronger than the D$_1$ line by a factor ranging from 1.0 to
2.0, for high and low column densities of \ion{Na}{1}, respectively
\citep[e.g.,][]{1948ApJ...108..242S,1973ApJ...182..481N,1988Obs...108...44S}.
The MW Na\,D doublet in our spectra exhibits the expected behaviour, with an EW ratio of D$_2$/D$_1$ $\approx 1.2$,
but the host-galaxy lines do not.
There are at least two distinct and overlapping Na\,D absorption doublets near the host-galaxy's rest
frame in the spectrum of SN~2015U.
Figure~\ref{fig:nad} shows them (as well as model fits) in detail. We model the absorption complex by overlaying
the modified Lorentzian emission-line profile (see \S\ref{sec:lineprofiles}) with two
doublets of Voigt absorption profiles (though we note that some of these lines may be unresolved and
saturated). D$_1$ and D$_2$ lines from a single doublet are forced
to have the same velocity properties and to be separated by 5.97\,\AA, but they are allowed independent strengths.
In the top panel of Figure~\ref{fig:nad} we show the result when we allow the relative strengths of the doublet (i.e.,\ D$_2$/D$_1$) to vary freely.
In the rest frame of the SN, the bluest doublet falls at $\lambda\lambda$5890.45, 5896.35 $\pm 0.066$\,\AA.
This is a blueshift of only $\sim25$\,km\,s$^{-1}$\ from the SN rest frame, and so this
doublet is likely to arise from the ISM of NGC~2388 along our sight line to SN~2015U.
The second (redder) doublet, however, falls at $\lambda\lambda$5895.80, 5901.69 $\pm$ 0.063\,\AA\ --- redshifted from the SN by almost 300\,km\,s$^{-1}$,
and even redshifted from the galaxy core by more than 100\,km\,s$^{-1}$. This is within the velocity range of NGC~2388's
rotation curve, but SN~2015U\ is located well away from the receding spiral arm and the bulk of the galaxy's receding material (see Figure~\ref{fig:finder}).
In addition, the second Na\,D doublet shows a strength ratio of D$_2$/D$_1$ $\approx 0.5$,
well outside the commonly observed values of $\sim1.0$--2.0 (in contrast, the blue doublet exhibits a reasonable D$_2$/D$_1$ $\approx 1.5$).
If, instead, we constrain the doublet ratio to D$_2$/D$_1$ $\geq 1.0$, we can also obtain a reasonable fit, though this introduces noticeable
discrepancies in the reddest part of the feature --- see the middle panel of Figure~\ref{fig:nad}.
In addition, this forces the doublet ratio of the bluer doublet outside of the range of normal values: D$_2$/D$_1$ $\approx 2.5$.
Adding more components to our model fit will not solve this quandary (though the data do
show a shoulder on the bluer doublet, suggesting a third absorption component).
The implications for the \ion{Na}{1} gas (and the dust) in NGC~2388 are not clear.
Given the spectral signatures of dense CSM
surrounding SN~2015U, it is natural to wonder whether some of the Na\,D absorption arises within the local CSM.
Variation in the Na\,D features over the timescale of the SN evolution would be a clear signature of local absorption \citep[e.g.,][]{2007Sci...317..924P};
however, as Figure~\ref{fig:HeI} shows, no such variation is apparent in our data.
The bottom panel of Figure~\ref{fig:nad} displays a cutout of the 2D DEIMOS spectrum around the Na\,D feature.
Both SN~2015U\ and NGC~2388 are clearly visible, and we also show the rotation curve of the galaxy via a cutout of the H$\alpha$ emission line.
The Na\,D absorption of NGC~2388's own ISM is seen against the stellar light of the galaxy, and
this galactic self-absorption roughly covers the velocity range
of the two components observed along the line of sight toward SN~2015U.
It appears that the anomalous red component described above obscures the host galaxy as well as the SN,
providing further evidence that this component is unlikely to arise within the CSM of SN~2015U.
\subsubsection{Spectropolarimetry}
\label{sec:specpolAnalysis}
We did not observe significant evolution in the polarization of SN~2015U\ between our three epochs, so we coadded all of the data to increase
the S/N; the results are illustrated in Figure~\ref{fig:specpol}. The polarization spectrum of SN~2015U\ appears to be dominated
by ISP associated with the host galaxy NGC~2388, showing a strong increase in continuum polarization toward shorter wavelengths with a value of
$P \approx 2.5$\% at 4600\,{\AA} and decreasing to $P \approx 1.0$\% near 7000\,{\AA}. This behaviour is dissimilar to what is typically observed for the
MW ISP, which exhibits a peak and turnover near 5500\,{\AA}. Instead, the observed behaviour is reminiscent of the ISP produced by the host galaxies of
several SNe~Ia, including SNe~1986G, 2006X, 2008fp, and 2014J \citep[][see their Figure 2]{2015A&A...577A..53P}. The continuous rise in $P$ beyond the
$B$ and $U$ photometric passbands has previously been interpreted as evidence for scattering by dust grains smaller than those characteristic of the MW disk ISM.
Interestingly, we observe a significant wavelength dependence for the position angle ($\theta$), which has a value of $\sim25^{\circ}$ at 4600\,{\AA}
and monotonically trends toward $\sim10^{\circ}$ near 7000\,{\AA}. In the MW, the ISP's position angle is generally flat. However,
a wavelength dependence in $\theta$ has been observed along particular lines of sight toward star-forming regions at distances greater than 0.6\,kpc
\citep[e.g.,][]{1965AJ.....70..579G,1966AJ.....71..355C}, and has been explained as the result of photons traversing multiple clouds or scattering media that
exhibit various sizes for the scattering particles as well as various orientations for the interstellar magnetic field.
A similar interpretation is plausible here: along the line of sight within NGC~2388 toward SN~2015U\ there are likely to be multiple separate components
of dusty scattering media having different grain sizes and/or different magnetic field orientations. This possibility is particularly interesting considering
the complex superposition of multiple \ion{Na}{1}\,D absorption doublets that we see in our high-resolution DEIMOS spectrum.
If the continuum flux spectrum of SN~2015U\ is devoid of broad SN features because of high optical-depth CSM, then one might suspect there also to be a separate and
distinct scattering component associated with this CSM, if it is dusty.
The subsequent re-scattering of this light as it traverses the dusty ISM of the host could provide a means for producing the observed wavelength dependence of $\theta$,
if there is a difference in grain sizes between these multiple scattering media.
Although this scenario is physically plausible, the spectropolarimetric data cannot discriminate between a CSM$+$ISM scattering combination
and multiple components of host ISM, and our analysis of the
total-flux \ion{Na}{1}\,D features indicates that they are likely associated with ISM (see \S\ref{sec:csm}).
Finally, there is a line-like feature near 5820\,{\AA} which might naively be interpreted as a signature of intrinsic polarization associated with SN~2015U\ and
with the \ion{He}{1} / \ion{Na}{1} transition. However, the MW \ion{Na}{1}\,D doublet falls at the same wavelength.
Although relatively weak compared to the redshifted host-galaxy Na\,D absorption, the results of our spectral arithmetic in the vicinity of this
poorly resolved doublet profile might have created a spurious artifact in the final coadded dataset mimicking the shape of a polarized line feature.
Indeed, in each individual epoch, the polarized spectra near this feature appear to suffer from systematic noise, so we are reluctant to attribute
this feature to the SN itself.
\begin{figure}
\includegraphics[width=\columnwidth]{specpol.pdf}
\caption{Our spectropolarimetric observations of SN~2015U. From top to bottom:
total-flux spectra near $V$-band peak and in higher resolution (for visual comparison),
$q$ and $u$ Stokes parameters,
$P$ with a fourth-order polynomial fit showing the overall trend,
and $\theta$ with a similar polynomial fit (after discarding outliers).
The spectropolarimetric data from three nights of observations have been coadded and binned to 25\,{\AA}.
\label{fig:specpol}}
\end{figure}
\subsection{Photometry}
\label{sec:phot_analysis}
The optical light curves of SN~2015U\ show that it was a remarkably luminous and rapidly evolving event ---
Figure~\ref{fig:phot_correct} shows our photometry corrected for host-galaxy dust reddening.
With a peak absolute magnitude of $\lesssim -19$\,mag at optical wavelengths,
a rise time of $\lesssim 10$\,d, a time above half-maximum of $t_{1/2} \approx 12$\,d, and a decline rate of nearly 0.2\,mag\,day$^{-1}$ after peak,
SN~2015U\ was more than a magnitude brighter than most stripped-envelope SNe and evolved much more rapidly \citep[e.g.,][]{2011ApJ...741...97D,2014ApJS..213...19B}.
\begin{figure}
\includegraphics[width=\columnwidth]{phot_host_corrected.pdf}
\caption{The KAIT and Nickel light curves (top) and colour curves (bottom) of SN~2015U, after
correcting for MW dust absorption and host absorption.
Nickel data are shown with diamonds and KAIT data with circles, and
the dates of our spectral observations are indicated in the top panel with dashed vertical lines.
Host-galaxy contamination becomes significant in our KAIT photometry around +10\,d, and so
we only show the Nickel data beyond that date in the lower panel (we include all data
in the upper panel). We apply vertical offsets to every passband except $R$ in the top panel, to enable comparisons.
\label{fig:phot_correct}}
\end{figure}
Though the first unfiltered ($clear$) KAIT detection of SN~2015U\ was on Feb.\,11, and it went undetected by KAIT on Feb.\,10 ($> 18.4$\,mag),
\citet{2015MNRAS.454.4293P} present detections from Feb.\,9 and 10 at $R = 18.62 \pm 0.26$ and $18.14 \pm 0.30$\,mag, respectively.
Unfortunately, the location does not appear to have been observed in the days prior and there are not deep upper limits constraining the
explosion date further. SN~2015U\ rose quite rapidly, so we adopt a tentative explosion date one day before the
first detection published by \citet{2015MNRAS.454.4293P}: $t_{\rm exp} \approx 57062$\,MJD (Feb.\,8). This provides us with
a rise time for SN~2015U\ of $t_{\rm rise} \approx 9$\,days.
\citet{2015IBVS.6140....1T} note that SN~2015U\ is among the most rapidly evolving SNe known, with a decline
rate similar to those of SNe 2002bj, 2005ek, and 2010X. We measure $\Delta M_{15} \approx 2.0$\,mag in the $R$ band,
but note that the decline rate increases ever more steeply after $\sim10$ days post-peak, and at all times the bluer passbands decline more rapidly than the red.
Simple linear fits indicate the following decline rates before and after $+10$\,d (in mag day$^{-1}$):
$B_{\rm early} = 0.110 \pm 0.007$, $V_{\rm early} = 0.099 \pm 0.005$, $V_{\rm late} = 0.28 \pm 0.07$,
$R_{\rm early} = 0.080 \pm 0.005$, $R_{\rm late} = 0.267 \pm 0.009$, $I_{\rm early} = 0.067 \pm 0.006$, and $I_{\rm late} = 0.26 \pm 0.04$
(uncertainties are statistical, and our data do not constrain the late $B$-band decline).
SN~2015U\ is one of the nearest SNe~Ibn to date \citep{2016MNRAS.456..853P}, but it is still relatively distant for direct progenitor studies
(and is obscured by the dust in NGC~2388). Regardless, the {\it HST} nondetections presented in
\S\ref{sec:phot} can be used to place interesting constraints on the SN's progenitor.
We compared these limits to the MIST stellar evolutionary tracks \citep{2016ApJ...823..102C} at solar metallicity generated in the
WFC3/infrared bandpasses
(negligible photometric differences exist between NICMOS/NIC2 and WFC3/IR for {\it F110W} and {\it F160W}).
Based on these tracks we can eliminate single-star progenitors with initial masses $M_{\rm ini} \gtrsim
9\, {\rm M}_{\odot}$ and $\lesssim 40\, {\rm M}_{\odot}$.
That is, the progenitor would have been either a low-mass star near the core-collapse limit or a highly massive evolved star, possibly in a luminous blue variable (LBV)
or Wolf-Rayet phase at the time of explosion. We did not interpret our upper limits with respect to existing binary evolution models, and we caution
that these results are somewhat dependent upon the uncertain properties of NGC~2388's dust population.
The structure near SN~2015U's position in the 1\,yr {\it HST} images (Figure~\ref{fig:finder}) may be due to clumpy star-forming regions, perhaps associated with SN~2015U.
NGC~2388 is a strongly star-forming and massive galaxy --- \citet{2015A&A...577A..78P} calculate an ongoing star-formation rate
of $\sim 40^{+9}_{-22}$\,M$_{\odot}$\,yr$^{-1}$ and a total stellar mass of 10$^{11.0 \pm 0.1}$\,M$_{\odot}$ --- and though SN~2015U\
is not obviously associated with the brightest star-forming regions of the galaxy, our spectra do show emission lines from nearby \ion{H}{2} regions.
The clump may also be an artifact of the intervening dust lanes in NGC~2388; unfortunately
there is essentially no colour information available from our images as the host-galaxy background dominates.
\subsection{Comparisons with Other Supernovae}
\begin{figure}
\includegraphics[width=\columnwidth]{phot_comparisons.pdf}
\caption{The extinction-corrected light curve of SN~2015U\ compared to light curves of SNe~2002bj and 2010X \citep{2010Sci...327...58P,2010ApJ...723L..98K},
the rapidly evolving SN~Ibn~LSQ13ccw \citep{2015MNRAS.449.1954P},
and one of the rapidly evolving events from the PS1 sample \citep{2014ApJ...794...23D}. We also show in black the $R$-band SN~Ib/c template from
\citet{2011ApJ...741...97D}.
For SN~2015U\ we include data from the {\it R} and {\it clear} passbands published here as well as the {\it r}-band photometry from \citet{2015MNRAS.454.4293P};
a vertical offset of $-0.25$\,mag was applied to the {\it r}-band data to match the $R$ and $clear$ bands, enabling visual comparison.
\label{fig:phot_comparisons}}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{spec_comparisons.pdf}
\caption{Spectra of SN~2015U\ compared with spectra of the prototypical SN~Ibn~2006jc \citep{2007ApJ...657L.105F},
the young SN~IIn~1998S \citep{2000ApJ...536..239L,2015ApJ...806..213S},
a previously unpublished spectrum of the transitional SN Ibn~2010al \citep{2010CBET.2223....1S,2015MNRAS.449.1921P},
the rapidly fading SNe~2002bj and 2010X \citep{2010Sci...327...58P,2010ApJ...723L..98K},
and one of the rapidly fading events from the
PS1 sample \citep{2014ApJ...794...23D}. Galactic emission features have been masked in the low-resolution spectra of
SN~2015U\ to facilitate comparisons.
\label{fig:spec_comparisons}}
\end{figure*}
In Figure~\ref{fig:phot_comparisons} we compare the light curves of SN~2015U\ to the $R$-band light curves of SN~2002bj \citep{2010Sci...327...58P},
which was very similar though $\sim1$\,mag fainter, and SN~2010X \citep{2010ApJ...723L..98K}, which also
matches well but was $\sim2.5$\,mag fainter. SNLS04D4ec and SNLS06D1hc, two of the rapidly rising transients discovered by the Supernova Legacy Survey (SNLS)
and presented by \citet{2016ApJ...819...35A}, also show similar light-curve behaviour but $\sim0.5$\,mag brighter than SN~2015U\ (see their Figure~2).
In Figure~\ref{fig:spec_comparisons} we compare the spectra of these events: SN~2002bj was hydrogen deficient (SN~IIb-like, though not a good spectroscopic
match to normal SNe~IIb) with a strong blue continuum and P-Cygni features at higher velocities than observed in SN~2015U, while SN~2010X showed no hydrogen
(SN~Ib/c-like, though again not a good spectroscopic match to normal SNe~Ib/c) with broad absorption features.
(Unfortunately, no spectra were obtained of the SNLS events.)
The light curves and spectra of SN~2015U\ are quite similar to those of the rapidly evolving and luminous
transients discovered in the PS1 dataset and presented by \citet{2014ApJ...794...23D}. Figure~\ref{fig:phot_correct}
includes the $g$ and $i$-band light curves of the relatively well-observed PS1-12bv from that sample;
it was discovered at $z = 0.405$, so the $g$ and $i$ passbands probe rest wavelengths around 3500\,\AA\ (about 100\,\AA\ shortward of $B$)
and 5350\,\AA\ (similar to $V$), respectively. Without the $0.5$\,mag offset used in Figure~\ref{fig:phot_correct}, the $i$-band light curve
of PS1-12bv would overlie the $V$ band of SN~2015U\ almost exactly, and the $g$-band curve would be somewhat brighter than the $B$ band
of SN~2015U. There are no clear detections of any emission features in the spectrum of PS1-12bv, but it and the other
events from the PS1 sample were discovered at very large distances, and high-S/N
spectra do not exist for those events. It is plausible that narrow emission lines of helium (and/or hydrogen) were present
but went undetected.
Figure~\ref{fig:spec_comparisons} shows that
SN~2015U\ shares the blue colour and prominent \ion{He}{1} emission features of the canonical
SN~Ibn~2006jc \citep[e.g.,][]{2007ApJ...657L.105F,2008MNRAS.389..113P}.
However, SN~2006jc was discovered post-peak and evolved more slowly than SN~2015U: the earliest extant spectrum of SN~2006jc
was taken after the event had faded $\sim1$\,mag from peak and the spectrum had already transitioned into a nearly nebular phase,
though a blue pseudocontinuum was apparent owing to ongoing interaction with the CSM \citep{2007ApJ...657L.105F}.
At later times, a second red continuum arose in SN~2006jc and the emission features became progressively more asymmetric and blueshifted,
evidence for dust formation with the SN system. Increased absorption in the optical passbands by this dust, with re-emission in the
infrared, provided an explanation for the rapid increase in the optical decline rate observed for SN~2006jc after $\sim$50 days
\citep{2008MNRAS.389..141M,2008ApJ...680..568S,2008MNRAS.389..113P}.
In contrast, SN~2015U\ presented a spectrum dominated by a single blue continuum through our last epoch of spectroscopy,
also taken after the event had faded $\sim1$\,mag from peak. The $B-R$ and $V-I$ colour curves of SN~2015U\ smoothly
trend redward from a few days before (optical) maximum brightness to $\sim$2\,mag out onto the rapidly fading tail
(at which point the SN fades below our detection threshold) --- see Figure~\ref{fig:phot_correct}.
The accelerating decline rate observed in the SN~2015U\ light curves does not appear to be due to dust formation: one
would expect a redward knee in the colour curves if it were. On the contrary, there is some evidence that the $B-R$ curve
of SN~2015U\ was beginning to flatten in our last few epochs, though note that $B$-band data become sparse.
We also show the light curve of the peculiar and rapidly fading SN Ibn LSQ13ccw in Figure~\ref{fig:phot_comparisons} \citep{2015MNRAS.449.1954P}.
LSQ13ccw rose to peak extremely quickly ($t_{\rm rise} \approx 5$\,d) and then faded rapidly for $\sim$10 days, similar to SN~2015U's behaviour.
However, it thereafter slowed in its decline, and the spectra of LSQ13ccw (not shown) exhibited both broad and narrow features (as do many, but not all, SNe~Ibn).
The knee in the light curve of LSQ13ccw is plausibly interpreted as evidence for ongoing energy
injection from ejecta/CSM interaction in the system --- no such knee is observed in the light curve of SN~2015U, which instead appears to be consistent
with a single shock-breakout/diffusion event.
The early-time spectrum of SN~2015U\ is similar to the very early spectra of the Type IIn SN~1998S
and the transitional Type IIn/Ibn SN~2010al, excepting the absence of hydrogen (though see \S\ref{sec:hydrogen?}
for a discussion of the tentative H$\delta$ feature in SN~2015U).
All three events show a smooth, blue, and (approximately) blackbody continuum.
The implied CSM velocity of SN~2015U\ is higher than in SN 1998S and SN 2010al,
and SN~2015U\ shows strong \ion{He}{1} lines while the other two events exhibit stronger \ion{He}{2}.
The \ion{N}{3}/\ion{C}{3} complex near 4500\,\AA\ is clearly detected in all three events ---
this feature dominates at very early times and disappears by peak (though that
evolution took place over different timescales for these SNe).
Despite the spectral similarities, these three SNe displayed diverse light-curve behaviour.
SN~1998S had a SN~IIL-like light curve with a long-lasting tail \citep[e.g.,][]{2000MNRAS.318.1093F},
while the evolution of SN~2010al was more similar to that of SN~2015U\ though less rapid
\citep{2015MNRAS.449.1921P,2016MNRAS.456..853P}. The $B-R$ colour curve of
SN~2015U\ (as shown in Figure~\ref{fig:phot_comparisons}) is also notably similar to that of SN~2010al,
with a slow trend toward redder colours for most of the SN's evolution and then either leveling off
or perhaps even becoming more blue again beyond +10\,d and +20\,d for SNe 2015U and 2010al, respectively
\citep[][see their Figure 4]{2015MNRAS.449.1921P}.
The $\sim20$-day timescale for SN~2015U's evolution and the increasing decline rate are reminiscent of the
light curves expected for low-$^{56}$Ni explosions in helium-dominated or oxygen-dominated envelopes ---
see, for example, the models of \citet{2011MNRAS.414.2985D} and \citet{2014MNRAS.438..318K}. In these models,
the recombination of helium or oxygen produces a dramatic and rapid drop in opacity within the ejecta and an inward-moving recombination front.
SN~2015U\ was much more luminous than the objects in these models, but perhaps it was an analogous event wherein
a recombination wave in the extended CSM produced a rapid fade from maximum luminosity as the
CSM cooled after shock breakout.
Similar situations have been observed in hydrogen-rich SNe:
there exists a subclass of SNe~IIn that shows light curves with a plateau likely produced via hydrogen recombination
within their extended CSM \citep[SNe 1994W, 2009kn, and 2011ht; e.g.,][]{2004MNRAS.352.1213C,2012MNRAS.424..855K,2013MNRAS.431.2599M}.
The narrow P-Cygni profiles and strong continuua of SN~2015U's spectra are clear indications of dense CSM.
The diversity of ways in which such CSM can affect the light curves of SNe has been explored in detail by several authors
motivated by observations of SNe~IIn/Ibn, superluminous SNe, and rapidly fading SNe
\citep[e.g.,][]{2010ApJ...724.1396O,2011ApJ...729L...6C,2014MNRAS.438..318K},
and comparisons of SN~2015U's light curve to these models implies that SN~2015U\ was a shock-breakout
event. The high luminosity indicates that shock breakout occurred at a large radius and that a significant
fraction of the SN's kinetic energy was converted into light, while the lack of a long-lasting
light-curve tail shows that the CSM surrounding SN~2015U\ was more shell-like than wind-like
(there is no interaction-powered tail) and that relatively little $^{56}$Ni was produced
(there is no radioactively powered tail).
The rapidly fading light curves of SNe~2002bj and 2010X are remarkably similar to those of SN~2015U, and we argue
that their spectral differences do not exclude the possibility that these three events were fundamentally quite similar.
By varying the radius of a putative opaque CSM shell in a simple model,
we can understand events spanning a range of luminosities and timespans for which the opaque CSM
reprocesses the SN flux.
Under the assumption that their light curves are all shock cooling curves without significant contribution from radioactive nickel,
the peak luminosities and timescales of these three SNe should be governed roughly by
\begin{equation*}
t_{\rm SN} \propto E^{-1/6} M^{1/2} R^{1/6} \kappa^{1/6} T^{-2/3}\, ,
\end{equation*}
\begin{equation*}
L_{\rm SN} \propto E^{5/6} M^{-1/2} R^{2/3} \kappa^{-1/3} T^{4/3}\, ,
\end{equation*}
following \citet{2014MNRAS.438..318K},
adapted from \citet{2009ApJ...703.2205K} for hydrogen-free SNe and based on the
analytic framework of \citet{1993ApJ...414..712P}. $E$ is the energy of the SN explosion,
$M$ is the mass ejected, and $R$ is the effective radius (in this case, the radius reached by a significant amount of mass ejected prior to the final explosion).
In particular, consider the peak-luminosity equation, assuming these explosions are similar in all their properties except for the effective
pre-SN radius $R$, which would be determined by the time before explosion
and the speed at which any pre-SN material was ejected.
Inverting the equation, the pre-SN effective radius $R \propto L_{\rm SN}^{3/2}$. If we assume that the ejecta velocities of all three objects are similar,
this means that the ejecta from the SN itself will pass through the surrounding CSM in a time $t_{\rm interact} \propto R$,
and the length of time after explosion we expect to see narrow lines from this interaction is proportional to $R$. Comparing the
relative peak luminosities, we find that, since SN~2015U\ exhibited narrow lines at least up to $\sim16.5$ days after explosion, SN\,2002bj should
have shown narrow lines at least until $\sim 5.5$ days after explosion and the even dimmer SN\,2010X would have shown narrow lines
at least until $\sim 0.52$ days. These times are well before the first spectra were taken of either SN 2002bj or SN 2010X. Typically,
we expect that to discover narrow lines in these rapidly fading SNe, we will either need to look at the brightest among them or catch them very early. The
luminosity (and timescale) may also depend on the explosion energies, ejected masses, and other properties of the SN, but these must be
disentangled with more sophisticated numerical approaches.
\subsection{Temperature and Luminosity Evolution}
\label{sec:temp}
The extreme and uncertain degree of host-galaxy dust reddening toward SN~2015U\ makes estimating bolometric
properties quite difficult. In \S\ref{sec:dust} we assume that the emission from SN~2015U\ is roughly blackbody
in spectral shape; here we further discuss this assumption.
It has long been known that the continua of young SNe~II exhibit ``diluted blackbody'' spectral energy distributions (SEDs), which (at optical wavelengths)
are similar to the Planck function at a lower temperature \citep[e.g.,][]{1986ApJ...301..220H,1996ApJ...466..911E}.
Though SN~2015U\ is certainly not a SN~II, it is continuum-dominated, and in the sections above we present evidence
that SN~2015U\ was shrouded in an optically thick CSM.
In addition,
\citet{2011MNRAS.415..199M} show that modeled shock breakouts from the hydrogen-dominated CSM around red supergiants exhibit
roughly blackbody SEDs.
\citet{2011ApJ...728...63R} explore shock breakouts from He or He/CO stellar envelopes in detail; for He-dominated stellar
envelopes, they show that the colour temperature of the system deviates from the photosphere's temperature by a (time-dependent) factor
of only $\sim20$\%, largely owing to diffusion effects as a helium recombination wave moves through the material.
Given the above discussion, we assume that SN~2015U's emission was roughly blackbody and
we estimate the bolometric properties by fitting a dust-reddened Planck spectrum to our multiband photometry.
Using our almost nightly observations
of SN~2015U\ in the {\it BVRI} passbands, we assembled the observed optical SEDs from $-5$\,d to +20\,d and
find the best-fit blackbody temperature, radius, and luminosity using MCMC maximum-likelihood methods.
As described in \S\ref{sec:dust}, there are large uncertainties in the properties of the host galaxy's dust, and
these uncertainties produce similarly large uncertainties in the absolute value of the temperature at any given time.
In addition, our optical photometry largely probes the Rayleigh-Jeans tail of SN~2015U's SED, so the implied
bolometric corrections are large and uncertain. Our MCMC-produced error bars include
these effects, and our photometric data do indicate significant temperature evolution (assuming $E(B-V)$ and $R_V$ do not change).
Because our data constrain the evolution of SN~2015U\ more
strongly than they constrain the absolute values of any given parameter, it is useful to consider the implications for a
few assumed values of $E(B-V)$.
Figure~\ref{fig:TLRt} shows the blackbody temperature, bolometric luminosity, and radius evolution of SN~2015U\ for
three different assumed values of $E(B-V)$, adopting $R_V = 2.1$. In all cases, the best-fit temperature decreases over time, with the temperature
falling rapidly before (optical) peak and then decreasing slowly thereafter.
The effective blackbody radius increases with a photospheric velocity of $v_{\rm phot} \approx 15,000$\,km\,s$^{-1}$\ until peak,
leveling off thereafter. Though this value of $v_{\rm phot}$ is similar to the characteristic velocities of ejecta in stripped-envelope SNe, our spectra
do not show any material moving that quickly and, given our interpretation of SN~2015U\ as a cooling shock-breakout event, we expect the photospheric radius to
be approximately constant throughout these observations.
In addition, Figure~\ref{fig:TLRt} may surprise the reader by indicating that the peak bolometric luminosity occurred several days before
the optical peak. Though the bolometric corrections are at their largest and most uncertain at early times,
and we unfortunately do not know of any observations constraining the ultraviolet emission of SN~2015U,
similar behaviour was observed from SN~Ibn~2010al with ultraviolet to near-infrared wavelength coverage \citep{2015MNRAS.449.1921P}.
In addition, we can constrain the temperature evolution of SN~2015U\ independently of the continuum shape
through the relative line strengths of detected emission lines in our spectra. Though detailed modeling is
beyond the scope of this paper, the pre-maximum detection and rapid fading of the \ion{N}{3}/\ion{C}{3}/\ion{He}{2} 4500\,\AA\ complex in our spectra of
SN~2015U\ brings to mind the spectral evolution of SN~1998S (see Figure~\ref{fig:spec_comparisons}). Detailed CMFGEN
models of SN~1998S while the 4500\,\AA\ feature was strong were
presented by \citet{2015ApJ...806..213S}, indicating a temperature
of $\sim$30,000\,K throughout most of the line-forming region. The agreement between this value and the pre-maximum
temperatures found for SN~2015U\ assuming $E(B-V) = 0.94$\,mag are heartening.
In Table~\ref{tab:intbolo} we present the integrated energy released and various values at $V$-band peak for
three assumed values of $E(B-V)$. Our uncertainty about the host galaxy's dust properties is the dominant source of error in this analysis,
and the spread over these three values of $E(B-V)$ indicates the range. All values are estimated through simple
polynomial fits to the curves shown in Figure~\ref{fig:TLRt}. To calculate $E_{\rm rad}$, we integrated from
day $-5$ to day +20 (the timespan of our almost nightly multiband photometric coverage), and for the other parameters
we list the value at $V$-band peak. Note that the bolometric luminosity peaks before the optical
maximum; again, see Figure~\ref{fig:TLRt}.
\begin{figure*}
\includegraphics[width=\textwidth]{TLRt.pdf}
\caption{The best-fit blackbody temperature, bolometric luminosity, and radius evolution of SN~2015U\ for three assumed values of $E(B-V)$
as a function of phase relative to $V$-band peak.
The leftmost column shows the evolution assuming a value of 0.5\,mag, the middle column assumes
our preferred value of 0.94\,mag, and the rightmost column corresponds to
a value of 1.05\,mag. In the lower panels, we show linear fits to the radius evolution up to
$V$-band peak with a grey dashed line.
All parameters are fitted using maximum-likelihood MCMC methods and error bars represent 95\% confidence intervals (but do not include
errors caused by uncertainty in the degree of host-galaxy dust reddening).
\label{fig:TLRt}}
\end{figure*}
\begin{table}
\begin{minipage}{\columnwidth}
\begin{center}
\caption{Bolometric Properties of SN~2015U}
\label{tab:intbolo}
\begin{tabular}{ l | c c c c }
\hline
$E(B-V)$ & $L_{\rm peak}$ & $E_{\rm rad}$ & $T_{\rm peak}$ & $R_{\rm peak}$ \\
(mag) & (erg\,s$^{-1}$) & (erg) & (K) & (cm) \\
\hline
0.5 & $7.3 \times 10^{42}$ & $9.3 \times 10^{48}$ & 8400 & $1.5 \times 10^{15}$ \\
0.94 & $5.9 \times 10^{43}$ & $7.1 \times 10^{49}$ & 19,000 & $8.7 \times 10^{14}$ \\
1.05 & $2.3 \times 10^{44}$ & $2.9 \times 10^{50}$ & 30,000 & $6.7 \times 10^{14}$ \\
\hline
\end{tabular}
\end{center}
\par
Bolometric luminosity, radiated energy, blackbody temperature, and blackbody radius for the three assumed values of $E(B-V)_{\rm host}$
shown in Figure~\ref{fig:TLRt}.
\end{minipage}
\end{table}
\subsection{Nickel Content}
A slowly declining light-curve tail, powered either via CSM interaction or $^{56}$Ni decay, has been detected in some SNe~Ibn,
but many members of this subclass (including SN~2015U) do not show one \citep[e.g.,][]{2007ApJ...657L.105F,2014MNRAS.443..671G,2016MNRAS.456..853P}.
For most SNe~I, Arnett's law can be used to estimate the total $^{56}$Ni synthesized by the explosion \citep{1982ApJ...253..785A}
based upon observed properties near the time of peak luminosity; however, that approach is not applicable for shock-breakout events.
If the opacity of the ejecta is well understood, the luminosity at late times can be used instead.
For example, \citet{2003ApJ...582..905H} analyzed Type IIP supernovae and,
given a luminosity $L_t$ at time $t$ and an explosion time $t_0$, estimated the $^{56}$Ni mass to be
\begin{multline*}
M_{\rm Ni} = (7.866 \times 10^{44})\, L_t \, \mathrm{exp} \left( \frac{(t - t_0)/(1+z) - 6.1}{111.26\,{\rm d}} \right)\, {\rm M}_{\odot}.
\end{multline*}
This equation assumes that $\gamma$-rays produced via radioactive decay are fully trapped and thermalized
by the ejecta, an assumption which (though reasonable for SNe IIP) has been shown not to be true for
most stripped-envelope SNe \citep[which fade faster than expected for complete trapping; e.g.,][]{2015MNRAS.450.1295W}.
The extreme CSM surrounding SN~2015U\ further complicates the issue, as it may be contributing luminosity
via ongoing interaction at late times and perhaps even trapping a higher fraction of the $\gamma$-rays than is normal for CSM-free SNe Ib/c.
Though a robust measure of the amount of nickel created by SN~2015U\ would require more sophisticated treatment,
we place a rough upper limit on the value by assuming that the ejecta+CSM system of SN~2015U\ completely traps any
radioactively produced $\gamma$-rays and that there is no luminosity contribution from ongoing interaction beyond
our last detection at $\sim 20$\,days
Our most constraining observation of a putative radioactively powered tail is a nondetection in the $R$ band at +36\,d.
Our last multiband photometric measurement of the temperature yields $T = 6000 \pm 1000$\,K at +18\,d (see \S\ref{sec:temp}).
We adopt that temperature to calculate the blackbody luminosity for a source at our nondetection threshold,
but we note that the temperature is still changing at this time, possibly affecting our results.
This calculation yields an upper limit of $M_{\rm Ni} \lesssim 0.02$\,M$_{\odot}$ ---
quite low for SNe~I
\citep[$M_{\rm Ni,Ia} \gtrsim 0.4$\,M$_{\odot}$, $M_{\rm Ni,Ib/c} \approx 0.2$\,M$_{\odot}$;][]{2000A&A...359..876C,2011ApJ...741...97D},
but within the range of values observed for SNe~II
\citep[$M_{\rm Ni,II} \approx 0.0016$--0.26\,M$_{\odot}$;][]{2003ApJ...582..905H}.
\subsection{Progenitor Mass-Loss Rate}
We adopt the simple shock-wind interaction model of \citet{2011ApJ...729L...6C} to estimate the properties of
the CSM surrounding SN~2015U. We assume that the CSM is spherically symmetric and follows a wind-like
density profile ($\rho \propto r^{-2}$), and we use the opacity of helium-dominated material in our calculations
($\kappa = 0.2$\,cm$^2$\,g$^{-1}$).
\citet{2014ApJ...780...21M} solve the \citet{2011ApJ...729L...6C} model
in terms of three observables: break-out radius $R_{\rm bo}$ (the radius at which radiation diffuses forward ahead of the shock), total radiated energy $E_{\rm rad}$, and light-curve rise time $t_{\rm rise}$.
We use $t_{\rm rise} = 9$\,days and, assuming $E(B-V)_{\rm host} = 0.94$\,mag, we adopt $E_{\rm rad} = 7.1 \times 10^{49}$\,erg from Table~\ref{tab:intbolo}.
For $R_{\rm bo}$ we take the first measured blackbody
radius from Figure~\ref{fig:TLRt}: $R_{\rm bo} = 5 \times 10^{14}$\,cm.
The resultant mass-loss estimate is very large, $\dot M \approx 1.2$\,M$_{\odot}$\,yr$^{-1}$.
However, in this model $\dot M \propto R_{\rm bo}^{-3}$, and there is reason to believe that our measurement of the blackbody
radius before peak brightness does not reflect the true $R_{\rm bo}$ (see \S\ref{sec:temp}). If we instead use $R_{\rm bo} = 9 \times 10^{14}$\,cm,
the blackbody radius at $V$-band peak, we calculate $\dot M \approx 0.2$\,M$_{\odot}$\,yr$^{-1}$.
Both of these values are several orders of magnitude above the most extreme mass-loss rates produced by steady winds from stars,
but they are not far from the time-averaged eruptive mass-loss rates from LBVs
\citep[which may well exhibit much higher instantaneous mass-loss rates;][]{2014ARA&A..52..487S}.
Observations of iPTF13beo, a recent SN~Ibn discovered by the intermediate Palomar Transient Factory (iPTF), implied an even higher (but short-lived)
mass-loss rate of $\dot M \approx 2.4$\,M$_{\odot}$\,yr$^{-1}$ immediately before the progenitor underwent core collapse
\citep[estimated via similar methods;][]{2014MNRAS.443..671G}.
If the CSM around SN~2015U\ was launched from the surface explosively rather than through a steady wind, the assumption that
$\rho_{\rm csm} \propto r^{-2}$ is suspect. In fact, the lack of an interaction-powered tail in the light curve of SN~2015U\
indicates that the density profile cannot be wind-like: \citet{2011ApJ...729L...6C} show that ongoing interaction with
a wind-like CSM powers a tail with $L \propto t^{-0.6}$, and SN~2015U\ fades more rapidly than that throughout its post-peak evolution.
The lack of any high-velocity features in our spectra argues that this outer CSM must be optically thick at least out to the radius
of the shock at the time of our last spectrum. Assuming $v_{\rm shock} \approx 20,000$\,km\,s$^{-1}$, the velocity of the
fastest material in unshrouded stripped-envelope SNe, this produces an estimate for the CSM extent of $R \gtrsim 3 \times 10^{15}$\,cm.
Adopting the average \ion{He}{1} velocity measured from our spectra (745\,km\,s$^{-1}$), material at that radius was launched 1--2\,yr before
core collapse. However, it is likely that the SN ejecta are slowed by its collision with the CSM, and so it may be more physically relevant to assume
the CSM extent to be $R \gtrsim 9 \times 10^{14}$\,cm (the blackbody radius at $V$-band peak); material at that radius
was launched $<1$\,yr before collapse.
It is not clear whether the extreme mass loss from SN~2015U's progenitor was episodic and brief
or sustained over a year or more, nor is it known whether the assumption of spherical symmetry is appropriate.
We leave a more thorough examination of these questions to future work.
\subsection{Constraints on Progenitor Variability}
The presence of dense CSM suggests a recent history of extreme mass loss and perhaps variability of the progenitor star. SN~Ibn 2006jc, for example, underwent a bright outburst
($M = -14$\,mag) about 2\,yr before becoming a genuine SN \citep{2006CBET..666....1N,2007ApJ...657L.105F,2007Natur.447..829P},
and there are multiple cases where SN~IIn progenitors have been detected in outburst in the years
prior to core collapse \citep[e.g.,][]{2014ApJ...789..104O}, though such outbursts do not appear to be
ubiquitous \citep[e.g.,][]{2015MNRAS.450..246B}.
KAIT has been monitoring NGC~2388 for almost
20\,yr and we searched this extensive dataset for evidence of pre-explosion variability.
We augment our unfiltered KAIT data with the publicly available PTF/iPTF images
of the field in the $r$ passband.\footnote{\url{http://irsa.ipac.caltech.edu/applications/ptf/}}
Examining 758 images with observed upper limits fainter than 17.0\,mag from
1998 Oct. through 2015 Feb. 10, we find no evidence for previous outbursts of the progenitor
based upon difference imaging (using one of our deepest single exposures as a template).
We stacked our images into rolling-window time bins 20\,d wide and subtracted templates to
search for evidence of previous outbursts of that timescale, but found none. We also stacked all images
together and searched for evidence of a progenitor, but found none. (Note
that we stacked the KAIT and PTF/iPTF images separately, and we did not perform difference imaging on these very deep stacks,
as we had no templates with which to compare.)
Figure~\ref{fig:pre-explosion} plots our 1$\sigma$ single-image nondetections and the observed light curve of SN~2015U.
Though we have regular imaging of NGC~2388 most years since 1998, the field was inaccessible to our
telescopes for several months every year, so there are significant gaps, especially
compared to the timescale of SN~2015U's light curve ($\sim20$\,d). The orange bars along the bottom of
Figure~\ref{fig:pre-explosion} mark every night at which more than 20 days had passed since
the previous upper limit, indicating timespans when a previous SN~2015U-like event could have occurred undetected.
Unfortunately, even the epochs that were covered by our monitoring campaigns do not yield stringent constraints on the
luminosity of a previous outburst, partially owing to the estimated $\sim1.5$\,mag of host-galaxy dust extinction along the line of sight toward SN~2015U.
The detected outburst of SN~2006jc's precursor reached a peak of $M_r \approx -14.1$\,mag \citep{2007Natur.447..829P}, and
such an outburst in SN~2015U's precursor would have probably remained undetected by our monitoring campaign.
\begin{figure}
\includegraphics[width=\columnwidth]{phot_pre-explosion.pdf}
\caption{Our 1$\sigma$ nondetections from KAIT unfiltered images (black) and
PTF/iPTF $R$ images (red). The light curve of SN~2015U\ is shown to the far right.
Timespans for which no upper limit had been obtained for at least 20 days are
marked in orange along the bottom.
\label{fig:pre-explosion}}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper we presented observations of SN~2015U, a highly extinguished, low-velocity, rapidly evolving, luminous, apparently hydrogen-free SN
which exploded in the strongly star-forming galaxy NGC~2388.
Though detailed modeling has yet to be performed, and the degree of host-galaxy dust interference is uncertain,
our data indicate that SN~2015U\ was a core-collapse SN with a peak powered by shock breakout
from a dense CSM rather than radioactive decay. We suggest that this CSM was not wind-like but was
instead created by at least one extreme episode of mass loss
($\dot M \approx 0.2$--1.2\,M$_{\odot}$\,yr$^{-1}$) within a few years of core collapse.
The CSM that surrounded SN~2015U\ was effectively hydrogen-free but was helium-rich; we also detect
features from nitrogen, iron, and probably carbon in our spectra.
No long-lasting light-curve tail was observed from radioactivity or from ongoing CSM interaction,
implying that SN~2015U\ produced a relatively small amount of $^{56}$Ni compared to normal SNe~Ib/c.
SN~2015U\ is a remarkably well-observed SN~Ibn, especially
among the rapidly fading subset of these events, and we find many similarities between it and other
SNe in the literature. Modern surveys indicate the existence of a class of blue, continuum-dominated, hydrogen-deficient, luminous, and rapidly evolving
SNe including those found by \citet{2014ApJ...794...23D} and \citet{2016ApJ...819...35A}; our analysis of SN~2015U\ implies fundamental
similarities between these events, a subset of the SNe~Ibn \citep[e.g.,][]{2016MNRAS.456..853P},
and other rapidly fading SNe of lower luminosity (i.e., SN 2002bj and SN 2010X).
While the exact progenitors of these events are quite uncertain, it is clear that they demand extreme mass-loss rates from their
stripped-envelope progenitor stars.
\section*{Acknowledgements}
The authors thank M.~Drout, R.~Foley, P.~Nugent, D.~Kasen, E.~Quataert,
D.~Poznanski, and O.~Fox for useful comments and discussions.
We are grateful to our referee, A.~Pastorello, for comments that have greatly improved this effort.
We thank the {\it HST}, Keck Observatory, and Lick Observatory staffs for their
expert assistance with the observations.
We also acknowledge the following Nickel 1\,m observers and KAIT checkers
for their valuable assistance with this work:
A.~Bigley, C.~Gould, G.~Halevi, K.~Hayakawa, A.~Hughes,
H.J.~Kim, M.~Kim, P.~Lu, K.~Pina, T.~Ross, S.~Stegman, and H.~Yuk.
A.V.F.'s supernova research group at U. C. Berkeley is supported by NSF
grant AST-1211916, the TABASGO Foundation, Gary and Cynthia Bengier,
and the Christopher R. Redlich Fund. Additional assistance is
provided by NASA/{\it HST} grants AR-14295 and GO-14149 from the
Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under
NASA contract NAS5-26555. J.M.S. is supported by an NSF Astronomy
and Astrophysics Postdoctoral Fellowship under award AST-1302771.
Some of the 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
NASA; the observatory was made possible by the generous financial
support of the W. M. Keck Foundation.
KAIT and its ongoing operation
were supported by donations from Sun Microsystems, Inc., the
Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the
NSF, the University of California, the Sylvia and Jim Katzman
Foundation, and the TABASGO Foundation. Research at Lick Observatory
is partially supported by a very generous gift from Google, as well as by
contributions from numerous individuals including
Eliza Brown and Hal Candee,
Kathy Burck and Gilbert Montoya,
David and Linda Cornfield,
William and Phyllis Draper,
Luke Ellis and Laura Sawczuk,
Alan and Gladys Hoefer,
DuBose and Nancy Montgomery,
Jeanne and Sanford Robertson,
Stanley and Miriam Schiffman,
Thomas and Alison Schneider,
the Hugh Stuart Center Charitable Trust,
Mary-Lou Smulders and Nicholas Hodson,
Clark and Sharon Winslow,
Weldon and Ruth Wood,
and many others.
This research was
based in part on data taken with the NASA/ESA {\it Hubble Space Telescope}, and
obtained from the Hubble Legacy Archive, which is a collaboration
between the Space Telescope Science Institute (STScI/NASA), the Space
Telescope European Coordinating Facility (ST-ECF/ESA), and the
Canadian Astronomy Data Centre (CADC/NRC/CSA). We also made use
of the NASA/IPAC Extragalactic Database (NED) which is operated by
the Jet Propulsion Laboratory, California Institute of Technology,
under contract with NASA.
\bibliographystyle{mnras}
|
2,877,628,089,549 | arxiv | \section{Introduction}
The different response of a molecule to right- and left- circularly polarized light can be measured experimentally and can be explained with a semiclassical theory of multipole oscillators induced by the electric and magnetic fields of the optical wave \cite{book2}. Thanks to advances in computational chemistry calculations of the optical activity of oriented molecules are today relatively easy to do using modern software packages that do quantum-mechanical calculations of molecular property tensors \cite{g09}. Measurements of optical activity started more than two centuries ago and modern chiroptical spectroscopy techniques that exploit the intrinsic chirality of circular polarized light are widely available in chemistry laboratories. However, the experimental study of the optical activity of oriented systems is still challenging, and active research is being carried to find reliable methods to measure the typically small chiroptical contributions embedded in the large optical anisotropy of oriented system. In last few years Mueller matrix spectroscopy has emerged as a promising and powerful technique for these type of measurements \cite{tesismia}. Therefore, it seems that we have reached a point where the experimental and theoretical approaches to study the optical activity of oriented molecules are mature enough to be compared. One additional factor that complicates the reconciliation between these two approaches is that theoretical calculations of optical activity are typically done assuming single molecules while, in most cases, measurements are performed in macroscopic media.
Traditionally, measurements of ensembles of molecules were always done in solution, in which small molecules tend to adopt random orientations. Therefore calculations were always made under the assumption of a large collections of molecules randomly oriented. However, in recent years there is a growing interest to study the richer anisotropic spectroscopic information provided by oriented molecules. There are many strategies to orient a molecule, specially if the molecules are large. The most evident strategy consist of crystallizing them and form a molecular crystal but, in some occasions, it is also possible to incorporate the molecules during the growth of a different crystal that acts as a host \cite{bartdye}. In these cases the optical properties not only depend on the individual molecules but also in the type of crystal lattice. Elongated molecular aggregates are typically orientable by flows \cite{flow} and some molecules can also be oriented by simple mechanical actions such as rubbing \cite{wong}. Other methods to orient molecules use lasers pulses or other forms of electric and magnetic fields \cite{gueri}. However, in this work, we will consider that there are no additional external fields affecting the molecules during the optical activity measurements.
In this paper we discuss the theoretical basis that permits the correlation between computations of the averaged molecular polarizability tensors and measurements. We show that the mathematical tools used to describe measurements and calculations naturally converge if a bianisotropic formulation of the material constitutive equations is considered. The key element of the analysis is to refer any measured or calculated optical activity to elements of the magnetoelectric tensor of gyration that is included in the bianisotropic constitutive equations. The results we find simplify the comparison between calculations and experiments.
\section{Bianisotropic constitutive equations and multipole theory}
The following form of bianisotropic constitutive equations was first given by Tellegen \cite{tellegen}:
\begin{subequations}\label{uno}
\begin{equation}
\mathbf{D}=\rtta{\varepsilon} \mathbf{E}+\rtta{\rho} \mathbf{H},
\end{equation}
\begin{equation}
\mathbf{B}=\rtta{\mu} \mathbf{H}+\rtta{\rho'} \mathbf{E},
\end{equation}
\end{subequations}
where frequency-domain fields are considered and the choice of the time dependence is given by $\exp(i\omega t)$. $\rtta{\varepsilon}$ is the permittivity dyadic, $\rtta{\mu}$ the permeability dyadic and $\rtta{\rho}$ and $\rtta{\rho'}$ are the two magnetoelectric dyadics \cite{malaka} and transmit the relation between the electric and magnetic field quantities $\mathbf{E}$, $\mathbf{H}$ and the flux quantities $\mathbf{D}$, $\mathbf{B}$. These four constitutive dyadics contain full information of the electromagnetic response of a bianisotropic medium. Lately, this form about the constitutive equations has attracted a lot of attention for the optical characterization of metamaterials \cite{advances}, because these materials typically have large magnetoelectric tensors. One common simplification for Eqs. \eqref{uno} is that the specific medium should be Lorentz-reciprocal \cite{lakabook1}. This implies:
\begin{equation}\label{const}
\rtta{\varepsilon}=\rtta{\varepsilon}^T, \quad \rtta{\mu}=\rtta{\mu}^T, \quad \rtta{\rho}=-\rtta{\rho}'^T,
\end{equation}
where the superscript $T$ indicates transposition. In this case the magnetoelectric dyadics, that in general are written as $\rtta{\rho}=\rtta{\chi}+i\rtta{\kappa}$ and $\rtta{\rho'}=\rtta{\chi}^T-i\rtta{\kappa}^T$, must satisfy $\rtta{\chi}=0$ and $\rtta{\kappa}\neq0$. Therefore, the constitutive equations for a bianisotropic reciprocal medium can be rewritten as:
\begin{subequations}\label{constitutivesimplied2}
\begin{equation}
\mathbf{D}=\rtta{\varepsilon} \mathbf{E}+i\rtta{\kappa} \mathbf{H},
\end{equation}
\begin{equation}
\mathbf{B}=\rtta{\mu} \mathbf{H}-i\rtta{\kappa}^T \mathbf{E}.
\end{equation}
\end{subequations}
It should be indicated that, in the presence of absorption, all the tensors in this constitutive equations, i.e. $\rtta{\varepsilon}$, $\rtta{\kappa}$ and $\rtta{\mu}$, become complex. For systems uni- and biaxial orthorhombic crystallographic symmetries the real and imaginary parts of the tensor have the same system of principal axes. However, the situation is more complicated if the tensors have no such system, as can happen for media with monoclinic and triclinic symmetries, as they can lead to apparent non-reciprocal optical response, despite being reciprocal media.
Multipole theory can be used to calculate the reciprocal magnetolectric coupling (now given by $\rtta{\kappa}$) between the electric and magnetic fields \cite{raab2}. Note that Eqs. \eqref{const} do not necessarily constraint $\rtta{\kappa}$ to be a symmetric tensor. However, natural optical activity is a Lorentz reciprocal effect and only it can be contributed by the symmetric part of the magnetoelectric tensor. In this work we only consider this part of the tensor as we are interested in natural optical activity and non-magnetic media. For a complete description of natural optical activity it has been shown that, within a semiclassical theory, in addition to the mean electric dipole induced by the electric field of light it is necessary to include: the electric dipole contribution by the time-derivative of the magnetic field of the light wave, the associated mean magnetic dipole induced by the time-derivative of the electric field, the electric dipole contribution induced by the electric field gradient of the electromagnetic wave and the electric quadrupole contribution induced by the electric field \cite{dunn}. When all these contributions are considered the oscillating induced moments (the electric dipole $\hat{\mu}_{\alpha}$, the magnetic dipole $\hat{m}_{\alpha}$ and the electric quadrupole $\hat{\Theta}_{\alpha\beta}$) are the real parts of the following complex expressions \cite{book2}:
\begin{subequations}\label{moments}
\begin{equation}
\hat{\mu}_\alpha=\hat{\alpha}_{\alpha\beta}\hat{E}_{\beta}+\hat{G}_{\alpha\beta}\hat{B}_{\beta}+\frac{1}{3}\hat{A}_{\alpha\beta\gamma}\nabla_{\beta}\hat{E}_{\gamma}+...,
\end{equation}
\begin{equation}
\hat{m}_\alpha=\hat{G}_{\beta\alpha}\hat{E}_{\beta}+...,
\end{equation}
\begin{equation}
\hat{\Theta}_{\alpha\beta}=\hat{A}_{\gamma\alpha\beta}\hat{E}_{\gamma}+...,
\end{equation}
\end{subequations}
where $\hat{E}_{\alpha}$ and $\hat{B}_{\alpha}$ the actual electric and magnetic fields of the optical wave and $\nabla_{\beta}\hat{E}_{\gamma}$ is the electric field gradient. $\hat{\alpha}_{\alpha\beta}$ is the electric dipole-electric dipole polarizability tensor, $\hat{G}_{\alpha\beta}$ is the electric dipole-magnetic dipole polarizability tensor and $\hat{A}_{\alpha\beta\gamma}$ is the electric dipole-electric quadrupole polarizability tensor. Higher order polarizabilities (e. g. electric octopole, magnetic quadrupole,...) have not been specified because they do not contribute to optical activity \cite{raabbook}. The hats $\hat{~}$ stress that these are complex quantities. Computational software has been developed to compute all these tensors quantum mechanically by applying time-dependent perturbation theory to molecular orbitals \cite{g09}. In these calculations the electric and magnetic dipole moments and the electric quadrupole moment are usually treated as microscopic quantities that apply to single molecules. However, in this paper we discuss the optical activity of a macroscopic medium and, therefore, all the moments included in Eqs. \eqref{moments} should be regarded as macroscopic moment densities, i. e. their statistical average multiplied by the number density. We are also assuming that local field effects are small enough to be neglected. From this point, we refer always to moment densities instead of molecular moments and to macroscopic fields instead of local fields.
In a nonmagnetic medium the molecular property tensors $\hat{\alpha}_{\alpha\beta}$, $\hat{G}_{\alpha\beta}$ and $\hat{A}_{\alpha\beta\gamma}$ must define the constitutive dyadics of the bianisotropic Eqs. \eqref{constitutivesimplied2} so that the optical response of an oriented molecule or crystal is calculated from them. Following the work of Graham and Raab \cite{raab1} we find the following relations:
\begin{subequations}\label{unodos}
\begin{equation}
(\rtta{\varepsilon})_{\alpha\beta}=\varepsilon_0\delta_{\alpha\beta}+\hat{\alpha}_{\alpha\beta},
\end{equation}
\begin{equation}
(\rtta{\mu})_{\alpha\beta}=\mu_0^{-1}\delta_{\alpha\beta},
\end{equation}
\begin{equation}\label{factor}
(\rtta{\kappa})_{\alpha\beta}=\frac{1}{2}[(-i\hat{G}_{\alpha\beta}-i\hat{G}_{\beta\alpha})+\frac{\omega}{3}(\varepsilon_{\beta\gamma\delta}\hat{A}_{\gamma\delta\alpha}+\varepsilon_{\alpha\gamma\delta}\hat{A}_{\gamma\delta\beta})],
\end{equation}
\end{subequations}
where $\varepsilon_{\beta\gamma\delta}$ is the Levi-Cevita operator and $\delta_{\alpha\beta}$ is the Kronecker delta. We are using the sum over indices Einstein convention and the notation is the same as in Refs. \cite{book2,dunn}. Eq. \ref{factor} can be found, apart from a numerical factor, in Ref. \cite{dunn}, but it is different from its equivalent appearing in Ref. \cite{raab1} because here we have disregarded nonreciprocal factors. As natural optical activity is a reciprocal optical phenomena only reciprocal contributions (time-even) need to be considered and the nonreciprocal factors (time-odd) can be safely neglected. The condition $\rtta{\rho}=-\rtta{\rho}'^T$ is automatically satisfied because $\hat{\alpha}_{\alpha\beta}$ as well as the sums ($\hat{G}_{\alpha\beta}+\hat{G}_{\beta\alpha}$) and ($\varepsilon_{\beta\gamma\delta}\hat{A}_{\gamma\delta\alpha}+\varepsilon_{\alpha\gamma\delta}\hat{A}_{\gamma\delta\beta}$) are all symmetric contributions. An additional difference compared to Ref. \cite{raab1} is that we get a 1/3 factor in front of the $A_{\gamma\delta\alpha}$ terms, as opposed to the 1/2 factor obtained by Graham and Raab. This change arises because they used a so-called primitive definition of the electric quadrupole moment compared to the more extended traceless definition used for example by Buckingham and Dunn \cite{dunn}. Both, the traceless and primitive definitions, are applicable because it has been shown that they allow for an origin independent description of the theoretical optical activity \cite{dunn,auts,raab2}, i.e. its final expression is not dependent on the arbitrary choice of coordinate origin.
\section{Optical activity in terms of the magnetolectric tensor of gyrotropy}
Eq. \eqref{factor} is specially important because provides the connection between the molecular polarizability tensors and the magnetoelectric tensor. This equation appears in the seminal publication of Buckingham-Dunn (Eq. [19] in Ref. \cite{dunn}) but, as far as we know, this work does not offer a clear interpretation about the meaning of $\kappa_{\alpha\beta}$. Apparently, it was introduced as an accessory equation for their calculation of optical activity but it was not highlighted as a main result of this publication.
The elements of the magnetoelectric tensor of gyrotropy $\kappa_{\alpha\beta}$ (from this point we omit the dyadic notation) are calculated expanding Eq. \eqref{factor}:
\begin{subequations}\label{kap}
\begin{equation}\label{one}
\kappa_{xx}=-i\hat{G}_{xx}+\frac{\omega }{3}(\hat{A}_{yzx}-\hat{A}_{zyx}),
\end{equation}
\begin{equation}\label{two}
\kappa_{yy}=-i\hat{G}_{yy}+\frac{\omega }{3}(\hat{A}_{zxy}-\hat{A}_{xzy}),
\end{equation}
\begin{equation}
\kappa_{zz}=-i\hat{G}_{zz}+\frac{\omega }{3}(\hat{A}_{xyz}-\hat{A}_{yxz}),
\end{equation}
\begin{equation}
\kappa_{xy}=\kappa_{yx}=-\frac{i}{2}(\hat{G}_{xy}+\hat{G}_{xy})+\frac{\omega }{6}(\hat{A}_{yzy}+\hat{A}_{zxx}-\hat{A}_{zyy}-\hat{A}_{xzx}),
\end{equation}
\begin{equation}
\kappa_{yz}=\kappa_{zy}=-\frac{i}{2}(\hat{G}_{yz}+\hat{G}_{zy})+\frac{\omega }{6}(\hat{A}_{zxz}+\hat{A}_{xyy}-\hat{A}_{xzz}-\hat{A}_{yxy}),
\end{equation}
\begin{equation}
\kappa_{xz}=\kappa_{zx}=-\frac{i}{2}(\hat{G}_{xz}+\hat{G}_{zx})+\frac{\omega }{6}(\hat{A}_{yzz}+\hat{A}_{xyx}-\hat{A}_{zzy}-\hat{A}_{yxx}),
\end{equation}
\end{subequations}
where $\hat{A}_{\alpha\beta\gamma}$ holds the symmetry property $\hat{A}_{\alpha\beta\gamma}=\hat{A}_{\alpha\gamma\beta}$ \cite{book2}. To relate this result with an experiment it is still necessary to investigate the relation between the optical activity measured in a given molecular direction and the components of $\kappa_{\alpha\beta}$. Optical activity is usually expressed in terms of circular birefringence CB (twice the optical rotation) and circular dichroism (CD). Both quantities can be measured independently and they can be combined into a complex expression given by
\begin{equation}\label{optact}
\mathrm{C}=\mathrm{CB}-i\mathrm{CD}=(\hat{n}_--\hat{n}_+)\frac{\omega l}{c},
\end{equation}
where $\hat{n}_-$ and $\hat{n}_+$ are, respectively, the complex refractive indices for left an right circularly polarized light. $l$ is the pathlength of the medium. A straight calculation of the values of $\hat{n}_-$ and $\hat{n}_+$ from the tensors $\varepsilon_{\alpha\beta}$, $\mu_{\alpha\beta}$ and $\kappa_{\alpha\beta}$ is in general a complicate task. The problem of light propagation (assuming planewaves) through a reciprocal bianosotropic medium is usually treated by eigenanalysis of a wave equation, which provides the refractive indices of the eigenpolarizations that propagate through the medium. For chiral isotropic medium or along the optic axis of a chiral uniaxial medium in which the eigenmodes are circularly polarized waves it can be easily shown that C depends only on the magnetoelectric parameter or tensor \cite{malaka}. However, the two eigenmodes, with complex refractive indices $\hat{n}_{\sigma}$ and $\hat{n}_{\delta}$, in general do not correspond with circularly polarized waves and, therefore, the values of $\hat{n}_-$ and $\hat{n}_+$ are not directly obtained.
With the help of the Jones formalism for polarization optics, we showed in Ref. \cite{recno} that optical activity ($\hat{n}_--\hat{n}_+$) could be calculated from the complex retardation between the eigenmodes $(\hat{n}_{\sigma}-\hat{n}_{\delta})$ with the following equation:
\begin{equation}\label{mybestequation}
\hat{n}_--\hat{n}_+=(\hat{n}_{\sigma}-\hat{n}_{\delta})i\frac{1+k_{\sigma}k_{\delta}}{k_{\sigma}-k_{\delta}}
\end{equation}
in which $k_{\sigma}$ and $k_{\delta}$ are given by the polarization of each eigenmode, namely $k_{\delta}=E_{\delta\perp}/E_{\delta\parallel}$ and $k_{\sigma}=E_{\sigma\perp}/E_{\sigma\parallel}$, where $E_i$ are electric field amplitudes. Note that in simple case of a chiral isotropic medium the eigenmodes are circularly polarized waves $k_{\sigma}=i$ and $k_{\delta}=-i$ and, obviously, $\hat{n}_--\hat{n}_+=\hat{n}_{\sigma}-\hat{n}_{\delta}$.
Using Eq. \eqref{mybestequation} it is possible to calculate the dependence of optical activity with the material's constitutive tensors of Eq. \eqref{constitutivesimplied2} for any direction of light propagation. In appendix A we show the results of this calculation for an uniaxial medium with crystallographic point group of symmetry 32, 422 or 622 and in which the optic axis is perpendicular to the direction of propagation of light. In practice, for media with lower symmetry it is very difficult to keep the eigenanalysis at an analytical level and it is more practical to do the calculation numerically. For constitutive tensors belonging to any crystal symmetry it is found, either analytically (as in Appendix A) or numerically, that C depends \emph{only} on the components of the tensor magnetoelectric tensor $\kappa_{\alpha\beta}$, so that this tensor fully describes the optical activity of the sample and that $\varepsilon_{\alpha\beta}$ is completely irrelevant to optical activity.
For any crystal class, the dependence of C with $\kappa_{\alpha\beta}$ follows a general rule: C can be calculated from $\kappa_{\alpha\beta}$ by adding the two components of the tensor that are perpendicular to the direction of propagation of the beam. This simple result was already suggested for some particular cases in \cite{recno}, but now we can generalized it with the following equations that apply to ensembles of molecules:
\begin{subequations}\label{orto}
\begin{equation}
C_{xx}=\frac{\omega}{c}\mu_0lN<\kappa_{yy}+\kappa_{zz}>,
\end{equation}
\begin{equation}
C_{yy}=\frac{\omega}{c}\mu_0lN<\kappa_{xx}+\kappa_{zz}>,
\end{equation}
\begin{equation}\label{cb1}
C_{zz}=\frac{\omega}{c}\mu_0lN<\kappa_{xx}+\kappa_{yy}>,
\end{equation}
\begin{equation}
C_{xy}=\frac{\omega}{c}\mu_0lN<\kappa_{zz}+\kappa_{yx}>,
\end{equation}
\begin{equation}
C_{xz}=\frac{\omega}{c}\mu_0lN<\kappa_{yy}+\kappa_{zx}>,
\end{equation}
\begin{equation}
C_{yz}=\frac{\omega}{c}\mu_0lN<\kappa_{xx}+\kappa_{zy}>,
\end{equation}
\end{subequations}
where the angular brackets $<>$ denote a statistical average. $N$ is the number of molecules per unit volume and $l$ is the pathlength. The subscript in $C_{ij}$ indicates the direction of measurement of optical activity, e.g. $C_{xy}$ corresponds to optical activity in the direction that bisects the $x$ and $y$ axes. As $\kappa_{ij}=\kappa_{ji}$, it immediately follows that $C_{ij}=C_{ji}$ because natural optical activity is a reciprocal phenomenon.
A qualitative understanding of the dependence of C with $\kappa_{\alpha\beta}$ is gained by realizing that a circularly polarized wave can be decomposed into the sum of two orthogonal linearly polarized components that are in the plane of polarization and have a phase difference of $90^{\circ}$. The sign of this phase difference determines the handedness of the wave. Only the tensor components that correspond to two these two orthogonal directions are able to contribute in a different way to left- and right- circularly polarized waves. The other non-vanishing tensor elements (either in the magnetoelectric or dielectric tensors) affect the absorption and refraction of light but they have exactly the same effect for left- and right- circularly polarized wave and, therefore, they do not contribute to C.
For example, to calculate the optical activity corresponding to light propagating in the z or -z direction of an oriented medium Eq.\eqref{cb1} is used in combination with Eqs. \eqref{one} and \eqref{two}:
\begin{equation}
C_{zz}=\frac{\omega}{c}\mu_0lN<-i\hat{G}_{xx}-i\hat{G}_{yy}+\frac{\omega}{3}(\hat{A}_{yzx}-\hat{A}_{xyz})>,
\end{equation}
where it has been used that $A_{zyx}=A_{zxy}$. According to Ref. \cite{dunn} at transparent frequencies $\hat{G}_{\alpha\beta}=-iG'_{\alpha\beta}$ and $\hat{A}_{\alpha\beta\gamma}=A_{\alpha\beta\gamma}$, where $G'_{\alpha\beta}$ and $A_{\alpha\beta\gamma}$ are real tensors, so it is possible to rewrite:
\begin{equation}
CB_{zz}=-\frac{\omega}{c}\mu_0lN<G'_{xx}+G'_{yy}+\frac{\omega}{3}(\hat{A}_{xyz}-\hat{A}_{yzx})>,
\end{equation}
which is the same result found in \cite{dunn}.
As a practical demonstration of the implications of Eqs. \eqref{orto}, we have plotted in Fig. \ref{figtensor} a visual representation of the value of CB as a function of the orientation of a molecule. In all cases we have considered an incoming linearly polarized light beam propagating along the z axis. Panels a, b and c correspond to a molecule with crystallographic point group of symmetry $\bar{4}2m$ or mm2 at three different orientations. The water (H$_2$O) molecule could be a good an example of this symmetry, which is characterized by a magnetoelectric tensor having only one independent component. Molecules belonging to this group do not have optical activity in solution because when all directions of the space are considered the magnetoelectric tensor averages to zero (which happens because the trace of the tensor matrix is always zero) \cite{kaminsky0}. However, for the orientations shown in b and c there exits CB and it, respectively, takes opposite signs. Panels d, e and f show a molecule with crystallographic point group of symmetry 3, 32 or 622 that has a magnetoelectric tensor with two different tensor components. In this example we have considered that these two components keep a relation -2:1. Note that the trace of the magnetoelectric tensor for molecules holding this ratio would be zero and, consequently, they would neither be optically active in solution. For each case the CB along the z axis is given by the real part of the sum of the tensor elements $\kappa_{xx}+\kappa_{yy}$, which is theoretically calculated using Eq. \eqref{cb1}.
\section{Comparison with the classical tensor description for optical activity}
The correspondence between calculated of molecular tensor and optical activity values is treated in other works considering another tensor, usually called gyration tensor or optical activity tensor, and typically represented by $g_{\alpha\beta}$ \cite{newnham,book2,raabcubic,veronica}. This tensor is based in the so-called equation of the normals:
\begin{equation}\label{eqNO}
(n^2-n^2_{01})(n^2-n^2_{02})=G^2,
\end{equation}
where $n$ gives the possible values of the refractive of the refractive index, for a given direction of the wave normal, $n_{01}$ and $n_{02}$ are the refractive indices of the eigenwaves propagating in the crystal in the absence of optical activity, and $G$ is the scalar gyration parameter
\begin{equation}
G=g_{\alpha\beta}N_{\alpha}N_{\beta}.
\end{equation}
$N_{\alpha}$ and $N_{\beta}$ are direction cosines of the wave normal. Eq. \eqref{eqNO} is typically approximated to $(n^2-\bar{n}^2)=G^2$ by assuming that the birefringence of the system is not too large ($n_{01}\cong n_{02}\cong \bar{n} =\sqrt{n_{01}n_{02}}$). Then if $G$ is assumed to be very small, the two solutions of this equation can be written as $n=\bar{n}\pm G/2n$ that correspond to the refractive indices for left and right circularly polarized waves. Then the optical activity [Eq. \eqref{optact}] at a given direction can be written as
\begin{equation}
C_{ij}=\frac{\omega l}{\bar{n}c}G= \frac{\omega l}{\bar{n}c}g_{\alpha\beta}N_{\alpha}N_{\beta},
\end{equation}
this equation is similar to Eqs. \eqref{orto}, but here the magnetoelectric tensor does not appear at all and and it has been replaced by a different tensor.
The description of optical activity based on the tensor $g_{\alpha\beta}$ is problematic not only because it works uniquely within the approximations that we have detailed above, but also because it arises from an incorrect formulation of the constitutive equations for optical activity. To the best of our knowledge Eq. \eqref{eqNO} was first proposed by Szivessy in 1928 \cite{szi} and it requires constitutive equations in which the permittivity tensor is perturbed by the gyration tensor. This formulation corresponds to the classic treatment of optical activity by M. Born \cite{born}, which neglects any magnetoelectric tensor and optical activity is then justified by modification of the permittivity tensor:
\begin{equation}
\varepsilon_{ik}=\varepsilon_{ik}^{(0)}-i\epsilon_{ikl}g_{lm}N_{m},
\end{equation}
where $\epsilon_{ikl}$ is the Levi-Civita symbol, $g_{lm}$ are the components of the optical activity tensor and $N_{m}$ are the direction cosines of the wave normal. This form of the constitutive equations is similar to the one used for describing magneto-optical phenomena (Faraday effect and magneto-optic Kerr effect) but, despite being quite widespread in classical crystal optics texts \cite{nye,newnham}, it is not suitable for natural optical activity as it violates fundamental principles that natural optical activity should preserve \cite{laka2,silvermangyro}.
The magnetoelectric tensor $\kappa_{\alpha\beta}$ corresponding every crystal class takes the same forms of symmetry as the gyration tensor $g_{\alpha\beta}$ (see for example the tables in Refs. \cite{nye, newnham}) because the same constraints based on Neumann's principle are applicable. However, the values of the non-vanishing elements of $\kappa_{\alpha\beta}$ bear no relation with those of $g_{\alpha\beta}$. In general, we strongly recommend to use $\kappa_{\alpha\beta}$ instead of $g_{\alpha\beta}$ to express the experimental or calculated optical activity of oriented systems.
\section{Conclusions}
The optical activity of anisotropic crystals or oriented molecules is adequately described by a reciprocal bianisotropic formulation of the constitutive equations. CB and CD for any molecular or crystallographic direction are defined by the symmetric magnetoelectric tensor of chirality $\kappa_{\alpha\beta}$ which, in turn, is expressed in terms of two molecular property tensors. These contain the effect of the mean electric and magnetic dipoles induced, respectively, by the time-derivative of the magnetic and electric fields of the light wave, as well as the electric dipole induced by the electric field gradient of the wave and the electric quadrupole induced by the electric field.
We have shown that the optical activity found by a light plane wave propagating along a certain direction of a molecule or a crystal is given by the sum of two components of $\kappa_{\alpha\beta}$ that are perpendicular to this direction of propagation. These two tensor components describe the change of polarization of light that is due to the special interaction between the electric and magnetic fields in systems with optical activity. With bianisotropic constitutive equations this change in polarization is only described by the magnetoelectric tensor and it is separable from the change in polarization due to the linear birefringence or linear dichroism typical of oriented systems. We expect that the use of the bianisotropic formalism will simplify the comparison between experiments and calculations and it will bring the field of optical activity in molecules to a closer connection with that of artificial metamaterials.
\section*{Acknowledgements}
The author acknowledges financial support from a Marie Curie IIF Fellowship (PIIF-GA-2012-330513 Nanochirality). He is also grateful to V. Murphy and B. Kahr for helpful discussions.
\newpage
|
2,877,628,089,550 | arxiv | \section{Introduction}
Vacuum asymptotically flat Robinson-Trautman spacetimes have been the subject
of much study since their discovery over thirty years ago.\R(RT60) They
possess some very nice features which make them amenable to analysis. They
are in some sense the simplest asymptotically flat solutions which exhibit
gravitational radiation, albeit of a fairly specialised nature. The feature
of the spacetime which simplifies analysis is that the full spacetime can be
built up from the solution of a fourth order parabolic equation on a 2+1
dimensional manifold. This is a consequence of the fact that the coordinate
system uses a retarded time coordinate, so that in effect initial data is
prescribed on a null hypersurface.
Early studies of the Robinson-Trautman equation related to behaviour of solutions of the
linearised equation.\R(FN67,Van87) Luk\'acs {\it et.al.}\R(LPPS) studied the
behaviour of solutions of the full nonlinear equation, using the concept of
Lyapunov stability to establish that global solutions, if they existed, would
converge to the static Schwarzschild equilibrium. A number of authors
subsequently focused on the existence of such solutions. Schmidt\R(Sch88)
showed local existence of solutions for sufficiently differentiable but
otherwise arbitrary initial data. Rendall\R(Ren88) showed global existence
for sufficiently small initial data, antipodally symmetric on the sphere, and
his proof was extended by Singleton\R(Sin90a) to remove the requirement of
antipodal symmetry. Finally Chru\'sciel\R(Chr91) proved semi-global existence
of solutions for arbitrary smooth initial data. Numerical studies by
Perj\'es\R(Per89) and Singleton\R(Sin90b) demonstrate vividly the evolution of
an initially perturbed spacetime to the steady state. It is interesting to
note that the evolution equations exhibit exponential divergence from
arbitrary initial data in the ``backwards'' time direction, and that no
solutions exist in this direction.\R(Chr91) It should be emphasised these
results, and the subsequent discussion, only apply to asymptotically flat Robinson-Trautman
spacetimes with regular topological $S^2$ surfaces: Perj\'es and
Hoenselaers\R(HoePer) have shown that there exists a class of Robinson-Trautman spacetimes
with cusp singularities which evolve to static C-metrics.
Thus the evolution of the Robinson-Trautman spacetimes would seem to be thoroughly
understood. We would expect then that the behaviour of physical features of
the spacetimes should not be too difficult to pin down. Near $\Scri+$, the
spacetimes, because of their algebraically degenerate structure, could
reasonably be interpreted as describing purely outgoing radiation around a
black hole source: e.g. the decaying tail of the radiation after the black
hole has formed (Note that there is no ingoing radiation, which means there
can be no backscattering or similar self-interaction of the radiation
field). However, in this paper, we will be studying the structure of past
apparent horizons in vacuum Robinson-Trautman spacetimes; thus we will focus our attention
on the behaviour of the ``white hole'' region of the spacetimes. Since these
spacetimes evolve to the Schwarzschild geometry, the future apparent horizons
coincide with the future event horizons at $u=\infty$ ($r=2m$ in Schwarschild
coordinates).
Our study of the apperent horizon structure of these spacetimes was initially
motivated by an investigation of the stability of the related class of
electrovac spacetimes, the Robinson-Trautman Einstein-Maxwell spacetimes, discussed elsewhere.\R(LC94)
The asymptotically flat vacuum Robinson-Trautman spacetimes turn out to have a particularly well behaved
apparent horizon which illustrates several general theorems concerning
apparent horizons. There are in fact very few well understood exact
solutions which can be used to illustrate such theorems -- there being few
examples of non-static black hole type solutions. The \afRobinson-Trautman spacetimes, in
some sense the simplest asymptotically flat spacetimes admitting
gravitational radiation, should be a useful example for the study of
radiating black hole spacetimes in general and many of the general theorems
relating to non-stationary spacetimes. The basic equations and features of
the spacetimes are outlined in Section 2. Section 3 contains the results
concerning the past apparent horizon and its properties. Some numerical
demonstration of these results is presented in Section 4.
\vfill\pagebreak
\section {Robinson-Trautman vacuum spacetimes}
\subsection{Basic equations and notation}
The line element for the Robinson-Trautman class of spacetimes is given by:
\begin{eqnarray
ds^2 = 2Hdu^2 + 2dudr - {2r^2d\zeta d\bar\zeta \over P^2 }
\label{eq:metric} \end{eqnarray
where $H= -r(\ln P),_{u} + \frac{1}{2}K - \frac{m}{r}$, and $K=\Delta(lnP)$.
The vacuum Robinson-Trautman spacetimes are
foliated by a two parameter family of 2-surfaces ${\cal S}_{u,r}$ which, for the
asymptotically flat case, have spherical topology.
The operator $\Delta = 2P^2\partial_{{\zeta}{\bar\zeta}}$ is the Laplacian on these
two-surfaces. For the vacuum spacetimes, $m=m(u)$ and we can use a coordinate
transformation on $u$ to make $m$ constant. We are interested in the $m>0$
case, as this gives rise to a spacetime with positive Bondi mass. The
spacetime is then determined by the evolution of $P = P(u,{\zeta},{\bar\zeta})$ on a
background two-sphere. The evolution equation is often written as
\begin{eqnarray
(\ln P),_u = -{1\over12m}\Delta K \label{eq:rteqn}
\end{eqnarray
This equation, the Robinson-Trautman equation, is also known in the
literature as the two-dimensional Calabi equation.\R(Tod89)
The steady state,
corresponding to the Schwarzschild solution, is given by
$P=P_0 = \sf(1,{\sqrt{2}})(a + b{\zeta} + \b b{\bar\zeta} + c{\zeta}{\bar\zeta})$, where $ac-b\b{b}=1$.
Note that the equilibrium
value of $P$ is not unique but includes a freedom corresponding to conformal
motions on the sphere. This freedom was a contributing factor to the
difficulty of proving existence of solutions of
the Robinson-Trautman equation.\R(Chr91) The condition $ac-b\b b=1$ normalises the Gaussian
curvature of $S^2$ to $K=1$.
The function $P$ determines the induced 2-metric on each ${\cal S}_{u,r}$, which is
given by
$$ g_{\cal S} = {2r^2 d{\zeta} d{\bar\zeta} \over P^2} = e^{2\Lambda}g_0 \label{eq:S2metric}$$
where $g_0$ is the metric of $S^2$, and $e^{-\Lambda} = {P\over P_0}$.
It is possible to ``factor out'' the background $S^2$ geometry and write the
Robinson-Trautman equation as
\begin{eqnarray
e^{2\Lambda} \Lambda,_u ={1\over 12m}\Delta_{0}{K} \label{eq:rteqn2}
\end{eqnarray
where $\Delta_{0}$ is the Laplacian on $S^2$. We thus solve the evolution equation
on a background sphere, say at $r=1$.
\vfill\pagebreak
\subsection{Conserved quantities and Lyapunov functionals}
Several conserved integrals are implied by the evolution equations.
First, we have the conservation of surface area of ${\cal S}_{u,r}$,
and the conservation of the ``irreducible mass'', as a consequence of this:
$$ A_{\cal S} = \int_{{{\cal S}_\infty}}{\vol1} = \int_{S^2}{e^{2\Lambda}\vol0} = 4\pi$$
$$ M_I = {1\over 4\pi} \int_{{{\cal S}_\infty}}{m\vol1} = {m\over 4\pi}
\int_{S^2}{e^{2\Lambda}\vol0}= m.\label{eq:MIrr}$$
The Gaussian curvature, in terms of $\Lambda$, takes the form $K= e^{-2\Lambda}( 1 -
\Delta_{0}\Lambda)$ which, making use of Stokes' theorem, immediately gives the
conservation of the Euler number for ${\cal S}_{u,r}$:
$$\chi_{{\cal S}} = {1\over 2\pi}\int_{S^2}{Ke^{2\Lambda}\vol0} =2.$$
Singleton\R(Sin90b) gave an expression for the Bondi-Sachs mass of the
spacetime:
\begin{eqnarray
M_B ={m\over4\pi}\int_{S^2}{e^{3\Lambda} \vol0} \label{eq:BMdef}
\end{eqnarray
which is manifestly positive and in fact is bounded below by
the irreducible mass, since by the H\"older inequality
$$ \left(\int_{S^2}\vol0\/\right)^{1\over 3\/}
\left(\int_{S^2} e^{3\Lambda}\vol0\/\right)^{2\over 3\/}
\geq \int_{S^2} e^{2\Lambda}\vol0 \label{eq:massbound} $$
That the Bondi mass is monotonically decreasing can be shown
by differentiating \E(BMdef) and using the Robinson-Trautman equation:
\begin{equation}
\frac{d}{du} M_B = -\frac{1}{4\pi}\int_{S^2}
e^\Lambda ({P_0}^2(e^{-\Lambda}),_{\bar\zeta}),_{\bar\zeta} ({P_0}^2(e^{-\Lambda}),_{\zeta}),_{\zeta}\vol0
\quad \leq 0 \label{eq:BMdecrease}
\end{equation}
These characteristics enabled Singleton\R(Sin90b) to show that the Bondi Mass
is a Lyapunov functional for the Robinson-Trautman evolution, complementing the earlier
work of Luk\'acs et. al\R(LPPS) who showed that the integral $\int_{S^2} K^2
e^{2\Lambda}\vol0$ is a Lyapunov functional for the Robinson-Trautman evolution. These Lyapunov
functionals played a key role in Chru\'sciel's semi-global existence
proof.\R(Chr91)
\vfill\pagebreak
\section {The Past Apparent Horizon}
\subsection{General Remarks}
Apparent horizons in black hole spacetimes have been the subject of much study
in recent times. The motivation for the study of the future or past apparent
horizons is that they provide a locally characterisable indication of the
presence of an event horizon (future) or particle horizon (past). Indeed, in
cases where the event horizon or particle horizon cannot be properly defined,
the apparent horizon may be the only useful definition of the ``surface'' of
the black hole or white hole.\R(Hay94) This arguably is the case in the Robinson-Trautman
spacetimes, where $\Scri-$ cannot be properly defined due to the instability
of the Robinson-Trautman equation in the negative $u$ direction.\R(Chr91) Apparent horizons
have been widely used in numerically generated spacetimes as an indicator of
the presence of a black hole,\R(NKO84,CY90) in lieu of being able to detect
the event horizon in a local way\footnote{ In recent work, Anninos
{\em et. al\/}\R(Ann94) have been able to locate the event horizon in numerically
generated spacetimes by integrating backwards from the stationary final state
of the black hole}. It is believed that the presence of an apparent horizon
indicates the presence of a nearby event horizon (where it exists), and it has
been shown that for all stationary black hole spacetimes the apparent and
event horizons coincide.\R(HE) Recent work has shown that the apparent
horizon of a black hole has ``thermodynamic'' properties: in particular it has
been shown that a future apparent horizon can only increase in area, while a
past apparent horizon can only decrease in area, obeying a law like that of
the first law of thermodynamics.\R(Col92,Hay94)
The Robinson-Trautman spacetimes, as they settle down to the Schwarzschild
solution as $u\rightarrow\infty$, would be expected to have a well behaved
future apparent horizon coinciding with the event horizon. Due to the
presence of outgoing radiation, we would expect the past apparent horizon to
be more complex. As a general (and non rigorous) consideration, it can be
seen quite clearly from the line element \E(metric) that the curvature
singularity at $r=0$ is a spacelike singularity i.e. that if the spacetime
were complete in the past null direction (if $\Scri-$ were complete) then
there would exist a particle horizon -- the past counterpart of the event
horizon (see \Fig(pd)). Effectively, the past singularity being spacelike
indicates that no null geodesics from $\Scri-$ can reach the singularity, if
the spacetime is to be causally well-behaved. Because the particle horizons
is a globally defined object, it does not exist in Robinson-Trautman spacetimes. However,
the spacelike nature of the past singularity suggests that there should always
be a past apparent horizon in these spacetimes, which we believe to be
causally well-behaved.
\begin{figure}[th]
\centering
\setlength{\unitlength}{0.007500in}%
\begin{picture}(0,0)%
\special{psfile=pendiag.eps}%
\end{picture}%
\begingroup\makeatletter\ifx\SetFigFont\undefined
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y
\gdef\SetFigFont#1#2#3{%
\ifnum #1<17\tiny\else \ifnum #1<20\small\else
\ifnum #1<24\normalsize\else \ifnum #1<29\large\else
\ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
\huge\fi\fi\fi\fi\fi\fi
\csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
\count@#1\relax \ifnum 25<\count@\count@25\fi
\def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
\expandafter\x
\csname \romannumeral\the\count@ pt\expandafter\endcsname
\csname @\romannumeral\the\count@ pt\endcsname
\csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(570,409)(160,305)
\put(400,615){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\H^{+}$}}}
\put(625,605){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\Scri+$}}}
\put(530,690){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$i^{+}$}}}
\put(450,425){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\H^{-}$}}}
\put(390,490){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}
${{\cal T}_{u_1}}^-$}}}
\put(620,365){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$(\Scri-$)}}}
\end{picture}
\caption{\ninerm Penrose diagram for the Robinson-Trautman spacetime. \label{f:pd}}
\end{figure}
\subsection{Apparent Horizon Equations}
The concept of an apparent horizon rests on the definition of marginally
trapped surface. We define an {\em outer past marginally trapped surface},
${\cal T^{-}}$, to be a spacelike two surface on which the ingoing future directed
congruence of orthogonal null geodesics has vanishing divergence, while the
outgoing future directed congruence of orthogonal null geodesics is
diverging. We define the {\em outer past apparent horizon}, $\cal H^{-}$, to be a
hypersurface $r={\Re}(u,{\zeta},{\bar\zeta})$ such that its intersection with each $u=u_0$
slice is an outer marginally past trapped two-surface ${{\cal T}_{u}}^{-}$, i.e. $\cal H^{-}$ is
foliated by the surfaces ${{\cal T}_{u}}^{-}$. The past apparent horizon is illustrated in
\Fig(pd).
The equations describing the apparent horizon are constructed as follow.
Consider a null tetrad $(l^a, n^a, m^a, \b m^a)$ for
the spacetime with $l^a$ tangent to a congruence of outgoing
shearfree diverging null geodesics, and $m^a$ and $\b m^a$ lying
on a family of two surfaces ${\cal T}_u$:
\begin{eqnarray}
l^a &=&\partial_r \nonumber\\
n^a &=& \partial_u + \left[{P^2 {\Re}_{\zeta} {\Re}_{\bar\zeta} \over
r^2} - H \right]\partial_r +
{P^2 {\Re}_{\zeta} \over r^2}\partial_{\zeta} + {P^2 {\Re}_{\bar\zeta} \over r^2}\partial_{\bar\zeta}
\nonumber\\ m^a &=& {P\over r}\partial_{\bar\zeta} + {P{\Re}_{\bar\zeta}
\over r}\partial_r \nonumber \\
{\bar m}^a &=& {P\over r}\partial_{\zeta} + {P{\Re}_{\zeta}
\over r}\partial_r \label{eq:tetrad}
\end{eqnarray}
where $H$ is as in \E(metric) and $r={\Re}(u,{\zeta},{\bar\zeta})$ defines the
surfaces ${\cal T}_u$.
The nonvanishing Newman-Penrose quantities are then given
by:
\begin{eqnarray}
\rho &=& -{1\over r}\/,\;\;\;
\tau = {\overline \pi} = {\overline \alpha}+\beta
= -{P\Re,_{\bar \zeta} \over r^2}\/,\;\;\;
\beta = -{P,_{\bar \zeta} \over 2r}\/, \nonumber \\
%
\lambda &=& (\partial_\zeta + \Re,_\zeta \partial_r)
\left({P^2 \Re,_\zeta \over r^2}\right)\/, \nonumber \\
%
\mu &=& -{1 \over 2r} \left[ K
- {2 \over r}(m + P^2 \Re,_{\zeta {\bar \zeta}})
+{1 \over r^2}({2P^2 \Re,_\zeta \Re,_{\bar \zeta}})\right]\/,
\nonumber \\
%
\gamma &=& -{1 \over 2} \left[ (\ln P),_{u},-
{1 \over r^2}(m + {P P,_\zeta \Re,_{\bar \zeta}}
- {P P,_{\bar \zeta} \Re,_\zeta})
+ {1 \over r^3}({2P^2 \Re,_\zeta \Re,_{\bar \zeta}}) \right]\/,
\nonumber \\
%
\nu &=& {P \over r}(\partial_\zeta + \Re,_\zeta \partial_r)
\left(\Re,_u -r(\ln P),_u + {K \over 2} -{m\over r}
+{P^2 \Re,_{\zeta} \Re,_{\bar \zeta} \over r^2}\right)\/, \nonumber
\\
%
\Psi_2 &=& -{m \over r^3}\/, \nonumber \\
\Psi_3 &=& -{P K,_\zeta \over 2r^2}
-{3 P m \Re,_\zeta \over r^4}
\/, \nonumber \\
\Psi_4 &=& -{[P^2 (\ln P),_{u \zeta}],_{\zeta} \over r}
+{(P^2 K,_\zeta),_\zeta \over 2r^2}
-{2P^2 K,_\zeta \Re,_\zeta \over r^3}
-{6 P^2 m (\Re,_\zeta)^2 \over r^5}
\label{eq:spcf}
\end{eqnarray}
{}From \E(tetrad) and \E(spcf) it can be seen that the null vector $l^a$ is
future directed, outward pointing, geodetic ($\kappa=0$), diverging ($\rho=-{1
\over r}$), null and orthogonal to the two-surface ${\cal T}_u$. That is,
$l^a$ is tangent to the outgoing congruence of null geodesics normal to ${\cal T}_u$.
The tetrad vector $n^a$ will be tangent at ${\cal T}_u$ to the ingoing congruence of
null geodesics normal to ${\cal T}_u$. The divergence of $n^a$ is given by
$Re(\mu)$ ($= \mu$ in this case). Thus the divergence of the ingoing
congruence of null geodesics is given on ${\cal T}_u$ by $\widetilde\mu$, where the tilde is
used here and subsequently to denote the restriction of an $r$-dependent
quantity to the hypersurface $r={\Re}(u,{\zeta},{\bar\zeta})$. If $\widetilde\mu=0$, ${\cal T}_u$ are
marginally past-trapped two-surfaces ${{\cal T}_{u}}^{-}$ and the hypersurface
$r={\Re}(u,{\zeta},{\bar\zeta})$ is the past apparent horizon $\cal H^{-}$. This condition gives us
the first horizon equation:
\begin{eqnarray
K - {2m \over \Re} - \Delta(\ln \Re) = 0
\label{eq:mu}
\end{eqnarray
This equation is attributed to Penrose\R(Pen73). Tod\R(Tod89)
examined it in more detail, proving that \E(mu) has a unique
$C^{\infty}$ solution given a $C^{\infty}$ background ${\cal S}_{u,r}$,
and also that the surface ${{\cal T}_{u}}^{-}$ defined by the solution is
in fact the outer boundary of past-trapped surfaces on $u=u_0$.
The other equations are found by examining the
embedding of the past apparent horizon in the spacetime; in particular
by considering the normal vector to the apparent horizon.
Let ${N}_{a}$
be a one-form on ${\cal H^{-}}$ defined by
${N}_{a} = -\Re,_u du + dr - \Re,_{\zeta} d\zeta - \Re,_{\bar\zeta}
d{\bar\zeta}\/$. Hence ${N}^{a}:=g^{ab}{N}_{b}$ is a
vector orthogonal to ${\cal H^{-}}$ and is given by
\begin{eqnarray}
{N}^a &=& {\widetilde n}^a - \left({\Re},_{u} + {\widetilde{H}} +
{P^2 \Re,_{\zeta} \Re,_{\bar \zeta} \over \Re^2}\right){\widetilde l}^a
\nonumber \\
&=& \partial_u - ({\Re},_{u} + 2{\widetilde{H}})\partial_r +
{P^2 \Re,_{\bar \zeta} \over \Re^2}\partial_\zeta +
{P^2 \Re,_\zeta \over \Re^2}\partial_{\bar \zeta}
\label{eq:normal}
\end{eqnarray}
Since the complex null vectors
${\widetilde m}^a$ and ${\widetilde{\overline m}}^a$ are tangent to
the two-sufaces ${{\cal T}_{u}}^{-}$ which foliate the hypersurface ${\cal H^{-}}$,
the null vectors ${\widetilde l}^a$ and
${\widetilde n}^a$ are orthogonal to ${{\cal T}_{u}}^{-}$. It can be seen that
the vector
\begin{eqnarray}
{Z}^a &=& {\widetilde n}^a + \left({\Re},_{u} + {\widetilde{H}} +
{P^2 \Re,_{\zeta} \Re,_{\bar \zeta} \over \Re^2}\right){\widetilde l}^a
\nonumber \\
&=&\partial_u + \left({\Re},_{u}+{2P^2 \Re,_\zeta \Re,_{\bar \zeta} \over
\Re^2} \right)\partial_r + {P^2 \Re,_{\bar \zeta} \over
\Re^2}\partial_\zeta + {P^2 \Re,_\zeta \over \Re^2}\partial_{\bar \zeta}
\label{eq:triad}
\end{eqnarray}
is orthogonal to ${N}^a$ and therefore is tangent
to the hypersurface ${\cal H^{-}}$. The ``magnitude'' of
${N}^a$ and ${Z}^a$,
from (\ref{eq:normal}) and (\ref{eq:triad}), is given by
\begin{eqnarray}
{N}_a{N}^a=-{Z}_a{Z}^a
&=&-2\left({\Re},_{u} + {\widetilde{H}} +
{P^2 \Re,_{\zeta} \Re,_{\bar \zeta} \over {\Re}^2}\right)
\label{eq:length}
\end{eqnarray}
Since from \E(mu) ${\widetilde{\mu}}=0$ on each ${\cal T}_u$,
the directional derivative of ${\widetilde{\mu}}$ along the
vector ${Z}^a$ tangent to ${\cal H^{-}}$ must vanish.
(\ref{eq:triad}) and (\ref{eq:length}) then imply
\begin{eqnarray
{Z}^a \nabla_a \widetilde{\mu} =
{\widetilde n}^a \nabla_a \widetilde{\mu} -
{1 \over 2}({N}_b{N}^b\,)
{\widetilde l}^a \nabla_a \widetilde{\mu} = 0 \label{eq:dd}
\end{eqnarray
\noindent Substituting \E(spcf) into
the Newman-Penrose equations\R(PR)
gives
\begin{eqnarray}
{\widetilde n}^a \nabla_a \widetilde{\mu}&=&
{\widetilde m}^a \nabla_a \widetilde{\nu} +
\widetilde{\nu}(-\widetilde{\tau} + \widetilde{\overline \alpha} +
3\widetilde{\beta}) + \widetilde{\pi} \widetilde{\overline \nu} -
\widetilde{\lambda} \widetilde{\overline \lambda} -
2\widetilde{\phi}_2 \widetilde{\overline \phi}_2 \nonumber \\
&=& -\partial_{\zeta}\!\left({P^2 \Re,_\zeta \over \Re^2}\right)
\partial_{\bar \zeta}\!\left({P^2 \Re,_{\bar\zeta} \over \Re^2}\right)
\nonumber\\
\nonumber \\
& &\qquad\qquad -{P^2 \over 2\Re}\left[\,\partial_{\zeta {\bar \zeta}}\!\left(
{{N}_a{N}^a \over \Re}\right) - ({N}_a{N}^a) \,\partial_{\zeta {\bar
\zeta}}\!\left({1 \over \Re}\right) \right]
\label{eq:np1} \\
{\widetilde l}^a \nabla_a \widetilde{\mu}&=&
{\widetilde m}^a \nabla_a \widetilde{\pi} +
\widetilde{\pi}(\widetilde{\overline \pi} - \widetilde{\overline \alpha} +
\widetilde{\beta}) + \widetilde{\Psi}_2 \nonumber \\
&=&{P^2 \over \Re} \partial_{\zeta{\bar \zeta}}\left({1 \over \Re}\right)
- {m\over \Re^3} \label{eq:np2}
\end{eqnarray}
In (\ref{eq:np1}) we have used
$\widetilde{\nu}=-{P \over 2\Re} ({N}_a{N}^a),_\zeta\,$ and $\widetilde\lambda =
(\frac{P^2{\Re},_{\zeta}}{{\Re}^2}),_{\zeta}$.
Combine (\ref{eq:dd}), (\ref{eq:np1}) and (\ref{eq:np2})
to give
$$
{1\over \Re}\Delta\left({{N}_a{N}^a \over \Re}\right)
+2 \widetilde{\Psi}_2({N}_a{N}^a)
= -4\widetilde{\lambda} \widetilde{\bar\lambda} \label{eq:h3}
$$
To summarise, then, the apparent horizon in \afRobinson-Trautman
spacetimes can be described
by the following equations:
\begin{eqnarray}
& &K - {2m\over {\Re}} =\Delta(\ln{\Re}) \label{eq:hor1}\\
& &{1\over {\Re}}\Delta\left({N_a N^a\over {\Re}}\right) - 2{m\over
{\Re}^3}(N_a N^a) = - 4\widetilde\lambda\widetilde{\b\lambda} \label{eq:hor2}\\
& &N_a
N^a = - 2{\Re}\left(\ln{{\Re}\over P}\right),_u - K + {2m\over {\Re}} +
- {2P^2{\Re},_{\zeta} {\Re},_{\bar\zeta} \over {\Re}^2}
\label{eq:hor3}
\end{eqnarray}
The $Re(\tilde\mu)=0$ equation \E(hor1) is the primary equation required to locate
the horizon. Tod's proof of existence and uniqueness of solutions for this
equation indicates that there must be a marginally trapped two-surface on
each slice $u=u_0$. However, in order for $\cal H^{-}$, the union of these
surfaces, to be considered an apparent horizon, we require that $\cal H^{-}$ be a
non-timelike hypersurface, i.e. that $N^a N_a$ is always non-negative. As
shown by Collins\R(Col92) and Hayward,\R(Hay94) this is equivalent to the
assumption that the directional derivative of the divergence of the ingoing
null geodesics in the direction of future pointing outgoing null geodesics is
negative. In fact, Hayward makes this assumption a part of the definition of
an outer apparent horizon, and thus does not require a spacetime to be
asymptotically flat in order for the outer apparent horizon to be selected.
In the Robinson-Trautman spacetime, which is asymptotically flat, we can prove that this
assumption is not necessary; using the maximum principle and \E(hor2) it is
straightforward to show that $N^aN_a\geq0$ (details of the proof will be given
elsewhere).
\subsection{Properties of the horizon}
Equation \E(hor1) can be written in the form
\begin{eqnarray
2me^{3\Lambda} = e^\Phi (1-\Delta_{0}\Phi) \label{eq:hor4}
\end{eqnarray
where $e^\Phi={\Re}{e^\Lambda}$. This gives a useful expression for the Gaussian
curvature of the apparent
horizon, $K_T = e^{-2\Phi}(1-\Delta_{0}\Phi)$. It is then straightforward
to show that the marginally trapped surfaces have the topology of $S^2$:
\begin{eqnarray
\chi_T = {1\over 2\pi}\int_{S^2} K_T e^{2\Phi}\vol0 =
{1\over 2\pi}\int_{S^2} [1-\Delta_{0}\Phi]\vol0 = 2.
\end{eqnarray
Furthermore, because $K_T e^{3\Phi} = 2me^{3\Lambda} $,
$K_T$ must be positive everywhere. Also, this relationship gives rise
to some additional interesting integral quantities:
\begin{eqnarray}
\int_{S^2} me^{2\Lambda} \vol0 = 4\pi m &\Leftrightarrow&\int_{S^2} K_T
e^{3\Phi-\Lambda} \vol0 = 8\pi m\nonumber\\
\int_{S^2} me^{3\Lambda-\Phi} \vol0 = 2\pi &\Leftrightarrow&\int_{S^2} K_T
e^{2\Phi} \vol0 = 4\pi \nonumber\\
\int_{S^2} me^{3\Lambda} \vol0 \geq 0 \qquad&\Leftrightarrow&\int_{S^2} K_T
e^{3\Phi} \vol0 \geq 0 \nonumber\\
\frac{d}{du}\int_{S^2} me^{3\Lambda} \vol0 \leq 0 \qquad
&\Leftrightarrow&\frac{d}{du}\int_{S^2} K_T e^{3\Phi} \vol0 \leq 0
\label{eq:mKiden}
\end{eqnarray}
This is suggestive of the fact that the Robinson-Trautman equation can be rewritten entirely
in terms of ``horizon quantities''. From \E(hor4), we have
$$e^{2\Lambda} = \left( {1\over 2m}e^\Phi(1-\Delta_{0}\Phi)\right)^{2\over3}$$
and
$$ \Delta_{0}\Lambda = {1\over 3} \Delta_{0}(\Phi + ln(1-\Delta_{0}\Phi)).$$
Using the Robinson-Trautman equation in the form
\begin{eqnarray
(e^{2\Lambda}),u = -{1\over 12m}\Delta_{0}(e^{-2\Lambda}[1-\Delta_{0}\Lambda]), \nonumber
\end{eqnarray
we can construct an equation in $\Phi$ alone. While rather messy,
this formulation of the Robinson-Trautman equation is physically
interesting as it describes in some sense the
``evolution'' of the apparent horizon of the white hole. The term
``evolution'' is used loosely here, since we expect the
past apparent horizon to be non-timelike.
The properties of the area of the apparent horizon,
$A_{\cal T} = \int_{S^2} e^{2\Phi} \vol0$, can also be
derived from the horizon equations, \E(hor1) to \E(hor3).
Tod proved the isoperimetric inequality, $ 16\pi M_B^2 \geq A_{\cal T}$,
for the area of the horizon.\R(Tod86)
In fact, $ 16\pi M_B^2 \geq A_{\cal T} \geq 16\pi m^2$, since, from the
H\"older Inequality and \E(mKiden),
\begin{eqnarray}
\left(\int_{S^2} e^{2\Phi}\vol0\right)^{1\over3}
\left(\int_{S^2} e^{3\Lambda-\Phi}\vol0\right)^{2\over3} &\geq&\int_{S^2}
e^{2\Lambda}\vol0\nonumber\\
\Rightarrow \qquad\qquad{A_{\cal T}}^{1\over3} \left({2\pi\over m}\right)^{2\over3}
&\geq& 4\pi\nonumber\\
\Rightarrow \qquad\qquad\qquad\qquad A_{\cal T} &\geq &16\pi m^2.\label{eq:arealb}
\end{eqnarray}
Since \E(hor2) implies that the apparent horizon must be a non-timelike
hypersurface, i.e. that $N^aN_a\geq 0$ always, we can show quite easily that
the $A_{\cal T}$ must always decrease. From \E(hor1) and \E(hor3), we have
\begin{eqnarray
(e^{2\Phi}),_u + \Delta_{0}(e^{\Phi-\Lambda}) = - e^{\Phi+\Lambda}(N^a N_a)
\end{eqnarray
Thus
\begin{eqnarray
\dduA_{\cal T} = \frac{d}{du}\int_{S^2} e^{2\Phi}\vol0 = -\int_{S^2} e^{\Phi+\Lambda}(N^a N_a)\vol0
\leq 0.
\end{eqnarray
Thus we have shown that the past apparent horizon $\cal H^{-}$ is a
non-timelike hypersurface foliated by past marginally trapped surfaces
${{\cal T}_{u}}^{-}$ whose surface area is a monotonically decreasing function of $u$.
\vfill\pagebreak
\section{Numerical modelling}
We are currently undertaking to solve the apparent horizon equations in
conjunction with the Robinson-Trautman equation numerically, and to use the numerical model
to investigate further some of the properties of the apparent horizon. Our
results at this stage demonstrate the nice properties of the horizon
equations. We are initially solving the axisymmetric equations: thus the
system is 1+1 dimensional only.
A full discussion of the numerical results will be presented elsewhere,
but we give here a brief introduction to the work.
\begin{figure}[htbp]
\epsfxsize=0.95\hsize
\epsfbox{plotf1.eps}
\epsfbox{plotR1.eps}
\caption{\ninerm Numerical evolution of axisymmetric Robinson-Trautman spacetime and
apparent horizon. (a) $f(u,x) \equiv {P\over P_0}$ (b)
${\Re}(u,x)$\label{f:poly}}
\end{figure}
\begin{figure}[htb]
\centering
\begin{picture}(0,0)%
\special{psfile=plotm1.eps}%
\end{picture}%
\setlength{\unitlength}{0.012500in}%
\begingroup\makeatletter\ifx\SetFigFont\undefined
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y
\gdef\SetFigFont#1#2#3{%
\ifnum #1<17\tiny\else \ifnum #1<20\small\else
\ifnum #1<24\normalsize\else \ifnum #1<29\large\else
\ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
\huge\fi\fi\fi\fi\fi\fi
\csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
\count@#1\relax \ifnum 25<\count@\count@25\fi
\def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
\expandafter\x
\csname \romannumeral\the\count@ pt\expandafter\endcsname
\csname @\romannumeral\the\count@ pt\endcsname
\csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(420,280)(40,505)
\end{picture}
\vspace{0.5cm}
\caption{\ninerm Monotonic decay of mass quantities in the numerical evolution
of the Robinson-Trautman spacetime.\label{f:mass}}
\end{figure}
In order to find the apparent horizon, we first solve the Robinson-Trautman equation to find
the ``background spacetime''. For this, we follow the work of
Singleton\R(Sin90b) and Prager and Lun,\R(Pra94) using exactly the
Crank-Nicolson algorithm described of the latter. Next, on each `time' slice
$u=u_0$, we solve \E(hor1) using Newton-Raphson iteration. That is, for the
values of $P$ (or $\Lambda$) determined by the Robinson-Trautman equation, \E(hor1) is solved to
find the function ${\Re}$ which describes the position of the marginally trapped
surface. Because the convergence of the Newton-Raphson method is entirely
dependent on making a good initial guess, we actually begin with the last
timestep of the evolution, where the marginally trapped surface is known to
be close to $r=2m$, and solve for each time slice in turn, going backwards to
$u=0$, taking as an initial guess for each slice the solution on the previous
slice.
The example shown has the initial condition $f \equiv {P\over P_0} = 1. +
0.1Y_{1,0} - 0.2Y_{2,0} - 0.3Y_{3,0}$, where $Y_{\ell,0}$ are spherical
harmonics restricted to the axisymmetric case (i.e. Legendre polynomials).
For convenience we set $12m= 1$. \Fig(poly)(a) shows a plot of $f$ against
$x=\cos{\theta}$ evolving through $u$. It can be seen that the system
settles down to equilibrium ($P=P_0$) very quickly. Note that the
equilibrium solution includes a component of the first harmonic in this case,
so $f$ is of the form $f=a + bx$. The function ${\Re}$, representing the
position of the marginally trapped surface ${{\cal T}_{u}}^{-}$ is plotted in
\Fig(poly)(b), also showing very smooth behaviour. ${\Re}$ is not subject to this
ambiguity in the final state, and settles down as it should to $r=2m$, which
corresponds to the Schwarzschild equilibrium.
In \Fig(mass) the ``horizon mass'', defined as
$M_T = \sqrt{\frac{A_{\cal T}}{16\pi}}$, is plotted as a function of $u$ with the
Bondi mass and the irreducible mass, demonstrating the monotonicity of the
horizon area, and the inequalities described above (\E(arealb)).
\section{Concluding Remarks}
In this paper we have demonstrated the existence of the past apparent
horizon in the vacuum \afRobinson-Trautman spacetimes, and described some of its
properties. Tod's proof of the existence of marginally past trapped
surfaces,\R(Tod89) together with our result that they foliate a
non-timelike hypersurface, allow us to call $\cal H^{-}$ a past apparent
horizon.
It is pleasing to find that the Robinson-Trautman spacetimes, which are the simplest
asymptotically flat radiating spacetimes, exhibit a well behaved apparent horizon
structure. In more general radiating spacetimes, we would not expect
the existence of apparent horizons to be guaranteed. This is
demonstrated by examples such as the electrovac Robinson-Trautman\R(LC94)
or the vacuum Bondi-Sachs spacetimes.
We have also shown that the surface area of the past apparent horizon
decreases monotonically with the retarded time $u$.
This result is a particular example of more general theorems about
apparent horizon dynamics\R(Col92,Hay94).
However, we have not required all the assumptions used in the proof
of those theorems.
We have illustrated a method for describing apparent horizons analytically
based on the construction of a suitable null tetrad for the spacetime - this
method could be easily applied to other cases. In a future paper we will
discuss further the numerical solution of the horizon equations, as a means of
investigating the physics of these spacetimes.
\section{Acknowledgements}
We would like to thank Dan Prager for helpful discussions and
the Australian Research Council for financial support for this work.
\section{References}
|
2,877,628,089,551 | arxiv | \section{ Introduction }
Let $\{ \xi_k\}_{k\in {\Bbb Z}_{+}}$ be a homogeneous Markov chain
defined on a probability space $(\Omega, {\cal F}, {\rm P})$.
Denote by ${\Bbb S}$ and ${\cal S},$ respectively, the phase-space
and its $\sigma-$algebra of measurable subsets. Further, denote by
$P(x,A),\ x\in{\Bbb S},\ A\in{\cal S}$ the transition probability
kernel of the chain. It means that for each $A\in {\cal S},\
P(x,A)$ is a non-negative measurable function on ${\Bbb S}$ while
for each $x\in{\Bbb S},\ P(x,A)$ is a probability measure on
${\cal S}.$ In what follows we assume that the chain is uniformly
ergodic. So, there exists a stationary distribution denoted by
$\pi.$
Consider the sequence of random variables
$X_0=f(\xi_0),\ldots,X_n=f(\xi_n)$ determined by a measurable
function $f\!\!:{\Bbb S}\to{\Bbb R}.$ In what follows we assume
that
\begin{equation}
\label{sigma}
\sigma^2={\rm E}_{\pi}[X_0^2] + 2\sum_{n=1}^{\infty} {\rm E}_{\pi}[X_0X_{n}]>0.
\end{equation}
There exists a huge literature concerning the limit theorems for
successive sums $ S_n=\sum_{i=1}^n X_i,\quad n=1,2,\ldots$. For
our purposes, it is enough to keep in mind only the works of S.
Nagaev (1957) and (1961) and the monograph by Sirazhdinov and Formanov (1979).
Despite the theory of limit theorems is well developed, some
settings seem to be set aside. For example, in Szewczak (2005)
it was shown that the Cram\'er method of conjugate distributions
assumes a special form of the local limit theorem that was not
considered before. The case studied in Szewczak (2005) concerns Markov
chains with a finite number of states. It worth noting that the
large deviation theorems, established there, proved to be very
useful in statistics of Markov chains (see A. Nagaev (2001) and (2002)).
The mentioned form of the local limit theorem means
the weak convergence of the measures
\begin{equation}\label{ea1}
Q_x^{(n)}(A\times B) = \sigma\sqrt{2\pi n}{\rm P}_{x}^{(n)}
\left[\,X_1+\ldots+X_n\in A,\,\xi_n\in B\,\right],
\end{equation}
where
\[{\rm P}_{x}^{(n)}(\xi_1\in A_1,\dots,\xi_n\in A_n)=\int_{A_1}\! P(x,{\rm
d}x_1)\int_{A_2}\! P(x_1,{\rm d}x_2)\dots\int_{A_n}\!
P(x_{n-1},{\rm d}x_n),\] $A_k\in{\cal S},\ k=1,\dots,n,\ B\in{\cal
S},\ A\in{\cal B}({\Bbb R})$ and $x\in{\Bbb S}.$
Define the linear operators
\begin{equation}\label{ea2}
({\bf K}_n g)(x) = \int P(x, {\rm d}x_1) \cdots \int
P(x_{n-1},{\rm d}x_n) g(x_n) K_n(x_1,\ldots, x_n);\quad g\in
L^{\infty}(\mu),
\end{equation}
where $K_n,$ $n=1,2,\ldots,$ are measurable kernels, and
$L^{\infty}(\mu)$ is the Banach space of measurable functions
equipped with the essential supremum norm
\[ \bigl\bracevert g \bigl\bracevert ={\rm ess}\sup_x\vert g(x)\vert =
\inf \{a;\, \mu\{x;\, \vert g(x)\vert > a \} = 0\}, \] $\mu$ is
the initial distribution, i.e. $\mu(A)={\rm P}[\,\xi_0\in A\,],\,$
$A\in{\cal S}.$
Various probability measures of interest can be represented as a
set indexed family of the operators (\ref{ea2}). If one puts
\[K_{n,A}(x_1,\ldots,x_n)= \sigma\sqrt{2\pi
n}I_A(f(x_1)+\cdots+f(x_n)),\quad g(x_n)= I_B(x_n),\, A\in{\cal
B}({\mathbb R}),
\]
then (\ref{ea1}) takes the form
\begin{equation}
\label{ea3}
Q_x^{(n)}(A\times B) = ({\bf K}_{n,A}I_B)(x).
\end{equation}
Similarly,
\begin{equation}
\label{ea4} {\rm P}_{x}[\frac{X_1+\cdots+X_n}{\sigma\sqrt{n}}\in
A, \xi_n\in B\,] = ({\bf K}_{n,A}g)(x)
\end{equation}
provided
\[K_{n,A}(x_1,\ldots,x_n)
=
I_A\left(\frac{f(x_1)+\cdots+f(x_n)}{\sigma\sqrt{n}}\right),\quad
g(x_n)= I_B(x_n).
\]
When $x$ is fixed the weak convergence of the measures
(\ref{ea3}) (or (\ref{ea4})) means a form of the classical local
limit (or central limit theorem). Naturally, we expect that the
measures (\ref{ea3}) weakly converge to $\lambda\times\pi$ while
(\ref{ea4}) converge to $\nu\times\pi$ where $\lambda$ is the
Lebesgue measure on ${\cal B}({\mathbb R})$ and
\[\nu(A)={{1}\over{\sqrt{2\pi}}}
\int_Ae^{-{{u^2}\over{2}}}{\rm d}u.\] Such statement can be
embedded into the following scheme of convergence.
Define
\begin{equation}
\label{epc}
\Vert {\bf K}\Vert_{+}=
\sup_{\{g\geq 0;\,g\in L^{\infty}(\mu), \,\bigl\bracevert
g\bigl\bracevert\leq 1\}}\!\!\!\! \bigl\bracevert {\bf K}g\bigl\bracevert.
\end{equation}
Consider a family of sequences $\{{\bf K}_{n,A}\},$ $A\in{\cal
A}\subset {\cal B}(\mathbb R).$ We say that a sequence ${\bf
K}_{n,A}$ is $L_{\cal A}^{\infty}(\mu)$-strongly convergent to
${\bf K}_A$ if
\begin{equation}
\label{da2}
\sup_{A\in{\cal A}}\Vert {\bf K}_{n,A} - {\bf K}_A\Vert_{+}\to 0
\qquad\mbox{as}\qquad n\to\infty.
\end{equation}
Let ${\cal A}=\{z\in{\mathbb R}\,\vert\,(-\infty,z\sigma)\}.$ If
the sequence of operators is defined as in (\ref{ea4}) then the
limit operator in (\ref{da2}) has the form
\[({\bf K}_Ag)(x)=\psi(x)\nu(A)\int_{\Bbb S}g(s)\mu( {\rm d} s)\]
where $\psi(x)\equiv 1.$ This fact is formally more general than
e.g. Th. 2.2 in Nagaev (1957) though its proof does not require
serious efforts. It is of much greater interest to establish the
{\em operator form} of the asymptotic expansions for the sequence
$\{{\mathbf K}_{n,z}\}$ determined by the kernels
\[K_{n,z}(x_1,\ldots,x_n) = I_{(-\infty, z)}
\left({{f(x_1)+\cdots+f(x_n)}\over{\sigma\sqrt{n}}}\right),\
z\in\Bbb R.\]
Such asymptotic expansions is basic goal of the present paper. The
paper is organized as follows. In Section 2 the main results are
stated. In Section 3 a new estimate for the so-called
characteristic operator in the neighborhood of zero is established
(Cf. Lemma 1.6 in Nagaev (1961)).
The proofs are given in Section 4.
\section{ The main results}
\noindent
In order to state the main results of the paper we have to
introduce the indispensable notation. We are going to establish an
asymptotic expansion of the form
\begin{equation}
\label{eb1}
\Vert {\bf K}_{n,z} -\sum_{m=0}^{k-2} n^{-{{m}\over{2}}}{\bf A}_{m,z}
\Vert_{+} = \, o(n^{-{{k-2}\over{2}}}),
\end{equation}
where ${\bf A}_{m,z}$ are linear operators defined on
$L^{\infty}(\mu),\ m=0,1,\ldots,\ z\in {\mathbb R}.$ The operators
${\bf A}_{m,z}$ are expressed through the Hermite polynomials
$H_k$ and certain derivatives of the so-called characteristic
operator
\[ \hat{\bold P}(\theta)(g)(x) = \int e^{i\theta f(y)} g(y) P(x, {\rm d}y), \]
where $g\in L^{\infty}(\mu).$ More precisely, let
$\lambda(\theta)$ be the principal eigenvalue of $\hat{\bold P}(\theta)$
and $\hat{\bold P}_1(\theta)$ be the projection on the
eigenspace corresponding to $\lambda(\theta).$ Assume that
$\hat{\bf P}(\theta)$ is $k$-times strongly differentiable at
$\theta=0$ and ${\bf P}=\hat{\bf P}(0)$ is $L^{\infty}$-regular
(or primitive), i.e. there exist $C>0$ and $\gamma,\,$
$0\leq\vert\gamma\vert<1,\,$ such that
\begin{equation}
\label{ae81}
\bigl\bracevert {\bold P}^n g - {\bold \Pi} g \bigl\bracevert \leq C
\vert\gamma\vert^n \bigl\bracevert g \bigl\bracevert,\qquad\qquad
g\in L^{\infty}(\mu),
\end{equation}
where \[({\bold \Pi}g)(x) = \psi(x)\int_{\Bbb S}g(s)\mu( {\rm d}
s)=\psi(x){\rm E}_{\pi}[g]\] (Cf. Gudynas (2000)).
Then $\hat{\bf P}_1(\theta)$ and $\ln{\lambda}(\theta)$ admit the following
MacLaurin expansions:
\[ \hat{\bold P}_1(\theta) = \sum_{m=0}^k {{(i\theta)^m}\over{m!}}
\hat{\bold P}_1^{(m)} +
o(\vert\theta\vert^k),\qquad\mbox{and}\qquad \ln{\lambda(\theta)}
= \sum_{m=0}^k {{(i\theta)^m}\over{m!}}\gamma_m +
o(\vert\theta\vert^k).
\]
Here, the operators $\hat{\bold P}_1^{(m)}$ can be explicitly
expressed in terms of ${\bold P}$ and ${\bold \Pi}$ (see Lemma
\ref{al6}). The coefficients $\gamma_m,$ $m=0,1,\ldots$ are
called {\em cumulants}. In what follows we assume $\gamma_1={\rm
E}_{\pi}[f]=0$ thus $\gamma_2=\sigma^2,$ where $\sigma^2$ is
defined by (\ref{sigma})
and $\gamma_3=\mu_3$ is defined
in Lemma 1.2 in Nagaev (1961).
Let $\frak N$ and $\frak n$ denote
the distribution function and the density function of the standard
normal law. Introduce the operators defined on
$L^{\infty}(\mu):\,$ ${\bf A}_{0,z} = {\mathfrak N}(z){\bf \Pi},\,
{\bf A}_{\nu,z}=\sum_{j=0}^{\nu} a_j(z){\hat{\bold P}_1^{(j)}},$
where
\[a_j(z) =- {\mathfrak n}(z)\!\!\!\!\!
\sum_{(k_1,k_2,\ldots,k_{\nu-j})\in {\cal K}_{\nu-j}} \!\!\!\!\!
a_{j,\nu-j}H_{\nu-1+2\sum\limits_{i=1}^{\nu-j} k_i}(z),\quad
a_{\nu}= - {\mathfrak n}(z)H_{\nu-1},
\]
\[a_{j,\nu-j} = {{1}\over{{j}!\sigma^{j}}}
\prod_{m=1}^{\nu-j} {{1}\over{k_m!}}
\Bigl({{\gamma_{m+2}}\over{(m+2)!\sigma^{m+2}}}\Bigl)^{k_m}\!\!\!,
\]
and ${\cal K}_m = \{(k_1,\ldots,k_m)\,;\, \sum_{i=1}^{m} ik_i=
m,\, k_i\geq 0, i=1,\ldots,m \}$. Thus, the operators
${\bf A}_{\nu,z}$ are well-defined provided $\hat{\bf P}(\theta)$ is
$k$-times strongly differentiable at $\theta=0$ and $\sigma>0$.
Let $r(\theta)$ be the spectral radius of $\hat{\bf P}(\theta)$.
It is well known that $r(\theta)$ inherits many principal
properties of the characteristic functions. In order to establish
asymptotic expansions (\ref{eb1}) we have to assume that
\begin{equation}
\label{eslat}
r(\theta)<1,\, \theta\neq 0,\qquad\mbox{and}\qquad
\limsup_{\vert\theta\vert\to\infty} r(\theta)<1.
\end{equation}
The second inequality in (\ref{eslat}) is analogous to the
well-known Cram\'er condition (C). As to the first one it
guarantees that the distributions of $\sum_{i=1}^n X_i$ for all
sufficiently large $n$ is non-lattice.
The operator form of asymptotic expansions implies such properties
of the considered Markov chain as strong differentiability of
$\hat{\bf P}(\theta),$ primitiveness and (\ref{eslat}). Of
course, one could simply assume that these properties take place.
Another way is to give a simply verified condition that guarantees
these properties. As such we take the following \vskip
2pt\noindent
{Condition $(\Psi)$}:\\
{\it there exist $\alpha>0$ and $\beta<\infty$ such that for every
Borel set $A$ of a positive measure $\mu$ we have
$\alpha\mu(A)\leq P(x,A) \leq \beta\mu(A)$ for $\mu$-a.a.
$\,x\in{\Bbb S}.$}
\vskip 2pt\noindent
This condition enables us to verify the required properties by the
initial distribution $\mu$. For example if Condition $(\Psi)$ is
fulfilled then (\ref{eslat}) takes place provided
$\mu_f=\mu\!\circ\! f^{-1}$ is non-lattice and
\begin{equation}
\label{ecra}
\limsup_{\vert\theta\vert\to\infty}\vert
{\widehat\mu_f}(\theta)\vert < 1,
\end{equation}
where $\widehat{\mu}_f=\int e^{i\theta f(y)}\mu({\rm d}y)$.
Moreover, if $\int\vert f(y)\vert^k\mu({\rm d}y)<\infty$ then
$\hat{\bf P}(\theta)$ is $k$-times strongly differentiable. It
should be noted (see the proof of Lemma 3.1 in Jensen (1991))
that Condition $(\Psi)$ implies $\sigma^2=\gamma_2>0$.
Now we, are able to state the main results.
\begin{theorem}\quad
\label{tn2} Let Condition $(\Psi)$ is fulfilled. If $\int \vert
f(x)\vert^k\mu({\rm d}x)<\infty,$ $k>3,$ and $\mu_f$ satisfies
(\ref{ecra}) then (\ref{eb1}) holds.
\end{theorem}
As in the case of asymptotic expansions for i.i.d. variables
(see Gnedenko and Kolmogorov, 1954, \S42, Th. 2)
the following statement does not
require the condition (\ref{ecra}).
\begin{theorem}\quad
\label{t4} Let Condition $(\Psi)$ is fulfilled. If $\int \vert
f(x)\vert^3\mu({\rm d}x)<\infty$ and $\mu_f$ is non-lattice then
(\ref{eb1}) holds with $k=3$.
\end{theorem}
In order to clarify the specificity of the limit theorems given
in the operator form consider two examples. First, let $\{\xi_k
\}$ be a finite state Markov chain, i.e. ${\mathbb
S}=\{1,\ldots,d\},$ $d\geq 3$. Denote by ${\mathbf P}$ the
transition matrix. The entries of ${\mathbf P}^{\nu}, \nu\geq 0,$
we denote by $p_{ij}^{(\nu)}, i,j\in{\mathbb S},$
$p_{ij}^{(0)}=\delta_{ij}$. For a real function $f$ on ${\mathbb
S}$ define the matrix ${\bold P}^{(1)}$ with the elements
$f(j)p_{ij},$ $i,j\in{\mathbb S}$. The following statement is of
independent interest.
\begin{corollary}\quad
\label{c1}
Suppose that transition matrix ${\mathbf P}$ is strictly positive.
If $f(\xi_0)$ is non-lattice and $\sum_{k=1}^d \pi_k f(k)=0$
then uniformly in $z\in\mathbb R$ the matrix
\[
({\rm P}[S_n< z\sigma\sqrt{n}\,;\,
\xi_n=j\,\vert\, \xi_0=i])_{i,j\in{\mathbb S}}
\]
is approximated by the matrix
\begin{equation}
\label{ee3}
{\mathfrak N}(z){\bold \Pi} + n^{-1/2}{\mathfrak n}(z)
({{{\mu}_3}\over{6{\sigma}^3}}(1-{z^2}){\mathbf \Pi}
- {{1}\over{\sigma}}\sum_{\nu\geq 0}
{{\bold \Pi}{\bold P}^{(1)}({\bold P}^{\nu} - {\bold \Pi})
+ ({\bold P}^{\nu} - {\bold \Pi})
{\bold P}^{(1)}{\bold \Pi}})
\end{equation}
with an error $o(n^{-1/2})$. Here,
\[{\bold \Pi}=\left(\begin{array}{llll}
\pi_1 & \pi_2 & \dots & \pi_d\\&\\
\dots & \dots & \dots & \dots\\&\\
\pi_1 & \pi_2 & \dots & \pi_d
\end{array}\right).\]
\end{corollary}
Another particular case of independent interest is covered by the
following statement.
\begin{corollary}\quad
\label{c2} Let ${\mathbb S}=[0,\ 1]$. Suppose that the transition
density $p(x,y)$ is such that $0<p_-\leq p(x,y)\leq p_+<\infty.$
If $f(\xi_0)$ is non-lattice and $\int f(u)\pi({\rm d}u)=0$ then
the linear operator (\ref{ee3}) $L^{\infty}_{\cal A}$-strongly
approximates the operator
$g\mapsto {\rm E}_{x}[I_{[S_n<z\sigma\sqrt{n}]}g(\xi_n)\,]$ with
an error $o(n^{-1/2})$. Here,
\[({\bold \Pi}g)(x)=\int_{\Bbb S}g(s)\mu( {\rm d} s)\psi(x).\]
\end{corollary}
Note that the classical scalar form of the presented statement is:
\begin{equation}
\label{esae}
{\rm P}_{\pi}[S_n< z\sigma\sqrt{n}\,] - {\mathfrak N}(z)
= n^{-1/2}{\mathfrak n}(z){{{\mu}_3}\over{6{\sigma}^3}}(1-{z^2})
+ o(n^{-1/2})
\end{equation}
(see e.g. Th. 2 in Nagaev (1961)).
The corollaries show that the
operator form of asymptotic expansions is much more sensitive to
the initial conditions than the scalar one. It should be
emphasized that the spectral method suggested by S. Nagaev
(see e.g. Nagaev, 1957)
remains efficient under this new setting though
requires some modification. Furthermore, the cumbersome
calculations, that are typical for asymptotic expansions, can be
implemented using the package {\em Maple.} This power software
proved to be very efficient for such purposes.
\section{ Characteristic operator}
\noindent
Given $m\in\Bbb N$
let us define operator ${\bf P}^{(m)}g = {\bf P}f^mg$.
The following lemma is an extension of the well-known result due to S. Nagaev
(Cf. Nagaev, 1961, pp. 71--75).
\begin{lemma}\quad
\label{al8}
Suppose that (\ref{ae81}) holds
and ${\bold P}^{(1)}$ is a bounded endomorphism.
Then there exists $\xi = \xi(C,\vert\gamma\vert,\Vert{\bold P}^{(1)}\Vert)$
such that for $\vert\theta\vert<\xi$,
\begin{equation}
\label{aae8}
\hat{\bold P}^n(\theta) =
\lambda^n(\theta)\hat{\bold P}_1(\theta)
+ \hat{\bold Q}_n(\theta) + ({\bold P}^n - {\bold \Pi})
\end{equation}
and $\vert\lambda(\theta)-1\vert<\delta$,
where
$\Vert \hat{\bold P}_1(\theta) - {\bold \Pi}\Vert = O(\mid\theta\mid),\,
\Vert\hat{\bold Q}_n(\theta)\Vert
= O(\kappa^n
\mid\theta\mid),$
$ \kappa = {1\over{3}} + {2\over{3}}\vert\gamma\vert,$
$\>\delta = {1\over{3}} - {1\over{3}}\vert\gamma\vert.$
\end{lemma}
\begin{proof} {\sc Lemma \ref{al8}}
Write
${\Gamma}_0 = \{\vert\zeta\vert = \kappa \},\,
{\Gamma}_1 = \{ \vert\zeta - 1 \vert = \delta \} $
and $D=\{\vert\zeta\vert\geq\kappa\}\cap\{\vert\zeta-1\vert\geq\delta\},$
where $\zeta\in \Bbb C$.
Denote by $\hat{\bold R}(\zeta, \theta)$ the resolvent
of $\hat{\bold P}(\theta)$ and set
${\bold R}(\zeta)=\hat{\bold R}(\zeta, 0)$.
Let $\xi = {{1}\over{2\Vert{\bold P}^{(1)}\Vert}}
\left({{1-\vert\gamma\vert}\over{3(3+C)}}\right)^2.$
Consequently for $\mid\theta\mid<\xi$, $\zeta\in D$
(see \S1 in Nagaev (1961)) we may define the projections
\begin{equation}
\label{esp} \hat{\bold P}_1(\theta) = {1\over{2\pi i}}
\oint_{\Gamma_1} \hat{\bold R}(\zeta, \theta) {\rm d}\zeta, \qquad
\hat{\bold P}_2(\theta) = {1\over{2\pi i}} \oint_{\Gamma_0}
\hat{\bold R}(\zeta, \theta) {\rm d}\zeta.
\end{equation}
Thus (\ref{aae8}) holds with
$\hat{\bold Q}_n(\theta) =
\hat{\bold P}^n(\theta)\hat{\bold P}_2(\theta)
- ({\bold P}^n - {\bold \Pi}). $
We see at once that
\[
\hat{\bold P}(\theta)\hat{\bold R}(\zeta, \theta) =
-{\bold I} + \zeta\hat{\bold R}(\zeta, \theta)
\]
therefore,
\[
\hat{\bold P}^n(\theta)\hat{\bold R}(\zeta, \theta) =
-\sum_{k=1}^n (\hat{\bold P}(\theta))^{n-k}{\zeta}^{k-1} +
{\zeta}^n\hat{\bold R}(\zeta, \theta).
\]
So it easily seen that
\begin{eqnarray*}
\Vert\hat{\bold Q}_n(\theta)\Vert
& = &
\Vert {1\over{2\pi i}}\oint_{\Gamma_0}
\zeta^n(\hat{\bold R}(\zeta, \theta) - {\bold R}(\zeta)) {\rm d}\zeta \Vert \\
& \leq & {1\over{2\pi}}\int_0^{2\pi}\kappa^n {{ 2(3(3+C))^3 \Vert
\hat{\bold P}(\theta) - {\bold P}\Vert}\over { (1 -
\vert\gamma\vert)^2(6(3+C) - 1 + \vert\gamma\vert)}} \kappa {\rm
d}\phi = O(\kappa^n \mid\theta\mid).
\end{eqnarray*}
Similarly we have
$\Vert \hat{\bold P}_1(\theta) - {\bold \Pi}\Vert = O(\mid\theta\mid).$
The proof is completed.
\end{proof}
The following lemma deals with the existence of the ``operator'' moments.
\begin{lemma}\quad
\label{al11}
If
\begin{equation}
\label{ena1} \lim_{L\to\infty} \bigl\bracevert \int_{\vert
f(y)\vert >L} \vert f(y)\vert^k P(x,{\rm d}y) \bigl\bracevert = 0
\end{equation}
then
\begin{equation}
\label{emac}
\hat{\bold P}(\theta) = \sum_{m=0}^k {{(i\theta)^m}\over{m!}}
{\bold P}^{(m)} + o(\vert\theta\vert^k),
\end{equation}
where ${\bold P}^{(m)}$ are bounded for $0\leq m\leq k$.
\end{lemma}
\begin{proof} {\sc Lemma \ref{al11}}
Indeed, we have
\begin{eqnarray*}
h^{-1}(i^{(k-1)}\hat{\bold P}^{(k-1)}(\theta + h)g
- i^{(k-1)}\hat{\bold P}^{(k-1)}(\theta)g) -
i^k \int e^{i\theta y}g(y) f^k(y) P(\,\cdot\,, {\rm d}y) \\
=
i^k \int e^{i\theta f(y)}g(y) f^k(y)\int_0^1 (e^{ihsf(y)} -1){\rm
d}s P(\,\cdot\,, {\rm d}y).
\end{eqnarray*}
Now, choose $L$ be sufficiently large positive number.
Since,
\begin{eqnarray*}
\lefteqn {
\inf \{ K\, ;\, \mu\{x\, ;\, \vert
\int (if(y))^k\int_0^1(e^{ihsf(y)} - 1){\rm d}s P(x,{\rm d}y)\vert > K \} = 0 \} }\\
& & \leq
\inf \{ K\, ;\, \mu\{x\, ;\, \vert
\int_{\vert f(y)\vert\leq L}
(if(y))^k\int_0^1(e^{ihsf(y)} - 1){\rm d}s P(x,{\rm d}y)\vert > K \} = 0 \} \\
& & \, +
\inf \{ K\, ;\, \mu\{x\, ;\, \vert
\int_{\vert f(y)\vert>L}
(if(y))^k\int_0^1(e^{ihsf(y)} - 1){\rm d}s P(x,{\rm d}y)\vert > K \} = 0 \}\\
& & \leq {{1}\over{2}}L^{k+1}\vert h\vert + 2\bigl\bracevert
\int_{\vert f(y)\vert >L} \vert f(y)\vert^k P(x,{\rm d}y)
\bigl\bracevert
\end{eqnarray*}
the lemma follows by the Taylor formula.
\end{proof}
The next lemma presents a series expansion for characteristic
projector.
\begin{lemma}\quad
\label{al6}
If a primitive operator ${\bf P}$ satisfies (\ref{emac}) then
\[ \hat{\bold P}_1(\theta) = \sum_{m=0}^k {{(i\theta)^m}\over{m!}}
\hat{\bold P}_1^{(m)} + o(\vert\theta\vert^k). \]
\end{lemma}
\begin{proof} {\sc Lemma \ref{al6}}
Put, for short
${\bold E}(\zeta) =
\sum_{n\geq 0}
({\bold P}^n - {\bold \Pi})\zeta^{-n-1},$
and ${\bold E} = {\bold E}(1)$.
In view of (1.10) in Nagaev (1957) and (\ref{emac})
we obtain for $\vert\theta\vert<\xi$
\[ \hat{\bold R}(\zeta, \theta) = {\bold R}(\zeta) +
\sum_{n\geq 1} {\bold R}(\zeta)(\sum_{m = 1}^k {\bold P}^{(m)}
{\bold R}(\zeta){{(i\theta)^m}\over{m!}})^n
+ o(\vert\theta\vert^k). \]
Hence taking in the above coefficient at $i\theta$ and using (\ref{esp})
we get for $k=1$
\begin{eqnarray*}
\hat{\bold P}_1^{(1)}
& = & {1\over {2{\pi}i}}\oint_{\Gamma_1}
({{{\bold \Pi}}\over{\zeta - 1}}+{\bold E}(\zeta)){\bold P}^{(1)}
({{{\bold \Pi}}\over{\zeta - 1}}+{\bold E}(\zeta)) {\rm d}\zeta \\
& = & {\bold \Pi}{\bold P}^{(1)}{\bold \Pi} {1\over
{2{\pi}i}}\oint_{\Gamma_1} {1\over{(\zeta - 1)^2}} {\rm d}\zeta +
{1\over {2{\pi}i}}\oint_{\Gamma_1} {\bold \Pi}{\bold P}^{(1)}
{{\bold E}(\zeta)\over{\zeta - 1}} {\rm d}\zeta
\\& & \quad
+ {1\over {2{\pi}i}}\oint_{\Gamma_1} {{\bold E}(\zeta)\over{\zeta
- 1}}{\bold P}^{(1)} {\bold \Pi} {\rm d}\zeta ={\bold \Pi}{\bold
P}^{(1)}{\bold E} + {\bold E}{\bold P}^{(1)}{\bold \Pi}
\end{eqnarray*}
by Cauchy's integral formula.
For $1 < m\leq k$ arguments are similar. We have
to replace every ${\bold R}(\zeta)$ by
${{{\bold \Pi}}\over{\zeta - 1}}+{\bold E}(\zeta)$ in
\[ \hat{\bold P}_1^{(m)} =
{{m!}\over{2\pi i}} \oint_{\Gamma_1}
\sum_{\nu_1+\nu_2+\ldots+\nu_l=m}\!\!\!\!\!\!\! {\bold
R}(\zeta){{{\bold P}^{(\nu_1)}}\over{\nu_1!}} {\bold
R}(\zeta){{{\bold P}^{(\nu_2)}}\over{\nu_2!}} \cdots {\bold
R}(\zeta){{{\bold P}^{(\nu_l)}}\over{\nu_l!}} {\bold R}(\zeta){\rm
d}\zeta,\,\,\nu_k\geq 1.\]
\end{proof}
Now, we are in a position to represent the principal eigenvalue
of the characteristic operator in a power series.
\begin{lemma}\quad
\label{al13}
If a primitive operator ${\bf P}$ satisfies (\ref{emac}) then
\[ \lambda(\theta) = 1 + {{(i\theta)}\over{1!}}\mu_1 +
{{(i\theta)^2}\over{2!}}\mu_2 +
{{(i\theta)^3}\over{3!}}\mu_3 +\cdots+
{{(i\theta)^k}\over{k!}}\mu_k + o(\vert\theta\vert^k). \]
\end{lemma}
\begin{proof} {\sc Lemma \ref{al13}}
It follows from (\ref{aae8}) that
\begin{equation}
\label{epi1} \pi\hat{\mathbf P}(\theta)\hat{\mathbf
P}_1(\theta)\psi =\lambda(\theta)\pi\hat{\mathbf P}_1(\theta)\psi.
\end{equation}
Denote $\hat\lambda^{(k)}=\pi\hat{\mathbf P}_1^{(k)}\psi.$
By virtue of (\ref{emac}) and Lemma \ref{al6}
\[ \lambda(\theta)\sum_{\nu=0}^k
\hat\lambda^{(\nu)}{{(i\theta)^{\nu}}\over{\nu!}} =
\sum_{m=0}^k
\Bigl(\sum_{\nu=0}^m {{m}\choose{\nu}}
\pi{\mathbf P}^{(\nu)}\hat{\mathbf P}_1^{(m-\nu)}\psi
{{(i\theta)^m}\over{m!}}\Bigl)
+ o(\vert\theta\vert^k). \]
Since $\lambda^{(0)}=\hat\lambda^{(0)}=1,$
$\lambda^{(1)}=\hat\lambda^{(1)}=0,$
and $\lambda^{(k)}$ exists so by the Leibniz formula
\begin{equation}
\label{encor}
\lambda^{(m)}= \mu_m =
\sum_{\nu=1}^m {{m}\choose{\nu}}
\pi{\mathbf P}^{(\nu)}\hat{\mathbf P}_1^{(m-\nu)}\psi
- \sum_{\nu=2}^{m-2} {{m}\choose{\nu}}
\lambda^{(\nu)}\hat\lambda^{(m-\nu)}.
\end{equation}
By (\ref{encor}) $\gamma_2=\lambda^{(2)}$,
for $m>2$ also use the equation (1.13) in Petrov (1996).
\end{proof}
The following theorem is the main result of the present Section.
\begin{theorem}\quad
\label{al3}
If a primitive operator ${\bold P}$ satisfies (\ref{emac}),
$k\geq 3$ and $\sigma^2>0$ then
there exists ${\eta}_k>0$ such that for $T_n = \eta_k{\sigma}\sqrt{n}$
and $\mid\theta\mid \leq T_n$ we have
\begin{eqnarray}
\label{ae7}
\lefteqn {
\Vert \hat{{\bold P}}^n({{\theta}\over{{\sigma}\sqrt{n}}})
- e^{-{{\theta^2}\over{2}}}
\Bigl(
\sum_{m=0}^{k-2}
\sum_{j=0}^{m} {{(i\theta)^j}\over{n^{{m}\over{2}}j!\sigma^j}}
{\frak P}_{m-j}(i\theta) \hat{\bold P}_1^{(j)}
\Bigl)
- ({\bold P}^n - {\bold \Pi})\Vert } \\
& & \qquad\qquad\qquad\leq
{{o(1)}\over{n^{{k-2}\over{2}}}}(\vert\theta\vert^{k-2}
+ \vert\theta\vert^{k-1} + \vert\theta\vert^{k} + \vert\theta\vert^{3(k-2)})
e^{-{{\theta^2}\over{4}}} + O({{\vert\theta\vert}\over{\sqrt{n}}}
\kappa^n), \nonumber
\end{eqnarray}
where
\[ {\frak P}_\nu(i\theta) =
\sum_{(k_1,k_2,\ldots,k_{\nu})\in {\cal K}_{\nu}}
\prod_{m=1}^{\nu} {{1}\over{k_m!}}
\left({{\gamma_{m+2}(i\theta)^{m+2}}\over{(m+2)!\sigma^{m+2}}}\right)^{k_m}
\!\!\!\!\!\!.\]
\end{theorem}
\begin{proof} {\sc Theorem \ref{al3}}
Let $0<\eta_3\leq\xi$ be such that
$\sup_{\vert\theta\vert\leq\eta_3}
\vert \lambda^{(3)}(\theta) - {\mu}_3 \vert
\leq \sigma^3. $
Put, for short
$T_n = \min\{{{{\sigma}^2}
\over{5({3\over{2}}\vert{\mu}_3\vert+{\sigma}^3)}}, {\eta_3} \}
{\sigma}\sqrt{n}. $
By Taylor's formula for $\mid\theta\mid \leq T_n$ we have
\begin{eqnarray*}
\vert \lambda({{\theta}\over{\sigma\sqrt{n}}})\vert
& \geq & 1 - {{\theta^2}\over{2n}}
- {{\vert\theta\vert^3(\vert\mu_3\vert + \sigma^3 + {1\over{2}}
\vert\mu_3\vert)}\over
{6n^{3\over2}\sigma^3}}
\geq
1 - {{T_n^2}\over{2n}}
- {{T_n^3({3\over{2}}\vert\mu_3\vert + \sigma^3)}\over
{6n^{3\over2}\sigma^3}} \\
& \geq &
1 - {{\sigma^6}\over{50({3\over{2}}\vert\mu_3\vert + \sigma^3)^2}}
- {{\sigma^6}\over{6\cdot 125({3\over{2}}\vert\mu_3\vert
+ \sigma^3)^2}}
> 1 - {2\over50} = {24\over25}\cdotp
\end{eqnarray*}
Hence for $\mid\theta\mid \leq T_n$ by Taylor's formula and Lemma \ref{al13}
\begin{eqnarray}
\label{aae10}
n\ln\lambda({{\theta}\over{\sigma\sqrt{n}}})
& = & - {{\theta^2}\over{2}} +
{{(i\theta)^3\mu_3}\over{6\sqrt{n}\sigma^3}}
+ {{(i\theta)^4\mu_4}\over{24n\sigma^4}}
- {{(i\theta)^4}\over{8n}} + \cdots \\
& & + {{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\gamma_k +
{{(i\theta)^k}\over{(k-1)!n^{{k-2}\over{2}}\sigma^k}} \int_0^1
(1-x)^{k-1} W_k({{x\theta}\over{\sigma\sqrt{n}}}){\rm d}x,
\nonumber
\end{eqnarray}
where
$W_k(x) = {{{\partial}^k}\over{\partial y^k}}\ln\lambda(y)\big
\vert_{y=x} - \gamma_k. $
Further, it is evident that we can insert $\eta_k\leq\eta_3$ in
$T_n =
\min\{{{{\sigma}^2}
\over{5({3\over{2}}\vert{\mu}_3\vert+{\sigma}^3)}}, {\eta_k} \}
{\sigma}\sqrt{n},$
such that for $\vert\theta\vert\leq T_n$ we have
\[
6\sigma^2
({{\vert\theta\vert^2}\over{24n\sigma^4}}\vert\gamma_4\vert +
\cdots +
{{\vert\theta\vert^{k-2}}\over{k!n^{{k-2}\over{2}}\sigma^k}}
(\vert\gamma_k\vert+c_k)) < 7, \] where $c_k = \sup_{x\in [0,1]}
\vert W_k(x)\vert$. Since
$W_k({{x\theta}\over{\sigma\sqrt{n}}})\rightarrow_n 0,$ so by the
Lebesgue dominated convergence theorem we get $\int_0^1
(1-x)^{k-1} W_k({{x\theta}\over{\sigma\sqrt{n}}}){\rm d}x = o(1).$
By virtue of (\ref{aae8}), Lemma \ref{al8} and (\ref{aae10}) we
obtain
\begin{eqnarray*}
\lefteqn {
\Vert \hat{{\bold P}}^n({{\theta}\over{\sigma\sqrt{n}}})
- e^{ - {{\theta^2}\over{2}} +
{{(i\theta)^3\mu_3}\over{6\sqrt{n}\sigma^3}}
+ {{(i\theta)^4\mu_4}\over{24n\sigma^4}}
- {{(i\theta)^4}\over{8n}} +
\cdots + {{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\gamma_k }
\hat{\bold P}_1({{\theta}\over{\sigma\sqrt{n}}})
-({\bold P}^n - {\bold \Pi})\Vert }
\\& & \leq
e^{ - {{\theta^2}\over{2}} +
\cdots +
{{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\gamma_k}
\left\vert \exp\{ {{(i\theta)^k\int_0^1 (1-x)^{k-1}
W_k({{x\theta}\over{\sigma\sqrt{n}}}){\rm d}x }
\over{(k-1)!n^{{k-2}\over{2}}\sigma^k}} \} - 1\right\vert O(1) \\
& & \qquad
+\, O({{\vert\theta\vert}\over{\sigma\sqrt{n}}}\kappa^n).
\end{eqnarray*}
By (\ref{aae10}) and the inequality
$ \vert e^x - 1\vert \leq \vert x\vert e^{\vert x\vert},$
we find that for $\mid\theta\mid\leq T_n$ we have
\[
\big\vert \exp\{
{{(i\theta)^k}\over{(k-1)!n^{{k-2}\over{2}}\sigma^k}}\!\int_0^1\!
(1-x)^{k-1} W_k({{x\theta}\over{\sigma\sqrt{n}}}) {\rm d}x\} -
1\big\vert \leq
{{o(1)\vert\theta\vert^k}\over{n^{{k-2}\over{2}}}}
\exp\{{{c_k\vert\theta\vert^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\}\cdotp
\]
Hence,
\begin{eqnarray*}
\lefteqn {
\Vert \hat{{\bold P}}^n({{\theta}\over{\sigma\sqrt{n}}})
- e^{ - {{\theta^2}\over{2}} +
{{(i\theta)^3\mu_3}\over{6\sqrt{n}\sigma^3}}
+ {{(i\theta)^4\mu_4}\over{24n\sigma^4}}
- {{(i\theta)^4}\over{8n}} + \cdots
+ {{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\gamma_k}
\hat{\bold P}_1({{\theta}\over{\sigma\sqrt{n}}})
-({\bold P}^n - {\bold \Pi})
\Vert } \\
& & \leq\exp\{ - {{\theta^2}\over{2}}
+ \cdots
+ {{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}(\vert\gamma_k\vert + c_k)\}
{{\vert\theta\vert^k}\over{n^{{k-2}\over{2}}}}o(1)
+ O({{\vert\theta\vert}\over{\sigma\sqrt{n}}}\kappa^n) \\
& & \leq
\exp\{ - {{\theta^2}\over{2}} + {{\theta^2}\over{2}}(
{1\over15}{{{3\over{2}}\vert\mu_3}\vert\over
{{3\over{2}}\vert\mu_3\vert + \sigma^3}} +
{7\over15}{{\sigma^3}\over{{3\over{2}}\vert\mu_3\vert
+ \sigma^3}}) \}
{{\vert\theta\vert^k}\over{n^{{k-2}\over{2}}}}o(1)
+ O({{\vert\theta\vert}\over{\sigma\sqrt{n}}}
\kappa^n) \\
& & \leq
o(1){{\vert\theta\vert^k}\over{n^{{k-2}\over{2}}}}
\exp\{ - {{\theta^2}\over{4}} \}
+ O({{\vert\theta\vert}\over{\sigma\sqrt{n}}}
\kappa^n).
\end{eqnarray*}
Thus expanding
$\exp\{{{(i\theta)^3\mu_3}\over{6\sqrt{n}\sigma^3}}
+ \cdots
+ {{(i\theta)^k}\over{k!n^{{k-2}\over{2}}\sigma^k}}\gamma_k\}$
and using Lemma \ref{al6} and Taylor's formula
for $\hat{\bold P}_1({{\theta}\over{\sigma\sqrt{n}}})$
we obtain (\ref{ae7}).
\end{proof}
The following lemma provides an estimate for the
iterates of characteristic operator (for
the proof see Lemma 1.5 in Nagaev (1961)).
\begin{lemma}\quad
\label{lna2}
Let Condition $(\Psi)$ is fulfilled.
Then for $n\geq 1$ and
$\bigl\bracevert g\bigl\bracevert\leq 1$
\[
\bigl\bracevert \hat{\bold P}^n(\theta)g\bigl\bracevert
\leq \left(\sqrt{1-{{\alpha^4}\over{2\beta}}
(1-\vert{\widehat\mu_f}(\theta)\vert^2)}\right)^{n-1}
\!\!\!\!\!\!\!\!\!. \]
\end{lemma}
\section{ Proofs }
\begin{proof} {\sc Theorems \ref{tn2} and \ref{t4}}
By virtue of Condition $(\Psi)$ and \S1 in Nagaev (1957)
${\bold P}$ is primitive in $L^{\infty}(\mu)$
(alternatively one can use Proposition 3.13 in Wu (2000)).
Moreover, it follows also that
\[ \lim_{L\to\infty}\bigl\bracevert
\int_{\vert f(y)\vert >L} \vert f(y)\vert^k P(x,{\rm d}y)
\bigl\bracevert \leq \beta\lim_{L\to\infty} \int_{\vert f(y)\vert
>L} \vert f(y)\vert^k \mu({\rm d}y) = 0
\]
so that (\ref{ena1}) holds.
Write
$F_{gn}(z) = F_{g(\cdot),n}(z) =
({\bf K}_{n,z}g)(\cdot),$ and
${G_{gn}}(z) = \sum_{m=0}^{k-2} n^{-{{m}\over{2}}} {\bf A}_{m,z}g$.
Let $K_{gn}(z)$ be the distribution function that assigns the
mass $({\bold P}^n - {\bold \Pi})g(\cdot)$ at $0$.
Put,
\[ H_{gn}(z) = G_{gn}(z) + K_{gn}(z),\>\,
\hat H_{gn}(\theta) = \int e^{i\theta x}{\rm d}H_{gn}(x), \>\,
\hat F_{gn}(\theta) = \int e^{i\theta x}{\rm d}F_{gn}(x).
\]
Note that
$\hat F_{gn}(\theta) = \hat{\bold P}^n({{\theta}\over{\sigma\sqrt{n}}})(g)$
and
\[\hat G_{gn}(\theta) =
\int e^{i\theta x}{\rm d}G_{gn}(x) = e^{-{{\theta^2}\over{2}}}
\Bigl(\sum_{m=0}^{k-2}{{1}\over{(\sqrt{n})^m}}\sum_{j=0}^{m}
{{1}\over{j!}} \big({{i\theta}\over{\sigma}}\big)^j{\frak
P}_{m-j}(i\theta) \hat{\bold P}_1^{(j)}g(\cdot) \Bigl)\cdot\]
Because of
\[ \bigl\bracevert H_{gn}(z+y) - H_{gn}(z)\bigl\bracevert \leq
\bigl\bracevert G_{gn}(z+y) - G_{gn}(z)\bigl\bracevert
+ \bigl\bracevert ({\bold P}^n - {\bold \Pi})g\bigl\bracevert, \]
\[ G_{gn}(z+y) - G_{gn}(z)
= y{{\partial }\over{\partial z}}G_{gn}(z) + {\rm sgn}(y)
\int\limits_{{{y-\vert y\vert}\over{2}}}^{{{y+\vert
y\vert}\over{2}}} ({{\partial}\over{\partial u}} G_{gn}(z+u) -
{{\partial }\over{\partial z}}G_{gn}(z)){\rm d}u \]
thus in view
of Th. 5.3 on pp. 146--147 in Petrov (1996)
and (\ref{ae81}) we have
\begin{eqnarray}
\label{ae41}
\bigl\bracevert F_{gn}(z) - G_{gn}(z)\bigl\bracevert
& \leq & \bigl\bracevert F_{gn}(z) - G_{gn}(z)
- K_{gn}(z) \bigl\bracevert +\, C\vert\gamma\vert^n
\bigl\bracevert g\bigl\bracevert \nonumber \\
& \leq &
{1\over{\pi}}\int_{\vert\theta\vert\leq T}
\bigl\bracevert\hat F_{gn}(\theta) - \hat H_{gn}(\theta)\bigl\bracevert
{{{\rm d}\theta}\over{\vert\theta\vert}} \\
& & \,
+ {3c^2({1\over{\pi}})\over{\pi T}}\sup_{z}
\bigl\bracevert{{\partial}\over{\partial z}} G_{gn}(z)\bigl\bracevert
+ C\vert\gamma\vert^n
\big(1 + {{2c({1\over{\pi}})}\over{\pi}}\big)
\bigl\bracevert g\bigl\bracevert. \nonumber
\end{eqnarray}
Now, since
$\sup_{z} \bigl\bracevert{{\partial}\over{\partial z}}
G_{gn}(z)\bigl\bracevert$
is bounded and $\vert\gamma\vert<1$
whence by (\ref{ae41}) for $T=n^k, k \geq 4$, we get
\begin{equation}
\label{efg} \bigl\bracevert F_{gn}(z) - G_{gn}(z)\bigl\bracevert
\leq {1\over{\pi}}\int_{\vert\theta\vert\leq n^k}
\bigl\bracevert\hat F_{gn}(\theta) - \hat
H_{gn}(\theta)\bigl\bracevert {{{\rm
d}\theta}\over{\vert\theta\vert}} + o({{\bigl\bracevert
g\bigl\bracevert}\over{n^{{k-2}\over{2}}}})\cdotp
\end{equation}
By virtue of Th. \ref{al3}
\begin{eqnarray}
\label{efh0}
\lefteqn
{ \int_{\vert\theta\vert\leq T_n}
\bigl\bracevert\hat F_{gn}(\theta) - \hat H_{gn}(\theta)\bigl\bracevert
{{{\rm d}\theta}\over{\vert\theta\vert}} } \\
& & \leq {{o(\bigl\bracevert
g\bigl\bracevert)}\over{n^{{k-2}\over{2}}}}
\int_{\vert\theta\vert\leq T_n} (\vert\theta\vert^{k-3} +
\vert\theta\vert^{k-1} + \vert\theta\vert^k +
\vert\theta\vert^{3k-7}) e^{-{{\theta^2}\over{4}}}{\rm d}\theta +
{{T_n^2}\over{\sqrt{n}}}O(\kappa^n) \nonumber\cdotp
\end{eqnarray}
This established, we have to show that
\begin{eqnarray}
\label{efh} \int_{T_n<\vert\theta\vert\leq n^k}
{{\bigl\bracevert\hat F_{gn}(\theta) - \hat
H_{gn}(\theta)\bigl\bracevert} \over{\vert\theta\vert}} {\rm
d}\theta \leq o({{\bigl\bracevert
g\bigl\bracevert}\over{n^{{k-2}\over{2}}}})\cdotp
\end{eqnarray}
For this observe that
\begin{eqnarray*}
\int_{T_n<\vert\theta\vert\leq n^k}\!\!\!\!\!\!\!\!\!
{{\bigl\bracevert\hat
G_{gn}(\theta)\bigl\bracevert}\over{\vert\theta\vert}} {\rm
d}\theta \leq 2\int_{T_n}^{\infty}\!\!\! e^{-{{\theta^2}\over{2}}}
\left\Vert \sum_{m=0}^{k-2} \sum_{j=0}^{m}
{{(i\theta)^j}\over{n^{{m}\over{2}}j!\sigma^j}} {\frak
P}_{m-j}(i\theta) \hat{\bold P}_1^{(j)} \right\Vert
\bigl\bracevert g\bigl\bracevert {{{\rm
d}\theta}\over{\vert\theta\vert}}
\end{eqnarray*}
and that by (\ref{ae81})
\begin{eqnarray*}
\int_{T_n<\vert\theta\vert\leq n^k} \bigl\bracevert \hat
H_{gn}(\theta) - \hat G_{gn}(\theta)\bigl\bracevert {{{\rm
d}\theta}\over{\vert\theta\vert}} \leq 2C_k\vert\gamma\vert^n
\bigl\bracevert g\bigl\bracevert\ln{n} = o({{\bigl\bracevert
g\bigl\bracevert}\over{n^{{k-2}\over{2}}}})\cdotp
\end{eqnarray*}
Further, by Lemma \ref{lna2} and (\ref{ecra})
there exists $\theta_0$ such that for any $\tau > \theta_0$
\[
n^{{k-2}\over{2}}\!\!\!\!\!\!
\int\limits_{T_n<\vert\theta\vert\leq n^k} \bigl\bracevert \hat
F_{gn}(\theta)\bigl\bracevert {{{\rm
d}\theta}\over{\vert\theta\vert}} \leq
n^{{k-2}\over{2}}\!\!\!\!\!\! \int\limits_{\tau\leq\vert
\theta\vert\leq n^k} \bigl\bracevert(\hat{{\bold P}}(\theta))^n
g\bigl\bracevert {{{\rm d}\theta}\over{\vert \theta\vert}} \leq
C_k n^{{k-2}\over{2}}e^{-cn} \bigl\bracevert g\bigl\bracevert
\ln{n}
\] which with the latter inequalities proves (\ref{efh}).
Consequently, the substitution of (\ref{efh0}) and (\ref{efh})
into (\ref{efg}) yields (\ref{eb1}). For the case $k=3$ set $T =
T_nr_n$ and choose a sequence $r_n\rightarrow \infty$ such that we
have
\[\int_{T_n<\vert\theta\vert\leq T_nr_n}
\bigl\bracevert \hat F_{gn}(\theta)\bigl\bracevert {{{\rm
d}\theta}\over{\vert\theta\vert}} =
\int_{T_n<\vert\theta\vert\sigma\sqrt{n}\leq T_n r_n}
\bigl\bracevert\big( \hat{{\bold P}}(\theta) \big)^n
g\bigl\bracevert {{{\rm d}\theta}\over{\vert \theta\vert}}
=\bigl\bracevert g\bigl\bracevert o(n^{-1/2}). \] This completes
the proof.
\end{proof}
\vskip 0pt\noindent
{\bf Acknowledgment.} The author thanks A. Nagaev for his comments
concerning this exposition.
\vskip 15pt\noindent
{\bf References}
\vskip 5pt\noindent
Gnedenko, B.V., Kolmogorov, A.N., 1954.
Limit Distributions for Sums of Independent Random Variables.
Addison-Wesley, Reading, Mass.
\vskip 2pt\noindent
Gudynas, P., 2000.
Refinements of the Central Limit Theorem for Homogene\-ous Markov
Chains, in: Yu. V. Prokhorov and V. Statulevi\v cius, eds.
Limit Theorems of Probability Theory. Springer, Berlin, pp.
165--183.
\vskip 2pt\noindent
Jensen, J.L., 1991.
Saddlepoint expansions for sums of Markov dependent variables on a
continuous state space.
Probab. Theory Related Fields 89, 181--199.
\vskip 2pt\noindent
Nagaev, A.V., 2001.
An asymptotic formula for the Bayes risk in discriminating between
two Markov chains. J. Appl. Probab. 38A, 131--141.
\vskip 2pt\noindent
Nagaev, A.V., 2002.
An asymptotic formula for the Neyman--Pearson risk in
dicriminating between two Markov chains.
J. Math. Sci. 111, 3592--3600.
\vskip 2pt\noindent
Nagaev, S.V., 1957.
Some limit theorems for stationary Markov chains.
Teor. Veroyatnost. i Primenen. 2, 389--416.
\vskip 2pt\noindent
Nagaev, S.V., 1961.
More exact statements of limit theorems for homogeneous Markov chains.
Teor. Veroyatnost. i Primenen. 6, 67--86.
\vskip 2pt\noindent
Petrov, V.V., 1996.
Limit Theorems of Probability Theory. Sequences of
Indenpendent Random Variables. Oxford Studies in Probability 4,
Oxford.
\vskip 2pt\noindent
Sirazhdinov, S.H., Formanov, S.K., 1979.
Limit Theorems for Sums of Random Vectors Connected in a
Markov Chain. FAN, Tashkent.
\vskip 2pt\noindent
Szewczak, Z.S. 2005.
A remark on large deviation theorem for Markov chain with finite
number of states.
Teor. Veroyatnost. i Primenen. 50 3, 612--622.
\vskip 2pt\noindent
Wu, L.M., 2000.
Uniformly Integrable Operators and Large Deviations for Markov
Processes. J. Funct. Anal. 172, 301--376.
\end{document}
|
2,877,628,089,552 | arxiv | \section{Introduction}
Arguably, individuals form groups to achieve more ambitious goals than they could achieve alone~\citep{allportInfluenceGroupAssociation1920,caporaelPartsWholesEvolutionary2001}. Yet, for groups to proverbially become more than the sum of their parts, they need to organize into effective structures: division of labor~\cite{rozasLoosenControlLosing2021}, structures of information flow and the structure of interpersonal relations~\citep{akdereEconomicsSocialCapital2008}, as well as a well internalized organizational culture are crucial for group performance.
Not surprisingly then, the structure of online collaboration teams and peer production systems, such as Free/Open Source Software (FOSS) or Wikipedia, have often been analyzed to pinpoint those structural properties that determine the quality of group work. Bipartite, collaboration networks (e.g.~co-editing of Wikipedia articles or co-contributions to files in FOSS projects) are the most likely targets of such analysis, as they show how contributors interact with each other when performing the actual tasks. For example, longer path lengths in co-edition networks of groups of editors on English Wikipedia are related to lower performance of the group~\citep{plattNetworkStructureEfficiency2018}. In FOSS projects, contributors self-organize into subgroups working on similar subsets of files in a bottom-up manner~\citep{palazziOnlineDivisionLabour2019}, by performing spontaneously self-chosen and self-defined tasks~\citep{chelkowskiInequalitiesOpenSource2016}, or self-select to tasks (e.g.~modules) in a well-defined hierarchy of labor division~\citep{mozillaRolesLeadership2021Special}.
However, a long-standing line of research on group work in offline settings shows that task-related structuring of the group is not enough for high performance. Task-related group interaction is accompanied by affiliation-related interaction~\citep{balesEquilibriumProblemSmall1953,balesHowPeopleInteract1955}. The task (instrumental) domain of group dynamics drives the group to differentiate, while the affiliation processes (affective domain) focus on integration and promote uniformity~\citep{balesHowPeopleInteract1955}.
Affiliation processes are important not only for the wellbeing of group members: group cohesion is positively related to performance~\citep{bealCohesionPerformanceGroups2003,evansGroupCohesionPerformance1991}. While instrumental, task-related activities can be organized by procedures and rules, social bonding within the group crucially relies on interpersonal communication. In organizations, coordination of work often happens in informal communication, alongside formal structures of management~\citep{beerDecisionControlMeaning1994}. Thus, the structure of interpersonal communication within online peer production groups - specifically, its cohesiveness - should affect the groups’ performance.
In this work, we focus on communication using direct messages within collaborating groups on the English Wikipedia. We used a mixed-methods approach - quantitative social network analysis and qualitative analysis of interviews - to analyze Wikipedia’s topic oriented editor groups: Wikiprojects. By reconstructing networks of direct messages and interviewing core members from selected projects we were able to elucidate how interpersonal interaction, and specifically affiliation related communication, impacts quality of group work. We show that high quality of group work is related to communication networks with strong ties, which also exhibit integrated connection structure, straying off the typical leader – followers, star-like motifs. We also found that direct messaging is associated with motivational messages and outreach, and with tapping on the group’s social capital. Taken together, our results suggest that communication structures that allow for complex integration of resources and knowledge distributed across specialized contributors within the group are conducive to group efficiency.
\section{Background and related work}
Quality of Wikipedia has been a topic in both public and academic discourse since its inception~\citep{jemielniakCulturalDiversityQuality2017,shafeeEvolutionWikipediaMedical2017,smith2020situating}. On the one hand, its popularity combined with lack of professional oversight over the content, raised concerns of possible misinformation of the general public~\citep{lanierDigitalMaoismHazards2006}. On the other hand, the process of bottom up encyclopedic content creation raised interest in self-organization possibilities afforded by new media~\citep{shirkyHereComesEverybody2008}. Thus, understanding how quality products emerge from bottom up social processes is both important for the public at large as well as for theoretical advancement of systems research.
With the growth of Wikipedia, and with refinement of its peer production process, the understanding of self-organized quality control has been accumulating~\citep{nemotoSocialCapitalIncreases2011,plattNetworkStructureEfficiency2018,qinInfluenceNetworkStructures2015}. Most importantly, Wikipedia’s peer review process and quality scores awarded to articles provide an opportunity to quantify the effects of distributed work and allow comparisons against a single scale of quality ratings.
\subsection{Quality in peer production}
For the average reader of Wikipedia, the quality grades of Featured Article (thereafter referred to as FA) or Good Article (GA) – a star or a plus sign, respectively, displayed at the top of the page – might be unnoticeable, but for Wikipedia editors they are cherished marks of accomplishment. Many editors list the FA and GAs they contributed to on their Wikipedia profile pages. The review process that awards those ratings is in many respects similar to academic reviews: the criteria for each rank of article quality are listed explicitly, and each submission is reviewed by independent editors whose remarks are then incorporated into the article. The difference lies in that at the end of the process, a discussion unfolds, and any editor can voice their opinion on whether the article is worthy of a particular quality mark; the ensuing decision is consensus based\footnote{%
\url{https://en.wikipedia.org/wiki/Wikipedia:Featured_article_review}
}.%
The universal criteria (well-known and applicable to all candidate articles) as well as independent review process makes these quality ratings suitable for a quality measurement of peer produced work.
The availability of such quality measures has spurred much investigation into what constitutes quality work. Interestingly, quality criteria differ among language versions of Wikipedia, pointing to possible differences in quality perceptions across cultures~\citep{jemielniakCulturalDiversityQuality2017}. On one hand, particular features of articles that predict well how close they are to being awarded a higher quality grade, can be identified. For example, classifiers trained on previously assessed articles can tell how close an article is to a certain quality rank~\citep{warncke2013tell}. On the other hand, the process of article production can also be investigated to pinpoint those properties that enable some articles to improve in quality much faster than others.
In standard organizations’ production process, quality is ensured, among others, through a careful division of labor among employees with appropriate expertise. In online peer production systems, contributors self-select to tasks~\citep{benklerCoasePenguinLinux2002}, and thus it has been suggested that composition of the editor base impacts quality of Wikipedia articles. \citet{sydow2017diversity} have found that articles edited by contributors of rich and broad background – i.e. those that in their history have edited many varied articles – are more likely to reach FA status. This might not only be an effect of broader subject expertise of these editors but also of a process of \enquote{creative abrasion} wherein editors with varied knowledge and experience engage in discussions to resolve their differences, which leads to article improvement~\citep{arazyInformationQualityWikipedia2011}. Whilst some abrasion might be positive and promote quality, too much of conflict can also be detrimental~\citep{arazyStayWikipediaTask2013}. It seems there is an optimal level of diversity of contributors working on an article and that it is moderated by the size of the group: smaller groups can be less diverse and still deliver high quality articles~\citep{robertCrowdSizeDiversity2015}.
Diversity might be a factor positively impacting quality, but combining diverse inputs requires processes of coordination, which are recognized as a crucial aspect of quality control in standard organizations~\citep{beerViableSystemModel1984}. In online social systems coordination is peer produced in a similar manner to product development~\citep{benklerCoasePenguinLinux2002}. That is, contributors self-select to perform coordination tasks. Not surprisingly, coordination plays a part in creating quality articles on Wikipedia: even such simple coordination as splitting into core and periphery roles (defined as editors who submit the bulk of content vs. editors who make minor additions) increases the chances for FA status~\citep{kitturHarnessingWisdomCrowds2008}. Interestingly, without such a split, simple increase in the number of editors did not lead to improved quality (ibid.).
Finally, an important factor impacting quality is engagement. In standard organizations motivation and conscientiousness can be promoted by various incentives, including financial ones. In online peer production other incentives are at play: ranging from reputation, to self-development, to socializing~\citep{oregExploringMotivationsContributing2008}. Engagement can be induced or sustained by interactions with other, well-socialized members of the community: welcoming messages, assistance and constructive criticism increase commitment of new editors on Wikipedia and increase the chances for their prolonged contribution~\citep{choiSocializationTacticsWikipedia2010}. Such motivational activities are mostly carried out by sending direct messages, i.e. editing a newcomer’s personal talk page~\citep{zhu2011identifying}. Whilst core community members send more such messages per person, in total peripheral members deliver the bulk of engagement related direct messaging, contributing to the shared leadership within Wikipedia~(ibid.). Interestingly, within the thematic oriented groups on Wikipedia (Wikiprojects) task related and feedback related direct messages are initiated more often by peripheral members while socializing aimed messages are sent more often by core members~(ibid.). This suggests that just as it happens in offline groups~\citep{balesHowPeopleInteract1955}, in online peer production communities social and task leadership is divided to a degree.
\subsection{Direct communication in group work}
Interpersonal, one-on-one communication not only affects motivation of group members but it is also the vehicle for much coordination in standard organizations. Whilst formal oversight and management are important for policy and top-level management, coordination of tasks is often carried out in informal meetings or communication~\citep{beerDecisionControlMeaning1994}. Dyadic relationships and communication play a role in maintaining motivation, engagement and cohesiveness of a group~\citep{lidenDyadicRelationships2016}, contributing to the affective domain within the group~\citep{balesHowPeopleInteract1955}. Through dyadic interactions, members of the group transmit emotional support and share task-related resources~\citep{omilion-hodgesContextualizingLMXWorkgroup2013}. Direct communication also warrants social learning, i.e. learning that occurs through observation and direct instruction of more experienced team members~\citep{singh2013social}.
In digital collaboration, personal messages have also been identified as impacting group work. For example, informal, direct communication among distributed workers helps to build bonds and ensures effective information flow~\citep{hindsUnderstandingConflictGeographically2005}. The importance of direct communication in distributed teams is corroborated by the fact that activity on user pages on the English Wikipedia continued to grow in spite of declining number of edits on article pages~\citep{nemotoSocialCapitalIncreases2011}. Moreover, disassortative direct communication network structures – i.e.~linkages between hubs and periphery of the networks – are prevalent among Wikipedia’s Wikiproject teams and mild disassortativity is related to higher quality of articles that these projects curate~\citep{rychwalskaQualityPeerProduction2020}.
\subsection{Summary and hypotheses}
The effectiveness of an online collaboration group – i.e.~the quality of its products – can~thus be related to both its task related structuration that ensures proper task performance and to interpersonal relations (affiliation processes) that allow coordination and promote engagement~\citep{balesEquilibriumProblemSmall1953}. The former has been extensively studied in both Wikipedia and FOSS by investigating division of labor and (co-)contribution patterns (e.g.~\cite{arazyFunctionalRolesCareer2015,plattNetworkStructureEfficiency2018}). Relation building and motivational structures, on the other hand, have been much less investigated; yet offline theory and research of group processes as well as initial investigations into the role of personal messaging indicate that it plays a crucial role in group effectiveness. Specifically, the content of direct communication has been shown to impact motivation to contribute~\citep{choiSocializationTacticsWikipedia2010} and to relate to relational leadership~\citep{zhu2011identifying}, however the structure of such communication may also play a role.
On Wikipedia this structure can be gleaned from direct messages sent between editors and analyzed as a network~\citep{rychwalskaQualityPeerProduction2020}. If indeed direct messages are a vehicle for building and maintaining engagement,
\textbf{(H1)} successful collaborating groups will exchange a larger volume of such messages than unsuccessful ones. Moreover, if they are a means for self-regulation and coordination
\textbf{(H2)}, direct communication structure of successful teams will allow for both local, specialized information sharing as well as global integration of information within the group~\citep{kleinEmergenceInformativeHigher2020}. Finally, if they serve as a vehicle for group integration
\textbf{(H3)} direct messages will link a larger part of the group (rather than some selected group members) within successful groups than unsuccessful ones. To test these three hypotheses we have designed an exploratory study into the communication structure of thematic groups on English Wikipedia: Wikiprojects, and into its impact on the effects of group work: quality of articles. For each project we have reconstructed a network of direct messaging and measured the related properties: average node strength, network degeneracy and determinism (components of effective information as defined by~\citet{kleinEmergenceInformativeHigher2020}), and fraction of project editors that take part in direct communication network. We related these network properties to the quality of articles within the relevant project’s scope and we found that direct communication networks with high effective information and strong links are related to more effective group work. To investigate in depth the role of such cohesive communication networks in Wikiprojects’ work and with results of our quantitative analysis in mind, we designed semi-structured interviews that we then carried out with selected members of two projects: the one that was most successful with regard to producing quality articles (Tropical Cyclones), and one that was the fastest growing project in 2020 (COVID-19). We found that direct messaging is crucial for feedback on individual work and that one-on-one communication networks play a part in tapping on the groups’ social capital.
\section{Methods}
We conducted a Thick Big Data analysis~\citep{jemielniakThickBigData2020} of collaboration on English Wikipedia. Thick Big Data is a novel mixed-method approach, allowing for combining quantitative analysis of large datasets with deep, qualitative inquiry. In our case, the analysis relied on combining qualitative interviews with heavily engaged Wikiproject participants and network analysis of direct communication within the projects.
\subsection{Wikiprojects}
As our unit of analysis we have chosen English Wikipedia’s Wikiprojects. These are groups of editors that work together on encyclopedic articles related to a specific topic\footnote{%
\url{https://en.wikipedia.org/wiki/Wikipedia:WikiProject}
}.%
For example, there is a project on Military History, on Michael Jackson or on the Simpsons. Projects vary in scope (i.e.~the number of articles they curate), depending both on the breadth of the subject as well as on its popularity among Wikipedians (the more editors are interested in a particular topic, the more articles they will create and the better the coverage of this topic on Wikipedia). There is one thing all projects share, though: they all strive to improve the quality of articles within their scope. To achieve this goal they use various methods of coordination - from discussion on article importance, to editathons and drives to improve articles, to templates and preparing manuals of style, among others. Each Wikiproject has a dedicated space on the so-called project namespace on Wikipedia. In there, the goals, rules and procedures, as well as current information on the project are displayed~(Fig.~\ref{fig:wp-diagram}). These spaces also include talk pages that are used for discussion of current issues within the project.
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{figures/wp-diagram}
\caption{Screenshots from the Wikipedia coordination space for Wikiproject Military History: (a.) the front page, delineating the goal of the project; (b.) fragment of a section with pending tasks for members to work on; (c.) quality statistics for the project; and (d.) fragment of the member list where Wikipedians can join the project.}
\label{fig:wp-diagram}
\end{figure}
\subsection{Data gathering and network construction}
Source code of all user talk pages on English Wikipedia was collected via Wikipedia API\footnote{%
\url{https://en.wikipedia.org/w/api.php}
during the first half of February 2021 (the process ended on February 11) using the \textit{CirrusDoc} query endpoint. Simultaneously, information on all quality assessments of articles available in the main encyclopedia (so-called main namespace) was extracted using the \textit{AllPages} query method of Wikipedia API (the process ended on February 12). Inconsistent naming of Wikiprojects was standardized. The extraction process spanned several days as in both cases a high level of auto-throttling was used in order not to consume too much of Wikipedia API bandwidth.
Next, we defined sets of Wikipedia editors associated with individual projects as all editors who left at least one signed message on one of the project pages (but not talk pages) belonging to a given Wikiproject. Crucially, project pages also include lists of project members which consist of (usually) user signatures. Hence, we included all editors properly signed on member lists as well as all who contributed in a detectable way to creation of project pages of a given Wikiproject. This second case is important as not all editors active within a project choose to sign up to member lists. We consciously did not include all editors who contributed to the encyclopedic articles curated by projects, as such lists would include many accidental editors, not collaborating within the projects.
Then, we used a discussion parser tailored for Wikipedia talk pages~\citep{talagaSztalWikitalkparserInitial2021} to extract all discussion threads from talk pages of all users who were identified as Wikiproject editors. The threads selected by the parser included template based messages (such as warnings, barnstars, welcome messages, etc.) but did not include mass messaging, such as newsletters.
Finally, the discussion threads were used to define undirected weighted links between pairs of editors who left messages on each other’s user pages with weights corresponding to the number of interactions between them. We chose to use undirected links as it is usually very hard to keep track of messages and their replies due to many different communication customs used on Wikipedia. For instance, in some cases an original question left on a user’s talk page may be answered by the user on that exact same page while in other cases the reply may be left on the talk page of the editor asking the original question. This is why we prefer to consider any message between two users as an instance of an undirected interaction between them.
We used the above approach to recreate a total of 1625 networks. However, for downstream analyses we used only a subset of projects with at least 5 editors being present in the direct communication network. This filtering step was necessary as most structural network metrics are not really meaningful for smaller networks as they are constrained to take values from only a very small and discrete set. Moreover, projects with no FA/GA pages were also excluded: having produced no quality output these projects may well comprise a different population and hence would not help us address our research questions on impact of interpersonal communication on group efficiency. After these two filtering steps the initial set of 1625 direct communication networks was reduced to 997 observations of undirected networks.
\subsection{Interviews: choice of respondents and procedure}
For the purpose of our qualitative part, we conducted four interviews: two with early participants from Wikiproject Covid-19, and two with early participants from Wikiproject Tropical Cyclones. We reached out to eight of the early editors listed on the Wikiprojects’ pages. We focused on those who are still active and who allow contact by email. Four people finally agreed to an interview. The interviews took about one hour each, and were conducted via Google Meet platform (with video). All interviews were recorded with consent and transcribed. The interviews were loosely structured and open-ended~\citep{ciesielskaQualitativeMethodologiesOrganization2018}. The questions were grouped into a few themes that reflected our hypotheses: engagement, project coordination, and communication within the project. The sample questions included:
\begin{itemize}
\item \textbf{Engagement and outreach}
\begin{itemize}
\item How did the project start?
\item What makes a successful wikiproject?
\item How are the new members recruited and on-boarded?
\end{itemize}
\item \textbf{Project coordination, division of tasks and governance}
\begin{itemize}
\item Why do you participate, and how do you choose what to do?
\item What does the leadership of the project look like?
\item What could you tell about the changes in coordination of the project over the years?
\end{itemize}
\item \textbf{Communication}
\begin{itemize}
\item How do project members communicate?
\end{itemize}
\end{itemize}
\begin{table}[h]
\centering
\caption{Interviewee list}
\begin{tabular}{lll}
User & Active since & Project \\
\hline
Interviewee 1 & 2007 & COVID-19 \\
Interviewee 2 & 2006 & Tropical Cyclones \\
Interviewee 3 & 2005 & Tropical Cyclones \\
Interviewee 4 & 2003 & COVID-19 \\
\hline
\end{tabular}
\label{tab:interviewees}
\end{table}
\section{Results – network analysis}
\subsection{Analytical strategy and measures}
Our hypotheses are centered on the impact of communication structure on the capacities of a group to deliver high quality products by collaboration. By reviewing extant findings on quality work in peer production we pinpointed that direct, one-on-one communication can bring in group integration, engagement and complex coordination patterns, all required to produce complex artifacts of high quality such as featured Wikipedia articles. However, these theoretical concepts can be measured in multiple ways and for the results to be meaningful a lot depends on proper operationalization.
\subsection{Group integration}
Following the results of~\citet{zhu2011identifying} we assume that both peripheral and core members are involved in leadership behaviors aimed at affiliation processes. Thus an important aspect of quality Wikiproject work would be to encourage communication between as many members of the group as possible. This can be measured as the fraction of project members that are included in the direct communication network.
\subsection{Engagement-building strong ties}
Following the work of~\citet{balesHowPeopleInteract1955} we assume that affective processes within a group are important for group effectiveness. Moreover we accept, after~\citet{oregExploringMotivationsContributing2008} that engagement of group members results in sustained, high quality contributions, and after~\citet{zhu2011identifying} that direct messages can be used to raise such engagement. Thus we might conclude that a simple network measure reflecting the number of communication links between project members (e.g. average node degree) would capture the extent to which such a network can transmit motivational messages. However, affective processes within a group are not the same as information transfer. For example, informing all group members about a task requires only one message per member. Not surprisingly, such processes on Wikipedia are automated – Wikiprojects use newsletter or mass-messaging to spread information broadly. However, raising motivation and building strong social ties requires reciprocity and perseverance; that is, multiple communication acts – messages – are conducive to a stronger relationship. Therefore, average volume of messages per node (i.e.~average node strength) – rather than average degree – would be a better measure of Wikiprojects’ capacity to build engagement.
\subsection{Complex coordination: Determinism, Degeneracy and Effective Information in networks}
Finally, following~\citet{nowakFunctionalSynchronizationEmergence2017} we assume that complex functions in social systems require complex coordination patterns between elements. Theoretical work by~\citet{tononi1998complexity} showed that structures that are capable of complex coordination need to have local specialization and global integration. Examples of such complex coordination can be found in brain imaging studies and neural modelling~\citet{tononi1994measure}. These concepts from information theory have recently been adapted into network measures of determinism (local specialization) and degeneracy (reverse of global integration)~\citep{kleinEmergenceInformativeHigher2020}. Both can be used to assess a social system’s capability to coordinate in a complex manner.
We follow~\citet{kleinEmergenceInformativeHigher2020} in our approach to quantifying structure and organization of the direct communication networks through the lenses of determinism, degeneracy and effective information. Here we review the main definitions.
A graph $G = (V, E)$ consists of a set of $n = |V|$ vertices and a set of $m = |E|$ links connecting selected pairs of vertices. Moreover, each link $(i, j) \in E$ has a weight $w(i, j) \in \mathbb{N}^+$ indicating the number of interactions between $i$ and $j$ and moreover $w(i, j) = w(j, i)$ for all $i, j = 1, \ldots, n$. Each graph can be represented by a $n \times n$ adjacency matrix $\mathbf{A}$ such that $a_{ij} = w(i, j) = a_{ji}$ if $(i, j) \in E$ and $0$ otherwise. Then, from any $\mathbf{A}$ a directed transition matrix $\mathbf{W}$ can be derived by dividing each row by the strength (sum of weights) of the corresponding node (rows with zeros only are left without change). This way values in each row are non-negative and sum up to $1$ so they can be interpreted as a valid probability distribution over possible next positions of a random walker starting at node $i$.
Then, determinism (local specialization of connections) of a graph G is defined as:
\begin{equation}\label{eq:det}
\ensuremath{\text{Det}}(G) = \log_2{n} - \frac{1}{n}\sum_{i=1}^n H(w_i)
\end{equation}
where $H(w_i) = \sum_j w_{ij}\log_2{w_{ij}}$ is Shannon entropy over the $i$-th row of the transition matrix $\mathbf{W}$ (isolated nodes are ignored in the calculation). Thus, determinism is the average certainty associated with the next move of a random walker placed randomly somewhere on the network. It is maximized when each node is connected to only one other node and minimized when each node connects to all other nodes. Hence, it is a measure of how specific local connections of nodes are on average.
Degeneracy of a graph G is defined quite similarly but in terms of Shannon entropy of the average over rows of W:
\begin{equation}\label{eq:deg}
\ensuremath{\text{Deg}}(G) = \log_2{n} - H\left(\frac{1}{n}\mathbf{1}_n^\top\mathbf{W}\right)
\end{equation}
where $\mathbf{1}_n$ is a vector of ones of length $n$ (again, isolated nodes are ignored). Thus, degeneracy is a measure of how concentrated the connectivity of $G$ is around a small subset of nodes. It is maximized when all nodes are linked to only a single hub and minimized when on average each node is visited equally often. In other words, it is a measure of the global integration of $G$. Example, artificial networks with representing structures of low and high degeneracy are presented in Fig. 2.
\begin{figure}[!t]
\centering
\includegraphics[width=.495\textwidth]{figures/ei_1-1}
\hfill
\includegraphics[width=.495\textwidth]{figures/ei_2-1}
\caption{%
Examples of networks with low and high measures of degeneracy; more examples can be found in~\citet{kleinEmergenceInformativeHigher2020}, figure 2.
}
\end{figure}
Both $\ensuremath{\text{Det}}$ and $\ensuremath{\text{Deg}}$ have well-defined maximum values of $\log_2{n}$ so they can be normalized in the range $[0, 1]$ simply by dividing by $\log_2{n}$. Last but not least, they allow to define a quantity known as effective information which describes the overall structure of $G$ in terms of local specificity and global integration:
\begin{equation}\label{eq:ei}
\ensuremath{\text{EI}}(G) = \ensuremath{\text{Det}}(G) - \ensuremath{\text{Deg}}(G)
\end{equation}
The above definition of effective information mimics closely the theory of integrated information of~\citet{tononi1998complexity} and can be seen as a measure of non-trivial, complex structural organization in networks.
\subsection{Group size}
Acknowledging previous work on quality in peer production that focused on task processes within a group – e.g.~the composition of teams in terms of expertise or size of the collaborating teams~\citep{wilkinson2007cooperation} – we chose the size of the contributor group (i.e.~number of project members) as a control variable. By controlling for size of the projects we can assess whether affiliation processes contribute to group efficiency when controlling for task-related processes.
\subsection{Quality of group work}
The quality of group work that we aimed to relate to the properties of communication networks can be approximated by the quality tags assigned to articles by the Wikipedia community. The two ranks that are assessed by the whole community with respect to the same criteria are FA and GA (other ranks can have different criteria depending on the Wikiproject and the review process for them is carried out within projects). The numbers of FAs and GAs for each project are highly correlated ($r_{\text{Pearson}} = .92, p < .0001$) and thus they can be combined into a single measure of quality. However, projects differ vastly in their scope – i.e.~the number of articles that they curate – and we can expect that larger projects would naturally have a larger number of quality articles. For example, more articles within a thematic area can draw in more contributors interested in the various subtopics, who would self-select to work on the relevant articles. Moreover, the number of articles within a project’s scope is the cap on the number of possible quality articles. Thus an absolute value of quality articles will lead to a quality ranking with the top filled with projects of very large scope.
On the other hand, normalizing the quality score by the number of articles within a project’s scope (i.e.~the quality score becomes the fraction of high quality articles within all project’s articles) privileges smaller projects. In the most extreme case, a project curating only one article can achieve a hundred percent efficiency in producing quality articles with relatively little effort. Since the distribution of projects’ scope is highly skewed (most curate few articles), using such normalization would unduly lower the quality ranking of the largest projects.
We can thus consider a family of quality scores indexed by a parameter $p \in [0, 1]$ such that the quality score for a project $x$ is $Q_p(x) = N_Q(x)/n^p$ where $N_Q(x)$ is the number of FA+GA pages curated by $x$. Within this family the two cases described above constitute the extreme cases. A~quality score with no normalization by the projects’ scope is given by $Q_0$ and quality score normalized with the number of articles within scope is given by $Q_1$ . To~construct our response variable we have chosen the middle case in this family of quality rankings: $Q_{1/2}$ . In~effect, we normalize the number of quality articles not by the absolute number of articles within a project’s scope but rather by the size class of the project’s scope. Indeed, the square root of size serves as variance-stabilizing transformation and as such attenuates the effects of the highly heterogeneous distribution of project sizes while not discarding this information completely. An additional confirmation of the validity of this approach is given by the fact that $Q$-score with $p=1/2$ has almost perfectly log-normal distribution. Table~\ref{tab:descriptives} lists descriptive statistics for the chosen measures.
\begin{table}[!h]
\centering
\caption{Descriptive statistics of project measures}
\begin{tabular}{lcc}
Variable & Mean (SD) & Median \\
\hline
Quality & 1.172 (1.684) & .652 \\
Fraction in communication network & .512 (.173) & .503 \\
Determinism & .761 (.093) & .782 \\
Degeneracy & .265 (.139) & .238 \\
Average connection strength & 30.094 (39.185) & 17.143 \\
Number of project members & 136.39 (249.98) & 61 \\
\hline
\end{tabular}
\label{tab:descriptives}
\end{table}
\subsection{Modelling quality}
To verify if group integration (fraction of project contributors within communication network), engagement building social ties (average node strength), and communication structure allowing for complex coordination (determinism and degeneracy) impact the quality of artifacts produced by peer production groups, we tested a linear regression model with all the above variables as predictors, quality score normalized by the square root of articles within scope as the response variable and number of editors listed within the project as control variable (Table~\ref{tab:model},~Model~1). Quality score, number of users within the project and average node strength were approximately log-normally distributed so we used their log transforms in the model. The model explained 23\% of variance in the quality scores of Wikiprojects ($F(5, 991) = 59.66, p < .001$). All predictors except the fraction of users within the communication network were significant. We tested a model (Table~\ref{tab:model},~Model~2) without the non-significant predictor, and the decrease in variance explained was not significant ($F(4, 991) = .178, p > .05$). The model without fraction of project members within the communication network also explained 23\% of the variance in quality scores ($F(4, 992) = 74.6, p < 0.001$).
\begin{table}[h]
\begin{threeparttable}
\centering
\caption{Effects of network structure on quality}
\begin{tabular}{lccc}
Predictor & Model 1 & Model 2 & Model 3 \\
\hline
Fraction in communication network & -.093 (.22) & & \\
Determinism & 1.313$^*$ (.54) & 1.33$^*$ (.54) & \\
Degeneracy & -.665$^*$ (.30) & -.71$^*$ (.28) & \\
Average connection strength (log) & .285$^{***}$ (.04) & .278$^{***}$ (.04) & .246$^{***}$ (.03) \\
Number of project members (log) & .315$^{***}$ (.04) & .32$^{***}$ (.04) & .347$^{***}$ (.03) \\
Effective information & & .667$^*$ (.28) \\
Constant & -3.369$^{***}$ (.38) & -3.41$^{***}$ (.37) & -2.994$^{***}$ (.18) \\
\hline
R$^2$ & .231 & .231 & .23 \\
\hline
\end{tabular}
\label{tab:model}
\begin{tablenotes}
\small
\item Note: Linear regression coefficients; standard errors in parentheses; $^{***}: p \leq .001$, $^*: p \leq .05$
\end{tablenotes}
\end{threeparttable}
\end{table}
Note that the regression coefficients for determinism and degeneracy are similar but with opposite signs. We can confirm this with a test of linear hypothesis: $H_0: b_{\text{deg}}+b_{\text{det}} = 0\ (F(2, 992) = 2.11, p > .05)$. The opposite impact of degeneracy (global integration) and determinism (local specialization) is indeed proposed to characterize complex systems. The measure of effective information, defined as $\ensuremath{\text{EI}}(G) = \ensuremath{\text{Det}}(G) – \ensuremath{\text{Deg}}(G)$~\citep{kleinEmergenceInformativeHigher2020,tononi1998complexity} describes how much information is integrated within a complex system and can be used as an estimate of the system’s complexity. It has even been suggested as an approximation of the level of consciousness of a system~\citep{tononi2012integrated}.
We tested a simplified model with degeneracy and determinism predictors replaced with effective information (Table~\ref{tab:model},~Model~3). All predictors were significant, the model explained 23\% of variance in quality scores $(F(3,993) = 98.65, p < .001)$, and in that was not significantly different from the model with degeneracy and determinism $(F(2, 992) = 2.11, p > .05)$. Example networks, depicting various levels of effective information are presented in Fig.~\ref{fig:wp-networks}. Thus we can conclude that effective information of the communication structure within Wikiprojects plays a significant role as a factor facilitating production of high-quality content on Wikipedia.
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{figures/plt2-1}
\caption{%
Sample Wikiproject communication structures, varying in degeneracy, determinism and position in the quality ranking.
}
\label{fig:wp-networks}
\end{figure}
\section{Results – interviews}
\subsection{Analytical strategy}
Our quantitative analysis of the structure of one-on-one communication networks within Wikiprojects indicated that quality of their collective output grows with the strength of the ties between Wikiproject members, with the specialization of local connections (i.e.~when project members have specific partners with whom they communicate more than with others), and with integration of information among all group members (i.e.~when the connections in the network do not coincide on a limited set of nodes but rather the information spreads among all parts of the communication network). To further verify these findings and to gain an in-depth understanding of why such communication structures form and in what way they contribute to group collaboration, we interviewed selected Wikiproject members. We focused our analyses of this qualitative data on the uses of direct communication within group work and its role in fostering engagement and in coordinating group work. The choice of an established, highly successful Wikiproject provided us with information on how direct communication contributes to success. Wikiproject Tropical Cyclones ranked as the project with highest quality of collective output. The choice of a fast-growing project allowed us to understand what processes lead to the formation of particular patterns of communication. Wikiproject COVID-19 was started on 15.03.2020 and by February 2021 had accumulated 1375 articles and gathered 274 participants.
\subsection{The role of direct communication}
At the moment of founding, a new project’s communication is concentrated within the project’s coordination page and thus easily accessible to all prospective members. In the case of our analyzed COVID-19 project, the quick growth in interest and participation resulted in a fast flow of information: \enquote{You also just saw lots of people contributing to those initial discussions on the main project talk page. There were so many discussions being started and ideas being floated (\ldots) that it's almost overwhelming to follow.} (Interviewee 1)
This flood of discussions resulted in diversification of communication channels. More specific discussions (pertaining e.g.~to the epidemics development in specific regions) migrated, in an organic manner, to the talk pages related to specific encyclopedic articles: \enquote{a lot of the discussion that you saw early on on the project talk page has migrated to the article space} (I1). This left the project’s main coordination space as a place for more general discussions: \enquote{the number of discussions being generated has of course declined a lot but it [is] still active. You still have editors asking thoughtful questions and soliciting feedback on specific pages.} and \enquote{\ldots I feel like it's still being used as a space for new information}. (I1) Interestingly, this early diversification of information points to an important role of direct messages posted on users’ talk pages.
Interviewee 1 said that he himself had little time to engage in interpersonal communication but that it was a common vehicle to provide feedback and corrections to the fast paced growth of contributions to the articles the project curated. \enquote{There were a lot of people who probably felt that their contributions to edit to Wikipedia were reverted [but] their additions were just relocated. (\ldots) Moving those requests to other spaces, I'm sure there was a lot of back and forth between editors on their talk page.} (I1) It seems that in this case direct messages were used as corrective feedback and explanation of actions taken by other editors. Importantly, \enquote{back and forth} exchanges took place, confirming our intuition that direct messages are used often not as a simple transfer of information but as a vehicle to build common understanding of the situation of group work: what are the accepted actions (norms) and what an article should look like (goals). In this, such messages might have reduced the discouragement of editors whose contributions were not accepted as they were but rather were moved to other articles.
Direct messages very often play the role of positive feedback as well. Such messages may take the form of a standardized text filled in with particular information (i.e.~a~filled in template), which can be pasted onto a user’s talk page using semi-automated tools such as the Twinkle Wikipedia add-on. A specific form of such templates are barnstars, acknowledgements and other awards which take the form of an image together with a personalized text message: \enquote{If someone writes a good article or like, they really show that initiative, they'll probably get a couple barnstars from a few different users.} (I3) More personalized feedback is delivered via messages written especially for the occasion. Interviewee 4: \enquote{I'd rather say thank you in my own words. I'd rather go and write, you know, thanks for your help with that. It was really useful.} In addition, the \enquote{thank you} function (an option available when viewing the edits contributed to a specific article) is an automated tool frequently used to give positive feedback, once an editor makes a good input (Interviewees 1 and 4). In contrast to personalized messages, such \enquote{thank yous} are not visible on the user talk page, but rather appear as a notification in the user profile. This gives editors a range of options to provide feedback that ranges from automated to very personalized, amplifying their capability to build social ties of different strength.
Negative feedback also ranges from more to less personal. Interviewee 4: \enquote{I also use twinkle to issue warnings and notifications. To people, I don't know. (\ldots) But for people I know I'm more likely to use a talk page message.} The templates available in the Twinkle tool allow users to provide depersonalized feedback and also to speed up the communication. This presumably allows to streamline task-related work. However, as the above quote shows, when there is already a relationship between users (a stronger social tie), a personalized message proves a better option: it might invite a reply and thus a deliberation on the shared understanding of norms and goals of the group. Thus, direct communication differs depending on the closeness of the interlocutors, with personalized messages inviting reciprocity and thus increasing the strength of the social tie.
The role of personalized, one-on-one communication is even more vivid when projects engage in soliciting new editors and in socializing them with the group. Communication on user talk pages is a way to establish sustainable relationships between more experienced users and newly joined members: \enquote{(\ldots) I tell them to leave me a talk page message [as a] part of their training. So again, they get the experience of doing that and I capture their username. So it's quite a cheating way of gathering everybody's username.} (I4). After leaving their name on each other's talk pages the editors are no longer anonymous strangers – they establish a personal, one-on-one channel of information exchange.
Experienced editors teach new project members about what needs to be edited and what makes a good edit. Interviewee 4 provides a detailed description of such training: \enquote{(\ldots) I train a lot of new people and I watch their early edits and I use welcome templates purely because it's a lot easier than trying to remember all the links that you want to give them for help pages and so on. (\ldots) And I always look for their first proper edits outside of their sandbox and leave a thank [you] with the thank [you] mechanism. Partly to thank them and encourage them but also so that they see how that works.}
Over time and with increasing volume of communication, relationships tend to transcend from task into more socio-emotional realm. This is accompanied by further diversification of communication channels. Interviewee 3: \enquote{A lot of it [is] user talk pages, there's a Facebook group among a couple users, we're all Facebook friends. (\ldots) there's an IRC Channel. There's also a Discord channel. So there's lots of different ways that the users will talk to each other. And it's become a bit of like socializing as well. Like there's some users I talk to almost every day. (\ldots) In a way, they've become almost a family of sorts.} Interviewee 4: \enquote{Off-Wiki friendships develop, and they obviously develop sometimes within projects. And you can tell people are making reference to things that they've talked about in email or that they met in a purple gun for a coffee or had a pizza or whatever}. Such strong, social bonds impact collaboration, for example: \enquote{I'm probably more likely to help if it's somebody I've met and work[ed] with previously than if it's just a name on a talk page, I'm not saying I wouldn't help and I'm not saying it's the only reason that I would help, but it's certainly encourages me to get involved.} (I4)
In sum, throughout a project's existence, collaboration is coordinated via various channels of communication that serve different communicative purposes. Direct messages on users’ talk pages often serve to invite and socialize new members and to provide feedback to encourage further contributions. Stronger relationships are enacted through more personalized messages, presumably to invite reciprocal and more prolonged exchanges.
\subsection{Organization and coordination of group work}
Interviewee 1 defines Wikiproject as \enquote{a single space where lots of editors (\ldots) come together and really work to bring some consistency across} thus underscoring the importance of coordination of work within the group. Good projects differ from bad projects in the extent to which they are able to maintain \enquote{an active community (\ldots), engagement with the wider community and an outreach} by organizing and announcing editathons, campaigns and competitions (I4). Thus, the distinction between successful and unsuccessful projects has a lot to do with how the project is connected to the wider Wikipedia community: \enquote{Good projects (\ldots) interact with the rest of the community to help people to understand the technicalities, to reach out to people who are working in that sort of area and involve them. And the bad projects see themselves as being the custodians, the ring fences, the owners of stuff} (I4). From this we can understand that good projects are able to tap on resources available in the community, while bad projects remain a specialized, closed group of contributors.
This understanding of success as an ability to mobilize distributed resources into a single effort is also reflected in how the work within projects is organized. When asked about division of labor and coordination of tasks, the interviewees underscore the bottom-up, organic manner in which such processes take place. In the newly formed COVID-19 project editors try to \enquote{find their niche, their dedicated space within the project (\ldots), saying: I want to focus on this or I'm gonna sign up to focus on [COVID-19] case counts in Italy or Spain (\ldots)} (I1). Similarly, our interviewee from the Tropical Cyclone projects reminiscens: \enquote{I~remember these users (\ldots) just took the initiative to make standardized info boxes, eventually getting standardized [hurricane] track maps, making sure that all the different hurricane articles look pretty much the same, that they were high quality, that they were referenced, that they weren’t using blogs for information.} (I3) In a sense, \enquote{the division of labor (\ldots) kind of became self-evident} (I3).
At the same time they notice that there is little in the way of formal roles or leaders that would serve as coordinators: \enquote{(\ldots) People just wanted to help organize the movement's efforts, (\ldots) so that editors weren't just thoroughly confused about where they were supposed to be adding new information. I wish I had a more clear answer for you on leadership. (\ldots) I'm sure there were people who stepped up and, you know, really took initiative.} (I1) \enquote{I haven't seen any sort of more formal structure. The only [projects] that tend to have a more formal structure are the ones where there is a user group off Wiki. So Wikiproject Medicine has the medicine foundation with a board and fundraising (\ldots)} (I4). In effect, the within project coordination becomes seamless and implicit, as users self-select to fill in where others left work to be done: \enquote{It's almost an informal collaboration, like we know what's expected so it's not exactly coordination} (I3).
In the case of established projects, this smooth collaboration might be an effect of experience, of editors internalizing their specific roles in the group effort. \enquote{Over [the] years we've collaborated and discussed (\ldots) and in the new [hurricane] season articles (\ldots) people can look back at other years and realize oh that's what we've done, so that's how it happens (\ldots).} (I3) It’s an effect of \enquote{past years’ work, coordination, reviewing the good articles, setting the standards, making sure that all the articles look the same}. (I3) The project works like well-oiled machinery, where the members are specialized in their tasks but at the same time rely on others to fulfill theirs.
Communication plays an important role in making such a system work. Project members specialized in certain areas (e.g.~making maps for hurricane tracking) naturally communicate with each other, but they also need to reach out farther - within and across projects - to tap on resources and skills they do not possess. Interviewee 4: \enquote{I go to the project page and find somebody, and then go and talk on their talk page. (\ldots) If I need to go and ask for some advice or get a second opinion (\ldots), you might have a source that I want to look something up for me.} Whilst direct messaging might be one of the choices for such advice seeking, other means of communication may be used. \enquote{I will private message them or leave them a message on their talk page. And in one or two cases, contact them off Wiki: a couple of them I know by email, or by Facebook. And I might just say, do you have this paper? Or I'm trying to find out this person's date of birth. Can you look it up in one of your sources for me?} (I4)
Such resource seeking might also happen on a project’s page to increase the visibility of the request or in situations when the editor does not have a personal relationship with anyone with fitting expertise. Interviewee 4: \enquote{you use your social capital, to use the jargon, to get help and to help other people. (\ldots) There are a few Wikiprojects that I go to when I have a question. (\ldots) If ever I'm writing a biography of somebody and they have some experience in the military (\ldots), the Military History project is a good place to go and check. (\ldots) Ask people to look things up in resources that they may have subscriptions to. (\ldots) Make sure I've got the terminology correct, things like that. (\ldots) There's a UK Railway project or rather a route and international railway project and that's simply a good place where you can go and get technical information and get people to look stuff up. Being sources for you.}
Project members also notify each other when they notice something that might be of interest to somebody. Interviewee 4: \enquote{I also tend to drop things into projects (\ldots) if I think they'll be of interest. So if I see something in the news and I can't write about it, but I think somebody might be interested and I know it's an active project in an area that I'm interested in, then I'll go and leave something on a talk page.}
Thus it seems members of Wikiprojects treat each other as a reliable source of information. Interviewee 2: \enquote{I've got access to Discord Channel, I've got access to the IRC, I can just generally ask: can you just take a look at this article? See what you think? [Am I going ] in the right direction. Am I wrong?} However, information exchange relies on previous experience with a particular Wikipedian and mutual trust. Interviewee 4: \enquote{If you (\ldots) use the [project] talk page to ask a question, you know which group of people will be replying and indeed, you know which ones you can rely on if you get an answer. (\ldots) You think: well, I've interacted with them often enough and seen their interaction with other people. Often up, I know they have a level of expertise and they're reliable and they [are] trustworthy in terms of the technical response. And indeed more generally.}
In sum, Wikiproject’s divide and integrate their work through a bottom-up self-selection to tasks. Experience in collaboration helps them develop procedures for implicit coordination of tasks and allows them to build strong ties that result in trust and reliance on each other. Moreover, project members specialize in specific tasks and learn where to seek advice and help when they need information or skills beyond their own expertise. Communication within and across projects helps to connect such specialists; however trust is crucial to rely on such sources. In this, Wikiproject’s display a high level of social capital: even if project members do not know something, they proverbially \enquote{know who knows} and are able - through their stronger (personal relationship) or weaker (through projects or initiatives they know) ties - to tap on this knowledge.
\subsection{Summary of results}
Our quantitative investigation into Wikiprojects’ direct communication structure showed that strength of the relationships within the group - as measured by the average number of messages sent over the social ties - correlates with higher quality products of collaboration. By interviewing selected members of an established and a growing Wikiproject we confirmed that direct messages can serve as a channel for socializing new project members and for sending positive and corrective feedback. Other popular means of communication within Wikipedia - such as project talk pages - are used for setting the agenda for future work or soliciting less personal feedback from a broader group of project members.
Moreover, our interviewees differentiated between automated and personalized messages: those framed as a standardized warning or acknowledgment of work were used for contacting distant or altogether unknown relations, while personalized messages were written for the strong and close connections. Thus, not only the volume of communication but also the form of messages can convey how close the interlocutors are. In sum, personalized, direct communication seems to complement more official, group communication channels by providing a vehicle to build strong social ties that facilitate affective processes within the group. In contrast, the fraction of group members that take part in direct communication did not play a part in group efficiency. We may conclude that what matters is the quality of relationships, not their span.
The other important result from the network analysis is that successful Wikiprojects have a direct communication structure that allows for complex coordination: for sharing of information locally, between project members clustered into small groups, and at the same time globally across such groups rather than by a single coordination center. In interviews, Wikiproject participants shared that they preferably seek advice from and offer help to those whom they trust and whose expertise they are aware of; thus they operate within a close circle of strong social ties. At the same time, they are skilled at using group level communication (e.g.~project talk pages) to acquire information that their closest relations do not possess. In effect, whilst locally they might communicate with a restricted number of collaborators, globally, it is easy for any resource or information to traverse the network. Thus both quantitative and qualitative analyses converge to show that complex patterns of information integration and coordination within Wikiprojects are related to better group performance.
\section{Discussion}
In our studies we attempted to verify how affiliation processes (affective domain) complement task-related coordination (instrumental domain) within groups collaborating online in the course of producing high quality output of collective effort. As argued by early influential research on group interaction~\citep{balesEquilibriumProblemSmall1953,balesHowPeopleInteract1955}, group dynamics must be studied in the context of these two conflicting developmental tendencies within the group, as they create inherent challenges that determine how the group will operate~\citep{balesAnalysisSmallGroup1950,balesPhasesGroupProblemsolving1951}.
The first type of challenges pertains strictly to the group task and how effective the group is in its achievement (instrumental domain). The second type of challenges relates to the extent to which group members are able to maintain close social ties and sustain group cohesion (expressive domain). Tensions caused by the instrumental challenges are reduced through division of resources and labor, and establishment of status and authority hierarchies within the group. On the other hand, tensions within the expressive domain induce behaviors that are focused on integration of opinions, and uniformity among group members. Moreover, according to~\citet{mcgrathGroupsInteractionPerformance1984}, affective processes have a regulatory (reinforcing and directing) function in relation to task behavior. Thus, a group is a system in a dynamical equilibrium between the conflicting drives for cohesion and differentiation~\cite{balesEquilibriumProblemSmall1953}. With the growing ubiquity of virtual collaboration - a tendency which the pandemics has strengthened and possibly calcified - it is important to understand how this balance can be achieved even when groups collaborate solely via new media. Wikipedia is a renowned success case of such technology mediated collaboration.
A growing body of work investigates task related coordination on Wikipedia. An influential study has found that implicit coordination - in the form of division of contributors into a highly productive core and periphery - was related to higher quality of articles, more so than the number of editors or explicit coordination in the form of discussions on the article talk pages~\citep{kitturHarnessingWisdomCrowds2008}. Other studies, however, found that numbers matter: both a higher number of editors, as well as a higher number of edits correlated with high quality~\citep{wilkinson2007cooperation}. An explanation of this discrepancy might be that it is important who comprises such a high number of editors. For example, varied background and diversified experience in editing~\citep{sydow2017diversity}, longer tenure, and willingness to perform multiple roles (functional tasks within article editing,~\citet{liuWhoDoesWhat2011}) positively impact quality of the articles on which editors work.
Explicit coordination has also been proven to matter. The structure of communication on Wikiproject talk pages impacts projects’ efficiency (measured as number of edits to articles): again, existence of a core of editors with high numbers of discussion posts and high values of betweenness centrality in the discussion network, was beneficial. Thus both implicit (division into core and periphery) as well as explicit (discussions) task related coordination is correlated with higher efficiency in pursuing collective goals. There is much less known, however, about the affiliation processes in online collaborative environments. Our study fills this gap.
First, interviews’ analysis showed that Wikipedians diversify their communication, depending on the communicative goal (task or affiliation related) and strength of relationship. The preferred task-related coordination space are talk pages - both of encyclopedic articles and of Wikiprojects. The affective processes, on the other hand, take place on user talk pages where Wikipedians post direct messages to one another, or outside of Wikipedia, on various social media sites. While such direct messages can have multiple goals (e.g.~they can also convey task related information) they are a dominant place to send general feedback (both positive and negative) which may help sustain engagement of contributors as well as socialize them with the collaborating group. Feedback messages were previously identified as types of leadership behaviors on Wikipedia - transactional leadership (positive feedback) and aversive leadership (negative feedback)~\citep{zhu2011identifying}. Our results enhance these findings by showing that feedback is preferentially provided via direct messaging.
Moreover, our quantitative network analysis also confirmed that direct communication impacts the effects of group work: high average volume of direct messages was correlated with higher quality of Wikiprojects’ output. Thus our results are in line with the so far scarce studies on effects of direct messaging on Wikipedia’s collaboration: that articles benefit from their contributors engaging in private messaging in addition to discussions on articles~\citep{parkCommunicationBehaviorOnline2016}, and that direct communication structure that allows linkages across and within contributors from core and periphery relates to higher quality of project output~\citep{rychwalskaQualityPeerProduction2020}. The results presented here go beyond these previous findings by showing that high intensity of affective processes is beneficial for effectiveness of online group collaboration. This result, combined with previous studies on task related coordination, shows that online groups, just like their offline counterparts, need to find a balance between affiliation and instrumental processes within the group. Understanding the interplay between affective and instrumental processes is important given that even on a such task focused community as Wikipedia, personal likes and dislikes may drive editing dynamics in a negative way (e.g.~into edit warring), wherein affiliation with particular others may be more important than the quality of contributions~\citep{lernerFreeEncyclopediaThat2020}.
Arguably, the most important and novel contribution of our paper, however, is that direct communication structure that allows for complex information integration within the group is related to higher quality of articles curated by Wikiprojects. That is, in successful projects, a particular individual’s contacts are relatively deterministic: she has a select group of relationships with whom she tends to communicate much more often than with others. On the other hand, in these successful projects, there is little determinism on the global, group-wide scale: if a particular piece of information or a resource is acquired by any member of the group, the next steps of its way across the group are on average quite unpredictable. In this way the effective mode of operation is that, locally, information is processed within circles of individuals and globally, there are sufficient connections distributed over the entire network to allow the information to spread rather than to be captured by a limited set of individuals. The interviews further elaborate on this result from network analysis: project members build strong trust and expertise based ties by history of previous interactions, but they also have multiple connections to others with different expertise through weaker relations. Such weak ties can exist within and across projects, as evidenced by our network analysis and interviews, respectively.
Our finding that global information integration (lack of a well determined sink for information or a finite coordination center) is beneficial for group effectiveness does not preclude the existence of a core of project contributors. Indeed, the distributions of both the number of messages sent as well as of degree centrality of Wikiprojects’ members tend to be highly skewed and heavy-tailed, suggesting that there are members who communicate more intensively than others and whom we might call “core”. What our results suggest, however, is that in online collaboration groups the core should not strive to be a pipeline for information and resource spread.
Specifically, globally integrated dynamics might be beneficial for more complex tasks, such as upgrading an article to a high quality rank. \citet{qinInfluenceNetworkStructures2015} have shown that the number of edits to articles within a projects’ scope is positively related to existence of leadership behaviors such as dominance in project talk page discussions (measured by betweenness centrality), which might suggest that centralized communication is favorable for Wikiprojects. However, mere adding of edits is a much simpler task than promoting an article to featured status. Indeed, \citet{nemotoSocialCapitalIncreases2011} have shown that promotion of articles to lower ranks of quality requires more centralized leadership (as measured by group degree centrality of direct communication networks) with less cohesive connections (measured by average clustering coefficient) than promotion to FA status. In this most complex task, cohesiveness was more important than centralized leadership. Our results show that this difference might be due to a larger press on global integration of information in such complex tasks. The practical implication of this finding is that in management of online collaboration, project members who are central in communication should strive to maintain redundant ties with the periphery, and encourage links across the periphery, so that the core of the group does not become a bottleneck in information integration or resource mobilization.
Such redundant connectivity across specialized local groups might be important specifically in online collaboration environments which are often characterized by high fluidity in membership. In standard organizations, maintenance of a skilled workforce can be achieved by, e.g.~financial incentives. Such solutions are not available in volunteer-based peer production and this often results in high turnover of contributors. For example, on Wikipedia, contributor career paths are volatile and unpredictable: editors switch between various functions and roles as they gain experience and self-select to favorite tasks. Some might take a long \enquote{Wikibreak} or even leave the community for good. At the same time the global proportions of functional roles are quite stable, implying that when individuals abandon certain roles, others take their place spontaneously~\citep{arazyTurbulentStabilityEmergent2016}. Our interviewees confirm that role uptake is organic and spontaneous. In such a working environment, no person in the direct communication network should be irreplaceable. Redundancy, not only in local, usually dense, ties but also in those weak, wide-spanning connections is critical to maintain sufficient levels of coordination in Wikiprojects, given the possibly fluctuating membership. In effect, our results suggest that complex tasks undertaken by online collaborating teams are benefited by communication structures that are characterized by specificity in local connections together with wide-reaching, integrative global links.
Such connectivity structure - and indeed the measures for effective information - might be related to high social capital of a group: \enquote{the sum of actual and potential resources embedded within, available through, and derived from the network of relationships possessed by an individual or social unit}~\citep[p.~243]{nahapietSocialCapitalIntellectual1998}. Social capital of a group is often defined in terms of bonding - i.e.~the cohesiveness of ties within a network. In offline teams an established network of close relationships within a social system increases commitment~\citep{brassBeingRightPlace1984,colemanFoundationsSocialTheory1994} and a higher density of connections leads to a higher likelihood of collective endeavors~\citep{putnamTuningTuningOut1995}. On Wikipedia, such social capital brought to an article and accumulated by previous collaboration, results in higher article quality~\citep{nemotoSocialCapitalIncreases2011}. Strong connections in a network of co-contributions to sentences within an article - also a measure of bonding capital - are also related to higher quality~\citep{liuWhoDoesWhat2011}. However, such capital is different from bridging capital of individuals that can be accrued by occupying a structural hole~\citep{jacksonTypologySocialCapital2020}.
Bridging capital is highest for individuals when they are unique intermediaries between parts of a network~\citep{jacksonTypologySocialCapital2020}. In offline studies on social capital an important role in communication networks is played by the so-called gatekeepers - a small number of key group members with expertise of how to find relevant information within a system~\citep{su2011multidimensional}. These gatekeepers are crucial for the group to work efficiently and that crucial position gives them power and control. Yet, for the group their uniqueness might be detrimental - they are a bottleneck in group coordination. As discussed above, in online systems that attempt complex tasks requiring specialized knowledge or skills, and which are characterized by high fluidity in membership, gatekeepers could prove disastrous: if they reduce their activity or simply are overwhelmed by amount of work, the whole group task (i.e.~FA~candidate article) might fail. Our results suggest that successful online teams are characterized by a communication structure that lacks any such gatekeepers. Future research might investigate whether complexity of the task (i.e.~comparison between different quality grades of articles), size of the team, as well as fluctuations in membership impact what communication structures are effective for an online peer production group.
\section{Limitations}
Without a doubt Wikipedia is one of the greatest success stories of collaborative knowledge production with no tangible financial incentives. As such it should be studied as this can reveal factors responsible for its unique growth and success. However, for the very same reasons our results should not be generalized to other similar platforms without additional considerations. Wikipedia may differ in important and unique ways from an average collaborative peer-production platform which can affect the scope of generalizability of our results. Studies similar to ours but based on data from other platforms need to be conducted in order to assess this.
Furthermore, in this study we focused exclusively on the structure of direct communication networks. It may be of interest to replicate similar analyses for other communication layers in the Wikipedia ecosystem, in particular discussions on article talk pages in the so-called main namespace as well as those on pages of individual Wikiprojects (project namespace). Moreover, correlations between communication structures at these different layers may provide additional insights and could be studied with methods developed for multilayer networks~\citep{boccalettiComplexNetworksStructure2006a}.
Last but not least, we used time-aggregated networks and even though this simplification was still enough to reveal several structural properties linked to quality of output of Wikiprojects it did not capture the full richness of our data. Therefore, it may be worthwhile to also study dynamic properties of communication within Wikiproject. However, one should also note that for many smaller projects available data may be too sparse to allow reliable dynamic analyses leading to significantly smaller sample sizes than the one we considered here.
\section{Acknowledgments}
This work was supported by the Polish National Science Centre through grant 2017/27/B/HS6/00626.
\printbibliography
\end{document}
|
2,877,628,089,553 | arxiv | \section{Introduction}
Our Galaxy presents a rich and contrasted X-ray landscape. Many different classes of Galactic objects can be the source of high energy emission and cover a wide range of spectral properties and X-ray luminosities. Active stellar coronae are by far the most numerous soft (kT\,$\sim$\,0.5\,keV) X-ray sources encountered at low Galactic latitudes. Their X-ray luminosities are in the range of 10$^{27}$ to 10$^{31}$\,erg\,s$^{-1}$. With $\rm L_{\rm X}$\ typically higher than 10$^{35}$\,erg\,s$^{-1}$, classical High and Low-Mass X-ray binaries (HMXBs, LMXBs) occupy the bright end of the Galactic X-ray luminosity function. However, X-ray surveys have also shown the presence of a large population of relatively hard Galactic sources with $\rm L_{\rm X}$\ in the range of 10$^{31}$ to a few 10$^{34}$\,erg\,s$^{-1}$. The nature and overall properties of these intermediate X-ray luminosity sources remains badly understood. Because of their faintness, these populations can only be studied in our Galaxy. Although CVs and coronally active binaries are likely to account for a large part of this population, many other kinds of objects have also been identified in this range of $\rm L_{\rm X}$, namely X-ray binary transients in quiescent states, magnetic OB stars or colliding wind massive binaries. Among the most interesting objects that could appear in this $\rm L_{\rm X}$\ regime are isolated compact remnants of early stellar formation accreting from the interstellar medium and low luminosity stages predicted by binary evolution models (e.g., Be + white dwarf, wind accreting neutron star binaries or precursors of LMXBs).
Low-luminosity hard X-ray sources are also likely to be important contributors to the Galactic ridge X-ray emission (GXRE; see e.g. \citealt{worrall1982}). The nature of this extended and apparently diffuse emission remains debated. Its X-ray spectrum displays a prominent emission line at 6.7\,keV and resembles that of a thin thermal plasma with temperatures of 5-10\,keV \citep{koyama1986}. However, such a hot interstellar medium cannot remain bound in the Galactic gravitational potential well and its presence would require that unrealistically powerful sources of hot plasma concur to replenish it continuously. The close similarity of the spatial distribution of the GXRE with near-infrared emission \citep{rev2006} also favours an explanation in terms of unresolved emission of many low-$\rm L_{\rm X}$\ sources. A deep Chandra observation of one of the brightest ridge emission close to Galactic Centre resolved at least 80\% of the diffuse emission into point sources at energies 6-7\,keV \citep{rev2009}. However, \cite{ebisawa2005} reach opposite conclusions based on another deep Chandra observation at $l$\,$\sim$\,28.5\degr.
The goal of this paper is to explore the nature of low to intermediate X-ray luminosity sources encountered in the Galaxy and shining in the hard (2-12\,keV) range. Our work is based on spectroscopic identifications obtained at the telescope and identifications derived from the cross-correlation of XMM-Newton serendipitous sources with large optical and infra-red archival catalogues.
\section{The X-ray content of the XMM-Newton Galactic Plane Survey}
The Survey Science Centre (SSC) of the XMM-Newton satellite has recently reported results from an optical campaign aiming at the identification of the brightest X-ray sources in the XGPS \citep{motch2010}. The $\sim$\,3\,deg$^{2}$ area surveyed is located at $l$\,=\,20\degr, $b$\,=\,0\degr\ \citep{hands2004}. Among the 29 hard (2-10\,keV; S/N\,$\geq$\,3) sources investigated optically, six are identified with massive stars possibly containing an accreting component or being powered by colliding winds, three are identified with CVs, two with low-mass X-ray binary candidates and six with stars. At $\rm F_{\rm X}$\ $\ga$ 10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$\ (2-10\,keV), a large fraction of the expected Galactic source population is positively identified. Active coronae account for $\sim$\,10\% of the expected number of Galactic sources in the hard band.
\section{2MASS and GLIMPSE identifications}
The advent of high quality photometric infra-red surveys covering large areas offers a unique opportunity to statistically identify and characterise hard Galactic X-ray sources. Using the method outlined in \cite{pineau2011}, we computed probabilities of identification with the 2MASS and GLIMPSE catalogues for all 2XMM-DR3 sources located within 3\degr\ from the Galactic plane. About one quarter of the $\sim$\,38,000 2XMM-DR3\footnote{The third release of the 2XMM catalogue was published in April 2010. see http://xmmssc-www.star.le.ac.uk/Catalogue/2XMMi-DR3/} low $b$ entries have a 2MASS match with an individual probability higher than 90\%. At this level, the expected number of spurious matches is $\sim$\,1.3\% \citep{motch2010}. Figs\,\ref{histo_pnhr2}\,and\,\ref{histo_pnhr3} show the distribution in EPIC pn hardness ratios of the 2XMM DR3 sources with and without 2MASS counterparts. Sources matching 2MASS entries are clearly much softer in X-rays. The peak in the HR2 histogram is consistent with optically thin thermal emission with kT\,$\sim$\,0.5\,keV undergoing an absorption with logN$_{\rm H}$\,$\sim$\,21.5 while the HR3 distribution indicates the presence of a harder X-ray component (kT\,$\sim$\,1\,keV). Most 2MASS identifications are thus likely active coronae. The few soft XMM-Newton sources without 2MASS entries are likely Me stars being too faint in the infrared to be listed in the 2MASS catalogue.
We show in Figs. \ref{Th_k_pnhr2} and \ref{Th_k_pnhr3} the distribution of EPIC pn HRs with the H-K colour. Since the intrinsic stellar H-K colour index remains within $-$0.1 to +0.1 from O to K spectral types and all luminosity classes \citep[see e.g.][]{covey2007}, the H-K colour mainly reflects interstellar absorption. No single X-ray energy distribution can account for the overall HR/E(H-K) relation. The most absorbed sources appear to be also the intrinsically hardest ones. The hotter X-ray temperature of young stellar coronae and active binaries combined with their higher luminosity make them detectable up to larger distances than older and X-ray softer stars. In particular, the bulk of the stars well detected above 2\,keV (i.e. appearing in Fig. \ref{Th_k_pnhr3}) have HR3 consistent with active binaries of the BY Dra or RS CVn type.
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true,angle=-90,bbllx=1.0cm,bburx=21.5cm,bblly=1cm,bbury=21.5cm]{motch_2011_01_fig01.ps}}
\caption{\footnotesize
Distribution of EPIC pn hardness ratio HR2 = ([1.0-2.0]\,--\,[0.5-1.0])/[0.5\,--\,2.0\,keV]) for $|b|<3$\degr\ 2XMM-DR3 sources with err(HR2)\,$\leq$\,0.1. Black: sources having a $\geq$\,90\% probability to be associated with a 2MASS entry; red: sources without any 2MASS entry within a combined X-ray + 2MASS 3$\sigma$ error radius.
}
\label{histo_pnhr2}
\end{figure}
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true,angle=-90,bbllx=1.0cm,bburx=21.5cm,bblly=1cm,bbury=21.5cm]{motch_2011_01_fig02.ps}}
\caption{\footnotesize
Same as Fig.\ref{histo_pnhr2} for EPIC pn HR3 = ([2.0-4.5]\,--\,[1.0-2.0])/[1.0\,-\,4.5\,keV].}
\label{histo_pnhr3}
\end{figure}
However, many very hard X-ray sources have high probability 2MASS identifications. Their number is significantly larger than expected from spurious matches. Assuming that their red H-K is due to interstellar absorption (N$_{\rm H}$\,$\sim$\,10$^{22}$ cm$^{-2}$ for E(H-K)\,=\,0.3) yield distances larger than 3\,kpc (for a mean particle density of $n$\,=\,1). Sources located above the $\Gamma$\,=\,2 powerlaw curve in the H-K versus HR3 diagram shown in Fig. \ref{Th_k_pnhr3} have a H magnitude of about 11 and a H-K around 0.45. At a distance of 3\,kpc, the absolute H magnitude is $\leq$\,-1.7 suggesting that these 2MASS identifications could well be massive stars. Their positions in the HR/H-K diagram are indeed comparable to those of HMXBs discovered by INTEGRAL and are akin to the Wolf-Rayet (WR) XGPS-14 \citep{motch2010}. Therefore, based on the optical and infrared spectroscopic identification work reported by \cite{motch2010} and by \cite{anderson2011}, we expect that many of these hard X-ray 2MASS identifications are low-$\rm L_{\rm X}$\ HMXBs, massive stars (either single or in wind colliding binaries) and $\gamma$-Cas analogs.
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true,angle=-90,bbllx=1.0cm,bburx=21.5cm,bblly=1cm,bbury=21.5cm]{motch_2011_01_fig03.ps}}
\caption{\footnotesize
H-K colour versus EPIC pn HR2 for all $|b|<3$\degr\ 2XMM-DR3 sources with err(HR2)\,$\leq$\,0.25, err(H-K)\,$\leq$\,0.1 and probability of 2MASS identification $\geq$\,90\%. The first three lower dark (red) lines show the expected E(H-K) versus HR2 relation assuming 2-T thermal emission from a 1.9\,Gyr, 300\,Myr and 30\,Myr active stars \citep{guedel1997}. The two hardest dark (red) relations correspond to AY Cet, a typical BY Dra binary \citep{dempsey1997} and to the RS CVn star WW Dra \citep{dempsey1993}. The two light (green) lines correspond to power laws with photon indices $\Gamma$ of 0 and 2. The RS CVn relation is almost superposed on the $\Gamma$\,=\,2 line. Positions of the WR star XGPS-14 \citep{motch2010}, (blue stars) and of a number of INTEGRAL HMXBs (big magenta squares) are also shown for comparison. Multiple detections are plotted.}
\label{Th_k_pnhr2}
\end{figure}
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true,angle=-90,bbllx=1.0cm,bburx=21.5cm,bblly=1cm,bbury=21.5cm]{motch_2011_01_fig04.ps}}
\caption{\footnotesize Same as Fig. \ref{Th_k_pnhr2} for EPIC pn HR3. WRs + INTEGRAL HMXBs are even better separated from coronal sources in this diagram.}
\label{Th_k_pnhr3}
\end{figure}
Because of the high density of Spitzer sources in the Galactic plane, the probability to find at random a relatively bright GLIMPSE source in the XMM error circle is rather large. Only 222 2XMM-DR3 sources, for which boresight correction could be carried out \citep[see][]{watson2009}, have a matching probability higher than 90\%. To this we add 103 sources with P$\geq$ 90\% resulting from the crosscorrelation with XMM observations for which no boresight correction was possible. These sources have on average larger error circles \citep{watson2009}. XMM-GLIMPSE sources with a high probability of identification have a distribution in hardness ratios similar to that seen for 2MASS. Many of the bright XMM-GLIMPSE sources do have identifications with known optically bright active coronae. Among the hardest sources we find several catalogued WR stars, HMXBs discovered by INTEGRAL and a number of unidentified sources sharing very similar X-ray and infrared properties.
\section{Hard X-ray sources with faint optical counterparts: - A Search for low $\rm L_{\rm X}$\ accreting black holes}
Optically faint hard X-ray sources are much less constrained than those associated with luminous objects such as the massive stars discussed in the two previous sections. Deep infrared observations have shown indeed that the majority of the hard X-ray sources detected in the Galactic Centre region are not associated with massive stars \citep{laycock2005}. Optical follow-up of ChaMPlane sources in the Galactic bulge \citep{koenig2008} suggests the presence of a large population of active binaries and young stellar objects with a small CV contribution. Our ESO-VLT optical observations of the brightest hard X-ray sources in a region located at $l\,=\,20$\degr\, $b\,=\,0$\degr\, \citep{motch2010} led to the identification of five CVs or LMXB candidates and of a few active coronae. This distribution is globally consistent with the locally determined X-ray luminosity function of faint point sources reported by \cite{sazonov2006}. However, we expect the contribution of the various kinds of hard X-ray emitters to strongly vary with Galactic position.
In addition, apart from toward the very central regions of the Galaxy, a large fraction of the hard X-ray sources detected at low Galactic latitude are background AGN. This extragalactic "contamination" depends sensitively on the direction of observation \citep{motch2006,hong2009}. A further difficulty arises from the fact that CV X-ray spectra resemble those of mildly absorbed AGN and in general cannot be efficiently preselected on the basis of hardness ratios only.
In order to increase our chances to select genuine Galactic X-ray sources for optical follow-up, we used the signature of the large photoelectric absorption imprinted on background AGN to achieve a high rejection rate for extragalactic sources. Our goals were twofold. First, investigate the nature of this optically faint hard X-ray Galactic population and second, constrain the surface density of low X-ray luminosity black holes (BH) in quiescent binaries, or being isolated and accreting from the interstellar medium.
Apart from three cases of long micro-lensing events possibly due to isolated black holes \citep{bennett2002,nucita2006} and perhaps a couple of massive unseen companions in X-ray quiet binaries, all established or candidate stellar mass black holes ($\sim$\,60) are in accreting binaries \citep[see][for a recent census]{ziolkowski2010}. However, the actual number of isolated black holes (IBHs) in the Galaxy could be of the order of 10$^{8}$ \citep{samland1998,sartore2010} or even higher. A fraction of them might accrete matter from the interstellar medium at a rate high enough to become detectable in the radio, optical or X-ray domains. IBH X-ray properties have been investigated in details by \cite{agol2002}. Their detectability depends sensitively on three parameters. The first one is the assumed spatial velocity distribution, which reflects the amplitude of the kicks received at birth, and is critical for Bondi-Hoyle accretion. \cite{jonker2004} have shown that BH X-ray binaries display scale heights comparable to those of neutron star binaries. This suggests that at birth BHs receive velocity kicks similar to those of neutron stars. The second key parameter is the efficiency of Bondi-Hoyle accretion, which can be quite significantly diminished in presence of magnetic fields. The accretion rate can be expressed as:
\begin{equation}
\dot{\rm M} = \lambda \ \frac{ 4 \pi G^{2} M^{2} \rho}{(v_{rel}^{2} + c_{s}^{2})^\frac{3}{2}}
\end{equation}
with $v_{rel}$ the relative velocity of the accreting object with respect to the ISM having sound speed $c_{s}$ and mass density $\rho$. \cite{perna2003} argue that the value of the dimensionless parameter $\lambda$ which determines the actual efficiency of Bondi-Hoyle accretion, and is usually assumed to be of the order of unity, could be as low as 10$^{-2}$ or even less. The last important parameter is the efficiency $\epsilon$ with which X-rays are generated in the accretion flow ($\rm L_{\rm X}$\,=\,$\epsilon \dot{\rm M} c^{2}$). The general agreement is that $\epsilon$ decreases sensitively at low $\dot{\rm M}$ due to the onset of radiatively inefficient accretion flows and as a result of an increasing fraction of accretion energy being transformed into kinetic energy of possible jets.
BH quiescent binaries display powerlaw X-ray spectra with photon index $\Gamma$ between 0.9 to 2.3 and X-ray luminosities in the range of 10$^{30}$ to 10$^{33}$\,erg\,s$^{-1}$\ \citep{kong2002,hameury2003}. Here we assume that IBHs exhibit similar X-ray spectra. As a best compromise between maximum distance range and survey area, we selected Galactic directions ($|b|$ $<$ 3\degr) toward which the total Galactic interstellar column is higher than 10$^{22}$cm$^{-2}$. For a typical ISM density of $n$\,$\sim$\,1\,cm$^{-3}$, this N$_{\rm H}$\ is reached at distances of $\sim$\,3\,kpc. At the time we started this project we relied on observations contained in the 1XMM catalogue to which we added several pointings extracted from the XMM-Newton archive. In order to obtain consistent HR definitions, we only considered the EPIC pn camera. The total area fulfilling these conditions in our source selection was 20 \,deg$^{2}$ at $\rm F_{\rm X}$\,$\geq$\,5\,10$^{-14}$\,erg\,cm$^{-2}$\,s$^{-1}$\ (0.2-12\,keV), taking into account overlapping exposures. This flux corresponds to a limiting X-ray luminosity of 6\,10$^{30}$\,erg\,s$^{-1}$\ at 1\,kpc. We then considered all sources displaying hardness ratios consistent within the errors with $\Gamma$\,=\,0.9--2.3 powerlaw energy distributions absorbed by N$_{\rm H}$\,$\leq$\,10$^{22}$cm$^{-2}$. Among the latter, we selected for optical follow-up 14 of the X-ray brightest sources having optical candidates fainter than B\,=\,18\footnote{A fraction of the accretion luminosity should be emitted in the optical and infrared domain via synchrotron emission of the hot accreting plasma} and being observable from ESO La Silla during Chilean winter time. All selected X-ray sources were visually checked and we discarded areas of high diffuse X-ray and optical emission as well as star forming regions. The set of hardness ratios used in the 1XMM provides less energy resolution in the soft bands than those used subsequently. Nevertheless, the resulting 1XMM based source selection was found to be consistent with that based on 2XMM HRs as shown in Fig.\ref{plot_P75_etc}. Source selection and validation have been done using the XCat-DB\footnote{http://xcatdb.u-strasbg.fr/} \citep{motch2009}, the official SSC interface to XMM catalogues developed in Strasbourg.
Optical observations were carried out with the ESO-NTT + EMMI instrument from July 31 to August 2, 2005. All spectra were obtained with Grism \# 5 through a 1.5\arcsec\ slit. This setup provided a spectral resolution of 1000 in the wavelength range of 3800 to 7000\AA.
We list in Tab. \ref{optids} a summary of the results of our optical identification work. Among these 14 sources we identified four cataclysmic variables, four Me stars and three active stars.
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true,angle=-90,bbllx=1.0cm,bburx=21.5cm,bblly=1cm,bbury=21.5cm]{motch_2011_01_fig05.ps}}
\caption{\footnotesize Distribution of candidate accreting black holes in the EPIC pn HR2/HR3 diagram (as defined in the 2XMM DR3 catalogue). The red lines define the area containing sources with X-ray energy distributions expected from accreting BHs undergoing N$_{\rm H}$\ $\leq$ 10$^{22}$cm$^{-2}$. Candidates extracted from the 2XMM DR3 are shown as black dots. Sources selected for optical follow-up observations are plotted in blue together with their HRs errors.}
\label{plot_P75_etc}
\end{figure}
\begin{table*}
\centering
\caption{Summary of optical identifications. AC\,=\,Active coronae earlier than M.}
\begin{minipage}{8.3cm}
\centering
\begin{tabular}{lccl}
\hline
\\
Source name & X-ray flux \footnote{In units of 10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$\ (0.2-12\,keV). Average of all detections.} & V mag & Identification \\
\\
\hline
\\
2XMM J135859.3-601518 & 1.87 & $>$ 22 & UNID \\
2XMM J154305.5-522709 & 7.93 & 20.8 & CV \\
2XMM J172803.0-350039 & 10.6 & 18.5 & Me \\
2XMM J174504.2-283552 & 0.58 & 17.5 & AC \\
2XMM J174512.9-290931 & 0.40 & 20.9 & Me \\
2XMM J175520.5-261433 & 2.07 & $>$ 22 & UNID \\
2XMM J180235.9-231332 & 2.87 & 20.0 & CV \\
2XMM J180243.0-224105 & 1.43 & 20.1 & Me \\
2XMM J180913.0-190535 & 2.39 & 22.4 & CV \\
2XMM J181003.2-212336 & 8.24 & 16.2 & AC \\
2XMM J181857.9-160208 & 1.06 & $>$ 22 & UNID \\
2XMM J182703.7-113713 & 0.65 & 17.3 & AC \\
2XMM J183228.1-102709 & 0.50 & 20.3 & Me \\
2XMM J185233.2+000638 & 0.62 & 20.4 & CV \\
\\
\hline
\end{tabular}\par
\vspace{-0.75\skip\footins}
\renewcommand{\footnoterule}{}
\end{minipage}
\label{optids}
\end{table*}
Only three XMM sources remain unidentified with candidate counterparts fainter than typically V\,=\,22. We note however, that our optical spectroscopic limit is not yet deep enough to rule out an identification with a high F$_{\rm X}$/F$_{\rm opt}$\ CV such as for instance 2XMM J183251.4-100106 ($\rm F_{\rm X}$\,=\,8.5 10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$, V\,=\,23.2; \citealt{motch2010}). From this optical campaign, we derive an upper limit of 0.2 BH deg$^{-2}$ at $\rm F_{\rm X}$\,=\,1.3\,10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$\ (0.2-12\,keV). This is $\ga$ 20 times the IBH surface density predicted by \cite{agol2002} for central regions of the Galaxy such as that covered by our optical targets ($b$\,=\,-1\degr,+2\degr\ and $l$\,=\,-49\degr,+33\degr). According to their model we expect 0.17 IBH in the total area surveyed at $\rm F_{\rm X}$\,=\,1.3\,10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$\ and less than 3 at a 10 times lower flux limit. It should also be stressed that our limiting N$_{\rm H}$\ de facto excludes from our sample all BH located at distances greater than a few kpc. This negative result clearly illustrates the difficulty of searching for Galactic IBHs in the X-ray domain where the background of AGN and the foreground of hard coronal emitters and CVs completely dominate the population. Using the population parameters of \cite{agol2002} our constraints can be expressed as $\lambda\,\times\,\epsilon\, \la\ 2\,10^{-4}\,/\,N_9 $ with $N_9$ the number of Galactic BHs in units of 10$^{9}$. For a reasonable value of $N_9$\,=\,0.2, we derive an upper limit of $\approx$\,10$^{-3}$ on the global (Bondi-Hoyle times X-ray) efficiency.
\section{Conclusions}
In this paper, we report on several efforts made to characterise the nature of the serendipitous XMM-Newton sources discovered in the Galactic plane. Dedicated optical follow-up observations and cross-correlation with archival catalogues reveal a significant number of hard X-ray emitting massive stars with luminosities in the range of $\sim$\,10$^{32}$ to $\sim$\,10$^{34}$\,erg\,s$^{-1}$. This population mainly consists of Wolf-Rayet stars, wind colliding binaries, HMXBs in quiescence and $\gamma$-Cas analogs. Young stars and active corona binaries of the BY Dra and RS CVn types also contribute significantly to the population of sources detected at energies above 2\,keV. The optically faintest Galactic hard X-ray sources are mostly identified with cataclysmic variables.
The typical XMM-Newton sensitivity allows us to constrain the nature of the hard X-ray sources of low- to intermediate- $\rm L_{\rm X}$ \ up to distances of a few kpc. It is therefore unclear whether these studies can be used to assess the nature of the unresolved population responsible for the Galactic Ridge X-ray emission mostly seen at $|l|$ $\la$\,50\degr.
Finally, we report on a first dedicated search for low-$\rm L_{\rm X}$\ black holes in quiescent binaries or in isolation and accreting from the interstellar medium. We derive an upper limit of 0.2 BH deg$^{-2}$ at $\rm F_{\rm X}$\,=\,1.3\,10$^{-13}$\,erg\,cm$^{-2}$\,s$^{-1}$\ (0.2-12\,keV) in directions of the central parts of the Galaxy. This observational limit implies that the efficiency $\epsilon$ with which X-rays are generated in the accretion flow is $\la$\,10$^{-3}$ if one assumes nominal Bondi-Hoyle mass accretion rates.
\bibliographystyle{aa}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.